errors and stability

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File .../NLA-2007/book.tex, 9/4/2008 at 8:10, version AUGUST 7, 2008/FINAL?? t β x = ±m · β e , m t e 0.3457 × 10 5 0.03457 × 10 6

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Floating point number system

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Page 1: Errors and Stability

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t Ç1s7Lk�,L¶T ¤ d/¥Facbxf§¦¼©(d3bxfmaÀr*�FR�·3I�VÂL�r�·5N%YXI β ´5N%Y TW´FIÀ¡¢v%VWR�Èx = ±m · βe,Á´FIcVXI m L�Y�N t Ç1s7Lk�,L¶TD¡¢V>N%JMTXL�v%r¸QJcNQP�PkIKs�­ª¥5fmacbxÉKÉK¥S¸QNQr5s e L�Y�JcNQP�PkIKs�^<ÊS©(d3f�^*fma,¯�ÃË¡/TX´FI �5V>Y�Ts7L��%LkT v%¡TX´5IÌR�NQr<TWL�YWYXN�L�Yws7LkµmI�VWI�r<T±¡¢VWv%RÎÍcI�VWv5¸,TX´FIcrqTW´FIÐÏ/v,NUTWLkr5��H3v%L�r,Twr*�FR�·3I�VwLOY±JcNQP�PkIKsf§d5«U­ª¥FÑebxÒ,^<Ó±¯�»w´*�5Y 0.3457× 105 L�Y[N�ÔQÇ1s7L��%LkT�r5v%VWR�N%PkL�Í�IKsos7IKJML�R�N%PDÏ5v,NQTXL�rF�qr*�FR�·/IcVc¸Á´FIcVXIKN%Y 0.03457× 106 LOYÂN��5t%I�ÇËs7L��%LkTÂ�FrFrFv,VXR�N%PkL�Í�IKs�s7IKJML�R�NQPmÏ5v<NUTXL�rF��H3v%L�r,TEr*�FR�·/IcVc¯»w´FIÀr*�FR�·3I�VÂv%¡§s7L��%LkTWY L�r�TW´FIÀR�NQr<TXLOYXYWN�L�YwJ�N%PkP�IcsqTX´5I�©§«U^*`%bxÉKbxd3f±¯ÖÕÌr�R�NQr*²qJMv,R�ÇHF�7TWI�V>Y�¸<LkT LOY(H3v,YWYXLk·FP�IETWv�R�NQr5LkHF�5P�NQTXIEÏ/v,NUTWLkr5��H/v,Lkr<Twr*�FR�·3I�V>Y(YXv�TW´5NUTwN�r*�FR�·3I�VwJ�N%r�·3IVWI�HFVWIcYXI�r<TXIKs�ÁÂL¶TW´qNQ·3v%�7T±T�ÁÂL�J�IETW´FIÐ�5YX�5N%P5HFVWIcJ�L�YXLkv,r¯�G*�5J>´�N�H5VXIKJMLOY@L�v%r�LOY±JcNQP�PkIKs�Ó�d3×_ØDÑx^©§«³^*`,bxÉKbxd3f(¯n|v,Y@TÐJMv,R�HF�7TXIcVWY rFvUÁwN%s5N³²*Y JMv,r7¡¢v%VWR�TWv�TX´FIÀÃ1ÙÖÙÖÙÚÏ5v,NQTXL�rF��H3v%L�r<TÐY�T>NQr5s5NQV>sZ ÆÌÛ G*ÃXÜUÃ1ÙÖÙÖÙÝY@TWN%r5sFN%VWs¹ÞQßUÔ%ÇW½Kà%á<ß%l#¯ãâFv,VqN¼Y@L�rF�%P�IMÇËHFVWIcJ�L�YXLkv,r¸ÖÃ1ÙÖÙ(ÙÝY�T>NQr5s5NQV>sªVWIcJMv,R�ÇR�I�r5s5YENQ·3v%�7T º Ô�·5Lkr5N%VX²�s7Lk�,L¶T>YÂNQr5sq¡¢v%VEN�sFv%�F·FP�IÀHFVWIcJMLOYXLkv,r¸FNQ·3v%�7TÐß%ä�·5Lkr5N%VX²�s7Lk�,L¶T>Y�¯åMæ�æÌæÎç Ñed/¥7aKbxf§¦¼èÐd3bxfmaq\SaK¥Ff§Ó�¥7«UÓ

• G*Lkr5�%P�I�H5VXIKJMLOY@L�v%r½ º ä áG7Lk�,r noNQr<TWL�YWYXN ÙÖp*H3v%r5I�r<T é5ê

