Практикум на ЭВМ. Методы приближения функции

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:::::::::::::::::::::::::::::: x1. : : : : x2. : : : : : : : : : : : : : : : : : : : : x3. : : : : : : x4. : : : : : : : : : : : : : : : : : : : : : : x5. # $ : : : : : : : : : x6. " &" : : x7. ( # $ : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : x8. : : : : : : x9. : : : : : : : : : : : : x10. + # $ : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : x11. ( : : : : : : : : : : : : : : : : : : : : : x12. ( : : : : : : : : : : : : : : : : : : : : : : : : : : : : : x13. ( : : : : : : : : x13.1. ,-./0,12/ 3/4/56 760890,9 60.8:;,7<=66 : : : : : : : : : x13.2. 7>,:6.? ;,-.:,806< :/37,@806< : : : : : : : : : : : : : : : x13.3. =802/ 2,7658-.1/ /:6A?8.658-26B ,;8:/=69 : : : : : : : : x13.4. 1<3C :/37,@806< ;, ?0,>,5780/? 8DEF81/ 6 60.8:;,7<=66 x14. & - : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : x15. - : : : : : : : : : : : : : x16. - (+ : : : : : : : : : : : : : : : : : : : : : : : : x16.1. ,-./0,12/ 3/4/56 760890,9 60.8:;,7<=66 : : : : : : : : : x16.2. E56-78068 ?/.:6=E -6-.8?E 3/4/56 760890,9 60.8:;,7<=66

6 7 7 10 11 12 17 18 20 22 23 27 31 33 35 36 39 40 41 43 44 45 46 47

x16.3. x16.4.

1,9-.1/ ;:6D76@/GH89 AI02=66 : : : : : : : : : : : : : : E56-78068 ;:/1,9 5/-.6 -6-.8?E 3/4/56 760890,9 60.8:;,7<=66 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : x17. ( (+ : : : : : : : : : : : : : : : : : x17.1. 0.8>:/7C0E8 -1,9-.1/ ?0,>,5780,1 8DEF81/ : : : : : : : x17.2. ,-./0,12/ 3/4/56 760890,9 60.8:;,7<=66 : : : : : : : : : x17.3. E56-78068 2,JAA6=680.,1 :/37,@806< : : : : : : : : : : : x17.4. 7>,:6.? 1E56-7806< 2,JAA6=680.,1 :/37,@806< : : : : : x17.5. =802/ 2,7658-.1/ /:6A?8.658-26B ,;8:/=69 : : : : : : : : x18. - + # + + : : : : : : : : : : : : : : : : x19. - # x19.1. DH/< -B8?/ : : : : : : : : : : : : : : : : : : : : : : : : : : : x19.2. 7>,:6.? 1E56-7806< 2,JAA6=680.,1 ?0,>,5780/ Pi : : : x19.3. I-,50/< 60.8:;,7<=6< 2ID658-26?6 ?0,>,5780/?6 K:?6./ x19.4. I-,50/< 60.8:;,7<=6< 2ID658-26?6 ?0,>,5780/?6 8--87< x19.5. I-,50/< 60.8:;,7<=6< 2ID658-26?6 ?0,>,5780/?6 ?8.,4,? 26?E : : : : : : : : : : : : : : : : : : : : : : : : : : : : x19.6. I-,50/< 60.8:;,7<=6< 2ID658-26?6 ?0,>,5780/?6 - 6-;,7C3,1/068? :/3487800EB :/30,-.89 : : : : : : : : : : : : : x19.7. 0.8:;,7<=6< 2ID658-26?6 -;7/90/?6 : : : : : : : : : : : : x19.8. ;:84878068 084,-./GH6B >:/0650EB I-7,169 : : : : : : : x19.8.1. ;:84878068 084,-./GH6B >:/0650EB I-7,169 ;, 6318-.0E? 30/5806:/0650EB I37/B : : : : : : : : : : : : : : : : x19.8.2. ;:84878068 084,-./GH6B >:/0650EB I-7,169 ;, 6318-.0E? 30/5806:/0650EB I37/B : : : : : : : : : : : : : : : : x19.8.3. \-.8-.1800E8" >:/0650E8 I-7,16< : : : : : : : : : x19.8.4. -7,168 \,.-I.-.16< I37/" 1 ;:6>:/0650EB I37/B x19.8.5. ,;,706.87C0E9 I387 1 ;:6>:/0650EB I37/B : : : x19.8.6. K2-.:/;,7<=6< 1 ;:6>:/0650EB I37/B : : : : : : : x20. : : : : x20.1. DH/< -B8?/ : : : : : : : : : : : : : : : : : : : : : : : : : : : x20.2. 7>,:6.? 1E56-7806< 2,JAA6=680.,1 ?0,>,5780/ Pi : : : x20.3. 0.8:;,7<=6< ;/:/D,7658-26?6 -;7/90/?6 : : : : : : : : : x20.4. ;:84878068 084,-./GH6B >:/0650EB I-7,169 : : : : : : :

3 50 52 56 56 58 59 62 64 64 67 67 67 68 69 70 71 72 74 74 75 76 76 79 80 80 80 81 82 84

4

x20.4.1. x20.4.2. x20.4.3. x20.4.4. x20.4.5. x20.4.6.

;:84878068 084,-./GH6B >:/0650EB I-7,169 ;, 6318-.0E? 30/5806:/0650EB I37/B : : : : : : : : : : : : : : : : ;:84878068 084,-./GH6B >:/0650EB I-7,169 ;, 6318-.0E? 30/5806:/0650EB I37/B : : : : : : : : : : : : : : : : \-.8-.1800E8" >:/0650E8 I-7,16< : : : : : : : : : -7,168 \,.-I.-.16< I37/" 1 ;:6>:/0650EB I37/B ,;,706.87C0E9 I387 1 ;:6>:/0650EB I37/B : : : K2-.:/;,7<=6< 1 ;:6>:/0650EB I37/B : : : : : : :

"& II. . )/0 1 2 . 3

3

:::::::::::::::::::::::::::: x1. # + + : : : : : : : : : : : : : : : : : : : x1.1. ,-./0,12/ 3/4/56 60.8:;,7<=66 .803,:0E?6 ;:,63184806,:6.? 60.8:;,7<=66 .803,:0E?6 ;:,63184806:/??0/< :8/763/=6< /7>,:6.?/ : : : : : : : : : : : : : x1.5. 0.8:;,7<=66 .803,:0E?6 ;:,63184806,57800,9 /;;:,2-6?/=66 : : : : : : : : : : : : : : : : : : : x1.5.1. 0.8:;,7<=6,00E9 ?0,>,5780 />:/0@/ : : : : : x1.5.2. /3487800E8 :/30,-.6 47< AI02=66 ?0,>6B ;8:8?800EB : : : : : : : : : : : : : : : : : : : : : : : : : x1.5.3. 0.8:;,7<=6,00/< A,:?I7/ CG.,0/ : : : : : : : x1.5.4. 0.8:;,7<=6< "416@IH6?6-<" ?0,>,5780/?6 : : x1.5.5. 0.8:;,7<=6< - 2:/.0E?6 I37/?6 : : : : : : : : : x1.5.6. /37,@8068 ;, ?0,>,5780/? 8DEF81/ : : : : : : x1.5.7. ;;:,2-6?/=6< ?0,>,5780/?6 8DEF81/ ?8.,4,? 0/6?80CF6B 21/4:/.,1 : : : : : : : : : : : : : : : : x1.6. 0.8:;,7<=66 .803,:0E?6 ;:,63184806,57800,9 /;;:,2-6?/=66 : : : : : : : : : : : : : : : x1.6.1. I-,50,-760890/< 60.8:;,7<=6< : : : : : : : : : : x1.6.2. I-,50,-760890/< /;;:,2-6?/=6< ?8.,4,? 0/6?80CF6B 21/4:/.,1 : : : : : : : : : : : : : : : : : : x1.6.3. I-,50,-2ID658-2/< 60.8:;,7<=6< : : : : : : : : : x1.6.4. I-,50,-21/4:/.650/< 60.8:;,7<=6< : : : : : : : x2. # + + + K : : : : : : : : : : : : : : : : : : :

84 87 87 88 90 90 91 91 91 93 94 95 96 96 97 97 98 99 99 100 102 102 104 105 108 111

5

x2.1. x2.2. x2.3.

,-./0,12/ 3/4/56 760890,9 60.8:;,7<=66 : : : : : ;,-,DE ;,-.:,806< .:6/0>I7<=66 ,D7/-.6 : : : : : :6D76@8068 AI02=69 1 .:8I>,7C0628 : : : : : : : : x2.3.1. ,-./0,12/ 3/4/56 760890,9 60.8:;,7<=66 x2.3.2. :6D76@8068 AI02=69 1 ;:,7C0,? >,7C0628 : : : : : : : : : : : : : : : : : : : : x2.3.3. :6D76@8068 760890E?6 AI02=6
:::: :::: :::: :::: .:8I:::: :::: :::: :::: :::: ::::

111 112 113 113 114 115 117 120 124 129

6

/-.,,:6.?,1, ;:847/>/8?EB 2 :8/763/=66 0/ K -.I480./? ?8B/062,-?/.8?/.658-2,>, A/2I7C.8./ 0/ 3/0<.6,:6.?,1 ;:61,46.-< 08,DB,46?,8 .8,:8.658-2,8 ,D,-0,1/068, -,,.18.-.1IGH68 :/-58.0E8 -,,.0,F806< 6 :82,?804/=66 ;, 6B ;:/2.658-2,?I ,-IH8-.17806G 0/ K (,:>/063/=6< ;:,=8--/ 1E56-78069, B:/0806< 4/00EB 6 :83I7C./.,1 1 ;/?<.6 K 6 ..;.). 0,>,,D:/368 /7>,:6.?,1 ,DP<-0<8.-<, - ,40,9 -.,:,0E, 08,DB,46?,-.CG ,D8-;856.C ;:8;,4/1/.8789 4,-./.,50E? 0/D,:,? 3/4/5 47< ;:,184806< 3/0<.69, / - 4:I>,9 -.,:,0E, @87/068? ;:,48?,0-.:6:,1/.C :/37650E8 ;,4B,4E 2 :8F806G 3/4/56 ;:6D76@806< AI02=69. 7>,:6.?E 6?8G. :/30IG .,50,-.C, 4/G. :/30IG >7/42,-.C ;:6D76@/GH89 AI02=66, 6?8G. :/30IG 5I1-.16.87C0,-.C 2 ;,>:8F0,-., 7I5F8>," ?8.,4/ ;:6D76@806< 08 -IH8-.1I8., 6 1ED,: /7>,:6.?/ 3/16-6. ,. 2,02:8.0,9 3/4/56. K.,. 1ED,: DI48. :/37650E?, 0/;:6?8:, 47< /;;:,2-6?/=66 /0/76.658-26 3/4/00,9 AI02=66 -7,@0,>, 164/ 6 47< ,D:/D,.26 :83I7C./.,1 63?8:8069 A63658-2,9 1876560E. ,4D,: /7>,:6.?,1 47< :/2.62I?/ 462.,1/7-<, 1 ,-0,10,?, 1,3?,@0,-.CG :8/763/=66 6B -.I480./?6 ;:6 -IH8-.1IGH6B :8-I:-/B 1:8?806 0/ K , 5., ;:6187, 2 ,.2/3I ,. :/--?,.:806< I-7,@0800EB ;,4B,4,1. /-.C ,;6-/00EB /7>,:6.?,1 1E.8-080/ 63 F6:,2,9 1E56-76.87C0,9 ;:/2.626 D,788 JAA82.610E?6 (6 D,788 -7,@0E?6) /7>,:6.?/?6, :/--?,.:8.C 2,.,:E8 1 2I:-8 ":/2.62I? 0/ K " 08 ;:84-./17<8.-< 1,3?,@0E?. 8? 08 ?8088, J.6 /7>,:6.?E ;:84-./17, 2:I>/ 3/4/5 6 127G580E 1 ;,-,D68. #,:?/ ,.58.0,-.6 -.I480.,1 ;, :/2.62I?I ;:631/0/ -.6?I76:,1/.C 2/2 :/316.68 ;:/2.658-26B 0/1E2,1 :8F806< ?/.8?/.658-26B 3/4/5 - ;,?,HCG 2,?;CG.8:/, ./2 6 -,34/068 ,;:8487800,>, 2:I>,3,:/ 1 ,D7/-.6 -IH8-.1IGH6B ?8.,4,1 :8F806< ;,-./17800,9 3/4/56. ,J.,?I 1 :/?2/B :/2.62I?/ -.I480./? ;:847/>/8.-< 2/2 :/3:/D,./.C ;:,>:/??I 0/ K , :8/763IGHIG 3/4/00E9 /7>,:6.?, ./2 6 ;6-C?800, ,.18.6.C B,.< DE 0/ ;,7,160I 1,;:,-,1 63 ;:847,@800,>, 1/:6/0./, -,-./17800,>, 63 1,;:,-,1, ;:6184800EB 1 2,0=8 ;,-,D6<. ,-0,18 0/-.,, ;,-,D6< 78@/. ?/.8:6/7E 782=69, 56./1F6B-< /1.,:,? 1 .858066 4-B 78. 1 :/?2/B A/2I7C./.610,>, 2I:-/ ":/2.62I? 0/ K ". J782.:,00,? 1/:6/0.8 ,0, I@8 D,788 5-.6 78. 6-;,7C3I8.-< ;:6 ;:,1848066 3/0<.69 -, -.I480./?6 1 46-;7890,? 27/--8. :847,@806<, 3/?85/06< 6 ,.?85800E8 ,;85/.26 ;:,-CD/ -,,DH/.C /1.,:I 0/ 2/A84:I 1E56-76.87C0,9 ?/.8?/.626.

7

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1.

I-.C 3/4/0/ AI02=6< f : X ! Y , f 2 F (X Y ) { 082,.,:,?I AI02=6,0/7C0,?I ;:,-.:/0-.1I, 0/D,: AI02=69 g1 : : : gn , gi : X ! Y , gi 2 F (X Y ), 6 0/D,: 760890EB AI02=6,0/7,1 1 : : : m 0/ ;:,-.:/0-.18 F (X Y ). :8DI8.-< ;,-.:,6.C /;;:,2-6?/=6G Pf AI02=66 f 164/

Pf =

n X j =1

j gj

(1)

./2IG, 5.,DE DE76 1E;,7080E 760890E8 I-7,16< 60.8:;,7<=66

i(Pf ) = i(f ) i = 1 : : : m: /4/5/ ;,-.:,806< ./2,9 Pf 0/3E1/8.-<

(2)

.

0.8:;,7<=6< ;, 0/D,:I ?0,>,5780,1. X = Ta b] Y = R F (X Y ) = C (Ta b]) gi (x) = xi i = 1 : : : n . 1

.%% 1.

,.,5850/< 60.8:;,7<=6<. X = Ta b] Y = C (Ta b]) i(f ) = f (xi), >48 x1 : : : xm { 0/D,: .,582 0/ Ta b]. .%% 2.

R1

X = Ta b] Y = R1 F (X Y ) = L1 (Ta b]) i(f ) = x1 : : : xm { 0/D,: .,582 0/ Ta b]. .%% 3.

F (X Y ) =

xR+1 i

x

i

f (x) dx , >48

6%$". /4/5/ 760890,9 60.8:;,7<=66 0/3E1/8.-< 2,::82.0,9, 8-76 88 :8F8068 (..8. 0/D,: 2,JAA6=680.,1 i ) -IH8-.1I8. 6 8460-.1800, 47< 1-<2,9 AI02=66 f 2 F (X Y ).

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8

D,30/56?

(n ) P n G(X Y ) = hgj ij=1:::n = j gj : = (j )j=1:::n 2 R F (X Y ) =1 jP m m V(X Y ) = hiii=1:::m = ii : = (i)j=i:::m 2 R F (X Y ): i=1

-76 AI02=66 g1 : : : gn 760890, 083/16-6?E 6 AI02=6,0/7E 1 : : : m 760890, 083/16-6?E, ..8. ,D:/3IG. D/36-E ;:,-.:/0-.1 G(X Y ) 6 V(X Y ) -,,.18.-.1800,, ., 3/4/5/ 760890,9 60.8:;,7<=66 ?,@8. DE.C ;,-./1780/ 1 J2161/780.0,9 A,:?8: 47< 3/4/00,9 f 2 F (X Y ) 0/9.6 g = Pf 2 G(X Y ) ./2IG, 5., (f ) = (g) 47< 1-8B 2 V(X Y ).

% 1.

g1 : : : gn

1 : : : m . 1. , m = n A = (i(gj )) i = 1 : : : m j = 1 : : : n

(3)

. 2. ,

Pf =

n X j =1

j gj

= (j )j=1:::n = A;1 ( (i(f ))i=1:::n ) :

(4)

3. ,

P : F (X Y ) ! G(X Y ) (.. P 2 = P , , Im P \ Ker P = 0 Pf = f " f 2 G(X Y ) ). . n 1. 3 (1), (2) -784I8., 5., Pf = P j gj <17<8.-< :8F8068? 3/4/56 760890,9 j =1 60.8:;,7<=66 .,>4/ 6 .,7C2, .,>4/, 2,>4/

7"#&

0n 1 X i @ j gj A = i(f ) 47< 1-8B i = 1 : : : m j =1

676 :I>6?6 -7,1/?6

n X j =1

j i(gj ) = i(f ) 47< 1-8B i = 1 : : : m:

n X j =1

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aij j = i(f ) 47< 1-8B i = 1 : : : m !"#$ $ %$

x1. '

9

() *+ , -- (',./

>48 aij = i(gj ) A = (aij ), ..8.

A = (i(f ))i=1:::m (5) >48 = (j )j=1:::n . 784,1/.87C0,, 3/4/5/ 760890,9 60.8:;,7<=66 2,::82.0/ .,>4/ 6 .,7C2, .,>4/, 2,>4/ -6-.8?/ 760890EB I:/108069 (5) 6?88. 8460-.1800,8 :8F8068 47< 1-<2,9 f 2 F (X Y ). 784,1/.87C0,, ?/.:6=/ A ,-IH8-.17<8. D682.610,8 -,,.18.-.168 ?8@4I ?0,@8-.1,? f(i(f ))i=1:::m : f 2 F (X Y )g Rm 6 ?0,@8-.1,? f : ; :8F8068 (5)g Rn:

/2 6318-.0, 63 2I:-/ ?/.8?/.658-2,>, /0/763/ (676 2I:-/ /7>8D:E) J., ,30/5/8. m = n 6 -IH8-.1,1/068 A;1 . (7< ;,70,.E 637,@806< 4/46? 4,2/3/.87C-.1, J.,>, A/2./. I-.C A 08,D:/.6?/. -76 m > n , ., -IH8-.1I8. 08.:616/7C0/< 760890/< 2,?D60/=6< -.,7D=,1 (n m)-?/.:6=E A , :/10/< 0I7G, ..8. -IH8-.1I8. 6= 0 n ./2,8, 5., A = 0. /2 2/2 fg g { 760890, 083/16-6?E, ., g = P g 6= 0. j j =1:::n

j =1

j j

784,1/.87C0,, 47< f 0 3/4/5/ 760890,9 60.8:;,7<=66 6?88. 41/ :8F806<: g 6 0. K., ;:,.61,:856. 88 2,::82.0,-.6. -76 m < n , ., -IH8-.1I8. 08.:616/7C0/< 760890/< 2,?D60/=6< -.:,2 (n m)-?/.:6=E A , :/10/< 0I7G, ..8. -IH8-.1I8.m 6= 0 ./2,8, 5., tAt = 0. /2 2/2 figi=1:::m { 760890, 083/16-6?E, ., = iP=1 ii 6= 0. 3 :/180-.1/ tAt = 0 ;,7I5/8? m m X X (gj ) = ii(gj ) = iaij = 0 i=1

i=1

47< 1-8B j = 1 : : : n . ,-2,7C2I { 080I781,9 AI02=6,0/7 0/ F (X Y ), ., -IH8-.1I8. f 2 F (X Y ) ./2/<, 5., (f ) 6= 0. ,>4/ 47< J.,9 AI02=66 f 3/4/5/ 760890,9 60.8:;,7<=66 08 6?88. :8F806<, ./2 2/2 (f ) 6= 0 = (gj ) 47< 1-8B j = 1 : : : n .) 2. .,:,8 I.18:@48068 .8,:8?E -784I8. 63 (5). 3. I-.C g = Pf , >48 f 2 G(X Y ). ,-2,7C2I fgj gj=1:::n { D/36- G(X Y ), n n ., f = P j gj , g = Pf = P j gj , ;:658? ;, 4,2/3/00,?I -1,9-.1I (4) j =1

j =1

0 0n 11 0n 1 X X = A;1 (i(f )) = A;1 @i @ j gj AA = A;1 @ j i(gj )A = j =1 j =1 0n 1 X = A;1 @ j aij A = A;1 (A) = j =1

..8. = 6 ;,.,?I g = f . ,J.,?I Pf = f 6 P 2 = P , ./2 2/2 Pf 47< 1-8B f 2 F (X Y ).

. .

2 G(X Y )

!"#$ $ %$

x2. 1

10

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2.

:6 1E56-78066 2,JAA6=680.,1 fj gnj=1 :/37,@806< AI02=66 Pf 2 n F (X Y ) ;, D/36-I fgj gnj=1 ;:,-.:/0-.1/ G(X Y ) F (X Y ), g = Pf = P j gj j =1 0863D8@0E 1E56-76.87C0E8 ;,>:8F0,-.6. B 176<068 0/ :83I7C./. { AI02=6G g B/:/2.8:63I8.-< 56-7,? ,DI-7,17800,-.6. I-.C 1 ;,4;:,-.:/0-.18 G(X Y ) F (X Y ) 118480/ 0,:?/ k kG , / 1 ;:,-.:/0-.18 Rn 3/A62-6:,1/0/ 082,.,:/< 0,:?/ k kR . 1848? n

n n P j gj P j gj j=1 G j=1 G M = max m = min 2R 2R kkR kkR : ,>4/ 47< 1-<2,>, 2 Rn n X mkkR j gj M kkR : j=1 G n

n

n

n

n

(1)

n

I-.C 1?8-., .,50,>, 182.,:/ 2,JAA6=680.,1 :/37,@806< ;,7I580 ;:6n P D76@800E9 b = + " . 784,1/.87C0,, 1?8-., AI02=66 g = j gj DI48? 6?8.C j =1

n n n n gb = P (j + "j )gj = P j gj + P "j gj = g + g" , >48 g" = P "j gj . 3 (1) 6?88? j =1

mk"kR

j =1

j =1

j =1

kg"kG M k"kR mkkR kgkG M kkR : .-G4/ m k"kR kg"kG M k"kR : M kkR kgkG m kkR 6%$". 6-7,? ,DI-7,17800,-.6 D/36-/ fgj gn j 0/3E1/8.-< n

n

n

n

n

n

n

n

=1

cond(gj ) = M : m 6-;,7C3,1/068? J.,>, ,;:8487806< 6 -,,.0,F8069 g" = gb ; g 6 " = b ; ;,-784088 08:/180-.1, ;8:8;6F8.-< 1 1648 1 kb ; kR kgb ; gkG cond(g ) kb ; kR : j cond(gj ) kkR kgkG kkR K., 08:/180-.1, I-./0/1761/8. -1<3C ?8@4I $ kgb ; gkG 6 $ & kgkG b ; kR k ' k kR . 8? D,7CF8 56-7, ,DI-7,17800,-.6 D/36-/, .8? BI@8 ,0 - 1E56-76.87C0,9 .,526 3:806<. n

n

n

n

n

n

. .

!"#$ $ %$

x3. (',./ 4- 5+, ,5 x

11

3.

,DH89 -B8?8 760890,9 60.8:;,7<=66 (-?. x 1) :/--?,.:6? -7I5/9: X = Ta b] Y = R1 F (X Y ) = C (Ta b]). I-.C 3/4/0E .,526 a = x1 < x2 < : : : < xn = b (0/3E1/8?E8 I37/?6 60.8:;,7<=66) 6 6318-.0E 30/5806< f (x1) : : : f (xn). ED8:8? AI02=66 n gi(x) li(x) = Q xx ;; xxj i = 1 : : : n (0/3E1/8?E8 J78?80./:0E?6 ?0,>,j =1j 6=i i j 5780/?6 />:/0@/) 6 AI02=6,0/7E i(f ) = f (xi) i = 1 : : : n . ,>4/ 3/4/5/ 760890,9 60.8:;,7<=66 ;:6?8. 164: 47< 3/4/00,9 (1 .,52/B xi i = 1 : : : n ) n AI02=66 f 0/9.6 L(x) = (Pf )(x) = P ili (x) ./2IG, 5., L(xi ) = f (xi) 47< 1-8B i=1 i = 1 : : : n . :,18:6?, 5., AI02=66 l1 : : : ln 760890, 083/16-6?E 6 AI02=6,0/7E 1 : : : n 760890, n083/16-6?E. 89-.16.87C0,, 8-76 -IH8-.1I8. ./2,8 2 n P P n R 6= 0, 5., ili(x) = 0 47< 1-8B x 2 Ta b], ., ili(xj ) = 0 47< i=1 i=1 1-8B j = 1 : : : n . ,-2,7C2I li(xj ) = ij , ., ,.-G4/ ;,7I5/8? j = 0 47< ./1-8B j = 1 : : : n , 5., ;:,.61,:856. 6= 0. 0/7,>650,, 8-76 -IH8-.1I8. n n P P n 2,8 2 R 6= 0, 5., ii(f ) = 0 47< 1-8B f 2 C (Ta b]), ., ii(lj ) = 0 i=1 i=1 47< 1-8B j = 1 : : : n . ,-2,7C2I i(lj ) = ij , ., ,.-G4/ ;,7I5/8? j = 0 47< 1-8B j = 1 : : : n , 5., ;:,.61,:856. 6= 0. /26? ,D:/3,?, ;:6?806?/ .8,:8?/ 1.1. /.:6=/ (1.3) { 8460650/< ( A = I ), ;,-2,7C2I i(lj ) = ij . 784,1/.87C0,, 3/4/5/ 760890,9 60.8:;,7<=66 2,::82.0/, 6 88 :8F8068 1 -67I (1.4) ;:84-./17<8.-< 1 1648

L(x) =

n X i=1

f (xi)li(x) =

n X i=1

f (xi)

n x;x Y j : x ; x i j j =1

(1)

j 6=i

, .8,:8?8 1.1 ;,-.:,800E9 ?0,>,5780 L { 8460-.1800E9 63 hliii=1:::n , I4,178.1,:,5780,1 -.8;806 n ; 1, I4,178.1,:,5780,1 -.8;806 n ; 1 6 I4,178.1,:<8. I-7,164/ L ; Lb 2 Pn;1 6 L(xi ) ; Lb (xi ) = f (xi) ; f (xi) = 0 47< 1-8B i = 1 : : : n , ..8. 6?88. n 0I789 1 .,52/B x1 : : : xn . 784,1/.87C0,, L ; Lb 0. (83 4,2/3/.87C-.1/). (

& " ) ' " xi const en=2 . 1.

. .

!"#$ $ %$

x4. *,4 *

12

(

8 " %

6%8

=806? 56-7, /:6A?8.658-26B ,;8:/=69, .:8DI8?EB 0/ 1E56-78068 30/5806< ?0,>,5780/ />:/0@/ 1 .,528 x ;, A,:?I78 (1). / 1E56-78068 2/@4,>, 63 li (x) .:8DI8.-< 2(n ; 1) ,;8:/=69 1E56./06<, 2(n ; 2) ,;8:/=69 I?0,@806< 6 ,40/ ,;8:/=6< 487806<. /26? ,D:/3,?, 47< 1E56-7806< l1(x) : : : ln(x) .:8DI8.-< 2n(n ; 1) ,;8:/=69 -7,@806<, 2n(n ; 2) ,;8:/=69 I?0,@806< 6 n ,;8:/=69 n 487806<. 7< 1E56-7806< L(x) = P f (xi )li(x) .:8DI8.-< 8H8 (n ; 1) ,;8:/=69 i=1 -7,@806< 6 n ,;8:/=69 I?0,@806<. DH88 56-7, ,;8:/=69: 2n2 + O(n) (n ! 1) /446.610EB 6 -.,7C2, @8 ?I7C.6;762/.610EB ,;8:/=69. 6-7, /:6A?8.658-26B ,;8:/=69, .:8DI8?EB 0/ 1E56-78068 30/5806< ?0,>,5780/ />:/0@/ 1 .,528 x ;, A,:?I78 (1), ?,@8. DE.C 78>2, I?80CF80, 1 41/ :/3/. 1. E56-76? Yi = f (xi) Qn 1 47< 1-8B i = 1 : : : n ( Yi ?,@0, B:/(xi ; xj ) j =1j 6=i 06.C 0/ ?8-.8 f (xi )). / J., .:8DI8.-< n(n ; 1) /446.610EB 6 -.,7C2, @8 ?I7C.6;762/.610EB ,;8:/=69. n 2. E56-76? '(x) = Q (x ; xi ). / J., .:8DI8.-< n /446.610EB 6 n ; 1 i=1 ?I7C.6;762/.610EB ,;8:/=69. n 3. E56-76? L(x) = '(x) P x ;Yi x . / J., .:8DI8.-< n ; 1 /446.610EB 6 i=1 i n + 1 ?I7C.6;762/.610EB ,;8:/=69. (3 J.,9 A,:?I7E ./2@8 1640,, 5., Yi ?,@0, 08 3/;,?60/.C, / 1E56-7<.C ;, ?8:8 08,DB,46?,-.6). DH88 56-7, ,;8:/=69: n2 + O(n) (n ! 1) /446.610EB 6 -.,7C2, @8 ?I7C.6;762/.610EB ,;8:/=69. E56-78068 ;, A,:?I78 (1) (6 ;8:1E?, 6 1.,:E? -;,-,D/?6) .:8DI8. D,7CF6B 1E56-76.87C0EB 3/.:/. 6 ;:61,46. 2 ;,<17806G D,7CF,9 1E56-76.87C0,9 ;,>:8F0,-.6. ,J.,?I DE76 :/3:/D,./0E 4:I>68 -;,-,DE 1E56-7806< 60.8:;,7<=6,00,>, ?0,>,5780/ />:/0@/. x

4.

I-.C 3/4/0E .,526 a = x1 < x2 < : : : < xn = b , AI02=6< f 2 C (Ta b]) 6 6318-.0E 30/5806< f (x1) : : : f (xn). 6%$". * AI02=66 f 0I781,>, ;,:<42/ 0/3E1/8.-< f (xi ) = f (xi ), :/3487800,9 :/30,-.CG 1->, ;,:<42/ 0/3E1/8.-< ; f (xi) , :/3487800,9 :/30,-.CG k ->, ;,:<42/ 0/3E1/8.-< f (xiZ xj ) = f (xxj ) ; x j

i

f (xiZ : : : Z xi+k ) = f (xi+1 Z : : : Z xi+xk ) ;;fx(xiZ : : : Z xi+k;1) : i+k

. .

i

(1)

!"#$ $ %$

x4. *,4 *

13

(

&& %7$"' %7 1. +

f (xiZ : : : Z xi+k ) =

i+k X j =i

iQ +k l=i l6=j

f (x j ) : (xj ; xl )

(2)

7"#&. :6 k = 0 :/180-.1, (2) 6?88. 164 f (xi ) = f (xi ) 6 ;,.,?I 18:0,. :6 k = 1 :/180-.1, (2) 6?88. 164

f (xi ) f (xiZ xi+1 ) = x f;(xxi ) + xf (xi;+1x) = f (xxi+1 ) ; i i+1 i+1 i i+1 ; xi 6 18:0, 1 -67I ,;:8487806< (1) :/3487800,9 :/30,-.6. I-.C :/180-.1, (2) 18:0, ;:6 082,.,:,? k . ,2/@8?, 5., ,0, 18:0, ;:6 k + 1 (.8? -/?E? 78??/ DI48. 4,2/3/0/ ;, 604I2=66). -67I ,;:8487806< (1) :/3487800,9 :/30,-.6 f (xiZ : : : Z xi+k ) = f (xiZ : : : Z xi+k+1)= f (xi+1 Z : : : Z xxi+k+1) ; i+k+1 ; xi 0 1 BBi+k+1 CC i +k X X B f (xj ) ; f (xj ) C CC = x 1 ;x B B i+Q k+1 iQ +k i+k+1 iB (xj ; xl ) j=i (xj ; xl ) C @j=i+1 A l=i+1 l6=j

l=i l6=j

E56-76? 2,JAA6=680.E ;:6 f (xi) f (xj ) j = i +1 : : : i + k f (xi+k+1), 6 ;,2/@8?, 5., ,06 -,1;/4/G. - -,,.18.-.1IGH6?6 2,JAA6=680./?6 1 :/180-.18 (2). 1. ,JAA6=680. ;:6 f (xi) :/180

0 1 BB CC BB CC 1 1 1 1 1 ; = = : B C i + k i + k i + k +1 Q Q xi+k+1 ; xi B x ; x C i i +k+1 Q (xi ; xl ) A (xi ; xl ) (xi ; xl ) @ l=i l6=i

l=i l6=i

l =i l6=i

2. ,JAA6=680. ;:6 f (xj ) j = i + 1 : : : i + k :/180

0 1 BB CC BB CC 1 1 1 ; B CC = i + k +1 i + k Q Q xi+k+1 ; xi B (xj ; xl ) A @ (xj ; xl ) l=i+1 l6=j

. .

l=i l6=j

!"#$ $ %$

x4. *,4 *

14

(

!

1 1 1 = ; i + k xi+k+1 ; xi Q (xj ; xl ) xj ; xi+k+1 xj ; xi = 1

=x

1 i+k+1

l=i+1 l6=j

;

1

+k xi iQ (xj l=i+1 l6=j

xi+k+1 ; xi 1 = : i + k +1 Q ( x ; x j i +k+1 )(xj ; xi ) ; xl) (xj ; xl ) l =i l6=i

3. ,JAA6=680. ;:6 f (xi+k+1) :/180 1 1 = xi+k+1 ; xi i+Qk+1 (x i+k+1 ; xl ) l=i+1 l6=i+k+1

i+Q k+1

1

(xi+k+1 ; xl )

:

l=i l6=i+k+1

/26? ,D:/3,?, 2,JAA6=680. ;:6 f (xj ) j = i : : : i + k + 1 :/180 1 i+Q k+1 (xj ; xl ) l=i l6=j

6 ;,.,?I

f (xiZ : : : Z xi+k+1) = 8??/ 4,2/3/0/. 2. * " .

7"#&

i+X k+1 j =i

f (xj ) : (xj ; xl )

i+Q k+1 l=i l6=j

f (xiZ : : : Z xi+k )

1E.82/8. 63 :/180-.1/ (2).

i s r i + k s 6= r

; f (xiZ : : : Z xs;1Z xs+1Z : : : Z xi+k ) : f (xiZ : : : Z xi+k ) = f (xi Z : : : Z xr;1Z xr+1Z : : : Z xi+xk ) ; s xr

7"#&. 83 ,>:/065806< ,DH0,-.6 DI48? -56./.C, 5., r < s . D,30/56? yi = xr yj+1 = xj j = i : : : r ; 1 yj = xj j = r + 1 : : : s ; 1 yj = xj+1 j = s : : : i + k ; 1 yi+k = xs: 0482-E .,582 y ;,7I5/G.-< 63 60482-,1 .,582 x ;I.8? 489-.16< ;8:8-./0,126

i i+ 1 ::: r ; 1 r r +1 ::: s ;1 s s+1 ::: i +k ! i+1 i+ 2 ::: r i r +1 ::: s ;1 i +k s ::: i+k ;1 3. , "

. .

!"#$ $ %$

x4. *,4 *

15

(

(;,4B,46. 7GD/< ,D:/.6?/< ;8:8-./0,12/, ./2/<, 5., yi = xr 6 yi+k = xs ). -67I -6??8.:650,-.6 :/3487800,9 :/30,-.6 6 ,;:8487806< (1) f (xi Z : : : Z xi+k ) = f (yiZ : : : Z yi+k) = f (yi+1Z : : : Z yi+yk ) ;;fy(yiZ : : : Z yi+k;1) = i+k i ; f (xiZ : : : Z xs;1Z xs+1Z : : : Z xi+k) : = f (xiZ : : : Z xr;1Z xr+1 Z : : : Z xi+xk ) ; x s

8??/ 4,2/3/0/.

r

f (xiZ : : : Z xi+k ) f , .. " 2 R1 (f + g)(xiZ : : : Z xi+k ) = f (xiZ : : : Z xi+k ) + g(xiZ : : : Z xi+k ):

7"#& 1E.82/8. 63 :/180-.1/ (2). D,30/56? 58:83 Lm 60.8:;,7<=6,00E9 ?0,>,5780 />:/0@/ (47< AI02=66 f ), ;,-.:,800E9 ;, .,52/? x1 : : : xm (-.8;806 m ; 1). 4. *

5. +

Lk+1(x) = Lk (x) + f (x1 Z : : : Z xk+1)(x ; x1 ) : : : (x ; xk ): (3)

7"#&. /--?,.:6? P (x) = Lk+1 (x) ; Lk (x) { ?0,>,5780 -.8;806 k . ,-2,7C2I P (xi) = Lk+1 (xi) ; Lk (xi ) = f (xi ) ; f (xi) = 0 47< 1-8B i = 1 : : : k (..8. x1 : : : xk 8-.C (1-8) 2,:06 ?0,>,5780/ P -.8;806 k ), ., P (x) = A(x ; x1 ) : : : (x ; xk ), A { ;,-.,<00/<, 30/58068 2,.,:,9 ?,@8. DE.C ,;:848780, 63 I-7,16< A(xk+1 ; x1) : : : (xk+1 ; xk ) = P (xk+1) = Lk+1(xk+1 ) ; Lk (xk+1) = k k x X Y k+1 ; xi = = f (xk+1) ; Lk (xk+1) = f (xk+1) ; f (xj ) j =1 i=1 xj ; xi k X

;

;

i6=j

; xi = ; xi

k Y = f (xk+1) + f (xj ) xxk+1 x xj xxk+1 j k+1 i=1 j j =1 i6=j Qk (x k k+1 xi ) X i =1 = f (xk+1) + f (xj ) kQ+1 = j =1 (xj xi) i=1 i6=j

; ;

0 1 B CC k B k X B CC Y 1 1 + f (xj ) kQ+1 =B f (xk+1) Qk B CC (xk+1 ; xi ) = B j =1 (xk+1 ; xi ) (xj ; xi) A i=1 @ i=1

. .

i=1 i6=j

!"#$ $ %$

x4. *,4 * =

kX +1 j =1

16

(

k f (xj ) Y (xk+1 ; xi) = f (x1 Z : : : Z xk+1)(xk+1 ; x1) : : : (xk+1 ; xk ) kQ +1 (xj ; xi ) i=1 i=1 i6=j

(1 ;,-78408? :/180-.18 ?E 6-;,7C3,1/76 78??I 1). 3 J.,>, I:/10806< 0/B,46?: A = f (x1 Z : : : Z xk+1). 784,1/.87C0,, Lk+1(x) ; Lk (x) = f (x1 Z : : : Z xk+1)(x ; x1 ) : : : (x ; xk ). I48? ,D,30/5/.C (2,>4/ J., 08 ;:61,46. 2 ;I./06=8) 58:83 Lk 60.8:;,7<=6,00E9 ?0,>,5780 />:/0@/ -.8;806 k ; 1, ;,-.:,800E9 ;, k .,52/? xi : : : xi+k;1 . ,>4/ 78??/ 5 ?,@8. DE.C 3/;6-/0/ 1 1648:

Lk+1 (x) = Lk (x) + f (xiZ : : : Z xi+k )(x ; xi) : : : (x ; xi+k;1)

(4)

>48 Lk+1 60.8:;,7<=6,00E9 ?0,>,5780 />:/0@/ -.8;806 k , ;,-.:,800E9 ;, k + 1 .,528 xi : : : xi+k . 6.

C (k)(Ty1 y2]) .

y1 = minfxi : : : xi+k g , y2 = maxfxi : : : xi+k g 2 Ty1 y2 ] ,

f

2

(k) (5) f (xiZ : : : Z xi+k ) = f k!( ) :

7"#&. ,-.:,6? 47< 47< AI02=66 f (x) 60.8:;,7<=6,00E9 ?0,>,5780 Lk+1 (x) -.8;806 k ;, k + 1 .,528 xi : : : xi+k (..8. Lk+1(xj ) = f (xj ), j = i i + 1 : : : i + k ), 47< 58>, 1,-;,7C3I8?-< (5):

Lk+1 (x) = Lk (x) + f (xiZ : : : Z xi+k )(x ; xi) : : : (x ; xi+k;1)

(6)

'(x) = f (x) ; Lk+1(x)

(7)

>48 Lk 60.8:;,7<=6,00E9 ?0,>,5780 />:/0@/ -.8;806 k ; 1, ;,-.:,800E9 ;, k .,52/? xi : : : xi+k;1 . 7< AI02=66 6?88? 1 -67I ;,-.:,806< Lk+1

'(xj ) = f (xj ) ; Lk+1(xj ) = f (xj ) ; f (xj ) = 0 47< 1-8B j = i : : : i + k

..8. ' 6?88. 08 ?8088 k + 1 0I7< 0/ ,.:8328 Ty1 y2]. , .8,:8?8 ,77< ;:,631,40/< '0 6?88. 08 ?8088 k 0I789 0/ ,.:8328 Ty1 y2]. , .8,:8?8 ,77<, ;:6?80800,9 2 AI02=66 '0 , AI02=6< '00 6?88. 08 ?8088 k ; 1 0I7< 0/ ,.:8328 Ty1 y2]. :,4,7@/< ./2 6 4/7CF8, 0/B,46?, 5., AI02=6< '(k) 6?88. ;, 2:/9089 ?8:8 ,460 0I7C 0/ ,.:8328 Ty1 y2]. D,30/56? 8>, : '(k) ( ) = 0 2 Ty1 y2]. -67I (7)

'(k)( ) = f (k)( ) ; L(kk+1) ( ) = 0

. .

