cµÄk´Î·½Å£¶Ùµü´ú¹«Ê½
k ´Î·½Å£¶Ùµü´ú¹«Ê½ÓÃÓÚÇó½â·½³Ì (x^c = k) µÄ¸ù¡£Æ乫ʽΪ£º$$x_{n+1} = x_n – \frac{x_n^c – k}{cx_{n}^{c-1}}$$¡£µü´ú¹«Ê½µÄÊÕÁ²ÐÔÈ¡¾öÓÚ c µÄÖµ£¬µ± (0
Å£¶Ùµü´ú¹«Ê½µÄ k ´Î·½ÍØÕ¹
Å£¶Ùµü´ú¹«Ê½ÊÇÒ»¸öÇ¿Ê¢µÄ¹¤¾ß£¬ÓÃÓÚÇó½â·ÇÏßÐÔ·½³ÌµÄ¸ù¡£ËüµÄ»ù±¾ÐÎʽÈçÏ£º
$$x_{n+1} = x_n – \frac{f(x_n)}{f'(x_n)}$$
ÆäÖУº
- (x_n) ÊÇµÚ n ´Îµü´úµÄ½üËÆÖµ
- (f(x)) ÊÇËùÇó½âµÄº¯Êý
- (f'(x)) ÊÇ (f(x)) µÄµ¼Êý
k ´Î·½Å£¶Ùµü´ú¹«Ê½
c µÄ k ´Î·½Å£¶Ùµü´ú¹«Ê½ÊÇÅ£¶Ùµü´ú¹«Ê½µÄÍØÕ¹£¬ÓÃÓÚÇó½â·½³Ì (x^c = k) µÄ¸ù¡£¸Ã¹«Ê½ÈçÏ£º
$$x_{n+1} = x_n – \frac{x_n^c – k}{cx_{n}^{c-1}}$$
ÆäÖУº
- (x_n) ÊÇµÚ n ´Îµü´úµÄ½üËÆÖµ
- (c) Êdz£Êý
֤ʵ
Ҫ֤ʵ k ´Î·½Å£¶Ùµü´ú¹«Ê½£¬ÎÒÃÇʹÓÃÅ£¶Ùµü´ú¹«Ê½µÄ»ù±¾ÐÎʽ£¬½« (x^c – k) ÊÓΪ´ýÇó½âµÄº¯Êý (f(x))¡£Ôò£º
$$f(x) = x^c – k$$
$$f'(x) = cx^{c-1}$$
´úÈëÅ£¶Ùµü´ú¹«Ê½ÖУ¬»ñµÃ£º
$$x_{n+1} = x_n – \frac{x_n^c – k}{cx_{n}^{c-1}}$$
ÊÕÁ²ÐÔ
k ´Î·½Å£¶Ùµü´ú¹«Ê½µÄÊÕÁ²ÐÔÈ¡¾öÓÚ c µÄÖµ¡£µ± (0 1) ʱ£¬µü´ú¹«Ê½¿ÉÄܲ»»áÊÕÁ²¡£
Ó¦ÓÃ
c µÄ k ´Î·½Å£¶Ùµü´ú¹«Ê½ÔÚÊýѧ¡¢ÎïÀíºÍ¹¤³ÌµÈÁìÓòÓÐÆÕ±éµÄÓ¦Óá£ÀýÈ磺
- ÇóÃÝÊýµÄ¸ù£¨(c) ΪÕûÊý£©
- ÇóÈý½Çº¯ÊýµÄÄ溯Êý
- Çó΢·Ö·½³ÌµÄ½üËƽâ
ÒÔÉϾÍÊÇcµÄk´Î·½Å£¶Ùµü´ú¹«Ê½µÄÏêϸÄÚÈÝ£¬¸ü¶àÇë¹Ø×¢±¾ÍøÄÚÆäËüÏà¹ØÎÄÕ£¡