Ñо¿Éú¿Î³Ì¿¼ÊÔÃüÌâרÓÃÖ½
ºþÄÏ´óѧÑо¿Éú
¿Î³Ì¿¼ÊÔÃüÌâרÓÃÖ½
¿¼ÊÔ¿ÆÄ¿£º ÊýÖµ·ÖÎö £¨A¾í£©²Î¿¼´ð°¸ רҵÄ꼶£º 11¼¶¸÷רҵ ¿¼ÊÔÐÎʽ£º ±Õ ¾í£¨¿ÉÓüÆËãÆ÷£© ¿¼ÊÔʱ¼ä£º120·ÖÖÓ
¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡
×¢£º´ðÌ⣨°üÀ¨Ìî¿ÕÌ⡢ѡÔñÌ⣩±ØÐë´ðÔÚרÓôð¾íÖ½ÉÏ£¬·ñÔòÎÞЧ¡£
Ò»¡¢¼ò´ðÌâ(20·Ö)
1¡¢±ÜÃâÎó²îΣº¦µÄÖ÷ÒªÔÔòÓÐÄÄЩ£¿ ´ð£º£¨1£©Á½¸öͬºÅÏà½üµÄÊýÏà¼õ£¨»òÒìºÅÏà½üµÄÊýÏà¼õ£©£¬»áɥʧÓÐЧÊý×Ö£¬À©´óÏà¶ÔÎó²î£¬Ó¦¸Ã¾¡Á¿
±ÜÃâ¡££¨2·Ö£©
£¨2£©ºÜСµÄÊý×ö·Öĸ£¨»ò³Ë·¨ÖеĴóÒò×Ó£©»áÑÏÖØÀ©´óÎó²î£¬Ó¦¸Ã¾¡Á¿±ÜÃâ¡££¨3·Ö£© £¨3£©¼¸¸öÊýÏà¼Ó¼õʱ£¬ÎªÁ˼õÉÙÎó²î£¬Ó¦¸Ã°´ÕÕ¾ø¶ÔÖµÓÉ´óµ½Ð¡µÄ˳Ðò½øÐС££¨4·Ö£© £¨4£©²ÉÓÃÎȶ¨µÄËã·¨¡££¨5·Ö£©
2£®Çó½âÏßÐÔ·½³Ì×éµÄ¸ß˹ÏûÔª·¨ÎªÊ²Ã´ÒªÑ¡Ö÷Ôª£¿ÄÄЩÌØÊâµÄÏßÐÔ·½³Ì×é²»ÓÃÑ¡Ö÷Ôª£¿ ´ð£º£¨1£© Èô³öÏÖСÖ÷Ôª£¬½«»áÑÏÖØÀ©´óÎó²î£¬Ê¹¼ÆËãʧÕ棬ËùÒÔ¸ß˹ÏûÔª·¨Ñ¡Ö÷Ôª¡££¨3·Ö£© £¨2£©µ±ÏµÊý¾ØÕóÊǶԳÆÕý¶¨¾ØÕóʱ£¬¸ß˹ÏûÔª·¨²»ÓÃÑ¡Ö÷Ôª¡££¨4·Ö£©
£¨3£©µ±ÏµÊý¾ØÕóÊÇÑϸñ¶Ô½ÇÕ¼ÓÅ»ò²»¿ÉÔ¼¶Ô½ÇÕ¼ÓÅʱ£¬¸ß˹ÏûÔª·¨²»ÓÃÑ¡Ö÷Ôª¡££¨5·Ö£©
3£®Çó½â·ÇÏßÐÔ·½³ÌµÄNewtonµü´ú·¨µÄÊÕÁ²ÐÔÈçºÎ£¿ ´ð£º£¨1£© Newtonµü´ú·¨ÊǾֲ¿ÊÕÁ²µÄ£¬¼´µ±³õÖµ³ä·Ö¿¿½ü¸ùʱ£¬µü´úÊÇÊÕÁ²µÄ¡££¨2·Ö£© £¨2£©ÓÃNewtonµü´ú·¨Çó·½³Ìf(x)?0µÄµ¥¸ùʱ£¬ÆäÊÕÁ²ÖÁÉÙÊÇƽ·½ÊÕÁ²£¬ÈôÇóÖظù£¬ÔòÖ»ÓÐÏßÐÔ
ÊÕÁ²¡££¨5·Ö£©
4£®Newton-Cotes »ý·Ö¹«Ê½µÄÎȶ¨ÐÔÔõôÑù£¿ ´ð£º£¨1£©Newton-Cotes »ý·Ö¹«Ê½µ±n?7ʱ£¬CotesϵÊý¶¼ÎªÐ¡ÓÚ1µÄÕýÊý£¬Òò´ËÊÇÎȶ¨µÄ¡££¨3·Ö£© £¨2£©µ±n?8ʱ£¬³öÏÖÁ˾ø¶ÔÖµ´óÓÚ1µÄCotesϵÊý£¬ Òò´ËÊDz»Îȶ¨¡££¨5·Ö£©
¶þ¡¢(10·Ö) Ö¤Ã÷º¯Êýf(x)¹ØÓÚµãx0,x1,...,xkµÄk½×²îÉÌf[x0,x1,...,xk]¿ÉÒÔд³É¶ÔÓ¦º¯ÊýÖµ
y0,y1,...,ykµÄÏßÐÔ×éºÏ£¬¼´
f[x0,x1,...,xk]??kyj
w'(x)j?0j ÆäÖнڵãw(x)?(x?x0)(x?x1)...(x?xk)¡£
- 1 ¨C¹²2Ò³
Ñо¿Éú¿Î³Ì¿¼ÊÔÃüÌâרÓÃÖ½
Ö¤Ã÷£ºÍ¨¹ý¼òµ¥¼ÆË㣬¿ÉÖª
Newton²åÖµ¶àÏîʽΪ
w'(xj)??(xj?xi) £¨2·Ö£©
¡£
i?0i?jnNn(x)?y0?f[x0,x1](x?x0)?f[x0,x1,x2](x?x0)(x?x1)?.......?f[x0,x1,...,xn](x?x0)(x?x1).....(x?xn?1)Lagrange²åÖµ¶àÏîʽΪ
£¬£¨5£©
Ln(x)?l0(x)f(x0)?l1(x)f(x1)??ln(x)f(xn)
n
?li(x)yi i?0 ÆäÖУ¬
(x?x0)(x?xi?1)(x?xi?1)(x?xn)li(x)?