!"#$ $ %$

x5. (',./ . 81, 2 ( 1 -67I (6)

17

L(kk+1) ( ) = k!f (xiZ : : : Z xi+k )

(./2 2/2 k -< ;:,631,40/< ?0,>,5780/ Lk -.8;806 k ; 1 :/10/ 0I7G, / k -< ;:,631,40/< ?0,>,5780/ Lk+1 -.8;806 k :/10/ 8>, -./:F8?I 2,JAA6=680.I, I?0,@800,?I 0/ k!). 3 ;,-78406B 41IB :/180-.1 ;,7I5/8?

f (k)( ) = k! f (xiZ : : : Z xi+k ) ,.2I4/ 1E.82/8. .:8DI8?,8 :/180-.1, (5). x

5. # $

I-.C .:8DI8.-< ;,-.:,6.C Ln(x) { 60.8:;,7<=6,00E9 ?0,>,5780 />:/0@/ -.8;806 n ; 1 ;, .,52/? x1 : : : xn . :84-./16? 8>, 1 1648

Ln(x) = L1 (x) + (L2 (x) ; L1 (x)) + : : : + (Ln(x) ; Ln;1(x)) 6 ;:6?806? 348-C 1 2/@4,? -7/>/8?,? 78??I 4.5. ,7I56?

Ln (x) = f (x1 ) + f (x1Z x2 )(x ; x1 ) + : : : + f (x1Z : : : Z xn)(x ; x1 ) : : : (x ; xn;1): (1) /2/< 3/;6-C 60.8:;,7<=6,00,>, ?0,>,5780/ />:/0@/ 0/3E1/8.-< - . /180-.1, (1) 5/-., 3/;6-E1/G. 1 1648

Ln(x) =

-

n X i=1

(x ; x1 ) : : : (x ; xi;1 )f (x1 Z : : : Z xi )

(2)

i;1 >48 (x ; x1 ) : : : (x ; xi;1 ) = Q (x ; xj ) -56./8.-< 1, 8-76 i = 1. j =1

"*% &'"< %6"<8* *" & #=

%

1. '" %7$"' %7 f (x1 Z : : : Z xi) i = 1 : : : n , I5/-.1IGH6B 1 ;:84-./178066 (2) 60.8:;,7<=6,00,>, ?0,>,5780/ 1 A,:?8 CG.,0/.

. .

!"#$ $ %$

x6. (',./ . ") 1;

." 5+,

18

,7C3I<-C ,;:84878068? (4.1) :/3487800EB :/30,-.89, 3/;,70<8? ./D76=I f (x1) n f (x1 Z x2) n . f (x2) n f (x1 Z x2Z x3 ) n . f (x2 Z x3) n f (x1 Z x2Z x3 Z x4) . . f (x3) n f (x2 Z x3Z x4 ) n

n . f (x3 Z x4) n f (x1 Z : : : Z xn) . . f (x4) n

... ... ... . .. f (xn;1)n f (xn;2Z xn;1 Z xn) f (xn;1 Z xn). f (xn) . .,7D=E 1E56-7
8 " %

6%8

1. / 1E56-78068 1-8B f (x1Z : : : Z xi) i = 1 : : : n ;, 637,@800,?I 1EF8 /7>,:6.?I 0/4, 1E56-76.C n(n ; 1)=2 :/3487800EB :/30,-.89 (J78?80.,1 ./D76=E). ,>7/-0, 6B ,;:8487806G (4.1) 47< 1E56-7806< 2/@4,9 .:8DI8.-< 2 1E56./06< 6 1 4878068. DH88 2,7658-.1, ,;8:/=69: n(n ; 1) 1E56./069 6 n(n ; 1)=2 4878069. 2. E56-78068 30/5806< 60.8:;,7<=6,00,>, ?0,>,5780/ 1 .,528 ;, A,:?I78 (3) .:8DI8. (n ; 1) I?0,@8069, (n ; 1) 1E56./069 6 (n ; 1) -7,@8069. .?8.6?, 5., 8-76 .:8DI8.-< 1E56-76.C 08-2,7C2, 30/58069 60.8:;,7<=6,00,>, ?0,>,5780/ (2) 1 :/37650EB .,52/B, ., :/3487800E8 :/30,-.6 f (x1 Z : : : Z xi ) i = 1 : : : n 1E56-7,5780/ 1 .,528 1E56-7
6. " ' "

I-.C 3/4/0E .,526 a = x1 < x2 < : : : < xn = b , 30/5806< f (x1) : : : f (xn) 6 =87,56-7800E9 ;/:/?8.: 1 k n ; 1. 0.8:;,76:IGH/< AI02=6< Pf -.:,6.-< ./26? ,D:/3,?, 5., 0/ 2/@4,? ,.:8328 Txi xi+1 ] i = 1 : : : n ; k ,0/ -,1;/4/8.

. .

!"#$ $ %$

x6. (',./ . ") 1;

." 5+,

19

- ?0,>,5780,? Pi -.8;806 k , ./26?, 5., Pi(xj ) = f (xj ) j = i : : : i + k (..8. Pi { 60.8:;,7<=6,00E9 ?0,>,5780 />:/0@/ -.8;806 k , ;,-.:,800E9 ;, .,52/? xi : : : xi+k ). / ,.:8328 Txn;k xn] 60.8:;,76:IGH/< AI02=6< Pf -,1;/4/8. Pn;k { 60.8:;,7<=6,00E? ?0,>,5780,? />:/0@/ -.8;806 k , ;,-.:,800E? ;, .,52/? xn;k : : : xn . , 60.8:;,7<=6,00,9 A,:?I78 CG.,0/

Pi(x) =

i+k X j =i

(x ; xi ) : : : (x ; xj;1)f (xiZ : : : Z xj ):

(1)

,JAA6=680.E Pi , i = n ; k n ; k ; 1 : : : 1 1E56-7,5780/ Pn;k : f (xn;k ) f (xn;k Z xn;k+1) : : : f (xn;k Z : : : Z xn): 3/ k(k +1) 1E56./069 6 k(k +1)=2 4878069 (;,-2,7C2I Pn;k { 60.8:;,7<=6,00E9 ?0,>,5780, ;,-.:,800E9 ;, k +1 .,528 xn;k : : : xn ). /.8? ./D76=/ 3/;,70<8.-< ;, 46/>,0/76: 1E56-7
. .

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20

( 81,4 2 (

3/ 2k 1E56./069 6 k 4878069 (J.I ;:,=84I:I 0/3E1/G. /7>,:6.?,? -416>/ ;:84-./17806< ?0,>,5780/ 1 A,:?8 CG.,0/), 6 4/788 f (xn;k;2) f (xn;k;2Z xn;k;1) : : : f (xn;k;2Z : : : Z xn;2) ::: ::: ::: ::: f (x1 ) f (x1Z x2 ) : : : f (x1Z : : : Z xk+1): -8>, 0/4, 3/;,706.C n ; k ; 1 46/>,0/789. / J., ;,.:8DI8.-< 2k(n ; k ; 1) 1E56./069 6 k(n ; k ; 1) 4878069. DH68 3/.:/.E 0/ 1E56-78068 1-8B 2,JAA6=680.,1: k(k +1)+2k(n;k ;1) = 2kn;k(k +1) 1E56./069 6 k(k +1)=2+k(n;k ;1) = kn ; k(k + 1)=2 4878069. '" 7< %6"%!=> !8 Pf & x ,-IH8-.17<8.-< ;, -784IGH8?I /7>,:6.?I. ;:8487<8? i 63 I-7,16< x 2 Txi xi+1 ]. -76 i > n ; k , ., ;,7/>/8? i = n ; k . ,>4/ (Pf )(x) = Pi(x). E56-78068 30/5806< ?0,>,5780/ (1) -.8;8066 k 1 .,528 x ,-IH8-.17<8.-< ;, -B8?8 ,:08:/ Pi(x) = f (xi) + (x ; xi)(f (xiZ xi+1 ) + (x ; xi+1 )(f (xiZ xi+1Z xi+2 ) + : : : + +(x ; xi+k;2 )(f (xiZ : : : Z xi+k;1) + (x ; xi+k;1)f (xiZ : : : Z xi+k ) |) :{z: :}) k;1

- 3/.:/.,9 k 1E56./069, k -7,@8069 6 k I?0,@8069.

7. ) # $ x

J.,? :/34878 ?E ;,7I56? ,=8026 ;,>:8F0,-.6 60.8:;,7<=66 - ;,?,HCG 60.8:;,7<=6,00,>, ?0,>,5780/ />:/0@/, 3/;6-/00,>, 1 A,:?8 CG.,0/.

f 2 C (Ta b]) fxi gni=1 { Ta b] , x 2 R1 f (x) = Ln(x) + (x ; x1) : : : (x ; xn )f (x1Z : : : xnZ x) Pn (x ; x ) : : : (x ; x )f (x Z : : : Z x ) . Ln (x) = 1 i;1 1 i

% 1.

(1)

i=1

. 83 ,>:/065806< ,DH0,-.6 ?,@0, -56./.C, 5., x 6= xj j = 1 : : : n , ;,-2,7C2I 1 .,52/B xj ;,>:8F0,-.C 60.8:;,7<=66 :/10/ 0I7G 6 J., -,,.18.-.1I8. (1). :6 n = 1 :/180-.1, (1) 6?88. 164 f (x) = f (x1) + (x ; x1)f (x1 Z x) 6 -;:/184761, 1 -67I ,;:8487806< (4.1) :/3487800,9 :/30,-.6. I-.C :/180-.1, (1) 4,2/3/0, ;:6 n = k : f (x) = Lk (x) + (x ; x1 ) : : : (x ; xk )f (x1 Z : : : xk Z x): (2)

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21

-67I 78??E (4.3)

f (x1 Z : : : Z xk Z xk+1) : f (x1 Z : : : Z xk Z xk+1 x) = f (x1 Z : : : Z xk Z xx) ; ;x k+1

.-G4/ 6?88?

f (x1 Z : : : Z xk Z x) = f (x1Z : : : Z xk Z xk+1) + (x ; xk+1)f (x1 Z : : : Z xk Z xk+1 x):

(3)

,4-./16? (3) 1 (2), ;,7I56?

f (x) = Lk (x)+(x ; x1) : : : (x ; xk )f (x1 Z : : : Z xk+1) +(x ; x1) : : : (x ; xk+1)f (x1 Z : : : Z xk Z xk+1 x):

(4)

-67I 78??E 4.5 -;:/184761, ;:84-./178068 (4.3)

Lk (x) + f (x1 Z : : : Z xk+1)(x ; x1) : : : (x ; xk ) = Lk+1 (x)

6 ;,.,?I (4) ;:606?/8. 164

f (x) = Lk+1(x) + (x ; x1 ) : : : (x ; xk+1 )f (x1Z : : : Z xk Z xk+1 x)

..8. :/180-.1, (1) -;:/184761, 47< n = k + 1. 8? -/?E? .8,:8?/ 4,2/3/0/ ;, 604I2=66.

f 2 C (n) (Ta b]) fxi gni=1 { Ta b] , x 2 Ta b] = (x) 2 Ty1 y2] ( y1 = y1(x) = minfx1 : : : xn xg , y2 = y2(x) = maxfx1 : : : xn xg) ,

% 2.

(n) f (x) = Ln (x) + (x ; x1 ) : : : (x ; xn) f n!( )

(5)

Ln(x) = P (x ; x1 ) : : : (x ; xi;1 )f (x1Z : : : Z xi) . n

i=1

. , .8,:8?8 1

7"#&

f (x) = Ln(x) + (x ; x1) : : : (x ; xn )f (x1Z : : : xnZ x):

(6)

, 78??8 4.6 -IH8-.1I8. = (x) 2 Ty1 y2] (>48 y1 = y1(x) = minfx1 : : : xn xg , y2 = y2(x) = maxfx1 : : : xn xg) ./2,8, 5., (n) f (x1 Z : : : Z xnZ x) = f n!( ) : ,4-./17<< J., 1 (6), ;,7I5/8? .:8DI8?,8 :/180-.1, (5).

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22

8.

. I-.C 3/4/0 0/D,: .,582 xi : : : xi+k (08 ,D<3/.87C0, :/37650EB). * AI02=66 f 0I781,>, ;,:<42/ 0/3E1/8.-< f (xi) = f (xi), :/3487800,9 :/30,-.CG k ->, ;,:<42/ 0/3E1/8.-< 81 > (k) (k ) > > < k! f (xi) 8-76 xi = : : : = xi+k 6 f 2 C f (xiZ : : : Z xi+k )= > f (xiZ : : : Z xr;1Z xr+1Z : : : Z xi+k ) ; f (xiZ : : : Z xs;1Z xs+1Z : : : Z xi+k ) > xs ; xr > : 8-76 -IH8-.1IG. xs 6= xr (1) (J., ,;:84878068 2,::82.0,, ..8. 08 3/16-6. ,. 1ED,:/ .,582 xr 6 xs , 1 -67I -6??8.:66 :/3487800,9 :/30,-.6 ;, -1,6? /:>I?80./?, 4,2/3/.87C-.1, -?. 78??I 4.3). 6%$"

f 2 C (k) , f (xiZ : : : Z xi+k ) k + 1 xi : : : xi+k .

1.

"

-

7"#&. :6 k = 0 :/3487800/< :/30,-.C f (xi ) -,1;/4/8. - f (xi ), 2,.,:/< 08;:8:E10/ ;, I-7,16G. I-.C 78??/ 4,2/3/0/ 47< :/3487800EB :/30,-.89 ;,:<42/ k ;1. /--?,.:6? ;:,631,7C0IG .,52I (xiZ : : : Z xi+k ) 2 Rk+1 . -76 -IH8-.1IG. i s r i + k ./268, 5., xs 6= xr , ., 1 -67I ,;:8487806< (1) :/3487800/< :/30,-.C f (xiZ : : : Z xi+k ) 08;:8:E10/ 2/2 760890/< 2,?D60/=6< 08;:8:E10EB (;, ;:84;,7,@806G 604I2=66) AI02=69: ; f (xiZ : : : Z xs;1Z xs+1Z : : : Z xi+k ) : f (xiZ : : : Z xi+k ) = f (xi Z : : : Z xr;1Z xr+1Z : : : Z xi+xk ) ; s xr -76 xi = : : : = xi+k , ., ;, ,;:8487806G (1) f (xiZ : : : Z xi+k ) = k1! f (k)(xi ): ,-2,7C2I ;, I-7,16G f (k) 08;:8:E10/, ., 47< 1-<2,>, " > 0 -IH8-.1I8. = (") > 0 ./2,8, 5., 47< 1-8B jxi ; j <

jf k (xi) ; f k ()j < ": ( )

( )

I-.C (x0i Z : : : Z x0i+k ) 2 Rk+1 { ;:,631,7C0/< .,52/, ./2/<, 5., jx0j ; xij < , j = i i + 1 : : : i + k . -67I 78??E 4.6 -IH8-.1I8. 2 Ty1 y2] (>48 y1 = minfx0i : : : x0i+k g , y2 = maxfx0i : : : x0i+k g ) ./2,8, 5., (k) f (x0iZ : : : Z x0i+k ) = f k!( ) :

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/2 2/2 ;, 1ED,:I (x0iZ : : : Z x0i+k ) 2 Rk+1 Ty1 y2] (xi ; xi + ), .,

jf (xiZ : : : Z xi k) ; f (xiZ : : : Z xi k)j = k1! jf k (xi) ; f k ()j < k1! " 0

+

0

+

( )

( )

5., 6 ,30/5/8. 08;:8:E10,-.C :/3487800,9 :/30,-.6 1 .,528 (xi Z : : : Z xi+k ). x

9.

I-.C 3/4/0E .,526 a = x1 < x2 : : : < xn = b 6 30/5806< f (i) (xj ) i = 0 : : : mj ; 1 j = 1 : : : n . :8DI8.-< ;,-.:,6.C 60.8:;,7<=6,00E9 ?0,>,5780 n Lm = Pf -.8;806 m ; 1 = P mi ; 1 ./2,9, 5., i=1

L(mi) (xj ) = f (i) (xj ) i = 0 : : : mj ; 1 j = 1 : : : n:

(1)

K.,. ?0,>,5780 0/3E1/G. , / 56-7/ m1 : : : mn { x1 : : : xn -,,.18.-.1800,. /2,9 ?0,>,5780 8460-.180080 -:846 1-8B ?0,>,5780,1 -.8;806 m ; 1. 89-.16.87C0,, 8-76 Qm { ?0,>,5780 -.8;806 m ; 1, I4,178.1,:,5780 Lm ; Qm -.8;806 08 1EF8 m ; 1 6?88. 1-8>, P mi = m i=1 0I789 (- I58.,? 2:/.0,-.6). ,J.,?I Lm ; Qm = 0.

% 1. /

(1), '

Lm (x) =

Lm ,

-

m X i=1

(x ; y1) : : : (x ; yi;1) f (y1Z : : : Z yi)

(2)

y1 = : : : = ym1 = x1 ym1 +1 = : : : = ym1 +m2 = x2 : : : yP =1;1 m +1 = : : : = ym = xn: (3) n i

i

. ,7,@6?

7"#&

x(ij") = xj + (i ; 1)" i = 1 : : : mj j = 1 : : : n

(4)

>48 0 < " < "0 "0 { ./2,8, 5., 1-8 .,526 fx(ij") g :/37650E 47< 1-8B " 2 (0 "0). ,-.:,6? ;, m .,52/? fx(ij") g 60.8:;,7<=6,00E9 ?0,>,5780 1 A,:?8 CG.,0/ Lm(") -.8;806 m ; 1 ./2, 5.,

L(m") (x"ij ) = f (x"ij ) 47< 1-8B i = 1 : : : mj j = 1 : : : n 0 < " < "0:

. .

(5)

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-67I 8460-.1800,-.6 ./2,>, ?0,>,5780/ (")

Lm (x) =

m X i=1

") ) f (y (")Z : : : Z y (") ) (x ; y1(")) : : : (x ; yi(;1 1 i

(6)

47< 1-<2,>, -;,-,D/ ;8:80I?8:,1/.C 0/D,:,? y1(") : : : ym(") 0/D,: fx(ij") g . ,J.,?I 0/? 4,-./.,50, 4,2/3/.C, 5., -,,.0,F806< (1) 1E;,7080E 47< .,526 x1 . ,7,@6? y1(") = x1("1) : : : ym("1) = xm1 1 ym("1) +1 = x(1"2) : : : ym("1) +m2 = x(m")2 2 : : : (7) (") (") ( ") (") yP ;1 m +1 = x1n : : : ym = xm n : n

=1

i

n

i

3 (7), (4), (3) 1E.82/8., 5., "lim yj(") = yj 47< 1-8B j = 1 : : : m . -67I 08;:8!0 :E10,-.6 :/3487800EB :/30,-.89 (78??/ 8.1) 1 :/180-.18 (6) ?,@0, ;8:89.6 2 ;:8487I ;, " ! 0 (;,7I561F69-< 1 :83I7C./.8 ?0,>,5780 ,D,30/5/8? Lm ): m X Lm (x) = (x ; y1) : : : (x ; yi;1) f (y1Z : : : Z yi) = i=1 m1 X = (x ; x1 )i;1f (|x1 Z :{z: : Z x1})+ i=1 i :/3 m X +(x ; x1 )m1 (x ; ym1 +1) : : : (x ; yi;1)f (y1Z : : : Z yi) = i=m1 +1

(8)

m1 (i;1) x ) X 1 (x ; x1 )i;1+ = f(i ; (1)! i=1 m X +(x ; x1 )m1 (x ; ym1 +1) : : : (x ; yi;1)f (y1Z : : : Z yi) i=m1 +1

(1 ;,-78408? :/180-.18 6-;,7C3,1/0, ,;:84878068 (8.1) :/3487800,9 :/30,-.6 2:/.0E?6 I37/?6). 3 J.,>, ;:84-./17806< -784I8. L(mi) (x1 ) = f (i) (x1 ) i = 0 : : : m1 ; 1 (;,-2,7C2I ;:,631,40E8 4, ;,:<42/ m1 ; 1 127G56.87C0, 1.,:,>, -7/>/8?,>, 1 (8) :/10E 0I7G 1 .,528 x1 ). 8? -/?E? 4,2/3/0,, 5., ?0,>,5780 164/ (2) I4,178.1,:<8. I-7,16,5780 0/3E1/8.-< 1 A,:?8 CG.,0/.

% 2. f 2 C (m) (Ta b]) (m = max mj ) fxi gn i=1 { j =1:::n Ta b] , x 2 R1 f (x) = Lm (x) + (x ; x1 )m1 : : : (x ; xn)m f (x| 1 Z :{z: : Z x1}Z : : : x| nZ :{z: : Z xn}Z x) (9) n

Lm (x)

. .

(2), (3).

m1

m

n

!"#$ $ %$

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25

(4 1*,

. /;6F8? :/180-.1, (9) 1 J2161/780.0,9 A,:?8 f (x) = Lm (x) + (x ; y1) : : : (x ; ym)f (y1Z : : : ymZ x) (10) >48 .,526 y1 : : : ym 3/4/G.-< (3), 6 DI48? 4,2/3E1/.C 1?8-., (9) :/180-.1, (10). :6 J.,? ?,@0, -56./.C, 5., x 6= yj j = 1 : : : m , ;,-2,7C2I 1 .,52/B yj ;,>:8F0,-.C 60.8:;,7<=66 :/10/ 0I7G 6 J., -,,.18.-.1I8. (10). /788 4,2/3/.87C-.1, ;,1.,:<8. 4,2/3/.87C-.1, .8,:8?E 7.1. :6 m = 1 :/180-.1, (9) 6?88. 164 (11) f (x) = f (y1) + (x ; y1)f (y1Z x) 6 -;:/184761, 1 -67I ,;:8487806< (8.1) :/3487800,9 :/30,-.6. I-.C :/180-.1, (9) 4,2/3/0, ;:6 m = k : f (x) = Lm (x) + (x ; y1) : : : (x ; yk )f (y1Z : : : yk Z x): (12) -67I 78??E (4.3) ; f (y1Z : : : Z ykZ yk+1) : f (y1Z : : : Z yk Z yk+1 x) = f (y1Z : : : Z yk Z xx) ; yk+1 .-G4/ 6?88? f (y1Z : : : Z yk Z x) = f (y1Z : : : Z yk Z yk+1) + (x ; yk+1)f (y1Z : : : Z yk Z yk+1 x): (13) ,4-./16? (13) 1 (12), ;,7I56? f (x) = Lk (x)+(x ; y1) : : : (x ; yk )f (y1Z : : : Z yk+1) (14) +(x ; y1) : : : (x ; yk+1)f (y1Z : : : Z yk Z yk+1 x): -67I .8,:8?E 2 -;:/184761, ;:84-./178068 Lk (x) + f (y1Z : : : Z yk+1)(x ; y1) : : : (x ; yk ) = Lk+1(x) 6 ;,.,?I (14) ;:606?/8. 164 f (x) = Lk+1(x) + (x ; y1) : : : (x ; yk+1)f (y1Z : : : Z yk Z yk+1 x) ..8. :/180-.1, (10) -;:/184761, 47< m = k + 1. 8? -/?E? .8,:8?/ 4,2/3/0/ ;, 604I2=66.

7"#&

f 2 C (m) (Ta b]) fxigni=1 { Ta b] , x 2 Ta b] = (x) 2 T1 2] ( 1 = 1(x) = minfx1 : : : xn xg , 2 = 2(x) = maxfx1 : : : xn xg) ,

(m) f (x) = Ln (x) + (x ; x1 )m1 : : : (x ; xn)m f m(! ) (15) Lm (x) (2), (3).

% 3.

n

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26

. 1848? 2/2 1 .8,:8?8 1 .,526 x(ij") , ;, 2,.,:E? ;,-.:,6? 60.8:;,7<=6,00E9 ?0,>,5780 1 A,:?8 CG.,0/ L(m") -.8;806 m ; 1, I4,178.1,:48 .,526 yj(") 3/4/G.-< (7)). , .8,:8?8 7.2 47< 1-<2,>, x 2 Ta b] -IH8-.1I8. (") = (")(x) 2 T1(") 2(")] (>48 1(") = 1(")(x) = minfy1(") : : : ym(") xg , 2(") = 2(")(x) = maxfy1(") : : : ym(") xg) ./2,8, 5., (m) (") f (x) = L(m") (x) + (x ; y1(")) : : : (x ; ym(")) f m(! ) : (16)

/2 ;,2/3/0, 1 .8,:8?8 1 "lim L(") (x) = Lm (x) (>48 Lm (x) 3/4/8.-< :/180-.1/!0 m ?6 (2), (3)). -67I 08;:8:E10,-.6 f (m) "lim f (m) ( (")) = f (m) ( ), >48 = "lim (") , !0 !0 ;:658? 2 T1 2], >48 1 -67I (3) 1 = 1(x) = minfx1 : : : xn xg , 2 = 2(x) = maxfx1 : : : xn xg). ,J.,?I, ;8:8B,4< 1 (16) 2 ;:8487I ;:6 " ! 0, ;,7I5/8? .:8DI8?,8 :/180-.1, (15).

7"#&

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7>,:6.? 1E56-7806< 60.8:;,7<=6,00,>, ?0,>,5780/ - 2:/.0E?6 I37/?6 Lm (x) 164/ (2), (3) .,. @8 -/?E9, 5., 47< 60.8:;,7<=6,00,>, ?0,>,5780/ 1 A,:?8 CG.,0/ D83 2:/.0EB I37,1 (-?. x 5), .,7C2, ;8:8-58. J78?80.,1 ./D76=E :/3487800EB :/30,-.89 ,-IH8-.17<8.-< ;, ,;:8487806G (8.1) :/3487800,9 :/30,-.6 - 2:/.0E?6 I37/?6. .%%. I-.C 1 n .,52/B x1 : : : xn 3/4/0E 30/5806< AI02=66 f (x): f (x1 ) : : : f (xn) 6 88 ;:,631,40,9 f 0(x): f 0(x1 ) : : : f 0(xn ) (..8. 1 ;,-./0,128 (1) 3/4/56 60.8:;,7<=66 - 2:/.0E?6 I37/?6 m1 = : : : = mn = 2). ,>4/ ./D76=/ :/3487800EB :/30,-.89 6?88. 164 f (x1) n f 0(x1) n . f (x1) n f (x1 Z x1Z x2 ) n . f (x1Z x2 ) n f (x1 Z x1Z x2 Z x2) . . f (x2) n f (x1 Z x2Z x2 ) n

. 0 f (x2) f (x1 Z x2Z x2 Z x3) n . .

f (x2) n f (x2 Z x2Z x3 ) n ... ... ... ... f (xn;1). f (xn;1Z xn;1 Z xn).

n n . f (xn;1Z xn)n f (xn;1Z xn;1 Z xnZ xn) . f (xn) n f (xn;1Z xn Z xn) . 0 f (xn) . f (xn) .,7D=E 1E56-7
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,2,05/066 1E56-78069 0/ ?8-.8 f (yi) DI48. 78@/.C f (y1Z : : : Z yi), >48 yi 3/4/G.-< (3). 8 " % 6%8, 08,DB,46?EB 47< ;,-.:,806< 60.8:;,7<=6,00,>, ?0,>,5780/ Lm (x) 164/ (2), (3), -784I8. 63 ,=8026 56-7/ ,;8:/=69, 08,DB,46?EB 47< ;,-.:,806< 60.8:;,7<=6,00,>, ?0,>,5780/ -.8;806 m ; 1 D83 2:/.0EB I37,1 (-?. x 5): .:8DI8.-< 08 D,788 m(m ; 1) 1E56./069 6 m(m ; 1)=2 4878069 (2,7658-.1, ,;8:/=69 DI48. ?80CF8 J.6B 1876560 63-3/ .,>,, 5., 5/-.C :/3487800EB :/30,-.89 1E56-7<8.-< 58:83 3/4/00E8 30/5806< ;:,631,40EB AI02=66).

10. # $

x

I-.C Ln(x) { ?0,>,5780 />:/0@/ AI02=66 f , ;,-.:,800E9 ;, .,52/? x1 : : : xn : n X Ln(x) = (x ; x1 ) : : : (x ; xi;1 )f (x1 Z : : : Z xi ): i=1

7< 082,.,:,>, k > 0 .:8DI8.-< 0/9.6 L(nk) (y) { k -G ;:,631,40IG Ln 1 082,.,:,9 .,528 y . -67I 8460-.1800,-.6 ?0,>,5780/ Ln (x) -.8;806 n ; 1, ;:606?/GH8>, 1 3/4/00EB .,52/B x1 : : : xn 3/4/00E8 30/5806< f (x1 ) : : : f (xn), ?0,>,5780 Ln(x) -,1;/4/8. - 60.8:;,7<=6,00E? ?0,>,5780,? 47< AI02=66 Ln(x) (-/?,>, -8D<):

Ln (x) =

n X i=1

(x ; x1 ) : : : (x ; xi;1 )Ln(x1 Z : : : Z xi):

(1)

-67I 8460-.1800,-.6 60.8:;,7<=6,00,>, ?0,>,5780/ Ln(x) -.8;806 n ; 1, ,0 -,1;/4/8. - 60.8:;,7<=6,00E? ?0,>,5780,?, ;:606?/GH6? 1 .,52/B y x1 : : : xn;1 30/5806< Ln(y) Ln(x1 ) : : : Ln(xn;1). , 60.8:;,7<=6,00,9 A,:?I78 CG.,0/ J.,. ?0,>,5780 6?88. 164

Ln(x) = Ln(y) + (x ; y) ,>4/

Ln (x)jx=y = 0

nX ;1 i=1

nX ;1 i=1

(x ; x1 ) : : : (x ; xi;1 )Ln(yZ x1Z : : : Z xi ):

(x ; x1 ) : : : (x ; xi;1)Ln (yZ x1Z : : : Z xi)

(2) (3)

6 .8? -/?E? 1E56-780/ 1-< ;:,631,40/< 1 .,528 y . "*% 1E56-7806< 30/5806< L0n (y ) 1 A,:?8 (3). /? .:8DI8.-< ;, 6318-.0E? 30/5806
Ln(y) Ln(x1 ) Ln(x1 Z x2 ) : : : Ln(x1Z : : : Z xn;1 )

. .

!"#$ $ %$

x10. )4+

28

, ' *)4A 5+, ,5

1E56-76.C

Ln(yZ x1) Ln(yZ x1Z x2) : : : Ln(yZ x1Z : : : Z xn;1) -;,7C3I8? 47< J.,>, /7>,:6.? -416>/ ;:84-./17806< ?0,>,5780/ 1 A,:?8 CG.,0/ (-?. x 6). ,D/16? .,52I y 6 30/58068 Ln (y) 1 ./D76=I :/3487800EB :/30,-.89 6 3/;,706? ;, ,;:8487806G :/3487800,9 :/30,-.6 Ln (yZ x1Z : : : Z xi;1 ) i = 1 : : : n ; 1 (4) Ln(yZ x1Z : : : Z xi) = Ln (x1Z : : : Z xi ) ; xi ; y J.I ./D76=I: Ln(y)

n

f (x1 )

. n

f (x2 )

. n

f (x3 )

. n

f (x4 ) ... f (xn;1)n

Ln (yZ x1)

n

f (x1 Z x2)

. n

f (x2 Z x3)

. n

f (x3 Z x4)

. n

. n

f (xn)

...

Ln(yZ x1Z x2)

n

f (x1Z x2 Z x3 )

. n

f (x2Z x3 Z x4 )

. n

...

f (xn;1 Z xn)

f (xn;2Z xn;1Z xn)

Ln(yZ x1Z x2Z x3 )

f (x Z x Z x Z x ) Ln(yZ x Z : : : Z xn

f (x Z : : : Z xn)

n

1

2

. ..

3

4

1

. n

.

;1

)

1

.

.

:I4,8?2,-.C 1E56-7806< L0n(y): 1. / 1E56-78068 n ; 1 :/3487800EB :/30,-.89 ;, A,:?I78 (4) .:8DI8.-< 2(n ; 1) 1E56./069 6 n ; 1 4878069. 2. / 1E56-78068 L0n (y) ;, A,:?I78 (3) ;, -B8?8 ,:08:/

L0n(y) = Ln(yZ x1) + (x ; x1 )(Ln(yZ x1Z x2 ) + (x ; x2 )(Ln(yZ x1Z x2Z x3 ) + : : : + +(x ; xn;3 )(Ln(yZ x1Z : : : Z xn;2 ) + (x ; xn;2)Ln (yZ x1Z : : : Z xn;1) |) :{z: :}) n;2

.:8DI8.-< n ; 2 1E56./069, n ; 2 -7,@8069 6 n ; 2 I?0,@8069. -8>, .:8DI8.-< 4(n ; 1) ; 2 /446.610EB ,;8:/=69 6 2(n ; 1) ; 1 ?I7C.6;762/.610EB ,;8:/=69. 7< 1E56-7806< L00n(y) /0/7,>650, ;:,4,7@/8?. -67I 8460-.1800,-.6 60.8:;,7<=6,00,>, ?0,>,5780/ Ln(x) -.8;806 n ; 1, ,0 -,1;/4/8. - 60.8:;,7<=6,00E? ?0,>,5780,?, ;:606?/GH6? 1 .,52/B y x1 : : : xn;2 30/5806< Ln (y) Ln(x1 ) : : : Ln(xn;2 ), / 1 .,528 y 6?8GH6? ;:,631,40IG, :/10IG L0n(y). , 60.8:;,7<=6,00,9 A,:?I78 CG.,0/ - 2:/.0E?6

. .

!"#$ $ %$

x10. )4+

29

, ' *)4A 5+, ,5

I37/?6 J.,. ?0,>,5780 6?88. 164 Ln (x) = n;2 = Ln(y) + (x ; y)Ln(yZ y) + (x ; y)2 P (x ; x1 ) : : : (x ; xi;1 )Ln(yZ yZ x1Z : : : Z xi ) = = Ln(y) + (x ; y)L0n(y) + (x ; y)

,>4/

Ln (x)jx=y = 2! 00

nX ;2 i=1

i=1 n ;2 P 2 i=1

(x ; x1 ) : : : (x ; xi;1 )Ln(yZ yZ x1Z : : : Z xi ):

(x ; x1 ) : : : (x ; xi;1)Ln (yZ yZ x1Z : : : Z xi)

(5) (6)

6 .8? -/?E? 1E56-780/ 2-< ;:,631,40/< 1 .,528 y . "*% 1E56-7806< 30/5806< L00n (y ) 1 A,:?8 (6). /? .:8DI8.-< ;, 6318-.0E? 30/5806
/2 6 1EF8 6-;,7C3I8? 47< J.,>, /7>,:6.? -416>/ ;:84-./17806< ?0,>,5780/ 1 A,:?8 CG.,0/ (-?. x 6). ,D/16? .,52I y 6 30/58068 Ln (y) 1 ./D76=I :/3487800EB :/30,-.89 6 3/;,706? ;, ,;:8487806G :/3487800,9 :/30,-.6 Ln(yZ yZ x1Z : : : Z xi;1) i = 1 : : : n ; 2 Ln(yZ yZ x1Z : : : Z xi) = Ln (yZ x1Z : : : Z xi) ; xi ; y (7) J.I ./D76=I: Ln (y) n L0n (y) n . Ln (y) n Ln(yZ yZ x1) n . Ln (yZ x1) n Ln(yZ yZ x1Z x2 Z ) . . f (x1 ) n Ln(yZ x1Z x2 ) n

. f (x1 Z x2) n Ln(yZ x1Z x2Z x3 ) Ln (yZ yZ x1Z : : : Z xn;2) . . . f (x2 ) n f (x1 Z x2Z x3 ) n

n f (x2 Z x3) . f ( x Z x Z x Z x ) Ln (yZ x1Z : : : Z xn;1) 1 2 3 4 n . . . f (x3 ) n f (x2 Z x3Z x4 ) n

n . f (x3 Z x4) n f (x1 Z : : : Z xn) . . f (x4 ) n

... ... ... . .. f (xn;1n) f (xn;2Z xn;1 Z xn) f (xn;1Z xn.) f (xn) .

. .

!"#$ $ %$

x10. )4+

30

, ' *)4A 5+, ,5

:I4,8?2,-.C 1E56-7806< L00n(y): 1. / 1E56-78068 n ; 2 :/3487800EB :/30,-.89 ;, A,:?I78 (7) .:8DI8.-< 2(n ; 2) 1E56./069 6 n ; 2 4878069. 2. / 1E56-78068 L00n (y) ;, A,:?I78 (6) ;, -B8?8 ,:08:/ L00n(y) = 2!(Ln(yZ yZ x1)+(x ; x1 )(Ln (yZ yZ x1Z x2 )+(x ; x2 )(Ln (yZ yZ x1Z x2 Z x3)+ : : : + +(x ; xn;4 )(Ln(yZ yZ x1Z : : : Z xn;3) + (x ; xn;3)Ln(yZ yZ x1Z : : : Z xn;2) )| :{z: :)} ) n;3

.:8DI8.-< n ; 3 1E56./069, n ; 3 -7,@8069 6 n ; 3 I?0,@8069. -8>, .:8DI8.-< 4(n ; 2) ; 2 /446.610EB ,;8:/=69 6 2(n ; 2) ; 1 ?I7C.6;762/.610EB ,;8:/=69. /--?,.:6? ?> "!. ,-78 l l = 0 : : : k ; 1 F/>,1 ,;6-/00,>, 1EF8 ;:,=8--/ ?0,>,5780 6?88. 164 l;1 (j ) nX ;l X Ln(x) = Lnj !(y) (x ; y)j + (x ; y)l (x ; x1 ) : : : (x ; xi;1 ) Ln(y| Z :{z: : Z y}Z x1Z : : : Z xi ) j =0 i=1

6 ?,@8. DE.C 1E56-780/

L(nl) (y) = l!

l

n;l X i=1

(x ; x1 ) : : : (x ; xi;1 ) Ln(y| Z :{z: : Z y}Z x1 Z : : : Z xi): l

7< ;8:8B,4/ 2 ,58:840,?I F/>I 0/4, 1E56-76.C Ln(y| Z :{z: : Z y}Z x1Z : : : Z xi ) i = l+1 1 : : : n ; l ; 1 ;, A,:?I7/? Ln (y| Z :{z: : Z y}Z x1 Z : : : Z xi) ; Ln(y| Z :{z: : Z y}Z x1Z : : : Z xi;1 ) l l+1 (8) Ln(y| Z :{z: : Z y}Z x1Z : : : Z xi) = xi ; y l+1

i = 1 : : : n ; l ; 1

/ 3/.8? (l+1)

Ln (y) = (l + 1)!

n; Xl;1 i=1

(x ; x1 ) : : : (x ; xi;1 ) Ln(y| Z :{z: : Z y}Z x1Z : : : Z xi ): l+1

(9)

:I4,8?2,-.C 1E56-7806< Ln (y): 1. / 1E56-78068 n ; l ; 1 :/3487800EB :/30,-.89 ;, A,:?I78 (8) .:8DI8.-< 2(n ; l ; 1) 1E56./069 6 n ; l ; 1 4878069. 2. / 1E56-78068 L(nl+1) (y) ;, A,:?I78 (9) ;, -B8?8 ,:08:/ L(nl+1) (y) = (l + 1)!(Ln(y| Z :{z: : Z y}Z x1 ) + (x ; x1 )(Ln(y| Z :{z: : Z y}Z x1 Z x2)+ (l+1)

l+1

l+1

+(x ; x2 )(Ln(y| Z :{z: : Z y}Z x1Z x2 Z x3)+ : : : +(x ; xn;l;3 )(Ln(y| Z :{z: : Z y}Z x1Z : : : Z xn;l;2)+ l+1

l+1

+(x ; xn;l;2 )Ln(y| Z :{z: : Z y}Z x1Z : : : Z xn;l;1) )| :{z: :)} ) l+1

. .

n;l;2

!"#$ $ %$

x11. 5+,4 + 4=)

31

.:8DI8.-< n ; l ; 2 1E56./069, n ; l ; 2 -7,@8069 6 n ; l ; 2 I?0,@8069. -8>, .:8DI8.-< 4(n ; l ; 2) ; 2 /446.610EB ,;8:/=69 6 2(n ; l ; 2) ; 1 ?I7C.6;762/.610EB ,;8:/=69. x

11. )

. 0,>,5780/?6 8DEF81/ 0/3E1/G. ?0,>,5780E Tn(x) -.8;806 n 0, 3/4/1/8?E8 -784IGH6?6 :822I:80.0E?6 -,,.0,F806 0: (2)

,JAA6=680. ;:6 -./:F8? 57808 ?0,>,5780/ Tn+1(x) ;,7I5/8.-< 63 -./:F8>, 5780/ ?0,>,5780/ Tn(x) I?0,@8068? 0/ 2x . 784,1/.87C0,, -./:F69 5780 ?0,>,5780/ Tn(x) ;:6 n > 0 8-.C 2n;1x . ,-2,7C2I 47< 1-<26B 6 cos cos = 21 (cos( + ) + cos( ; )), ., ;:6 7GD,? cos((n + 1) ) = 2 cos cos(n ) ; cos((n ; 1) ): ,7,@61 = arccos x , ;,7I56? ;:6 x 2 T;1 1] cos((n + 1) arccos x) = 2x cos(n arccos x) ; cos((n ; 1) arccos x) ..8. AI02=6< 'n(x) = cos(n arccos x) I4,178.1,:<8. -,,.0,F806 0 6 x 2 T;1 1]. /2 2/2 '0(x) = cos(0 arccos x) = 1, '1(x) = cos(1 arccos x) = 1 ;:6 x 2 T;1 1], ..8. ,0/ I4,178.1,:<8. (1), ., Tn(x) 'n(x): Tn(x) = cos(n arccos x) n 0 x 2 T;1 1]: (3) 3 ;:84-./17806< (3) 1E.82/G. -784IGH68 -1,9-.1/ ?0,>,5780,1 8DEF81/: 1) jTn(x)j 1 ;:6 x 2 T;1 1]: 2) ,526 J2-.:8?I?/ ?0,>,5780/ Tn(x) 0/ T;1 1], ..8. .,526 xbm , jTn(xbm )j = 1 8-.C xbm = cos m n m = 0 1 : : : n ;:658? Tn (xbm ) = cos m = (;1)m : 3) I76 ?0,>,5780/ Tn(x) 0/ T;1 1] 8-.C !

