Вычислительные методы для физиков. Часть 1: Аппроксимация функций, численное дифференцирование: Учебное пособие

FBGBKL?JKL...

Author: Курганский С.И. | Дубровский О.И. | Куркина Л.И.

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FBGBKL?JKL?J:PBB :JKL
Nbabq_kdbcnZdmevl_l DZn_^jZnbabdbl\_j^h]hl_eZ

U>EYNBABDH< Q:KLV :iijhdkbfZpbynmgdpbcqbke_ggh_^bnn_j_gpbjh\Zgb_

>eyklm^_glh\dmjkZki_pbZevghklb-NbabdZb dmjkZki_pbZevghklb-Fbdjhwe_dljhgbdZbiiijb[hju

KhklZ\bl_eb ijhnKBDmj]Zgkdbc ^hpHB>m[jh\kdbc ^hpEBDmjdbgZ

2

:IIJHDKBF:PBYNMGDPBC AZ^ZqZbgl_jiheypbb < nbabd_ fZl_fZlbd_ l_ogbd_ wdhghfbd_ qZklh ijboh^blky klZedb\Zlvky kh ke_^mxs_c aZ^Zq_c G_dhlhjZy nmgdpby y = f ( x ) aZ^ZgZ lZ[ebqgh l_ \ ^bkdj_lguo lhqdZo x0 , x1 , K , x n ba\_klgu khhl\_lkl\mxsb_ agZq_gbynmgdpbb f (x 0 ), f (x1 ), K , f (x n ) Lj_[m_lkyhij_^_eblvagZq_gb_wlhc nmgdpbb \ g_dhlhjuo lhqdZo x hlebqguo hl agZq_gbc Zj]mf_glZ nbdkbjh\Zgguo \ lZ[ebp_ Ih^h[gZy aZ^ZqZ \hagbdZ_l b \ g_kdhevdh bghc kblmZpbbGZdhfivxl_j_lj_[m_lky\uqbkeblvh^gmblm`_keh`gmxnmgdpbx y = f ( x ) \hq_gv[hevrhfqbke_lhq_dhlj_adZ [a , b] Ij_^eZ]Z_lky\uqbkeblv wlm nmgdpbx \ hlghkbl_evgh g_[hevrhf qbke_ lhq_d \u[jZgguo gZfb ih k\h_fmmkfhlj_gbxZagZq_gby\hklZevguolhqdZojZkkqblu\ZlvihdZdbf-lh ijhkluf nhjfmeZf bkihevamy bgnhjfZpbx h[ wlbo ba\_klguo agZq_gbyo Ih^h[gu_ ijZdlbq_kdb_ aZ^Zqb nhjfZebamxlky dZd fZl_fZlbq_kdZy aZ^ZqZ bgl_jihebjh\Zgbybgl_jiheypbb NhjfZevghwlZaZ^ZqZhij_^_ey_lkylZd. ImklvgZhlj_ad_ [a , b] ba\_klguagZq_gbyg_dhlhjhcnmgdpbb y = f ( x ) \ n + 1jZaebqguolhqdZo x 0 , x1 , K , x n yi = f (xi )

(i = 0, , K , n) .

Lj_[m_lky ijb[eb`_ggh k lhc beb bghc kl_i_gvx lhqghklb hij_^_eblv agZq_gb_nmgdpbb\lhqd_ x . Lhqdb x i gZau\Zxlky maeZfb bgl_jiheypbb < ^Zevg_cr_f [_a h]jZgbq_gbyh[sghklb[m^_fkqblZlv x i −1 < x i (i = 1, , K , n) . ?keb lhqdZ x \ dhlhjhc g_h[oh^bfh \uqbkeblv f ( x ) ijbgZ^e_`bl hlj_adm [ x 0 , x n ] lh ihklZ\e_ggZy aZ^ZqZ gZau\Z_lky kh[kl\_ggh aZ^Zq_c bgl_jiheypbb beb bgl_jiheypb_c \ madhf kfuke_ \ ijhlb\ghf kemqZ_ __ gZau\ZxllZd`_wdkljZiheypb_c Ijbbgl_jihebjh\Zgbb\hagbdZ_ljy^aZ^Zq \u[hjgZb[he__m^h[gh]h kihkh[Z ihkljh_gby bgl_jihebjmxs_c nmgdpbb hp_gdZ ih]j_rghklb ijb aZf_g_ f ( x ) bgl_jihebjmxs_c nmgdpb_c hilbfZevguc \u[hj maeh\ bgl_jiheypbb^eyihemq_gbyfbgbfZevghcih]j_rghklb Ebg_cgZybihebghfbZevgZybgl_jiheypby

3 JZkkfhljbfkgZqZeZh[sbcih^oh^dj_r_gbxaZ^Zqbbgl_jiheypbb< ijhp_kk_ j_r_gby wlhc aZ^Zqb kljhyl nmgdpbx ϕ (x) dhlhjZy \ lhqdZo x 0 , x1 , K , x n ijbgbfZ_lagZq_gby f (x 0 ), f (x1 ), K , f (x n ) GZwlhfhkgh\Zgbb kqblZxl qlh b \ hklZevguo lhqdZo hlj_adZ bgl_jiheypbb [a , b] ϕ (x) ijb[eb`_ggh ij_^klZ\ey_l f (x) < ^Zevg_cr_f m`_ \f_klh f (x) jZ[hlZxl k ϕ (x) =_hf_ljbq_kdb wlh hagZqZ_l ihkljh_gb_ djb\hc gZ iehkdhklb ijhoh^ys_c q_j_a lhqdb k dhhj^bgZlZfb (x 0 ; \ ) , (x1 ; \ ) , ..., (x n ; \ Q ) . H^gZdh m`_ ba ]_hf_ljbq_kdbo khh[jZ`_gbc ykgh qlh q_j_a ^Zggu_ lhqdb fh`gh ijh\_klb [_kqbke_ggh_ fgh`_kl\h jZaebqguo djb\uo jZaebqgufb kihkh[Zfb Kj_^b kihkh[h\ bgl_jihebjh\Zgby gZb[he__ jZkijhkljZg_gguf b mihlj_[bl_evguf gZ ijZdlbd_ y\ey_lky kihkh[ ebg_cgh]h bgl_jihebjh\Zgby dh]^Z bgl_jihebjmxsZy nmgdpby ϕ (x) bs_lky \ \b^_ ebg_cghc dhf[bgZpbb ba\_klguo\u[bjZ_fuogZfbnmgdpbc n

ϕ (x) = ∑ ai ϕ i (x) , i=0

]^_ ϕ i ( x ) - ba\_klgu_ nmgdpbb Dhwnnbpb_glu ai hij_^_eyxlky ba mkeh\by kh\iZ^_gby ϕ (x) k f (x) \maeZobgl_jiheypbb x0 , x1 , K , x n :

∑ ai ϕ i ( x j ) = f ( x j ) ( j = 0, , K , n) . n

i=0

Ihemqbeb kbkl_fm n + 1 ebg_cguo Ze]_[jZbq_kdbo mjZ\g_gbc gZ n + 1 dhwnnbpb_gl ai J_rb\ __ gZc^_f dhwnnbpb_glu ai b l_f kZfuf bgl_jihebjmxsmx nmgdpbx ϕ (x) LZdhckihkh[ijbdhlhjhfdhwnnbpb_glu ai hij_^_eyxlky g_ihkj_^kl\_gguf j_r_gb_f wlhc kbkl_fu gZau\Z_lky f_lh^hfg_hij_^_e_gguodhwnnbpb_glh\ GZb[he__ bamq_gguf b rbjhdh ijbf_gy_fuf gZ ijZdlbd_ deZkkhf bgl_jihebjmxsbonmgdpbcy\ey_lkyfgh`_kl\hZe]_[jZbq_kdbofgh]hqe_gh\
ϕ i (x ) = x i

(i = 0, , K , n) .

Fgh]hqe_gubf_xlhq_\b^gu_^hklhbgkl\Z-boe_]dh\uqbkeylvkdeZ^u\Zlv i_j_fgh`Zlv bgl_]jbjh\Zlv ^bnn_j_gpbjh\Zlv b li Ijbf_g_gb_ fgh]hqe_gh\ \ l_hjbb bgl_jihebjh\Zgby hkgh\Zgh gZ ZiijhdkbfZpbhgghc l_hj_f_<_c_jrljZkkZ L_hj_fZ ([_a ^hdZaZl_evkl\Z ?keb f - g_ij_ju\gZy gZ dhg_qghf aZfdgmlhf hlj_ad_ [a , b] nmgdpby lh ^ey ex[h]h ε > 0 kms_kl\m_l ihebghf p k (x ) kl_i_gb k lZdhcqlh

4 max f (x ) − p k (x ) < ε .

x ∈[ a , b ]

BlZd bgl_jihebjmxsZy nmgdpby ϕ (x) aZibku\Z_lky \ \b^_ fgh]hqe_gZ p n ( x ) kl_i_gb n n

ϕ ( x ) = pn ( x ) = ∑ ai x i = a 0 + a1 x + a 2 x 2 +K + a n x n , i=0

b ^ey hij_^_e_gby g_ba\_klguo dhwnnbpb_glh\ ai g_h[oh^bfh j_rblv kbkl_fmebg_cguoZe]_[jZbq_kdbomjZ\g_gbc

∑ ai x ij = f ( x j ) ( j = 0, , K , n) . n

i=0

Fh`gh^hdZaZlvqlhhij_^_ebl_evwlhckbkl_fuhij_^_ebl_ev
( )

det x ij = ∏ ∏ i> j

( xi − x j ) .

