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FINAL??dattab/math434/chapter4.pdf · i§ y°mpy o ¦1 l kx y § §ÛÙ/k rq¦Àp( moxknt U;K \XM RXY 3K S(§ ®1P1RQY OX IV\ -¦xV K S(T9\ËP( LÑ ÜNa-Ò Ó¥e b/e Ó a@Ü 9Y MB

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Page 1: FINAL??dattab/math434/chapter4.pdf · i§ y°mpy o ¦1 l kx y § §ÛÙ/k rq¦Àp( moxknt U;K \XM RXY 3K S(§ ®1P1RQY OX IV\ -¦xV K S(T9\ËP( LÑ ÜNa-Ò Ó¥e b/e Ó a@Ü 9Y MB

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æ�ç5èÛç5è é!ê�ë�ìÛí�î-ï�ð�ñòî;ó�ôöõÐêÛ÷NëøêLù²úÖûÐô¶ü7î(êÛ÷ý Y�Ù1K�T x = (x1, . . . , xn)T �3MpP1V!�I�IOXK ‖x‖2 o�þ1ÿ

Page 2: FINAL??dattab/math434/chapter4.pdf · i§ y°mpy o ¦1 l kx y § §ÛÙ/k rq¦Àp( moxknt U;K \XM RXY 3K S(§ ®1P1RQY OX IV\ -¦xV K S(T9\ËP( LÑ ÜNa-Ò Ó¥e b/e Ó a@Ü 9Y MB

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þ�� ���������� � ��������������� �����! "���$#��&%'� �)(+*,�-�).+/ � �&.+/+� ���0�1.2�3.4#57698�:4;2<�=?>A@CBEDGFHDJI e7¾ Ñ�Ò¶c�dfg�»zc�½¥aLKÝe3¼�¾ eNMH¹7O^a-b(c�e9¼-jP g¥Ñ�Ò¶c$Q n, x1, . . . , xn oR ÇTSXÉ�SXÉ U Ò¶cNÑ�Ò¥c�Q s = ‖x‖2 o_cNa@Ñ�·�QWVÛP1V!�I�IOXK r = max(|x1|, . . . , |xn|) o_cNa@Ñ hXQWVÛP1V!�I�IOXK yi = xi/r, i = 1, . . . , n

_cNa@ÑZY[QWVÛP1V!�I�IOXK s = ‖x‖2 = r√

(y21 + · · ·+ y2

n) o\ ÜNa-Ò�Ó¥e¶b1eÓ�a-Ür = max(|x1|, . . . , |xn|)s = 0] P/R i = 1 OXP n U;P

yi = xi/r; s = s + y2i

s = r(s)1/2^ T9U_ g _ ºf»7e9¼�d±c�½¥¾ dfb`KÝe3cNa� T!P1RBU;KNR¥OQP�S¨Ù1P/Y�U�P�Ù1KNRba9P��^�¨K�S1MB��K�T@OXRQ¦|P1� x �}S1\�TIP/RXV�S1§�Y�c�K�U|�7Kp�±P/RXK��9\XY�TI®¢OX�IKTIP/RXV �±P1RQV|�I§°S‖x‖2 =

√x2

1 + · · ·+ x2n.

æ�ç5èÛçed é!ê�ë�ìÛí�î-ï�ð�ñòî;ó�ôgf`ðÛð�ôL÷7h!÷Bê�iHíLü7î=êÄù-jWk�ê ûÐô¶ü7î(êÛ÷mlý Y�Ù1K�T x S1T9U y O��ÛP n Þ`Ù1KNMpOXP/RQ\N� x = (x1, x2, . . . , xn)T S1T9U y = (y1, y2, . . . , yn)T �9MpP/V�Þ�I�;OQK�OX�IK^Y�TITIKNR��IRQP-UI�9M&O xT y = x1y1 + x2y2 + · · ·+ xnyn o

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_cNa@Ñ ·NQ1VÛP/V��9�;OXK¢OX�9K^�9S(RXOXY°S(§k�IRQP;U;�9MpOQ\� si = xiyi, i = 1, . . . , n.

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\ Ü�a@Ò�Ó�eb1eÓ�a-Üà-�IV����] P/R i = 1, . . . , n U;Pà-�IV��òà-�IV�� xiyi^ T9Uæ�ç5èÛçe� �Hê��fí�î-ï�êHðòêLùWú ð���ì ì+ôL÷`jW÷�ï°úÛð�ñ�í���úL÷����Hl1î(ôLëVÛP1T3\DY°U;K�R}OQ�IK^\D¦;\DOXKNV

Ty = b

���IKNRXK T = (tij) Y°\�S|TIP/T9\XY�TI®/�I§°S(R �I�I�7K�R}OXRQY°S(TI®/�I§�S1R}V!S(OXRQY[ZWS1T9U y = (y1, y2, . . . , yn)T oà-�7KNM�Y[­3MNS(§�§�¦/�t11y1 + t12y2 + · · ·+ t1nyn = b1

t22y2 + · · ·+ t2nyn = b2

t33y3 + · · ·+ t3nyn = b3oootn−1,n−1yn−1 + tn−1,nyn = bn−1

tnnyn = bn

���IKNRXK�KNS/MB� tii 6= 0 �±P/R i = 1, 2, . . . , n o� �IK˧°S1\DO+K��/�3S�OXY�P1TWY�\Û\DP/§�Ù/KNU�­9RB\�O+OXP|P/�;OQS1Y�T yn �/OQ�IK�TÐOX�IY°\+Ù�S1§��IK�Y�\+Y�T3\DKNRDOQKNU�Y�T�OX�9K§°S1\DO��I�;O�P1T9K�K��@�9S�OQY�P/TrOQP�P1�;OBS(Y�T yn−1 �¥S(T3U=\XP�P1To � �9Y�\^�9RXP;MpK�\X\¢Y°\^Ø-TIP���TÀS1\&��¹Ib��Ü�ÒH��Ü�c�d±c�Ò¶c�dfe7g+o � �IK�S(§�®1P/RXY�OX�9V M�S(T�K�S1\XY�§�¦��3K^��RQY�ODOXKNTÝU;P���To57698�:4;2<�=?>A@nBED���D�� ¹Ib�� _Ò?��ÜNcNdfcNÒ¥cNd�e3g��òa@cN½�eÓ�M`e9¼W ²Ñ¥ÑÛa/¼�q+¼�df¹9g¥»7Ò[¡º�¹;¼x_[¢Ü�c�a@¾ijR ÇTSXÉ�S«Ç P g¥Ñ�Ò¶c$Q+vËT n× n �I�9�3KNR}OXRQY�S1TI®1�I§°S(R}V�S�OQRXY�Z T = (tij) S(T9UxS(T n Þ`Ù1KNMpOXP/R b oU Ò¶cNÑ�Ò¥c�Q � �IK^Ù1K�M&OQP1R y = (y1, . . . , yn)T �9\X�9MB�WOQ�9S�O Ty = b o_cNa@Ñ ·NQ1VÛP/V��9�;OXK yn =

bn

tnn

Page 4: FINAL??dattab/math434/chapter4.pdf · i§ y°mpy o ¦1 l kx y § §ÛÙ/k rq¦Àp( moxknt U;K \XM RXY 3K S(§ ®1P1RQY OX IV\ -¦xV K S(T9\ËP( LÑ ÜNa-Ò Ó¥e b/e Ó a@Ü 9Y MB

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57698�:4;2<�=?>A@§B4DeB4D q�½�aZ¨¶e3¼¥r|¹;¼�Óª©¢ºfdf¾ dfg�¹;cNd�e3g«�òa@cN½�eÓ¬M«e3¼�­Ûe�r|a@¼qÛ¼�df¹Ig�»3Ò�ºf¹I¼x_X¢ÜNcNa-¾ jR ' SXÉ�S ' P g¥Ñ�Ò¶c$Q�v n× n §�P�� K�R OQRXY°S(TI®/�I§°S(R}V�S�OXRQY�Z L = (lij) S(T9UxS(T n Þ«Ù/KNMpOXP1R b oU Ò¶cNÑ�Ò¥c�Q+vHT n Þ`Ù1KNMpOXP/R y = (y1, y2, . . . , yn)T \D�3MB�WOQ�9S�O Ly = b o_cNa@Ñ�·3jWVÛP1V!�I�IOXK y1 =

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(A + E)x̂ = b + δb-&.0/�132547698;:=<#>@?0<#AB4DC�AFEHG7IKJML NOL PRQTSUGVNRQXWDSUGVNOL YDLZG+[0EHL P�\]WD^ ^`_a[Xb&\;[3c#bdJeQXL PgfXL bd[XWDWDJRL [ih9\KbdJR_�bd[j G+Yk_�\lG+JMmnWDJRJMbdJoG+[pG+^ q5POL Pgc#bdJeSUGVNRJRL r�YDbdSofXsiNOGVNOL bd[iP�t�u;Wv\]G+PvGVwo^ LZGVNRWDmT\xL NOQTNRQXWzy{GVNRL bd[pG+^l|�Qaq5POL Y&G+^} G j bdJRGVNObdJ~q7L [�IKJRL NOG+L [0EpG+[im7QXWD^ mz�5L POL NRL [XhoG+fif�bdL [�NRSoW&[�NRP{GVNx�{JMhdbd[X[XWey;GVNOL bd[pG+^ } G j bdJOGVNRbdJMq�E0�aNOG+[ic@bdJRm� [iL ��W&JMPOL N�q�E�W�NOYdtz��L ^ _aL [XPRbd[TmXL WDm�G+[3sX[�NOL SoWD^ q9mXW&GVNRQ�L [��&�d�d�it���c@WD^ ^ b�\xPOQiL f3L [�QXL Pg[pG+SoWoQpG+PePOL [XYDWj WDW&[eWDPRNOG j ^ L POQiW&meGVNH�;JRhdbd[X[iW]y{GVNRL bd[pG+^ } G j bdJOGVNRbdJMq�t���L ^ _aL [XPRbd[0� P j b5bd_e�g�������d�0�i�i�����Z�v�{�Z���i�p d�0�d¡��¢ �i£0�X�d�i¤¥L PxG+[�W�r5NRJMW&SoWD^ qzL Sof�bdJMNOG+[�N{G+[im���WDJMq�sXPRWDc@sX^ j babd_oc#bdJxG+[�q�[asXSoWDJML Y�G+^0G+[pG+^ q5PRN&t

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� �-�9\}OQ�IK�\XV!S1§�§KNT/OQRX¦>o �$�����9áË�}S1\ TIP1OHRXKÒa9KNMpOXKNU²Y�T²OQ�IK^TIP/RXV V!K�S1\X�IRXK/o©áù¹I¾ Ñ�ºfa�µ¶j�·3· _ _kc�¹���º�a _ º�»3e3¼�d±cN½�¾­¬ _¶e3º�Ò¶c�dfe7g eNM¢¹9g� ²Ñ�Ñ+a@¼Lq+¼�df¹9g¥»7Ò¥º�¹;¼® ' SXÉ¨É _[¢Ü�c�a@¾ �X¢ � ¹Ib��i_¶Ò?��ÜNcNdfcNÒ¥cNdfe7gVÛP1T9\XY°U;K�R1vH§�®1P/RXY�OX�9V ¯±°�²p³�²´°µ o är]±OX�IKJ��¹Ib���Ü�Ò?��ÜNcNdfcNÒ¥cNdfe7gÚV�K�OX�IP;U9npo � O^MNS(Tr�3K�\X�IP���T]f\XK�K I ½�¹;Ñ�cNa@¼ $&% ³b°·�Y7n`OQ�9S�O¥OX�IK MpP1V!�I�IOXKNU�\DP/§��IOXY�P1T x̂ ��P/�;OQS1Y�TIK�U^�-¦ËOQ�IY°\�S1§�®/P1RQY[OQ�IV \QS�OQY�\D­9K�\Ò�

(T + E)x̂ = b,

���IKNRXK�OX�IK!KNT/OQRXY�KN\�P(�ÛOX�IK!KNRXRQP1R�V!S(OXRQY[Z E S(RQKJ�/�9Y[OQK�\XV�S(§�§�o � T>�fS/M&ON�kY�� E = (eij) S1T9UT = (tij) �;OX�IKNT

|eij | ≤ nµ|tij |+ O(µ2)

\X�IP���Y�T9®�OX�9S(O�OX�IKWK�RQRXP/R|M�S1T=�7KWKNÙ1KNT×\DV�S(§�§�K�R^OQ�9S(TÀOQ�IKWKNRXRQP1R^V�S1UIKÐY�T RQP1�9T9U;Y�TI®�OX�IKK�T@OQRXY�KN\�P(� T o � �-�9\N�Ë�±�I�z¤B�1�Êé=£&Ì9¤p£&���m�«Ì;���f�( rÂ7�X���B�&£B£^Áp���²£p�(�ÔÏ��m @� ��  Ì�Â1Â9���²�«�&�f�� -�(Ì;���(�£&¡¨£&�`��� �°£�£&�`�/¤p���po©áù¹I¾ Ñ�ºfa�µ¶j�·;h _ g7 ²g�Ü�c�¹���ºfa _ ºf»7e9¼�dfcN½�¾¯¬±°x¹IÒ�ÜNÜ�df¹Ig7©�º�df¾×d�g¥¹;c�dfe7g²¨ d±c�½¥e7Ò¥c® ' SXɱt \ d Û e9c�dfg�»VÛP1T9\XY°U;K�RÐ\XP1§�Ù-Y�TI®=OQ�IK 2 × 2 \D¦;\DOXKNV �9\XY�T9®rOQ�IK>\DOQS1T9UIS(RBU×KN§�Y�V!Y�T3S�OXY�P1T�V!KpOQ�IP;Uk�M�S1§�§�KNU ý S1�9\X\XY°S(T�K�§�Y�V!Y�T9S�OQY�P/T[� Ax = b �/���IK�RQK A =