(2 m ; 1) m = 1 2 : : : n: xm = cos 2n ,-2,7C2I 0/9480E n 0I789 ?0,>,5780/ Tn(x) -.8;806 n , ., 4:I>6B 0I789 I 08>, 08. 6 Tn(x) = 2n;1(x ; x1) : : : (x ; xn ): 6%$"

. .

!"#$ $ %$

x11. 5+,4 + 4=)

32

. 1848?

6%$"

T n(x) = 2n1;1 Tn (x) = xn + : : :

1.

Pn(x) = xn + : : : { n

$ &-

, 1,

1 : max j P ( x ) j

max j T ( x ) j = n n x2 ;11] x2 ;11] 2n;1 . :84;,7,@6? ;:,.610,8, ..8. -IH8-.1I8. ./2,9 ?0,>,5780 -.8;806 n -, -./:F6? 2,JAA6=680.,?, :/10E? 1, 5., jPn(x)j < 1 47< 1-8B x 2 T;1 1]. /-?,.:6? Q (x) = T (x) ; P (x) { ?0,>,5780 n n n;1 2n;1 n n -.8;806 08 1EF8 n ; 1, ;,-2,7C2I T n(x) = x + : : : , Pn(x) = x + : : : . E56-76? 30/2 ?0,>,5780/ Qn;1 1 .,52/B xbm , m = 0 1 : : : n : 1 m sign Qn;1(xbm ) = sign(T n(xbm ) ; Pn(xbm )) = sign (;1) 2n;1 = = sign (;1)m n1;1 = (;1)m 2 (;,-2,7C2I jPn(xbm )j < 2n1;1 ). ,458:208?, 5., Qn;1(xbm ) 6= 0, m = 0 1 : : : n . /26? ,D:/3,?, 0/ ,.:8328 Txbm xbm+1 ], m = 0 1 : : : n ; 1 AI02=6< Qn;1(x) ?80<8. 30/2 6 ;,J.,?I 6?88. 2,:80C, ;:60/478@/H69 60.8:1/7I (xbm xbm+1 ), m = 0 1 : : : n ; 1. 784,1/.87C0,, ?0,>,5780 Qn;1 -.8;806 08 1EF8 n ; 1 6?88. n :/37650EB 2,:089 6 ;,.,?I Qn;1 0. 40/2,, ;, 4,2/3/00,?I Qn;1 (xbm ) 6= 0, m = 0 1 : : : n . ,7I5800,8 ;:,.61,:8568 4,2/3E1/8. 78??I. 6%$". 0,>,5780 T n (x) 0/3E1/8.-< ?0,>,5780,? -.8;806 n , 0/ ,.:8328 T;1 1]. I-.C Ta b] 8-.C 3/4/00E9 ,.:83,2. 6%$". 1848? ,.,D:/@806< y : T;1 1] ! Ta b] y(x) = (a + b) ;2 (a ; b)x (4) 2 y ; ( b + a ) x : Ta b] ! T;1 1] x(y) = b ; a 6 1$ Ta b]

! bTn(y) = Tn(x(y)) = Tn 2y ; (b + a) : b;a

7"#&

Pn(x) = xn + : : :

I76 ?0,>,5780/ Tbn(y) (1-8 ;:60/478@/. ,.:832I Ta b]) 8-.C ym = y(xm) = a +2 b + b ;2 a cos (2m2n; 1) m = 1 2 : : : n:

. .

(5)

!"#$ $ %$

x12. */ . '5=

( (',./

33

2 n 2n;1 b ./:F69 2,JAA6=680. Tn(y) 8-.C 2n;1 = 2 n . b;a (b ; a) 6%$". ,7,@6?

! a)n T (y) = (b ; a)n T 2y ; (b + a) : eTn(y) = (b ;2n;1 n 2 22n;1 n b;a 2. 2 Ten (y ) , , .. Pn (y ) = y n + : : : n $ & , 1,

a) : en(y)j = (b ;2n;1 max j P ( y ) j

max j T n y2 ab] y2 ab] 2

7"#&. .18:@48068 1E.82/8. 63 78??E 1 ;I.8? 3/?80E ;8:8?800,9. n

12. ) x

606?636:I8? ;,>:8F0,-.C 60.8:;,7<=66 AI02=66 f 0/ ,.:8328 Ta b] - ;,?,HCG 60.8:;,7<=6,00,>, ?0,>,5780/ />:/0@/ Ln -.8;806 n ; 1 3/ -58. 1ED,:/ I37,1 60.8:;,7<=66 x1 : : : xn . , .8,:8?8 7.2 (n) f f (x) ; Ln(x) = n!( ) (x ; x1 ) : : : (x ; xn) (1) >48 x1 : : : xn x 2 Ta b], 2 Ta b]. 784,1/.87C0,, (n) (2) kf ; LnkC( ab]) fn! k(x ; x1) : : : (x ; xn)kC( ab]) C ( ab]) >48 kgkC( ab]) = xmax jg(x)j: 2 ab]

606?636:I8? ;:/1IG 5/-.C, 1ED:/1 ?0,>,5780 Pn(x) = (x ; x1 ) : : : (x ; xn ) 0/6?8088 I27,0, :/180-.1/ 1E.82/8., 5., .,526 x1 : : : xn <17,5780/ Ten(x) 0/ ,.:8328 Ta b]. -67I (11.5) xm = ym = a +2 b + b ;2 a cos (2m2n; 1) m = 1 2 : : : n: (3)

. .

!"#$ $ %$

x12. */ . '5=

34

( (',./

, 78??8 11.2 ;:6 ./2,? 1ED,:8 I37,1

kPnkC ab (

(b ; a) ]) = 22n;1

n

6 ,=802/ (2) 6?88. 164

n kf ; LnkC ab n1! f n C ab (b2;n a) : (

( )

])

(

])

(4)

2 ;1

,7I5800/< ,=802/ (4) 08I7I5F/8?/. n 89-.16.87C0,, 1,3C?8? f (x) = anxn + : : : + a0 = P am xm { ?0,>,5780 m=0 -.8;806 n . ,>4/ f n( ) = ann! 6 ,=802/ ;,>:8F0,-.6 (1) ;:6?8. 164

f (x) ; Ln (x) = an(x ; x1 ) : : : (x ; xn ) = anPn(x)

>48 Pn(x) { ?0,>,5780 (0/ ,.:8328 Ta b]) -, -./:F6? 2,JAA6=680.,?, :/10E? 1. , 78??8 11.2 ;,7I5/8?

kf ; LnkC ab = janjkPnkC ab janj (b2;n a)

n

(

])

(

])

2 ;1

a)n = 1 f (n) C ( ab]) (b ; n! 22n;1

5., 6 ,30/5/8. 08I7I5F/8?,-.C ,=8026 (4). ./2, 47< ?606?63/=66 ;,>:8F0,-.6 1 2/58-.18 I37,1 x1 : : : xn 60.8:;,7<=66 0/4, D:/.C 0I76 (3) ?0,>,5780/ 8DEF81/ Ten(x). /2,9 1ED,: I7I5F/8. ./2@8 2/58-.1800,8 ;,1848068 ;,>:8F0,-.6 ;:6 n ! 1. 1 .%% (D83 4,2/3/.87C-.1/). /--?,.:6? AI02=6G f (x) = 1 + 25x2 6 ,.:83,2 Ta b] = T;1 1]. 7< :/10,?8:0, :/-;:8487800EB 0/ ,.:8328 Ta b] .,582 x1 : : : xn (..8. xi = a + (i ; 1)h , h = nb ;; a1 , i = 1 : : : n ) ;,>:8F0,-.C 60.8:;,7<=66 kf ; LnkC ( ;11]) ! 1 ;:6 n ! 1 . 7< .,582 x1 : : : xn { 0I789 (3) ?0,>,5780/ 8DEF81/ Ten(x) ;,>:8F0,-.C 60.8:;,7<=66 kf ; LnkC ( ;11]) ! 0 ;:6 n ! 1 . ,50IG B/:/2.8:6-.62I ;,184806< ;,>:8F0,-.6 60.8:;,7<=66 ;, 0I7,5780/ 8DEF81/ Ten(x) 4/G. -784IGH68 I.18:@4806<. n = Q x ; xj j =1 xi ; xj

% 1. ( ). ( & -

" ) ' li (x)

2 ln n + 1 .

. .

xm = ym

(3) $-

j 6=i

!"#$ $ %$

x13. *, ' 5+, + 4=) ;

35

% 2. $ f Ln ' f ) ' Ln n 1 , (3) 1$,

Pn

;1

;

dist (f Pn;1 ) kf ; LnkC ( ab]) 2 + 2 ln n distC (f Pn;1) { n ; 1 , distC (f Pn;1) = P ;1inf kf ; Pn;1kC( ab]): 2P ;1 n

n

f 2 C (r) Ta b] n > r + 1 ,

! r b ; a !r

6(3 e ) b ; a (r) distC (f Pn;1) 1 + r n ; 1 ! f Z 2(n ; 1 ; r) :

% 3 ( A).

!(gZ h) = supfjg(x) ; g(y)j : x y 2 Ta b] jx ; yj h g

.

. /--?,.:6? AI02=6G f (x) = jxj , 2 (0 1) 0/ ,.:8328 T;1 1]. ,>4/ !(f Z h) = h . , .8,:8?8 3 ;:6 r = 0, n > 1 ;,7I5/8? !

2 = 6 : distC (f Pn;1) 6! f Z 2(n ; 1) (n ; 1) , .8,:8?8 2 2 6 kf ; LnkC( ;11]) 2 + ln n (n ; 1) : q /;:6?8:, 47< f (x) = jxj 2 6 kf ; LnkC( ;11]) 2 + ln n (n ; 1)1=2 nln1=n2 ;:6 n ! 1: .%%

x

13. )

I-.C n { 3/4/00,8 =87,8 56-7,, Ta b] { 3/4/00E9 ,.:83,2, x1 : : : xn { :/37650E8 .,526, ;:60/478@/H68 ,.:832I Ta b]. 1848? ? ;:,-.:/0-.18 C (Ta b]) 46-2:8.0,8 \-2/7<:0,8" ;:,631848068: 47< 1-<26B f g 2 C (Ta b]) ;,7,@6?

hf gin = mX f (xm)g(xm): n

=1

. .

(1)

!"#$ $ %$

x13. *, ' 5+, + 4=) x 13.1.

36

.& 7$ " %6"<8

/--?,.:6? 3/4/5I ;,-.:,806< :/37,@806<

Pf (x) =

nX ;1 i=0

i Tbi(x)

(2)

AI02=66 f 2 C (Ta b]) ;, ?0,>,5780/? 8DEF81/ 0/ ,.:8328 Ta b]:

! 2 x ; ( b + a ) Tbi (x) = Ti i = 0 1 : : : n ; 1 b;a ./2, 5.,DE DE76 1E;,7080E 760890E8 I-7,16< 60.8:;,7<=66: Db E Db E Ti Pf n = Ti f n i = 0 1 : : : n ; 1:

D E i(f ) = Tbi f n

1. 2

Tb0 Tb1 : : : Tbn;1

(3) (4)

,

.

. :84;,7,@6?, 5., ?0,>,5780E Tb0 Tb1 : : : Tbn;1 760890, n;1 3/16-6?E, ..8. -IH8-.1I8. 2 Rn , 6= 0 ./2,8, 5., Pn;1(x) = P iTbi (x) = 0 i=0 47< 1-8B x 2 Ta b]. /2 2/2 deg Tbi = i , ., -./:F69 2,JAA6=680. ?0,>,5780/ n;1 Pn;1(x) = P i Tbi(x) :/180 -./:F8?I 2,JAA6=680.I ?0,>,5780/ Tbj (x) (>48 j = i=0 maxfj : j 6= 0g ), I?0,@800,?I 0/ j . ,-2,7C2I ?0,>,5780 Pn;1(x) 0, ., 8>, -./:F69 2,JAA6=680. :/180 0. 784,1/.87C0,, j = 0, 5., ;:,.61,:856. .,?I, 5., j = maxfj : j 6= 0g . ,7I5800,8 ;:,.61,:8568 4,2/3E1/8. ;8:1,8 I.18:@48068 78??E. D E :84;,7,@6?, 5., AI02=6,0/7E i(f ) = Tbi f n 760890, 3/16-6?E, ..8. n;1 -IH8-.1I8. 2 Rn , 6= 0 ./2,8, 5., = P ii = 0. :I>6?6 -7,1/?6,

7"#&

i=0 D b E nP;1 b (f ) = i Ti f n = iTi f = 0 47< 1-<2,9 f 2 C (Ta b]). ,3C?8? i=0 i=0 n n x ; xi Q f (x) = lk (x) = x ; x . ,-2,7C2I lk (xj ) = kj , ., i=1 k i i6=k *nX + X ;1 n nX ;1 n nX ;1 X b 0 = (lk ) = iTi lk = iTbi(xj )lk (xj ) = i Tbi(xj ) kj = nP ;1

i=0

n

j =1 i=0

=

nX ;1 i=0

j =1 i=0

iTbi (xk ) 47< 1-8B k = 1 2 : : : n:

n;1 784,1/.87C0,, ?0,>,5780 Pn;1(x) = P iTbi(x) -.8;806 08 1EF8 n ; 1 6?88. i=0 nP ;1

n 0I789 6 ;,.,?I Pn;1(x) 0, ..8. i Tbi(x) = 0 47< 1-8B x . -67I 4,i=0 2/3/00,9 1EF8 760890,9 083/16-6?,-.6 ?0,>,5780,1 Tb0 Tb1 : : : Tbn;1 ;,7I5/8?

. .

!"#$ $ %$

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37

= 0. ,7I5800,8 ;:,.61,:8568 - ;:84;,7,@8068? 6= 0 4,2/3E1/8. 1.,:,8 I.18:@48068 78??E.

p2n .

% 1. ( ). (

1$

Tb0 Tb1 : : : Tbn;1

$

3 78??E 1 6 .8,:8?E 1.1 1E.82/8., 5., 8-76 ?/.:6=/

D E A = (aij ) aij = Tbi Tbj n i j = 0 1 : : : n ; 1

(5)

,D:/.6?/, ., 3/4/5/ 760890,9 60.8:;,7<=66 ;, ?0,>,5780/? 8DEF81/ (3) 2,::82.0/ 6 88 :8F8068 6?88. 164 (2), >48 2,JAA6=680.E :/37,@806<

= A;1

D

E Tbi f n i=01:::n;1 :

(6)

,4D8:8? .,526 x1 : : : xn ./2, 5.,DE ?/.:6=/ A DE7/ 78>2, ,D:/.6?/. 2.

xm = ym = a +2 b + b ;2 a cos (2m2n; 1) m = 1 2 : : : n Tbn

(7)

n . Db b E Db b E n Db b E Ti Tj n = 0 i 6= j Tj Tj n = 2 j 6= 0 T0 T0 n = n: !

2 x ; ( b + a ) b 47< 1-8B x 2

7"#&. /2 2/2 Tj (x) = cos j arccos b;a Ta b], .,

!

! n n Db b E X X (2 m ; 1) (2 m ; 1) Ti Tj n = Tbi(xm )Tbj (xm ) = cos i 2n cos j 2n : m=1 m=1 1$

,-;,7C3I8?-< A,:?I7,9 cos cos = 1 (cos( ; ) + cos( + )): 2

!

! n n Db b E X X 1 (2 m ; 1) 1 (2 m ; 1) Ti Tj n = 2 cos (i ; j ) 2n + 2 cos (i + j ) 2n = m=1 m=1 = F (i ; j ) + F (i + j )

. .

(8)

!"#$ $ %$

x13. *, ' 5+, + 4=) >48

38

!

n X F (k) = 21 cos k (2m2;n 1) = m=1

( ) ( )! n 1 X 1 k (2 m ; 1) k (2 m ; 1) = 2 2 exp i 2n + exp ;i 2n = m=1

( ) ( )! n X 1 ik (2 m ; 1) ik (2( ; ( m ; 1)) ; 1) = exp + exp = 4 m=1 2n 2n ( ) n X ik (2 m ; 1) 1 exp = =4 2n m=;(n;1) ( ) X

( )!m n 1 ik exp ik = = 4 exp ; 2n n m=;(n;1) ( ) ( )!;n X

( )!m 2n 1 ik ik = 4 exp ; 2n exp n exp ik n m=1

,-2,7C2I ., ;:6 k 6= 0

8 2n < q ;1 X m > q = > q q ; 1 ;:6 q 6= 1 : 2n m=1 ;:6 q = 1 2n

) ( ) ik ik)g ; 1 = 0 ; 2n exp f;ikg exp ikn exp (f2ik exp n ;1 ( ) ./2 2/2 exp f2ikg = 1 47< 1-8B k . :6 k = 0 q = exp ik = 1 6 n F (k) = 41 exp

./2,

(

F (0) = 14 1 1 2n = n2 :

8 > < 0 ;:6 k 6= 0 F (k) = > n : 2 ;:6 k = 0

,4-./17<< J., 30/58068 1 (8), ;,7I5/8?

Db b E Ti Tj n = 0 + 0 = 0 ;:6 i 6= j Db bE n Ti Ti n = 2 + 0 = n2 ;:6 i 6= 0 Db b E n n T0 T0 n = 2 + 2 = n

. .

!"#$ $ %$

x13. *, ' 5+, + 4=)

39

8??/ 4,2/3/0/. 3 78??E 2 ;,7I5/8?, 5., ;:6 1ED,:8 .,582 x1 : : : xn 2/2 1 (7) ?/.:6=/ A 1 (5) 8-.C n n A = diag n 2 : : : 2 (9) 6 6-2,?,8 :/37,@8068 (2), (6) 6?88. 164

Pf (x) =

nX ;1 i=0

x 13.2.

iTbi (x)

Db E Ti f i = D b b En i = 0 1 : : : n ; 1: Ti Ti n

(10)

"*% 6%< %7"A<

/;6F8? 1E:/@806< (10) 47< i 1 1648 ./D76=E 0 = n1 ( Tb0 (x1)f (x1 ) + Tb0 (x2 )f (x2) + : : : + Tb0(xn )f (xn) ) 1 = n2 ( Tb1 (x1)f (x1 ) + Tb1 (x2 )f (x2) + : : : + Tb1(xn )f (xn) ) 2 = n2 ( Tb2 (x1)f (x1 ) + Tb2 (x2 )f (x2) + : : : + Tb2(xn )f (xn) ) (11) ... ... ... ... ... n;1 = n2 ( Tbn;1(x1 )f (x1 ) + Tbn;1(x2 )f (x2 ) + : : : + Tbn;1(xn)f (xn) ) 3 :822I:80.0EB A,:?I7 (11.1), (11.2) 47< ?0,>,5780,1 8DEF81/ Tn 0/ ,.:8328 T;1 1] 3/?80,9 ;8:8?800EB (11.4)

;,7I5/G.-< ! :822I:80.0E8 A,:?I7E 2 x ; ( b + a ) 47< ?0,>,5780,1 8DEF81/ Tbn(x) = Tn 0/ ,.:8328 Ta b]: b;a

Tb0 (x) = 1 Tb1(x) = 2x ;b ;(b a+ a) (12) bTn+1(x) = 2 2x ; (b + a) Tbn (x) ; Tbn;1(x) ;:6 n > 0: b;a ,7C3I<-C J.6?6 A,:?I7/?6, DI48? 1E56-7<.C 1 -I??/B (11) -.,7D=E -781/ 0/;:/1,, / 2/@4E9 -.,7D8= { -18:BI 1063. D,30/56? gij = Tbi(xj )f (xj ) zj = 2 cos (2jn; 1) i = 0 : : : n ; 1 j = 1 : : : n: , A,:?I7/? (12) 47< 1-8B j = 1 2 : : : n 1E56-7
. .

!"#$ $ %$

x13. *, ' 5+, + 4=)

40

K.6 A,:?I7E ;,31,7
0 := n1 0 i := n2 i i = 1 2 : : : n ; 1:

(15)

,-78 .,>,, 2/2 2,JAA6=680.E i 1E56-780E, 30/58068 ;:6D76@/GH8>, n;1 ?0,>,5780/ 1 .,528 Pf (x) = P iTbi(x) 1E56-7<8.-< - 6-;,7C3,1/068? A,:i=0 ?I7 (12)

Tb0(x) = 1 Tb1 (x) = z=2 Tbi(x) = z Tbi;1(x) ; Tbi;2 (x) >48 z = 2 2x ; (b + a) . b;a

x 13.3.

Pf (x) = 0 Tb0(x) + 1Tb1 (x) Pf (x) := Pf (x) + iTbi(x) i = 2 : : : n ; 1:

8 "& %

(16)

6%8

7< 1-8B j = 1 2 : : : n 0/? .:8DI8.-< ;:,6318-.6 1E56-7806< 1) , A,:?I7/? (13). / J., .:8DI8.-< n + O(1) ?I7C.6;762/.610EB 6 -.,7C2, @8 /446.610EB ,;8:/=69. 2) , A,:?I7/? (14). / J., .:8DI8.-< n + O(1) /446.610EB ,;8:/=69. DH88 2,7658-.1, ,;8:/=69 47< 1E56-78069 ;, A,:?I7/? (13) 6 (14) 47< 1-8B j = 1 2 : : : n : n2 +O(n) ?I7C.6;762/.610EB 6 2n2 +O(n) /446.610EB ,;8:/=69. 7< 1E56-7806< 2,JAA6=680.,1 i ;, A,:?I7/? (15) .:8DI8.-< 8H8 n + O(1) ?I7C.6;762/.610EB ,;8:/=69. 784,1/.87C0,, -I??/:0,8 2,7658-.1, ,;8:/=69, 08,DB,46?EB 47< 1E56-7806< 2,JAA6=680.,1 i , :/10, n2 + O(n) ?I7C.6;762/.610E? 6 2n2 + O(n) /446.610E? ,;8:/=6, ?0,>,5780/ Pf 1 .,528 ;, A,:?I7/? (16) .:8DI8.-< 2n + O(1) ?I7C.6;762/.610EB 6 -.,7C2, @8 /446.610EB ,;8:/=69.

. .

!"#$ $ %$

x13. *, ' 5+, + 4=) x 13.4.

41

&<7# %7"A< 6 *" C?'D& %6"<8

:6 ;,-.:,8066 ?0,>,5780/ Pf 08 .:8D,1/7,-C, 5.,DE ,0 -,1;/4/7 - AI02=689 f 1 2/26B-DE ., 06 DE7, .,52/B (.:8D,1/7,-C 76FC 1E;,708068 760890EB I-7,169 60.8:;,7<=66 (4)). 40/2,, ,2/3E1/8.-<, 5., ;,-.:,800E9 1EF8 ;:6D76@/GH69 ?0,>,5780 Pf ?,@0, :/--?/.:61/.C 2/2 60.8:;,7<=6,00E9 ?0,>,5780. 3. $

(10) (7) -

&" ", ..

(Pf )(xm) = f (xm ) m = 1 2 : : : n

(17)

, , ) ', 1$ Tn .

b

. E56-76? ?0,>,5780 (10) 1 .,52/B xm , m = 1 2 : : : n :

7"#&

Db E Tj f (Pf )(xm) = D b b En Tbj (xm ) = j =0 Tj Tj n nX ;1 n X = D b 1b E Tbj (xi )f (xi)Tbj (xm ) = j =0 Tj Tj n i=1 n nX ;1 X = f (xi) D b 1b E Tbj (xi)Tbj (xm ) i=1 j =0 Tj Tj n nX ;1

/2,? ,D:/3,?, 47< 4,2/3/.87C-.1/ (17) (/ 30/56., 6 78??E) 0/? 4,-./.,50, ;,2/3/.C, 5., nX ;1 D b 1b E Tbj (xi)Tbj (xm ) = im j =0 Tj Tj n ..8. ;1 2 Tb (x )Tb (x ) = 1 + nX im n j=1 n j i j m 676 nX ;1 (18) Tbj (xi )Tbj (xm ) = n2 im + 12 : j =0 ?88?

!

! n;1 bTj (xi)Tbj (xm ) = X cos j (2i ; 1) cos j (2m ; 1) : 2n 2n j =0 j =0

nX ;1

. .

!"#$ $ %$

x13. *, ' 5+, + 4=) ,-;,7C3I8?-< A,:?I7,9 cos cos = 1 (cos( ; ) + cos( + )): 2

! nX

! n ;1 ;1 nP ;1 X 1 ( i ; m ) 1 ( i + m ; 1) Tbj (xi)Tbj (xm ) = 2 cos j n + 2 cos j = n j =0 j =0 j =0 = S (i ; m) + S (i + m ; 1) >48

! nX

( ) ( )! nX ;1 ;1 jk 1 ijk ; ijk 1 S (k) = 2 cos n = 4 exp n + exp n = j =0 j =0 0 ( )1 nX ;1 1 A= exp ijk = 4 @1 + n j =;(n;1) 0 ( )1 n X 1 = @1 ; exp fikg + exp ijk A = 4 n j =;(n;1) 0 ( ) 2X ( )!j 1 n;1

1 ik ( n ; 1) ik A = 4 @1 ; (;1)k + exp ; exp n n j =0 ,-2,7C2I 8 2n > 2X n;1 < q ; 1 ;:6 q 6= 1 j q => q;1 : 2n j =0 ;:6 q = 1 ., ;:6 k 6= 0

42

(19)

0 1 ( ) BB CC 1 1 ik ( n ; 1) exp f 2 ik g ; 1 k B CC = 1 ; (;1)k ( ) S (k) = 4 B1 ; (;1) + exp ; n @ A 4 exp ik ; 1 n ( ) ./2 2/2 exp f2ikg = 1 47< 1-8B k . :6 k = 0 q = exp ik = 1 6 n S (0) = 1 (1 ; 1 + 2n) = n ./2,

8 > < S (k) = > :

4

2

1 1 ; (;1)k ;:6 k 6= 0 4n ;:6 k = 0 2 ,4-./17<< J., 30/58068 1 (19), ;,7I5/8? ;:6 i 6= m nX ;1 Tbj (xi)Tbj (xm ) = S (i ; m) + S (i + m ; 1) = 14 1 ; (;1)i;m + j =0 + 1 1 ; (;1)i+m;1 = 1 + 1 = 1 4 4 4 2

. .

!"#$ $ %$

x14. 1

43

+-5+,. '' / .

(;,-2,7C2I 56-7/ i ; m 6 i + m ; 1 6?8G. :/30IG 58.0,-.C). :6 i = m nX ;1 j =0

Tbj (xi )Tbj (xm ) = S (i ; m) + S (i + m ; 1) = n2 + 41 1 ; (;1)2i;1 = n2 + 21 :

8? -/?E? A,:?I7/ (18) 6 (1?8-.8 - 089 78??/) 4,2/3/0/. -8 :/--?,.:800E8 1EF8 /;;:,2-6?/=66 Pf AI02=66 f 6?876 164: 0/ ,.:8328 Ta b] AI02=6< f ;:6D76@/8.-< ?0,>,5780,? Pf = Lm -.8;806 m . :/106? J.,. -;,-,D -, -784IGH6?: :/34876? ,.8:3,2 Ta b] 0/ k 5/-.89 6 0/ 2/@4,9 63 06B ;:6D7636? ?0,>,5780,? Ln -.8;806 n . /2/< /;;:,2-6?/=6< 0/3E1/8.-< 2I-,50,-?0,>,57800,9.

14. '

- x

=8026 ;,>:8F0,-.6 1-8B :/--?,.:800EB :/088 ?8.,4,1 6?8G. 164

kf ; Pf kC const(n) distC (f Pn ) (>48 Pf { /;;:,2-6?/=6< ?0,>,5780,? -.8;806 n ; 1, Pn { ;:,-.:/0-.1, ?0,>,5780,1 -.8;806 n ; 1). , .8,:8?8 @82-,0/ (-?. .8,:8?I 12.3) ;:6 n > r+1

!r

! b ; a b ; a r kf ; Pf kC const(r) n ; 1 ! f Z 2(n ; 1 ; r) : ;1

;1

( )

784,1/.87C0,, ,=802/ ;,>:8F0,-.6

!r

! b ; a b ; a kf ; Pf kC const(n) const(r) n ; 1 ! f (r)Z 2(n ; 1 ; r) : 187658068 .,50,-.6 ;:6 A62-6:,1/00,9 18765608 ?,4I7< 08;:8:E10,-.6 AI02=66 f (r) ?,@8. DE.C 4,-.6>0I., 76D, I187658068? n , 76D, I?80CF8068? b ; a Z :/3D68068 ,.:832/ Ta b] 0/ k 5/-.89 4/8. .,. @8 JAA82., 5., 6 6-;,7C3,1/068 ?0,>,5780,1 -.8;806 kn . :86?IH8-.1/ :/3D6806< ,.:832/ ;8:84 I187658068? -.8;806 ?0,>,5780/: 1) :8DI8.-< :8F/.C (1 ,DH89 -B8?8 3/4/56 760890,9 60.8:;,7<=66) k -6-.8? :/3?8:/ n 1?8-., ,40,9 :/3?8:/ kn . 2) :,-.,? -.8;806 ?0,>,5780/ n :/-.8. ,DI-7,17800,-.C D/36-/, / ;:6 :/3D68066 ,.:832/ ,0/ 08 ?80<8.-<, ./2 2/2 :/3?8:0,-.C D/36-/ n (-.8;80C ?0,>,5780/ 0/ 2/@4,? 63 ,.:832,1 :/3D6806<) 08 63?80<8.-< - :,-.,? 2,7658-.1/ ,.:832,1 :/3D6806< k .

. .

!"#$ $ %$

x15. 1 x

44

+-, -. (',./ .

15. -

I-.C 0/ ,.:8328 Ta b] 3/4/0E .,526 a = x1 < x2 < : : : < xn = b 6 30/5806< f (x1 ) f (x2) : : : f (xn) 082,.,:,9 AI02=66 f 2 C (Ta b]). :8DI8.-< ;,-.:,6.C ;:6D76@8068 AI02=66 f 2I-,50,-760890,9 AI02=689 I2f , -,1;/4/GH89 - f 1 .,52/B xi , i = 1 2 : : : n . :I>6?6 -7,1/?6, .:8DI8.-< ;,-.:,6.C AI02=6G I2f ./2IG, 5., 47< 1-8B i = 1 2 : : : n ; 1 0/ ,.:8328 Txi xi+1] AI02=6< I2 f <17<8.-< 760890,9 AI02=689, 30/5806< 2,.,:,9 1 .,52/B xi 6 xi+1 -,1;/4/G. -, 30/5806
-

kf ; I f kC ab 18 h kf kC ab 2

(

])

2

00

(

])

h = i=1max (x ; xi ): 2:::n;1 i+1

. , .8,:8?8 7.2 ;,7I5/8? 00 kf ; Pf kC( x x +1]) 2!1 x2 max j ( x ; x )( x ; x (2) i i +1 )jkf kC ( x x +1 ]) : x x +1] #I02=6< g(x) = (x ; xi )(x ; xi+1 ) = x2 ; (xi + xi+1 )x + xixi+1 0 47< 1-8B x 2 Txi xi+1] 6 6?88. 0I7C ;:,631,40,9 (xi + xi+1 )=2, ;:60/478@/H69 ,.:832I Txi xi+1]. ,-2,7C2I g(xi) = g(xi+1) = 0, ., xi + xi+1 xi+1 ; xi xi ; xi+1 1 = = (x ; x )2 : max jg(x)j = g x2 x x +1 ] 2 2 2 4 i+1 i ,4-./17<< J., 1 (2), ;,7I5/8? kf ; Pf kC( x x +1]) 81 (xi+1 ; xi)2 kf 00kC( x x +1]): 784,1/.87C0,, 1 h2 kf 00k kf ; I2f kC( ab]) = i=1max k f ; Pf k C ( ab]) : C ( x x +1 ]) 2:::n;1 8 8??/ 4,2/3/0/. 6%$". D,30/56? 58:83 S2 760890,8 ;:,-.:/0-.1, 08;:8:E10EB 7,?/0EB 76069 0/ ,.:8328 Tx1 xn] = Ta b] - 637,?/?6 x2 < x3 < : : : < xn;1 .

7"#&

i

i

i

i

i

i

i

i

i

i

i

i

. .

i

i

!"#$ $ %$

x16. ( 2= A )()

45

2

2. f C (Ta b]) , $ -

distC (f S2 ) kf ; I2f kC ( ab]) 2distC (f S2):

7"#&. -67I 8460-.1800,-.6 60.8:;,7<=6,00,>, ?0,>,5780/ 0/ 2/@4,? 63 ,.:832,1 Txi xi+1], i = 1 2 : : : n ; 1 ;,7I5/8? 47< 1-<2,9 f 2 S2 I2f = f (3) 581640,, 5., kf ; I2f kC( ab]) ginf2S2 kf ; gkC( ab]) = distC (f S2):

-67I 760890,-.6 I2f 0/ Txi xi+1 ] kI2f kC( ab]) = i=1max j(I f )(xi)j = i=1max jf (xi)j kf kC( ab]) 2:::n 2 2:::n ..8. 47< 1-<2,9 f 2 C (Ta b])

kI f kC ab kf kC ab : 2

(

])

(

])

(4)

?88? 47< 1-<2,9 g 2 S2 kf ; I2f kC( ab]) = k(f ; g) + (g ; I2f )kC( ab]): -;,7C3I< (3) 6 760890,-.C I2 , ;:,4,7@/8? kf ; I2f kC( ab]) = k(f ; g) ; I2(f ; g)kC( ab]) k(f ; g)kC( ab]) + kI2(f ; g)kC( ab]): ;,?,HCG (4) 0/B,46? kf ; I2f kC( ab]) k(f ; g)kC( ab]) + k(f ; g)kC( ab]) = 2k(f ; g)kC( ab]) 47< 1-<2,9 g 2 S2 . 784,1/.87C0,, kf ; I2f kC( ab]) ginf2S k(f ; g)kC( ab]) = 2distC (f S2):

8??/ 4,2/3/0/. x

2

16. - )-

;:84E4IH8? :/34878 ?E ;,-.:,676 2I-,50,-760890IG AI02=6G, -,1;/4/GHIG - AI02=689 f 1 3/4/00EB .,52/B. 40/2,, 8-76 88 30/5806< 6318-.0E 082,.,:,9 ;,>:8F0,-.CG, ., J., ?,@8. 4/.C ;:6D76@8068 ;7,B,>, 2/58-.1/. ,;6-E1/8?,? 06@8 /7>,:6.?8 D763,-.C AI02=66 6 88 ;:6D76@806< 63?8:<8.-< 1 60.8>:/7C0,9 0,:?8, 6, -784,1/.87C0,, ;:6D76@/GH/< AI02=6< DI48. ?8088 5I1-.16.87C0,9 2 1ED:,-/? 1 30/5806
. .

!"#$ $ %$

x16. ( 2= A )() x 16.1.

46

.& 7$ " %6"<8

I-.C .:8DI8.-< ;,-.:,6.C ;:6D76@8068 AI02=66 f 2I-,50,-760890,9 AI02=689 L2 f 2 S2 ./2,8, 5.,

kf ; L f kL2 ab ! min 2

..8.

(

kf ; L f kL2 ab 2

>48

kgkL2 ab (

])

(

= (g g)

1=2 L2( ab])

])

])

= ginf kf ; gkL2( ab]) 2S 2

Zb

(u v)L2( ab]) = u(x)v(x) dx: a

1848? D/36- ;:,-.:/0-.1/ S2

Hj 2 S2 Hj (xi) = ij i = 1 2 : : : n j = 1 2 : : : n:

-<2/< g 2 S2 :/-27/4E1/8.-< ;, J.,?I D/36-I -784IGH6? ,D:/3,?

g(x) =

(1)

n X

j =1

g(xj )Hj (x):

(2) (3)

89-.16.87C0,, 1 -67I (2) 6?88? 47< gb(x) = P g(xj )Hj (x): gb(xi) = g(xi), j =1 b i = 1 2 : : : n . 784,1/.87C0,, g(x) = I2g { 2I-,50,-760890/< 60.8:;,76:IGH/< AI02=6<. , 4,2/3/00,?I 1 78??8 15.2 47< g 2 S2 1E;,7080, (15.3). 784,1/.87C0,, g = gb . /26? ,D:/3,?, 3/4/5/ (1) ?,@8. DE.C -A,:?I76:,1/0/ 1 -784IGH8? 1648: 0/9.6 (j )j=12:::n ./268, 5., AI02=6< n

(L2f )(x) =

n X j =1

j Hj (x)

(4)

I4,178.1,:<8. -,,.0,F806G

n X = ginf kf ; gkL2( ab]): f ; j Hj 2S2 j =1 L2 ( ab])

(5)

318-.0,, 5., 1 812764,1,? ;:,-.:/0-.18 (1 0/F8? -7I5/8 { L2 (Ta b])) ?606?I? (5) :8/763I8.-< 0/ AI02=66 (4), <17,0/780 ;,4;:,-.:/0-.1I S2 . ,-2,7C2I fHigni=1 -,-./17,0/7C0,-.6 S2 J2161/780.0, I-7,16G ,:.,>,0/7C0,-.6 1-8? Hi , i = 1 : : : n : (f ; L2f Hi)L2 ( ab]) = 0 47< 1-8B i = 1 : : : n

. .

!"#$ $ %$

x16. ( 2= A )()

47

..8.

(L2f Hi)L2 ( ab]) = (f Hi)L2 ( ab]) i = 1 : : : n: (6) -7,16< (6) ;:84-./17
676

j =1 n Zb X j =1 a

(Hj Hi)L2( ab]) j = (f Hi)L2 ( ab]) i = 1 : : : n

Zb

Hj (x)Hi(x) dx j = f (x)Hi(x) dx i = 1 : : : n:

(7)

a

,. @8 :83I7C./. (7) ?,@8. DE.C ;,7I580 63 I-7,16<

2 n X F () = f ; j Hj j =1

L2 ( ab])

0 1 n n X X = @f ; j Hj f ; j Hj A j =1

j =1

L2 ( ab])

! min

;I.8? 46AA8:80=6:,1/06< AI02=66 F ():

0 0 1 1 n n X @F = @ @ @f ; X j Hj A f ; j Hj A + @i @ j =1 j =1 L2 ( ab]) 0 0 1 1 n n X @ @f ; X + @f ; j Hj @ j Hj AA = j =1 j =1 L2 ( ab]) 0 1 n X = @;2Hi f ; j Hj A j =1

L2 ( ab])

6 ;:6:/1061/06< 2 0I7G ;:,631,40,9 @F 1 .,528 ?606?I?/: @i @F = 0 i = 1 : : : n: @i

x 16.2.

'" %8' ' 7$ " %6"<8

E56-76? ?/.:6=I -6-.8?E (7). /2 2/2 supp Hi = Txi;1 xi+1] ;:6 i = 2 3 : : : n ; 1 supp H1 = Tx1 x2 ] supp Hn = Txn;1 xn] .,

. .

(Hj Hi)L2 ( ab]) = 0 ;:6 ji ; j j > 1: !"#$ $ %$

x16. ( 2= A )()

48

,J.,?I 1 i -,? I:/108066 -6-.8?E (7) ,.7650E ,. 0I7< 08 D,788 3-B 2,JAA6=680.,1: (Hi;1 Hi)L2( ab]) ;:6 i;1 6 i = 2 3 : : : n (Hi Hi)L2( ab]) ;:6 i 6 i = 1 2 : : : n (Hi+1 Hi)L2( ab]) ;:6 i+1 6 i = 1 2 : : : n ; 1:

1. E56-76? 47< i = 1 2 : : : n ; 1 (Hi Hi+1)L2 ( ab]) =

Z supp Hi \supp Hi+1

Hi(x)Hi+1 (x) dx =

xZ +1 i

x

Hi(x)Hi+1(x) dx:

i

/2 2/2 Hi , Hi+1 2 S2 { 2I-,50, 760890E8 6 Hi(xi ) = 1, Hi(xi+1 ) = 0, Hi+1 (xi) = 0, Hi+1(xi+1 ) = 1, .,

Hi(x) = xx ;; xxi+1 ;:6 x 2 Txi xi+1] i i+1 x ; x Hi+1(x) = x ; ix ;:6 x 2 Txi xi+1] i+1

i

784,1/.87C0,, (Hi Hi+1)L2( ab]) =

1

xZ +1

1

x xZ +1

(xi+1 ; xi)2

= (x

i+1

; xi )

2

i

(x ; xi )(xi+1 ; x) dx =

i

i

(x ; xi )((xi+1 ; xi ) + xi ; x) dx =

x xZ +1 i

= (x 1; x )2 (xi+1 ; xi )(x ; xi) ; (x ; xi )2 dx = i+1 i x 1 1 1 1 3 3 = (x ; x )2 2 (xi+1 ; xi ) ; 3 (xi+1 ; xi ) = 6 (xi+1 ; xi) i+1 i i

i

2. /?80,9 60482-,1 i := i ; 1, i + 1 := i 1 ;:84E4IH89 A,:?I78 ;,7I5/8? 47< i = 2 3 : : : n (Hi;1 Hi)L2( ab]) = 16 (xi ; xi;1 ) 3. E56-76? 47< i = 2 3 : : : n ; 1 (Hi Hi)L2 ( ab]) =

xZ +1 i

x ;1 i

. .