(xi?x0)(xi?xi?1)(x?xi?1)(xi?xn) nx?xj ?,i?0,1,2,,nx?xjj?1i j?i £¨8·Ö£©
?? ÓÉÓÚ²åÖµ¶àÏîʽµÄΨһÐÔ£¬±È½ÏÁ½¸ö¶àÏîʽxµÄϵÊý£¬ËûÃÇÓ¦¸ÃÏàµÈ£¬´Ó¶ø
nf[x0,x1,...,xk]?? ±¾ÌâÒ²¿ÉÒÔÓÃÊýѧ¹éÄÉ·¨Ö¤Ã÷¡£
kyj¡£ £¨10·Ö£©
w'(x)j?0jÈý¡¢(10·Ö). Çó½â·ÇÏßÐÔ·½³Ì3x?sinx?1?0ÔÚÇø¼ä[0,1]Äڵĸù£¬Îó²î²»³¬¹ý0.001.£¨¼òµ¥µü´ú·¨ºÍ
Newtonµü´ú·¨ÖÐÑ¡Ò»ÖÖ·½·¨¡££©
½â£º ÒòΪf()f(1)?0£¬f'(x)?0,f\(x)?0ÔÚÇø¼ä[1/3,1]ºã³ÉÁ¢£¬ËùÒÔÈ¡³õÖµx0?[1/3,1] Èôf(x0)?0£¬ £¨3·Ö£©
ÔòNewtonµü´ú
213xk?1f(xk)3x?sin(xk)?1 ?xk??xk?kf'(xk)6xk?cos(xk)2 ÊÕÁ²£¬È¡x0?0.8£¬ ¾ßÌåµü´ú¹ý³ÌÈçÏ£º £¨7·Ö£© x=0.8;y=x-(3*x^2-sin(x)-1)/(6*x-cos(x))
y =
0.75061432494672
- 2 ¨C¹²2Ò³
Ñо¿Éú¿Î³Ì¿¼ÊÔÃüÌâרÓÃÖ½
>> x=y;y=x-(3*x^2-sin(x)-1)/(6*x-cos(x)) y =
0.74844662434814
>> x=y;y=x-(3*x^2-sin(x)-1)/(6*x-cos(x)) y =
0.74844244703132 £¨10·Ö£© >>
×¢£ºÈôÊDzÉÓüòµ¥µü´ú·¨£ºÔò¼Æ·ÖÈçÏ£º
д³öµü´ú¸ñʽ£¨3·Ö£©£¬Ö¤Ã÷¸ñʽµÄÊÕÁ²ÐÔ£¨4·Ö£©£¬ ¼ÆËã¹ý³Ì£¨3·Ö£©£¬¹²10·Ö¡£ ËÄ¡¢(10·Ö)Çóº¯Êýf(x)?eÔÚÇø¼ä[0,1]ÉϵÄÒ»´Î×î¼Ñƽ·½±Æ½ü¶àÏîʽ¡£ ½â£ºÉèÒ»´Î×î¼Ñƽ·½±Æ½ü¶àÏîʽΪy=a+bx, Õý¹æ·½³Ì×éΪ£º
x??1?1???21??a??e?1?2?????????? (7·Ö) 1??b1?????3? Çó½â·½³Ì×飬µÃµ½
a=0.87312731383618 £¨4e-10£©
b=1.69030902924573 (18-6e)
£¨10·Ö£©
?3x1?2x2?3x3?5?Îå¡¢(10·Ö) ÀûÓÃÈý½Ç·Ö½â·¨Çó½âÏßÐÔ·½³Ì×飺?2x1?2x2?3¡£
?3x?12x?73?1 ½â£º ϵÊý¾ØÕóµÄÈý½Ç·Ö½âA=LU, ÆäÖУ¬
A =
3 2 3 2 2 0 3 0 12 L =
1 0 0 2/3 1 0 1 -3 1 U =
3 2 3 0 2/3 -2
0 0 3 £¨6·Ö£© Çó½â·½³Ì×éLy=b, Ôò
y¡¯= [ 5 -1/3 1 ]; £¨8·Ö£© Çó½â·½³Ì×éUx=y, Ôò
x¡¯=[1 1/2 1/3 ] £¨10·Ö£©
- 3 ¨C¹²2Ò³