< kbem gZrbo ij_^iheh`_gbc x i −1 < x i hg hlebq_g hl gmey Ke_^h\Zl_evgh ^ZggZy kbkl_fZ bf__l j_r_gb_ b wlh j_r_gb_ _^bgkl\_ggh_ L_f kZfuf ^hdZau\Z_lky kms_kl\h\Zgb_ b _^bgkl\_gghklv bgl_jiheypbhggh]h fgh]hqe_gZ LZdbf h[jZahf aZ^ZqZ ihkljh_gby bgl_jiheypbhggh]h ihebghfZ kl_i_gb n ih n + 1 maemy\ey_lkyh^ghagZqghc H^gZdhg_ihkj_^kl\_ggh_gZoh`^_gb_dhwnnbpb_glh\ ai j_r_gb_fwlhc kbkl_fu ^Z`_ ^ey g_[hevrbo n ij_^klZ\ey_l ljm^gmx \ l_ogbq_kdhf hlghr_gbb aZ^Zqm Ijh[e_fZ aZdexqZ_lky \ lhf qlh kbkl_fZ iehoh h[mkeh\e_gZqlhijb\h^blddZlZkljhnbq_kdhfmbkdZ`_gbxdhwnnbpb_glh\ ai \uqbkebl_evghc ih]j_rghklvx Ih wlhc ijbqbg_ h[uqgh ijbf_gyxl ^jm]b_ nhjfuaZibkbbgl_jiheypbhggh]hfgh]hqe_gZbkihkh[u_]hihkljh_gby Bgl_jiheypbhggucfgh]hqe_gEZ]jZg`Z ;m^_f bkdZlv bgl_jiheypbhgguc fgh]hqe_g \ \b^_ ebg_cghc dhf[bgZpbbfgh]hqe_gh\kl_i_gb n : n

pn ( x ) = y 0 l0 ( x ) + y1l1 (x )+K + y n ln ( x ) = ∑ yi li (x ) . i =0

Ijbwlhfihlj_[m_fqlh[udZ`^ucfgh]hqe_g li (x) h[jZsZeky\gmev\h \k_o maeZo bgl_jiheypbb aZ bkdexq_gb_f h^gh]h i -]h ]^_ hg ^he`_g jZ\gylvky_^bgbp_l_

5 0, _ k ebi ≠ j = δ ij . li x j =  1, _ k ebi = j

( )

LZd dZd li (x) - fgh]hqe_g kl_i_gb n h[jZsZxsbcky \ gmev \ lhqdZo x 0 , x1 , K , x i −1 , x i +1 , K , x n lhhgfh`_l[ulvaZibkZg\\b^_ li (x) = A(x − x 0 )(x − x1 )K (x − xi −1 )(x − xi +1 )K (x − x n ) .

>eyhij_^_e_gbydhgklZglu A mql_fqlh li (xi ) = 1l_

A( xi − x 0 )( xi − x1 )K ( xi − xi −1 )( xi − xi +1 )K ( xi − x n ) = 1 .

Hlkx^ZgZoh^bf A bhdhgqZl_evghihemqZ_f

(x − x 0 )(x − x1 )K (x − xi −1 )(x − xi +1 )K (x − x n ) . (xi − x 0 )(xi − x1 )K (xi − xi −1 )(xi − xi +1 )K (xi − x n )

li ( x ) =

LZdbfh[jZahf n

pn ( x ) = ∑ yi i =0

(x − x 0 )(x − x1 )K (x − xi −1 )(x − xi +1 )K (x − x n ) . (xi − x 0 )(xi − x1 )K (xi − xi −1 )(xi − xi +1 )K (xi − x n )

G_ljm^gh \b^_lv qlh agZq_gby wlh]h fgh]hqe_gZ \ maeZo bgl_jiheypbb kh\iZ^ZxlkaZ^ZggufbagZq_gbyfbnmgdpbb>_ckl\bl_evgh

( )

n

( )

n

pn x j = ∑ yi li x j = ∑ yi δ ij = y j . i =0

i =0

Wlm nhjfm aZibkb bgl_jiheypbhggh]h bgl_jiheypbhgguffgh]hqe_ghfEZ]jZg`Z

fgh]hqe_gZ

gZau\Zxl

Bgl_jiheypbhggucfgh]hqe_gEZ]jZg`Z^eyjZ\ghhlklhysbomaeh\ JZkkfhljbfkemqZcdh]^ZagZq_gby xi y\eyxlkyjZ\ghhlklhysbfbl_ x1 − x 0 = x 2 − x1 = K x n − x n −1 = h .

x − x0 = t lhihemqbf h (x − x 0 )(x − x1 )K (x − xi −1 )(x − xi +1 )K (x − x n )

?keb\\_klbh[hagZq_gb_ li ( x ) =

( xi

=

− x 0 )(x i − x1 )K ( xi − xi −1 )( xi − x i +1 )K ( xi − x n )

[

][

]

th(th − h)K th − (i − 1)h th − (i + 1)h K (th − nh)

[

ih(i − 1)hK h(− h)K −(n − i )h

]

=

=

6 =

t (t − 1)K (t − n) (−1)

(t − i )

n −i

i !(n − i )!

= (−1)

n

t(t − 1)K (t − n) n!

Cni . (−1) t −i i

BlZd\kemqZ_jZ\ghhlklhysbomaeh\bgl_jiheypbhggucfgh]hqe_gEZ]jZg`Z bf__l\b^ pn ( x ) = pn ( x 0 + th) = (−1)

n

t (t − 1)K (t − n) n!

Cni ∑ (−1) t − i yi . i =0 n

i

HklZlhqgucqe_gbgl_jiheypbhgghcnhjfmeuEZ]jZg`Z ?keb \k_ \uqbke_gby ijh\_^_gu lhqgh lhbgl_jiheypbhggucihebghf EZ]jZg`Z pn (x ) \ maeZo bgl_jiheypbb \ lhqghklb kh\iZ^Z_l k aZ^Zggufb agZq_gbyfbnmgdpbb f (x ) :kh\iZ^Zxleb pn (x ) b f (x ) \hklZevguolhqdZo hlj_adZ bgl_jihebjh\Zgby" ?keb kZfZ f (x ) y\ey_lky Ze]_[jZbq_kdbf fgh]hqe_ghf kl_i_gb g_ \ur_ n lh bf__l f_klh lh`^_kl\_ggh_ kh\iZ^_gb_ l_ pn ( x ) = f (x ) \h \k_o lhqdZo < ijhlb\ghf kemqZ_ \ lhqdZo hlebqguo hl maeh\bgl_jihebjh\ZgbyjZaghklv f (x ) − pn (x ) hlebqgZhlgmeyWlZjZaghklv _klv ih]j_rghklv bgl_jiheypbb b gZau\Z_lky hklZlhqguf qe_ghf bgl_jiheypbhgghcnhjfmeu?_g_h[oh^bfhhp_gblv >ey hp_gdb ih]j_rghklb bgl_jiheypbb fu ^he`gu kmablv deZkk bgl_jihebjm_fuonmgdpbclZddZdijhba\hevgZynmgdpbykh\iZ^Zyk f (x ) \ maeZobgl_jiheypbbfh`_ldZdm]h^ghhlebqZlvkyhlg__\hklZevguolhqdZo GZeh`bf gZ f (x ) ke_^mxsb_ h]jZgbq_gby ;m^_f kqblZlv qlh bgl_jihebjm_fZynmgdpby f (x ) h[eZ^Z_lgZhlj_ad_bgl_jihebjh\Zgby [a , b] g_ij_ju\gufb ijhba\h^gufb ^h ihjy^dZ n \dexqbl_evgh b kms_kl\m_l n+1 f ( ) x gZ a , b LZdb_h]jZgbq_gby\uihegyxlky^ey[hevrbgkl\ZkemqZ_\

()

[

]

kdhlhjufbijboh^blkyklZedb\ZlvkygZijZdlbd_Fh`gh^hdZaZlvqlh\wlhf kemqZ_ n +1 f ( ) (ξ ) f ( x ) − pn ( x ) = (x − x 0 )(x − x1 )K (x − x n ) , (n + 1)!