Hi (x) dx = 2

Zx

i

x ;1 i

Hi (x) dx + 2

xZ +1 i

x

Hi2(x) dx:

i

!"#$ $ %$

x16. ( 2= A )()

49

?88?:

Zx x ; xi;1 !2 1 (x ; x ) dx = Hi (x) dx = 3 i i;1 x x ;1 xi ; xi;1 !2 xZ +1 xZ +1

x ; x 1 1 ;(x ; x )3 = i+1 2 Hi (x) dx = dx = i i+1 xi ; xi+1 (xi ; xi+1 )2 3 x x xZ ;1 i

i

2

i

i

i

i

i

i

= 31 (xi+1 ; xi)

784,1/.87C0,, (Hi Hi)L2 ( ab]) = 1 (xi ; xi;1 ) + 1 (xi+1 ; xi ) = 1 (xi+1 ; xi;1 ) 3 3 3 4. E56-76?

Zx2 x ; x2 2 (H1 H1)L2( ab]) = H (x) dx = dx = 1 (x2 ; x1 ) 3 x1 x1 x1 ; x2 Zx2

5. E56-76?

2 1

Zx x ; xn;1 !2 1 (x ; x ) dx = (Hn Hn)L2( ab]) = Hn(x) dx = 3 n n;1 x ;1 x ;1 xn ; xn;1 Zx

n

n

2

n

n

./2, -6-.8?/ (7) 6?88. 164: 1 (x ; x ) + 1 (x ; x ) = Zx2 H (x)f (x) dx 3 2 1 1 6 2 1 2 x1 1

1 (x ; x ) + 1 (x ; x ) + 1 (x ; x ) = Z +1H (x)f (x) dx 6 i i;1 i;1 3 i+1 i;1 i 6 i+1 i i+1 x ;1 i i = 2 3 : : : n ; 1 x 1 (x ; x ) + 1 (x ; x ) = Z H (x)f (x) dx 6 n n;1 n;1 3 n n;1 n x ;1 n x

i

i

n

n

-76 ;,7,@6.C x0 = x1 , xn+1 = xn , ., J.I -6-.8?I ?,@0, 3/;6-/.C 1 D,788 2,?;/2.0,? 1648 1 (x ; x ) + 1 (x ; x ) + 1 (x ; x ) = Z+1H (x)f (x) dx i = 1 : : : n 6 i i;1 i;1 3 i+1 i;1 i 6 i+1 i i+1 x ;1 i (8) ( 0 6 n+1 1B,4<. -G4/ - 0I781E?6 2,JAA6=680./?6). x

i

i

. .

!"#$ $ %$

x16. ( 2= A )()

50

. /.:6=/ A = (aij ) 2 Mn 0/3E1/8.-< ?/.:6=89 - 8-76

6%$"

,

jaiij X jaij j n

i = 1 2 : : : n:

j =1 j 6=i

/.:6=/ -6-.8?E (8) <17<8.-< .:8B46/>,0/7C0,9 - 46/>,0/7C0E? ;:8,D7/4/068?. 6-.8?E - ./26?6 ?/.:6=/?6 ?,@0, :8F/.C ,DE50E? ?8.,4,? /I--/ D83 1ED,:/ >7/10,>, J78?80./.

x 16.3.

&& 6%?"A=>

2 C (Ta b]) , ' kL f kC ab 3kf kC ab :

1. f

(

2

])

(

!8

L2 f

-

])

. /2 2/2 Hj (xi) = ij , ., 47< 2I-,50,-760890,9 AI02-

7"#&

=66 (4)

n X kL2f kC( ab]) = i=1max j ( L H ( x ) max j j: 2 f )(xi )j = max j j i = i=1 :::n i=1:::n :::n i j =1

D,30/56?

jj j = i max j j = kL f kC ab : :::n i 2

=1

(

])

(9)

?0,@6? j -,8 I:/108068 -6-.8?E (8) 0/ x ;6 x , ;,7I56? j +1 j ;1 6

j;1j;1 + 2j + j+1j+1 = x

j +1

>48

3 (10) 6?88?

j +1

j ;1

j +1

j

; xj ; xj ; xj

j;1 = xxj ;;xxj;1 j+1 = xxj+1

xZ +1

Hj (x)f (x) dx

(10)

j;1 + j+1 = 1:

(11)

;1 x

;1

j

;1

x +1 Z 6 Hj (x)f (x) dx + jj;1j;1 + j+1j+1j: 2jj j xj+1 ; xj;1 x ;1 j

(12)

j

/--?,.:6? -67I (9)

jj

jj ;1

. .

;1

j;1 + j+1j+1j j;1jj;1j + j+1jj+1j:

j;1 + j+1j+1j j;1jj j + j+1jj j = (j;1 + j+1)jj j !"#$ $ %$

x16. ( 2= A )() -67I (11)

jj

51

j;1 + j+1j+1j jj j: ,4-./17<< J., 1 (12), 0/B,46? x +1 Z 6 Hj (x)f (x) dx + jj j: 2jj j xj+1 ; xj;1 x ;1 ;1

j

j

.-G4/ ;,7I5/8?

xZ +1

6

jj j xj ; xj +1

,-2,7C2I

j

;1 x

j

;1

Hj (x) dx kf kC ( x ;1x +1]): j

(13)

j

xZ +1 Zx x ; xj;1 Hj (x) dx = x ; x dx + xx ;;xxj+1 dx = j ;1 j j +1 x ;1 x ;1 j x xZ +1 j

j

j

j

j

j

= 1 (xj ; xj;1) + 1 (xj+1 ; xj ) = 1 (xj+1 ; xj;1 ) 2 2 2 ., 63 (9) 6 (13) ;,7I5/8?

kL f kC ab = jj j 3kf kC x ;1x +1 3kf kC ab : 2

(

8??/ 4,2/3/0/.

])

(

j

j

])

(

])

2

2. f C (Ta b]) , $ -

$"

distC (f S2) kf ; L2 f kC ( ab]) 4distC (f S2 ):

7"#&. /2 2/2 ;:,82=6< J78?80./ 63 ;,4;:,-.:/0-.1/ 0/ J., ;,4;:,-.:/0-.1, :/10/ J.,?I J78?80.I, .,

L2 f = f 47< 1-8B f 2 S2 :

(14)

581640,, 5.,

kf ; L f kC ab ginfS2 kf ; gkC ab = distC (f S ): ?88? 47< 1-<2,9 g 2 S kf ; L f kC ab = k(f ; g) + (g ; L f )kC ab : 2

(

])

(

2

2

])

2

2

(

])

2

])

(

-;,7C3I< (14) 6 760890,-.C L2 , ;:,4,7@/8?

kf ; L f kC ab = k(f ; g) ; L (f ; g)kC ab k(f ; g)kC ab + kL (f ; g)kC ab : 2

(

. .

])

2

(

])

(

])

2

(

])

!"#$ $ %$

x16. ( 2= A )()

52

;,?,HCG 78??E 1 0/B,46?

kf ; L f kC ab k(f ; g)kC ab + 3k(f ; g)kC ab = 4k(f ; g)kC ab 47< 1-<2,9 g 2 S . 784,1/.87C0,, kf ; L f kC ab ginfS2 4k(f ; g)kC ab = 4distC (f S ): 2

(

])

(

])

(

])

(

])

2

2

8??/ 4,2/3/0/.

(

])

2C

(2)

(

2

])

2

(Ta b]) , $ -

$" 3.

f

kf ; L f kC ab 21 h kf kC ab 2

(

])

2

00

(

])

h = i=1max (x ; xi ): 2:::n;1 i+1 . -67I 78??E 15.2 distC (f S2) kf ; I2 f kC ( ab]):

7"#&

, 78??8 15.1 -67I 78??E 2

distC (f S2) kf ; I2 f kC ( ab]) 81 h2 kf 00kC ( ab]):

kf ; L f kC ab 4 distC (f S ) 21 h kf kC ab : 2

(

])

2

2

00

(

])

8??/ 4,2/3/0/.

x 16.4.

'" 6%& ' 7$ " %6"<8

7< 1E56-7806< ;:/1,9 5/-.6 -6-.8?E (8) .:8DI8.-< I?8.C 1E56-7<.C 60.8>:/7E 164/ x

Z+1 i

x ;1

Hi(x)f (x) dx i = 1 2 : : : n:

i

DE50, ,;6-/00E9 1EF8 /7>,:6.? ;:6?80 n . ,>4/ ;,7/>/G., 0/;:6?8:, xZ +1 i

x ;1 i

. .

Hi(x)f (x) dx

xZ +1 i

x ;1

Hi(x)(I2 f )(x) dx i = 1 2 : : : n

(15)

i

!"#$ $ %$

x16. ( 2= A )()

53

>48 I2f { 2I-,50,-760890/< 60.8:;,76:IGH/< AI02=6<, ;,-.:,800/< ;, .,52/? y1 y2 : : : yN . 0.8>:/7E 1 ;:/1,9 5/-.6 ;:6D76@800,>, :/180-.1/ (15) 1E56-7:/7C0/< AI02=6< ;:84-./17<8. -,D,9 ?0,>,5780 08 1EF8 1.,:,9 -.8;806. ,>:8F0,-.C, 10,-6?/< ;:6 3/?8008 781,9 5/-.6 (15) 0/ ;:/1IG, ?,@8. DE.C ,=8080/ - ;,?,HCG 78??E 15.1: x +1 xZ +1 Z Hi(x)f (x) dx ; Hi(x)(I2 f )(x) dx 81 h2 kf 00kC ( ab])Ci x ;1 x ;1 >48 xZ +1 Ci = Hi(x) dx = 12 (xi+1 ; xi;1 ) h x ;1 6 h = i=1max (x ; xi ): 2:::n;1 i+1 ,J.,?I i

i

i

i

i

i

x +1 xZ +1 Z Hi(x)f (x) dx ; Hi(x)(I2f )(x) dx 81 h3kf 00kC ( ab]) x ;1 x ;1 ,2/@8?, 5., I-7,168 N > n 1/@0,. :84;,7,@6?, 5., N = n , ..8. , AI02=66 f 0/? 6318-.0E 76FC 88 30/5806< f (x1 ) f (x2) : : : f (xn) 1 .,52/B x1 x2 : : : xn . 1. E56-76? 47< i = 1 2 : : : n ; 1 xZ +1 Hi(x)f (x) dx: (16) i

i

i

i

i

x

i

:6D7636? AI02=6G f 0/ ,.:8328 Txi xi+1 ] 760890,9 AI02=689 { 60.8:;,7<=6,00E? ?0,>,5780,? />:/0@/ L(2i) ;8:1,9 -.8;806, ;,-.:,800E? ;, .,52/? xi 6 xi+1 : ; f (xi) : L(2i) (x) = f (xi) + (x ; xi)f (xiZ xi+1 ) = f (xi) + (x ; xi) f (xxi+1) ; i+1 xi :6 J.,? 1 -67I .8,:8?E 7.2 47< 1-8B x 2 Txi xi+1] -;:/184761, :/180-.1, f (x) ; L(2i) (x) = 2!1 (x ; xi )(x ; xi+1 ) f 00( (x)) >48 (x) 2 Txi xi+1 ]. 784,1/.87C0,, xZ+1 xZ+1 xZ +1 1 (i) Hi(x)f (x) dx ; Hi(x)L2 dx 2! (x ; xi )(x ; xi+1 ) f 00( (x)) dx x x x i

i

i

i

i

i

xZ +1 1 00 2 kf kC( x x +1]) (x ; xi)(xi+1 ; x) dx: x i

i

i

i

. .

!"#$ $ %$

x16. ( 2= A )() ,-2,7C2I xZ +1

(x ; xi )(xi+1 ; x) dx =

i

x

xZ +1

(x ; xi)((xi+1 ; xi) ; (x ; xi )) dx = x 1 = (xi+1 ; xi)3 ; 1 (xi+1 ; xi)3 = 1 (xi+1 ; xi )3 2 3 6

i

.,

54

i

(17)

i

xZ+1 xZ +1 1 (i) Hi(x)f (x) dx ; Hi(x)L2 dx 12 (xi+1 ; xi )3 kf 00kC ( x x +1]): i

x

i

i

x

i

(18)

i

i

,J.,?I ?E ?,@8? ;:6D7636.C 60.8>:/7 (16) -784IGH6? -;,-,D,?: xZ +1 i

x

Hi(x)f (x) dx

i

xZ +1 i

x

Hi(x)L(2i) (x) dx

(19)

i

- ,F6D2,9 (18). ?88?

Hi(x)L(2i) dx = xx ;; xxi+1 (f (xi ) + (x ; xi)f (xi Z xi+1)) dx = i i+1 x 0 1 xZ +1 xZ +1 1 = x ; x @f (xi ) (xi+1 ; x) dx + f (xiZ xi+1 ) (x ; xi )(xi+1 ; x) dxA xZ +1 i

i

i

i+1

i

i

x

x

i

i

;,?,HCG (17) ;:,4,7@/8?: xZ +1 i

x

i

; f (xi) 1 (xi+1 ; xi)3 Hi(x)L dx = x ; x f (xi) 21 (xi+1 ; xi )2 + f (xxi+1 ) ; 6 i+1 i i+1 xi 1 = (xi+1 ; xi) 2 f (xi) + 16 (f (xi+1) ; f (xi)) = 61 (xi+1 ; xi)(2f (xi) + f (xi+1 )): 1

(i) 2

!

./2, 1 -67I (19), (18) ;,7I5/8?: xZ +1 i

x

i

Hi(x)f (x) dx 61 (xi+1 ; xi )(2f (xi) + f (xi+1 ))

(20)

- ,F6D2,9

xZ+1 1 1 Hi(x)f (x) dx ; 6 (xi+1 ; xi)(2f (xi) + f (xi+1 )) 12 (xi+1 ; xi)3 kf 00kC ( x x +1]): i

x

i

i

. .

i

(21)

!"#$ $ %$

x16. ( 2= A )()

55

2. 0/7,>650, 1E56-76? 47< i = 2 3 : : : n Zx Hi(x)f (x) dx 16 (xi ; xi;1 )(2f (xi) + f (xi;1)) (22) x ;1 - ,F6D2,9 Zx Hi(x)f (x) dx ; 1 (xi ; xi;1 )(2f (xi) + f (xi;1)) 1 (xi ; xi;1)3 kf 00kC ( x x ]): ;1 x 12 6 ;1 (23) E56-76? ;:/1IG 5/-.C -6-.8?E (8): xR2 1. #,:?I7E 47< H1 (x)f (x) dx ;,7I5/G.-< 63 (20), (21) ;:6 i = 1. x1 2. E56-76? 47< i = 2 3 : : : n ; 1 i

i

i

i

i

i

(Hi f )L2( ab]) =

xZ +1 i

x ;1

Hi(x)f (x) dx =

i

Zx

i

x ;1

Hi(x)f (x) dx +

i

xZ +1 i

x

Hi(x)f (x) dx:

i

-67I (20), (21), (22), (23) (Hi f )L2 ( ab]) 16 (xi+1 ; xi )(2f (xi) + f (xi+1)) + 16 (xi ; xi;1 )(2f (xi) + f (xi;1 )) = 16 (xi ; xi;1)f (xi;1 ) + 13 (xi+1 ; xi;1 )f (xi) + 16 (xi+1 ; xi )f (xi+1) - ,F6D2,9 (Hi f )L2 ( ab]) ; 1 1 1 ; 6 (xi ; xi;1)f (xi;1) + 3 (xi+1 ; xi;1)f (xi) + 6 (xi+1 ; xi)f (xi+1) 121 (xi+1 ; xi)3kf 00kC( x x +1]) + 121 (xi ; xi;1)3kf 00kC( x ;1x ]) 121 kf 00kC( x ;1x +1]) (xi+1 ; xi)3 + (xi ; xi;1)3 xR 3. #,:?I7E 47< Hn(x)f (x) dx ;,7I5/G.-< 63 (22), (23) ;:6 i = n ; 1. x ;1 -76 ;,7,@6.C x0 = x1 , xn+1 = xn , ., A,:?I7E 47< ;:/1,9 5/-.6 -6-.8?E (8) ?,@0, 3/;6-/.C 1 D,788 2,?;/2.0,? 1648: (Hi f )L2( ab]) 1 (xi ; xi;1 )f (xi;1 )+ 1 (xi+1 ; xi;1 )f (xi)+ 1 (xi+1 ; xi )f (xi+1) (24) 6 3 6 >48 ,F6D2/ ;:6D76@806< (Hi f )L2( ab]) ; ; 61 (xi ; xi;1)f (xi;1) + 13 (xi+1 ; xi;1)f (xi) + 16 (xi+1 ; xi)f (xi+1) (25) 121 (xi+1 ; xi)3 + (xi ; xi;1)3 kf 00kC( x ;1x +1]) i

i

i

i

i

i

n

n

i

. .

i

!"#$ $ %$

x17. ( 2= A )() -76 ,D,30/56.C

56

h = i=1max (x ; xi ) 2:::n;1 i+1

., ,F6D2I (25) ?,@0, 3/;6-/.C 1 1648

(Hi f )L2( ab]) ; 1 1 1 ; 6 (xi ; xi;1)f (xi;1) + 3 (xi+1 ; xi;1)f (xi) + 6 (xi+1 ; xi)f (xi+1)

61 h kf kC x ;1x +1 3

00

(

i

i

(26)

])

3 164/ -6-.8?E (8) 6 88 ;:/1,9 5/-.6 (24) ;,7I5/8?, 5., i = f (xi), ..8. ;:6D76@8068 ;, ?8.,4I 0/6?80CF6B 21/4:/.,1 -,1;/4/8. - ;:6D76@8068?, ;,-.:,800E? 2I-,50,-760890,9 60.8:;,7<=689: L2 f = I2f . x

17. ) )

/4/5/ ;,-.,806< ;:6D76@/GH89 AI02=66 Pf 2 Pn;1 , 0/6D,788 D7632,9 2 3/4/00,9 AI02=66 f 1 0,:?8 L2 (..8. ?606?636:IGH89 AI02=6,0/7 kf ; Pf kL2 ) <17<8.-< -7,@0,9 - 1E56-76.87C0,9 .,526 3:806<. 40/2,, 3/4/5/ ?606?63/=66 AI02=6,0/7/ kf ; Pf k , >48 k k { 0,:?/ ;:,-.:/0-.1/ L2 - 082,.,:E? 18-,?, ?,@8. DE.C :8F80/ 4,-./.,50, 78>2,.

x 17.1.

*%"#' && *"& C?'D&

1. 2 1$

Ta b]

Tb0 Tb1 : : : Tbn;1

Pn

;1

(. (13.3)) n 1 .

;

. 60890/< 083/16-6?,-.C ?0,>,5780,1 Tb0 Tb1 : : : Tbn;1 4,-

7"#&

2/3/0/ 1 78??8 13.1. ,2/@8?, 5., 1-<269 ?0,>,5780 Pn;1 2 Pn;1 ?,@8. DE.C ;:84-./1780 1 1648 760890,9 2,?D60/=66 ?0,>,5780,1 Tb0 Tb1 : : : Tbn;1 . ;:84876? ;, .,52/? (13.7) 46-2:8.0,8 \-2/7<:0,8" ;:,631848068 (13.1) 6 ;,-.:,6? :/37,@8068 (13.10) ;, ?0,>,5780/? 8DEF81/ Tb0 Tb1 : : : Tbn;1 47< AI02=66 f = Pn;1 . -67I 78??E 13.3 Pn;1(xi ) = Pf (xi), i = 1 2 : : : n . ,-2,7C2I Pn;1 6 Pf <17,5780/?6 -.8;806 08 1EF8 n ; 1 6 -,1;/4/G. 1 n .,52/B, ., Pn;1 Pf . ./2, 1-<269 ?0,>,5780 Pn;1 2 Pn;1 ;:84-./1780 1 1648 760890,9 2,?D60/=66 ?0,>,5780,1 Tb0 Tb1 : : : Tbn;1 (;:658? 2,JAA6=680.E :/37,@806< 1E56-780E 0/?6 <10,, -?. (13.10)). 8??/ 4,2/3/0/.

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$ :

Tb0 Tb1 : : : Tbn;1

57

Ta b]

-

Zb

Zb bi(y)Tbj (y) b2 T 2 2 q q T0 (y ) dy = 2 ij i + j 6= 0 dy = : a (b ; y )(y ; a) a (b ; y )(y ; a)

7"#&. ,2/@8? 10/5/78, 5., ?0,>,5780E T0 T1 : : : Tn;1 0/ ,.:8328 T;1 1] I4,178.1,:
>48

Z Z 1 1 = cos((i ; j ) ) d + cos((i + j ) ) d = 20 20 = 21 (I (i ; j ) + I (i + j ))

9 8 1 > > sin( k ) = 0 k = 6 0 = < k 0 I (k) = cos(k ) d = > Z > = k0: : 1 d = k = 0 0 0 Z

784,1/.87C0,,

81 2 2 Z1 Ti (x)Tj (x) 1 1 p1 ; x2 dx = 2 ij + 2 i;j = <: 2 ij ii =+jj=6=0 0 ;1 487/8? 348-C 3/?80I ;8:8?800EB (11.4): x = x(y), x : Ta b] ! T;1 1], x(y) = 2y ; (b + a) : b;a Zb Z1 Ti(x)Tj (x) p1 ; x2 dx = r 21y;(b+a) 2 Tbi(y)Tbj (y) b ;2 a dy a 1; ;1 b;a

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r

58

1 q(b ; a)2 ; (2y ; (b + a))2 = = 2y;(b+a) 2 b ; a 1

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.,

Z1 Ti (x)Tj (x) Zb Tbi (y)Tbj (y) p1 ; x2 dx = q(b ; y)(y ; a) dy a ;1

3 (1) ;,7I5/8? .:8DI8?E9 :83I7C./.. 8??/ 4,2/3/0/.

x 17.2.

.& 7$ " %6"<8

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Zb

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a

q u(y)v(y) dy (b ; y)(y ; a)

kuk

= Tu u]1=2

L2 (Ta b]) = f f : kf k < 1g:

Zb

dy <1 a (b ; y )(y ; a) 6?88. ?8-., 127G58068 C (Ta b]) L2 (Ta b]). , ?8.,4I 0/6?80CF6B 21/4:/.,1 47< AI02=66 f 2 L2 (Ta b]) .:8DI8.-< ;,-.:,6.C ;:6D76@8068 Pf 2 Pn;1 ./2,8, 5., kf ; Pf k = g2Pinf;1 kf ; gk :

q

n

-67I 78??E 1 ?,@0, 6-2/.C Pf 1 1648

Pf =

nX ;1 j =0

j Tbj :

(2)

;1 784,1/.87C0,, 0/4, 0/9.6 fj gnj=0 ./268, 5.,

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59

812764,1,? ;:,-.:/0-.18 L2 Ta b] ?606?I? kf ; Pf k :8/763I8.-< D b En;10/ AI02=66 Pf , <17,0/780 ;,4;:,-.:/0-.1I Pn;1 . /2 2/2 Tbj j=0 { D/36- Pn;1 , ., I-7,168 ,:.,>,0/7C0,-.6 Pn;1 J2161/780.0, ,:.,>,0/7C0,-.6 Tbi , i = 0 1 : : : n ; 1:

h i f ; Pf Tbi = 0 i = 0 1 : : : n ; 1

..8.

h

i h i Pf Tbi = f Tbi i = 0 1 : : : n ; 1: (3) -7,16< (3) ;:84-./17
-67I 78??E 2

hb bi h i Tj Ti = ij Tbi Tbi i j = 0 1 : : : n ; 1

;,J.,?I 63 (4) ;,7I5/8?

h bi f Ti i = h b b i i = 0 1 : : : n ; 1 Ti Ti

..8.

h i h i 0 = 1 f Tb0 i = 2 f Tbi

x 17.3.

'" F

i = 1 2 : : : n ; 1

(5)

8& %7"A<

7< 1E56-7806< 2,JAA6=680.,1 :/37,@806< (5) .:8DI8.-< 1E56-7<.C 60.8>:/7E h b i Zb f (y)Tbi(y) dy f Ti = q ( b ; y )( y ; a ) a ;a, 1848? .,526 a = y0 < y1 < : : : < yN = b (0/;:6?8:, yj = a + jh , h = b N j = 0 1 : : : N , N > 0, 5/-., N n ). ,>4/ y ;1 Z +1 b h b i NX q f (y)Ti(y) dy f Ti = (b ; y)(y ; a) j =0 y j

(6)

j

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x17. ( 2= A )()

60

:6D7636? AI02=6G

g(y) = f (y)Tbi(y) (7) 0/ ,.:8328 Tyj yj+1] 760890,9 AI02=689 { 60.8:;,7<=6,00E? ?0,>,5780,? />:/0@/ L(2j) ;8:1,9 -.8;806, ;,-.:,800E? ;, .,52/? yj 6 yj+1 : g(yj ) : L(2j)(y) = g(yj ) + (y ; yj )g(yj Z yj+1) = g(yj ) + (y ; yj ) g(yyj+1) ; j +1 ; yj :6 J.,? 1 -67I 78??E 15.1 -;:/184761, 08:/180-.1, kg ; L(2j)kC( y y +1]) 18 (yj+1 ; yj )2 kg00kC( y y +1]): 784,1/.87C0,, y +1 yZ +1 (j ) Z q g(y) dy ; q L2 dy (b ; y)(y ; a) y (b ; y)(y ; a) y yZ +1 (8) 18 (yj+1 ; yj )2 kg00kC( y y +1]) q(b ; y1)(y ; a) dy = y 1 = (yj+1 ; yj )2 kg00kC ( y y +1])Cj (a b) 8 >48 yZ +1 1 Cj (a b) = q dy: (b ; y)(y ; a) y yZ +1 dy -784IGH6? -;,,J.,?I ?E ?,@8? ;:6D7636.C 60.8>:/7 q g(y) (b ; y)(y ; a) y -,D,?: yZ +1 yZ +1 (j ) g ( y ) q dy q L2 (y) dy ( b ; y )( y ; a ) ( b ; y )( y ; a ) y y - ,F6D2,9 (8). ,4-./17<< J., 1 (6), 0/B,46? y (j ) ;1 Z +1 h b i NX q L2 (y) dy (9) f Ti (b ; y)(y ; a) j =0 y - ,F6D2,9 h i NX y (j ) ;1 ;1 Z +1 NX 1 (y ; y )2 kg00k L2 (y) f Tbi ; q dy j +1 j C ( y y +1 ]) Cj (a b) (b ; y)(y ; a) j=0 8 j =0 y N ;1 18 kg00kC( ab]) h2 X Cj (a b) = j =0 1 2 b = 8 C (a b) h k(f Ti)00kC ( ab]) (10) j

j

j

j

j

j

j

j

j

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j

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j

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61

Zb

1 dy: (b ; y)(y ; a)

h = j=0max (y ; yj ) C (a b) = q 1:::N ;1 j +1 a

E56-76?

(j ) Z +1 g(yj ) + (y ; yj )g(yj Z yj+1) L 2 (y ) q q dy = dy = ( b ; y )( y ; a ) (b ; y)(y ; a) y y yZ +1 yZ +1 dy = (g(yj ) ; yj g(yj Z yj+1)) q + g(yj Z yj+1) q y dy (b ; y)(y ; a) (b ; y)(y ; a) y y (11) /2 2/2 Z Z b;a dx Z dx dy 2 q p p1 ; x2 = arcsin x = = b;a 1 ; x2 (b ; y)(y ; a) y= b+a + b;a x 2

yZ +1

y

j

j

j

j

j

j

j

j

2

.,

aj =

yZ +1 j

y

j

/2 2/2

2

y +1 dy 2 y ; ( b + a ) : q = arcsin b ; a y (b ; y)(y ; a) j

(12)

j

Z b+2 a + b;2 a x b;2 a dx Z b + a dx ydy q p = = b;a p1 ; x2 2 1 ; x2 + (b ; y)(y ; a) y= b+a + b;a x 2 2 2 Z b ; a x dx b + a b ; a ; p1 ; x2 p = + arcsin x + 2 1 ; x2 2 2 ., y +1 v yZ +1 !

u 2 u bj = q y dy = b + a aj ; b ; a t1 ; 2y ; (b + a) : (13) 2 2 b;a (b ; y)(y ; a) y y Z

j

j

j

j

,4-./17<< (12) 6 (13) 1 (11), ;,7I5/8? yZ +1

(j ) L q 2 (y) dy = (g(yj ) ; yj g(yj Z yj+1))aj + g(yj Z yj+1)bj = (b ; y)(y ; a) y ; g(yj) = = aj g(yj ) + (bj ; aj yj )g(yj Z yj+1)) = aj g(yj ) + (bj ; aj yj ) g(yyj+1) ; j +1 yj !

aj yj g(y ) + bj ; aj yj g(y ) = aj yj+1 ; bj g(y ) + bj ; aj yj g(y ) = aj ; ybj ; ; j j y y ; y j+1 y ;y y ; y j+1 j

j

j +1

. .

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62

cj = ayj yj+1;;ybj dj = ybj ; ;aj yyj : j +1

,>4/

j

j +1

(14)

j

yZ +1

(j ) L q 2 (y) dy = cj g(yj ) + dj g(yj+1): (b ; y)(y ; a) y ,4-./17<< J., 1 (9), 0/B,46? ;1 NX ;1 N h b i NX X f Ti (cj g(yj ) + dj g(yj+1)) = cj g(yj ) + dj;1g(yj ) = j

j

j =0

= c0g(y0) +

NX ;1 j =1

j =0

j =1

(cj + dj;1)g(yj ) + dN ;1g(yN )

-67I (7) 6?88? NX ;1 h bi f Ti c0f (y0)Tbi(y0) + (cj + dj;1)f (yj )Tbi (yj ) + dN ;1f (yN )Tbi(yN ) j =1

D,30/56?

u0 = c0f (y0) uj = (cj + dj;1)f (yj ) j = 1 2 : : : N ; 1 uN = dN ;1f (yN ) (15)

(,.?8.6?, 5., uj 08 3/16-<. ,. i ). ,>4/ N h bi X f Ti uj Tbi (yj )

(16)

j =0

- ,F6D2,9 (-?. (10))

N h b i X 1 f Ti ; uj Tbi (yj ) 8 C (a b) h2 k(f Tbi)00 kC ( ab]) j =0

x 17.4.

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63

I48? 1E56-7<.C J.6 -I??E /0/7,>650, 2,JAA6=680./? :/37,@806< ;, ?0,>,5780/? 8DEF81/. /;6F8? 1E:/@806< (18) 47< i 1 1648 ./D76=E 0 = 1 ( u0Tb0 (y1) + u1Tb0 (y2) + : : : + uN Tb0 (yN ) ) 1 = 2 ( u0Tb1 (y1) + u1Tb1 (y2) + : : : + uN Tb1 (yN ) ) (19) 2 = 2 ( u0Tb2 (y1) + u1Tb2 (y2) + : : : + uN Tb2 (yN ) ) ... ... ... ... ... n;1 = 2 ( u0Tbn;1 (y1) + u1Tbn;1(y2) + : : : + uN Tbn;1(yN ) ) ,7C3I<-C :822I:80.0E?6 A,:?I7/?6 (13.12), DI48? 1E56-7<.C 1 -I??/B (19) -.,7D=E -781/ 0/;:/1,, / 2/@4E9 -.,7D8= { -18:BI 1063. D,30/56? gij = uj Tbi (yj ) zj = 2 2yj ;b ;(ba+ a) i = 0 : : : n ; 1 j = 0 1 : : : N:

, A,:?I7/? (13.12) 47< 1-8B j = 0 1 : : : N 1E56-7,, 2/2 2,JAA6=680.E i 1E56-780E, 30/58068 ;:6D76@/GH8>, n;1 ?0,>,5780/ 1 .,528 Pf (y) = P iTbi(y) 1E56-7<8.-< - 6-;,7C3,1/068? A,:i=0 ?I7 (13.12) Tb0 (y) = 1 Tb1(y) = z=2 Pf (y) = 0Tb0 (y) + 1 Tb1(y) (23) Tbi (y) = z Tbi;1 (y) ; Tbi;2(y) Pf (y) := Pf (y) + i Tbi(y) i = 2 : : : n ; 1: >48 z = 2 2y ;b ;(b a+ a) .

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7< 1-8B j = 0 1 : : : N 0/? .:8DI8.-< ;:,6318-.6 1E56-7806< ;, A,:?I7/? (12), (13), (14), (15) 6 1E56-76.C 1-8 2,JAA6=680.E uj . K., ,-IH8-.17<8.-< 3/ 760890,8 ;, N 56-7, /:6A?8.658-26B ,;8:/=69: O(N ). 7< 1-8B j = 0 1 : : : N 0/? .:8DI8.-< ;:,6318-.6 1E56-7806< 1) , A,:?I7/? (20). / J., .:8DI8.-< n + O(1) ?I7C.6;762/.610EB 6 -.,7C2, @8 /446.610EB ,;8:/=69. 2) , A,:?I7/? (21). / J., .:8DI8.-< n + O(1) /446.610EB ,;8:/=69. DH88 2,7658-.1, ,;8:/=69 47< 1E56-78069 ;, A,:?I7/? (20) 6 (21) 47< 1-8B j = 0 1 : : : N : (N + 1)n + O(N ) ?I7C.6;762/.610EB 6 2(N + 1)n + O(N ) /446.610EB ,;8:/=69. 7< 1E56-7806< 2,JAA6=680.,1 i ;, A,:?I7/? (22) .:8DI8.-< 8H8 n + O(1) ?I7C.6;762/.610EB ,;8:/=69. 784,1/.87C0,, -I??/:0,8 2,7658-.1, ,;8:/=69, 08,DB,46?EB 47< 1E56-7806< 2,JAA6=680.,1 i , :/10, Nn + O(N + n) ?I7C.6;762/.610E? 6 2Nn + O(N + n) /446.610E? ,;8:/=6, ?0,>,5780/ Pf 1 .,528 ;, A,:?I7/? (23) .:8DI8.-< 2n + O(1) ?I7C.6;762/.610EB 6 -.,7C2, @8 /446.610EB ,;8:/=69. G 1.

/-., 1ED6:/G. N = n + 1 6 .,526

y0 = a yj = a +2 b + b ;2 a cos (2j2n; 1) j = 1 2 : : : n yn+1 = b

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65

Rb jf 00(x)j1=2 dx < 1 4 a

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Zb Zx i ; 1 00 1= 2 jf (x)j dx = n ; 1 jf 00(x)j1=2 dx i = 2 : : : n ; 1: i

a

a

$ f - f " x1 x2 : : : xn ,

$

kf ; I f kC ab 2

(

])

= O(n;2) (n ! 1):

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Z1 0

jf (x)j 00

1=2

Z1 1=4 1 1 1 x ;3=4 dx = 2 x dx = 2 1=4 = 2 < 1 Z 0 0

i ; 1 4 i ; 1 4) 63 I-7,16< jf (x)j dx = 2xi = n ; 1 2 ;,7I5/8? xi = n ; 1 , i = a 2 3 : : : n ; 1, x1 = a = 0, xn = b = 1. /948? kf ; I2f kC ab] = xmax jf (x) ; (I2f )(x)j = 2 ab] !

(1) f ( x i+1 ) ; f (xi ) = i=1max max f ( x ) ; f ( x ) + ( x ; x ) i i 2:::n;1 x2 x x +1 ] xi+1 ; xi Zx

i

1=2

00

i

D,30/56?

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f ( x i+1 ) ; f (xi ) gi(x) = f (x) ; f (xi ) + (x ; xi) x ; x : i+1

i

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f (xi) : f 0(x) = f (xxi+1) ; ;x i+1

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pxi ; pxi ! p p = g(x) = x ; xi + (x ; xi ) x ; x i i

! p p p p 1 1 1 p = ( xi + xi ) ; xi + ( xi + xi ) ; 4xi p 2 4 xi + pxi = p p p = 12 (pxi ; xi ) ; 14 (pxi ; xi ) (pxi + 3 xi ) px 1+ px = i i p

p ! p xi ; xi xi + 3pxi p 1 p = ( xi ; xi ) 2 ; p =1 p p 4 xi + xi 4 xi + pxi 7< AI02=66 f (x) =

+1 +1

+1

2

+1

0

0

+1

+1 +1

+1

+1

+1

+1

+1

+1

+1

+1

./2,

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+1

1 max j g ( x ) j = x2 x x +1 ] 4 i

+1 2

i

2 xi+1 ; pxi pxi+1 + pxi :

p

i ; 1 4 :6 xi = n ; 1 ;,7I5/8? !

i 2 ; i ; 1 2 2 2 2 2 1 n ; 1 n ; 1 1 1 ( i ; ( i ; 1) ) = max j g ( x ) j = = 2 2 2 2 2 x2 x x +1 ] 4 i i;1 4 (n ; 1) i + (i ; 1) + n;1 n;1

! 2 1 1 (2 i ; 1) 1 1 1 = 4 (n ; 1)2 i2 + (i ; 1)2 = 4 (n ; 1)2 2 ; i2 + (i ; 1)2 i

i

14 (n ;1 1)

2

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I-.C 3/4/0E .,526 a = x1 < x2 < : : : < xn = b 6 30/5806< f (x1 ) f (x2) : : : f (xn). 0.8:;,76:IGH/< AI02=6< Pf -.:,6.-< ./26? ,D:/3,?, 5., 0/ 2/@4,? ,.:8328 Txi xi+1], i = 1 : : : n ; 1 ,0/ <17<8.-< ?0,>,5780,? Pi -.8;806 3, ./26?, 5.,

9 Pi(xi+1 ) = f (xi+1) = i = 1 : : : n ; 1 Pi0(xi+1 ) = di+1

Pi(xi) = f (xi) Pi0(xi ) = di

(1)

>48 di , i = 1 : : : n { -1,D,40E8 ;/:/?8.:E, .,. 676 60,9 -;,-,D 1ED,:/ 2,.,:EB ,;:8487<8. ?8.,4 2I-,50,9 60.8:;,7<=66 2ID658-26?6 ?0,>,5780/?6. ,7I5800/< AI02=6< Pf -,1;/4/8. - f 1 .,52/B xi , i = 1 : : : n 6 47< 7GD,>, 0/D,:/ ;/:/?8.:,1 di Pf 2 C (1) (Ta b]).

x 19.2.

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Pi

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Pi(x) = a1i + a2i(x ; xi ) + a3i(x ; xi )2 + a4i (x ; xi)2 (x ; xi+1 )

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f (xi ) f (xi )

n

di

n

. n

f (xiZ xi+1 ) . n f (xi+1 ) . n di+1 . f (xi+1 )

. .

f (xi Z xi+1) ; di xi+1 ; xi

n

di+1 ; f (xi Z xi+1) . xi+1 ; xi .

di + di+1 ; 2f (xiZ xi+1) (xi+1 ; xi )2

!"#$ $ %$

x19. (',./ . 1

+- 1 + 81 / .

68

.,7D=E 1E56-7
i+1

i

Pi(x) = c1i + c2i (x ; xi) + c3i(x ; xi)2 + c4i(x ; xi)3

(2)

3/;6F8?

(x ; xi)2 (x ; xi+1 ) = (x ; xi )3 ; (xi+1 ; xi )(x ; xi )2 ,.2I4/ ;,7I5/8?

c1i = a1i = f (xi) c2i = a2i = di

c3i = a3i ; a4i(xi+1 ; xi ) = 3f (xiZ xxi+1 ) ;; 2xdi ; di+1 i+1 i d i + di+1 ; 2f (xi Z xi+1 ) c4i = a4i = (xi+1 ; xi )2

x 19.3.

(3)

!< %6"<8< !? *" H%

I-.C 1 .,52/B a = x1 < x2 < : : : < xn = b 6318-.0E 30/5806< AI02=66: f (x1 ) : : : f (xn) 6 88 ;:,631,40,9 f 0(x1 ) : : : f 0(xn). ,>4/ ;/:/?8.: di 1 ,DH89 -B8?8 60.8:;,7<=66 2I-,50,-2ID658-26?6 AI02=6/8.-< :/10E?

di = f 0(xi )

i = 1 2 : : : n:

-0,10E8 && 6%?"A=>

!8

(4)

Pf :

1) Pf <17<8.-< 2I-,50,-?0,>,57800,9 AI02=689, 30/5806< 2,.,:,9 3/16-<. .,7C2, ,. 7,2/7C0,>, ;,184806< AI02=66 f 6 f 0 , 6?800,: 30/5806< Pf 0/ ,.:8328 Txi xi+1 ] 3/16-<. .,7C2, ,. 30/58069 AI02=66 f 6 88 ;:,631,40,9 f 0 1 .,52/B xi 6 xi+1 . 2) 8.,4 ;:6D76@806< <17<8.-< 760890E?: P (f + g) = Pf + Pg , P (f ) = Pf .

. .

!"#$ $ %$

x19. (',./ . 1 2C

69

(Ta b]) , $

8 1.

f

+- 1 + 81 / .

(4)

kf ; Pf kC ab 4!1 21 h kf kC ab (

])

4

4

(4)

(

])

h = i=1max (x ; xi ): 2:::n;1 i+1

(5)

. , .8,:8?8 7.2 ;,7I5/8? j(x ; xi)2(x ; xi+1)2jkf (4)kC( x x +1]): (6) kf ; Pf kC( x x +1]) 4!1 x2 max x x +1 ] #I02=6< g(x) = (x ; xi )(x ; xi+1 ) = x2 ; (xi + xi+1 )x + xixi+1 0 47< 1-8B x 2 Txi xi+1] 6 6?88. 0I7C ;:,631,40,9 (xi + xi+1 )=2, ;:60/478@/H69 ,.:832I Txi xi+1]. ,-2,7C2I g(xi) = g(xi+1) = 0, ., xi + xi+1 2 xi+1 ; xi xi ; xi+1 2 1 2 g = = 2 (xi+1 ; xi)4: max j g ( x ) j = x2 x x +1 ] 2 2 2 4 ,4-./17<< J., 1 (6), ;,7I5/8? kf ; Pf kC( x x +1]) 4!1 214 (xi+1 ; xi)4 kf (4)kC( x x +1]): 784,1/.87C0,, 1 1 h4 kf (4) k kf ; Pf kC( ab]) = i=1max k f ; Pf k C ( x x +1 ]) C ( ab]) : 2:::n;1 4! 24 8??/ 4,2/3/0/.