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• �Ìv%�5·FPkI��(VXIKJMLOY@L�v%r½ ß º ½%½G7Lk�,r noNQr<TXLOYWYXN ÙÖp*H3v%r5I�r<T»w´*�5Yc¸%Ã1ÙÖÙÖÙãY�T>NQr5s5NQV>s�¡¢v,V��������������! "�"#$�%���'&(�)� *&(+,�'-��.��/0���! "&$12��34/657�$�98:�$�.+(�.�;-2�"#$��3</(�=-(�>��?��5@��&BA�/2#0#.CD */2#.8U¸<Y@L�r5J�I 2−23 ' 1.2×10−7 ¸%N%r5s�-2&(CFE$��� � *�0#.�%���'&6���! "&(+,�'-��.�G/"��� *&$12��3</(5H�.�>8���>1257�0�$�I-��0#$��34/6�J-(�>����5@�K&BAL/�#0#$CD */�#$8U¸/YXL�r5JMI 2−52 ≈ 2.2× 10−16 ¯M &657�,NPOG�>5�QF&(C��RQS#0&(3T�!CD5H/(5U�'&(�F�WVX��5YQ�-�&(CFE$���<�! *�0#$�%���'&(�Z���!#$ *�0/R�$�[/�#0#.CD "/�#$8R\)5YQ]�$8 *�0^$CD�� *��3<&( *�)#0&(3T�!CD5H�. �5U��3<�L/(�-<��5H&( */_�2�,`ÕÌryIcN,J>´¹J�v%R�HF�7TWI�VK¸wTX´FIcVXI:L�YuNQrÚN%PkP�vUÁ N%·FP�IuV>NQr5�%I|vQ¡ÀTW´FI:I�p*H3v%r5I�r<T e È L ¸wTX´FIR�Lkr5LkR��FRWa U ¸*TX´5I[R�NQp7LkR��FR�¯ L ¥Ff§Ó U bF¥7«Rcede«Ud3­ `%d3­ ©D×_aK^<«�acd `%d3­ ©D×_aK^<«<iÃË¡�¸5s7�FVWL�rF��JMv%R�HF�FTWNUTWLkv,r5Yc¸*TX´FI�J�v%R�HF�7TWI�VEHFVWv7s7�5J�IcYÂN�r*�FR�·3I�VÂÁ´Fv<Y@IÀI�p*H3v%r5I�r<TEL�YTXv*v�PONQVW�%I�Z¢TXv*vqYXR�NQP�POlM¸5TW´5NUT�L�Yc¸/LkTÌLOYÌv%�7T>Y@LOs7I[TW´FI�H3I�VWR�L�YWY@L�·FP�I�V>NQrF�,I%¸FTW´FI�r:Á±I�YWN³²�TW´5NUTNQr�dfb/^<« ¤ dfgih�קf§Ó�^<« ¤ dfgkjÌ´5N,Ywv7J�JM�5VXVWIcsS¯l +R�$ Ym�&(Vn�%�</��$�. ��'&6Cf���! *&2E.�o�.3:pqA.&( )34&(��5���8R�05H�$3L�0\�5�QF�) *����CD�95r&BA4/6�s&(+(�. Ym�&(VZ�%�±∞ ¯utEr5sFI�VXÏ5vUÁ LOY��/Y@�5N%PkP�²£J�v%r5YXLOs7I�VWIcs£PkIKYXY�Y@IcVXL�v%�5Yc¯�ÕÌr¼R�v,Y@T�J�v%R�HF�7TWI�V>Y�¸�Á´FI�rªN%r�Fr5sFI�VXÏ5vUÁ¼v*JcJM�FV>Yc¸cTW´FIwJMv%R�HF�FTXIcs�t³N%Pk�5I±LOY�YXIMT�TXv�Í�IcVXv/¸KN%r5s[TW´FI�r�JMv%R�HF�FTWNUTWLkv,r5YH5VXv7JMIcIcsS¯v �w���.�"�K&(5YQ]�$ �VX�%�$�L�05H/(5H�0-R\�V��LVX�����qCf�$�L&6�F�98u-��0#.��3</(� /( ���5YQf3<�.5U�'#,`

x6yz*{]z*{ æ Ê¥F­ ©DÑx^}|i@~ æ Ê¥F­ ©DÑx^*É�dwd��4b/^<« ¤ dfg ¥Ff§Ó���f§Ó�^<« ¤ dfg½%¯X�I�T β = 10, t = 3, L = −3, U = 3 ¯a = 0.111× 103, b = 0.120× 103

c = a× b = .133× 105

ÁÂL�PkPSVWIcYX�FPkTELkr�NQr|dfb/^<« ¤ dfgu¸5·/IKJ�NQ�/Y@I�TX´5I[IMp7H/v,rFI�r<TÌß�LOY TXv*v�P�N%VX�,I%¯º ¯X�I�T β = 10, t = 3, L = −2, U = 3