]^_ ξ ∈[a , b] ?kebfZdkbfZevgh_agZq_gb_wlhcijhba\h^ghcgZhlj_ad_ [a , b] n +1 jZ\gh max f ( ) ( x ) = M n +1 lh x ∈[ a , b ] f ( x ) − pn ( x ) =

M n +1 (x − x 0 )(x − x1 )K (x − x n ) . (n + 1)!

7 Wlb ^\Z \ujZ`_gby fh]ml kem`blv hp_gdhc hldehg_gby pn (x ) hl f (x ) _keb n+1 ijhba\h^gZy f ( ) x fh`_l[ulvhp_g_gZ

()

8 JZa^_e_ggu_jZaghklbbbok\hckl\Z NhjfZfgh]hqe_gh\EZ]jZg`ZbaysgZh^gZdhkms_kl\mxl^jm]b_[he__ wnn_dlb\gu_ ih qbkem hi_jZpbc nhjfu aZibkb bgl_jiheypbhggh]h fgh]hqe_gZ Ij_`^_ q_f i_j_clbdbojZkkfhlj_gbx\\_^_fgh\h_ihgylb_jZa^_e_ggu_jZaghklb Imklv fu bf__f g_dhlhjmx ihke_^h\Zl_evghklv maeh\ bgl_jiheypbb x 0 , x1 , K , x n nmgdpbb f (x ) >ey wlhc nmgdpbb b maeh\ \uqbkebf \k_\hafh`gu_hlghr_gby f ( x1 ) − f ( x 0 ) x1 − x 0 f ( x 2 ) − f ( x1 ) x 2 − x1

= f ( x 0 ; x1 ) ; = f ( x1 ; x 2 ) ;

K

f ( x n ) − f ( x n −1 ) x n − x n −1

= f (x n −1 ; x n ) .

LZdb_ hlghr_gby gZau\Zxl jZa^_e_ggufb jZaghklyfb i_j\h]h ihjy^dZ Ihemqb\bofufh`_fihkljhblvjZa^_e_ggu_jZaghklb\lhjh]hihjy^dZ f ( x1 ; x 2 ) − f ( x 0 ; x1 ) x2 − x0 f ( x 2 ; x 3 ) − f ( x1 ; x 2 ) x 3 − x1

= f ( x 0 ; x1 ; x 2 ) ; = f ( x1 ; x 2 ; x 3 ) ;

K

f ( x n −1 ; x n ) − f ( x n − 2 ; x n −1 ) xn − xn− 2

= f ( x n − 2 ; x n −1 ; x n ) .

k -]h
(

)

ihjy^dZhij_^_eyxlkyihnhjfme_

(

) (

f x j ; x j +1 ; K ; x j + k − f x j −1 ; x j ; K ; x j + k −1 x j + k − x j −1

)= f

( x j −1 ; x j ;K ; x j + k ) .

H[uqgh jZa^_e_ggu_ jZaghklb jZkiheZ]Zxl \ \b^_ lZ[ebpu ke_^mxsbf h[jZahf

9 x0

f (x 0 )

x1

f ( x1 )

x2

f (x 2 )

x3

f (x 3 )

x4

f (x 4 )

f ( x 0 ; x1 )

f ( x 0 ; x1 ; x 2 )

f ( x1 ; x 2 ) f (x2 ; x 3 ) f (x 3 ; x 4 )

f ( x1 ; x 2 ; x 3 ) f (x 2 ; x 3 ; x 4 )

f ( x 0 ; x1 ; x 2 ; x 3 )

f ( x 0 ; x1 ; x 2 ; x 3 ; x 4 )

f ( x1 ; x 2 ; x 3 ; x 4 )

Fh`gh ^hdZaZlv qlh jZa^_e_ggZy jZaghklv k -]h ihjy^dZ jZ\gZ j+k f (xi ) f x j ; x j +1 ; K ; x j + k = ∑ . − − − − − x x x x x x x x x x K K ( )( ) i= j i j i j +1 i i −1 i i +1 i j+k

(

)

(

)(

)

(

)

Bgl_jiheypbhggucfgh]hqe_gGvxlhgZ^eyg_jZ\guoijhf_`mldh\ Imklv x 0 , x1 , K , x n - maeu bgl_jiheypbb nmgdpbb f (x ) Z p k (x) bgl_jiheypbhgguc fgh]hqe_g EZ]jZg`Z ihkljh_gguc ^ey wlhc nmgdpbb ih maeZf x 0 , x1 , K , x k Lh]^Z

[

] [

]

[

]

p n (x) = p 0 (x) + p1 (x) − p 0 (x) + p 2 (x) − p1 (x) +K + p n (x) − p n−1 (x) . JZkkfhljbf ex[mx jZaghklv klhysmx \ ijZ\hc qZklb p k (x) − p k −1 (x) Wlh fgh]hqe_gkl_i_gb k Hgh[jZsZ_lky\gmev\lhqdZo x 0 , x1 , K , x k −1 Ihwlhfm p k ( x ) − p k −1 ( x ) = A(x − x 0 )( x − x1 )K ( x − x k −1 ) ,

]^_ A - ihklhyggZy >ey __ hij_^_e_gby \ ihke_^g_f jZ\_gkl\_ iheh`bf x = x k Ijbwlhfihemqbf f (x k ) − p k −1 ( x k ) = A( x k − x 0 )( x k − x1 )K ( x k − x k −1 ) .

Hlkx^ZgZoh^bf A=

(x k

f (x k )

− x 0 )(x k − x1 )K (x k − x k −1 )

−

(x k − x 0 )(x k − x1 )K (x k − x i −1 )(x k − x i +1 )K (x k − x k −1 ) ∑ f (xi ) (x − x )(x − x )K (x − x )(x − x )K (x − x ) i i i i −1 i i +1 i k −1 0 1 − i=0 = (x k − x 0 )(x k − x1 )K (x k − x k −1 ) k −1

10 f ( xi )

k

=∑

i = 0 ( xi

− x 0 )( xi − x1 )K ( xi − xi −1 )( xi − xi +1 )K ( xi − x k )

= f ( x 0 ; x1 ; K ; x k )

bke_^h\Zl_evgh

p k ( x ) − p k −1 ( x ) = (x − x 0 )( x − x1 )K ( x − x k −1 ) f ( x 0 ; x1 ;K ; x k ) .

LZdbfh[jZahf

p n ( x ) = f ( x 0 ) + ( x − x 0 ) f (x 0 ; x1 ) + ( x − x 0 )(x − x1 ) f (x 0 ; x1 ; x 2 ) + +K +( x − x 0 )( x − x1 )K ( x − x n −1 ) f (x 0 ; x1 ; K ; x n ) .

WlZ nhjfZ aZibkb bgl_jiheypbhggh]h fgh]hqe_gZ gZau\Z_lky bgl_jiheypbhgguf fgh]hqe_ghf GvxlhgZ ^ey g_jZ\guo ijhf_`mldh\ ^ey bgl_jihebjh\Zgby \i_j_^ HgZ [he__ m^h[gZ ^ey ijZdlbq_kdbo jZkq_lh\ q_f nhjfmeZEZ]jZg`Zblj_[m_lf_gvr_]hqbkeZZjbnf_lbq_kdbohi_jZpbcWlhl fgh]hqe_gbkihevam_llhevdh\_jogxxkljhdmlZ[ebpujZa^_e_gguojZaghkl_c ohly ^ey \uqbke_gby we_f_glh\ wlhc kljhdb g_h[oh^bfu b \k_ hklZevgu_ we_f_glulZ[ebpu qlhiha\hey_lwdhghfblvhi_jZlb\gmxiZfylvdhfivxl_jZ :gZeh]bqgh g_keh`gh ihemqblv fgh]hqe_g GvxlhgZ ^ey bgl_jihebjh\ZgbygZaZ^ p n ( x ) = f ( x n ) + ( x − x n ) f (x n −1 ; x n ) + ( x − x n )(x − x n −1 ) f (x n − 2 ; x n −1 ; x n ) + +K +( x − x n )( x − x n −1 )K ( x − x1 ) f (x 0 ; x1 ; K ; x n ) ,

bkihevamxsbcebrvgb`gxxkljhdmlZ[ebpujZa^_e_gguojZaghkl_cH[_wlb nhjfmeuih\k_fiZjZf_ljZfkh\_jr_gghjZ\ghp_ggu Bgl_jiheypbhggZyko_fZWcld_gZ ?_ ijbf_gyxl lh]^Z dh]^Z lj_[m_lky gZclb g_ h[s__ ZgZeblbq_kdh_ \ujZ`_gb_ ^ey p n ( x ) Z ebrv _]h agZq_gb_ ijb dhgdj_lghf agZq_gbb x Ih wlhcko_f_agZq_gb_bgl_jiheypbhggh]hfgh]hqe_gZ^eydZdh]h-lhagZq_gby x gZoh^blky iml_f ihke_^h\Zl_evgh]h ijbf_g_gby _^bghh[jZagh]h ijhp_kkZ JZkkfhljbf\ujZ`_gb_ p01 (x) =

y0 y1

x0 − x x1 − x

x1 − x 0

.