7"#& i

i

i

i

i

i

i

i

i

i

i

x 19.4.

i

i

i

!< %6"<8< !? *" )"<

I-.C 1 .,52/B a = x1 < x2 < : : : < xn = b 6318-.0E 30/5806< AI02=66 f (x1 ) : : : f (xn). /:/?8.: di , i = 2 3 : : : n ; 1 1 ,DH89 -B8?8 60.8:;,7<=66 2I-,50,-2ID658-26?6 AI02=6/8.-< :/10E? 30/5806G 1 .,528 xi ;:,631,40,9 ?0,>,5780/ P(i) 1.,:,9 -.8;806, -,1;/4/GH8>, - f 1 .,52/B xi;1 xi xi+1 . , 60.8:;,7<=6,00,9 A,:?I78 CG.,0/ P(i) (x) = f (xi;1) + (x ; xi;1 )f (xi;1Z xi ) + (x ; xi;1 )(x ; xi)f (xi;1 Z xiZ xi+1 ) 6 P(0i) (xi) = f (xi;1 Z xi) + (xi ; xi;1 )f (xi;1 Z xiZ xi+1) = = f (xi;1 Z xi) + (xi ; xi;1 ) f (xiZ xi+1 ) ; f (xi;1 Z xi) = xi+1 ; xi;1 = (xi+1 ; xi;1)f (xi;1 Z xi) + (xi ;x xi;1;)fx(xi Z xi+1) ; (xi ; xi;1 )f (xi;1Z xi) = i+1 i;1 ( x ; x ) f ( x Z x ) + ( x ; x ) f ( x i i;1 i Z xi+1 ) = i+1 i i;1 i xi+1 ; xi;1

. .

!"#$ $ %$

x19. (',./ . 1

+- 1 + 81 / .

70

784,1/.87C0,,

+ (xi ; xi;1 )f (xiZ xi+1 ) i = 2 3 : : : n ; 1: di = P(0i)(xi ) = (xi+1 ; xi)f (xi;1xZ xi) ; i+1 xi;1 (7) 84,-./GH68 30/5806< d1 6 dn 1 >:/0650EB I37/B 1ED6:/G.-<, 6-B,4< 63 .,9 676 60,9 4,;,706.87C0,9 60A,:?/=66 , AI02=66 f (-?. 06@8). -0,10E8 && 6%?"A=> !8 Pf : 1) Pf <17<8.-< 2I-,50,-?0,>,57800,9 AI02=689, 30/5806< 2,.,:,9 3/16-<. .,7C2, ,. 7,2/7C0,>, ;,184806< AI02=66 f , 6?800,: 30/5806< Pf 0/ ,.:8328 Txi xi+1 ] 3/16-<. .,7C2, ,. 30/58069 AI02=66 f 1 .,52/B xi;1 xi xi+1 xi+2 . 2) 8.,4 ;:6D76@806< <17<8.-< 760890E?: P (f + g) = Pf + Pg , P (f ) = Pf . 2.

(D83 4,2/3/.87C-.1/). f 2 C (3) (Ta b]) , -

$

9

kf ; Pf kC x2x ;1 const h kf kC(3) ab (

h

n

])

3

(

])

(5).

,>:8F0,-.C /;;:,2-6?/=66 0/ ,.:832/B Tx1 x2] 6 Txn;1 xn] 3/16-6. ,. -;,-,D/ 1ED,:/ 30/58069 d1 6 dn . -76 .,526 xi :/-;:848780E 0/ ,.:8328 Ta b] :/10,?8:0,: xi = a + (i ; 1)h , h = (b ; a)=(n ; 1), ., A,:?I7E (7) ;:6,D:8./G. ,-,D800, ;:,-.,9 164: di = 12 (f (xi;1Z xi) + f (xiZ xi+1)) = f (xi+1 ) 2;h f (xi;1 ) i = 2 3 : : : n ; 1: G 1.

x 19.5.

!< %6"<8< !? *" $ '

K.,. ?8.,4 ;:6D76@806< 6-;,7C3I8.-< 47< D,:CDE - 1ED:,-/?6 ;:6D76@/GH89 AI02=66, 2,.,:E8 ;,<17:8F0,-.CG. I-.C 1 .,52/B a = x1 < x2 < : : : < xn = b 6318-.0E 30/5806< AI02=66 f (x1 ) : : : f (xn). -67I 78??E 4.6 :/3487800/< :/30,-.C f (xi;1 xi ) <17<8.-< ;:6D76@8068? 2 f 0(xi ) -781/, / f (xi xi+1 ) <17<8.-< ;:6D76@8068? 2 f 0(xi) -;:/1/. ?8.,48 26?E J.6 ;:6D76@806< I-:8407/42,-.C AI02=66 0/ -,-8408? ,.:8328. 2,05/.87C0/< A,:?I7/ 47<

. .

!"#$ $ %$

x19. (',./ . 1

+- 1 + 81 / .

71

,;:8487806< ;/:/?8.:/ di 1 ,DH89 -B8?8 60.8:;,7<=66 2I-,50,-2ID658-26?6 AI02=6 wi+1f (xi;1Z xi) + wi;1 f (xiZ xi+1) 8-76 wi2+1 + wi2;1 6= 0 > < w + w (8) di = (x ; x )fi(+1x Z xi;1 > i +1 i i ;1 i ) + (xi ; xi;1 )f (xi Z xi+1 ) > 8-76 wi+1 = wi;1 = 0 : xi+1 ; xi;1 >48 i = 3 4 : : : n ; 2 6 wj = jf (xj Z xj+1) ; f (xj;1Z xj )j : 84,-./GH68 30/5806< d1 d2 6 dn;1 dn 1 ;:6>:/0650EB I37/B 1ED6:/G.-<, 6-B,4< 63 .,9 676 60,9 4,;,706.87C0,9 60A,:?/=66 , AI02=66 f (-?. 06@8). -0,10E8 && 6%?"A=> !8 Pf : 1) Pf <17<8.-< 2I-,50,-?0,>,57800,9 AI02=689, 30/5806< 2,.,:,9 3/16-<. .,7C2, ,. 7,2/7C0,>, ;,184806< AI02=66 f , 6?800,: 30/5806< Pf 0/ ,.:8328 Txi xi+1 ] 3/16-<. .,7C2, ,. 30/58069 AI02=66 f 1 .,52/B xi;2 xi;1 xi xi+1 xi+2 xi+3 . 2) 8.,4 ;:6D76@806< 08 <17<8.-< 760890E?: P (f + g) 6= Pf + Pg , B,.< P (f ) = Pf . 3. (D83 4,2/3/.87C-.1/). f 2 C (2) (Ta b]) , $

: f Pf C ( x3xn;2]) const h2 f C (2)( ab])

k; k

kk

h

(5). ,>:8F0,-.C /;;:,2-6?/=66 0/ ,.:832/B Tx1 x3] 6 Txn;2 xn] 3/16-6. ,. -;,-,D/ 1ED,:/ 30/58069 d1 d2 6 dn;1 dn .

x 19.6.

!< %6"<8< !? *" 6"#7& %7$"' %7

K.,. ?8.,4 ;:6D76@806< 6-;,7C3I8.-< 47< ;,7I5806< ;:6D76@/GH89 AI02=66 Pf , ;8:1/< 6 1.,:/< ;:,631,40/< 2,.,:,9 ,>:/06580E :/3487800,9 :/30,-.CG AI02=66 f (..8. ;:6D76@8068? 2 f 0 ). 8D,7CF68 .:8D,1/06< 2 >7/42,-.6 ;:6D76@/8?,9 AI02=66 f ;,31,74/ 30/5806< AI02=66 1 .,52/B 3/4/0E - 082,.,:,9 ;,>:8F0,-.CG. I-.C 1 .,52/B a = x1 < x2 < : : : < xn = b 6318-.0E 30/5806< AI02=66 f (x1 ) : : : f (xn). /:/?8.: di , i = 2 3 : : : n ; 1 1 ,DH89 -B8?8 60.8:;,7<=66 2I-,50,-2ID658-26?6 AI02=6/8.-< :/10E? 8 > sign f (xiZ xi+1) min fjf (xi;1Z xi)j jf (xiZ xi+1)jg > < di = > 8-76 sign f (xi;1 Z xi) = sign f (xi Z xi+1) (9) > :0 60/58

. .

!"#$ $ %$

x19. (',./ . 1

72

+- 1 + 81 / .

7I5/9 sign f (xi;1Z xi) 6= sign f (xiZ xi+1) -,,.18.-.1I8. -6.I/=66, 2,>4/ 1 .,528 xi ;:6D76@8068 f (xi;1 Z xi) 2 f 0 -781/ >,1,:6., 5., AI02=6< 1,3:/-./8. (IDE1/8.) 0/ ,.:8328 Txi;1 xi ], / ;:6D76@8068 f (xiZ xi+1 ) 2 f 0 -;:/1/ I.18:@4/8., 5., AI02=6< -,,.18.-.1800, IDE1/8. (1,3:/-./8.) 0/ ,.:8328 Txi xi+1]. 784,1/.87C0,, 0/ ,.:8328 Txi;1 xi+1] ;:,6-B,46. 63?808068 B/:/2.8:/ ?,0,.,00,-.6 AI02=66 6 ?E ;,7/>/8? f 0 (xi) di = 0. -7I5/8 sign f (xi;1 Z xi) = sign f (xiZ xi+1 ) B/:/2.8: ?,0,.,00,-.6 AI02=66 0/ ,.:8328 Txi;1 xi+1 ] 08 63?80<8.-< 6 1 2/58-.18 di 1ED6:/8.-< ;:6D76@8068 2 f 0 - 0/6?80CF6? ?,4I78?. 84,-./GH68 30/5806< d1 6 dn 1 >:/0650EB I37/B 1ED6:/G.-<, 6-B,4< 63 .,9 676 60,9 4,;,706.87C0,9 60A,:?/=66 , AI02=66 f (-?. 06@8). -0,10E8 && 6%?"A=> !8 Pf : 1) Pf <17<8.-< 2I-,50,-?0,>,57800,9 AI02=689, 30/5806< 2,.,:,9 3/16-<. .,7C2, ,. 7,2/7C0,>, ;,184806< AI02=66 f , 6?800,: 30/5806< Pf 0/ ,.:8328 Txi xi+1 ] 3/16-<. .,7C2, ,. 30/58069 AI02=66 f 1 .,52/B xi;1 xi xi+1 xi+2 .

2) 8.,4 ;:6D76@806< 08 <17<8.-< 760890E?: P (f + g) = 6 Pf + Pg , B,.< P (f ) = Pf . (D83 4,2/3/.87C-.1/). f

2C

(2)

(Ta b]) ,

$

" 4.

kf ; Pf kC x2x ;1 const h kf kC(2) ab (

h

k(Pf ) kC ab 23 i k(Pf ) kC ab 6 i 0

00

(

(

])

max

max

=23:::n;2

2

])

(

])

(5). & :

jf (xiZ xi )j ( max xxi ; ;x xi jf (xiZ xi )j xxi ;;xxi jf (xi i i i i

=12:::n;1

])

n

+1

+2

;1

+1

2

+1

;1

;1

;1

Z xi )j

2

)

,>:8F0,-.C /;;:,2-6?/=66 0/ ,.:832/B Tx1 x2] 6 Txn;1 xn] 3/16-6. ,. -;,-,D/ 1ED,:/ 30/58069 d1 6 dn .

x 19.7.

%6"<8< !? 6"

I-.C 1 .,52/B a = x1 < x2 < : : : < xn = b 6318-.0E 30/5806< AI02=66 f (x1 ) : : : f (xn). 6%$". ;7/90,? ;,:<42/ m 0/3E1/8.-< AI02=6<, 2,.,:/< 0/ 2/@4,? ,.:8328 Txi xi+1 ], i = 1 2 : : : n ; 1 -,1;/4/8. - ?0,>,5780,? Pi -.8;806 m 6 <17<8.-< AI02=689 27/--/ C (m;1) (Tx1 xn]).

. .

!"#$ $ %$

x19. (',./ . 1

+- 1 + 81 / .

73

0.8:;,7<=6< 2ID658-26?6 -;7/90/?6 (m = 3) 6-;,7C3I8.-< 47< ;,7I5806< 0/6D,788 >7/42,9 ;:6D76@/GH89 AI02=66 Pf 1 :/--?/.:61/8?,? 27/--8 2I-,50,-2ID658-26B ?0,>,5780,1. /:/?8.:E di , i = 1 2 : : : n 1 ,DH89 -B8?8 60.8:;,7<=66 2I-,50,2ID658-26?6 AI02=6,5780/ Pi 1 1648 (2) 0/B,46? (11) Pi00;1(xi ) = 2c3i;1 + 6c4i;1(xi ; xi;1 ) Pi00(xi) = 2c3i: ,4-./17<< J.6 30/5806< 1 (10), 6?88? 2c3i;1 + 6c4i;1(xi ; xi;1 ) = 2c3i i = 2 3 : : : n ; 1: ;,?,HCG (3) 0/B,46? 3f (xi;1Z xi) ; 2di;1 ; di + 3(x ; x ) di;1 + di ; 2f (xi;1Z xi) = i i;1 xi ; xi;1 (xi ; xi;1 )2 = 3f (xiZ xi+1 ) ; 2di ; di+1 : xi+1 ; xi ..8. (xi+1 ; xi )(3f (xi;1Z xi) ; 2di;1 ; di + 3di;1 + 3di ; 6f (xi;1Z xi)) = = (xi ; xi;1 )(3f (xiZ xi+1 ) ; 2di ; di+1) :8,D:/3I8? J., :/180-.1,: (xi+1 ; xi)(di;1 + 2di ; 3f (xi;1Z xi)) = (xi ; xi;1 )(3f (xiZ xi+1) ; 2di ; di+1) 676 (xi+1 ; xi )di;1 + 2(xi+1 ; xi )di + 2(xi ; xi;1)di + (xi ; xi;1 )di+1 = = 3f (xi;1Z xi)(xi+1 ; xi) + 3f (xiZ xi+1)(xi ; xi;1 ) 784,1/.87C0,, (xi+1 ; xi)di;1 + 2(xi+1 ; xi;1 )di + (xi ; xi;1 )di+1 = = 3f (xi;1Z xi )(xi+1 ; xi ) + 3f (xiZ xi+1)(xi ; xi;1) (12) i = 2 3 : : : n ; 1: K., -6-.8?/ 63 n ; 2 760890EB I:/108069 ,.0,-6.87C0, n 086318-.0EB d1 d2 : : : dn . 7< 88 3/?E2/06< .:8DI8.-< 8H8 41/ I:/10806<, 2,.,:E8 -.:,<.-<, 6-B,4< 63 .,9 676 60,9 4,;,706.87C0,9 60A,:?/=66 , AI02=66 f (-?. 06@8). 6-.8?/ (12) <17<8.-< -6-.8?,9 I:/108069 - .:8B46/>,0/7C0,9 ?/.:6=89 (-.:,>6?) 46/>,0/7C0E? ;:8,D7/4/068? 6 88 ?,@0, :8F/.C ?8.,4,? /I--/ D83 1ED,:/ >7/10,>, J78?80./. -0,10E8 && 6%?"A=> !8 Pf :

. .

!"#$ $ %$

x19. (',./ . 1

74

+- 1 + 81 / .

1) Pf <17<8.-< 087,2/7C0,9 2I-,50,-?0,>,57800,9 AI02=689, 30/5806< 2,.,:,9 3/16-<. ,. ;,184806< AI02=66 f 1 2/@4,9 .,528 x1 x2 : : : xn . 2) 8.,4 ;:6D76@806< <17<8.-< 760890E?: P (f + g) = Pf + Pg , P (f ) = Pf . ,>:8F0,-.C /;;:,2-6?/=66 2ID658-26?6 -;7/90/?6 (1 -67I 087,2/7C0,-.6 -;7/90/) 3/16-6. ,. 1ED,:/ 4,;,706.87C0EB I:/108069 47< 3/?E2/06< -6-.8?E (12) 6 -,-./17<8. ,. O(h2) 4, O(h4), >48 h ,;:8487<8.-< :/180-.1,? (5).

x 19.8.

6%$" $=> *%' !"&

?8.,4/B, 637,@800EB 1 x 19.4 6 x 19.6, .:8DIG.-< 41/ 4,;,706.87C0EB I-7,16< 47< ,;:8487806< ;/:/?8.:,1 d1 6 dn , 1 ?8.,48 63 x 19.5 .:8DIG.-< 58.E:8 4,;,706.87C0EB I-7,16< 47< ,;:8487806< ;/:/?8.:,1 d1 d2 6 dn;1 dn . ?8.,48 ;:6D76@806< 2ID658-26?6 -;7/90/?6 (x 19.7) .:8DIG.-< 41/ 4,;,706.87C0EB I:/10806< 47< 3/?E2/06< -6-.8?E 760890EB I:/108069. -;,7C3IG.-< 08-2,7C2, ?8.,4,1 ;,-.:,806< 4,;,706.87C0EB I-7,169, 2,.,:E8 1ED6:/G.-< 1 3/16-6?,-.6 ,. .,>,, 2/2/< 4,;,706.87C0/< 60A,:?/=6< 6318-.0/ , ;:6D76@/8?,9 AI02=66.

x 19.8.1.

6%$" $=> *%' !"& 6 7&' 7< 6%& 6%7&$ !8 & *%' !7"

I-.C 6318-.0E 30/5806< f 0(x1 ) 6 f 0(xn) ;:,631,40,9 f 0 AI02=66 f 1 >:/0650EB I37/B x1 6 xn . ,>4/ ;,7/>/8?

d1 = f 0(x1 )

dn = f 0(xn ):

(13)

?8.,4/B, 637,@800EB 1 x 19.4, x 19.5 6 x 19.6, :/180-.1/ (13) 4/G. 084,-./GH68 30/5806< ;/:/?8.:,1 d1 6 dn . ?8.,48 ;:6D76@806< 2ID658-26?6 -;7/90/?6 (x 19.7) :/180-.1/ (13) <17,0/7C0,-.6 -6-.8?E (12) 0/4, -487/.C -,,.18.-.1800, ;8:1E? 6 ;,-78406? I:/10806, -;,-,D/ 1ED,:/ >:/0650EB I-7,169 4/8. -;7/90 Pf , 2,.,:E9 0/3E1/G. !$"#' 6". 8I7I5F/8?E8 ,=8026 8>, ;,>:8F0,-.6 kf ; Pf kC( ab]) 3845 h4kf (4)kC( ab]) kf 0 ; (Pf )0kC( ab]) 241 h3kf (4)kC( ab]) >48 h ,;:8487<8.-< :/180-.1,? (5).

. .

!"#$ $ %$

x19. (',./ . 1

+- 1 + 81 / .

75

x

19.8.2. 6%$" $=> *%' !"& 6 7&' 7< &% 6%7&$ !8 & *%' !7"

I-.C 6318-.0E 30/5806< f 00(x1 ) 6 f 00(xn) ;:,631,40,9 f 00 AI02=66 f 1 >:/0650EB I37/B x1 6 xn . ,>4/ ;,.:8DI8? ,. ;:6D76@/GH89 AI02=66 Pf (Pf )00(x1 ) = f 00(x1 ) (Pf )00(xn) = f 00(xn) ..8. P100(x1 ) = f 00(x1 ) Pn00;1(xn) = f 00(xn): (14) 3 (11) 0/B,46? P100(x1 ) = 2c31 Pn00;1(xn ) = 2c3n;1 + 6c4n;1(xn ; xn;1 ): ,4-./17<< J.6 30/5806< 1 (14), 6?88? 2c31 = f 00(x1 ) 2c3n;1 + 6c4n;1(xn ; xn;1) = f 00(xn ): ;,?,HCG (3) 0/B,46? 2 3f (x1Z x2 ) ; 2d1 ; d2 = f 00(x1 ) x2 ; x1 2 3f (xn;1Z xn) ; 2dn;1 ; xdn ;+ x3(dn;1 + dn ; 2f (xn;1Z xn)) = f 00(xn ):

..8.

n

n;1

2 3f (x1Z xx2);;x2d1 ; d2 = f 00(x1 ) 2 dn;1 + 2xdn ;; x3f (xn;1Z xn) = f 00(xn ): 2

1

/;6F8? J.6 1E:/@806< 1 1648

n

n;1

2d1 + d2 = 3f (x1Z x2 ) ; 1 f 00 (x1)(x2 ; x1 ) 2 (15) 1 00 dn;1 + 2dn = 3f (xn;1Z xn) + 2 f (xn)(xn ; xn;1) ?8.,4/B, 637,@800EB 1 x 19.4 6 x 19.6, :/180-.1/ (15) ;,31,748 .:8DIG.-< 4 ;/:/?8.:/ d1 d2 6 dn;1 dn , :/180-.1/ (15) ;:84-./17,0/7C0,-.6 -6-.8?E (12) 0/4, -487/.C -,,.18.-.1800, ;8:1E? 6 ;,-78406? I:/10806
. .

!"#$ $ %$

x19. (',./ . 1 x 19.8.3.

+- 1 + 81 / .

76

\&'" *%' !"&<

K.,. ?8.,4 6-;,7C3I8.-<, 2,>4/ 062/2,9 4,;,706.87C0,9 60A,:?/=66 , ;:6D76@/8?,9 AI02=66 f 08.. ,>4/ ;,.:8DI8? ,. ;:6D76@/GH89 AI02=66 Pf (Pf )00(x1 ) = 0 (Pf )00(xn) = 0 ..8. P100(x1 ) = 0 Pn00;1(xn) = 0: 3 I:/108069 (15) ;:6 f 00(x1 ) = 0 6 f 00 (xn) = 0 ;,7I5/8? 2d1 + d2 = 3f (x1Z x2) dn;1 + 2dn = 3f (xn;1Z xn) K.6 I:/10806< 6-;,7C3IG.-< ./2 @8, 2/2 I:/10806< (15). -76 I ;:6D76@/8?,9 AI02=66 f 00(x1 ) 6= 0 676 f 00 (xn) 6= 0, ., ;:,631,7 1 1ED,:8 >:/0650EB I-7,169 ;:61,46. 2 ;,1EF800,9 ;,>:8F0,-.6 1 ;:6>:/0650EB I37/B (;,:<42/ O(h2)), 5., ,-,D800, ;7,B, ,.:/@/8.-< 0/ ;:6D76@8066 -;7/90/?6, >48 1 -67I 8>, 087,2/7C0,-.6 J./ ;,>:8F0,-.C DI48. 1, 1-8B .,52/B ,.:832/.

x 19.8.4.

"& \!&< !7"" & 6%*%' !7"

K.,. ?8.,4 6-;,7C3I8.-<, 2,>4/ 062/2,9 4,;,706.87C0,9 60A,:?/=66 , ;:6D76@/8?,9 AI02=66 f 08.. ,>4/ ;,.:8DI8? ,. ;:6D76@/GH89 AI02=66 Pf P1 P2 Pn;2 Pn;1 (16) ..8. ?0,>,5780 P1 P2 60.8:;,76:I8. f 0/ Tx1 x3 ] ;, .,52/? x1 x2 x3 : P1(x1 ) = f (x1 ) P1(x2) = f (x2 ) P1(x3 ) = f (x3 ) 6 P10 (x3 ) = d3 , / ?0,>,5780 Pn;2 Pn;1 60.8:;,76:I8. f 0/ Txn;2 xn] ;, .,52/? xn;2 xn;1 xn : Pn;1(xn;2) = f (xn;2) Pn;1(xn;1) = f (xn;1) Pn;1(xn) = f (xn) 6 Pn0 ;1(xn;2 ) = dn;2 . 784,1/.87C0,, ?0,>,5780 P1 P2 -.:,6.-< 2/2 60.8:;,7<=6,00E9 ?0,>,5780 .:8.C89 -.8;806, ;:606?/GH69 1 .,52/B x1 x2 x3 30/5806< f (x1) f (x2) f (x3) 6 6?8GH69 1 .,528 x3 ;:,631,40IG, :/10IG d3 Z ?0,>,5780 Pn;1 Pn;2 -.:,6.-< 2/2 60.8:;,7<=6,00E9 ?0,>,5780 .:8.C89 -.8;806, ;:606?/GH69 1 .,52/B xn;2 xn;1 xn 30/5806< f (xn;2) f (xn;1) f (xn) 6 6?8GH69 1 .,528 xn;2 ;:,631,40IG, :/10IG dn;2 . /26? ,D:/3,?, 1?8-., n ; 1 ?0,>,5780/ Pi i = 1 2 : : : n ; 1 6 n ;/:/?8.:,1 di i = 1 2 : : : n 6?88? n ; 3 ?0,>,5780/ P1 P2 P3 P4 : : : Pn;3 Pn;2 Pn;1 6 n ; 4 ;/:/?8.:/ d3 d4 : : : dn;2 . 8? -/?E? ;:,D78?/ >:/0650EB I-7,169 1 ?8.,4/B, 637,@800EB 1 x 19.4, x 19.5 6 x 19.6, -0<./, ;,-2,7C2I ;/:/?8.:E di i = 3 4 : : : n ; 2 ,;:8487
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?8.,48 ;:6D76@806< 2ID658-26?6 -;7/90/?6 (x 19.7) 1,3?,@0E 41/ ;,4B,4/. 1. 3 I-7,16< Pf 2 C (2) (Ta b]) ;,7I5/8? Pi00;1(xi ) = Pi00(xi ) i = 3 4 : : : n ; 2 { n;4 I:/10806< ,.0,-6.87C0, n;4 086318-.0EB d3 d4 : : : dn;2 . :/10806< 47< i = 4 5 : : : n ; 3 -,1;/4/G. - 1E;6-/00E?6 :/088 I:/10806,-< 164/ ?0,>,5780,1 P1 P2 , Pn;2 Pn;1 . 2. I48? 3/4/1/.C P1 P2 : : : Pn;1 2/2 :/0CF8 0/D,:,? d1 d2 : : : dn . E;,70806< I-7,16< (16) 4,DP8?-<, ;,.:8D,1/1 08;:8:E10,-.6 (Pf )000 1 .,52/B x2 6 xn;1 , ..8. P1000(x2 ) = P2000(x2 ) Pn000;2(xn;1 = Pn000;1(xn;1) (17) (I-7,16< (16) 6 (17) J2161/780.0E 47< ?0,>,5780,1 .:8.C89 -.8;806, ;,-2,7C2I 1 -67I (10) I P1 6 P2 , Pn;2 6 Pn;1 -,1;/4/G. 1-8 ;:,631,40E8 ,. 0I781,>, 4, .:8.C8>, ;,:<42/ 127G56.87C0,). ,,.0,F806< (17) 4/G. 41/ 084,-./GH6B 47< 3/?E2/06< -6-.8?E (12) I:/10806<. 3 ;:84-./17806< ?0,>,5780/ Pi 1 1648 (2) 0/B,46?, 5., -,,.0,F806< (17) J2161/780.0E (18) c41 = c42 c4n;2 = c4n;1 /) /--?,.:6? ;8:1,8 I:/108068 (18). ;,?,HCG (3) 0/B,46? d1 + d2 ; 2f (x1 Z x2) = d2 + d3 ; 2f (x2 Z x3) (x2 ; x1 )2 (x3 ; x2 )2 676 (x3 ; x2 )2d1 + (x3 ; x2 )2d2 ; (x2 ; x1 )2d2 ; (x2 ; x1)2 d3 = = 2f (x1Z x2 )(x3 ; x2 )2 ; 2f (x2 Z x3)(x2 ; x1 )2 ..8. (x3 ; x2 )2d1 + (x3 ; x1 )(x3 ; 2x2 + x1 )2 d2 ; (x2 ; x1 )2d3 = = 2f (x1Z x2 )(x3 ; x2 )2 ; 2f (x2 Z x3)(x2 ; x1 )2 -76 4,D/16.C J., I:/108068 2 -6-.8?8 (12), ., 0/:IF6.-< 88 .:8B46/>,0/7C0,-.C. ,J.,?I ;:6D/16? 2 J.,?I I:/10806G ;8:1,8 I:/108068 -6-.8?E (12) (..8. ;:6 i = 2): (x3 ; x2 )d1 + 2(x3 ; x1)d2 + (x2 ; x1 )d3 = = 3f (x1 Z x2)(x3 ; x2 ) + 3f (x2Z x3 )(x2 ; x1 )

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I?0,@800,8 0/ (x2 ; x1 ), ;,7I56? (x3 ; x2 )(x3 ; x2 + x2 ; x1 )d1 + (x3 ; x1 )(x3 ; 2x2 + x1 + 2(x2 ; x1 ))d2 = = f (x1 Z x2)(x3 ; x2 )(2(x3 ; x2 ) + 3(x2 ; x1 )) + f (x2 Z x3)(x2 ; x1 )2 676 (x3 ; x2 )d1 + (x3 ; x1 )d2 = 2 = f (x1Z x2 )(x3 ; x2 )(2x3 +xx2;;x3x1 ) + f (x2 Z x3)(x2 ; x1 ) : 3

1

K., I:/108068 4,D/17<8.-< 2 -6-.8?8 (12) 1 2/58-.18 ;8:1,>,. D) /--?,.:6? 1.,:,8 I:/108068 (18). ;,?,HCG (3) 0/B,46? dn;2 + dn;1 ; 2f (xn;2Z xn;1) = dn;1 + dn ; 2f (xn;1Z xn) (xn;1 ; xn;2)2 (xn ; xn;1 )2 676 (xn ; xn;1)2 dn;2 + (xn ; xn;1)2 dn;1 ; (xn;1 ; xn;2)2dn;1 ; (xn;1 ; xn;2 )2dn = = 2f (xn;2Z xn;1)(xn ; xn;1)2 ; 2f (xn;1Z xn)(xn;1 ; xn;2 )2 ..8. (xn ; xn;1)2 dn;2 + (xn ; xn;2)(xn ; 2xn;1 + xn;2)2 dn;1 ; (xn;1 ; xn;2)2 dn = = 2f (xn;2Z xn;1)(xn ; xn;1)2 ; 2f (xn;1Z xn)(xn;1 ; xn;2 )2 -76 4,D/16.C J., I:/108068 2 -6-.8?8 (12), ., 0/:IF6.-< 88 .:8B46/>,0/7C0,-.C. ,J.,?I 1E5.8? J., I:/108068 63 ;,-78408>, I:/10806< -6-.8?E (12) (..8. ;:6 i = n ; 1): (xn ; xn;1)dn;2 + 2(xn ; xn;2 )dn;1 + (xn;1 ; xn;2)dn = = 3f (xn;2Z xn;1)(xn ; xn;1) + 3f (xn;1Z xn)(xn;1 ; xn;2 ) I?0,@800,>, 0/ (xn ; xn;1 ), ;,7I56? (xn ; xn;2)(2(xn ; xn;1 ) ; (xn ; 2xn;1 + xn;2))dn;1+ +(xn;1 ; xn;2 )(xn ; xn;1 ; (xn;1 ; xn;2 ))dn = 2 = f (xn;2Z xn;1)(xn ; xn;1 ) + f (xn;1 Z xn)(xn;1 ; xn;2 )(3(xn ; xn;1 )+2(xn ; xn;2 )) 676 (xn ; xn;2)dn;1 + (xn;1 ; xn;2 )dn = f ( x n;2 Z xn;1 )(xn ; xn;1 )2 + f (xn;1 Z xn )(xn;1 ; xn;2 )(3xn ; xn;1 ; 2xn;2 ) = : x ;x n

n;2

K., I:/108068 4,D/17<8.-< 2 -6-.8?8 (12) 1 2/58-.18 ;,-78408>,.

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K.,. ?8.,4 6-;,7C3I8.-<, 2,>4/ 062/2,9 4,;,706.87C0,9 60A,:?/=66 , ;:6D76@/8?,9 AI02=66 f 08.. 1848? 4,;,706.87C0E8 .,526 x0 < x1 6 xn+1 > xn 6 30/5806< ;:6D76@/8?,9 AI02=66 f (x0 ) 6 f (xn+1) 1 J.6B .,52/B. 8? -/?E? ;:,D78?/ >:/0650EB I-7,169 1 ?8.,4/B, 637,@800EB 1 x 19.4 6 x 19.6, -0<./, ;,-2,7C2I ;/:/?8.:E di i = 1 2 : : : n .8;8:C ,;:8487
P1(x0 ) = f (x0 )

Pn;1(xn+1) = f (xn+1):

3 ;:84-./17806< (2) ?0,>,5780/ Pi 6 164/ (3) 8>, 2,JAA6=680.,1 1640,, 5., 1 J.6 I:/10806< 1B,4<. -,,.18.-.1800, d1 d2 6 dn;1 dn . 784,1/.87C0,, J.6 I:/10806< ?,@0, 4,D/16.C 2 -6-.8?8 (12) 1 2/58-.18 ;8:1,>, 6 ;,-78408>, I:/108069 - -,B:/08068? 88 .:8B46/>,0/7C0,>, 164/. 2. 0,>,5780 P1 60.8:;,76:I8. f 0/ Tx0 x2] ;, .,52/? x0 x1 x2 : P1(x0 ) = f (x0 ) P1(x1 ) = f (x1 ) P1(x2 ) = f (x2) 6 P10 (x2 ) = d2 , / ?0,>,5780 Pn;1 60.8:;,76:I8. f 0/ Txn;1 xn+1] ;, .,52/? xn;1 xn xn+1 : Pn;1(xn;1 ) = f (xn;1) Pn;1(xn) = f (xn) Pn;1(xn+1) = f (xn+1) 6 Pn0 ;1(xn;1) = dn;1 . 784,1/.87C0,, ?0,>,5780 P1 -.:,6.-< 2/2 60.8:;,7<=6,00E9 ?0,>,5780 .:8.C89 -.8;806, ;:606?/GH69 1 .,52/B x0 x1 x2 30/5806< f (x0 ) f (x1) f (x2) 6 6?8GH69 1 .,528 x2 ;:,631,40IG, :/10IG d2 Z ?0,>,5780 Pn;1 -.:,6.-< 2/2 60.8:;,7<=6,00E9 ?0,>,5780 .:8.C89 -.8;806, ;:606?/GH69 1 .,52/B xn;1 xn xn+1 30/5806< f (xn;1) f (xn) f (xn+1) 6 6?8GH69 1 .,528 xn;1 ;:,631,40IG, :/10IG dn;1 . /26? ,D:/3,?, 1?8-., n ;/:/?8.:,1 di i = 1 2 : : : n 6?88? 6 n ; 2 ;/:/?8.:/ d2 d3 : : : dn;1 . 3 I-7,16< Pf 2 C (2) (Ta b]) ;,7I5/8?

Pi00;1(xi ) = Pi00(xi )

i = 2 3 : : : n ; 1

{ n;2 I:/10806< ,.0,-6.87C0, n;2 086318-.0EB d2 d3 : : : dn;1 . :/10806< 47< i = 3 4 : : : n ; 2 -,1;/4/G. - 1E;6-/00E?6 :/088 I:/10806,-< 164/ ?0,>,5780,1. 3. E56-7<8? ;:6D76@8068 2 f 0(x1 ) 6 f 0(xn ) /0/7,>650, ;:6D76@806G 2ID658-26?6 ?0,>,5780/?6 8--87< (-?.x 19.4). :I>6?6 -7,1/?6, ?E 1E56-7<8? d1 6 dn ;, A,:?I7/? (7) ;:6 i = 1 6 i = n . K.6 I:/10806< 4,D/17, 6 ;,-78408>, I:/108069 - -,B:/08068? 88 .:8B46/>,0/7C0,>, 164/.

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0/5806< f (x0) 6 f (xn+1 ) ?,>I. 0/B,46.C-< 63 -,,D:/@8069 ;8:6,4650,-.6, 58.0,-.6 6 ..;.. -76 062/2,9 4,;,706.87C0,9 60A,:?/=66 , ;:6D76@/8?,9 AI02=66 f 08., ., ?,@0, 6-;,7C3,1/.C J2-.:/;,7<=6G, 0/;:6?8:, 760890IG: x0 = x1 ; (x2 ; x1 ) f (x0 ) = f (x1 ) ; (f (x2) ; f (x1)) xn+1 = xn + (xn ; xn;1 ) f (xn+1) = f (xn ) + (f (xn) ; f (xn;1 )):

x 19.8.6.

H%6"<8< & 6%*%' !7"

K.,. ?8.,4 6-;,7C3I8.-<, 2,>4/ 062/2,9 4,;,706.87C0,9 60A,:?/=66 , ;:6D76@/8?,9 AI02=66 f 08.. 2/58-.18 ;:6D76@800,>, 30/5806< f 0(x1) f 0(xn) 76D, f 00(x1 ) f 00(xn) D8:I.-< 30/5806< Q0k (x1 ) Rk0 (xn) 76D, Q00k (x1 ) Rk00(xn), >48 Qk (x) { 60.8:;,7<=6,00E9 ?0,>,5780 />:/0@/ -.8;806 k ; 1, ;,-.:,800E9 ;, .,52/? x1 x2 : : : xk , Rk (x) { 60.8:;,7<=6,00E9 ?0,>,5780 />:/0@/ -.8;806 k ;1, ;,-.:,800E9 ;, .,52/? xn;k+1 xn;k+2 : : : xn , k 3. ,7I5800E8 30/5806< f 0 676 f 00 1 .,52/B x1 6 xn 6-;,7C3IG.-< 2/2 I2/3E1/7,-C 1EF8 (-?. ;,-.:,8068 >:/0650EB I-7,169 ;, 30/5806:/0650EB I37/B). x

20.

x 20.1.

?><

I-.C 3/4/0E .,526 a = x1 < x2 < : : : < xn = b 6 30/5806< f (x1 ) f (x2) : : : f (xn). 1848? 4,;,706.87C0E8 .,526 1 2 : : : n+1 ./2, 5.,DE 1 < a = x1 < 2 < x2 < 3 < x3 < : : : < xn;1 < n < xn = b < n+1: /;:6?8:, ?,@0, ;,7,@6.C i = xi;12+ xi i = 2 3 : : : n: 0.8:;,76:IGH/< AI02=6< Pf -.:,6.-< ./26? ,D:/3,?, 5., 0/ 2/@4,? ,.:8328 Ti i+1], i = 1 : : : n ,0/ <17<8.-< ?0,>,5780,? Pi -.8;806 2, ./26?, 5., 9 = Pi(xi) = f (xi) i = 1 : : : n (1) Pi(i) = vi Pi(i+1) = vi+1 >48 vi , i = 1 2 : : : n + 1 { -1,D,40E8 ;/:/?8.:E, .,. 676 60,9 -;,-,D 1ED,:/ 2,.,:EB ,;:8487<8. ?8.,4 2I-,50,9 60.8:;,7<=66 ;/:/D,7658-26?6 ?0,>,5780/?6. ,7I5800/< AI02=6< Pf -,1;/4/8. - f 1 .,52/B xi , i = 1 : : : n 6 47< 7GD,>, 0/D,:/ ;/:/?8.:,1 vi Pf 2 C (Ta b]).

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,JAA6=680.E ?0,>,5780/ Pi , 3/;6-/00,>, 1 A,:?8

Pi

Pi(x) = a1i + a2i(x ; i) + a3i(x ; i)(x ; xi )

?,>I. DE.C 1E56-780E ;, 60.8:;,7<=6,00,9 A,:?I78 CG.,0/ (-?. ,;:84878068 (4.1) :/3487800EB :/30,-.89):

vi f (xi)

n

. n

f (xi) ; vi xi ; i

n

vi+1 ; f (xi ) . i+1 ; xi .

vi+1 ; f (xi) ; f (xi) ; vi i+1 ; i i+1 ; xi xi ; i 1

!

vi+1 .,7D=E 1E56-7
! v 1 i+1 ; f (xi ) f (xi ) ; vi a3i = ; ; x ; x ; i+1

i

i+1

i

i

i

7< ;:84-./17806< 1 1648

Pi(x) = c1i + c2i(x ; i) + c3i(x ; i)2

(2)

3/;6F8?

(x ; i)(x ; xi) = (x ; i)(x ; i + i ; xi ) = (x ; i)2 ; (xi ; i)(x ; i)

,.2I4/ ;,7I5/8?

c1i = a1i = vi c2i c3i

! f ( x x i ) ; vi i ; i vi+1 ; f (xi ) f (xi ) ; vi = a2i ; (xi ; i)a3i = ; i+1 ; i i+1 ; xi ; xi ; i x ; i i

! v 1 i+1 ; f (xi ) f (xi ) ; vi = a3i = ; ; x ; x ; i+1 i i+1 i i i (3)

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0.8:;,7<=6< ;/:/D,7658-26?6 -;7/90/?6 6-;,7C3I8.-< 47< ;,7I5806< 0/6D,788 >7/42,9 ;:6D76@/GH89 AI02=66 Pf 1 :/--?/.:61/8?,? 27/--8 2I-,50,-21/4:/.650EB ?0,>,5780,1. /:/?8.:E vi , i = 1 2 : : : n + 1 1 ,DH89 -B8?8 60.8:;,7<=66 2I-,50,21/4:/.650E?6 AI02=6, ;,:<42/ ..8. Pf 2 C (1) (Ta b]) 676

Pi0;1 (i) = Pi0(i)

i = 2 3 : : : n:

(4)

3 ;:84-./17806< ?0,>,5780/ Pi 1 1648 (2) 0/B,46?

Pi0;1(i) = c2i;1 + 2(i ; i;1)c3i;1

Pi0(i) = c2i:

(5)

;,?,HCG (3) 1E56-76?

/ ./2@8

! f ( x x i ) ; vi i ; i vi+1 ; f (xi ) f (xi ) ; vi c2i = x ; ; ; ; x ; x ; = i i

i i !i+1 i i+1 i = 1 + xi ; i f (xi ) ; vi ; xi ; i vi+1 ; f (xi)

i+1 ; i xi ; i! i+1 ; i i+1 ; xi 1 + 1 xi ; i = ;vi x ; ; v + i+1 ( i i i+1 ; i i+1 ; i )(i+1 ; xi ) !