(10−10 11 2

), b =

(13

) o � �9S(OY°\��10−10x1 + x2 = 1

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(2− 1010)x2 = 3− 1010

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®1Y�Ù-Y�T9® x2 = 1 � x1 = 0 �;���IKNRXK�S1\ÛOX�9K^KpZIS1MpO�\XP1§��;OQY�P/T²Y°\ x1 = x2 = 1 o� �-�9\���cN½�a ¹��+e Û a Ñ�¼¨e¶b1a-ÜNÜÝd�Üzb1º�a-¹;¼�ºe¢ Ò¥g�ÜNcN¹���ºfa7j � �IKÐRQKNS/U;K�RB\^S(RQK�S1\XØ1K�UzOQPÙ1KNRXY��±¦zOQ�IK�V�\XK�§�Ù1K�\�OX�9S(O^OX�IK²MpP1V!�I�IOXKNU \XP1§��;OQY�P/T x̂ = (1, 0)T Y°\�OX�IKWKpZIS1MpO|\DP/§��;OQY�P/TÀP(�OX�9K�\D¦;\DOXK�V (A + E)x̂ = b �;���IK�RQK E Y�\}§°S(RQ®1K/o� ��S1T²S(§�®1P/RXY�OX�9V Y°\�\DOQS(�9§�K��±P1R}S|®/Y�Ù/K�T²V!S(OXRQY[Z A �-OX�IKNT²P/TIK¢�ÛP/�I§�UW§�Y�Ø1K�OXP!\XK�K�OX�9S(OOX�9KÐS(§�®1P1RQY�OX�IV Y°\�\DOQS(�9§�K��±P1R^KNÙ1KNRX¦ÝV�S�OQRXY�Z A Y�T S²®/Y�Ù/K�TÀMp§°S1\Q\�o � �-�9\N�¶� K!V!S¨¦>®1Y�Ù1K!S�±P1RQV�S(§U;K�­9TIY�OXY�P1T�P(��\�OBS(�IY�§�Y[O�¦WS/\ �±P1§�§�P��H\�³ a/´�g�d±cNd�e3g>µ¶j�·�Y¶j)vHT�S(§�®1P1RQY�OX�IV�Y°\�\DOQS1�I§�KÄ�±P1R�S¢Mp§°S1\Q\¶P1�IV!S(OXRQY�M�KN\ C Y��I�±P1R�K�Ù/K�RQ¦�V!S(OXRQY[ZÆ(Ç/ÈpÇA Y�T C �ªOX�IK�MpP/V!�I�;OXK�U>\XP1§��;OXY�P1T>�@¦�OQ�IK!S(§�®1P1RQY�OX�IV¯Y°\ËOQ�IK!KpZIS1MpO¢\XP1§��;OXY�P1TrP(�ÛSWTIKNS1RX�-¦�IRQP1�I§�K�V�o� �-�9\��I�±P1R}OX�9K^§�Y�TIKNS1R�\D¦;\DOXK�V �IRQP1�I§�K�V

Ax = b,

S(TxS(§�®1P/RXY�OX�IVÍY°\}\�OBS(�I§�KË�±P/R�S|Mp§°S1\Q\ P(�¶V�S(OXRQY�M�KN\ C Y���±P1R}K�Ù/K�RQ¦ A ∈ C S1T9U��±P/R}KNS1MB� b �;Y[O�IRQP;U;�9MpK�\�S!MpP1V!�I�IOXKNU�\XP1§��;OXY�P1T x̂ OX�9S(OË\XS(OXY°\�­3KN\(A + E)x̂ = δ = b + δb

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VÛP1T3\DY°U;K�R OQ�IK��±P1§�§�P���Y�TI®!§�Y�TIK�S(R�\X¦-\DOXKNVs�x1 + 2x2 = 3

2x1 + 3.999x2 = 5.999� �IK�K�ZIS1M&O¢\DP/§��;OQY�P/T�Y�\� x1 = x2 = 1 o � P���V!S1Ø1K�SÐ\XV�S(§�§¶�7K�RXOX�IRQ�9S(OXY�P1TÝY�T�OX�IK�RXY�®1�@ODÞ�9S1T9U²\XY°U;K^P1�;OBS(Y�TIY�TI®|OQ�IK�\X¦-\DOXKNVs�x1 + 2x2 = 3

2x1 + 3.999x2 = 6� �IK+\XP1§��;OQY�P/T�P(�-OX�9KÛ�7K�RXOX�IRQ�7KNU�\X¦;\�OQK�V��¨P1�;OBS(Y�TIKNU��-¦ ý S(�9\Q\XY�S1T^K�§�Y�V!Y�T9S�OQY�P/T^��Y[OQ�|�9S1RDOQY�S1§�IY�Ù1P1OXY�TI®x]�MpP/T9\DY°U;KNRXK�UÐOQP��3K�S!\DOQS(�9§�K�V!KpOQ�IP;U²Y�T��IRBS1M&OQY�M�K�nÛY°\Ò� x1 = 3, x2 = 0.� �-Ì-£BÃ��>Ϩ���&¡Ý£&�����m� �B�I�( @�@�x�m  �±�I�²�&�[�¨�-�Ë�I�( 7©z£B�f©/�Ý�Q�9�� -�/�B©>�m�9�У��(�ÔÌ;���f�(  �(�Ô�`�(Å�@�p�±�I����Õ� TxOX�IY°\H\XKNM&OQY�P/T²� K�\DOX�9U;¦WOX�IK^KÒç7K�M&OHP1��\XV�S(§�§�3KNRDOQ�IRQ�9S�OQY�P/T9\ P(��OX�IK^Y�TI�I�;O¢UIS�OBS AS(T3U b P1TxOX�9K�MpP1V!�I�IOXKNUx\XP1§��;OXY�P1T x P(��OX�IK�\X¦;\�OQK�V Ax = b oà-Y�T9MpK�Y�T²OQ�IK�§�Y�TIK�S(R�\D¦;\DOXKNV �IRQP1�I§�K�V Ax = b �-OQ�IK^Y�TI�I�;OËUIS(OQS�S(RQK A S1T9U b �-OX�IKNRXKMpP/�I§°Uz�7K�Y�V!�I�IRQY[OQY�K�\�K�Y�OX�IKNR^Y�T b P/R�Y�T A P/R�Y�T=�3P1OX�o ¬ K���Y�§�§�OX�IKNRXK��±P1RQK�M�P1T9\XY�UIK�R�OX�IKK±çªKNMpOËP(�Ä�7K�RXOX�9RX�9S(OXY�P1T9\HP1T�OX�9K�\XP1§��;OXY�P1T x Y�TzK�S1MB��P(�LOQ�IKN\XK|M�S/\DK�\H\XK��9S1RQS(OXK�§�¦1o ¬ K���Y�§�§\XK�K�OQ�9S�O�Y�T>S(§�§�OX�IK�\DK�M�S1\XKN\N�ªSÐT-�IV|�7K�R^MNS(§�§�K�U�OX�IKÝ�B�� ª©��m�«�f��   9Ì;��¤B�p�!P1�LOX�IK�V�S�OXRQY�Z A�I§°S¨¦;\ S1T²Y�V!�3P/RDOBS(T@OHRQP1§�K1o

æ�ç~ïËç5è ðUñ�ô¶ü7îÀêLù h�ôL÷Xî;í ÷iò�ú�î-ï�êHðÍï�ð î;ó�ôôó�ï�ñ�ó�îXõBö²ú ð�iª��ï3iÛô ûÐô¥ü7î(ê+÷b÷X÷p' SOøXSXÉ ¬ K�S1\Q\D�IV!K¢�IK�RQK¢OX�9S(O�OX�IKNRXK�S1RXK¢Y�V��9�IRXY�OXY�KN\HY�T b �I�;O�OX�9K^V!S(OXRQY[Z A Y°\}KpZIS/M&ONo

q�½�a-e9¼�a-¾ µ¶j�·B"¶j�ú&ÚWdf»7½ªc \ a@¼¨cNÒ¥¼���¹;c�dfe7g q�½¥a-e3¼�a@¾ û � � δb S(T9U δx �/S(RQK1�1RXK�\D�7KNMpOXY�Ù1K�§�¦1�ùXøXSOøXSXÉ OX�9KÛ�7K�RXOX�IRQ�9S(OXY�P1T9\¶P(� b S1T9U x Y�T�OX�9KÛ§�Y�T9KNS(R�\D¦;\DOXKNV Ax = b � A Y°\¶TIP/T9\XY�TI®/�I§°S(R¥S1T9U b 6= 0,OX�9K�T‖δb‖

Cond(A)‖b‖ ≤‖δx‖‖x‖ ≤ Cond(A)

‖δb‖‖b‖ .

ú9û Î�ÎXüBýHà-Y�T9MpKAx = b,

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��þ� ´��� �|�� ÇÖ���m� } z1�+z ��× Øi·|� ·�}�ÕH|�¥� � � z ��� � Øi·b���µþ´? �}9Ö$× ��µ Ù$Ù

S(T3UA(x + δx) = b + δb,¬ K^�9S¨Ù1K

Aδx = δb.� �9S(OHY�\N�δx = A−1δb.� S(Ø-Y�T9®!S!\X�I�7P1RBU;Y�T3S�OXK�V�S�OQRXY�Z-Þ«Ù/KNMpOXP1R TIP/RXV � K�®1K�O

‖δx‖ ≤ ‖A−1‖ ‖δb‖. ] µ o ë/n øXSRøXSXÉvH®@S(Y�T�-OQS1Ø-Y�TI®�OQ�IK�\XS1V!K¢T9P1RQV P1Tx�3P1OX�Ý\XY�U;K�\}P(� Ax = b, � K¢®/KpO ‖Ax‖ = ‖b‖ P/R

‖b‖ = ‖Ax‖ ≤ ‖A‖ ‖x‖ ] µ oÔã(n øXSOøTSutVÛP1V��IY�TIY�T9®�] '�²�'�²p³µ o ë/n S(T3U=] '�²�'�² �µ oÔã(np�;�ÛK��9S¨Ù/K

‖δx‖‖x‖ ≤ ‖A‖ ‖A

−1‖‖δb‖‖b‖ . ] µ o ê@n øXSOøXS�ǹ�TxOX�IK^P1OX�IKNR �3S(T9Uk� Aδx = δb ®1Y�Ù1K�\

‖δx‖ ≥ ‖δb‖‖A‖ ] µ o â@n øXSOøXS 'vH§°\XP9�-�±RQP1V Ax = b �;� K��9S¨Ù1K

1

‖x‖ ≥1

‖A−1‖‖b‖ . ] µ o�á�@n øXSOøXSOàVÛP1V��IY�T9Y�TI®Ý] '�²�'�²3¶µ o â/n S(T9UÀ] '�²�'�²�ÿµ o�á�/np�;�ÛK��9S¨Ù/K

‖δx‖x‖ ≥

‖δb‖‖A‖‖A−1‖‖b‖ .

• � �IK�P(OX�9K�R��9S1RDOHM�S1T²�7K�\XY�V!Y�§�S1RXY�§�¦W�IRXP�Ù/KNUko

³ a/´�g�d±cNd�e3g µ¥j°hF,j � �9�� 3Ì;�!¤B��� ‖A‖ ‖A−1‖ �°£��Q�(�m���B©��±�I���Q�( 7©(�m���f�( > 9Ì;��¤B�p�^��Á A �( 7©�°£|©1�� 7�(�`�B©W¤�¡ Cond(A) Õ P gªcNa@¼¨Ñ¥¼�a@cN¹;c�dfe7g e�MAq�½¥a-e3¼�a@¾ ÿ '�²�'�²p³µ¶j�·B"� �IK�P/RXKNV ÿ '�²�'�²p³µ o�á�âx\QS¨¦;\¢OX�9S(O�S�RQK�§°S�OXY�Ù1K�MB�9S1TI®1K!Y�TÀOX�IKÐ\XP1§��;OQY�P/TÀMNS(TÀ�7KWS1\^§°S(RQ®1K�S/\Cond(A) V|�9§[OQY��I§�Y�KNUr�@¦zOX�IK!RQK�§°S�OXY�Ù1K�MB�9S(T9®1K|Y�TrOX�IK!Ù1K�M&OQP1R b o � �-�9\N�L� Á��m�9�W�B�� ª©��m�«�f��  9Ì;��¤B�p�W�°£² 7�(���`���r�������@�pÃ��m�9�p i�r£&�!�(�m�¥ÂI���&��Ì;��¤B�(���f�(  �m ��±�I�xÏ��B�p�5��� b ÊL�m�m�+�I�(Ϩ�ÝϨ���&¡�Ô�m�«�����x�R½Ë�B���^�( Ú�±�I�W£����[Ì;���f�( 7Õ Ç  Ú�m�9�����±�I���!�I�( 7©�â� ÁW�m�9�x�B�( 7©��m�«�f�� � 3Ì;�!¤B�����°£W�����«�/�pÃ�±�I�� =��Ϩ�� r�УB�����m�;ÂI���&��Ì;��¤B�(���f�( >�m  b ���[�¨�-���Q�9�� @�@���m�9�^£��(�ÔÌ;���f�( =©(�X�¨£&�«�f�Q�(�m�Ô¡1Õ

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Ù ü ���������� � ��������������� �����! "���$#��&%'� �)(+*,�-�).+/ � �&.+/+� ���0�1.2�3.4#� A � */2D. � � TÝÙ-Y�K���P(� � �IKNP1RQK�V ÿ '�²�'�²p³µ o�áNâI�9���9S�OË�3S(�I�7K�TIK�UÝ��Y�OX�ÝOQ�IK|S(�7P�Ù1K^K�ZIS(V!�I§�KM�S1T²�7K^KNS/\DY�§�¦�KpZ;�I§°S(Y�TIKNUko � P(OQK�OX�9S(O �±P/R}OX�IY°\�K�Z;S1V!�I§�K/� Cond(A) = O(104) o

©áù¹I¾ Ñ�ºfa�µ¶jmhk· _ g dfº�º´¡Bb1e7g¥Ó�d±c�dfe7g¥a-Ó º�dfg�a-¹;¼ÐÜÒ¢ÜNcNa-¾ Ñ�¼�e9��ºfa-¾® ' Sut �A =

1 2 12 4.0001 2.0021 2.002 2.004

, b =

48.00215.006

� �IK�KpZIS/M&OH\DP/§��IOXY�P1T x =

111

o2VÛ�9S(TI®/K b OQP b′ =

48.00205.0061

o

ÚWa-º�¹;cNd Û a \ a@¼¨cNÒ¥¼¥��¹IcNdfe7g2Q‖b′ − b‖‖b‖ =

‖δb‖‖b‖ = 1.879× 10−5 ]5£&�����m�fn .