Wlhfgh]hqe_gi_j\hckl_i_gbhlghkbl_evgh x ?]hagZq_gb_ijb x = x 0 p01 (x 0 ) =

y0 y1

0 x1 − x 0 x1 − x 0

= y0 .

11 LhqghlZd`_ijb x = x1 p01 (x1 ) = y1 LZddZdbgl_jiheypbhggucfgh]hqe_g i_j\hc kl_i_gb ijbgbfZxsbc \ lhqdZo x 0 b x1 agZq_gby y 0 b y1 _^bgkl\_gguc lh p01 ( x ) b y\ey_lky bgl_jiheypbhgguf fgh]hqe_ghf ihkljh_ggufih^\mfwlbflhqdZfLhqghlZd`_fufh`_f\uqbkeblv p 12 ( x ) , p 23 ( x ) bl^khhl\_lkl\_gghihlhqdZf x1 b x 2 , x 2 b x 3 bl^Wlb\ujZ`_gby e_]dh\uqbkeyxlkygZW
p01 ( x ) x 0 − x p12 ( x ) x 2 − x x 2 − x0

.

Wlh-fgh]hqe_g\lhjhckl_i_gbhlghkbl_evgh x Ijyfhcih^klZgh\dhce_]dh m[_^blvkyqlh_]hagZq_gby\lhqdZo x 0 , x1 b x 2 jZ\gukhhl\_lkl\_ggh y 0 , y1 b y 2 Ke_^h\Zl_evgh p012 ( x ) y\ey_lky bgl_jiheypbhgguf fgh]hqe_ghf ihkljh_ggufihlj_fwlbflhqdZf
p01K k −1 ( x ) x 0 − x 1 x k − x 0 p12K k (x ) x k − x

[m^_l bgl_jiheypbhgguf fgh]hqe_ghf EZ]jZg`Z ijbgbfZxsbf \ lhqdZo x 0 , x1 , K , x k khhl\_lkl\_ggh agZq_gby y 0 , \ 1 , K , \ k qlh we_f_glZjgh ^hdZau\Z_lky ih f_lh^m fZl_fZlbq_kdhc bg^mdpbb Ijb wlhf ihjy^hd b gmf_jZpby maeh\ agZq_gby g_ bf_xl Bgl_jiheypbhgguc ijhp_kk Wcld_gZ oZjZdl_jbam_lky k\hbf _^bghh[jZab_f b e_]dh j_Zebam_lky gZ dhfivxl_j_ DZ`^ucfgh]hqe_g p012K k (x ) ihemqZ_lkyba p01K k −1 (x ) b p12K k ( x ) lhqghlZd `_dZdb p01 ( x ) ihemqZ_lkyba y 0 b y1 . H^gZdh ijb kjZ\g_gbb wlhc ko_fu k bgl_jiheypbhgguf fgh]hqe_ghf GvxlhgZfh`ghhlf_lblvqlhijb\uqbke_gbbihke_^g_]hlj_[m_lkyf_gvr__ qbkehZjbnf_lbq_kdbohi_jZpbc
ω n ( x ) = ( x − x 0 )(x − x1 )K (x − x n ) .

DZd fu \b^_eb jZg__ hldehg_gb_ pn (x ) hl f (x ) hij_^_ey_lky n+1 n+1 \_ebqbgZfb f ( ) (ξ ) b ω n ( x ) ?keb h \_ebqbg_ f ( ) (ξ ) fu fh`_f kdZaZlv ebrv \ dZdbo ij_^_eZo hgZ aZdexq_gZ lh \_ebqbgm ω n ( x ) \ g_dhlhjuo kemqZyofufh`_fbaf_gylvihgZr_fm`_eZgbxbaf_gyymaeubgl_jiheypbb

12 x i IhklZ\bf ke_^mxsmx aZ^Zqm DZd gm`gh \u[jZlv maeu bgl_jiheypbb x i

^ey lh]h qlh[u max ω n ( x ) [ue [u gZbf_gvrbf Ykgh qlh ^ey lZdh]h x ∈[ a , b ] \u[hjZ maeh\ gZbf_gvr_c [m^_l b ih]j_rghklv bgl_jiheypbb >ey hl\_lZ gZ wlhl\hijhkjZkkfhljbffgh]hqe_guQ_[ur_\Z Fgh]hqe_gQ_[ur_\Z Tn (x ) hij_^_ey_lkylZd Tn (x ) = cos[n arccos x ] ,

Ijb n = 0

x ≤1 .

T0 (x ) = cos 0 = 1 .

Ijb n = 1

T1 (x ) = cos(arccos x ) = x .

Ijb n = 2

T2 ( x ) = cos[2 arccos x ] = 2 cos 2 (arccos x ) − 1 = 2 x 2 − 1 .

>Ze__balh`^_kl\Z

[

]

[

]

cos (n + 1)θ = 2 cos θ cos(nθ ) − cos (n − 1)θ ,

iheZ]Zy θ = arccos x ihemqbf j_dmjj_glgh_ khhlghr_gb_ ^ey fgh]hqe_gh\ Q_[ur_\Z Tn +1 ( x ) = 2xTn +1 ( x ) − Tn −1 ( x ) .

LZdbf h[jZahf Tn (x ) ^_ckl\bl_evgh y\eyxlky fgh]hqe_gZfb kl_i_gb n Ba j_dmjj_glghcnhjfmeuihke_^h\Zl_evghgZoh^bf T3 ( x ) = 4 x 3 − 3x , T4 ( x ) = 8x 4 − 8x 2 + 1 , T5 ( x ) = 16x 5 − 20x 3 + 5x , ............. Hlf_lbf g_kdhevdh ihqlb hq_\b^guo k\hckl\ fgh]hqe_gh\ Q_[ur_\Z dhlhju_fu[m^_fbkihevah\Zlv\^Zevg_crbo\udeZ^dZo Dhwnnbpb_glijbklZjr_ckl_i_gb x jZ\_g 2 n−1 .

2) Tn ( x ) dZd fgh]hqe_g kl_i_gb n bf__l jh\gh n dhjg_c GZc^_f bo ba mjZ\g_gby cos[n arccos x ] = 0 .

Hlkx^Z

(2i + 1)π π . (2i + 1) beb xi = cos 2n 2 >Z\Zy i agZq_gby 0, 1, K n − 1 ihemqbf n jZaebqguo dhjg_c fgh]hqe_gZ Tn ( x )
13 AZf_lbf lZd`_ qlh max Tn (x) gZ hlj_ad_ [−1, ] jZ\_g b ^hklb]Z_lky \ mπ (m = 0, 1, K n) . n + 1lhqd_ x m = cos n Imklv l_i_jv \ dZq_kl\_ hlj_adZ bgl_jihebjh\Zgby [a , b] fu bf__f hlj_ahd [−1, ]
(2i + 1)π 2 (n + 1)

(i = 0, 1, K n) .

Lh]^Z

ω n (x) = (x − x 0 )(x − x1 )K (x − x n ) =

Tn+1 (x) 2n

.

Ke_^h\Zl_evgh 1 max ω n (x) = n . x∈[ −1, ] 2 Fh`gh ^hdZaZlv qlh ijb ex[hf ^jm]hf \u[hj_ maeh\ bgl_jiheypbb wlZ \_ebqbgZfh`_llhevdh\hajZklb

LZdbf h[jZahf _keb h]jZgbqblvky hlj_adhf [−1, ] lh ω n ( x ) [m^_l bf_lv gZbf_gvr__ \hafh`gh_ agZq_gb_ k\h_]h fZdkbfmfZ ijb mkeh\bb qlh \ dZq_kl\_ maeh\ bgl_jihebjh\Zgby \aylu dhjgb fgh]hqe_gZ Q_[ur_\Z < wlhf kemqZ_hp_gdZih]j_rghklbbgl_jihebjh\Zgbyijbh[j_lZ_l\b^ M f (x) − p n (x) = n n +1 . 2 (n + 1)!

?keb bgl_jihebjh\Zgb_ nmgdpbb f (x ) ijhba\h^blky gZ ijhba\hevghf hlj_ad_ [a , b] _]h fh`gh i_j_\_klb \ hlj_ahd [−1, ] ebg_cghc aZf_ghc i_j_f_gghc 1 x = (b − a )x '+(b + a ) , 2 1 x' = [2 x − b − a ] . b−a Hp_gdZih]j_rghklb^eywlh]hkemqZylZdh\Z

[

]

M n +1 (b − a ) f (x ) − p n ( x ) = (n + 1)! 2 2n+1

n +1

KieZcg-bgl_jiheypby

.

14 ?_ gZau\Zxl lZd`_ dmkhqgh-fgh]hqe_gghc ZiijhdkbfZpb_c Kmlv kieZcg-bgl_jiheypbb aZdexqZ_lky \ ke_^mxs_f Hlj_ahd bgl_jihebjh\Zgby [a, b] lhqdZfb x 0 = a, x1 , K , x n = b jZa[b\Z_lky gZ n hlj_adh\ [ x i −1 , x i ] (i = 1, 2, K , Q ) ijbq_f dZd h[uqgh ba\_klgu agZq_gby yi = f (x i ) (i = 0, , K , Q ) .

GZah\_f kieZcghf S m ( x ) ihjy^dZ m nmgdpbx y\eyxsmxky fgh]hqe_ghfkl_i_gb m gZdZ`^hfbahlj_adh\ [ x i −1 , x i ] S m (x ) = pim ( x ) = ci 0 + ci1 x + ci 2 x 2 +K + cim x m

(x i −1 ≤ x ≤ x i ) ,

ijbgbfZxsmxagZq_gby y i \maeZobgl_jiheypbbl_ pim (x i −1 ) = y i −1 , pim ( x i ) = y i ,

(i = 1, 2, K , n)

bh[eZ^Zxsmxg_ij_ju\gufbijhba\h^gufb^hihjy^dZ m − 1 \dexqbl_evgh \h\k_o\gmlj_ggbomaeZobgl_jiheypbb (k ) (k ) pim ( x i ) = pi +1,m (x i ) , (i = 1, 2, K , n − 1) ,

(k = 1, 2, K , m − 1) . IhkljhblvkieZcg–agZqblhij_^_eblv\k_dhwnnbpb_glu cij gZ\k_o n

hlj_adZo
dhlhjuf ^he`gu m^h\e_l\hjylv fgh]hqe_gu pim (x ) ih kms_kl\m y\eyxlky mjZ\g_gbyfb ^ey hij_^_e_gby wlbo dhwnnbpb_glh\ _eh \ lhf qlh dmkhqguc fgh]hqe_g lj_lv_c kl_i_gb ij_^klZ\ey_l kh[hc fZl_fZlbq_kdmx fh^_ev ]b[dh]h lhgdh]h kl_j`gy ba mijm]h]h fZl_jbZeZ ?keb lZdhc kl_j`_gv aZdj_iblv \ ^\mo khk_^gbo maeZo bgl_jiheypbb k aZ^Zggufb m]eZfb gZdehgh\ lh f_`^m lhqdZfb aZdj_ie_gby kl_j`_gv ijbf_l nhjfm fbgbfbabjmxsmx _]h ihl_gpbZevgmx wg_j]bx Imklv nhjfZ kl_j`gy hibku\Z_lky nmgdpb_c S ( x ) Ba nbabdb ba\_klgh qlh 4 mjZ\g_gb_ k\h[h^gh]h jZ\gh\_kby bf__l \b^ S ( ) x = 0 Hlkx^Z ke_^m_l qlh

() f_`^m dZ`^hc iZjhc khk_^gbo maeh\ nmgdpby S ( x ) y\ey_lky fgh]hqe_ghf

lj_lv_c kl_i_gb < ^Zevg_cr_f fu [m^_f jZkkfZljb\Zlv lhevdh kieZcgu lj_lv_]hihjy^dZihwlhfmbg^_dk[m^_fhimkdZlv

Imklv i_j\Zy ijhba\h^gZy nmgdpbb S ( x ) \ lhqdZo x 0 , x1 , K , x n jZ\gZ khhl\_lkl\_ggh m0 , m1 , K , mn gZdehgukieZcgZ Lh]^ZgZhlj_ad_ [ x i −1 , x i ]

15 fgh]hqe_glj_lv_ckl_i_gb pi (x ) \lhqdZo xi −1 , x i ijbgbfZ_lagZq_gbyjZ\gu_ khhl\_lkl\_ggh yi −1 , yi b bf__l \ wlbo lhqdZo ijhba\h^gmx khhl\_lkl\_ggh jZ\gmx mi −1 , mi Fh`gh^hdZaZlvqlhlZdhcfgh]hqe_g^he`_g[ulvaZibkZg\ \b^_ pi (x ) =

(x i

[

− x ) 2( x − x i −1 ) + hi 2

hi3 +

(x i

− x ) ( x − xi −1 )

]y

i −1

2

hi2

mi −1 −

(x − xi −1 )2 [2(xi

+

− x ) + hi

hi3

(x − xi −1 )2 (xi

− x)

hi2

]y

i

+

mi ,

]^_ hi = x i − x i −1 -^ebgZ i -]hhlj_adZ

BlZdqlh[uihkljhblvkieZcggZ\k_fhlj_ad_ [a , b] gm`ghhij_^_eblv _]h gZdehgu mi \h \k_o maeZo bgl_jiheypbb Ijb wlhf Z\lhfZlbq_kdb h[_ki_qb\Z_lky g_ij_ju\ghklv i_j\hc ijhba\h^ghc kieZcgZ \h \k_o lhqdZo >ey hlukdZgby \_ebqbg mi bkihevam_f mkeh\b_ g_ij_ju\ghklb \lhjhc ijhba\h^ghc \h \gmlj_ggbo maeZo bgl_jiheypbb Ijh^bnn_j_gpbjh\Z\ ^\Z`^uihke_^g__jZ\_gkl\hihemqbf pi'' (x ) =

[

2 2( x − x i −1 ) − 4(xi − x ) + hi

+

[

hi3

]y

]m

2 (x − xi −1 ) − 2( xi − x ) hi2

i −1

i −1

+

+

[

2 2(xi − x ) − 4( x − x i −1 ) + hi hi3

[

]y

i

+

]m

2 2(x − xi −1 ) − ( xi − x ) hi2

i

IjbjZ\gb\Z_f\dZ`^hf\gmlj_gg_fmae_bgl_jiheypbb xi (i = 1, 2, K , n − 1) agZq_gby\lhjuoijhba\h^guo\uqbke_gguo\e_\hfbijZ\hfhlmaeZhlj_adZo pi'' (x i ) = pi''+1 ( xi ) ,

(i = 1, 2, K , n − 1) .

Ih^klZ\b\ kx^Z y\guc \b^ \lhjhc ijhba\h^ghc ihke_ ijhkluo ij_h[jZah\ZgbcgZoh^bf  yi − yi −1 yi +1 − yi  1 1 1  1   , + = + mi −1 + 2 + m m 3  i i +1 2 2 hi hi +1  hi hi +1  h h   i i +1

(i = 1, 2, K , n − 1) . Fu ihemqbeb kbkl_fm n − 1 ebg_cguo Ze]_[jZbq_kdbo mjZ\g_gbc k n + 1 g_ba\_klguf mi (i = 0, 1, K , n) G_^hklZxsb_ ^\Z mjZ\g_gby ihemqZxl ba djZ_\uomkeh\bc>Z^bfljb\ZjbZglZlZdbomkeh\bc ?keb aZ^Zgu agZq_gby i_j\hc ijhba\h^ghc \ djZcgbo lhqdZo y 0' b y n' lh d ihemq_gghckbkl_f_^h[Z\ey_f^\ZmjZ\g_gby m0 = y 0' ,

16 mn = y n' . < g_dhlhjuo kemqZyo [u\Zxl ba\_klgu agZq_gby \lhjhc ijhba\h^ghc \ djZcgbolhqdZo y 0'' b y n''
y1 − y 0 − h1 y 0'' , h1 yn − yn −1 + hn y n'' . h1

Ijb k\h[h^ghf aZdj_ie_gbb dhgph\ kieZcgZ fu ^he`gu ijbjZ\gylv gmex djb\bagm ebgbb \ djZcgbo lhqdZo LZdZy nmgdpby gZau\Z_lky k\h[h^guf dm[bq_kdbf beb _kl_kl\_gguf kieZcghf HgZ h[eZ^Z_l k\hckl\hf fbgbfZevghc djb\bagu l_ hgZ kZfZy ]eZ^dZy kj_^b \k_o bgl_jiheypbhgguo nmgdpbc ^Zggh]h deZkkZ Ba mkeh\by gme_\hc djb\bagu \ djZcgbo lhqdZo ke_^m_l jZ\_gkl\h gmex \ gbo \lhjuo ijhba\h^guo ihwlhfm ^h[Z\ey_f ^\Z mjZ\g_gby y − y0 , 2m0 + m1 = 3 1 h1 mn −1 + 2mn = 3

y n − y n −1 . hn

DjZ_\u_ mkeh\by - fh`gh dhf[bgbjh\Zlv l_ \ e_\hf b ijZ\hf djZcgbomaeZo\u[bjZlvbog_aZ\bkbfh Ijb\k_ojZkkfhlj_gguo\ZjbZglZodjZ_\uomkeh\bcihemq_ggZykbkl_fZ n + 1 ebg_cguo Ze]_[jZbq_kdbo mjZ\g_gbc k n + 1 g_ba\_klguf mi (i = 0, 1, K , n) bf__l _^bgkl\_ggh_ j_r_gb_ FZljbpZ wlhc kbkl_fu lj_o^bZ]hgZevgZ l_ g_gme_\u_ we_f_glu gZoh^ylky ebrv gZ ]eZ\ghc b ^\mo khk_^gbo k g_c ^bZ]hgZeyo >ey j_r_gby lZdbo kbkl_f h[uqgh bkihevamxl f_lh^ijh]hgdbdhlhjuc[m^_ljZkkfhlj_g\jZa^_e_ Ih^[hjwfibjbq_kdbonhjfmeF_lh^gZbf_gvrbod\Z^jZlh\ >h kbo ihj fu jZkkfZljb\Zeb aZ^Zqm bgl_jiheypbb Bgl_jihebjm_fZy nmgdpby f ( x ) [ueZ aZ^ZgZ \ dhg_qghf qbke_ lhq_d hlj_adZ [a , b] x 0 , x1 , K , x n bbgl_jihebjmxsZynmgdpby\maeZobgl_jiheypbb\lhqghklb kh\iZ^Z_l k agZq_gbyfb nmgdpbb f ( x ) H^gZdh bgl_jiheypby \ g_dhlhjuo kemqZyo klZgh\blky g_m^h[ghc b ^Z`_ ijhklh g_ijb_fe_fhc Wlh dZkZ_lky ij_`^_\k_]hkemqZydh]^ZagZq_gbynmgdpbbih^\_j`_gudZdbf-lhhrb[dZf gZijbf_j hrb[dZf baf_j_gby Lh]^Z wlb hrb[db [m^ml \g_k_gu \ bgl_jihebjmxsmx nmgdpbx b l_f kZfuf bkdZayl bklbggmx dZjlbgm

17 Wdki_jbf_glZevgu_hrb[dbihboijhbkoh`^_gbxfh`ghmkeh\ghjZa[blvgZ ljb]jmiiukbkl_fZlbq_kdb_kemqZcgu_b]jm[u_ Kbkl_fZlbq_kdb_ hrb[db h[uqgh ^Zxl hldehg_gb_ \ h^gm klhjhgm hl bklbggh]hagZq_gbybaf_jy_fhc\_ebqbguHgbfh]ml[ulvihklhyggufbbeb aZdhghf_jgh baf_gylvky ijb ih\lhj_gbb hiulZ bo ijbqbgu b oZjZdl_j ba\_klgu Kbkl_fZlbq_kdb_ hrb[db fh]ml [ulv \ua\Zgu mkeh\byfb wdki_jbf_glZ \eZ`ghklvx l_fi_jZlmjhc kj_^u b li iehohc j_]mebjh\dhc baf_jbl_evghc ZiiZjZlmju gZijbf_j kf_s_gb_f mdZaZl_evghc klj_edb hl gme_\h]hiheh`_gby bl^Wlbhrb[dbfh`ghmkljZgblvgZeZ^dhcZiiZjZlmju beb \\_^_gb_f khhl\_lkl\mxsbo ihijZ\hd < ^Zevg_cr_f [m^_f kqblZlv qlh kbkl_fZlbq_kdb_hrb[db\hiulguo^Zgguom`_bkdexq_gu KemqZcgu_hrb[dbhij_^_eyxlky[hevrbfqbkehfnZdlhjh\dhlhju_g_ fh]ml [ulv mkljZg_gu beb ^hklZlhqgh lhqgh mql_guijbbaf_j_gbyobebijb h[jZ[hld_ j_amevlZlh\ Hgb bf_xl kemqZcguc g_kbkl_fZlbq_kdbc oZjZdl_j ^Zxlhldehg_gb_hlkj_^g_c\_ebqbgu\lmb^jm]mxklhjhguijbih\lhj_gbb baf_j_gbcbg_fh]ml[ulvmkljZg_gu\wdki_jbf_gl_dZd[ulsZl_evghhggb ijh\h^beky K \_jhylghklghc lhqdb aj_gby fZl_fZlbq_kdh_ h`b^Zgb_ kemqZcghchrb[dbjZ\ghgmexWlhagZqblqlhhgbfh]ml[ulvmf_gvr_gu^h kdhevm]h^ghfZeh]hagZq_gbyiml_ffgh]hdjZlgh]hih\lhj_gbyhiulZ =jm[u_ hrb[db y\gh bkdZ`Zxl j_amevlZlu baf_j_gbc hgb qj_af_jgh [hevrb_ b h[uqgh ijhiZ^Zxl ijb ih\lhj_gbb hiulZ =jm[u_ hrb[db kms_kl\_ggh \uoh^yl aZ ij_^_eu kemqZcghc hrb[db ihemq_ggu_ ijb klZlbklbq_kdhc h[jZ[hld_ Baf_j_gby k lZdbfb hrb[dZfb hl[jZku\Zxlky b \ jZkq_lijbhdhgqZl_evghch[jZ[hld_j_amevlZlh\baf_j_gbcg_ijbgbfZxlky LZdbfh[jZahf\wdki_jbf_glZevguo^Zgguo\k_]^Zbf_xlkykemqZcgu_ hrb[dbBomf_gvr_gb_^hijb_fe_fhc\_ebqbguih\lhj_gb_fwdki_jbf_glZ ^Ze_dhg_\k_]^Zp_e_khh[jZaghihkdhevdmfh]mlihlj_[h\ZlvkyagZqbl_evgu_ fZl_jbZevgu_ b beb \j_f_ggu_ j_kmjku AgZqbl_evgh ^_r_\e_ b [uklj__ fh`gh \ jy^_ kemqZ_\ ihemqblv mlhqg_ggu_ ^Zggu_ khhl\_lkl\mxs_c fZl_fZlbq_kdhc h[jZ[hldhc bf_xsboky j_amevlZlh\ baf_j_gbc < qZklghklb fh`gh gZclb aZdhg jZkij_^_e_gby hrb[hd baf_j_gbc gZb[he__ \_jhylguc ^bZiZahg baf_g_gby bkdhfhc \_ebqbgu ^h\_jbl_evguc bgl_j\Ze b ^jm]b_ iZjZf_lju JZkkfhlj_gb_ \k_o wlbo \hijhkh\ \uoh^bl aZ jZfdb ^Zggh]h ihkh[by A^_kv fu h]jZgbqbfky ebrv hij_^_e_gb_f k\yab f_`^m bkoh^guf iZjZf_ljhf x b\_ebqbghc y gZhkgh\Zgbbj_amevlZlh\baf_j_gbc Imklv bamqZy g_ba\_klgmx nmgdpbhgZevgmx aZ\bkbfhklv y hl x \ j_amevlZl_wdki_jbf_glh\fuihemqbeblZ[ebpmagZq_gbc

x y

x0

x1

y0

y1

K K

xn yn

18 AZ^ZqZkhklhbl\lhfqlh[ugZclbijb[eb`_ggmxaZ\bkbfhklv y = ϕ (x ) ,

agZq_gby dhlhjhc ijb x = x i (i = 0, 1, K n) fZeh hlebqZxlky hl hiulguo ^Zgguo yi LZdZyijb[eb`_ggZynmgdpbhgZevgZyaZ\bkbfhklvihkljh_ggZygZ hkgh\_wdki_jbf_glZevguo^ZgguogZau\Z_lkywfibjbq_kdhcnhjfmehc AZ^ZqZ ihkljh_gby wfibjbq_kdhc nhjfmeu hlebqZ_lky hl aZ^Zqb bgl_jihebjh\Zgby =jZnbd wfibjbq_kdhc nhjfmeu \hh[s_ ]h\hjy g_ ijhoh^bl q_j_a aZ^Zggu_ lhqdb (xi , yi ) dZd \ kemqZ_ bgl_jiheypbb Wlh ijb\h^bl d lhfm qlh wdki_jbf_glZevgu_ ^Zggu_ \ g_dhlhjhc kl_i_gb k]eZ`b\Zxlky Z bgl_jiheypbhggZy nhjfmeZ ih\lhjbeZ [u \k_ hrb[db bf_xsb_ky\wdki_jbf_glZevguo^Zgguo Ihkljh_gb_ wfibjbq_kdhc nhjfmeu \dexqZ_l \ k_[y ^\Z wlZiZ ih^[hj h[s_]h \b^Z nhjfmeu b hij_^_e_gb_ gZbemqrbo qbke_gguo agZq_gbc kh^_j`Zsboky\g_ciZjZf_ljh\ H[sbc \b^ nhjfmeu qZklh ba\_kl_g ba nbabq_kdbo khh[jZ`_gbc GZijbf_j ^ey mijm]hc kj_^u k\yav f_`^m gZijy`_gb_f σ b hlghkbl_evghc ^_nhjfZpb_c ε hij_^_ey_lkyaZdhghf=mdZ σ = Eε ]^_ E -fh^mevmijm]hklb AZ^ZqZ k\h^blky d hij_^_e_gbx qbkeh\h]h agZq_gby _^bgkl\_ggh]hiZjZf_ljZ E ?keb `_ oZjZdl_j aZ\bkbfhklb g_ba\_kl_g lh \b^ wfibjbq_kdhc nhjfmeu fh`_l [ulv ijhba\hevguf H^gZdh ij_^ihql_gb_ h[uqgh hl^Z_lky gZb[he__ ijhkluf nhjfmeZf k gZbf_gvrbf qbkehf iZjZf_ljh\ ijb mkeh\bb bo ^hklZlhqghclhqghklb Ijhkl_cr_c b hq_gv qZklh bkihevam_fhc wfibjbq_kdhc nhjfmehc y\ey_lkyebg_cgZyaZ\bkbfhklv y = a 0 + a1 x . QZklh d g_c fh]ml [ulv k\_^_gu b ^jm]b_ aZ\bkbfhklb dh]^Z bo ]jZnbd \ ^_dZjlh\hc kbkl_f_ dhhj^bgZl g_ y\ey_lky ijyfhc ebgb_c GZijbf_j wdkihg_gpbZevgZy aZ\bkbfhklv m^_evghc we_dljhijh\h^ghklb kh[kl\_ggh]h ihemijh\h^gbdZ σ hlZ[khexlghcl_fi_jZlmju T −