1 1 x i ; i + f (xi ) x ; + ; + ( ; )( ; x ) i i i+1 i+1 !i

i i+1 i !

1 ; 1 1 + 1 ; v + = ;vi x ; i+1 i i i+1 ; i i+1 ; xi i+1 ; i !

1 1 + f (xi ) x ; + ;x i

i+1

i

! v i+1 ; f (xi ) f (xi ) ; vi c3i = ; ; x ; x ; = i+1 i i i

i+1 i

!! 1 1 1 1 1 = ; vi x ; + vi+1 ; x ; f (xi) x ; + ; x : i+1 i i i i+1 i i i i+1 i 1

6

i

c2i + 2( i+1 ; i)c3i = !

! 1 1 1 1 = ;vi x ; + ; ; vi+1 ; x ; ; + i i+1 i i+1 i

i i+1 i ! 1 + 1 + f (xi) x ; ;x + i

. .

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i+1

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!! 1 1 1 1 + 2 vi + vi+1 ; f (xi) xi ; i + i+1 ; xi = x i ; i i+1 ; xi !

!

1 1 1 1 = vi xi ; i ; i+1 ; i + vi!+1 i+1 ; xi + i+1 ; i ; ; f (xi) x ;1 + 1; x i

i

i+1

i

;,?,?,HCG J.6B :/180-.1 6 (5) 0/B,46?

1

1

!

1

1

!

Pi (i) = ;vi x ; + ; ; vi+1 ; x ; ; + i i i+1 i i+1 i i+1 i ! 1 1 + f (xi) x ; + ; x 0

i

6

i

i+1

i

!

1

1 + 1 P (i) = vi;1 x ; ; ;1 + vi i;1 i;1 i i;1 ! i ; xi;1 i ; i;1 ; f (xi;1) x ;1 + ;1x 0 i;1

i;1

i;1

i

!

!

;

(7)

i;1

,4-./17<< (6) 6 (7) 1 (4), ;,7I5/8? 47< 1-8B i = 2 3 : : : n

(6)

!

1 1 + 1 + 1 + 1 ; v i;1 + xi;1 ; i;1 i ; i;1

i ; xi;1 i ;! i;1 xi ; i i+1 ; i vi + + 1; x ; 1; vi+1 = i+1 ! i i+1 i !

1 1 1 1 f (xi;1) + x ; + ; x f (xi) = x ; + ;x i;1 i;1 i i;1 i i i+1 i (8) K., -6-.8?/ 63 n ; 1 760890EB I:/108069 ,.0,-6.87C0, n + 1 086318-.0EB v1 v2 : : : vn+1 . 7< 88 3/?E2/06< .:8DI8.-< 8H8 41/ I:/10806<, 2,.,:E8 -.:,<.-<, 6-B,4< 63 .,9 676 60,9 4,;,706.87C0,9 60A,:?/=66 , AI02=66 f (-?. 06@8). 6-.8?/ (8) <17<8.-< -6-.8?,9 I:/108069 - .:8B46/>,0/7C0,9 ?/.:6=89 (-.:,>6?) 46/>,0/7C0E? ;:8,D7/4/068? 6 88 ?,@0, :8F/.C ?8.,4,? /I--/ D83 1ED,:/ >7/10,>, J78?80./. -0,10E8 && 6%?"A=> !8 Pf : 1) Pf <17<8.-< 087,2/7C0,9 2I-,50,-?0,>,57800,9 AI02=689, 30/5806< 2,.,:,9 3/16-<. ,. ;,184806< AI02=66 f 1 2/@4,9 .,528 x1 x2 : : : xn . 1

2) 8.,4 ;:6D76@806< <17<8.-< 760890E?: P (f + g) = Pf + Pg , P (f ) = Pf .

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84

,>:8F0,-.C /;;:,2-6?/=66 ;/:/D,7658-26?6 -;7/90/?6 (1 -67I 087,2/7C0,-.6 -;7/90/) 3/16-6. ,. 1ED,:/ 4,;,706.87C0EB I:/108069 47< 3/?E2/06< -6-.8?E (8) 6 -,-./17<8. ,. O(h1) 4, O(h3), >48 h ,;:8487<8.-< :/180-.1,? (19.5).

x 20.4.

6%$" $=> *%' !"&

-;,7C3IG.-< 08-2,7C2, ?8.,4,1 ;,-.:,806< 41IB 4,;,706.87C0EB I:/108069 47< 3/?E2/06< -6-.8?E (8), 2,.,:E8 1ED6:/G.-< 1 3/16-6?,-.6 ,. .,>,, 2/2/< 4,;,706.87C0/< 60A,:?/=6< 6318-.0/ , ;:6D76@/8?,9 AI02=66.

x 20.4.1.

6%$" $=> *%' !"& 6 7&' 7< 6%& 6%7&$ !8 & *%' !7"

I-.C 6318-.0E 30/5806< f 0(x1 ) 6 f 0(xn) ;:,631,40,9 f 0 AI02=66 f 1 >:/0650EB I37/B x1 6 xn . ,3?,@0E 41/ J2161/780.0EB ;,4B,4/. 1. 1/ 4,;,706.87C0EB I:/10806< ;,7I5/G.-< 63 I-7,169

P10 (x1 ) = f 0(x1 )

Pn0 (xn) = f 0(xn ):

E56-76?

Pi0(xi ) = c2i + 2(xi ; i)c3i = !

! 1 1 1 1 = ;vi xi ; i + i+1 ; i ; !vi+1 i+1 ; xi ; i+1 ; i + 1 + 1 + f (xi) xi ; i i+1 ; xi +

!! 1 1 1 1 x i ; i + 2 ; vi x ; + vi+1 ; x ; f (xi) x ; + ; x i+1 i i i i+1 i i i i+1 i

,-2,7C2I x i ; i 1 = 1 ; 1 i+1 ; i i+1 ; xi i+1 ; xi! i+1 ; i xi ; i 1 + 1 i+1 ; i = xi ; i = 1 i+1 ; i xi ; i i+1 ; xi i+1 ; i (i+1 ; xi)(xi ; i) i+1 ; xi ., !

!

1 1 1 1 0 Pi (xi ) = ;vi x ; + ; ; vi+1 ; x ; ; + i i+1 i i+1 i

i i+1 i ! 1 + 1 + f (xi) x ; + i i i+1 ; xi

. .

!"#$ $ %$

x20. (',./ . ' , +

+ 2vi+1

+1

1

!

; ; 2f (xi) 1 = !i ; x i i ; i ! i ; xi

i ; i 1 ; 1 1 ; 1 = vi + vi + i ; i xi ; i i ; xi i ; i ! 1 1 + f (xi) x ; ; ; x i i i i + 2vi

1

85

',-

1

+1

+1

+1

+1

+1

+1

+1

+1

1/ 4,;,706.87C0EB I:/10806< ;,7I5/G.-< 63 I-7,169

Pi0(xi ) = f 0(xi )

i = 1 n:

K.6 :/180-.1/ <17,0/7C0,-.6 -6-.8?E (8) 0/4, -487/.C -,,.18.-.1800, ;8:1E? 6 ;,-78406? I:/108064/ 1 -6-.8?I I:/108069 (8) v1 6 vn+1 08 1,94I., / P1 6 Pn -.:,<.-< ;, 60.8:;,7<=6,00,9 A,:?I78 CG.,0/ - 2:/.0E?6 I37/?6: ?0,>,5780

P1(x) = c11 + (x ; x1 )c21 + (x ; x1 )2 c31

-.:,6.-< 63 I-7,169

P1(x1 ) = f (x1 ) P10 (x1 ) = f 0(x1 ) P1 (2) = v2 5., 4/8.

c11 = f (x1) c21 = f (x1 ) c31 =

1

0

2

/0/7,>650,, ?0,>,5780

;x

1

! v2 ; f (x1) ; f 0(x ) Z 1 2 ; x1

Pn(x) = c1n + (x ; n)c2n + (x ; n)2c3n

-.:,6.-< 63 I-7,169

Pn(n) = vn P1(xn) = f (xn) P10 (xn) = f 0(xn ) 5., 4/8.

! f ( x 1 f ( x n ) ; vn n ) ; vn 0 0 c1n = vn c2n = 2 x ; ; f (xn ) c3n = x ; f (xn) ; x ; : n n n n n n 3 I-7,16< Pf 2 C (1) (Ta b]) ;,7I5/8?

Pi0;1 (i) = Pi0(i)

. .

i = 2 3 : : : n:

!"#$ $ %$

x20. (',./ . ' , +

',-

86

{ n ; 1 I:/10806< ,.0,-6.87C0, n ; 1 086318-.0EB v2 v3 : : : vn . :/10806< 47< i = 3 4 : : : n ; 1 -,1;/4/G. - 1E;6-/00E?6 :/088 I:/10806,-< 164/ ?0,>,5780,1 P1 6 Pn;1 . 1. E;6F8? 0,1,8 ;8:1,8 I:/108068 -6-.8?E (8). ?88?

! v ; f ( x ) 2 1 P10 (2) = c21 + 2(2 ; x1 ) = f 0(x1 ) + 2 ; x ; f 0(x1 ) 2 1 = 2 v2;;f (xx1) ; f 0(x1 ) 2

1

1E:/@8068 47< P (2) ;,7I5/8? 63 (6) ;:6 i = 2:

!

! 1 1 1 1 0 P2 (2) = ;v2 x ; + ; ; v3 ; x ; ; + 2

2 3 2 ! 3 2 3 2 1 + 1 + f (x2 ) x ; 2 3 ; x2 : 2 784,1/.87C0,, I:/108068 6?88. 164 !

!

1 1 1 1 2 v2 ; x + x ; + ; + v3 ; x ; ; = 2 1 2 2 3 2 3 2

3 2 ! 2 1 1 = f (x1) ; x + f (x2 ) x ; + ; x + f 0(x1 ): 2 1 2 2 3 2 2. E;6F8? 0,1,8 ;,-784088 I:/108068 -6-.8?E (8). ?88? ) ; vn ; f 0(x ) Pn0 (n) = c2n = 2 f (xxn; n n n 1E:/@8068 47< Pn0 ;1(n) ;,7I5/8? 63 (7) ;:6 i = n :

!

! 1 1 1 1 0 Pn;1(n) = vn;1 x ; ; ; + vn ; x + ; ; n;1 n;1 n n;1 n n;1 n n;1

! 1 1 ; f (xn;1) xn;1 ; n;1 + n ; xn;1 784,1/.87C0,, I:/108068 6?88. 164

!

! 1 1 1 1 2 vn;1 x ; ; ; + vn ; x + ; + x ; = n;1 n;1 n n;1 n n;1 !n n;1 n n

1 1 + + f (xn) 2 ; f 0(xn): = f (xn;1) xn;1 ; n;1 n ; xn;1 xn ; n 0 2

. .

!"#$ $ %$

x20. (',./ . ' , +

',-

87

x

20.4.2. 6%$" $=> *%' !"& 6 7&' 7< &% 6%7&$ !8 & *%' !7"

I-.C 6318-.0E 30/5806< f 00(x1 ) 6 f 00(xn) ;:,631,40,9 f 00 AI02=66 f 1 >:/0650EB I37/B x1 6 xn . ,>4/ ;,.:8DI8? ,. ;:6D76@/GH89 AI02=66 Pf (Pf )00(x1 ) = f 00(x1 ) (Pf )00(xn) = f 00(xn) ..8. P100(x1 ) = f 00(x1 ) Pn00(xn ) = f 00(xn): 3 (2) 0/B,46? Pi00(xi ) = 2c3i =

!! 2 1 1 1 1 = i+1 ; i vi xi ; i + vi+1 i+1 ; xi ; f (xi) xi ; i + i+1 ; xi : (9) ,>4/ 4,;,706.87C0E8 I:/10806< Pi00(xi ) = f 00(xi ) i = 1 n: K.6 :/180-.1/ <17,0/7C0,-.6 -6-.8?E (8) 0/4, -487/.C -,,.18.-.1800, ;8:1E? 6 ;,-78406? I:/10806
x 20.4.3.

\&'" *%' !"&<

K.,. ?8.,4 6-;,7C3I8.-<, 2,>4/ 062/2,9 4,;,706.87C0,9 60A,:?/=66 , ;:6D76@/8?,9 AI02=66 f 08.. ,>4/ ;,.:8DI8? ,. ;:6D76@/GH89 AI02=66 Pf (Pf )00(x1 ) = 0 (Pf )00(xn) = 0 ..8. P100(x1 ) = 0 Pn00(xn ) = 0: 3 -,,.0,F8069 (9) ;,7I5/8? 41/ 4,;,706.87C0EB I:/10806< Pi00(xi ) = 0 i = 1 n: K.6 :/180-.1/ <17,0/7C0,-.6 -6-.8?E (8) 0/4, -487/.C -,,.18.-.1800, ;8:1E? 6 ;,-78406? I:/10806:/0650EB I-7,169 ;:61,46. 2 ;,1EF800,9 ;,>:8F0,-.6 1 ;:6>:/0650EB I37/B.

. .

!"#$ $ %$

x20. (',./ . ' , + x 20.4.4.

',-

88

"& \!&< !7"" & 6%*%' !7"

K.,. ?8.,4 6-;,7C3I8.-<, 2,>4/ 062/2,9 4,;,706.87C0,9 60A,:?/=66 , ;:6D76@/8?,9 AI02=66 f 08.. ,>4/ ;,.:8DI8? ,. ;:6D76@/GH89 AI02=66 Pf

P1 P2 Pn;1 Pn

(10)

..8. ?0,>,5780 P1 P2 60.8:;,76:I8. f 0/ T1 3] ;, .,52/? x1 x2 : P1(x1 ) = f (x1 ) P1(x2 ) = f (x2) 6 P1 (3) = v3 , / ?0,>,5780 Pn;1 Pn 60.8:;,76:I8. f 0/ Tn;1 n+1] ;, .,52/? xn;1 xn : Pn(xn;1 ) = f (xn;1 ) Pn(xn) = f (xn) 6 Pn(n;1) = vn;1 . 784,1/.87C0,, ?0,>,5780 P1 P2 -.:,6.-< 2/2 60.8:;,7<=6,00E9 ?0,>,5780 1.,:,9 -.8;806, ;:606?/GH69 1 .,52/B x1 x2 3 30/5806< f (x1) f (x2) v3 Z ?0,>,5780 Pn Pn;1 -.:,6.-< 2/2 60.8:;,7<=6,00E9 ?0,>,5780 1.,:,9 -.8;806, ;:606?/GH69 1 .,52/B n;1 xn;1 xn 30/5806< vn;1 f (xn;1) f (xn). /26? ,D:/3,?, 1?8-., n ?0,>,5780,1 Pi i = 1 2 : : : n 6 n + 1 ;/:/?8.:,1 vi i = 1 2 : : : n + 1 6?88? n ; 2 ?0,>,5780/ P1 P2 P3 P4 : : : Pn;2 Pn;1 Pn 6 n ; 3 ;/:/?8.:/ v3 v4 : : : vn;1 . ?8.,48 ;:6D76@806< ;/:/D,7658-26?6 -;7/90/?6 1,3?,@0E 41/ ;,4B,4/. 1. 3 I-7,16< Pf 2 C (1) (Ta b]) ;,7I5/8?

Pi0;1(i) = Pi0(i)

i = 3 4 : : : n ; 1

{ n;3 I:/10806< ,.0,-6.87C0, n;3 086318-.0EB v3 v4 : : : vn;1 . :/10806< 47< i = 4 5 : : : n ; 2 -,1;/4/G. - 1E;6-/00E?6 :/088 I:/10806,-< 164/ ?0,>,5780,1 P1 P2 , Pn;1 Pn . /). E;6F8? 0,1,8 ;8:1,8 I:/108068 -6-.8?E (8). , 60.8:;,7<=6,00,9 A,:?I78 CG.,0/

P1 (x) = a11 + (x ; x1 )a21 + (x ; x1 )(x ; x2 )a31

>48

a11 = f (x1) a21 = f (x1 Z x2 ) a31 =

1

3

,J.,?I

;x

1

! v3 ; f (x2) ; f (x Z x ) : 1 2 3 ; x2

P10 (3) = a21 + (23 ; x1 ; x2)a31 =

! 2 v 3 ; x1 ; x2 3 ; f (x 2 ) ; f (x1Z x2) = = f (x1Z x2 ) + ; x 3 1 3 ; x2

! 1 1 x2 f (x Z x ): = ; x + ; x (v3 ; f (x2 )) ; 3 ; ;x 1 2 3

. .

1

3

2

3

1

!"#$ $ %$

x20. (',./ . ' , +

',-

89

E:/@8068 47< P30 (3) ;,7I5/8? 63 (6) ;:6 i = 3: !

!

1 1 1 1 P30 (3) = ;v3 x ; + ; ; v4 ; x ; ; + 3

3 4 3 ! 4 3 4 3 1 + 1 + f (x3 ) x ; 3 4 ; x3 : 3 784,1/.87C0,, I:/108068 6?88. 164

!

! 1 1 1 1 1 1 v3 ; x + ; x + x ; + ; + v4 ; x ; ; = 3 4 3 ! 4 3

1 3 2 3 ! 3 4 3 1 + 1 3 ; x2 f (x Z x ): + = f (x2 ) ;1 x + ;1 x + f (x3 ) x ; 3 4 ; x3 3 ; x1 1 2 3 1 3 2 3 D). E;6F8? 0,1,8 ;,-784088 I:/108068 -6-.8?E (8). , 60.8:;,7<=6,00,9 A,:?I78 CG.,0/

Pn(x) = a1n + (x ; n;1)a2n + (x ; n;1)(x ; xn;1 )a3n

>48

1 (f (x Z x ) ; a ) : ) ; vn;1 a = a1n = vn;1 a2n = f (xxn;1; 3n n;1 n 2n xn ; n;1 n;1 n;1 ,J.,?I Pn0 (n;1) = a2n + (n;1 ; xn;1 )a3n = a2n ; xxn;1;;n;1 (f (xn;1 Z xn) ; a2n) = n n;1 xn ; 2n;1 a ; xn;1 ; n;1 f (x Z x ) = = xn;1x+ ; 2n n;1 n x ; n n ;1 n n ;1

! 1 1 (f (xn;1 ) ; vn;1 ) ; xxn;1;;n;1 f (xn;1 Z xn): = x ; + x ; n;1 n;1 n n;1 n n;1 E:/@8068 47< Pn0 ;2(n;1) ;,7I5/8? 63 (7) ;:6 i = n ; 1:

! 1 1 0 Pn;2(n;1) = vn;2 x ; ; ; + n;2 n;2 n;1 n;2

! 1 1 + vn;1 n ;1 ; xn;2 + n;1 ; n;2 ;! ; f (xn;2) x ;1 + ;1 x n;2

n;2

n;1

n;2

784,1/.87C0,, I:/108068 6?88. 164

! 1 1 vn;2 x ; ; ; + n;2 n;2 n;1 n;2

. .

!"#$ $ %$

x20. (',./ . ' , +

1

1

!

1 +vn;1 + + + n;1 ; xn;2 n;1 ; n;2 xn;1 ; n;1 ! xn ; n;1 = 1 1 + + = f (xn;2) x n;2 ; n;2 n;1 ; xn;2 !

1 xn;1 ; n;1 f (x Z x ): 1 ; +f (xn;1) x ; + x ; xn ; n;1 n;1 n n;1 n;1 n n;1 2. I48? 3/4/1/.C P1 P2 : : : Pn 2/2 :/0CF8 0/D,:,? v1 v2 : : : vn+1 . E;,70806< I-7,16< (10) 4,DP8?-<, ;,.:8D,1/1 08;:8:E10,-.6 (Pf )00 1 .,52/B 2 6 n , ..8. P100(2) = P200(2) Pn00;1(n) = Pn00(n) (11) (I-7,16< (10) 6 (11) J2161/780.0E 47< ?0,>,5780,1 1.,:,9 -.8;806, ;,-2,7C2I 1 -67I (4) I P1 6 P2 , Pn;1 6 Pn -,1;/4/G. 1-8 ;:,631,40E8 ,. 0I781,>, 4, 1.,:,>, ;,:<42/ 127G56.87C0,). ,,.0,F806< (11) 4/G. 41/ 084,-./GH6B 47< 3/?E2/06< -6-.8?E (8) I:/10806<. 3 ;:84-./17806< ?0,>,5780/ Pi 1 1648 (2) 0/B,46?, 5., -,,.0,F806< (11) J2161/780.0E c31 = c32 c3n;1 = c3n (12)

x 20.4.5.

1

90

',-

6""#' !7" & 6%*%' !7"

K.,. ?8.,4 6-;,7C3I8.-<, 2,>4/ 062/2,9 4,;,706.87C0,9 60A,:?/=66 , ;:6D76@/8?,9 AI02=66 f 08.. 1848? 4,;,706.87C0E8 .,526 x0 < x1 6 xn+1 > xn 6 30/5806< ;:6D76@/8?,9 AI02=66 f (x0) 6 f (xn+1) 1 J.6B .,52/B. ,>4/ ;,7/>/8? 1 = x0 6 n+1 = xn+1 v1 = f (1) vn+1 = f (n+1): K.6 :/180-.1/ <17,0/7C0,-.6 -6-.8?E (8) 0/4, -487/.C -,,.18.-.1800, ;8:1E? 6 ;,-78406? I:/10806
x 20.4.6.

H%6"<8< & 6%*%' !7"

K.,. ?8.,4 6-;,7C3I8.-<, 2,>4/ 062/2,9 4,;,706.87C0,9 60A,:?/=66 , ;:6D76@/8?,9 AI02=66 f 08.. 2/58-.18 ;:6D76@800,>, 30/5806< f 0(x1) f 0(xn) 76D, f 00(x1 ) f 00(xn) D8:I.-< 30/5806< Q0k (x1 ) Rk0 (xn) 76D, Q00k (x1 ) Rk00(xn), >48 Qk (x) { 60.8:;,7<=6,00E9 ?0,>,5780 />:/0@/ -.8;806 k ; 1, ;,-.:,800E9 ;, .,52/? x1 x2 : : : xk , Rk (x) { 60.8:;,7<=6,00E9 ?0,>,5780 />:/0@/ -.8;806 k ;1, ;,-.:,800E9 ;, .,52/? xn;k+1 xn;k+2 : : : xn , k 2. ,7I5800E8 30/5806< f 0 676 f 00 1 .,52/B x1 6 xn 6-;,7C3IG.-< 2/2 I2/3E1/7,-C 1EF8 (-?. ;,-.:,8068 >:/0650EB I-7,169 ;, 30/5806:/0650EB I37/B).

. .

!"#$ $ %$

91

II.

! " #$%& '(%)*+ % ,- %%"0.8:;,7<=6< AI02=69 f : \ Rn ! R1 I27/4E1/8.-< 1 ,DHIG -B8?I 3/4/56 760890,9 60.8:;,7<=66. , J./ -B8?/ .:8DI8. ,D:/H806< ?/.:6=E (I.1.3) 6 ;,.,?I 47< ,40,?8:0,>, -7I5/< 1EF8 DE7 ;:847,@80 :<4 ?8.,4,1, 1 2,.,:EB J./ ?/.:6=/ 08 D,788, 58? .:8B46/>,0/7C0/. 7< ?0,>,?8:0EB ,D7/-.89 ;,-.:,8068 B,:,F, ,D:/.6?,9 ?/.:6=E (I.1.3) ;:,631,46.-<, 1 ,-0,10,?, 41I?< ?8.,4/?6: 1) 60.8:;,76:,1/068? .803,:0E?6 ;:,63184806
1. # - x

K.,. ?8.,4 60.8:;,7<=66 ;,31,7<8. 0/6D,788 ;:,-., -18-.6 3/4/5I 60.:8;,7<=66 AI02=66 ?0,>6B ;8:8?800EB, 3/4/00,9 1 ;/:/77878;6;848, 2 3/4/58 60.8:;,7<=66 AI02=69 ,40,9 ;8:8?800,9.

x 1.1.

.& 7$ %6"<8 7%' 6%7&$< % %%

I-.C 3/4/0/ AI02=6< f : Z = X Y ! R1 , f 2 F (Z R1) { 082,.,:,?I AI02=6,0/7C0,?I ;:,-.:/0-.1I, 0/D,:E AI02=69 u1 : : : un , ui : X ! R1 , ui 2 F (X R1) 6 v1 : : : vm , vi : Y ! R1 , vi 2 F (Y R1) 6 0/D,:E 760890EB AI02=6,0/7,1 1 : : : n 0/ ;:,-.:/0-.18 F (X R1) 6 1 : : : m 0/ ;:,-.:/0-.18 F (Y R1). :8DI8.-< ;,-.:,6.C /;;:,2-6?/=6G Pf AI02=66 f 164/

Pf =

n X m X

i=1 j =1

ij wij

./2IG, 5.,DE DE76 1E;,7080E 760890E8 I-7,16< 60.8:;,7<=66 ij (Pf ) = ij (f ) i = 1 : : : n j = 1 : : : m

. .

(1)

!"#$ %C %C

x1. (',./ . (*4 ' *) . >48

92

wij = ui vj ..8. wij (x y) = ui(x)vj (y) ij = i j ..8. ij (f ) = i(j (fx(y))) = j (i(fy (x))) fx(y) = f (x y) 2 F (Y R1) fy (x) = f (x y) 2 F (X R1) f 2 F (Z R1) i = 1 : : : n j = 1 : : : m x 2 X y 2 Y: D,30/56? (X ) n n U = huiii=1 = iui : 2 Rn V = hvj imj=1 i=1

V = hiini=1 M = hj inj=1 W = U V = fw = u v : u 2 U v 2 V g :

% 1.

1. u1 : : : un ,

1 : : : n

A = (i(uj )) ij=1:::n

4 2. v1 : : : vm ,

(2)

1 : : : m

B = (i(vj )) ij=1:::m

(3)

. 1. ; wij = ui vj , W nm . 2. < . 3. /

Pf =

n X m X

i=1 j =1

W = U V

ij wij

,

(4)

;1

; = (ij )(i=1:::n j=1:::m) = A;1 (ij (f ))(i=1:::n j=1:::m) B t : (5)

7"#&. -<269 J78?80. w 2 W = U V ;:84-./17<8.-< 1 1648

w=

n X m X

i=1 j =1

ij wij

89-.16.87C0,, w 2 U V 8-.C w = u v , ..8. w(x y) = u(x)v(y). /2 2/2 u1 : : : un -,-./17
u(x) =

. .

n X i=1

iui(x)

v(y) =

m X

j =1

j vj (y):

!"#$ %C %C

x1. (',./ . (*4 ' *) . ,J.,?I

w(x y) =

n X m X i=1 j =1

93

ij ui(x)vj (y):

:,18:6?, 5., J., ;:84-./178068 8460-.1800, (J., 6 DI48. ,30/5/.C, 5., wij = ui vj ,D:/3IG. D/36- U V ). /--?,.:6?

0n m 1 n m X X XX ij k (ui)l (vj ) = kl(w) = (k l )(w) = (k l ) @ ij ui vj A = i=1 j =1 i=1 j =1 n X m n X m X X = ij akiblj = akiij blj = (A;B t )kl i=1 j =1

i=1 j =1

, I-7,16G ?/.:6=E A 6 B ,D:/.6?E, -784,1/.87C0, ; = A;1(kl (w))k=1:::n l=1:::m(B t);1 (6) .-G4/ 1E.82/8., 5., 2,JAA6=680.E ij ,40,30/50, ,;:8487
x 1.2.

8 " %

6%8

=806? .:I4,8?2,-.C 1E56-78069 ;, A,:?I78 (5). 1) ;,?,HCG, 0/;:6?8:, LU {:/37,@806< ?E ?,@8? 1E56-76.C A;1 - 3/.:/.,9 O(n3) /:6A?8.658-26B ,;8:/=69. 2) ;,?,HCG, 0/;:6?8:, LU {:/37,@806< ?E ?,@8? 1E56-76.C B ;1 - 3/.:/.,9 O(m3) /:6A?8.658-26B ,;8:/=69. 3) /.:6=/ T = A;1(kl (f )) 1E56-7<8.-< ;, ,;:8487806G ;:,63184806< ?/.:6= - 3/.:/.,9 O(n2m) /:6A?8.658-26B ,;8:/=69. 4) /.:6=/ ; = T (B t );1 1E56-7<8.-< ;, ,;:8487806G ;:,63184806< ?/.:6= - 3/.:/.,9 O(nm2) /:6A?8.658-26B ,;8:/=69. /26? ,D:/3,?, ,DH88 2,7658-.1, /:6A?8.658-26B ,;8:/=69, .:8DI8?EB 47< ;:,184806< 1E56-78069 ;, A,:?I78 (5), :/10, O(n3 + n2m + nm2 + m3 ). G 1. 8;,-:84-.1800,8 ,;:84878068 2,JAA6=680.,1 ; = (ij ) 63 -6-.8?E n X m X

i=1 j =1

k (ui)l (vj )ij = kl (f )

k = 1 2 : : : n l = 1 2 : : : m

63 n m 760890EB I:/108069 ,.0,-6.87C0, n m 086318-.0EB ij .:8DI8. 2 (n m)3 3 -7,@8069 6 -.,7C2, @8 I?0,@8069 ;, ?8.,4I /I--/. /;:6?8:, ;:6 n = m J., 4/8. O(n6) /:6A?8.658-26B ,;8:/=69, 1 ., 1:8?<, 2/2 A,:?I78 (5) O(n3).

. .

!"#$ %C %C

x1. (',./ . (*4 ' *) . x 1.3.

94

"*% %6"<8 7%' 6%7&$<

3 .8,:8?E 1 1E.82/8. -784IGH69 /7>,:6.? 1E56-7806< 6-2,?EB 2,JAA6=680.,1 ; = (ij )(i=1:::n j=1:::m) AI02=66 Pf . D,30/56? fkl = kl (f ), fl = (f1l : : : fnl)t { 182.,: 63 J78?80.,1 fkl , F = Tf1 : : : fm ] { n m ?/.:6=/, -,-./17800/< 63 182.,:,1 fk . ,>4/ A,:?I7/ (5) I.18:@4/8., 5., ; = A;1 F (B t);1: /--?,.:6? ?/.:6=I

T = A;1 F = A;1Tf1 : : : fm] = TA;1 f1 : : : A;1fm ]: -67I :83I7C././ (I.1.4) .8,:8?E (I.1.1) 8-76 b = (b1 : : : bn)t { ;:,631,7C0E9 182.,:, ., 182.,: c = A;1 b 8-.C 182.,: 2,JAA6=680.,1 60.8:;,76:IGH89 AI02n =66 u = P ciui ./2,9, 5., k (u) = bk , k = 1 2 : : : n . 784,1/.87C0,, k -9 i=1 -.,7D8= ?/.:6=E T 8-.C 182.,: 2,JAA6=680.,1 60.8:;,76:IGH89 AI02=66, ;,-.:,800,9 ;, k -?I -.,7D=I fk ?/.:6=E F . /--?,.:6? ?/.:6=I

t

; = T (B t);1 = B ;1 T t : -76 ,D,30/56.C T t = Tt1 : : : tn] { -.,7D=E ti 2 Rm ?/.:6=E T t , ., B ;1T = TB ;1 t1 : : : B ;1tn]. -67I :83I7C././ (I.1.4) .8,:8?E (I.1.1) 8-76 b = (b1 : : : bm )t { ;:,631,7C0E9 182.,:, ., 182.,: c = B ;1b 8-.C 182.,: 2,JAA6=680.,1 60m .8:;,76:IGH89 AI02=66 u = P cj uj ./2,9, 5., l(u) = bl , l = 1 2 : : : m . j =1 784,1/.87C0,, l -/< -.:,2/ ?/.:6=E ; 8-.C 182.,: 2,JAA6=680.,1 60.8:;,76:IGH89 AI02=66, ;,-.:,800,9 ;, l -,9 -.:,28 tl ?/.:6=E T . /.:6=E T 6 ; ,DE50, B:/0<.-< 0/ ?8-.8 ?/.:6=E F . J.,? -7I5/8 ,;6-/00E9 1EF8 /7>,:6.? ?,@8. DE.C -A,:?I76:,1/0 1 -784IGH8? 1648. 1) 7< 1-8B k = 1 2 : : : n 3/?806.C k -9 -.,7D8= fk ?/.:6=E F 0/ 182.,: n k 2,JAA6=680.,1 60.8:;,76:IGH89 AI02=66 u = P kiui ./2,9, 5., i=1 i(fk ) = fik = i(u), i = 1 2 : : : n . 2) 7< 1-8B l = 1 2 : : : m 3/?806.C l -IG -.:,2I tl ?/.:6=E F 0/ 182.,: m l 2,JAA6=680.,1 60.8:;,76:IGH89 AI02=66 v = P lj vj ./2,9, 5., j =1 j (tl ) = tjl = j (u), j = 1 2 : : : m .

. .

!"#$ %C %C

x1. (',./ . (*4 ' *) . x 1.4.

95

.%*%< %"78< "*%

I-.C ;,4;:,>:/??/ INTERP X(LU, N, ALPFA) ;, 182.,:I

LU = (i(u)) (i=1:::N)

30/58069 AI02=6,0/7,1 i 0/ 082,.,:,9 AI02=66 u 2 F (X R1) -.:,6. 182.,: ALPFA = (i ) (i=1:::N) 2,JAA6=680.,1 60.8:;,76:IGH89 AI02=66 PX u = PN u ;, 082,.,:,?I ?8.,4I 1 760890,9 60.8:;,7<=66 AI02=69 63 F (X R1). i=1 i i I-.C ;,4;:,>:/??/ INTERP Y(MV, M, BETA) ;, 182.,:I

MV = (j (v)) (j=1:::M)

30/58069 AI02=6,0/7,1 i 0/ 082,.,:,9 AI02=66 v 2 F (Y R1) -.:,6. 182.,: BETA = (i) (j=1:::M) 2,JAA6=680.,1 60.8:;,76:IGH89 AI02=66 PY v = PM v ;, 082,.,:,?I ?8.,4I 2 760890,9 60.8:;,7<=66 AI02=69 63 F (Y R1). j =1 j j I-.C f 2 F (X Y R1). D,30/56? N M ?/.:6=I

NF = ( (i j )(f ) )(i=1:::N j=1:::M)

-2,?/< N M ?/.:6=/ 2,JAA6=680.,1 GAMMA = ; AI02=66 Pf NX M X (Pf )(x y) = ij ui(x)vj (y) i=1 j =1

1E56-7<8.-< -784IGH6? /7>,:6.?,? 1. 7< i = 1 N 1E;,70<.C INTERP X(i{9 -.,7D8= NF, N, i{9 -.,7D8= GAMMA) 2. 7< j = 1 M 1E;,70<.C INTERP Y(j{< -.:,2/ GAMMA, M, j{< -.:,2/ GAMMA) -76 ?8.,4 1 676 2 .:8DIG. 47< -1,8>, ;:,184806< ,D:/H806< 082,.,:,9 ?/.:6=E, ., J./ ?/.:6=/ DI48. ,40,9 6 .,9 @8 ;:6 2/@4,? 1E3,18 ;,4;:,>:/??E INTERP X 6 INTERP Y. 784,1/.87C0,, ;:6 ;8:1,? ,D:/H8066 2 ;,4;:,>:/??8 INTERP X 676 INTERP Y ?,@0, ;,-.:,6.C, 0/;:6?8:, LU{:/37,@8068 J.,9 ?/.:6=E 6 -,B:/06.C J., :/37,@8068 4, -784IGH8>, 1E3,1/ ;,4;:,>:/??E 1, 1-;,?,>/.87C0,? ?/--618. 3/16-6?,-.6 ,. 6-;,7C3I8?,>, <3E2/ ;:,>:/??6:,1/06< 76D, ;I02. 1 76D, ;I02. 2 J.,>, /7>,:6.?/ ?,@8. DE.C ,-IH8-.1780 D83 4,;,706.87C0EB 1E56-76.87C0EB 3/.:/. 0/ 1E3,1 ;,4;:,>:/?? 60.8:;,7<=66 ;, ;8:8?800,9 X 676 Y.

. .

!"#$ %C %C

x1. (',./ . (*4 ' *) .

96

/;:6?8:, 0/ <3E28 FORTRAN, 1 2,.,:,? ?/--61E B:/0<.-< ;, -.,7D=/?, ;8:1E9 ;I02. /7>,:6.?/ ?,@0, 3/;6-/.C 1 1648 DO I=1,N CALL INTERP X( NF(1,I), N, GAMMA(1,I) ) ENDDO / <3E28 C, 1 2,.,:,? ?/--61E B:/0<.-< ;, -.:,2/?, 1.,:,9 ;I02. /7>,:6.?/ ?,@0, 3/;6-/.C 1 1648 for( j=0Z j < MZ j++ ) INTERP Y( GAMMATj], M, GAMMATj] )Z

x 1.5.

%6"<8 7%' 6%7&$< & "! *" 66%8

I-.C f = f (x y) { AI02=6< 41IB ;8:8?800EB, 3/4/00/< 0/ ;:,7C0628 Ta b] Tc d]. I-.C ./2@8 3/4/0E .,526 a = x1 < x2 < : : : < xn = b 6 c = y1 < y2 < : : : < ym = d , 6 6318-.0E 30/5806< f (xi yj ), i = 1 2 : : : n , j = 1 2 : : : m AI02=66 f 1 .,52/B (xi yj ). :,48?,0-.:6:I8? ;:6?808068 .8,:8?E 1 47< :/--?,.:800EB :/088 ?8.,4,1 ;:6D76@806<.

x 1.5.1.

%6"<8' *" *%A

I-.C 2/@4E9 63 ?8.,4,1 ;:6D76@806< ;, ;8:8?800E? x 6 y <17<8.-< /;;:,2-6?/=689 ?0,>,5780,? />:/0@/ (-?. x I.3). :6D76@/GH/< AI02=6< (;, 2/@4,9 63 ;8:8?800EB) 1E56-7<8.-< ;, A,:?I78 (I.3.1). 1848? 1-8 I5/-.1IGH68 1 ;,-./0,128 3/4/56 760890,9 60.8:;,7<=66 .803,:0E?6 ;:,63184806
i (u) = u(xi) i = 1 2 : : : n j (v) = v(yj ) j = 1 2 : : : m:

(7)

/.:6=E A 1 (2) 6 B 1 (3) { 8460650E8. /36- 1 ;:,-.:/0-.18 AI02=69 { .803,:0EB ;:,631848069: n m x;x y;y Y Y k l wij (x y) = ui(x)vj (y) = x ; x y ; y k j l k=1k6=i l=1l6=j i i = 1 2 : : : n j = 1 2 : : : m: /36- 1 ;:,-.:/0-.18 AI02=6,0/7,1 { .803,:0EB ;:,631848069:

ij (f ) = i(j (f )) = f (xi yj ) i = 1 2 : : : n j = 1 2 : : : m:

. .

!"#$ %C %C

x1. (',./ . (*4 ' *) .

97

, .8,:8?8 1 ;:6D76@/GH/< AI02=6< 6?88. 164 (4), >48 2,JAA6=680.E ij 1E56-78 ;:6D76@/GH/< AI02=6< 6?88. 164

Pf =

x 1.5.2.

n X m X i=1 j =1

f (xi yj )

m x;x y;y Y k l : x ; x y ; y k j l k=1k6=i l=1l6=j i n Y

7$"' %7 $"<

!8 * 6%'

/3487800E8 :/30,-.6, 1184800E8 1 x I.8, <17
f (xiZ : : : Z xi+k yj Z : : : Z yj+l) = = (f (xiZ : : : Z xi+k ))(yj Z : : : Z yj+l) = (f (yj Z : : : Z yj+l))(xiZ : : : Z xi+k ): /180-.1, 348-C -;:/184761, 1 -67I ;:84-./17806< (I.4.2). K., ,;:84878068 J2161/780.0, -784IGH8?I /0/7,>I ,;:848706< (I.8.1). 6%$". I-.C 3/4/0 0/D,: .,582 xi : : : xi+k , yj : : : xj +l (08 ,D<3/.87C0, :/37650EB). * AI02=66 f 0I781,>, ;,:<42/ 0/3E1/8.-< f (xi yj ) = f (xi yj ), :/3487800,9 :/30,-.CG (k + l)->, ;,:<42/ 0/3E1/8.-<

f (xiZ : : : Z xi+k yj Z : : : Z yj+l) 8 k > 1 @ f (x y Z : : : Z y ) 8-76 x = : : : = x 6 f 2 C (k+l) > j +l i i+k > k! @xk i j > > (f (xiZ : : : Z xr;1Z xr+1Z : : : Z xi+k yj Z : : : Z yj+l); > > > ;f (xiZ : : : Z xs;1Z xs+1Z : : : Z xi+k yj Z : : : Z yj+l))=(xs ; xr) > > > < 8-76 -IH8-.1IG. xs 6= xr => l 1 @ f (x Z : : : Z x y ) 8-76 y = : : : = y 6 f 2 C (k+l) > > i+k j j j +l > l! @yl i > > (f (xiZ : : : Z xi+k yj Z : : : Z yr;1Z yr+1Z : : : Z yj+l); > > > ;f (xiZ : : : Z xi+k yj Z : : : Z ys;1Z ys+1Z : : : Z yj+l))=(ys ; yr ) > > : 8-76 -IH8-.1IG. ys 6= yr

x 1.5.3.

%6"<8<

%!" #=

I-.C 2/@4E9 63 ?8.,4,1 ;:6D76@806< ;, ;8:8?800E? x 6 y <17<8.-< /;;:,2-6?/=689 ?0,>,5780,? />:/0@/, 2,.,:E9 1E56-7<8.-< ;, A,:?I78 CG.,0/ (-?. x I.5). :6D76@/GH/< AI02=6< (;, 2/@4,9 63 ;8:8?800EB) 1E56-7<8.-< ;, A,:?I78 (I.5.2).