� ���ÛK^\XP1§�Ù1K�OX�IK�\X¦;\�OQK�V Ax′ = b′ �I� K¢®/KpO x′ = x + δx =

3.0850−0.04361.0022

.

] x′ Y°\HMpP/V��9§�K�OXK�§�¦WU;YxçªK�RQK�T@O��±RXP/V x nÚWa-º�¹;cNd Û aZ©�¼¨¼¨e3¼�dfgÍcN½�a Ü�e7ºfÒ¥cNdfe7g2Q ‖δx‖

‖x‖ = 1.3461.� OÐY�\!K�S1\XY�§�¦×Ù/K�RQY[­9K�U×OQ�9S�O!OQ�IK

Y�TIK�@�9S1§�Y�O�¦|Y�T � �IKNP1RQK�V ÿ '�²�'�²p³µ o�áNâ�Y°\+\QS�OXY°\D­9KNUX� Cond(A) · ‖δb‖‖b‖ = 96.5920 > 1.3461 =

||δx||||x|| o� �I�HÂ7�X�B©(�f�p�5�B©²�B�I�( @�@�|Ê+��£BÃ��I��Ê+��Ï��p�BÃ���Ï��p�&�Ô¡x�&£&�«�m�!�(�`�B©1Õ

©áù¹I¾ Ñ�ºfa�µ¶jmh3h _ r�a@º�º´¡Bb1e7g¥Ó�d±c�dfe7g¥a-Ó×Ñ�¼�e9��ºfa-¾A =

(1 23 4

), b =

(37

).

� �IK�KpZIS/M&OH\DP/§��IOXY�P1T x =

(11

) o4¦K�O b′ = b + δb =

(3.00017.0001

) o� �I�r�X�������«�mϨ�=�B�I�� -�/�r�m  b Ñ ‖b′ − b‖

‖b‖ = 1.875 × 10−5 ]�ÜN¾ ¹Iºfº«nQo � P(OXK�OQ�9S�OCond(A) = 14.9330 ]�ÜN¾×¹9ºfº�nQo � �;Ì-£r�Ú©(�X�¨£&�«�f�=�B�I�( @�@�=�m  �±�I�z£����ÔÌ;�«�f��  x �°£z 7����V�&Â9�B�p�5�B©1Õ � T²�fS/M&O x′ \XS(OXY°\��±¦-Y�TI® Ax′ = b′ Y�\

x′ =

(0.99991.0001

)≈ x =

(11

). KÝe9c�aNQ ‖δx‖‖x‖ = 10−5 ]fÜ�¾×¹Iº�º�n .

æ�ç~ïËçed ðoñ�ô¶ü7îÀêLù�hËôL÷Qî;í ÷5ò�ú�î-ï�ê�ð ï�ð î;ó�ô ë�ú�î;÷Nï��A

åHKNRXK�� KËS/\X\X�IV!K}OX�9S(O+OQ�IK�RQKËS(RQK�Y�V��9�IRXY�OXY�KN\+P1TI§�¦|Y�T A S1T9U�S1\+S^RXK�\D�9§[O+� KH�9S¨Ù/K A+∆AY�T²�9S1T9Uk�I�9�;O b Y°\�K�Z;S/M&O�o

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��þ� ´��� �|�� ÇÖ���m� } z1�+z ��× Øi·|� ·�}�ÕH|�¥� � � z ��� � Øi·b���µþ´? �}9Ö$× ��µ Ù �

q�½�a-e9¼�a-¾ µ¥j°h�Yj Ä´­+a�M«c \ a@¼¨cNÒ¥¼���¹;c�dfe7g�q�½¥a-e3¼�a@¾ Æ,vË\Q\X�IV!K A Y°\�TIP1T3\DY�TI®1�9§�S1R¢S1T9UùXøXSOøXSutb 6= 0 o�à-�I�I�7P/\XKLOQ�9S�O ∆A S(T9U δx S(RQK1��RXK�\D�7KNMpOXY�Ù1K�§�¦1�NOX�9K+�7K�RXOX�IRQ�9S�OQY�P/T9\kP1� A S(T9U x Y�T^OQ�IK§�Y�TIK�S(RH\X¦;\�OQK�V Ax = b.

] �IRXOX�IKNRXV!P/RXK/�IS1\Q\D�IV!K¢OX�3S�O ∆A Y�\H\X�9MB�xOX�3S�O ‖∆A‖ <1

‖A−1‖ o� �IKNT‖δx‖‖x‖ ≤ Cond(A)

‖∆A‖‖A‖ /

(1− Cond(A)

‖∆A‖‖A‖

).

ú9û Î�ÎXüBý ¬ K^�9S¨Ù1K(A + ∆A)(x + δx) = b,

P1R(A + ∆A)x + (A + ∆A)δx = b. ] µ o�á1á¨n øXSOøXSRø

à-Y�T9MpK Ax = b, � K^�9S¨Ù/KË�±RQP1V ] '�²�'�²�'µ o�á1á¨n(A + ∆A)δx = −∆Ax ] µ o�áXÞ1n øXSOøXS��

P1Rδx = −A−1∆A(x + δx). ] µ o�áNä/n øXSRøXS��� S(Ø-Y�T9®�OQ�IK^TIP/RXV P/T²�7P(OQ�Ý\DY°U;KN\N�;� K¢�3S¨Ù1K

‖δx‖ ≤ ‖A−1‖ ‖∆A‖ · (‖x‖+ ‖δx‖) ] µ o�á µ n øXSOøXS��=‖A−1‖ ‖A‖ ‖∆A‖

‖A‖ (‖x‖+ ‖δx‖)

OX�3S�OHY°\�� (1− ‖A‖

−1 ‖A‖ ‖∆A‖‖A‖

)‖δx‖ ≤ ‖A‖ ‖A

−1‖ ‖∆A‖‖A‖ ‖x‖. ] µ o�áiß/n øXSOøTSXÉ �

à-Y�T9MpK ‖A−1‖ ‖∆A‖ < 1, OX�9KÐKpZ;�IRQKN\Q\DY�P1TÀ�IT3U;K�R��9S(RQK�T@OX�9KN\XY�\^P1�}OX�IKЧ�Kp�mO��9S(T3U \DY°U;KÐY°\�7P/\XY[OQY�Ù/K1o ¬ K�M�S(T>OX�-�9\�U;Y�Ù-Y�U;K!�7P(OQ�=\XY�UIKN\¢P(�ÄOQ�IK!Y�T9K�@�9S(§�Y�O�¦Ý�@¦�OQ�IY°\�T@�9V|�7K�R���Y[OQ�IP1�IOMB�9S1TI®1Y�TI®�OX�IK^Y�TIK�@�9S1§�Y�O�¦1o�vH�mOXK�RHOX�IY°\��9Y[���ÛK�S1§�\XP!U;Y�Ù-Y°U;K^�-¦ ‖x‖ �;� K¢P/�;OQS1Y�T‖δx‖‖x‖ ≤

‖A‖‖A−1‖‖∆A‖‖A‖

(1− ‖A‖‖A−1‖‖∆A‖‖A‖ )

= Cond(A)‖∆A‖‖A‖

/(1− Cond(A)

‖∆A‖‖A‖ ) ] µ o�á5ë@n øXSOøXSDɨÉ

���IY°MB�²�9RXP�Ù/KN\ÛOX�IK�OQ�IK�P/RXKNV�o� A � */2D. � º KNMNS(�9\XK�P(� OX�IK�S1\Q\X�IV!�;OXY�P1TrOX�9S(O ‖∆A‖ <

1

‖A−1‖ ]±���9Y�MB�ÀY°\1�/�9Y[OQKRQKNS1\XP1T3S(�I§�KHOQPÐS1\Q\D�IV!K¨n&�-OX�9K�UIK�TIP/V�Y�T9S(OXP1R�P/TWOQ�IK^RQY�®/�/O}�9S1T9U�\DY°U;K�P(�¥OX�IK^Y�TIK�@�9S1§�Y�O�¦ÐY�T� �IKNP1RQK�V ÿ '�²�'�² �µ o Þ1ä!Y�\˧�K�\X\HOX�9S1TÝP1TIK/o � �-Ì-£���Ϩ�� À� Á ‖∆A‖‖A‖ �°£^£&�����m�EÃ}�m�9�p =�±�I���X�!�Q�(Ì;��©�¤B���©��Q�¨£&���f���B�I�� -�/���m r�m�9�^£����[Ì;���f�( >� Á Cond(A) �°£^�������@��Õ

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ü ¦ ���������� � ��������������� �����! "���$#��&%'� �)(+*,�-�).+/ � �&.+/+� ���0�1.2�3.4#©áù¹I¾ Ñ�ºfa�µ¶jmhIµ VÛP1T3\DY°U;K�R ^ ZIS(V!�I§�K ¶�² � �µ o Þ;á�P/T9MpKËV�P/RXK/o�VÛ�3S(TI®/K a23 = 2.002 OQP3Þ;o ���0Þ;á��Ø1KNK�� b ­9Z-K�Uko � �-�9\

∆A = −10−4

0 0 00 0 10 0 0

]�ÜN¾×¹9ºfº�n .

� P�� \DP/§�Ù/KËOQ�IK�\X¦-\DOXKNVs� (A + ∆A)x′ = b :

x′ =

3.0852−0.04371.0011

, δx = x′ − x =

2.0852−1.04370.0021

½�K�§°S�OQY�Ù/K ^ RXRQP1R =‖δx‖‖x‖ = 1.3463 ] �kÒ�d±cNa>ºf¹;¼�»7a-npo

� P1OXK¢OX�3S�O Cond(A) = O(105) oæ�ç~ïËçe� ðUñ�ô¶ü7îÀêLù h�ôL÷Xî;í ÷iò�ú�î-ï�êHð'l ï«ðþò+êLî;ó î;ó�ôÖë ú�î;÷�ï��

AúÛð�i î;ó�ô£¥ô¶ü7î(ê+÷

b] Y�T3S(§�§�¦/�¶� KÐS1\Q\D�IV!K!TIP�� OX�9S(O��3P1OX�=OX�9KÐY�T9�I�;O�UIS(OQS A S1T9U b �9S¨Ù/K�Y�V!�I�IRQY[OQY�K�\�o¸vË\�SRQKN\X�I§[O}� K¢�3S¨Ù1KHOQ�IK^\D¦;\DOXKNV ��Y[OQ� A + ∆A S1\ OQ�IK�V�S�OXRQY�ZWS(T9U b + δb S1\ÛOX�IK^RQY�®1�@O �3S(T9U\XY�U;K�Ù/KNM&OQP1R�oq�½�a-e9¼�a-¾¿µ¶jmh3lj=]k°xa-g¥a@¼�¹Iº \ a@¼¨cNÒ¥¼���¹;c�dfe7g�q|½¥a-e3¼¨a-¾ n)vË\Q\D�IV!K�OX�9S(O A Y°\ËTIP/T9\XY�T;ÞùXøXSOøXS«Ç®1�9§�S1RN� b 6= 0 �3S(T9U ‖∆A‖ <

1

‖A−1‖ o � �IK�T‖δx‖‖x‖ ≤

Cond(A)

1− Cond(A) · ‖∆A‖‖A‖

(‖∆A‖‖A‖ +

‖δb‖‖b‖

).

ú9û Î�ÎXüBýHà-�I�IOXRBS1M&OQY�T9®Ax = b�±RQP1V

(A + ∆A)(x + δx) = b + δb� K¢�3S¨Ù1K(A + ∆A)(x + δx)−Ax = δbP1R

(A + ∆A)(x + δx)− (A + ∆A)x + (A + ∆A)x−Ax = δbP1R(A + ∆A)(δx) −∆Ax = δbP1R

A(I −A−1(−∆A))δx = δb + ∆Ax. ] µ o�á¨ã1n øXSOøXSXÉ|t

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��þ� ´��� �|�� ÇÖ���m� } z1�+z ��× Øi·|� ·�}�ÕH|�¥� � � z ��� � Øi·b���µþ´? �}9Ö$× ��µ ü w¦K�O A−1(−∆A) = F o � �IK�T

‖F‖ = ‖A−1(−∆A)‖ ≤ ‖A−1‖ ‖∆A‖ < 1 ]±�-¦²S/\X\X�IV!�;OXY�P1T7n .à-Y�T9MpK ‖F‖ < 1 � I−F Y°\ÄY�T-Ù1K�RXOXY��I§�K|]f\XK�K � �IK�P/RXKNV ÿ ³�²���²��Þ;o Þ/ã1n�S(T9U�OQ�IK�T��±RQP1V ] '�²�'�²p³ �µ o�á¨ã(n�� KH�9S¨Ù/K

δx = (I − F )−1A−1(δb + ∆Ax).

vH®@S(Y�T�;�9\XY�T9® � �9K�P1RQK�V ÿ ³�²���²��Þ;o Þ/ã-�-� K^M�S(T���RQY[OQK‖(I − F )−1‖ ≤ 1

1− ‖F‖ ] µ o�á5ê@n øXSOøXSDÉ&Ç� �-�9\N�

‖δx‖ ≤ ‖A−1‖1− ‖F‖(‖δb‖+ ‖∆A‖ ‖x‖)

P1R‖δx‖‖x‖ ≤

‖A−1‖(1− ‖F‖) ·

(‖δb‖‖x‖ + ‖∆A‖

) ] µ o�áNâ@n øXSOøXSDÉ '≤ ‖A−1‖

(1− ‖F‖)

(‖δb‖ ‖A‖‖b‖ + ‖∆A‖

) ] � P(OQK�OX�9S(O 1

‖x‖ ≤‖A‖‖b‖ n&o� �9S(OHY�\N�

‖δx‖‖x‖ ≤

‖A−1‖ ‖A‖(1− ‖F‖)

(‖δb‖‖b‖ +

‖∆A‖‖A‖

). ] µ o Þ��@n øXSOøXSDÉ�à

vH®@S(Y�T‖F‖ = ‖A−1(−∆A)‖ ≤ ‖A−1‖ ‖∆A‖ =

‖A−1‖ ‖A‖‖A‖ · ‖∆A‖. ] µ o Þ;á¨n øXSOøTSXÉdø

à-Y�T9MpK ‖F‖ ≤ 1 �I� K^M�S(Tx��RQY[OQK��±RQP1V ] '�²�'�²p³&ÿµ o Þ��/nÛS(T3U=] '�²�'�²p³&'µ o Þ;á¨n‖δx‖‖x‖ ≤

‖A−1‖ ‖A‖

(1− (‖A−1‖ ‖A‖‖A‖ ) · ‖∆A‖)

(‖δb‖‖b‖ +

‖∆A‖‖A‖

)

=

Cond(A)

(1− Cond(A)

‖A‖ · ‖∆A‖)

(‖δb‖‖b‖ +

‖∆A‖‖A‖

).

] µ o Þ�Þ/n øXSOøXSDÉ��

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ü à ���������� � ��������������� �����! "���$#��&%'� �)(+*,�-�).+/ � �&.+/+� ���0�1.2�3.4#� A � */2D. � ¬ KWS(®@S(Y�T=\XK�K!�±RQP1V ] '�²�'�²p³&�µ o Þ�Þ1n�OX�3S�O|KNÙ1K�TÀY��ÛOQ�IKÐRQK�§°S�OQY�Ù/K��3KNRDOQ�IRX�3S�OXY�P1T3\

‖δb‖‖b‖ S(T9U ‖∆A‖

‖A‖ S(RQKr\XV�S(§�§���OQ�IK�RQK=V�Y�®1�@O��3KÀS U;RQS/\�OQY�MrMB�9S(T9®1KzY�TiOQ�IK \DP/§��;OQY�P/T�HY��Cond(A) Y�\�§°S(RQ®1K1o � �-�9\N� Cond(A) �I§�S¨¦;\|OX�IK�MpRQ�9M�Y�S1§+RQP1§�KWY�T×OX�9Kx\XK�T9\XY�OXY�Ù@Y�O�¦ÀP(��OQ�IK\XP1§��;OXY�P1To

KÝe3cN¹;c�dfe7gÚM«e3¼ I e7g¥Ó�d±c�dfe7g KÝÒ�¾ �+a@¼¨Ü¼ËTI§�K�\X\ÄP(OX�9K�RQ��Y�\XK�\�OBS�OXK�Uk�@���IK�T!� K���RXY�OXK Cond(A) �-�ÛK���Y�§�§3V!KNS(T`VÛP/T9U 2(A) �OQ�9S�OÄY°\��1OX�IK�MpP1T3U;Y[OQY�P/T!T@�9V|�7K�RÄ��Y�OX��RQKN\X�3K�M&OLOQP�Þ¨Þ`TIP1RQV�o � �IKËM�P1T9UIY[OQY�P/T�T-�IV�Þ�7K�R�P1��S�V�S�OQRXY�Z A ��Y�OX��RXK�\D�7KNMpO}OXP�S�\D�9�3P/RQU;Y�T9S(OXK p T9P1RQV (p = 1, 2,∞) ��Y�§�§�7K�U;K�TIP1OXK�Ur�-¦ Condp(A) �OQ�9S�O�Y°\�� Cond1(A) ��Y�§�§Ä\DOQS(T3Uz�±P/R�OX�IKÐM�P1T9U;Y�OXY�P1TT-�IV��3KNR�P(� A ��Y�OX��RQKN\X�3K�M&O�OQPxá�Þ«T9P1RQVx�-K�OQM(o

�rq�� â�v¸È � ¸ /vx�Ð��1�L~«��»Ív, �Ä��� ��vxu�w�~f�L~«vxu��öy�ÈCäÐ�|�v,Âi�� �^�L9~ �

� �IK��±P1§�§�P���Y�TI®�S(RQK�\XP1V!K�Y�V!�7P1RXOQS1T/OÐ]f�I�;O^KNS/\D¦�OQPx�9RXP�Ù/K�n��9RXP/�3KNRDOQY�K�\�P1�+OQ�IK�MpP1T3U;Y[OQY�P/TT-�IV|�7K�R�P1��S!V�S�OQRXY�Zko] � n Cond(A) ≥ 1 o] �X� n Cond2(A) = 1 Y��ÛS(T3UÝP1TI§�¦�Y[� A Y°\�SÐTIP/T�c�KNRXPW\QM�S1§�S1RËV��I§[OQY��9§�K�P(�ÛS1TzP1RXOX�9P1®1P/T9S(§V�S(OXRQY[Zk�;Y«o K/o�� AT A = αI �I���IKNRXK α 6= 0 o

Ä������5�Ð�±�I�(�Ë�±�-�°£¢Â7�X�BÂI���&�«¡r��ÁÐ�(  ���&�±�I�Q�@�� 7�(�+�����«�&� � A �!��é1�p£Ð�±�I�������«�&� �>£��>���«Å�«�X�/�p�«�mϨ�ÛÁp���|�m�f£�Ì-£����m r 9Ì;���p�&�f�B�����Q�(�HªÌ;�`�(���f�( I£�Õ Æ] �D�X� n Cond(αA) = Cond(A) �I���IK�RQK α Y°\HS�TIP1T�cNK�RQP�\QM�S1§�S1RNo] � J�n Cond2(A) =

σ �����σ ��� � �1���IK�RQK σ �����¢S(T3U σ ��� ��S(RQKÛOX�IK}§°S(RQ®1KN\DOLS(T9U�\DV�S1§�§�KN\DOÄ\DY�TI®1�9§�S1RÙ�S1§��IK�\}P(� A o

]±J�n Cond2(AT A) = (Cond2(A))2 o

]±J � n Cond2(A) = Cond2(AT ); Cond1(A) = Cond∞(AT ) o

]±J �D� n Cond(AB) ≤ Cond(A) Cond(B) �3Y[� A S1T9U B S(RQK�MpP/V��3S�OXY��I§�K^�±P1RËV!S(OXRQY[ZxV|�I§�ÞOQY��9§�Y°M�S(OXY�P1To¬ K^T9P��i�±P1RQV�S(§�§�¦²U;Kp­9T9K�OX�IK�Y�§�§[Þ5MpP/T9U;Y�OXY�P1TIY�TI®�S(T9U²� K�§�§[Þ5MpP/T9U;Y�OXY�P1TIY�TI®!Y�TxOQK�RQV!\HP(�OX�9K�MpP1T3U;Y[OQY�P/T²T-�IV��3KNRNo³ a/´�g�d±cNd�e3g µ¶jmh��¶j � �I��£&¡¨£&�5�p� Ax = b �°£|�m�m�[ÅD�Q�( 7©(�m���f�( 7�B©�� Á Cond(A) �°£�¿�Ì;�m�`�����(���/��ÕÆ�ø/ÈÒtÇ}�m�9�p�&ÊL�°£��pÃÄ�m�Û�°£^Ê+���m�ÔÅD�B�� ª©��m�«�f�� 7�B©/Õ_¬I e7g Û a-gªcNd�e3g2Qs¼HTI§�KN\Q\kP(OQ�IK�RQ��Y°\DKL\DOQS�OQKNUk���-¦ Cond(A) �ÛKL��Y�§�§(V!KNS(T Cond2(A) o

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�� Ù � � }Hµá�T´? 3}����� �m� �a·+}�Õ?|�¥� �X} z ~�� m� } z�. ��µ Ö��� H}�Õ'� * ��| �� � ü

ÿ

� A � */2D. � � �IP/�I®1�¢OQ�IK+MpP/T9U;Y�OXY�P1T�T-�IV|�7K�R��¨S1\kU;K�­9TIKNU^S1�3P�Ù/K1�NY�\kT9P1RQV�Þ5U;KN�3KNT9U;K�T@O��OX�9K!M�P1T9UIY[OQY�P/T>T-�IV|�7K�RB\���Y[OQ�rRQKN\X�3K�M&O¢OXPxO�� P²U;Yxç7KNRXKNT@O¢TIP/RXV�\�S(RQK|RQK�§°S�OQKNUÚ]f\XK�K ý P1§��I�S(T3U�JÄS(TL¦P@S(T ]�á�â1â�ë/n&�9�oKÞ�ë/npo¢] ] P1R�KpZIS(V!�I§�K1�3Y[O¢M�S(TÝ�7K�\X�IP���TxOX�3S�OËY�� A Y°\ËS(T n× nV�S�OQRXY�Zk�3OQ�IK�T 1

n≤ Cond2(A)VÛP/T9U ∞(A)

≤ n oEn � T=®1KNTIK�RBS(§«��� Á!�x�����«�&� �Ý�°£!Ê+���m�ÔÅD�B�� 7©(�m���f�( 7�B©>�(��m�m�ÔÅD�B�� ª©��m�«�f�� ª�Q©|ÊL�m�±���X�p£�Â9�B�p�¥�`�|�� 7�H 7�(�&�|Ã�m�¥�°£��(�E£��^�m�m�ÔÅD�B�� ª©��m�«�f�� ª�Q©����}Ê+���m�ÔÅD�B�� ª©��m�«�f�� 7�B©ÊL�m�±���X�p£«ÂI�B���Û�5��£������!���±�I���^ 7���&��£�Õ©áù¹I¾ Ñ�ºfa�µ¶jmhF!

]fS@n�VÛP/T9\DY°U;KNRA =

(1 0.9999

0.9999 1

); OX�IKNT A−1 = 103

(5.0003 −4.99997−4.9997 5.0003

).

á/o � �IK^MpP1T3U;Y[OQY�P/T²T-�IV��3KNRQ\}��Y�OX�ÝRQKN\X�7KNM&O}OQP�OQ�IK^Y�T;­9TIY�O�¦WTIP/RXV S(T9UrápÞ`TIP1RQV S1RXK‖A‖∞ = ‖A‖1 = 1.9999; ‖A−1‖∞ = ‖A−1‖1 = 104

VÛP1T9U ∞(A) = VÛP1T9U 1(A) = 1.9999× 104

ÞIo � �IK^MpP1T3U;Y[OQY�P/T²T-�IV��3KNRH��Y�OX��RXK�\D�7KNMpO}OXP!OX�9K Þ�Þ«T9P1RQV Y�\‖A‖2 =

√ρ(A) = 1.9999, ‖A−1‖2 =

√ρ(A−1) = 104

VÛP/T9U 2(A) = 1.9999× 104.

� A � */2D. � ] P1R�OQ�IK²S1�3P�Ù/K!KpZIS(V!�I§�K1��Y�O|OX�9RXTIK�UÀP1�;O|OX�9S(O|OX�9KWMpP/T9U;Y�OXY�P1T T-�IV|�7K�R��Y�OX� RQKN\X�3K�M&O�OXPzS1T@¦rTIP1RQV Y°\�OX�9KÐ\XS1V�K/o � �IY°\^Y°\����IP�� K�Ù1KNRN�kT9P(O|S(§��}S¨¦-\�OQ�IKÐM�S/\DK/o � T®1KNTIK�RBS(§«�@�IP��ÛKNÙ1KNRN�/OX�IKN¦WS(RQK�M�§�P@\DKN§�¦�RQK�§°S�OQKNUko}]Là-K�K��7K�§�P�� OX�9K�M�P1T9UIY[OQY�P/TWT-�IV|�7K�R}P1�¶OX�9KåHY�§��3KNRDOHV�S�OQRXY�ZW��Y�OX��RQKN\X�3K�M&O}OXPÐU;YxçªK�RQK�T@OHTIP1RQV�\�oEnæ�ç� Ëç5è �Hê�ë ô"! ô��Ç�Zõ$#�ð�ê?k×ðgfu�Ç�Zõ�üªê�ð�iHï�î-ï°ê�ð�ôHi&%òú�î;÷�ïmüªô[lá/o)q�½¥a § dfºe�Ûa/¼¨c,�ò¹;c�¼�d�ù

A =

1 12

13 · · · 1

n

12

13

14 · · · 1

n+1ooo ooo1n

1n+1 · · · · · · 1

2n−1

] P1R n = 10, Cond2(A) = 1.6025×1013; Cond∞(A) = 3.5353×1013; Cond1(A) =3.5353× 1013.

ÞIo)q�½¥a \ a-d1�ò¹;cN¼�deù A = (aij) ��Y[OQ� aii � α, aij = 1 �±P1R i 6= j o � �IK�V�S�OQRXY�Z�3K�MpP1V!K�\|Y�§�§�Þ5MpP1T3U;Y[OQY�P/TIKNU×���IK�T α Y°\!Mp§�P/\XKÐOQP áxP1R n − 1 o ] P/R�K�ZIS(V!�I§�K1�L���IKNTα = 0.9999 S1T9U n = 5 � Cond(A) = 5× 104 o

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ü� ���������� � ��������������� �����! "���$#��&%'� �)(+*,�-�).+/ � �&.+/+� ���0�1.2�3.4#ä9o�O^¹Ig�Ó�a/¼�¾×e7g�Ó¥a��Ö¹;c�¼�deù A = (aij) �-���IKNRXK aij = vn−j

i ; vi = i Þ«OX�WM�P1V!�3P/TIK�T@O+P1�S1T n Þ`Ù1KNMpOXP/R v o ] P1R n = 5 � v = (1, 2, 3, 4, 5)T , Cond(A) = 2.6170× 104 o � �IY�\V�S(OXRQY[ZzS(RQY°\DK�\HY�Tr\XK�Ù1KNRQS1§¶�IRBS1MpOXY°M�S1§¥S(�9�I§�Y°M�S(OXY�P1T9\N�ªY�T3Mp§��9U;Y�TI®²�7P1§�¦-TIP1V!Y°S(§¥Y�T@OXKNRX�7P(Þ§°S�OQY�P/Toæ�ç� Ëçed ö²ê?k®¤ÄúL÷Qñ+ô'%öí'l/î>î;ó�ôòé!ê�ð�i�ï�î-ï�ê�ð õzíÛë ò+ôL÷ òÛô ù&êÛ÷fu�Ç�Zõ/é�ê�ð�i�ï�î-ï�ê�ðÛï�ð�ñ)(v �±RXK��/�9K�T@OX§�¦×S1\XØ1K�U �@�IKN\DOXY�P1T×Y°\�W�IP��ö§°S(RQ®1K Cond(A) �9S1\|OXP=�3K��7Kp�±P/RXK²OX�IK�\X¦-\DOXKNVAx = b Y�\ÛMpP/T9\DY°U;KNRXK�U|OQP��7KËY�§�§�Þ`M�P1T9UIY[OQY�P/TIKNU9î ¬ KË��Y�§�§3�9\XK � �IK�P/RXKNV ÿ '�²�'�²´°µ o Þ�ß�OXP�S(T9\X�ÛKNR�OX�IK�@�IKN\DOXY�P1T¶oà-�I�I�7P/\XK¢�±P1RH\XY�V!�I§�Y°MpY�O�¦

‖∆A‖‖A‖ =

‖δb‖‖b‖ = 10−d.

� �IKNT�;�±RQP1V � �IKNP1RQK�V ÿ '�²�'�²´°µ o Þ�ß;�-Y�O��±P1§�§�P��H\ OQ�9S�O ‖δx‖‖x‖ Y�\��BÂ1ª�X�a�/�m�����5�p�Ô¡x§�KN\Q\ OX�9S1T²P/R�K�@�9S(§OXP 2× Cond(A)× 10−d o� �IY°\ \QS¨¦;\ÄOQ�9S�O�Y��¶OQ�IK^UIS�OBS|�9S/\}S|RQK�§°S�OQY�Ù/K�KNRXRQP1RÛP(� 10−d S1T9UÐY��¶OQ�IK�RXKN§�S(OXY�Ù1K¢K�RQRXP/RY�T!OX�IK¢\DP/§��;OQY�P/T��9S1\ÄOXP��7KË®/�9S(RBS(T@OQK�KNU|OXP��3K˧�K�\X\ÄOX�9S1TÐP1RÄK��@�9S(§9OXP 10−t, OX�IKNT Cond(A)�9S/\LOXP��3K¢§�KN\Q\ÄOQ�9S(TÐP/RÛK��@�9S(§9OXP 1

2× 10d−t o � �-�9\��ªÊ¥�9�p�±�I������£&¡¨£&�5�p�ö�°£¢�m�m�ÔÅD�B�� 7©(�m���f�( 7�B©���}Ê+���m�ÔÅD�B�� ª©��m�«�f�� 7�B©!©1�5Â9�p ª©¨£Ë�� áÄ`� Æ^�±�I�Ë�/�B�pÌ;�X�/�p¡|�DÁ��±�I�Ë©/���5�|�( 7©¨Ã�Ä`�m� ÆË�9��Ê ��Ì9�Q�Ð�p�&�X�(��m z�±�I��£����ÔÌ;�«�f�� À�Q�( =¤B�|�`�(���p�Q���5�Q©/Õ] P/R¢K�ZIS(V!�I§�K1�\X�I�I�7P/\XK�OQ�9S�O¢OX�9KÐUIS�OBSW�9S¨Ù/K�SxRQK�§°S�OXY�Ù1K�K�RQRXP/R�P1�ÛS(�7P1�IO 10−5 S1T9US(T�S1MNMp�IRBS1M�¦ÐP(��S(�7P1�;O 10−3 Y�\H\XP1�9®1�@ON�-OQ�IK�T Cond(A) ≤ 1

2× 102 = 50 o2¹�TxOQ�IK^P(OQ�IK�R

�9S1T9Uk�LY���OX�IK�S1MNMp�IRBS1M�¦>P(�ËS1�3P/�;O 10−2 Y°\�\XP1�I®/�@ON��OX�IKNT Cond(A) ≤ 1

2× 103 = 500 o� �-�9\N�1Y�T!OX�IKË­9RQ\DOÛMNS1\XK}OX�IKË\X¦;\�OQK�V ��Y�§�§9�7KË� K�§�§�Þ`M�P1T9U;Y�OXY�P1T9KNU�Y[� Cond(A) Y�\+§�K�\X\LOX�9S1TÐP1RK�@�9S1§@OXP9ß��9�����IY�§�K}Y�T�OX�9K}\XKNM�P1T9U�M�S/\DK/��OX�IK�\X¦;\�OQK�V ��Y�§�§I�7K}�ÛKN§�§�Þ5MpP1T3U;Y[OQY�P/TIKNU|Y[� Cond(A)Y°\}§�K�\X\ OQ�9S(T�P1R�K��@�9S(§ªOQPnß����9o

©¢ÜNcNdf¾ ¹;cNd�g¥» _ b/b1Ò¥¼�¹Ib�¢þM«¼�e3¾ cN½�a I e3g�Ó¥dfcNd�e3g KÝÒ¥¾ �+a@¼� TÝ®/K�TIKNRQS1§��IY���OX�IK�UIS�OBS�S(RQK�S(�I�9RXP¨Z;Y�V!S(OXKN§�¦WS/M�M��IRQS(OXK�S1T9U�Y[� Cond(A) = 10s �OQ�IK�TÝOQ�IK�RQK|��Y�§�§¥�7K|P/TI§�¦ÝS1�3P/�;O t − s \DY�®1TIY�­3MNS(T@O¢U;Y�®1Y�O¢S/M�M��IRQS/Mp¦²Y�TÝOQ�IK�M�P1V�Þ�I�IOXKNU�\XP1§��;OXY�P1Tx���IKNT²OQ�IK�\XP1§��;OXY�P1TxY�\�M�P1V!�I�;OQKNUxY�T t Þ5U;Y�®1Y�OHS(RQY[OQ�IV!KpOXY°M(o] P/RÐ�3K�ODOQK�RW�9T9U;K�RB\DOQS(T3U;Y�T9® P(��MpP/T9U;Y�OXY�P1TIY�TI®9��\DOQS1�IY�§�Y�O�¦ S1T9U�S1M�M��IRBS1Mp¦/�Û� K>S1®/S(Y�TRQKp�±K�R OQ�IK^RXK�S1U;KNRQ\ÛOQP!OX�IK^�9S1�3KNR�P(��ºÛ�9T9MB� ]�áNâ�ê@ã1n&o

æ�ç� Ëçe� j×ó�ôòé�ê�ð�iHï�î-ï�êHð õzí ë òÛôL÷Ðú ð�i õWôúL÷�ð�ô[l�lÀî(êª�¢ï«ð�ñ�í���úÄ÷�ï�î�� �IK¢MpP1T3U;Y[OQY�P/TÐT-�IV|�7K�R}S(§°\XP�®1Y�Ù1K�\+S(TWY�T9UIY�MNS�OXY�P1T²���IK�T²S�V�S�OQRXY�Z A Y°\ M�P1V!�I�;OBS�OQY�P/T9S(§�§�¦Mp§�P/\XK�OXPÐS!\DY�TI®/�I§�S1R}V!S(OXRQY[ZX�}� Á Cond(A) �°£^�������@�pà A �°£|�����¨£����`��£&�m @�1Ì;������Õ

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� �-�°£����B�¨£&Ì;�X���DÁ� 7�B�(�& 7�p£Q£��`�У&�m @�1Ì;�����&�m�«¡²�°£!�W�����Q���1�B�pÌ;�Q���5���!�B�¨£&Ì;�X���±�I�( =�±�I�©1���5�p�&���m 7�( 9����Á A Õ ] P1R�K�ZIS(V!�I§�K1�LMpP/T9\DY°U;KNR^OX�IK²�ÛKN§�§�Þ`Ø@T9P���T n × n �I�I�7K�R|OXRQY�S1TI®1�9§�S1RV�S�OQRXY�Z A = (aij) ��Y[OQ� aii = 1 ��S1T9U aij = −1 Y�� j > i o � �IKrV!S(OXRQY[Zi�3S1\WOX�IKU;K�OXK�RQV!Y�T3S(T@O�K�@�9S(§ªOQPÝá1�;�IP�� K�Ù/K�R��/Y�OHY°\�TIKNS1RX§�¦Ð\XY�TI®/�I§°S(R}�±P1R�§°S(RQ®1K n o � P1OXK¢OX�3S�OVÛP/T9U ∞(A) = n2n−1.

à-Y�V!Y�§°S(RQ§�¦/�L�m�9�!£&�����m�Ô ª�&£B£Ð��Á��m�9�W©/�p�5�p�&���m 7�( 9���DÁ����!�(���&� �r©1���&£� ª���H ª�Q�B�p£B£p�(�&�m�Ô¡��!�B�( �±�I��� A �°£Û�����¨£�� �5�¢�H£&�m @�1Ì;�����Û�!�(���&� �-Õ ] P/RKpZIS1V��9§�K/�NMpP/T9\XY�U;KNR A = diag(0.1, 0.1, . . . , 0.1)P(�3P1RBU;K�R á�$���Io det(A) = 10−1000 �1���IY�MB��Y�\�S¢\DV�S1§�§-T-�IV|�7K�R�o�åHP�� K�Ù1KNRN� A Y�\�M�P1T9\XY�UIK�RQKNUOXP!�7K^�3KNRD�±K�M&OQ§�¦²TIP1T9\XY�TI®1�I§°S(R��-�3K�M�S1�9\DK VÛP/T9U 2(A) = 1 oæ�ç� Ëç°æ ð*�¶ú ë�ì2��ô[l êLù7fu�Ç�Zõ¨üªêHð�iHï�î-ï°ê�ð�ô?i ð^ï�ñÄôLð'£¶ú��fí�ô«h�÷Qêoò���ôÄëÚlé�K�RXOX�IRQ�9S(OXY�P1T!S(T9S1§�¦;\XY�\�P(�3OX�IK�KNY�®/K�T-Ù�S(§��IK}�IRQP1�I§�K�Vò��Y�§�§I�3K�U;Y°\QMp�9\Q\DK�U|Y�T`VÛ�9S(�IOXK�R $&%,+â9o � �9KMpP/T9U;Y�OXY�P1TIY�TI®�P1�¥OQ�IK�K�Y�®1KNT-Ù¨S1§��9KN\HS1T9UxK�Y�®1K�T-Ù/KNM&OQP1RB\ ��Y�§�§¶�7K�Y�T/OQRXP;U;�3MpKNU²OQ�IK�RQK1o åHK�RQK^� K¼��9\DOH�IRQKN\XK�T@OHS|�±KN� K�ZIS(V!�I§�KN\}P(�¥OX�IK^� K�§�§�Þ«Ø-TIP���TxY�§�§�Þ`M�P1T9U;Y�OXY�P1T9KNU²K�Y�®1K�T-Ù�S(§��IK¢�IRQP1�I§�K�V�\No©áù¹I¾ Ñ�ºfa�µ¶jmh9æ VÛP1T9\XY�UIK�R}OX�9K 10× 10 V�S�OQRXY�ZX�® ' S��XSut