Eg

σ = e 2 kT eh]Zjbnfbjh\Zgb_fijb\h^blkydebg_cghc\dhhj^bgZlZo 1 T , ln σ . I_j_c^_f l_i_jv d jZkkfhlj_gbx \lhjh]h wlZiZ ihkljh_gby wfibjbq_kdhc aZ\bkbfhklb ;m^_f kqblZlv qlh lbi wfibjbq_kdhc nhjfmeu \u[jZg?_fh`ghaZibkZlv\\b^_ y = ϕ (x, a 0 , a1 , K , a m ) ,

19 ]^_ ϕ -ba\_klgZynmgdpby a 0 , a1 , K , a m -g_ba\_klgu_ihklhyggu_iZjZf_lju (m < n) . AZ^ZqZ khklhbl \ lhf qlh[u hij_^_eblv lZdb_ agZq_gby wlbo iZjZf_ljh\ijbdhlhjuowfibjbq_kdZynhjfmeZ^Z_lgZbemqr__ijb[eb`_gb_ d ^Zgghc nmgdpbb agZq_gby dhlhjhc \ lhqdZo x0 , x1 , K , x n jZ\gu khhl\_lkl\_ggh y 0 , y1 , K , y n . A^_kv g_ klZ\blky mkeh\b_ dZd \ kemqZ_ bgl_jiheypbb kh\iZ^_gby agZq_gbc wfibjbq_kdhc nmgdpbb \ lhqdZo xi ϕ (xi , a 0 , a1 , K , a m ) k hiulgufb ^Zggufb yi JZaghklv f_`^m wlbfb agZq_gbyfb hldehg_gb_ h[hagZqbfq_j_a ε i :

ε i = ϕ (xi , a 0 , a1 , K , a m ) − yi ,

(i = 0, 1, K n) .

AZ^ZqZ gZoh`^_gby gZbemqrbo iZjZf_ljh\ a 0 , a1 , K , a m k\h^blky lZdbf h[jZahf d g_dhlhjhc fbgbfbaZpbb hldehg_gbc ε i Kms_kl\m_l g_kdhevdh kihkh[h\ j_r_gby wlhc aZ^Zqb GZb[he__ mihlj_[bl_evguf y\ey_lky f_lh^ gZbf_gvrbod\Z^jZlh\ F_lh^ gZbf_gvrbo d\Z^jZlh\ AZibr_f kmffm d\Z^jZlh\ hldehg_gbc ^ey\k_olhq_d x0 , x1 , K , x n : n

S=∑

i=0

ε i2

n

[

= ∑ ϕ (xi , a 0 , a1 , K , a m ) − yi i= 0

]2

.

S y\ey_lky nmgdpb_c g_ba\_klguo iZjZf_ljh\ a 0 , a1 , K , a m ijbq_f S ≥ 0 . IZjZf_lju a 0 , a1 , K , a m wfibjbq_kdhc nhjfmeu [m^_f gZoh^blv ba mkeh\by fbgbfmfZ nmgdpbb S = S(a 0 , a1 , K , a m ) < wlhf khklhbl f_lh^ gZbf_gvrbo d\Z^jZlh\ < l_hjbb \_jhylghkl_c ihdZau\Z_lky qlh ihemq_ggu_ lZdbf kihkh[hf agZq_gby iZjZf_ljh\ gZb[he__ \_jhylgu _keb hldehg_gby ε i ih^qbgyxlkyghjfZevghfmaZdhgmjZkij_^_e_gby Ihkdhevdm iZjZf_lju a 0 , a1 , K , a m \uklmiZxl \ jheb g_aZ\bkbfuo i_j_f_gguo nmgdpbb S gZc^_f bo ba mkeh\by jZ\_gkl\Z gmex qZklguo ijhba\h^guonmgdpbb S ihwlbfiZjZf_ljZf\lhqd_fbgbfmfZ

 ∂S  ∂a = 0 0   ∂S =0  a ∂ .  1 K   ∂S  ∂a = 0 m  L_f kZfuf fu ihemqbeb kbkl_fm m + 1 mjZ\g_gbc k m + 1 g_ba\_klgufb a 0 , a1 , K , a m J_rb\__fh`ghgZclbhilbfZevgu_agZq_gbywlboiZjZf_ljh\

20 Y\guc \b^ wlhc kbkl_fu hij_^_ey_lky dhgdj_lguf \b^hf wfibjbq_kdhc aZ\bkbfhklb JZkkfhljbf ijbf_g_gb_ f_lh^Z gZbf_gvrbo d\Z^jZlh\ ^ey qZklgh]h kemqZygZb[he__rbjhdhijbf_gy_fh]hgZijZdlbd_
ϕ ( x ) = a 0 + a1 x + a 2 x 2 +K + a m x m , Lh]^ZnhjfmeZ^eykmffud\Z^jZlh\hldehg_gbcijbf_l\b^ n

(

S = ∑ a 0 + a1 xi +K + a m xim − yi i =0

)

2

,

hldm^Z ^bnn_j_gpbjh\Zgb_f ihemqZ_f kbkl_fm ebg_cguo Ze]_[jZbq_kdbo mjZ\g_gbchlghkbl_evghg_ba\_klguoiZjZf_ljh\ a 0 , a1 , K , a m : n n n  m + + + + = K a n a x a x 1 ) 1∑ i ∑ yi  0( m∑ i i =0 i =0 i =0  n n n  n a 0 ∑ xi + a1 ∑ x i2 +K + a m ∑ xim+1 = ∑ xi yi  i =0 i =0 i =0 i =0   n n n  n m m m+1 2m a 0 ∑ xi + a1 ∑ x i +K + a m ∑ xi = ∑ xi yi  i =0 i =0 i =0 i =0 Qbke_ggu_ f_lh^u j_r_gby lZdbo kbkl_f [m^ml jZkkfhlj_gu \ jZa^_e_ J_rb\ ^Zggmx kbkl_fm l_f beb bguf f_lh^hf gZc^_f iZjZf_lju a 0 , a1 , K , a m bl_fkZfufihemqbfwfibjbq_kdmxnhjfmem

GZdhg_p \ ijhkl_cr_f kemqZ_ ebg_cghc aZ\bkbfhklb ihemqZxsmxky kbkl_fm ^\mo mjZ\g_gbc k ^\mfy g_ba\_klgufb fh`gh jZaj_rblv \ ZgZeblbq_kdhf\b^_bihemqblvke_^mxsb_\ujZ`_gby^eyiZjZf_ljh\ a0 =

a1 =

n

n

n

n

i =0

i =0

i=0

i =0

∑ xi ∑ xi yi − ∑ xi2 ∑ xi yi  n  2 (n + 1)∑ xi −  ∑ xi   i =0  i =0 n

,

2

n

n

n

i =0

i =0

i =0 2

(n + 1)∑ xi yi − ∑ xi ∑ yi n



n



(n + 1)∑ xi2 −  ∑ xi  i =0

 i =0



.

21

QBKE?GGH?>BNN?J?GPBJH<:GB? D qbke_gghfm ^bnn_j_gpbjh\Zgbx ijb[_]Zxl dh]^Z g_h[oh^bfh \uqbkeblv ijhba\h^gmx nmgdpbb f (x) aZ^Zgghc lZ[ebqgh beb bf_xs_c hq_gvkeh`gh_ZgZeblbq_kdh_\ujZ`_gb_bnn_j_gpbjmy wlh lh`^_kl\h k jZa \ ij_^iheh`_gbb qlh f (x) b ϕ (x) bf_xl ijhba\h^gu_ ^h k-]h ihjy^dZ ihemqbf f ( k ) ( x) = ϕ ( k ) ( x) + R ( k ) ( x ).

L d aZ ijb[eb`_ggh_ agZq_gb_ f ( k ) ( x) [_j_lky ϕ ( k ) ( x) lh ih]j_rghklv nhjfmeu qbke_ggh]h ^bnn_j_gpbjh\Zgby k-]h ihjy^dZ jZ\gZ k-hc ijhba\h^ghchklZlhqgh]hqe_gZbgl_jiheypbhgghcnhjfmeu Nhjfmeuqbke_ggh]h^bnn_j_gpbjh\ZgbygZhkgh\_ bgl_jiheypbhggh]hihebghfZGvxlhgZ^eyg_jZ\ghhlklhysbomaeh\ JZkkfhljbfbgl_jiheypbhggmxnhjfmemGvxlhgZ^eynmgdpbb f (x) [_a hklZlhqgh]hqe_gZ f ( x ) ≈ f ( x0 ) + ( x − x0 ) f ( x0 , x1 ) + ... + ( x − x0 )...( x − xn −1 ) f ( x0 , x1,..., xn ). H[hagZqbf x − xi = α i bijh^bnn_j_gpbjm_fh[_qZklbwlh]hjZ\_gkl\Zih x : f ' ( x) ≈ f ( x 0 , x1 ) + (α 0 + α 1 ) f ( x 0 , x1 , x 2 ) + (α 0α 1 + α 0α 2 + α 1α 2 ) f ( x 0 , x1 , x 2 , x 3 ) + ... ... + (α 0α 1 ...α n − 2 + α 0α 1 ...α n −3α n −1 + ... + α 1α 2 ...α n −1 ) f ( x 0 , x1 ,...x n ).

>bnn_j_gpbjmy _s_ jZa ihemqbf ijb[eb`_ggh_ agZq_gb_ ^ey \lhjhc ijhba\h^ghc f ' ' ( x) ≈ 2[ f ( x 0 , x1 , x 2 ) + (α 0 + α 1 + α 2 ) f ( x 0 , x1 , x 2 , x 3 ) + ... ... + (α 0α 1 ...α n −3 + α 0α 1 ...α n − 4α n − 2 + ... + α 2α 2 ...α n −1 ) f ( x 0 , x1 ,...x n )].

22 f

(k )

( x) ≈ k![ f ( x 0 , x1 , x k ) + (α 0 + α 1 + ... + α k ) f ( x 0 , x1 ,..., x k +1 ) + + (α 0α 1 + α 0α 2 + ... + α k α k +1 ) f ( x 0 , x1 ,..., x k + 2 ) + ... ... + (α 0α 1 ...α n − k −1 + ...α k α k +1 ...α n −1 ) f ( x 0 , x1 ,...x n )].

>eyk > n

f

(k )

( x ) = 0.

2.Nhjfmeuqbke_ggh]h^bnn_j_gpbjh\Zgby^eykemqZyjZ\ghhlklhysbo maeh\ Ihemqbf nhjfmeu qbke_ggh]h ^bnn_j_gpbjh\Zgby \ kemqZ_ dh]^Z maeu bgl_jiheypbb nmgdpbb f (x) jZkiheh`_gu jZ\ghf_jgh gZ hlj_ad_ [a, b]. JZkklhygb_ f_`^m ex[ufb ^\mfy maeZfb xi b x i −1 h[hagZqbf h Imklv agZq_gby nmgdpbb f (x) ba\_klgu \ lj_o lhqdZo n = 2): ϕ (x) Bgl_jihebjmxsmx nmgdpbx f ( x 0 ) = y 0 , f ( x1 ) = y1 , f ( x 2 ) = y 2 . ij_^klZ\bf\\b^_d\Z^jZlbqgh]hihebghfZGvxlhgZLh]^Z f ( x ) = ϕ ( x) + R( x) = = f ( x 0 ) + f ( x 0 , x1 )( x − x 0 ) + f ( x 0 , x1 , x 2 )( x − x 0 )( x − x1 ) + R( x) .

(2.1)

Ijh^bnn_j_gpbjm_f\ujZ`_gb_ ihx: f ' ( x ) = f ( x 0 , x1 ) + f ( x 0 , x1 , x 2 )(2 x − x 0 − x1 ) + R ' ( x ).

A^_kv

(2.2)

f ( x 0 , x1 ) = ( f ( x1 ) − f ( x 0 )) /( x1 − x 0 ) = ( y1 − y 0 ) / h,

f ( x 0 , x1 , x 2 ) = ( f ( x1 , x 2 ) − f ( x 0 , x1 )) /( x 2 − x 0 ) = ( y 2 − 2 y1 + y 0 ) /(2h 2 ).

(2.3)

Ih^klZ\bf \ f ' ( x ) = ( y1 − y 0 ) / h + ( y 2 − 2 y1 + y 0 )(2 x − x 0 − x1 ) /( 2h 2 ) + R ' ( x).

(2.4)

HklZlhqgucqe_g\nhjfme_GvxlhgZ Rn ( x) =

f

( n +1)

(ξ ) n ∏ ( x − x i ), (n + 1)! i =0

ξ ∈ [x 0 , x n ] .

>eybgl_jiheypbbihlj_fmaeZfn Hlkx^Ze_]dhihemqblvijhba\h^gmx hklZlhqgh]hqe_gZ\nhjfme_

23 R ' ( x) =

f ' ' ' (ξ ) [( x − x 0 )( x − x1 ) + ( x − x1 )( x − x 2 ) + ( x − x 0 )( x − x 2 )]. 6

GZc^_f f ' ( x) \ maeZo bgl_jiheypbb Ijb x = x 0 ba nhjfmeu ihemqZ_fagZq_gb_i_j\hcijhba\h^ghcnmgdpbb f (x) \lhqd_ x 0 : Ijb x = x1 : ijb x = x 2 :

f ' ( x 0 ) = (−3 y 0 + 4 y1 − y 2 ) /(2h) + h 2 f ' ' ' (ξ ) / 3.

(2.5)

f ' ( x1 ) = (− y 0 + y 2 ) /(2h) − h 2 f ' ' ' (ξ ) / 6,

(2.6)

f ' ( x 2 ) = ( y 0 − 4 y1 + 3 y 2 ) /( 2h) + h 2 f ' ' ' (ξ ) / 3. (2.7) :gZeh]bqgh fh`gh gZclb ijhba\h^gu_ nmgdpbb f (x) bkihevamy agZq_gby ^Zgghcnmgdpbb\[hevr_fdhebq_kl\_maeh\bgl_jiheypbbLZd_kebaZ^Zgu agZq_gby nmgdpbb \ q_luj_o maeZo bkihevah\Zgb_ f ( x 0 ) = y 0 , f ( x1 ) = y1 , f ( x 2 ) = y 2 , f ( x 3 ) = y 3 , bgl_jiheypbhggh]h ihebghfZ GvxlhgZ n ijb\h^bl d ke_^mxsbf nhjfmeZf^eyi_j\hcijhba\h^ghc f ' ( x 0 ) = (−11 y 0 + 18 y1 − 9 y 2 + 2 y 3 ) /(6h) − h 3 f

( IV )

(ξ ) / 12,

f ' ( x1 ) = (−2 y 0 − 3 y1 + 6 y 2 − y 3 ) /(6h) + h 3 f

( IV )

f ' ( x 2 ) = (11 y 0 − 6 y1 + 3 y 2 + 2 y 3 ) /(6h) − h 3 f

( IV )

f ' ( x 3 ) = (−2 y 0 + 9 y1 − 18 y 2 + 11 y 3 ) /(6h) + h 3 f

(ξ ) / 4,

(ξ ) / 12,

( IV )

(ξ ) / 4.

GZb[he__ijhklu_blhqgu_nhjfmeuihemqZxlky^eyq_lguon \kj_^gbo lhqdZoIhwlhfmgZijZdlbd_ih\hafh`ghklbke_^m_lijbf_gylvbf_gghbo GZijbf_j nmgdpby f (x) aZ^ZgZ \ lhqdZo x 0 , x1 , x 2 ,..., x n G_h[oh^bfh \uqbkeblv ijhba\h^gmx f ' ( x) \ wlbo `_ lhqdZo bkihevamy lj_olhq_qgmx ko_fm - GZb[he__ lhqgh_ j_r_gb_ [m^_l ihemq_gh _keb ^ey ijhba\h^ghcnmgdpbb\]jZgbqguolhqdZo x 0 b x n bkihevah\Zlvnhjfmeu f ' ( x 0 ) ≈ (−3 y 0 + 4 y1 − y 2 ) /( 2h), Z f ' ( x n ) ≈ ( y n − 2 − 4 y n −1 + 3 y n ) /( 2h),

Z

Z f ' ( xi ) ]^_i = 1, 2, ... , n-\uqbkeylvihnhjfme_ f ' ( x i ) ≈ ( y i +1 − y i −1 ) /(2h). Z

24

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25 DmjdbgZEZjbkZB\Zgh\gZ J_^Zdlhj;mgbgZL>