. .

!"#$ %C %C

x1. (',./ . (*4 ' *) .

98

1848? 1-8 I5/-.1IGH68 1 ;,-./0,128 3/4/56 760890,9 60.8:;,7<=66 .803,:0E?6 ;:,63184806
ui(x) = vj (y) =

iY ;1

(x ; xk ) i = 1 2 : : : n

k=1 jY ;1 l=1

(y ; yl)

j = 1 2 : : : m

(348-C -56./8.-<, 5., ;:,631848068 :/10, 1, 8-76 18:B069 ;:8487 ?80CF8 06@08>,). #I02=6,0/7E 3/4/G.-< 1 (7). 40/2,, I4,D088 :/--?,.:8.C 4:I>IG -6-.8?I AI02=6,0/7,1, 2,.,:/< 4/8. .I @8 ;:6D76@/GHIG AI02=6G. -67I (I.10.1) 1 2/58-.18 AI02=6,0/7,1 ?,@0, 13<.C i(u) = u(x1Z : : : Z xi) i = 1 2 : : : n j (v) = v(y1Z : : : Z yj ) j = 1 2 : : : m: 7< J.,9 -6-.8?E AI02=6,0/7,1 ?/.:6=E A 1 (2) 6 B 1 (3) { 8460650E8. /361 ;:,-.:/0-.18 AI02=69 { .803,:0EB ;:,631848069: ;1 iY ;1 jY wij (x y) = ui(x)vj (y) = (x ; xk )(y ; yl) k=1 l=1 i = 1 2 : : : n j = 1 2 : : : m: /36- 1 ;:,-.:/0-.18 AI02=6,0/7,1 { .803,:0EB ;:,631848069: ij (f ) = i(j (f )) = f (x1 Z : : : Z xi y1Z : : : Z yj ) i = 1 2 : : : n j = 1 2 : : : m: , .8,:8?8 1 ;:6D76@/GH/< AI02=6< 6?88. 164 (4), >48 2,JAA6=680.E ij 1E56-78 ;:6D76@/GH/< AI02=6< 6?88. 164 ;1 n X m iY ;1 jY X Pf (x y) = f (x1Z : : : Z xi y1Z : : : Z yj ) (x ; xk )(y ; yl ) (8)

i=1 j =1

k=1 l=1

E56-78068 :/3487800EB :/30,-.89 ,-IH8-.17<8.-< -,>7/-0, x 1.4 ;:6?808068? /7>,:6.?/ 63 x I.5 ;, 2/@4,9 63 ;8:8?800EB x 6 y .

x 1.5.4.

%6"<8< "$&A!><" *"

I-.C 2/@4E9 63 ?8.,4,1 ;:6D76@806< ;, ;8:8?800E? x 6 y <17<8.-< 60.8:;,7<=689 "416@IH6?6-<" ?0,>,5780/?6 (-?. x I.6) -.8;806 k 6 l -,,.18.-.1800,. K.6 ?8.,4E <17,5780,? />:/0@/, 3/;6-/00E? 1 A,:?8 CG.,0/. ,J.,?I ;:6D76@/GH/< AI02=6< -.:,6.-<, ./2 @8, 2/2 1 ;:84E4IH8? :/34878.

. .

!"#$ %C %C

x1. (',./ . (*4 ' *) . x 1.5.5.

99

%6"<8< %' !7"

I-.C 2/@4E9 63 ?8.,4,1 ;:6D76@806< ;, ;8:8?800E? x 6 y <17<8.-< /;;:,2-6?/=689 ?0,>,5780,? />:/0@/ - 2:/.0E?6 I37/?6 (-?. x I.9). 8,:8?/ I.9.1 ;,31,7<8. :/--?/.:61/.C ./2IG /;;:,2-6?/=6G /0/7,>650, ,;6-/00,9 1EF8 D83 2:/.0EB I37,1.

x 1.5.6.

7"A 6 *" C?'D&

I-.C 2/@4E9 63 ?8.,4,1 ;:6D76@806< ;, ;8:8?800E? x 6 y <17<8.-< /;;:,2-6?/=689 ?0,>,5780/?6 8DEF81/ (-?. x I.13). :6D76@/GH/< AI02=6< (;, 2/@4,9 63 ;8:8?800EB) 1E56-7<8.-< ;, A,:?I78 (I.13.10). 1848? 1-8 I5/-.1IGH68 1 ;,-./0,128 3/4/56 760890,9 60.8:;,7<=66 .803,:0E?6 ;:,63184806
! bTi(x) (x) = Ti 2x ; (b + a) i = 0 1 : : : n ; 1

b;a ! (d + c) j = 0 1 : : : m ; 1 Tbj(y) (y) = Tj 2y ;d ; c

(9)

#I02=6,0/7E 3/4/G.-< 1 (I.13.4):

D E(x) i(u) = Tbi(x) u n i = 0 1 : : : n ; 1 D E(y) j (v) = Tbi(y) v m j = 0 1 : : : m ; 1

>48

hf ginx

( )

=

n X i=1

f (xi)g(xi)

hf gimy

( )

=

m X j =1

f (yj )g(yj ):

7< J.,9 -6-.8?E AI02=6,0/7,1 ?/.:6=E A 1 (2) 6 B 1 (3) { 46/>,0/7C0E8 (-?. (I.13.9)): A = diag n n2 : : : n2 2 Mn B = diag m m2 : : : m2 2 Mm : /36- 1 ;:,-.:/0-.18 AI02=69 { .803,:0EB ;:,631848069:

wij (x y) = ui(x)vj (y) = Tbi(x)(x)Tbj(y) (y) i = 0 1 : : : n ; 1 j = 0 1 : : : m ; 1: /36- 1 ;:,-.:/0-.18 AI02=6,0/7,1 { .803,:0EB ;:,631848069: n X m X xy) ij (f ) = i (j (f )) = wij (xi yj )f (xi yj ) hwij f i(nm

(10)

i=1 j =1

i = 0 1 : : : n ; 1 j = 0 1 : : : m ; 1

. .

!"#$ %C %C

x1. (',./ . (*4 ' *) . >48

n X m X

hf ginm = i (xy)

3 78??E I.13.2 ;,7I5/8? hwij wkli(nmxy) = 0 hwij wij i(nmxy) = nm4 hwij wij i(nmxy) = nm2 hw00 w00i(nmxy) = nm

=1 j =1

100

f (xi yj )g(xi yj ):

;:6 i 6= j 6 k 6= l ;:6 i 6= 0 6 j 6= 0

;:6 i 6= 0 j = 0 676 i = 0 j 6= 0

..8. D/36- wij ,:.,>,0/780 ,.0,-6.87C0, -2/7<:0,>, ;:,63184806< h i(xy) . , .8,:8?8 1 ;:6D76@/GH/< AI02=6< 6?88. 164 (4), >48 2,JAA6=680.E ij 1E56-7
xy) ij (f ) = hwij f i(nm hwij wij i(nmxy) hwij wiji(nmxy) : 6.,>8 ;:6D76@/GH/< AI02=6< 6?88. 164 (xy) nX ;1 m ;1 X h w ij f inm w : Pf (x) = (xy) ij i=0 j =0 hwij wij inm E56-78068 2,JAA6=680.,1 :/37,@806< ,-IH8-.17<8.-< -,>7/-0, x 1.4 ;:6?808068? /7>,:6.?/ (I.13.11) ;, 2/@4,9 63 ;8:8?800EB x 6 y .

ij =

x 1.5.7.

66%8< *" C?'D& $ #D &$%&

I-.C 2/@4E9 63 ?8.,4,1 ;:6D76@806< ;, ;8:8?800E? x 6 y <17<8.-< /;;:,2-6?/=689 ?0,>,5780/?6 8DEF81/, ;,-.:,800,9 ?8.,4,? 0/6?80CF6B 21/4:/.,1 (-?. x I.17). :6D76@/GH/< AI02=6< (;, 2/@4,9 63 ;8:8?800EB) 1E56-7<8.-< ;, A,:?I78 (I.17.2) - 2,JAA6=680./?6 (I.17.5). 1848? 1-8 I5/-.1IGH68 1 ;,-./0,128 3/4/56 760890,9 60.8:;,7<=66 .803,:0E?6 ;:,63184806
h i(x) i(u) = u Tbi(x) i = 0 1 : : : n ; 1 h i(y) j (v) = v Tbj(y) j = 0 1 : : : m ; 1

>48

Zd f (y)g(y) f ( x ) g ( x ) ( y ) dx Tf g] = q dy: Tf g] = q c (d ; y )(y ; c) a (b ; x)(x ; a) (x)

. .

Zb

!"#$ %C %C

x1. (',./ . (*4 ' *) .

101

7< J.,9 -6-.8?E AI02=6,0/7,1 ?/.:6=E A 1 (2) 6 B 1 (3) { 46/>,0/7C0E8 (-?. 78??I I.17.2): A = diag 2 : : : 2 2 Mn B = diag 2 : : : 2 2 Mm : /36- 1 ;:,-.:/0-.18 AI02=69 { .803,:0EB ;:,631848069: (10). /36- 1 ;:,-.:/0-.18 AI02=6,0/7,1 { .803,:0EB ;:,631848069:

h i(xy) ij (f ) = i(j (f )) = Tbi(x)Tbj(y) f = Twij f ](xy) i = 0 1 : : : n ; 1 j = 0 1 : : : m ; 1

>48

Zb Zd

f (x y)g(x y) dxdy: ( b ; x )( x ; a )( d ; y )( y ; c ) a c 784,1/.87C0,, ;:6D76@/GH/< AI02=6< Pf 2 Pn;1 (x) Pm;1 (y) I4,178.1,:<8. -784IGH6? 760890E? I-7,16
Tf g]

q

2,.,:E8 <17,?8:0E? /0/7,>,? I-7,169 (I.17.3). -67I 4,2/3/00,>, 1 x I.17 J., ,30/5/8., 5., ;:6D76@/GH/< AI02=6< Pf 2 Pn;1 (x) Pm;1 (y) :8/763I8. ?606?I?

kf ; Pf k Tf ; Pf f ; Pf ] xy ! min 1 ;:,-.:/0-.18 1-8B AI02=69 Pf 2 Pn (x) Pm (y). (

2

3 78??E I.17.2 ;,7I5/8?

Twij wkl](xy) = 0 2 Twij wij ](xy) = 4 2 Twij wij ](xy) = 2 Tw00 w00](xy) = 2

;1

)

;1

;:6 i 6= j 6 k 6= l

;:6 i 6= 0 6 j 6= 0

;:6 i 6= 0 j = 0 676 i = 0 j 6= 0

..8. D/36- wij ,:.,>,0/780 ,.0,-6.87C0, -2/7<:0,>, ;:,63184806< T ](xy) . , .8,:8?8 1 ;:6D76@/GH/< AI02=6< 6?88. 164 (4), >48 2,JAA6=680.E ij 1E56-7
. .

!"#$ %C %C

x1. (',./ . (*4 ' *) .

102

6.,>8 ;:6D76@/GH/< AI02=6< 6?88. 164

Pf (x) =

nX ;1 m ;1 X

Twij f ](xy) w : (xy) ij i=0 j =0 Twij wij ]

E56-78068 2,JAA6=680.,1 :/37,@806< ,-IH8-.17<8.-< -,>7/-0, x 1.4 ;:6?808068? /7>,:6.?/ (I.17.19) ;, 2/@4,9 63 ;8:8?800EB x 6 y .

x 1.6.

%6"<8 7%' 6%7&$< & "! !-*" 66%8

-7I5/8, 2,>4/ 2/@4E9 63 ?8.,4,1 60.8:;,7<=66 ;, ;8:8?800E? x 6 y <17<8.-< 2I-,50,-?0,>,57800E?, ,;6-/00E9 1EF8 /7>,:6.? 60.8:;,7<=66 .803,:0E?6 ;:,63184806,7C0628 Ta b] Tc d]. I-.C ./2@8 3/4/0E .,526 a = x1 < x2 < : : : < xn = b 6 c = y1 < y2 < : : : < ym = d , 6 2/@4E9 63 ?8.,4,1 60.8:;,7<=66 ;, ;8:8?800E? x 6 y ./2,1, 5., 60.8:;,76:IGH/< AI02=6< 0/ 2/@4,? 63 ,.:832,1 Txi xi+1 ], Tyj yj+1] -.:,6.-< 083/16-6?, (,. 4:I>6B ,.:832,1 Txk xk+1], Tyl yl+1], k 6= i , l 6= j ). ,>4/ ?/.:6=E A;1 6 B ;1 1 (5) DI4I. D7,50,-46/>,0/7C0E?6, 6 63 .8,:8?E 1 1E.82/8., 5., 60.8:;,76:IGH/< AI02=6< 0/ ;:,7C0628 Txi xi+1 ] Tyj yj+1] -.:,6.-< 083/16-6?, (,. 4:I>6B ;:,7C062,1 Txk xk+1] Tyl yl+1], k 6= i , l 6= j ). ,J.,?I .8,:8?I 1 ?,@0, ;:6?80<.C 083/16-6?, 0/ 2/@4,? 63 ;:,7C062,1 Txi xi+1] Tyj yj+1]. K., ,D-.,<.87C-.1, ;,31,7<8. ./2@8 :/-F6:6.C ,D7/-.C ;:6?806?,-.6 /7>,:6.?/: AI02=6< f ?,@8. DE.C 3/4/0/ 08 .,7C2, 0/ ;:,7C0628 Ta b] Tc d], / 6 0/ 7GD,9 ,D7/c.6, ;:84-./16?,9 1 1648 ,DP8460806< 08;8:82:E1/GH6B-< ;:,7C062,1 S (Txi xi+1 ] Tyj yj+1]). ij

x 1.6.1.

!-"< %6"<8<

I-.C 2/@4E9 63 ?8.,4,1 ;:6D76@806< ;, ;8:8?800E? x 6 y <17<8.-< 2I-,50,-760890,9 60.8:;,7<=689 (-?. x I.15). ,>7/-0, -2/3/00,?I 1EF8 4,-./.,50, :/--?,.:8.C ;:6D76@8068 0/ ;:,7C0628 Txi xi+1]Tyj yj+1]. :6D76@/GH/< AI02=6< (;, 2/@4,9 63 ;8:8?800EB) 1E56-7<8.-< ;, A,:?I78 (I.15.1). 1848? 1-8 I5/-.1IGH68 1 ;,-./0,128 3/4/56 760890,9 60.8:;,7<=66 .803,:0E?6 ;:,63184806
i1(u) = u(xi) i2(u) = u(xi+1) j1(v) = v(yj ) j2(v) = v(yj+1):

. .

!"#$ %C %C

x1. (',./ . (*4 ' *) .

103

/.:6=E A 1 (2) 6 B 1 (3) { 8460650E8. /36- 1 ;:,-.:/0-.18 AI02=69 { .803,:0EB ;:,631848069: wijkl(x y) = uik(x)vjl (y) k l = 1 2: /36- 1 ;:,-.:/0-.18 AI02=6,0/7,1 { .803,:0EB ;:,631848069: ijkl(f ) = ik (jl(f )) k l = 1 2 ..8. ij11(f ) = f (xi yj ) ij12(f ) = f (xi yj+1) ij21(f ) = f (xi+1 yj ) ij22(f ) = f (xi+1 yj+1): , .8,:8?8 1 ;:6D76@/GH/< AI02=6< 6?88. 164 (4):

Pf =

2 X 2 X

k=1 l=1

ijklwijkl

>48 2,JAA6=680.E ijkl 1E56-78 ;:6D76@/GH/< AI02=6< 6?88. 164 Pf = f (xi yj ) xx ;; xxi+1 yy ;; yyj+1 +f (xi yj+1) xx ;; xxi+1 y y ;;yjy i i+1 j j +1 i i+1 j +1 j x ; x y ; y x ; x y ; y i j +1 i j +f (xi+1 yj ) xi+1 ; xi yj ; yj+1 +f (xi+1 yj+1) xi+1 ; xi yj+1 ; yj : ,-.:,800IG ;:6D76@/GHIG AI02=6G 5/-., 0/3E1/G. . -7I5/8 2I-,50,-760890,9 60.8:;,7<=66 .,. @8 :83I7C./. ?,@0, ;,7I56.C, ;:6?80<< .8,:8?I 1 0/ 1-89 ,D7/-.6 ,;:8487806< AI02=66 f . (x) 6%$". D,30/56? 58:83 S2 760890,8 ;:,-.:/0-.1, 08;:8:E10EB 7,?/0EB 76069 0/ ,.:8328 Tx1 xn] = Ta b] - 637,?/?6 x2 < x3 < : : : < xn;1 Z / 58:83 S2(y) { 760890,8 ;:,-.:/0-.1, 08;:8:E10EB 7,?/0EB 76069 0/ ,.:8328 Ty1 ym] = Tc d] - 637,?/?6 y2 < y3 < : : : < ym;1 . 1848? D/36-E ;:,-.:/0-.1/ S2(x) , S2(y) (-?. (I.16.2))

Hi(x) 2 S2(x) Hi(x) (xk ) = ik i k = 1 2 : : : n Hj(y) 2 S2(y) Hj(y) (yl ) = jl j l = 1 2 : : : m:

-<2/< u 2 S2(x) 6 1-<2/< v 2 S2(y) :/-27/4E1/8.-< ;, J.,?I D/36-I -784IGH6? ,D:/3,? (-?.( I.16.3)):

u(x) =

. .

n X

k=1

(x)

u(xk )Hk (x) v(y) =

m X l=1

v(yl)Hl(y) (y):

!"#$ %C %C

x1. (',./ . (*4 ' *) .

104

2/58-.18 AI02=6,0/7,1 1ED6:/8? i(u) = u(xi), i = 1 2 : : : n , j (v) = v(yj ), j = 1 2 : : : m . ,>4/ ?/.:6=E A 1 (2) 6 B 1 (3) { 8460650E8Z D/361 ;:,-.:/0-.18 AI02=69 { .803,:0EB ;:,631848069: wij (x y) = Hi(x) (x)Hj(y) (y), i = 1 2 : : : n , j = 1 2 : : : m Z D/36- 1 ;:,-.:/0-.18 AI02=6,0/7,1 { .803,:0EB ;:,631848069: ij (f ) = f (xi yj ), i = 1 2 : : : n , j = 1 2 : : : m . , .8,:8?8 1 ;:6D76@/GH/< AI02=6< 6?88. 164 (4), >48 2,JAA6=680.E ij 1E56-78 ;:6D76@/GH/< AI02=6< 6?88. 164 n X m X Pf = f (xi yj )Hi(x)(x)Hj(y) (y): i=1 j =1

x 1.6.2.

!-"< 66%8< $ #D &$%&

I-.C 2/@4E9 63 ?8.,4,1 ;:6D76@806< ;, ;8:8?800E? x 6 y <17<8.-< 2I-,50,-760890,9 /;;:,2-6?/=689, ;,-.:,800,9 ?8.,4,? 0/6?80CF6B 21/4:/.,1 (-?. x I.16). J.,? -7I5/8 I4,D088 ;:6?80<.C .8,:8?I 1 0/ 1-89 ,D7/-.6 ,;:8487806< AI02=66 f . 1848? 1-8 I5/-.1IGH68 1 ;,-./0,128 3/4/56 760890,9 60.8:;,7<=66 .803,:0E?6 ;:,63184806
ui(x) Hi(x)(x) i = 1 2 : : : n vj (y) Hj(y)(y) j = 1 2 : : : m: #I02=6,0/7E i(u) = u Hi(x) L2 ( ab]) i = 1 2 : : : n j (v) = v Hj(y) L2( cd]) j = 1 2 : : : m:

/.:6=E A 1 (2) 6 B 1 (3) { .:8B46/>,0/7C0E8, -?. (I.16.8). /36- 1 ;:,-.:/0-.18 AI02=69 { .803,:0EB ;:,631848069: wij (x y) = Hi(x)(x)Hj(y) (y) i = 1 2 : : : n j = 1 2 : : : m: /36- 1 ;:,-.:/0-.18 AI02=6,0/7,1 { .803,:0EB ;:,631848069: ij (f ) = i(j (f )) = f Hi(x)Hj(y) L2 ( ab] cd]) i = 1 2 : : : n j = 1 2 : : : m: 784,1/.87C0,, ;:6D76@/GH/< AI02=6< Pf L2 f 2 S2(x) S2(y) I4,178.1,:<8. -784IGH6? 760890E? I-7,16
. .

!"#$ %C %C

x1. (',./ . (*4 ' *) .

105

2,.,:E8 <17,?8:0E? /0/7,>,? I-7,169 (I.16.6). -67I 4,2/3/00,>, 1 x I.16 J., ,30/5/8., 5., ;:6D76@/GH/< AI02=6< Pf = L2 f 2 S2(x) S2(y) :8/763I8. ?606?I? kf ; L2f kL2( ab] cd]) ! min 1 ;:,-.:/0-.18 1-8B AI02=69 L2 f 2 S2(x) S2(y) . , .8,:8?8 1 ;:6D76@/GH/< AI02=6< 6?88. 164 (4), >48 2,JAA6=680.E ij 1E56-7,:6.? 63 x 1.4, / 47< I-2,:806< 1E56-78069 47< ?/.:6= A 6 B -.:,<.-< 6B LU -:/37,@806<.

x 1.6.3.

!-!?< %6"<8<

I-.C 2/@4E9 63 ?8.,4,1 ;:6D76@806< ;, ;8:8?800E? x 6 y <17<8.-< 2I-,50,-2ID658-2,9 60.8:;,7<=689 (-?. x I.19). ,>7/-0, -2/3/00,?I 1EF8 4,-./.,50, :/--?,.:8.C ;:6D76@8068 0/ ;:,7C0628 Txi xi+1] Tyj yj+1]. :6D76@/GH/< AI02=6< (;, 2/@4,9 63 ;8:8?800EB) 1E56-7<8.-< ;, A,:?I78 (I.19.2), >48 2,JAA6=680.E 3/4/G.-< (I.19.3). 1848? 1-8 I5/-.1IGH68 1 ;,-./0,128 3/4/56 760890,9 60.8:;,7<=66 .803,:0E?6 ;:,6318480648 1876560E di , dj ,;:8487
. .

!"#$ %C %C

x1. (',./ . (*4 ' *) .

106

784,1/.87C0,,

ci (c1i c2i c3i c4i)t = A;1 (hi) (f (xi ) di f (xi+1) di+1)t >48

0 1 1 0 0 0 BB CC BB 0 1 0 0 C CC B ;1 B C A (h) = B 3 BB ; h3 ; h22 h33 ; h12 CCC B@ 2 C 1 ;2 1 A h3 h2 h3 h2 ;1 ;1 ;1 /26? ,D:/3,?, A;1 i = A (xi+1 ; xi ), Bj = A (yj +1 ; yj ). (:I>6?6 -7,1/?6, 4 2,JAA6=680.E i = (ik )tk=1:::4 2 R4 ;:6D76@/GH89 AI02=66 Pu = P ik uik , k=1 I4,178.1,:48 i(u) = (ik (u))k=1:::4 2 R4 . ,JAA64 =680.E j = (jl)tl=1:::4 2 R4 ;:6D76@/GH89 AI02=66 Pv = P jlvjl , I4,178l=1 .1,:48 j (v) = (jl(v))l=1:::4 2 R4 ). /36- 1 ;:,-.:/0-.18 AI02=69 { .803,:0EB ;:,631848069:

wijkl(x y) = uik (x)vjl(y) = (x ; xi)k;1(y ; yj )l;1 k l = 1 2 3 4: /36- 1 ;:,-.:/0-.18 AI02=6,0/7,1 { .803,:0EB ;:,631848069:

ijkl(f ) = ik (jl(f )) k l = 1 2 3 4:

(11)

, .8,:8?8 1 ;:6D76@/GH/< AI02=6< 6?88. 164 (4):

Pf =

4 X 4 X

k=1 l=1

ijklwijkl =

4 X 4 X

k=1 l=1

ijkl(x ; xi)k;1(y ; yj )l;1

;1 ;1 >48 2,JAA6=680.E ijkl 1E56-7
;ij = A;1(xi+1 ; xi )Fij (A;1(yj+1 ; yj ))t

>48 ;ij = (ijkl)kl=1:::4 2 M4 (R), Fij = (ijkl(f ))kl=1:::4 2 M4(R).

. .

!"#$ %C %C

x1. (',./ . (*4 ' *) . '"

107

!8"& { 7%' 6%7&$

-7I5/8 2I-,50,9 60.8:;,7<=66 2ID658-26?6 ?0,>,5780/?6 K:?6./ (-?.

x I.19.3) ?/.:6=/ Fij 6?88. 164 0 BB BB BB Fij = B BB BB BB @

@f (xi yj ) @f (xi yj+1) f (xi yj ) f ( x i yj +1) @y @y @f (xi yj ) @ 2 f (xi yj ) @f (xi yj+1) @ 2 f (xi yj+1) @x @x@y @x @x@y yj+1) i+1 yj ) f (xi+1 yj ) @f (x@y f (xi+1 yj+1) @f (xi+1 @y @f (xi+1 yj ) @ 2 f (xi+1 yj ) @f (xi+1 yj+1) @ 2 f (xi+1 yj+1) @x @x@y @x @x@y

1 CC CC CC CC CC CC CA

..8. 1 2/@4,9 .,528 (xi yj ), i = 1 2 : : : n , j = 1 2 : : : m (DE.C ?,@8., 3/ xi yj ) , 6-27G58068? >:/0650EB) 4,7@0E DE.C 3/4/0E 30/5806< f (xi yj ), @f (@x @f (xi yj ) , @ 2 f (xi yj ) . @y @x@y -7I5/8 2I-,50,9 60.8:;,7<=66 2ID658-26?6 ?0,>,5780/?6 8--87< (-?. x I.19.4), 2I-,50,9 60.8:;,7<=66 2ID658-26?6 ?0,>,5780/?6 ?8.,4,? 26?E (-?. x I.19.5), 2I-,50,9 60.8:;,7<=66 2ID658-26?6 ?0,>,5780/?6 - 6-;,7C3,1/068? :/3487800EB :/30,-.89 (-?. x I.19.6), 60.8:;,7<=66 2ID658-26?6 -;7/90/?6 (-?. x I.19.7) 1E:/@806< (I.19.7), (I.19.8), (I.19.9), (I.19.12) (4,;,70800E8 -,,.18.-.1IGH6?6 >:/0650E?6 I-7,16,0/7C0IG ?/.:6=I). D,30/56? 58:83 G(x) (;:6 xi := xi ) 6, -,,.18.-.1800,, G(y) (;:6 xi := yi ), ?/.:6=I, 1E:/@/GHIG :8F8068 -6-.8?E (I.19.7), (I.19.8), (I.19.9) 676 (I.19.12) 58:83 182.,:/ F (x) = (f (xi))i=12:::n (-,,.18.-.1800,, F (y) = (f (yj ))j=12:::m ), ./2, 5., D(x) = G(x) F (x) , D(y) = G(y) F (y) , >48 D(x) = (di)i=12:::n ;:6 xi := xi , D(y) = (dj )j=12:::m ;:6 xi := yi . /.:6=/ G(x) 6?88. :/3?8: n n , G(y) { :/3?8: m m , 6 I56.E1/8. -,,.18.-.1IGH68 >:/0650E8 I-7,16<. 8;8:C ?E ?,@8? -A,:?I76:,1/.C /7>,:6.? 1E56-7806< AI02=6,0/7,1 ijkl(f ) 1 (11). I-.C F = (fij ) ,D,30/5/8. n m ?/.:6=I 30/58069 f (xi yj ). E56-76? (,460 :/3): n m ?/.:6=I F (x) = (fij(x)) G(x) F , n m ?/.:6=I F (y) = (fij(y) ) F (G(y) )t , n m ?/.:6=I F (xy) = (fij(xy) ) G(x)F (G(y) )t .

. .

!"#$ %C %C

x1. (',./ . (*4 ' *) . 0/5806< AI02=6,0/7,1 Fij = (ijkl(f ))kl=1234 1 (11):

0 BB fij BB f (x) Fij = B BB ij BB fi+1j @ (x) fi+1j

fij(y) fij(xy) y) fi(+1 j xy) fi(+1 j

fij+1 ) fij(x+1 fi+1j+1 x) fi(+1 j +1

fij(y)+1 fij(xy+1) y) fi(+1 j +1 xy) fi(+1 j +1

108

1 CC CC CC CC CA

%*78< %< $' & 6< H

+:/08068 60A,:?/=66 , AI02=66 6 88 ;:6D76@8066 I4,D0, ,-IH8-.16.C -784IGH6? ,D:/3,?. B,40E8 4/00E8 { 30/5806< AI02=6,0/7,1 (ijkl(f )), i = 1 2 : : : n , j = 1 2 : : : m , k l = 1 2 3 4, :/3?8H/G.-< 1 3-B ?8:0,? ?/--618 F = (fijk ), i = 1 2 : : : n , j = 1 2 : : : m , k = 1 2 3 4 :/3?8:/ (n m 4), >48 fij1 = ij11(f ) = f (xi yj ) fij2 = ij12(f ) fij3 = ij21(f ) fij4 = ij22(f ): /.:6=/ Fij 1EF8 ;,7I5/8.-< 63 J78?80.,1 4-B ?8:0EB 182.,:,1 Fij , Fi+1j , Fij+1 , Fi+1j+1 . ,JAA6=680.E ;:6D76@/GH89 AI02=66 { (ijkl), i = 1 2 : : : n , j = 1 2 : : : m , k l = 1 2 3 4, 1E56-748 ijkl = ijkl .

x 1.6.4.

!-&$%< %6"<8<

I-.C 2/@4E9 63 ?8.,4,1 ;:6D76@806< ;, ;8:8?800E? x 6 y <17<8.-< 2I-,50,-21/4:/.650,9 60.8:;,7<=689 (-?. x I.20), ..8. 118480E .,526

1(x) < a = x1 < 2(x) < x2 < 3(x) < x3 < : : : < xn;1 < n(x) < xn = b < n(x+1)

6 ) 1(y) < c = y1 < 2(y) < y2 < 3(y) < y3 < : : : < ym;1 < m(y) < xm = d < m(y+1 6 60.8:;,76:IGH/< AI02=6< ,40,9 ;8:8?800,9 P (x)f ( P (y)f ) -.:,6.-< ./x) ], i = 1 2 : : : n (T (y) (y) ], 26? ,D:/3,?, 5., 0/ 2/@4,? ,.:8328 Ti(x) i(+1 i i+1 i = 1 2 : : : m ) ,0/ <17<8.-< ?0,>,5780,? Pi(x) ( Pj(y) ) -.8;806 2, ./26?, 5., 9 = Pi(x)(xi ) = f (xi) i = 1 2 : : : n (12) x) ) = v (x) Pi(x)(i(x)) = vi(x) Pi(x)(i(+1 i+1 676, -,,.18.-.1800,, 9 = Pj(y) (yj ) = f (yj ) j = 1 2 : : : m (13) Pj(y) (j(y) ) = vj(y) Pj(y) (j(y+1) ) = vj(y+1)

. .

!"#$ %C %C

x1. (',./ . (*4 ' *) .

109

>48 vi(x) , i = 1 2 : : : n+1, vj(y) , j = 1 2 : : : m+1 { -1,D,40E8 ;/:/?8.:E, .,. 676 60,9 -;,-,D 1ED,:/ 2,.,:EB ,;:8487<8. ?8.,4 2I-,50,9 60.8:;,7<=66 21/4:/.650E?6 ?0,>,5780/?6. ?8.,48 60.8:;,7<=66 ;/:/D,7658-26?6 -;7/90/?6 J.6 ;/:/?8.:E ;,7I5/G.-< 2/2 :8F8068 -6-.8?E 760890EB I:/108069 (I.20.8). ,>7/-0, -2/3/00,?I 1EF8 4,-./.,50, :/--?,.:8.C ;:6D76@8068 0/ ;:,7C0628 Ti(x) i(+1 j j +1 ?800EB) 1E56-7<8.-< ;, A,:?I78 (I.20.2), >48 2,JAA6=680.E 3/4/G.-< (I.20.3). 1848? 1-8 I5/-.1IGH68 1 ;,-./0,128 3/4/56 760890,9 60.8:;,7<=66 .803,:0E?6 ;:,63184806
#I02=6,0/7E x) i1(u) = vi(x) i2(u) = u(xi) i3(u) = vi(+1 j1(v) = vj(y) j2(v) = v(yj ) j3(v) = vj(y+1) : #,:?I7E (I.20.3) ;:84-./1748 0 1 1 0 0 BB 1 1 1 CCC 1 1 1 B C A;1(h z) = B BB ; h + h + z h + z ; z ; h + z C CA @ 1 1 1 1 1 1 1 ;h + z h + z h+z h h+z z (y) (y) (x) (x) ;1 ;1 ;1 /26? ,D:/3,?, A;1 i = A (xi ; i i+1 ; xi ), Bj = A (yj ; j j +1 ; yj ). (:I>6?6 -7,1/?6, 2,JAA6=680.E i = (ik )tk=1:::3 2 R3 ;:6D76@/GH89 AI02=66 4 Pi(x)u = P ik uik , I4,178.1,:48 164/ ik (P (x)u) = ik (u), 1E56-7
. .

!"#$ %C %C

x1. (',./ . (*4 ' *) .

110

i(u) = (ik (u))k=1:::3 2 R3 . ,JAA6=680.E j = (jl)tl=1:::3 2 R3 ;:6D76@/GH89 3 AI02=66 Pj(y) v = P jlvjl , I4,178.1,:48 j (v) = (jl(v))l=1:::3 2 R3 ). /36- 1 ;:,-.:/0-.18 AI02=69 { .803,:0EB ;:,631848069: wijkl(x y) = uik (x)vjl (y) = (x ; i(x))k;1(y ; j(y) )l;1 k l = 1 2 3: /36- 1 ;:,-.:/0-.18 AI02=6,0/7,1 { .803,:0EB ;:,631848069:

ijkl(f ) = ik (jl(f )) k l = 1 2 3: , .8,:8?8 1 ;:6D76@/GH/< AI02=6< 6?88. 164 (4): 3 X 3 3 X 3 X X Pf = ijklwijkl = ijkl(x ; i(x) )k;1(y ; j(y) )l;1 k=1 l=1

(14)

k=1 l=1

(x) (x) ;1 >48 2,JAA6=680.E ijkl 1E56-7
x) ; x )F (A;1 (y ; (y) (y) ; y ))t ;ij = A;1 (xi ; i(x) i(+1 i ij j j j j +1

>48 ;ij = (ijkl)kl=1:::3 2 M3 (R), Fij = (ijkl(f ))kl=1:::3 2 M3(R). '"

!8"& { 7%' 6%7&$

B,40E?6 4/00E?6 /7>,:6.?/ <17:/0650E8 I-7,16<. 8;8:C ?E ?,@8? -A,:?I76:,1/.C /7>,:6.? 1E56-7806< AI02=6,0/7,1 ijkl(f ) 1 (14). I-.C F = (fij ) ,D,30/5/8. n m ?/.:6=I 30/58069 f (xi yj ). E56-76? (,460 :/3): (n + 1) m ?/.:6=I F (x) = (fij(x)) G(x) F , n (m + 1) ?/.:6=I F (y) = (fij(y) ) F (G(y) )t ,

. .

!"#$ %C %C

x2. (',./ . +4 D,(

111

(n + 1) (m + 1) ?/.:6=I F (xy) = (fij(xy) ) G(x) F (G(y) )t . 0/5806< AI02=6,0/7,1 Fij = (ijkl(f ))kl=123 1 (14):

0 (xy) (x) (xy) BB fij fij fij+1 Fij = B BB fij(y) fij fij(y)+1 @ (xy) (x) (xy) fi+1j fi+1j fi+1j+1

1 CC CC CA

(15)

%*78< %< $' & 6< H

+:/08068 60A,:?/=66 , AI02=66 6 88 ;:6D76@8066 I4,D0, ,-IH8-.16.C -784IGH6? ,D:/3,?. B,40E8 4/00E8 { 30/5806< AI02=6,0/7,1 (ijkl(f )), i = 1 2 : : : n , j = 1 2 : : : m , k l = 1 2 3 4, B:/0<.-< 1 58.E:8B 1184800EB 1EF8 ?/.:6=/B F 2 Mnm , F (x) 2 Mn+1m , F (y) 2 Mnm+1 , F (xy) 2 Mn+m+1 . /.:6=/ Fij = (ijkl(f ))kl=123 ;,7I5/8.-< -,>7/-0, (15). ,JAA6=680.E ;:6D76@/GH89 AI02=66 { (ijkl), i = 1 2 : : : n , j = 1 2 : : : m , k l = 1 2 3, 1E56-748 ijkl = ijkl . x

2. # - /

K.,. ?8.,4 60.8:;,7<=66 ;,31,7<8. ;:6D7636.C AI02=66 ?0,>6B ;8:8?800EB, 3/4/00,9 1 ,D7/-.
x 2.1.

.& 7$ " %6"<8

I-.C 3/4/0/ AI02=6< f : D ! R1 , >48 D R2 { ,>:/065800/< ,D7/-.C - 4,-./.,50, >7/42,9 >:/06=89, f 2 F (D R1) { 082,.,:,?I AI02=6,0/7C0,?I ;:,-.:/0-.1I. :84-./16? ,D7/-.C D 1 1648 ,DP8460806< 2,0850,>, 56-7/ .:8I>,7C062,1 N k D , 08 6?8GH6B ,DH6B 10I.:8006B .,582: D0 S k D . ,-.:,8068 k=1 ./2,>, 0/D,:/ .:8I>,7C062,1 (0/3E1/8?,>, ,D7/-.6 D ) 1 ,DH8? -7I5/8 <17<8.-< ,580C -7,@0,9 3/4/589 6 47< 88 :8F806< -IH8-.1I8. D,7CF,8 56-7, ?8.,4,1. 6@8 ?E :/--?,.:6? 08-2,7C2, -/?EB ;:,-.89F6B 63 06B. /788, 0/ 2/@4,? .:8I>,7C0628 k AI02=6< f ;:6D76@/8.-< .8? 676 60E? ?8.,4,?, ..8. -.:,6.-< AI02=6< Pk f , I4,178.1,:,7C062/B 4,7@0E DE.C -,>7/-,1/0E: 8-76 (x y) 2 k1 T k2 { .,52/ ,DH89 >:/06=E .:8I>,7C062,1, ., Pfk1 (x y) = Pfk2 (x y). 0/58068 ;:6D76@/GH89 AI02=66

. .

!"#$ %C %C

x2. (',./ . +4 D,(

112

Pf 1 .,528 (x y) 2 D0 1E56-7<8.-< -784IGH6? ,D:/3,?: ,;:8487<8.-< k , ./2,8, 5., (x y) 2 k , 6 ;, ,;:8487806G ;,7/>/8.-< Pf (x y) = Pk f (x y). K.,. ;,4B,4 <17<8.-< ,D,DH8068? 2I-,50,-?0,>,57800,9 /;;:,2-6?/=66 0/ -7I5/9 AI02=69 ?0,>6B ;8:8?800EB. ,3?,@80 4:I>,9 -;,-,D ;,-.:,806< ;:6D76@/GH89 AI02=66. / ,D7/-.6 0 D ,;:8487,7C062,1 k (,. J.,>, -1,9-.1/ D/36-0EB AI02=69 ;,F7, 0/31/068 \ & "). /;:6?8:, 1ED6:/8.-< .,52/ (xl yl) 2 D0 6 D/36-0/< AI02=6< 'l (x y) ;,7/>/8.-< :/10,9 ?0,>,5780I 3/4/00,9 -.8;806 1 2/@4,? 63 .:8I>,7C062,1 k , ./26B, 5., (xl yl) 2 k . ,-./7C0EB .:8I>,7C062/B 'l (x y) ;,7/>/8.-< :/10,9 0I7G. -76 AI02=6,0/7E l 1 3/4/58 760890,9 60.8:;,7<=66 8-.C 30/5806< AI02=66 1 .,528: l (f ) = f (xl yl), ., ;:6 ./2,? ;,-.:,8066 D/36-0EB AI02=69 -I@8068 ;:6D76@/GH89 AI02=66 Pf 0/ .:8I>,7C062 k 6 ,;6-/00/< 1EF8 AI02=6< Pk f -,1;/4/G..

x 2.2.

6?' 6%< %*!"<8 ?"

6@8 ?E ,;6F8? 08-2,7C2, ;:,-.89F6B -;,-,D,1 ;,-.:,806< .:6/0>I7<=66 ,>:/065800,9 ,D7/-.6 D - 4,-./.,50, >7/42,9 >:/06=89. 1) D7/-.C D ,>:/06580/, ;,J.,?I -IH8-.1I8. ;:,7C062 Ta b] Tc d], ./2,9, 5., D Ta b]Tc d]. /@4/< 63 -.,:,0 J.,>, ;:,7C062/ :/3D61/8.-< 0/ :/10E8 5/-.6 .,52/?6 xi = a + ihx , i = 0 1 : : : n , hx = (b ; a)=n , yj = a + jhy , j = 0 1 : : : m , hy = (d ; c)=m . k <17<8.-< .:8I>,7C062,? - 18:F60/?6 1 .,52/B Qk1 , Qk2 , Qk3 , 8-76 1E;,7080, ,40, 63 58.E:8B I-7,169:

Qk1 = (xi yj ) 2 D Qk1 = (xi yj ) 2 D Qk1 = (xi yj ) 2 D Qk1 = (xi+1 yj ) 2 D

Qk2 = (xi+1 yj ) 2 D Qk2 = (xi yj+1) 2 D Qk2 = (xi+1 yj ) 2 D Qk2 = (xi+1 yj+1) 2 D

Qk3 = (xi+1 yj+1) 2 DZ Qk3 = (xi+1 yj+1) 2 DZ Qk3 = (xi yj+1) 2 DZ Qk3 = (xi yj+1) 2 D:

(1)

-7,168 k D ?,@8. DE.C 08 1E;,7080, 1 ;:6>:/0650EB .:8I>,7C062/B. ,S 7I5800/< .:6/0>I7<=6< DI48. .,50, /;;:,2-6?6:,1/.C ,D7/-.C (..8. D = k ), k 8-76 D <17<8.-< ?0,>,I>,7C062,? - 18:F60/?6 1 .,52/B (xi yj ), -.,:,0E 2,.,:,>, 76D, ;/:/7787C0E ,-,7 =4. 2) E;,70<8? ,;6-/00IG 1 ;I02.8 1) .:6/0>I7<=6G. 7< ;,1EF806< .,50,-.6 /;;:,2-6?/=66 ,D7/-.6 :/--?,.:6? 1-8 ;:,7C0626 k 6 D - 18:F60/?6 Qk1 = (xi yj ), Qk2 = (xi+1 yj ), Qk3 = (xi+1 yj+1), Qk4 = (xi yj+1), ./26?6, 5., B,.< DE ,40/ 63 .,582 Qkl 2 D . 416>/068? .,582 Qkl ! Q0kl , l = 1 2 3 4 0/ 0/6?80CFIG 1,3?,@0IG 1876560I ;, 0/;:/17806G 2 >:/06=8 @D ,D7/-.6 D 4,D61/8?-< .,>,, 5.,DE 76D, 1-8 0,1E8 .,526 Q0kl 2= D , 76D, 47<

. .