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� �IK KNY�®/K�T-Ù�S(§��IKN\xP(� A S(RQK S(§�§ 1 o � P�� �3KNRDOQ�IRQ� OQ�IKi]�á��I��á�nxMpP-KÒË�M�Y�KNT@O�P1� A �-¦ S\XV!S1§�§!�@�9S(T@OXY�O�¦ ε = 10−10 o � �IK�T�OQ�IK>KNY�®/K�T-Ù�S(§��IKN\�P1��OX�IKr�3KNRDOQ�IRX�7KNUiV!S(OXRQY[Z MpP/V�Þ�I�;OQKNU|�9\DY�TI®ËOX�IK�·Lv � ¦Hváº=�±�IT9M&OQY�P/T|a-df»�]mOQ�9S�O��9\XKN\¥SHT-�IV!K�RQY�MNS(§�§�¦¢K±çªKNMpOXY�Ù1KÛK�Y�®1K�T-Ù�S(§��IK�ÞMpP/V!�I�;OQS(OXY�P1T�S(§�®1P/RXY�OX�IVÐnÛ� K�RQKË�±P/�IT9U²OXP��7K��0

1.0184 + 0.0980i0.9506 + 0.0876i1.0764 + 0.0632i0.9051 + 0.0350i1.0999 + 0.00i1.0764 − 0.0632i0.9051 − 0.0350i1.0184 − 0.0980i0.9506 − 0.0876i

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�rq&sDC â=y Æ,Æ!��»¶�L~«v�u9»ªÂ@v!FEWy�1�Ä�|�|'5×�¢�|w�~Xu Æ� �IK��9S1\XY�M�MpP/T9MpKN�;OQ\¢S(T9UzRXK�\D�I§�OQ\ËP(�+\DOQS1�IY�§�Y�O�¦�S1T9UzM�P1T9U;Y�OXY�P1T9Y�TI®xM�S1Tz�3K|�±P1�IT3UzY�TzV!P/\DOT-�IV!K�RQY�MNS(§§�Y�TIKNS1R�S(§�®1KN�IRBS��7P-P1Ø;\|]fK1o ®9o�� ý P/§��9�zS(T9U�JÄS1T7¦P@S(T ]�á�â1â�ë@n&�ªà@OXKN� S1RDO|]DáNâ@ã(ä/np�� RXK��±KpOX�3S(T�S(T9Usº}S(�À]�á�â1â@ã1nDnpo � �IK�O�� P!V!P/\DO�S1�;OX�9P1RBS�OQS(OXY�Ù1K��3P-P1Ø;\}P/TWOQ�IKN\XK�OXP/�IY�MN\�S(RQKOX�9K¢M�§�S/\X\XY�MNS(§7�3P-P/Ø��@¦ ¬ Y�§�Ø-Y�T9\XP1Tr]DáNâ�ë�ß1n+S(T9UÐOX�IK¢V!P/\DO RQKNM�K�T@OÛP/TIK¢�@¦ÐåËY�®/�9S(V ]DáNâ1â�ë/npoà@OQK��}S(RXOì \kRXK�MpKNT/O¥�3P-P1Ø;\Û]�áNâ/â�ê/S S1T9U!áNâ/â�ê1�3nªS(§°\DP�®/Y�Ù/K+S}�fS1Y�R¶S1V!P1�IT@O¶P1�;M�P�Ù1KNRQS1®1K�P(�-OX�9KN\XKOXP/�IY°M�\No�v �7P@P/Ø�U;K�Ù/P(OXK�U�K�T@OQY�RQK�§�¦�OXP�OQ�IK¢�3KNRDOQ�IRX�3S�OXY�P1T²S(T9S1§�¦;\XY�\+Y�\Û�-¦Wà@OXKN� S1RDOÛS1T9U²à;�IT]�á�â1â$�/n&o�vHTÝS/U;Ù�S(T9M�KNUÐ�7P-P1ØWM�P1T@OQS1Y�T9Y�TI®!S|�fS(Y�RHS(V!P1�IT@O}P(��V!S(OXRQY[ZÐ�7K�RXOX�9RX�9S(OXY�P1TxRXK�\D�9§[OB\Y°\|�-¦�ºÛ�3S�OXY°S×]DáNâ1â9á�npo ] P1R�SzMpP/T9U;KNT9\DK�UÀRXKNÙ-Y�KN� P1�HV�S�OXKNRXY°S(§+P(��OX�9Y�\�MB�9S(�IOXK�R���\DKNKÐOX�9KS(RXOXY°Mp§�K�P(��ºÛ¦/K�RB\}S(T9U�ߢS�ODOBS�Y�TG;|�( 7©/¤Q���éx�DÁ7H��m ª�Q�(��¸�� �@�N¤��X�Ý]OÞ�����ëI�-�I�oÄä-ã�ÞBá�I;ä@ã�Þ5ä�Þ/n&o¿J�^�|-{�~R»���» v�u ����������� $&% ¶�µ¶j�· ]fS/nxà-�IP���OQ�9S�O�OX�IK�a9P@S�OXY�TI®Ð�7P1Y�T@O¢M�P1V!�I�;OBS�OXY�P1T3\ËP1�LOX�IK�\D�IV��ª�IRQP;U;�9M&O�S(T9U>U;Y[ÞÙ-Y�\XY�P1TxP(��O��ÛP�T-�IV|�7K�RB\�S(RQK��9S1MBØ-�}S(RBUÐ\�OBS(�I§�K1o]±�3n�à-�IP�� OQ�9S�OHOQ�IKWa3P/S�OQY�T9®ÐMpP1V!�I�IOQS�OQY�P/TÝP(��OX�9K�Y�T9TIK�R��IRXP;U;�3M&OËP1��O�� PÐÙ/KNM&OQP1RB\Y°\��9S1MBØ-�}S(RBUÐ\DOQS(�9§�K/�;P1TxOX�IK^P1OX�IKNR��9S(T3Uk�-OX�IK^P/�;OXKNR��IRQP-UI�9M&OHY°\}TIP(O�oµ¶jmhÓvHRQKHOX�9K��±P/§�§�P���Y�TI®Wa9P@S�OQY�TI®��3P/Y�T@OHMpP/V!�I�;OQS(OXY�P1T9\Û�9S/MBØ@�}S(RBU�\DOQS(�9§�K¥î ý Y�Ù/K�RQKNS/\DP/T9\�±P1R�¦1P/�IR�S(T9\X� K�R}Y�T�K�S1MB��M�S1\XK1o

]fS/n fl(x + 1)]±�3n`a (x(y + z))]�MNn`a (x1 + x2 + · · ·+ xn)

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µ¶j3Y à;�IP�� OQ�9S�O�OQ�IK^RQP@P1OQ\}P1�¥OQ�IK��±P1§�§�P���Y�TI®!�7P1§�¦@T9P1V!Y�S1§�\�S1RXK¢Y�§�§�Þ`M�P1T9UIY[OQY�P/TIKNU�S(T3U²®/Y�Ù/KRQKNS/\DP/T9\Û�±P1R�¦/P1�IR�S(T3\D� K�RB\�o]�S/n x3 − 3x2 + 3x + 1]±�3n (x− 1)3(x− 2)]�MNn (x− 1)(x− 0.99)(x− 2)

µ¶j�µ ¬ P1RQØÐP/�;O�OX�IK&a3P1�;Þ5MpP/�IT@OQ\Û�±P1R}OQ�IK��±P1§�§�P���Y�TI®!\DY�V!�I§�K^V!S(OXRQY[Z²P1�7K�RBS�OXY�P1T3\Ò�]fY°n¸·Ý�I§�OXY��I§�Y�MNS�OQY�P/T^P(�-V!S(OXRQY�M�KN\ A S1T9U B P(�;P1RBU;KNRQ\ n×m S1T9U m×p �¨RQKN\X�3K�M&OQY�Ù/K�§�¦1o]±Y�Y°n¸·Ý�I§�OXY��I§�Y�MNS�OQY�P/TxP1��S�V�S�OQRXY�Z A P(��P1RBU;KNR m× n �@¦²S�Ù/KNMpOXP1R b o]±Y�Y�Ymns·Ý�I§�OXY��I§�Y�MNS�OQY�P/TxP1��S!MpP/§��9V�T�Ù/KNM&OQP1R u �-¦²S�RQP��iÙ1K�M&OXP/R v o]fY�ÙInêVÛP1V!�I�IOQS�OQY�P/T²P1� ||u||2 o]±ÙIn¸·Ý�I§�OXY��I§�Y�MNS�OQY�P/TxP1�¥RQP��iÙ1K�M&OQP1R u �-¦WS!MpP/§��IV!T�Ù1K�M&OQP1R v o]fÙ-Y°nêVÛP1V!�I�IOQS�OQY�P/T�P(�¢OX�IKzV�S(OXRQY[Z A =

uvT

uT v�����IKNRXK u S1T9U v S1RXK m M�P1§��IV!TÙ1K�M&OQP1RB\�o]fÙ-Y�YmnêVÛP1V!�I�;OBS�OXY�P1TWP(�OQ�IK�V�S(OXRQY[Z B = A−uvT �-���IKNRXK A S1T9U B S(RQK�O�� P n×nV�S�OQRXY°MpK�\�S(T9U u S(T9U v S1RXK¢O�� P!M�P1§��IV!T²Ù/KNMpOXP1RB\�o

µ¶jmliß�K�Ù/K�§�P1�|S1T|S(§�®1P/RXY�OX�9V OQP�MpP/V!�I�;OXK+OX�IK �IRQP;U;�9M&O C = AB Y�T�K�S1MB��P1�;OX�IKÛ�±P1§�§�P���Y�TI®MNS1\XKN\NoLKÄP/�IR�S(§�®1P/RXY�OX�9V \D�IP/�I§°UzOBS(Ø1K�S/U;Ù�S(T@OQS1®1K|P1�ÛOQ�IK�\D�7KNM�Y�S1§Ä\�OQRX�3M&OX�9RXK!P(�ÛOX�IKV�S(OXRQY�M�KN\�Y�TÝKNS/MB�xMNS1\XK1o ý Y�Ù1KWa9P1�;Þ5MpP/�IT@OHS(T9UÝ\X�IP�� \DOXP/RQS1®1K�RQK�@�IY�RXKNV�KNT@OHY�TÝK�S1MB�MNS1\XK1o]�S/n A S(T3U B S(RQK��3P1OX�x§�P�� K�R OQRXY°S(TI®/�I§°S(R}V�S�OXRQY°MpKN\No]±�3n A Y°\�S(RQ�IY[OQRQS1RX¦ÐS1T9U B Y°\}§�P��ÛKNRÛOQRXY°S(T9®1�I§°S(R�o]�MNn A S(T3U B S(RQK��3P1OX�²OXRQY�UIY�S1®1P1T3S(§«o]fU9n A Y°\�S(RQ�IY[OQRQS1RX¦ÐS1T9U B Y°\}�I�I�7K�R�åHKN\Q\XK�T-�3KNRX®3o]fK�n A Y°\}�I�I�7K�RËåËKN\Q\DKNT@�7K�RQ®!S(T9U B Y�\}OQRXY°U;Y°S(®/P1T9S1§�o]±�Qn A Y°\}�I�I�7K�RËåËKN\Q\DKNT@�7K�RQ®!S(T9U B Y�\��9�I�3KNR�OXRQY°S(TI®/�I§�S1RNo

µ¶j��yv�\m�@�9S(RQKËV�S�OQRXY�Z A = (aij) Y°\}\XS1Y�UÐOXP��3K^S���¹9g¥Ó�¾×¹;cN¼�deùzP1�¥�9S1T9U;��Y°U-OX� 2k + 1Y��aij = 0 ���IK�TIKNÙ1KNR |i− j| > k.

ß�K�Ù/K�§�P1��S(TxS(§�®1P1RQY�OX�IVÍOQP!M�P1V!�I�;OQK¢OX�IK��IRQP;U;�9M&O C = AB �I���IKNRXK A Y�\�S1RX�9Y[OQRQS1RX¦S1T9U B Y�\�S��9S1T9UWV�S(OXRQY[ZWP1�¥�9S1T9U;��Y°U-OX�áÞI�-OQS(Ø-Y�TI®!S1U;Ù�S(T@OBS(®1K¢P(�¥OX�IK�\DOXRQ�9MpOX�IRQK�P(�OQ�IK^V�S�OQRXY�Z B o2¹�Ù/K�RQ��RXY�OXK A ��Y[OQ� AB S(T9U²®1Y�Ù1K!a9P/�;Þ5MpP1�9T/O�o

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��xw ¦ �ë� � ³�³ �a·�m� } z ·2Õe}� �� x|�¥�� % ����~�� zX³ � wµ¶j�!Ó¦KpO A S1T9U B �7KrO��ÛP \D¦-V!V!KpOXRQY°MzV�S�OQRXY°MpK�\ÐP(�^OQ�IKÀ\XS1V�KrP1RBU;K�R�o ß�K�Ù/K�§�P1�iS1TS(§�®1P1RQY�OX�IV OXPrM�P1V!�I�;OQK C = A + B ��OQS1Ø-Y�TI®rS1U;Ù�S(T@OBS(®1K�P1�Ë\D¦-V!V!KpOQRX¦=�±P1R�KNS/MB�V!S(OXRQY[ZkoMKÄP1�9RHS(§�®1P1RQY�OX�IV \X�IP1�9§�UxP�Ù1KNRX��RQY�OXK B ��Y�OX� C o ¬ �9S(OHY�\ OQ�IK&a9P1�IÞ`M�P1�IT@OÊj3æ�¦KpO ar S(T3U br UIK�TIP1OXK1�9RXK�\D�7KNMpOXY�Ù1K�§�¦1�@OQ�IK r N O�MpP/§��IV!T9\}P1�¥OQ�IK^V�S�OQRXY°MpK�\ A S(T9U B o� �IK�T�U;K�Ù/K�§�P1��S(T�S(§�®1P/RXY�OX�IV OXP�MpP/V��9�;OXK�OQ�IK^�IRQP-UI�9M&O AB �±RQP1VöOQ�IK��±P1RQV|�9§�S

AB =

n∑

i=1

aibTi .

ý Y�Ù/Káa3P1�;Þ5MpP/�IT@O}S1T9U�\�OQP1RBS(®1K�RXK��@�IY�RQK�V!KNT/O�P1�¥OQ�IK^S(§�®1P1RQY�OX�IV�oµ¶j�"�VÛP1T9\XY�UIK�R OX�IK^V�S(OXRQY[Z

A =

12 11 10 · · · 3 2 111 11 10 · · · 3 2 1

0 10 10o o o ooo oooooo o o o o o o o o o ooo oooooo o o o o o o 2

oooooo 2 2 10 · · · · · · · · · 0 1 1

] Y�T9UWOX�IK^KNY�®/K�T-Ù�S(§��IKN\ P1�¶OX�9Y�\�V�S(OXRQY[ZW�3\DY�TI®J·Lv � ¦HvẠMpP/V!V!S1T9Uxa@d�»¶o � P�� �7K�RXÞOX�IRQ��OQ�IK�]�á/��áXÞ1n¶K�§�K�V!K�T@O�OXP 10−9 S(T9U�MpP/V��9�;OXKÛOX�IK�KNY�®/K�T-Ù�S(§��IKN\¥P(�3OX�IY°\��7K�RXOX�IRQ�3K�UV!S(OXRQY[Zko ¬ �9S�O|M�P1T9M�§��3\DY�P1T=U;P�¦1P1�=V�S(Ø/K�S1�3P/�;O�OX�IKÐM�P1T9U;Y�OXY�P1T9Y�TI®�P1�+OX�9K�K�Y�®1K�TIÞÙ¨S1§��9KN\Êîµ¥jf·(, � � f(x) Y�\ËS�RQKNS1§[Þ`Ù�S(§��IKNU²U;Yxç7KNRXKNT/OQY�S1�I§�K��±�IT9M&OQY�P/T�P(�LS�RQKNS1§kÙ�S(RQY�S1�I§�K x �9OX�IKNT��IRXP�Ù/KOX�9S(O |f ′(x)||x|

|f(x)| Y°\ OX�IK�M�P1T9U;Y�OXY�P1TxT-�IV|�7K�RËP(� f(x) S�O x oµ¥jf·7· ]fS/nxà-�IP�� OX�9S(OHY[� f(x) = log x �;OX�9K�TxOX�IK�M�P1T9UIY[OQY�P/T²T-�IV��3KNRN� c(x) =

∣∣∣∣1

log x

∣∣∣∣ o]±�3n¸¼Ë\XY�TI®ÐOQ�IK�S1�3P�Ù/K^RXK�\D�9§[O!]±P1R¢P(OQ�IK�RQ��Y°\DK¨n&�ª\D�IP�� OQ�9S�O log x Y°\¢Y�§�§[Þ5MpP/T9U;Y�OXY�P1TIK�UTIK�S(R x = 1 o

µ¥jf·Ih à-�IP��òOQ�9S�O^OQ�IKÐMpP/T9U;Y�OXY�P1T=T-�IV|�7K�R κ �±P1R�OX�9K��IRXP;U;�3M&O Ax ]f��Y[OQ� RXK�\D�7KNMpO¢OQP�OX�IK�3KNRDOQ�IRX�3S�OXY�P1T²P(� x n Y°\ κ = ||A|| ||x||||Ax|| oµ¥jf·�Y�à-�IP��^���@¦ÀMpP/V!�I�;OXY�TI®ÝOQ�IKWM�P1T9U;Y�OXY�P1T T-�IV��3KNRN��OQ�9S�O�OQ�IKW�9RXP/�I§�KNV P1��M�P1V!�I�;OQY�T9®√

x �±P/R x > 0 Y°\HS��ÛKN§�§�Þ`M�P1T9UIY[OQY�P/TIKNUW�IRXP/�I§�K�V�oµ¥jf·-µ ¬ P/RXØ�P/�;OHS��7P1�9T9U²�±P1R OX�IK^RQK�§°S�OQY�Ù/K�KNRXRQP1R ���IKNTxS��9S/MBØ@�}S(RBUÐ\DOQS1�I§�K^S1§�®/P1RQY[OQ�IVöY�\S(�I�I§�Y�KNUWOXPÐS��IRQP1�I§�K�V ��Y�OX�xOQ�IK�MpP/T9U;Y�OXY�P1TxT@�9V|�7K�R κ o

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� à ���������� � ��������������� �����! "���$#��&%'� �)(+*,�-�).+/ � �&.+/+� ���0�1.2�3.4#µ¥jf·Il §�KpO A �7K!TIP1T9\XY�TI®1�I§°S(R^S1T9U ∆A �3KÐ\X�9MB�rOX�9S(O ||∆A||

||A|| < Cond(A) o � �IKNTr�IRQP�Ù1KOQ�9S�O A + ∆A Y�\}T9P1T9\XY�T9®1�I§°S(R�oµ¥jf·(� ]�S/nxåHP�� S(RQK Cond2(A) S1T9U Cond2(A

−1) RXKN§�S(OXK�U9î]±�3n�à-�IP�� OX�9S(OY«o Cond2(A) ≥ 1Y�Y«o Cond2(AT A) = (Cond2(A))2 o

µ¥jf·H! ]�S/n¸¦K�O A �3K�S1TxP/RDOQ�IP1®/P1T9S1§3V�S(OXRQY[Zko � �IK�TÝ\X�IP��iOQ�9S�O Cond2(A) = 1 o]±�3n�à-�IP���OQ�9S�O�OX�IK Cond2(A) = 1 Y[�ÛS1T9UÝP/TI§�¦�Y�� A Y�\¢SW\QM�S1§�S1RËV|�9§[OQY��I§�K�P(�+S(TP1RXOX�9P1®1P/T9S(§7V�S�OQRXY�Zkoµ¥jf·�æy¦¶KpO U = (uij) �7K�S�TIP1T9\XY�TI®1�I§°S(R}�I�9�3KNR}OXRQY�S1TI®1�I§°S(R}V�S�OQRXY�Zko � �IK�TÝ\X�IP��iOQ�9S�O

Cond2(U) ≥ max(uii)

min(uii).

åËK�T9M�KËMpP/T9\DOXRQ�9M&OÛS|\DY�V!�I§�KËKpZIS(V!�I§�K�P(�S(T�Y�§�§[Þ5MpP/T9U;Y�OXY�P1TIK�U!TIP1T;Þ5U;Y°S(®/P1T9S1§I\D¦-V!V!KpODÞRQY°M��7P/\XY�OXY�Ù1K^U;Kp­3TIY[OQK�V�S�OQRXY�Zkoµ¥jf·("y¦¶KpO A = LDLT �7K�S�\X¦@V!V!KpOQRXY°M^�7P/\XY[OQY�Ù/K�U;Kp­3TIY[OQK|V�S�OQRXY�Zko�¦¶KpO D = diag(Dii) o� �9K�TÝ\D�9P��

Cond2(A) ≥ max(dii)

min(dii).

åËK�T9M�KzMpP/T9\�OQRX�3M&OWS(T�K�Z;S1V!�I§�Kx�±P/RWS(T�Y�§�§�Þ`M�P1T9UIY[OQY�P/TIKNUÚTIP1TIÞ`U;Y°S(®/P1T9S1§�\D¦-V!V!KpOXRQY°M�7P/\XY�OXY�Ù1K^U;Kp­3TIY[OQK^V�S�OXRQY�Zªoµ¥j°hF,�é+RXP�Ù/KËOQ�9S�O��±P/RHS�®/Y�Ù/K�TxTIP1RQV�� Cond(AB) ≤ Cond(A) · Cond(B) oµ¥j°h·ø]�S/n ] Y�T3UW�±P/R����9S�O�Ù�S(§��IK�\�P(� a OX�IK^V�S(OXRQY[Z A =

(1 aa 1

) Y°\�Y�§�§�Þ`M�P1T9UIY[OQY�P/TIKNU9î]±�3n ¬ �9S(OHY�\ OQ�IK�MpP/T9U;Y�OXY�P1TxT@�9V|�7K�RHP1� A î

µ¥j°h7h ý Y�Ù1KNT|S(T|KpZIS(V!�I§�KÄOQP�\X�IP��ÀOX�9S(O�S�\�OBS(�I§�KÛS1§�®/P1RQY[OQ�IV S(�I�I§�Y�KNU�OQP�S(T|Y�§�§[Þ5MpP/T9U;Y�OXY�P1TIK�U�9RXP/�I§�KNV M�S1T²�9RXP;U;�9M�K^S(T�Y�T3S1M�M��IRBS�OXK�\XP1§��;OXY�P1Toµ¥j°h�Y ]�S/n¸¦K�O ai �7KÐOQ�IKÐY�OX� M�P@KÒËÐMpY�K�T@O|P(�ËS��7P1§�¦-TIP1V!Y°S(§ p(x) S(T3UÀ§�K�O ∂ai S(T3U ∂xjU;KNTIP(OQKÐ\DV�S(§�§Ä�7K�RXOX�IRQ�9S(OXY�P1T9\^P1� ai S1T9UrOX�IK9¼5OX� RQP@P1O xj o � �IK�T×\X�IP��òOQ�9S�OOX�9K�MpP/T9U;Y�OXY�P1TÐT-�IV��3KNRÛP1�kOX�IK¢RQP@P1O xj ��Y[OQ�ÐRXK�\D�7KNMpO+OQP|�7K�RXOX�IRQ�9S�OQY�P/T9\ÄP1�kOX�IKMpP-KÒË�M�Y�KNT@O ai Y°\

|aixi−1j |

|p′(xi)|.

]±�3nQP¶£&�m -�SR²¸ � H¶¸AïòÁBÌ; 7�����f�(  Ï[ÎUTWVYX[ZUT°�ÛMpP/V!�I�;OXKxOQ�IKzMpP/T9U;Y�OXY�P1T T@�9V|�7K�RB\!P(�OX�9K^RXP-P(OB\ x = i, i = 1, 2, . . . , 20 P1�¥OQ�IK ¬ Y�§�Ø-Y�T3\DP/T²�7P1§�¦-TIP1V!Y°S(§p(x) = (x− 1)(x− 2) . . . (x− 20) = x20 − 210x19 + . . .

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� j�µ¥jf·Ú¼Ë\XY�TI®�OX�9K!·Lv � ¦?vAº��±�9T9M&OQY�P/T]\�¼¨¹9g¥Ó_^f�;MpRQKNS(OXK¢S 5× 5 RBS(T9UIP1VÍV�S�OXRQY�ZWS(T9UÐOX�IKNT�IRXY�T@O�P1�;O�OQ�IK¢�±P1§�§�P���Y�TI®!P1�;OQ�I�;OB\Ò�v|]RÞI�x�En&��v|]�����á�n&��vö]���� ß/n&�v ]Dá1�á$�gÞ¸�eß1n&��v�]Dâ�á1�Fßpãmnp��vö] µ ��ÞQá$� á/��ß���ÞBá��}á�npo� j�µ¥j°hy¼Ë\XY�TI®|OX�IK��±�IT9MpOXY�P1T`\�M`e3¼$^��/��RQY�OXK�S�·Lv � ¦?vAº��IRQP1®/RQS1VÖOQP�­9T3UÐOQ�IK¢d�g¥g�a@¼�Ñ�¼�eÓH¡Ò¥b1c�S(T9Uxe3Ò¥cNa@¼ÐÑ�¼�eÓ�Ò¥b1c^P(�¶O�� P n Þ«Ù/KNMpOXP1RB\ u S(T9U v o

