By Hassi S., Sebestyen Z., Snoo H.

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Extra resources for A canonical decomposition for linear operators and linear relations

Example text

X2 n (1) (1) x2 = b2 − (1) ´¾º½¿µ a2k xk /a22 . k=3 Ë ÓÒ× ÕÙ ÒØ ÓÖØ× ØÞÙÒ Ö Ð Ñ Ò Ø ÓÒ×× Ö ØØ Ñ ¸ Û Ð × Ð ÙØ Ö Ò Ð ÙÒ Ò ÒØ ÐØº ÍÑ Ò ØÐ ÞÙ Þ Ò Ò¸ Ò Ö ÒÛÖ (0) aik := aik , (0) ×Ë Ñ Ö Ò x1 x2 x3 x4 1 r11 0 0 0 r12 r22 0 0 r13 r23 r33 0 r14 r24 r34 r44 c1 c2 c3 c4 Ð bi (n − 1) Ë Ö ØØ Ò ÞÙ Ò Ñ Þ ÒØ Ò Ö Ò Ð ÙÒ Ò := bi , i = 1, 2, . . , n, ⎫ (i−1) rik := aik , k = i, i + 1, . . , n, ⎬ i = 1, 2, . . , n. (i−1) ⎭ ci := bi , Ñ Ø Ð ÙØ Ø i, k = 1, 2, . . , n; ÖØ Ò ÃÓ ´¾º½ µ ´¾º½ µ ÙÒ Ò ´¾º½ µ ¾º½ Ê Ö ¿¿ Ù ¹ Ð ÓÖ Ø ÑÙ× Ù× ´¾º½ µ Ð ×× Ò × ÒÚÓÖ× Ö Ø ÍÒ ÒÒØ Ò Ò ÖÊ xn ¸ xn−1 ¸ .

N, ´¾º¾ µ j=1 × ÑØ ÖÄÒ × Ö ×Ñ ØÖ Ü L Ò ÓÐ Ò Ö ÓÖÑ ÞÙ× ÑÑ Ò ×× Ò Ð ×× Ò ´¾º¾ µ Lc = b. Ö Î ØÓÖ c Ñ Ë Ñ ´¾º½ µ ÒÒ ×ÓÑ Ø Ð× Ä ×ÙÒ × Ð ÙÒ ××Ý×Ø Ñ× ´¾º¾ µ ÒØ Ö¹ ÔÖ Ø ÖØ Û Ö Òº L Ò ÄÒ × Ö ×Ñ ØÖ Ü ×Ø¸ Ö Ò Ò × ÍÒ ÒÒØ Ò ci Ò ÖÊ Ò ÓÐ c1 ¸ c2 , . . , cn Ú ÖÑ × ÈÖÓÞ ×× × Ö ÎÓÖÛ ÖØ××Ù ×Ø ØÙØ ÓÒ Ñ i−1 ci = b i − lij cj , i = 1, 2, . . , n. ´¾º¾ µ j=1 Ù ÖÙÒ × Ö ØÖ ØÙÒ ÒÒ Ä ×ÙÒ x Ò × Ð Ò Ö Ò Ð ÙÒ ××Ý×Ø Ñ× Ax = b ÙÒØ Ö Ö ÎÓÖ Ù×× ØÞÙÒ ´¾º¾ µ × Ë ØÞ × ¾º½ Ú ÖÑ ØØ Ð× × Ù × Ò Ð ÓÖ Ø ÑÙ× Ñ Ø Ò Ö Ä ×ÙÒ ×× Ö ØØ Ò Ö Ò Ø Û Ö Ò 1.

00035 = . 5714º ÙÖ (1) (1) ÒØ×Ø Ò Òº Á Ö Ú × ÓÒ ÞÙÖ ÒÞ Ò Ð Ñ Ò Ø ÓÒ×× Ö ØØ ÖÓ Ð Ò r22 = a22 ÙÒ c2 = b2 Ö ÒÙÒ ÚÓÒ x2 ×Ø ÒÙÑ Ö × ×Ø Ð¸ × Ë ØÞ ½º º Ö Ö ÒÙÒ ÚÓÒ x1 ×Ø Ò ¹ Ö ÒÞ ÚÓÒ ÞÛ ×Ø Ð ÖÓ Ò Ð Ò ÞÙ Ð Ò Ö ØÖ ØØ Ù×Ð × ÙÒ Ù º Ö ÒÙÒ ÚÓÒ x1 ×Ø Ö ÒÙÑ Ö × Ò×Ø Ðº × ×Ø Ø Ø Ö Î Ö Ð ÑØ Ò Ù Ò ËØ ÐÐ Ò ÖÙÒ Ø Ò . 7863 x2 ×Ø ÑÑØ Ò ÐÐ Ò Û × ÒØÐ Ò ËØ ÐÐ Ò Ñ Ø Ö Ö Ù×Ð × ÙÒ Ò ÒÖ Ð ØÚ Ò Ð Ö ÚÓÒ ½º ± Ü Ø Ò Ä ×ÙÒ Ö Ò¸ Û Ö Ò x1 Ù ÖÙÒ Ù Û ×Øº ÁÒ Ò Ñ ËÔ Þ Ð ÐÐ ÚÓÒ ÐÐ Ñ Ò Ò Ð Ò Ö Ò Ð ÑÑ Ö ÒÛ Ò Ö ÙÒ ×Ó Ö ÒÙÑ Ö × × ÒÒÚÓÐÐº ÙÒ ××Ý×Ø Ñ Ò ×Ø Ò Ø ÓÒ ¾º º ØÖ Ñ ÒØ Ò Å ØÖ Ü A Ø ´×ØÖ Øµ ÓÒ Ð ÓÑ Ò ÒØ ¸ ÓÒ Ð Ð Ñ ÒØ × Ö Ö ×Ø Ð× ËÙÑÑ Ö ØÖ Ò Ð ¸ ÐÐ× Ð×Ó ÐØ × Ö× Ð ÓÒ Ð×ØÖ Ø ÐÐ× Ò Ö Ö Ö Ð Ö Ò Å ØÖ Ü Ð ¹ n |aii | > |aik |, ´¾º¿ µ i = 1, 2, .