!"#$ %C %C

x2. (',./ . +4 D,(

113

2/2,9-76D, .:,926 Q0kl (676 41IB .:,82) DE7, 1E;,7080, ,40, 63 I-7,169 (1). :8I>,7C0626 - ./26?6 18:F60/?6 Q0kl 2 D 4,D/17<8? 2 .:6/0>I7<=66. S,7I5800/< .:6/0>I7<=6< DI48. .,50, /;;:,2-6?6:,1/.C ,D7/-.C (..8. D = k ), k 8-76 D <17<8.-< ?0,>,I>,7C062,? - 18:F60/?6 1 .,52/B (xi yj ). 3) :84;,7,@6?, 5., ?,@0, ;,-.:,6.C ;:8,D:/3,1/068 (>7/42,8 6 D83 ,-,DEB .,582), ;8:81,4I7<=6G (0/;:6?8:, ;:,7C062), 1 :/--?/.:61/8?IG ,D7/-.C D . ,>4/ 18:F60E .:8I>,7C062,1 .:6/0>I7<=66 ,D7/-.6 D ?,>I. DE.C ;,7I580E 2/2 ,D:/3E 18:F60 .:6/0>I7<=66 ,D7/-.6 G ;:6 J.,? ;:8,D:/3,1/066. 4) :84;,7,@6?, 5., ,D7/-.C D ?,@0, :/3:83/.C ;:I7<=6G. -76 18:F60E .:8I>,7C062,1, 78@/H68 0/ 7606I7<=69 -,,.18.-.1800, -,1;/4/G., ., ,DP84608068 J.6B .:6/0>I7<=69 4/8. .:6/0>I7<=6G 1-89 ,D7/-.6 D .

x 2.3.

.%?"A

!8 & %!*"#

/--?,.:6? 3/4/5I ;:6D76@806< AI02=66 f = f (x y), 3/4/00,9 1 .:8I>,7C0628 .

x 2.3.1.

.& 7$ " %6"<8

I-.C 2/@4/< -.,:,0/ .:8I>,7C062/ :/3D6./ 0/ l :/10EB 5/-.89, 6 58:83 .,526 :/3D6806< (127G5/< 18:F60E .:8I>,7C062/) ;:,18480E ;:, -.,:,0/?. D,30/56? 58:83 T ?0,@8-.1,, -,-.,,7C0628 . 6-7, J.6B .,582 :/10, n = l + 1k = (l + 1)(l + 2)=2. I48? ,D,30/5/.C 6B 58:83 Q1 (x1 y1), : : : , k=1 Qn(xn yn). /--?,.:6? 3/4/5I ;,-.:,806< ?0,>,5780/ Pf -.8;806 l X ij Pf (x y) = aij x y (2) i+j l

;:606?/GH8>, 1 .,52/B Qk (xk yk ) 3/4/00E8 30/5806< Pf (xk yk ) = f (xk yk ) k = 1 2 : : : n: (3) -7,16< (3) ;:84-./17,5780 Pf ?,@0, 1E;6-/.C <10, 6 1 :/--?,.:800,? 1EF8 ,DH8? -7I5/8. 40/2,, J., 1E:/@8068 4,-./.,50, -7,@0, 6 0/ ;:/2.628 6-;,7C3I8.-< :842,. 6@8 ?E 1E;6F8? 164 ?0,>,5780/ Pf 1 -7I5/8, 2,>4/ .:8I>,7C062

. .

!"#$ %C %C

x2. (',./ . +4 D,(

114

{ ;:,7C0E9 -, -.,:,0/?6, ;/:/7787C0E?6 ,-
x 2.3.2.

.%?"A

!8 & 6%<!*"# %!*"#

8F6? ;,-./17800IG 1EF8 3/4/5I 760890,9 60.8:;,7<=66 1 -7I5/8, 2,>4/ .:8I>,7C062 { ;:,7C0E9 -, -.,:,0/?6, ;/:/7787C0E?6 ,-, 18:F60/?6 <17,7C062 ,D:/3,1/0 4878068? ;:,7C062/ Ta b] Tc d] 8>, 46/>,0/7CG). I-.C a = x1 < x2 < : : : < xn;1 < xn = b c = y1 < y2 < : : : < ym;1 < ym = d 6 6318-.0E 30/5806< AI02=66 f 1 .,52/B (xi yj ) 2 . :8DI8.-< ;,-.:,6.C ;:6D76@/GHIG AI02=6G Pf 164/ (2), I4,178.1,:,:6.?/ ;:6D76@806< ?E DI48? -56./.C, 5., 6318-.0E 30/5806< AI02=66 f 1 .,52/B (xi yj ) 2= (0/;:6?8:, DI48? -56./.C f :/10,9 0I7G 1 J.6B .,52/B). ,>4/ 47< AI02=66 f ;:6?806?E /7>,:6.?E /;;:,2-6?/=66 .803,:0E?6 ;:,63184806,7C0628 Ta b] Tc d]. /? I4,D0, 6-;,7C3,1/.C ;:6D76@8068 60.8:;,7<=6,00E? ?0,>,5780,? />:/0@/ 1 A,:?8 CG.,0/, ..8. ;:6D76@/GHIG AI02=6G (1.8), 2,.,:IG ?E 3/;6F8? 1 -784IGH8? 1648 i;1 j ;1 f (x1Z : : : Z xi y1Z : : : Z yj ) Q Q (x ; xk )(y ; yl ) k=1 l=1 ij : (x y )2 i;1 j ;1 P + f (x Z : : : Z x y Z : : : Z y ) Q Q (x ; x )(y ; y )

Pf (x y) =

P i

j

ij : (x y )2= i

i

1

1

j

j

k

k=1 l=1

l

(5)

(348-C -56./8.-< 3/?20I.E? ?0,@8-.1,?). , ;,-.:,806G, J./ AI02=6< I4,178.1,:<8. I-7,16G Pf (xi yj ) = f (xi yj ) 47< 1-8B (xi yj ) 2 Ta b]Tc d]. /?8.6?, 5., ;1 iY ;1 jY

(x ; xk )(y ; yl ) = 0

k=1 l=1 (x y) = (xp yq )

47< 1-8B .,582 p = 1 2 : : : i ; 1 q = 1 2 : : : j ; 1: 784,1/.87C0,, 1.,:/< -I??/ 1 (5) :/10/ 0 47< 1-8B (x y) = (xi yj ) 2 (;,-2,7C2I 47< 1-<26B (xi yj ) 2= , (xp yq ) 2 1E;,7080, 76D, xi > xp , 76D, yj > yq ). ,J.,?I AI02=6<

Pf (x y) =

X ij : (x y i

. .

j

)2

f (x1Z : : : Z xi y1Z : : : Z yj )

;1 iY ;1 jY

k=1 l=1

(x ; xk )(y ; yl )

(6)

!"#$ %C %C

x2. (',./ . +4 D,(

115

I4,178.1,:<8. I-7,16 xk ;:6 i > k 6 yj > yl ;:6 j > l , ., 1-8 :/3487800E8 :/30,-.6 f (x1Z : : : Z xi y1Z : : : Z yj ) 1 (6) 1E56-7
Pf (x y) =

X

i+j n+1

x 2.3.3.

f (x1 Z : : : Z xi y1Z : : : Z yj )

;1 iY ;1 jY

k=1 l=1

.%?"A "'

(x ; xk )(y ; yl ) !8<

/--?,.:6? ,;6-/00IG 1EF8 3/4/5I 760890,9 60.8:;,7<=66 1 5/-.0,? -7I5/8 l = 1 1 (2), ..8. ;:6D76@8068 760890,9 AI02=689. .%?"A %!*"#

D,30/56? 18:F60E .:8I>,7C062/ 58:83 (x1 y1), (x2 y2), (x3 y3). /? .:8DI8.-< ;,-.:,6.C ;:6D76@/GHIG AI02=6< 164/ Pf (x y) = x + y + I4,178.1,:
,JAA6=680.E , , ?,>I. DE.C ;,7I580E 63 I:/10806< ;7,-2,-.6, ;:,B,4, 637,@806< 0/? I4,D088 118-.6 D/36-0E8 AI02=66 3/4/56 760890,9 60.8:;,7<=66 6 ;,-.:,6.C Pf 2/2 6B 760890IG 2,?D60/=6G. D,30/56? x ; u y ; v L(u1 v1 )(u2 v2) (x y) = u ; u1 v ; v1 = (x ; u1)(v2 ; v1) ; (y ; v1 )(u2 ; u1) (8) 2 1 2 1 { 760890/< AI02=6<, ,D:/H/GH/<-< 1 0I7C 0/ ;:
. .

!"#$ %C %C

x2. (',./ . +4 D,( / ./2@8

116

'i(x y) = i((xx yy)) i = 1 2 3:

(9)

i i i

?88?:

'i(xj yj ) = ij i j = 1 2 3: #I02=66 'i(x y), i = 1 2 3 1,3C?8? 1 2/58-.18 D/36-0EB 1 3/4/58 760890,9 60.8:;,7<=66, ..8. ;:6D76@/GHIG AI02=6G Pf DI48? 6-2/.C 1 1648 3 X Pf (x y) = i'i(x y) i=1

>48 2,JAA6=680.E i ,;:8487
Pf (x y) =

3 X

i=1

f (xi yi)'i(x y):

0/5806< ;:6D76@/GH89 AI02=66 Pf 0/ -.,:,08 .:8I>,7C062/ (0/;:6?8:, T(x1 y1) (x2 y2)]) ,;:8487,7C0626 1 6 2 6?8G. ,DHIG -.,:,0I, ., 30/5806< ;:6D76@/GH6B AI02=69, ;,-.:,800EB 1 2/@4,? 63 06B, -,1;/4/G. 1 .,52/B J.,9 ,DH89 -.,:,0E. K., ;,31,7<8. -.:,6.C 63 ;:6D76@/GH6B AI02=69 Pk f , ,;:8487800EB 1 .:8I>,7C062/B k , 2I-,50,-760890IG /;;:,2-6?/=6G Pf , ,;:8487800IG 0/ N .:6/0>I7<=66 S k , ;, ;:/167I: Pf (x y) = Pk f (x y) ;:6 (x y) 2 k . k=1

.%?"A %*!"<8

,-.:,800IG 1EF8 ;:6D76@/GHIG AI02=6< Pf , ,;:8487800IG 0/ .:6/0>IN 7<=66 D0 = S k , ?,@0, ;,7I56.C ./2@8 -784IGH6? -;,-,D,?. k=1 6%$". D,30/56? 58:83 V1 (D 0 ) 760890,8 ;:,-.:/0-.1, 08;:8:E10EB 1 D0 AI02=69, <17,7C0628 i D0 = SN . k k=1 1848? D/36- ;:,-.:/0-.1/ V1 (D0): (1) 0 '(1) i 2 V1 (D ) 'i (xj yj ) = ij i j = 1 2 : : : M1

(10)

>48 f(xj yj )gj=12:::M1 { ?0,@8-.1, 18:F60 .:8I>,7C062,1 k D0 . #I02=66 (1) '(1) i 5/-., 0/3E1/G. ? . 0/76.658-26 AI02=6G 'i ?,@0, 3/4/.C -784IGH6? ,D:/3,?. I-.C k , l = 1 2 : : : ni { .:8I>,7C0626, 6?8GH68 .,52I (xi yi) 1 2/58-.18 18:F60E. D,30/56? 58:83 'k (x y) 760890IG l

l

. .

!"#$ %C %C

x2. (',./ . +4 D,(

117

AI02=6G, ,;:8487800IG 1 .:8I>,7C0628 k , :/10IG 1 1 .,528 (xi yi) 6 0 1 ,-./7C0EB 18:F60/B k . /268 AI02=66 ?E ;,-.:,676 :/088, -?. (9). ,>4/ l

l

8 > < 'k (x y) 8-76 (x y) 2 k (1) 'i (x y) = > n 8-76 (x y) 2= S k : 0 l=1 l

l

i

l

-<2/< g 2 V1(D0) :/-27/4E1/8.-< ;, D/36-I 63 AI02=69 f'(1) i (x y )gi=12:::M1 -784IGH6? ,D:/3,?:

g(x y) =

M1 X i=1

g(xi yi)'(1) i (x y ):

89-.16.87C0,, 1 -67I (10) 6?88? 47< gb(x y) = P g(xi yi)'(1) i (x y ): gb(xj yj ) = i=1 g(xj yj ), j = 1 2 : : : M1 . 784,1/.87C0,, gb(x y) <17<8.-< 1 2/@4,? .:8I>,7C0628 k 760890,9 60.8:;,76:IGH89 AI02=689. -67I 2,::82.0,-.6 3/4/56 760890,9 60.8:;,7<=66 60.8:;,7<=6,00E9 ?0,>,5780 8460-.180080. 784,1/N .87C0,, g = gb 1 2/@4,? .:8I>,7C0628 k , 6, 30/56., 1 D0 = S k . k=1 /--?,.:6? 3/4/5I 0/B,@4806< ;:6D76@/GH89 AI02=66 Pf 2 V1(D0), -,1;/4/GH89 - AI02=689 f 1 .,52/B f(xj yj )gj=12:::M1 : M1

Pf (xj yj ) = f (xj yj ) j = 1 2 : : : M1 :

(11)

#I02=6G Pf DI48? 6-2/.C 1 1648

Pf (x y) =

M1 X i=1

i'(1) i (x y )

>48 2,JAA6=680.E i ,;:8487
Pf (x y) =

x 2.3.4.

M1 X i=1

f (xi yi)'(1) i (x y ):

.%?"A &$%

!8<

/--?,.:6? ,;6-/00IG 1EF8 3/4/5I 760890,9 60.8:;,7<=66 1 5/-.0,? -7I5/8 l = 2 1 (2), ..8. ;:6D76@8068 21/4:/.658-2,9 AI02=689.

. .

!"#$ %C %C

x2. (',./ . +4 D,(

118

.%?"A %!*"#

D,30/56? 18:F60E .:8I>,7C062/ 58:83 (x1 y1), (x2 y2), (x3 y3), -8:8460E 8>, -.,:,0 { 58:83 (x4 y4) (x12 y12) (x5 y5) (x23 y23) (x6 y6) (x13 y13)

>48 xkl = (xk + xl )=2, ykl = (yk + yl )=2. /? .:8DI8.-< ;,-.:,6.C ;:6D76@/GHIG AI02=6G 164/ (2) I4,178.1,:
Pf (xk yk ) = f (xk yk ) k = 1 2 : : : 6:

(12)

,-.:,6? D/36-0E8 AI02=66 3/4/56 760890,9 60.8:;,7<=66. 1848? AI02=66

1 (x y) = L(x2 y2)(x3 y3) (x y) 2 (x y) = L(x1 y1)(x3 y3)(x y) 3 (x y) = L(x1 y1)(x2 y2) (x y) 4 (x y) = L(x1 2 y1 2)(x1 3 y1 3) (x y) 5 (x y) = L(x1 2 y1 2 )(x2 3 y2 3) (x y) 6 (x y) = L(x1 3 y1 3)(x2 3 y2 3) (x y)

(>48 AI02=66 L ,;:848780E 1 (8)), / ./2@8

i(x y) = i((xx yy)) i = 1 2 : : : 6: i i i

,7,@6?

'1(x y) = 1 (x y)4(x y) '2(x y) = 2 (x y)5(x y) '3(x y) = 3 (x y)6(x y) '4(x y) = 1 (x y)2(x y) '5(x y) = 2 (x y)3(x y) '6(x y) = 1 (x y)3(x y):

(13)

?88?:

'i (xj yj ) = ij i j = 1 2 : : : 6: #I02=66 'i(x y), i = 1 2 : : : 6 1,3C?8? 1 2/58-.18 D/36-0EB 1 3/4/58 760890,9 60.8:;,7<=66, ..8. ;:6D76@/GHIG AI02=6G Pf DI48? 6-2/.C 1 1648 6 X Pf (x y) = i'i(x y) i=1

>48 2,JAA6=680.E i ,;:8487
Pf (x y) =

. .

6 X

i=1

f (xi yi)'i(x y):

!"#$ %C %C

x2. (',./ . +4 D,(

119

0/5806< ;:6D76@/GH89 AI02=66 Pf 0/ -.,:,08 .:8I>,7C062/ (0/;:6?8:, T(x1 y1) (x2 y2)]) ,;:8487,7C0626 1 6 2 6?8G. ,DHIG -.,:,0I, ., 30/5806< ;:6D76@/GH6B AI02=69, ;,-.:,800EB 1 2/@4,? 63 06B, -,1;/4/G. 1 .,52/B J.,9 ,DH89 -.,:,0E. K., ;,31,7<8. -.:,6.C 63 ;:6D76@/GH6B AI02=69 Pk f , ,;:8487800EB 1 .:8I>,7C062/B k , 2I-,50,-21/4:/.658-2IG /;;:,2-6?/=6G Pf , N ,;:8487800IG 0/ .:6/0>I7<=66 S k , ;, ;:/167I: Pf (x y) = Pk f (x y) ;:6 k=1 (x y) 2 k . .%?"A %*!"<8

,-.:,800IG 1EF8 ;:6D76@/GHIG AI02=6< Pf , ,;:8487800IG 0/ .:6/0>IN 7<=66 D0 = S k , ?,@0, ;,7I56.C ./2@8 -784IGH6? -;,-,D,?. k=1 6%$". D,30/56? 58:83 V2 (D 0 ) 760890,8 ;:,-.:/0-.1, 08;:8:E10EB 1 D0 AI02=69, <17,7C0628 i D0 = SN . k k=1 1848? D/36- ;:,-.:/0-.1/ V2 (D0): (2) 0 '(2) i 2 V2 (D ) 'i (xj yj ) = ij i j = 1 2 : : : M2

(14)

>48 f(xj yj )gj=12:::M2 { ?0,@8-.1, 18:F60 6 -:8406B .,582 -.,:,0 .:8I>,7C062,1 k D0 .

0/76.658-26 AI02=6G '(2) i ?,@0, 3/4/.C -784IGH6? ,D:/3,?. I-.C k , l = 1 2 : : : ni { .:8I>,7C0626, ./268, 5., (xi yi) 2 k . D,30/56? 58:83 'k (x y) 21/4:/.658-2IG AI02=6G, ,;:8487800IG 1 .:8I>,7C0628 k , :/10IG 1 1 .,528 (xi yi) 6 0 1 ,-./7C0EB 18:F60/B 6 -8:8460/B -.,:,0 k . /268 AI02=66 ?E ;,-.:,676 :/088, -?. (13). ,>4/ l

l

l

l

l

8 > < 'k (x y) 8-76 (x y) 2 k (2) 'i (x y) = > n 8-76 (x y) 2= S k : 0 l

l

i

l=1

l

-<2/< g 2 V2(D0) :/-27/4E1/8.-< ;, D/36-I 63 AI02=69 f'(2) i (x y )gi=12:::M2 -784IGH6? ,D:/3,?: M2 X g(x y) = g(xi yi)'(2) i (x y ): i=1

89-.16.87C0,, 1 -67I (14) 6?88? 47< gb(x y) = P g(xi yi)'(2) i (x y ): gb(xj yj ) = i=1 g(xj yj ), j = 1 2 : : : M2 . 784,1/.87C0,, gb(x y) <17<8.-< 1 2/@4,? .:8I>,7C0628 k 21/4:/.658-2,9 60.8:;,76:IGH89 AI02=689. -67I 2,::82.0,-.6 3/4/56 M2

. .

!"#$ %C %C

x2. (',./ . +4 D,(

120

760890,9 60.8:;,7<=66 60.8:;,7<=6,00E9 ?0,>,5780 8460-.180080. 784,1/N .87C0,, g = gb 1 2/@4,? .:8I>,7C0628 k , 6, 30/56., 1 D0 = S k . k=1 /--?,.:6? 3/4/5I 0/B,@4806< ;:6D76@/GH89 AI02=66 Pf 2 V2(D0), -,1;/4/GH89 - AI02=689 f 1 .,52/B f(xj yj )gj=12:::M2 :

Pf (xj yj ) = f (xj yj ) j = 1 2 : : : M2 :

(15)

#I02=6G Pf DI48? 6-2/.C 1 1648

Pf (x y) =

M2 X i=1

i'(2) i (x y )

>48 2,JAA6=680.E i ,;:8487
Pf (x y) =

x 2.3.5.

M2 X i=1

f (xi yi)'(2) i (x y ):

.%?"A !?

!8<

/--?,.:6? ,;6-/00IG 1EF8 3/4/5I 760890,9 60.8:;,7<=66 1 5/-.0,? -7I5/8 l = 3 1 (2), ..8. ;:6D76@8068 2ID658-2,9 AI02=689. .%?"A %!*"#

D,30/56? 18:F60E .:8I>,7C062/ 58:83 (x1 y1), (x2 y2), (x3 y3), .,526, 487, -.:,0E 0/ .:6 :/10E8 5/-.6, { 58:83 (x4 y4) (x121 y121) (x5 y5) (x122 y122) (x6 y6) (x231 y231) (x7 y7) (x232 y232) (x8 y8) (x131 y131) (x9 y9) (x132 y132)

>48 xkls = ((3 ; s)xk + sxl )=3, ykls = ((3 ; s)yk + syl )=3, =80.: .<@8-.6 .:8I>,7C062/ { 58:83 (x10 y10) (x123 y123) >48 x123 = (x1 + x2 + x3 )=3, y123 = (y1 + y2 + y3)=3. /? .:8DI8.-< ;,-.:,6.C ;:6D76@/GHIG AI02=6G 164/ (2) I4,178.1,:
Pf (xk yk) = f (xk yk ) k = 1 2 : : : 10:

(16)

,-.:,6? D/36-0E8 AI02=66 3/4/56 760890,9 60.8:;,7<=66.

. .

!"#$ %C %C

x2. (',./ . +4 D,(

121

1848? AI02=66 1 (x y) = L(x2 y2)(x3y3 )(x y) 2 (x y) = L(x1 y1)(x3 y3)(x y) 3 (x y) = L(x1 y1)(x2y2 )(x y) 4 (x y) = L(x1 2 1 y1 2 1 )(x1 3 1 y1 3 1) (x y) 5 (x y) = L(x1 2 2 y1 2 2 )(x1 3 2 y1 3 2) (x y) 6 (x y) = L(x1 2 2 y1 2 2 )(x2 3 1 y2 3 1) (x y) 7 (x y) = L(x1 2 1 y1 2 1 )(x2 3 2 y2 3 2) (x y) 8 (x y) = L(x1 3 2 y1 3 2 )(x2 3 2 y2 3 2) (x y) 9 (x y) = L(x1 3 1 y1 3 1 )(x2 3 1 y2 3 1) (x y) (>48 AI02=66 L ,;:848780E 1 (8)), / ./2@8 i(x y) = i((xx yy)) i = 1 2 : : : 10: i i i ,7,@6? '1(x y) = 1(x y)4(x y)5(x y) '2 (x y) = 2(x y)6(x y)7(x y) '3(x y) = 3(x y)8(x y)9(x y) '4 (x y) = 1(x y)2(x y)5(x y) '5(x y) = 1(x y)2(x y)7(x y) '6 (x y) = 2(x y)3(x y)7(x y) (17) '7(x y) = 2(x y)3(x y)9(x y) '8 (x y) = 1(x y)3(x y)5(x y) '9(x y) = 1(x y)3(x y)9(x y) '10 (x y) = 1(x y)2(x y)3(x y): ?88?: 'i(xj yj ) = ij i j = 1 2 : : : 10: #I02=66 'i (x y), i = 1 2 : : : 10 1,3C?8? 1 2/58-.18 D/36-0EB 1 3/4/58 760890,9 60.8:;,7<=66, ..8. ;:6D76@/GHIG AI02=6G Pf DI48? 6-2/.C 1 1648

Pf (x y) =

10 X

i=1

i'i(x y)

>48 2,JAA6=680.E i ,;:8487
Pf (x y) =

10 X

i=1

f (xi yi)'i(x y):

0/5806< ;:6D76@/GH89 AI02=66 Pf 0/ -.,:,08 .:8I>,7C062/ (0/;:6?8:, T(x1 y1) (x2 y2)]) ,;:8487,7C0626 1 6 2 6?8G. ,DHIG -.,:,0I, ., 30/5806< ;:6D76@/GH6B AI02=69, ;,-.:,800EB 1 2/@4,? 63 06B, -,1;/4/G. 1 .,52/B J.,9 ,DH89 -.,:,0E. K., ;,31,7<8. -.:,6.C 63 ;:6D76@/GH6B AI02=69 Pk f , ,;:8487800EB 1 .:8I>,7C062/B k , 2I-,50,-2ID658-2IG /;;:,2N -6?/=6G Pf , ,;:8487800IG 0/ .:6/0>I7<=66 S k , ;, ;:/167I: Pf (x y) = k=1 Pk f (x y) ;:6 (x y) 2 k .

. .

!"#$ %C %C

x2. (',./ . +4 D,(

122

.%?"A %*!"<8

,-.:,800IG 1EF8 ;:6D76@/GHIG AI02=6< Pf , ,;:8487800IG 0/ .:6/0>IN 7<=66 D0 = S k , ?,@0, ;,7I56.C ./2@8 -784IGH6? -;,-,D,?. k=1 6%$". D,30/56? 58:83 V3 (D 0 ) 760890,8 ;:,-.:/0-.1, 08;:8:E10EB 1 D0 AI02=69, <17,7C0628 i D0 = SN . k k=1 1848? D/36- ;:,-.:/0-.1/ V3 (D0):

(3) 0 '(3) (18) i 2 V3 (D ) 'i (xj yj ) = ij i j = 1 2 : : : M3 >48 f(xj yj )gj=12:::M3 { ?0,@8-.1,, -,-.,,7C062,1 k D0 (..8. ?0,@8-.1, ;,-.:,800EB 1EF8 .,582 (xl yl ) 2 = k , l = 1 2 : : : 10, k = 1 2 : : : N ).

0/76.658-26 AI02=6G '(3) i ?,@0, 3/4/.C -784IGH6? ,D:/3,?. I-.C k , l = 1 2 : : : ni { .:8I>,7C0626, ./268, 5., (xi yi) 2 k . D,30/56? 58:83 'k (x y) 2ID658-2IG AI02=6G, ,;:8487800IG 1 .:8I>,7C0628 k , :/10IG 1 1 .,528 (xi yi) 6 0 1 ,-./7C0EB 1ED:/00EB 47< 60.8:;,7<=66 .,52/B (xj yj ) 2 k . /268 AI02=66 ?E ;,-.:,676 :/088, -?. (17). ,>4/ 8 > < 'k (x y) 8-76 (x y) 2 k (3) 'i (x y) = > n 8-76 (x y) 2= S k : 0 l

l

l

l

l

l

l

i

l=1

l

-<2/< g 2 V3(D0) :/-27/4E1/8.-< ;, D/36-I 63 AI02=69 f'(3) i (x y )gi=12:::M3 -784IGH6? ,D:/3,?: M3 X g(x y) = g(xi yi)'(3) i (x y ): i=1

89-.16.87C0,, 1 -67I (18) 6?88? 47< gb(x y) = P g(xi yi)'(3) i (x y ): gb(xj yj ) = i=1 g(xj yj ), j = 1 2 : : : M3 . 784,1/.87C0,, gb(x y) <17<8.-< 1 2/@4,? .:8I>,7C0628 k 2ID658-2,9 60.8:;,76:IGH89 AI02=689. -67I 2,::82.0,-.6 3/4/56 760890,9 60.8:;,7<=66 60.8:;,7<=6,00E9 ?0,>,5780 8460-.180080. 784,1/N .87C0,, g = gb 1 2/@4,? .:8I>,7C0628 k , 6, 30/56., 1 D0 = S k . k=1 /--?,.:6? 3/4/5I 0/B,@4806< ;:6D76@/GH89 AI02=66 Pf 2 V3(D0), -,1;/4/GH89 - AI02=689 f 1 .,52/B f(xj yj )gj=12:::M3 : Pf (xj yj ) = f (xj yj ) j = 1 2 : : : M3 : (19) #I02=6G Pf DI48? 6-2/.C 1 1648

Pf (x y) =

. .

M3

M3 X i=1

i'(3) i (x y )

!"#$ %C %C

x2. (',./ . +4 D,(

123

>48 2,JAA6=680.E i ,;:8487
Pf (x y) =

. .

M3 X i=1

f (xi yi)'(3) i (x y ):

!"#$ %C %C

124

,. . )(. 1. ,-./0,12/ 3/4/56 760890,9 60.8:;,7<=66 (). 8,:8?/ , 2,::82.0,-.6 . :84-./178068 :8F806< . 2. DI-7,17800,-.C D/36-/. =802/ ,.0,-6.87C0,9 ;,>:8F0,-.6 760890,9 2,?D60/=66 J78?80.,1 D/36-/ 58:83 ,.0,-6.87C0IG ;,>:8F0,-.C 2,JAA6=680.,1 J.,9 760890,9 2,?D60/=66. 3. 0.8:;,7<=6,00E9 ?0,>,5780 />:/0@/. >, 8460-.1800,-.C. =802/ 56-7/ /:6A?8.658-26B ,;8:/=69, 08,DB,46?,>, 47< 1E56-7806< 30/5806< 60.8:;,7<=6,00,>, ?0,>,5780/ />:/0@/ 1 .,528. 4. ;:84878068 :/3487800EB :/30,-.89. #,:?I7/ 47< 08;,-:84-.1800,>, ;:84-./17806< :/3487800,9 :/30,-.6 58:83 30/5806< AI02=66. 6??8.:650,-.C :/3487800,9 :/30,-.6 2/2 AI02=66 -1,6B /:>I?80.,1. 60890,-.C :/3487800,9 :/30,-.6 2/2 AI02=6,0/7/. 5. E56-78068 60.8:;,7<=6,00,>, ?0,>,5780/ />:/0@/, ;,-.:,800,>, ;, k .,52/?, 58:83 60.8:;,7<=6,00E9 ?0,>,5780 />:/0@/, ;,-.:,800E9 ;, k-1 .,528. 0.8:;,7<=6,00/< A,:?I7/ CG.,0/. 6. 7>,:6.? 1E56-7806< 2,JAA6=680.,1 60.8:;,7<=6,00,>, ?0,>,5780/ 1 A,:?8 CG.,0/. :>/063/=6< ;:,=8--/ 1E56-78069 6 B:/0806< ;:,?8@I.,50EB :83I7C./.,1. =802/ 56-7/ /:6A?8.658-26B ,;8:/=69. 7>,:6.? 1E56-7806< 30/5806< 60.8:;,7<=6,00,>, ?0,>,5780/ 1 A,:?8 CG.,0/ 1 .,528. =802/ 56-7/ /:6A?8.658-26B ,;8:/=69. 7. 0.8:;,7<=6< "416@IH6?6-<" ?0,>,5780/?6 (/7>,:6.? -416>/ ;:84-./17806< 60.8:;,7<=6,00,>, ?0,>,5780/ 1 A,:?8 CG.,0/). =802/ 56-7/ /:6A?8.658-26B ,;8:/=69. 8. =802/ ;,>:8F0,-.6 60.8:;,7<=6,00,9 A,:?I7E CG.,0/ - ;:84-./178068? ,-./.,50,>, 5780/ 1 1648 ;:,63184806< :/3487800,9 :/30,-.6 6 A62-6:,1/00,>, ?0,>,5780/. 9. =802/ ;,>:8F0,-.6 60.8:;,7<=6,00,9 A,:?I7E CG.,0/ - ;:84-./178068? ,-./.,50,>, 5780/ 1 1648 ;:,63184806< ;:,631,40,9 AI02=66 1 082,.,:,9 .,528 6 A62-6:,1/00,>, ?0,>,5780/.

. .

'%% "!E

'5 1

125

10. :84-./178068 :/3487800,9 :/30,-.6 58:83 ;:,631,40IG AI02=66. ;:84878068 :/3487800EB :/30,-.89 - 2:/.0E?6 I37/?6. 8;:8:E10,-.C :/3487800,9 :/30,-.6 2/2 AI02=66 -1,6B /:>I?80.,1. 11. ,-./0,12/ 3/4/56 60.8:;,7<=66 - 2:/.0E?6 I37/?6. 460-.1800,-.C 60.8:;,7<=6,00,>, ?0,>,5780/. 8,:8?/ , ;,-.:,8066 60.8:;,7<=6,00,>, ?0,>,5780/ 1 A,:?8 CG.,0/. 12. =802/ ;,>:8F0,-.6 60.8:;,7<=66 - 2:/.0E?6 I37/?6 - ;:84-./178068? ,-./.,50,>, 5780/ 1 1648 ;:,63184806< :/3487800,9 :/30,-.6 6 A62-6:,1/00,>, ?0,>,5780/. 13. 7>,:6.? 1E56-7806< 2,JAA6=680.,1 60.8:;,7<=6,00,>, ?0,>,5780/ 2:/.0E?6 I37/?6. :>/063/=6< ;:,=8--/ 1E56-78069 6 B:/0806< ;:,?8@I.,50EB :83I7C./.,1. =802/ 56-7/ /:6A?8.658-26B ,;8:/=69. 7>,:6.? 1E56-7806< 30/5806< 60.8:;,7<=6,00,>, ?0,>,5780/ - 2:/.0E?6 I37/?6 1 .,528. =802/ 56-7/ /:6A?8.658-26B ,;8:/=69. 14. 7>,:6.? 1E56-7806< 30/58069 ;:,631,40EB ?0,>,5780/ 1 A,:?8 CG.,0/ 1 .,528. =802/ 56-7/ /:6A?8.658-26B ,;8:/=69. 15. 0,>,5780E 8DEF81/. 822I:80.0/< A,:?I7/. 0/76.658-2/< A,:?/, 0I76 ?0,>,5780,1, J2-.:8?I?E. 8,:8?/ , ?0,>,57808, 0/6?8088 I27,0:8F0,-.6 60.8:;,7<=66 3/ -58. 1ED,:/ I37,1 60.8:;,7<=66. 0.8:;,7<=6< ;, 0I7,5780,1 8DEF81/. 8I7I5F/8?,-.C ,=8026 ;,>:8F0,-.6 1 :/--?/.:61/8?,? 27/--8 ,=80,2 ,-./.,50,>, 5780/. =802/ -063I 6 -18:BI ;,>:8F0,-.6 60.8:;,7<=66 ;, 0I7,5780,1 8DEF81/ 58:83 ;,>:8F0,-.C 0/67I5F8>, :/10,?8:0,>, ;:6D76@806< (D83 4,2/3/.87C-.1/). 8,:8?/ @82-,0/ (D83 4,2/3/.87C-.1/). 17. /37,@8068 ;, ?0,>,5780/? 8DEF81/ - 6-;,7C3,1/068? 46-2:8.0,>, -2/7<:0,>, ;:,63184806<. :.,>,0/7C0,-.C ?0,>,5780,1 8DEF81/ ;, J.,?I -2/7<:0,?I ;:,63184806G. 18. 7>,:6.? 1E56-7806< 2,JAA6=680.,1 :/37,@806< ;, ?0,>,5780/? 8DEF81/. :>/063/=6< ;:,=8--/ 1E56-78069 6 B:/0806< ;:,?8@I.,50EB :83I7C./.,1. =802/ 56-7/ /:6A?8.658-26B ,;8:/=69. 7>,:6.? 1E56-7806< 30/5806< /;;:,2-6?6:IGH8>, ?0,>,5780/ 1 .,528. =802/ 56-7/ /:6A?8.658-26B ,;8:/=69. 19. 8,:8?/ , -,1;/48066 60.8:;,7<=6,00,>, ?0,>,5780/, ;,-.:,8000,>, ;, 0I7,5780,1 8DEF81/, - /;;:,2-6?6:IGH6? ?0,>,5780,?, ;,7I5800E? :/37,@8068? ;, ?0,>,5780/? 8DEF81/ - 6-;,7C3,1/068? 46-2:8.0,>, -2/7<:0,>, ;:,63184806<.

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126

20. :86?IH8-.1/ 2I-,50,{?0,>,57800,9 /;;:,2-6?/=66. I-,50,{760890/< 60.8:;,7<=6<. =802/ ;,>:8F0,-.6. =802/ ;,>:8F0,-.6 58:83 ;,>:8F0,-.C 0/67I5F8>, :/10,?8:0,>, ;:6D76@806< 1 27/--8 08;:8:E10EB 7,?/0EB 76069. 21. I-,50,-760890/< /;;:,2-6?/=6< ;, ?8.,4I 0/6?80CF6B 21/4:/.,1. ,-.:,8068 D/36-/, ;,-./0,12/ 3/4/56, 1E56-78068 ?/.:6=E -6-.8?E. 22. =802/ ;,>:8F0,-.6 2I-,50,-760890,9 /;;:,2-6?/=66 ;, ?8.,4I 0/6?80CF6B 21/4:/.,1 58:83 ;,>:8F0,-.C 0/67I5F8>, :/10,?8:0,>, ;:6D76@806< 1 27/--8 08;:8:E10EB 7,?/0EB 76069. 23. ;;:,2-6?/=6< ?0,>,5780/?6 ;, ?8.,4I 0/6?80CF6B 21/4:/.,1 1 ;:,-.:/0-.18 -, -;8=6/7C0E? -2/7<:0E? ;:,631848068?. ,-.:,8068 D/36-/ 63 ?0,>,5780,1 8DEF81/, 8>, ,:.,>,0/7C0,-.C ,.0,-6.87C0, J.,>, -2/7<:0,>, ;:,63184806<, 1E56-78068 2,JAA6=680.,1 /;;:,2-6?6:IGH89 AI02=66. 24. 8,:8?/ , 2I-,50,-760890,9 60.8:;,7<=66 08>7/426B AI02=69 0/ -;8=6/7C0EB -8.2/B (D83 4,2/3/.87C-1/). ,2/3/.87C-.1, .8,:8?E 1 -7I5/8 AI02=66 "2,:80C 63 ?,4I7< x" 0/ ,.:8328 T;1 1]. 25. DH/< -B8?/ 2I-,50,9 60.8:;,7<=66 2ID658-26?6 ?0,>,5780/?6. 7>,:6.? 1E56-7806< 2,JAA6=680.,1 60.8:;,76:IGH6B 2ID658-26B ?0,>,5780,1. 26. I-,50/< 60.8:;,7<=6< 2ID658-26?6 ?0,>,5780/?6 K:?6./. ED,: ;/:/?8.:,1. :,-.89F68 -1,9-.1/ ;:6D76@/GH89 AI02=66. =802/ ;,>:8F0,-.6 (D83 4,2/3/.87C-.1/). 27. I-,50/< 60.8:;,7<=6< 2ID658-26?6 ?0,>,5780/?6 8--87<. ED,: ;/:/?8.:,1. :,-.89F68 -1,9-.1/ ;:6D76@/GH89 AI02=66. =802/ ;,>:8F0,-.6 (D83 4,2/3/.87C-.1/). 28. I-,50/< 60.8:;,7<=6< 2ID658-26?6 ?0,>,5780/?6 ?8.,4,? 26?E. ED,: ;/:/?8.:,1. :,-.89F68 -1,9-.1/ ;:6D76@/GH89 AI02=66. =802/ ;,>:8F0,-.6 (D83 4,2/3/.87C-.1/). 29. I-,50/< 60.8:;,7<=6< 2ID658-26?6 ?0,>,5780/?6 - 6-;,7C3,1/068? :/3487800EB :/30,-.89. ED,: ;/:/?8.:,1. :,-.89F68 -1,9-.1/ ;:6D76@/GH89 AI02=66. =802/ ;,>:8F0,-.6 (D83 4,2/3/.87C-.1/). 30. ;:84878068 -;7/90/. 0.8:;,7<=6< 2ID658-26?6 -;7/90/?6. ED,: ;/:/?8.:,1, ;,-.:,8068 ?/.:6=E -6-.8?E. :,-.89F68 -1,9-.1/ ?/.:6=E -6-.8?E 6 ;:6D76@/GH89 AI02=66. =802/ ;,>:8F0,-.6 (D83 4,2/3/.87C-.1/).

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!.!(. 1. .. /B1/7,1, .. 642,1, .. ,D87C2,1. 6-7800E8 ?8.,4E. ., "/I2/", 1987. 2. . 8 I:. :/2.658-2,8 :I2,1,4-.1, ;, -;7/90/?. ., "/46, 6 -1<3C", 1985. 3. $.. /1C<7,1, .. 1/-,1, .. 6:,F065802,. 8.,4E -;7/90{AI02=69. ., "/I2/", 1980. 4. .. .85260, $.. IDD,.60. ;7/90E 1 1E56-76.87C0,9 ?/.8?/.628. ., "/I2/", 1976.

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