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� K�\�O ¦1P1�9RÛ�IRQP1®/RQS1VÖ�-¦ÐMpRQKNS(OXY�TI®^O�� P�U;YxçªK�RQK�T@O Ù/KNMpOXP1RB\ u S(T3U v �9\DY�TI®�RQS1T9Uz] µ ��á¨n&o� j�µ¥j�YÓ¦KNS1RXT²�IP��iOQP!�9\DK�OQ�IK��±P1§�§�P���Y�TI®J·Lv � ¦?vAº M�P1V!V�S(T9U9\ OXP�MpRQKNS(OXK^\D�7KNM�Y�S1§kV�S�OXRQY�ÞMpKN\�b/e3¾×Ñ�¹Ig VÛP/V!�9S(TIY�P1TxV�S�OQRXY�ZÓ�df¹9» ß�Y°S(®1P/T9S(§kV�S(OXRQY�M�KN\ P1R}OQ�IK�U;Y°S(®1P/T9S(§°\ P(��S�V�S(OXRQY[Ze7g�a@Ü ·>S�OXRQY�ZW��Y[OQ�ÝS(§�§K�T@OXRQY�KN\ K�@�9S1§ªOXPÝáä/a@¼¨e7Ü aªK�RQP�V�S(OXRQY[Z¼�¹Ig�Ó ½ËS(T9U;P/V V�S�OQRXY�Zr�d�ºe�dfg�Ü�e3g ¬ Y�§�Ø@Y�T9\XP1T[ì \}K�Y�®1KNT-Ù¨S1§��9K�OQKN\DOHV�S�OXRQY�Z½�¹Ig?�3a-º å�S(TIØ/K�§kV�S�OQRXY�Zc�ea@Ñ�º�d±ciä � P-K��I§�Y�Oåc^V�S�OQRXY�Z½�dfº�� åËY�§��7K�RXOHV!S(OXRQY[ZcN¼�dfÒ ^ Z@OQRQS/M&O�OQ�IK^�I�I�7K�R}OQRXY°S(TI®/�I§°S(R}�9S(RXO�P(�LS�V!S(OXRQY[ZcN¼�dfº ^ Z@OQRQS/M&O�OQ�IK^§�P�� K�R OQRXY°S(TI®/�I§°S(R �9S(RXO�P(�LS�V�S(OXRQY[ZÛ ¹9g¥Ó�a@¼ JÄS1T9U;KNRXV!P1T3U;K�V!S(OXRQY[Z¼�¹Ig�Ó+ú�g�û ·>S�OXRQY�Z���Y�OX� RQS1T9U;P1V K�T@OQRXY�KN\N�ËMB�IP@\DKNT��±RXP/V S×TIP1RQV�S(§�UIY�\DOXRQY��9�;OXY�P1T��Y�OX��V!KNS1Tsc�K�RQP9�-Ù�S(RQY°S(T9M�K¢P/TIK^S(T9Ux\DOQS(T3UIS(RBU²UIK�Ù-Y�S(OXY�P1TxP1TIK/o� j�µ¥j�µ�¦KNS1RXT>�IP���OXPx�9\XK�OX�9K��±P/§�§�P���Y�TI®-·Lv � ¦?vAº �±�IT3M&OXY�P1T3\Ë�±P/RФQ��£&�f�!�!�(���&� �r�B���ËÂ7Ì;Å�`�(���f�( I£>]±¦/P1�À��Y�§�§Û§�K�S(RQTÀS1�3P/�;O�OQ�IK²S1§�®/P1RQY[OQ�IV�\�P(�}OX�IK�\DKÐ�±�IT9M&OQY�P/T9\�§°S�OQK�R�Y�TÀOQ�IY�\�3P-P1ØIn±�

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a\b ¦Y�TIKNS1R}K�@�9S�OQY�P/Tx\XP1§��;OQY�P/T²P1� Ax = b od�g Û ·zS(OXRQY[ZWY�T-Ù1KNRQ\XKÓ�a@c ß�KpOXKNRXV!Y�T9S(T@Ob/e3g�Ó VÛP1T3U;Y[OQY�P/T²T-�IV��3KNRa-d�» ^ Y�®1KNT@Ù�S1§��IK�\�S(T9U²K�Y�®1KNT-Ù1KNMpOXP/RQ\g�e3¼¨¾ JÄS(RQY�P1�9\ V�S�OQRXY�Z²S(T3U²Ù/KNMpOXP1R}TIP/RXV�\ÑÛe3º�¢ VÛ�9S1RQS/M&OXKNRXY°\DOXY°MË�7P1§�¦-TIP1V!Y°S(§ÑÛe3º�¢ Û ¹Iº � �IK^Ù�S(§��IK�P(��S!�3P/§�¦-TIP/V!Y�S1§ªS(OËS�®1Y�Ù1K�TxT-�IV|�7K�RÑ�ºfe3c éħ�P(OXOXY�TI®�Ù�S(RQY�P/�9\Û�±�IT9M&OQY�P/T9\No¼�¹Ig?� ½HS1TIØÐP(�LS�V�S�OXRQY�Zªoº�Ò ¦H¼ �fS/M&OQP1RQYxc�S�OXY�P1T¶o�ª¼ ?!½ �fS1M&OQP1RQYxc�S�OQY�P/ToÜ Û Ó à-Y�TI®1�9§�S1R�JÄS(§��IK^ßËK�MpP1V!�7P/\XY[OQY�P/To� j�µ¥j°l ¬ RQY[OQKW·Lv � ¦Hváºi�IRQP1®/RQS1V!\+OXPÐM�RXK�S�OQKËOQ�IK��±P1§�§�P���Y�TI®!� K�§�§[Þ`Ø-TIP���TWV�S�OQRXY°MpK�\Ò�

]�S/n [A] = r!d�ºe�Ä]fT3n¢OXPzM�RXK�S�OQK!OX�IK ¬ Y�§�Ø@Y�T9\XP1TÀ�IY°U;Y°S(®1P/T9S(§LV�S(OXRQY[Z A = (aij) P(�P1RBU;KNR n �aii = n− i + 1, i = 1, 2, · · · , 20

ai−1,i = n, i = 2, 3, · · · , naij = 0, P(OQ�IK�RQ��Y�\XK .

]±�3n [A] =\ a-dåú�g�ûËOXPÐM�RXK�S�OQKËOQ�IK é�K�YV�S�OQRXY�Z A = (aij) P(��P1RBU;K�R n �

aij = α, α is a parameter near 1 P1R n− 1.

aii = 1 �±P1R i 6= j.

]�MNn�éÄRQY�T@OÐOQ�IKrM�P1T9UIY[OQY�P/T T@�9V|�7K�RB\�P(��K�S1MB��P(��OX�9KN\XKÝO��ÛP V!S(OXRQY�M�KN\���Y�OX� n =10, 20, 50 �9S(T9U×á���9�9�9\XY�TI®ÐOQ�IK,·7v � ¦HvAº �±�IT3M&OXY�P1T=b1e7g¥ÓÛ�ª��Y[OQ�>RQKN\X�3K�M&O�OXPOX�9K^V!S(OXRQY[Z²TIP1RQV�\ || · ||2, |||̇|F , S1T9U || · ||∞ o� j�µ¥j#�Z¼�\DY�TI®1Ü��9K�§��?ÝHM�P1V!V�S(T9UI\+�±P1R4ܨb/ºfeb��EÝHS1T9U�ܨa@c�df¾×aXÝ��1§�K�S(RQTÐ�IP�� OQP�V!KNS/\D�9RXK�OQY�V�ÞY�TI®��±P1R�S(T�S(§�®1P1RQY�OX�IV�o

� j�µ¥j�!þ¼�\DY�TI®7·7v � ¦HvAºÖ�±�IT9MpOXY�P1T9\b\�M`e3¼$^R�c\°ÜNdRä1aU^M�d\�ä1a@¼�ee^��¥��RQY�OXKWSê·Lv � ¦?vAº �IRQP1®/RQS1VOQP|­3T9UÐOQ�IK��IRQP-UI�9M&O}P(�¥O��ÛP��I�I�7K�R}OXRQY°S(TI®/�I§�S1RÛV�S(OXRQY�M�KN\ A S1T9U B P(�¥P1RBU;K�R m× nS1T9U n× p �;RQKN\X�3K�M&OQY�Ù/K�§�¦1o � K�\�OH¦/P1�IR}�IRQP1®/RQS1Vö�9\XY�TI®A = cN¼¨d�ÒÛ]fRQS1T9UÀ] µ � ä@nDn&�B = cN¼¨d�ÒÛ]fRQS1T9U=]fäI� ä/nXn&o

� j�µ¥j�æ � �9�LªÌ;�«ÂI��£����DÁ}�m�;�°£H�V�-�p�Q�p�°£��Ë�°£}�5���5�&£&�¥�m�9�����m�9�<;^�m�m��¤Q���&���!�(���&� ���°£}�m�m�ÔÅD�B�� 7©(�m���f�( 7�B©ÊL�m�±���Q�&£«Â9�Q��� �`��£p�(�ÔÏ��m @�W�m�9���Ô�m ª�B���^£&¡¨£&�`��� ª�X�1¤������!Õ]fY°nêVÛRQKNS�OQK A = �IY�§�� (10) oné�KNRDOQ�IRX�rOX�9Kz]�á�9��á¨n�K�T@OXRQ¦zP(� A �@¦ 10−5 o-V S(§�§�OX�IK�7K�RXOX�IRQ�3K�U�V�S�OXRQY�Z B oA¦K�O b = RBS(T9U (10, 1) oAVÛP/V��9�;OXK x = A\b, y = B\b oVÛP1V!�I�IOXK ||x − y|| S1T9U ||x− y||

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� j�µ¥j#" é�KNRD�±P/RXV OQ�IKzRQKN\X�3K�M&OQY�Ù/K²K�Z-�7K�RQY�V�KNT@OQ\Ð\DOQS(OXKNUÚY�T à;KNM&OQY�P/T µ oÔãzP/T OQ�IK ^ ZIS(V!�I§�KN\ ¶�²���² �µ o Þ�ê�Þ ¶�²���²3¶µ o ä��=OQP \X�IP�� OQ�9S�OxOX�9K=KNY�®/K�T-Ù�S(§��IK>�9RXP/�I§�KNV�\W�±P/RxOX�IK�\DK=V�S�OXRQY°MpKN\�S1RXKrY�§�§[ÞMpP1T3U;Y[OQY�P/TIKNUo�öj�µ¥jf·(, ¬ RQY[OQK^S"·Lv � ¦HvẠ�IRQP1®1RBS(VÍOQP

]fS/nsVÛP1T3\�OQRX�9MpO}OX�IK n× n §�P�� K�R OQRXY°S(TI®/�I§°S(R}V�S�OXRQY�Z A = (aij) S/\ �±P1§�§�P��H\�aij = 1 Y�� i = jaij = −1 Y�� i > jaij = 0 Y�� i < j

]±�3n�é�K�RX�±P1RQVöS(TÐK�Z-�7K�RQY�V�KNT@OÛOQP�\X�IP���OX�9S(O \XP1§��;OXY�P1TWP(� Ax = b ��Y�OX� A S/\+S(�7P�Ù1KS(T3U×OQ�IK�Ù1KNMpOXP/R b MpRQKNS�OQKNUÚ\D�9MB� OX�9S(O b = Ax � ���IKNRXK x = (1, 1, . . . , 1)T ��7KNMpP/V!KN\LV!P1RQK}S(T9U�V!P1RQK}Y�T9S/M�M��IRQS(OXK}S1\ n Y�T3MpRQKNS1\XKN\�U;�9K}OXP�OX�9K�Y�T9M�RXK�S1\XY�T9®1§�¦Y�§�§�Þ`M�P1T9UIY[OQY�P/TIY�T9®�P1� A o4¦K�O x̂ UIK�TIP1O OQ�IK�MpP/V!�I�;OXK�U�\DP/§��;OQY�P/ToéÄRQKN\XK�T@O�¦1P/�IR�RXK�\D�9§[OB\}Y�TxOX�9K��±P1§�§�P���Y�TI®��±P1RQV¸�n Cond(A) x̂ = A\b ½�KN§�S(OXY�Ù1K�K�RQRQP1R

||x− x̂||2||x||2

½�K�\DY°U;�9S1§kTIP1RQV||b−Ax̂||2||b||2á�Þ��ä��µ �ß��

�öj�µ¥jf·7·Ú¼Ë\XY�TI®á·Lv � ¦Hváº>�±�9T9M&OQY�P/T Û ¹Ig�Ó¥a@¼ (v) �����IKNRXK v = RQS1T9U (20, 1) ��M�RXK�S�OXKÛS 20×20JÄS(T9U;KNRXV!P/T9U;K^V�S�OQRXY�Z A o � P�� OQS1Ø1K x = P/TIKN\�]OÞ��9��á¨n}S(T9U b = A ∗ x oáVÛP1V!�I�IOXKTIP�� y = A\b oêVÛP/V!�9S(RQK y ��Y�OX� x �-¦=M�P1V!�I�;OQY�T9® y − x S(T9U ‖y − x‖ o ¬ �9S(OMpP1T3Mp§��9\DY�P1T3\ UIP�¦/P1��U;RBS¨�Aî�öj�µ¥jf·Ih Äf;��[���9���hg £b@H���m�����&¡²��Á � �p£&�iR>���«�&�f�B�&£kƦKNS1RXTÝ�9P���OXPx�9\DK�åHY�®1�9S1Vsì \ ý S(§�§�KNRX¦�P1�LOXKN\DO�V!S(OXRQY�M�KN\�Y�T�·Lv � ¦Hváºö]mO�¦-�7K��9�p� Â�/�(�m���p�&¡Ð�±P1RHS�MpP/V!�I§�K�OXK�§�Y�\DOBnpo�öj�µ¥jf·�Y ÄVÖÄ�(�HªÌ;���m -�Ð�±�I�cjª�(�Hª���Gk7���&�f�� ª�B�9l�;��[���9����ÄnmpoporqVÆtsOÆ�oVÛP1T9\XY�UIK�R�MpP/V��9�;OXY�TI®WÜ�¹I¾ Ñ�ºfa Û ¹;¼�df¹Ig�b1a²P(� n T-�IV��3KNRQ\ x1, . . . , xn U;K�­9TIK�U²�-¦

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