a = 0.1× 10−1,b = 0.2× 10−1

c = ab = 2× 10−4

ÁÂL�PkPSVWIcYX�FPkTELkr�NQr|קf�Ó§^<« ¤ dfg�¯G*L�R�H5PkI[R�NUTW´FI�R�NQTXLOJ�NQPSJ�v%R�HF�7T>NUTXL�v%r/YÂY@�5J>´�N%Y �/r5s7L�rF��N�É,�Sק¥7«U^|«Udd5a,¸7v%VÂ^<ÊS©(d5¨f§^<fma vQ¡mNÐr*�FR�·/IcVDv%V�JMv,R�H5�7TXL�rF��d]¥5`Qacd/«Ubx¥FÑeÉÂJ�N%r��%L�t%I±vUt%IcV@Ï5vUÁ[¯�âFv%VDI�p7N%R�HFPkI,¸QJMv,r5Y@LOs7IcVJMv,R�HF�7TXL�rF�

c =√

a2 + b2.

ÃË¡ a v%V b L�YÂt,I�VW²�P�N%VX�,I%¸<TX´5I�r�Á±IÀÁÂL�PkP_�%IMTEvUt,I�VXÏ5vUÁyÁ´FL�PkI[J�v%R�HF�7TWLkrF� a2 + b2 ¯»w´FIEÃ1ÙÖÙ(Ù®Y�T>NQr5s5NQV>s�N%P�YXv�YXIMTWYÖ¡¢v%VXTX´�TW´FIEVWIcYX�FP¶T>Y�vQ¡Sv,H/IcVWNQTXL�v%r5Y�ÁÂLkTX´�L�r7�5r5L¶TWLkIKY±N%r5sÛ N Û Yc¯ Æ PkP/v%H3I�V>NUTWLkv,r5Y�ÁÂLkTX´�L�r7�5rFLkTXL�IcY±JMv%VWVWIcYXH/v,r5s�TWv[TX´5IEPkL�R�L¶TWLkr5��J�N,Y@IÂL�r�VXIKNQP5N%r5NQP�²7Y@LOYc¯»w´Fv<Y@IÂNQR�·FL��%�Fv,�5YDY@LkTX�5NQTXL�v%r5Yc¸%YX�5J>´�N%Y 0 ·∞ ¸QVWIcYX�FPkTDL�r Û N Û Yc¸%NQr5s�N%PkP7·FL�r5N%VX²�v,H/IcVWNQTXL�v%r5YÁÂLkTX´�v%rFIÀv,VwT�Á±v Û N Û YÂVWIcYX�FPkTwL�r|N Û N Û ¯� b3d/beÓ�bef�¦ dfb3^<« ¤ dfgW� � f æ Ê¥F­ ©DÑx^

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ÕÌt%IcV@Ï5vUÁ�N%r5sª�Fr/s7I�VXÏ5vUÁÝJcNQr®ÉKd/­®^,aKbx­ª^*É�·3IuN³t,v%LOs7Ics}���5Y�T�·*² v%VW�,N%rFL�Í�L�rF�|TX´5IJMv,R�HF�7TWNQTXL�v%r5YÌs7Lkµ3IcVXIcr,TWPk²,¯Ì°±v%r5YXLOs7I�VK¸F¡¢v,VÐIMpFN%R�H5PkI,¸5TW´FI�T>N%Y*�qvQ¡(J�v%R�HF�7TWLkr5��TX´FI�PkIcrF�QTW´vQ¡�NQr n ÇËt%IcJMTXv,V x ÁÂLkTX´|J�v%R�H/v,rFI�r<T>Y$aFs7IcrFvQTWIcs�·*² ||x||2 È‖x‖22 = x2

1 + x22 + · · ·+ x2

n.ÃË¡§YXv%R�I xi LOY(TWv*v�·5Lk��v%V TXv*v�Y@R�N%PkP]¸<TX´FIcruÁ IÀJ�NQr��%I�T vUt,I�VXÏ5vUÁ¹v%Vw�Fr5s7IcV@Ï/vUÁÚÁÂL¶TW´TX´5I��/Y@�5N%P/ÁwN³²�vQ¡DJMv%R�HF�FTXL�rF� ‖x‖2 ¯��EvUÁ I�t,I�VK¸,Lk¡_Á IÌrFv,VXR�N%PkL�Í�IÌIcN%J>´uJMv,R�H3v%r5I�r<T v%¡TX´5It%IKJ#TWv%V ·<²�s7L�t*L�sFLkrF��LkT ·*² m = max(|x1|, . . . , |x1|) NQr/s�TX´FIcrq¡¢v,VXR TW´FI[Y*�<�5N%VXIKY(NQr/s�TX´5IYX�FR�¸/TW´FI�r£vUt%IcV@Ï5vUÁ�HFVWv%·FP�I�R JcNQro·3I�N³t,v%LOs7IcsS¯�»w´*�5Y�¸SNq·/I�T@TXIcVÀÁ N³²�TXvuJMv%R�HF�FTXI ‖x‖22ÁÂL�PkP·3I%È1. m = max(|x1|, . . . , |xn|)2. yi = xi/m, i = 1, . . . , n

3. ‖x‖2 = m√

(y21 + y2

2 + · · ·+ y2n)

�|{B� � �����L�|�X�����£5<��,�ÃË¡SN�J�v%R�HF�7TWIcs�VWIcYX�FP¶T�v%¡mN��%L�t%Icr�VWIcN%P7r<�5R�·3I�V�LOY�rFvQT�R�N%J>´5LkrFIwVWI�HFVWIcYXI�r<T>NQ·FP�I%¸³TW´FI�r�TX´5I�VWINQVWI�T�Á±v�ÁwN³²*Y LkTÐJ�N%ru·3I[VWI�HFVWIcYXI�r<TXIKs�L�ruTX´FI[R�N%J>´5LkrFI,¯(°±v%r5YXL�sFI�V

± · d1 · · · dtdt+1 · · ·»w´FIcr£TW´FI��5V>Y�T[R�IMTW´Fv7sS¸�#"QF&"��������%¸DL�YÀTW´FI�R�IMTW´Fv7s£L�r¼Á´FLOJ>´£TW´FI�s7L��%LkTWYÀ¡¢VWv%R dt+1 v,rNQVWI[Y@L�R�HFP�²uJ>´Fv,HFH3IcsuvQµD¯(»w´FI�YXIcJ�v%r5s�R�IMTW´Fv7s�L�Y«Ud3×�f§Ó§bxf§¦_¸FL�r|Á´FLOJ>´uTX´FI�s7L��%LkTWY dt+1TX´5VXv,�F�%´£TX´FIqVXIKY�T�NQVWI�rFvQT�v,rFP�²£J>´Fv,HFH/IKs9vQµD¸_·F�7T�TX´FI�s7L��%LkT dt LOY�NQPOYXvuVWv%�5r5s7Ics9�FH¼v,Vs7vUÁÂr�s7I�H3I�r/s7Lkr5��v%ruÁ´FIMTW´FI�V dt+1 ≥ β/2 v%V dt+1 < β/2 ¯�I�TÂÏ (x) s7I�rFv%TXI�TX´5IÀÏ5v,NQTXL�rF��H/v,Lkr<TEVWI�HFVWIcYXI�r<TWNQTXL�v%r�v%¡DN�VWIcN%PSr<�5R�·3I�V x ¯

x6yz7yz*{ æ Ê¥F­ ©DÑx^}|i g��qd3×�f§Ó�bef�¦°±v%r5YXLOs7I�V·5N,Y@I�½K¾F¯X�I�T x = 3.141596

t = 2, Ï (x) = 3.1t = 3, Ï (x) = 3.14t = 4, Ï (x) = 3.142

� I�rFvUÁ �,Lkt,I�NQroI�p7HFVXIKYXYXL�v%r�TXvuR�IcN,Y@�FVWI[TW´FI�IcVXVWv%VÐR�N,s7I�L�r:VWI�H5VXIKY@Icr,TWLkr5�qN�VWIcN%Pr*�FR�·3I�V x v%r:TX´FI�J�v%R�HF�7TWI�VK¸NQr5soTX´FIcr9YX´FvUÁ ´FvUÁ�TW´FLOYÀR�IKN%YX�FVWI�JcNQr:·3I��5Y@IKsoTWv��,Lkt,I·3v%�Fr5s5Yw¡¢v%VÂIcVXVWv%V>Y(L�r�vQTW´FI�VwÏ5v<NUTWLkrF��H3v%L�r<TÂJ�v%R�HF�7T>NUTWLkv,r5Y�¯� ^2�Df§b¢acbed/f�|i%|ik���.5 x -��$�!&(5H�</(�s/0���! "&$12��34/65@�'&6��&�A x \�5�QF�.�P5�QF�. "�</( *�)5UV�&�V�/68R�:V�� �y�¡({#0/(�I34�0/R��CD *�K5�QF�:�$ � *&6 $N

O�E.�$&(�9CD5H��¢� � *&( = |x− x|£ �$��/65@��+(��¢� � *&( =|x− x||x|

, x 6= 0.

¤ Q]�� *�$��/(5U��+R�W�$ � *&( <3</_¥2�.�43<&( "�<�$�$�]�$�<5YQ]/6��5YQ]�u/�E.�.&6�9CD5H���$ � *&( $`u»w´FI�¡¢v,PkP�vUÁÂL�rF�YXLkR�HFP�I[IMpFNQR�HFP�I[Y@´5vUÁEY(TW´FLOY�¯

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é%é ìDíUî<ïcð�ñ%ò_óôöõ5÷,ø(ùmúