數(shù)值分析：7-5 數(shù)值微分-new

1、第七章第七章 微積分的數(shù)值計(jì)算方法微積分的數(shù)值計(jì)算方法 7.5 數(shù)值微分?jǐn)?shù)值微分 h h 0 0f f( (x x + + h h) ) - - f f( (x x) )由由導(dǎo)導(dǎo)數(shù)數(shù)定定義義 f f ( (x x) ) = = l li immh h1 1、差差商商型型求求導(dǎo)導(dǎo)公公式式（ 1 1） 向向 前前 差差 商商 公公 式式 f f( (x x + + h h) ) - - f f( (x x) ) f f ( (x x) ) h h（ 2 2） 向向 后后 差差 商商 公公 式式 f f( (x x) ) - - f f( (x x - - h h) ) f f ( (x x) )

2、h h（ 3 3） 中中 心心 差差 商商 公公 式式（ 中中 點(diǎn)點(diǎn) 方方 法法） f f( (x x + + h h) ) - - f f( (x x - - h h) ) f f ( (x x) ) 2 2h h x-h x x+hBCAT f(x)差商型求導(dǎo)公式的余項(xiàng)差商型求導(dǎo)公式的余項(xiàng) ( ( 3 3 ) ) 2 22 2 由由 T T a a y y l l o o r r 公公 式式f f ( () )f f ( ( x x + + h h ) )- - f f ( ( x x ) ) f f ( ( x x ) )- -= = - -h h = = O O ( ( h h ) )

3、h h2 2f f ( () )f f ( ( x x ) )- - f f ( ( x x - - h h ) ) f f ( ( x x ) )- -= =h h = = O O ( ( h h ) )h h2 2f f ( ( x x + + h h ) )- - f f ( ( x x - - h h ) )f f( () ) f f ( ( x x ) )- -= = - -h h= = O O ( ( h h ) )2 2 h h6 6從截?cái)嗾`差的角度看，步長(zhǎng)越小，計(jì)算結(jié)果越準(zhǔn)確；從截?cái)嗾`差的角度看，步長(zhǎng)越小，計(jì)算結(jié)果越準(zhǔn)確；從舍入誤差的角度來(lái)看，步長(zhǎng)不宜太小。從舍入誤差的角度來(lái)看

4、，步長(zhǎng)不宜太小。2、插值型求導(dǎo)公式、插值型求導(dǎo)公式i ii in n 若若 已已 知知 函函 數(shù) f f( (x x ) )在在 a a , , b b 內(nèi) n n +1 1 個(gè) 節(jié) 點(diǎn)點(diǎn) ( (x x , , f f( (x x ) ) )( (i i = = 0 0 , ,1 1 , , , n n ) )， 可可 用用 其其 插 值 多多 項(xiàng) 式式 P P ( (x x ) )的的 導(dǎo) 數(shù) 近近似似 函函 數(shù) f f( (x x ) )的的 導(dǎo) 數(shù) 。( (n n+ +1 1) )n nn nn n+ +1 1( (n n+ +1 1) )( (n n+ +1 1) )n n+ +1 1

5、n nn n+ +1 1f f( ( ) ) 由由 R R ( (x x) ) = = f f( (x x) ) - - P P ( (x x) ) = = ( (x x) )( (n n + + 1 1) )! ! ( (x x) )f f( ( ) )d d f f ( (x x) ) - - P P ( (x x) ) = = ( (x x) ) + +f f( ( ) )( (n n + + 1 1) )! !( (n n + + 1 1) )! ! d dx x( (n n+ +1 1) )( (n n+ +1 1) )n ni in ni in n+ +1 1i ii ij jj

6、j= =0 0j j i i 對(duì)對(duì)任任意意x x a a, ,b b ，因因 未未知知，故故上上式式很很難難估估計(jì)計(jì)誤誤差差，但但若若只只求求某某個(gè)個(gè)節(jié)節(jié)點(diǎn)點(diǎn)上上的的導(dǎo)導(dǎo)數(shù)數(shù)值值，誤誤差差可可估估計(jì)計(jì)。f f( ( ) )f f( ( ) )f f ( (x x ) )- -P P ( (x x ) ) = = ( (x x ) ) = =( (x x - - x x ) )( (n n+ +1 1) )! !( (n n+ +1 1) )! !因因此此，插插值值型型求求導(dǎo)導(dǎo)公公式式通通常常用用于于求求節(jié)節(jié)點(diǎn)點(diǎn)處處導(dǎo)導(dǎo)數(shù)數(shù)的的近近似似值值。兩兩 點(diǎn)點(diǎn) 公公 式式0 00 01 11 11 10

7、 00 01 11 10 01 10 01 11 10 01 10 01 1 設(shè)設(shè) 給給 出出 兩兩 節(jié)節(jié) 點(diǎn)點(diǎn) ( (x x , , f f( (x x) ) ), ,( (x x , , f f( (x x ) ) ), ,記記 x x- - x x= = h hx x - - x xx x - - x x有有 P P ( (x x) ) = =f f( (x x) ) + +f f( (x x ) ). .x x- - x xx x- - x x1 1 P P ( (x x) ) = = - -f f( (x x) ) + + f f( (x x ) ) h h1 10 01 10 01

8、 11 11 10 0 1 10 01 10 01 1 1 11 11 10 02 21 1 P P( (x x) ) = = f f( (x x) ) - - f f( (x x) ) , , h h1 1 P P( (x x) ) = = f f( (x x) ) - - f f( (x x) ) ；h h帶帶 余余 項(xiàng)項(xiàng) 的的 兩兩 點(diǎn)點(diǎn) 公公 式式 是是 ：1 1h hf f ( (x x) ) = = f f( (x x) ) - - f f( (x x) ) - -f f ( ( ) ), ,h h2 21 1h hf f ( (x x) ) = = f f( (x x) ) -

9、- f f( (x x) ) + +f f ( ( ) ). .h h2 2三三 點(diǎn)點(diǎn) 公公 式式0 01 10 02 20 01 12 22 20 00 01 10 02 20 02 20 01 11 12 21 10 01 12 22 20 02 21 10 02 20 00 01 12 2 設(shè)設(shè)已已給給出出三三個(gè)個(gè)節(jié)節(jié)點(diǎn)點(diǎn)x x , ,x x = = x x + +h h, ,x x = = x x + + 2 2h h上上的的函函數(shù)數(shù)值值( (x x - - x x ) )( (x x - - x x ) )P P ( (x x) ) = =f f( (x x ) )( (x x -

10、- x x ) )( (x x - - x x ) )( (x x - - x x ) )( (x x - - x x ) )( (x x - - x x ) )( (x x - - x x ) )+ +f f( (x x ) )+ +f f( (x x ) ), ,( (x x - - x x ) )( (x x - - x x ) )( (x x - - x x ) )( (x x - - x x ) )令令x x = = x x + + t th h, ,則則1 1P P ( (x x + + t th h) ) = =( (t t - -1 1) )( (t t - - 2 2) )f

11、 f( (x x ) )- - t t( (t t - - 2 2) )f f( (x x ) )2 21 1+ +t t( (t t - -1 1) )f f( (x x2 2) ), ,2 20 00 01 12 22 20 00 01 12 22 21 10 02 22 22 20 01 12 21 1上上 式式 對(duì)對(duì) t t求求 導(dǎo)導(dǎo) ： P P( ( x x+ + t th h ) ) = = ( ( 2 2 t t - - 3 3 ) ) f f( ( x x) )2 2 h h- - ( ( 4 4 t t - - 4 4 ) ) f f( ( x x) ) + + ( ( 2

12、2 t t - - 1 1 ) ) f f( ( x x) ) . .1 1 P P( ( x x) ) = = - - 3 3 f f( ( x x) ) + + 4 4 f f( ( x x) ) - - f f( ( x x) ) ; ;2 2 h h1 1P P( ( x x) ) = = - - f f( ( x x) ) + + f f( ( x x) ) ; ; （ 中中 點(diǎn)點(diǎn) 公公 式式 ）2 2 h h1 1P P( ( x x) ) = = f f( ( x x) ) - - 4 4 f f( ( x x) ) 3 3 f f( ( x x) ) . .2 2 h h2 2

13、0 00 01 12 22 21 10 02 22 22 20 01 12 2帶帶 余余 項(xiàng)項(xiàng) 的的 三三 點(diǎn)點(diǎn) 求求 導(dǎo)導(dǎo) 公公 式式 ：1 1h hf f ( (x x ) ) = = - -3 3f f( (x x ) )+ + 4 4f f( (x x ) )- - f f( (x x ) ) + +f f ( ( ) ); ;2 2h h3 31 1h hf f ( (x x ) ) = = - -f f( (x x ) )+ + f f( (x x ) ) - -f f ( ( ) ); ; （ 中中 點(diǎn)點(diǎn) 公公 式式 ）2 2h h6 61 1h hf f ( (x x ) )

14、= = f f( (x x ) )- - 4 4f f( (x x ) ) 3 3f f( (x x ) ) + +f f ( ( ) ). .2 2h h3 3m( (k k) )( (k k) ) 可可 利利 用用 插插 值值 多多 項(xiàng)項(xiàng) 式式 ， 建建 立立 高高 階階 數(shù)數(shù) 值值 微微 分分 公公 式式 ：f f P P( (x x) ), ,k k = = 1 1, ,2 2, ,2 20 00 01 12 2 2 20 00 01 12 22 2 2 21 11 11 11 12 22 2 ( (4 4) )1 11 11 11 12 21 1例例：對(duì)對(duì) P P ( (x x +

15、+ t th h) ) = = ( (2 2t t - - 3 3) )f f( (x x ) )2 2h h- -( (4 4t t - - 4 4) )f f( (x x ) )+ + ( (2 2t t - -1 1) )f f( (x x ) ) . . 再再對(duì)對(duì)t t求求導(dǎo)導(dǎo)，1 1有有 P P ( (x x + + t th h) ) = =( (f f( (x x ) )- - 2 2f f( (x x ) )+ + f f( (x x ) ) , ,h h1 1 P P ( (x x ) ) = =( (f f( (x x - -h h) )- - 2 2f f( (x x )

16、 )+ + f f( (x x + + h h) ) , ,h h帶帶余余項(xiàng)項(xiàng)的的二二階階三三點(diǎn)點(diǎn)公公式式：1 1h hf f ( (x x ) ) = =( (f f( (x x - -h h) )- - 2 2f f( (x x ) )+ + f f( (x x + + h h) ) - -f f( ( ) ). .h h1 12 2同樣，針對(duì)同樣，針對(duì)m也可擴(kuò)展，如五點(diǎn)插值求積公式。也可擴(kuò)展，如五點(diǎn)插值求積公式。3、 樣樣 條條 求求 導(dǎo)導(dǎo)( (k k) )( (k k) )( (k k) )( (k k) )4 4- -k k 三三次次樣樣條條函函數(shù)數(shù)S S( (x x) )及及其其一

17、一、二二階階導(dǎo)導(dǎo)數(shù)數(shù)均均一一致致收收斂斂于于被被插插值值函函數(shù)數(shù)f f( (x x) )及及其其一一、二二階階導(dǎo)導(dǎo)數(shù)數(shù)，故故用用樣樣條條函函數(shù)數(shù)的的導(dǎo)導(dǎo)數(shù)數(shù)近近似似函函數(shù)數(shù)導(dǎo)導(dǎo)數(shù)數(shù)f f( (x x) ) S S ( (x x) )( (k k = = 1 1, ,2 2, ,) )不不僅僅可可靠靠性性好好，且且可可計(jì)計(jì)算算非非節(jié)節(jié)點(diǎn)點(diǎn)處處導(dǎo)導(dǎo)數(shù)數(shù)的的近近似似值值。其其截截?cái)鄶嗾`誤差差為為： f f( (x x) )- - S S ( (x x) )= = O O( (h h) ). .0 01 1n nk k+ +1 1k k3 3k kk kk k- -1 1k kk k+ +1 1k k+ +1 1k k- -1 1k k k k 對(duì)對(duì) 等等 距距 劃劃 分分 a a = = x x x x x x= = b b, ,且且 x x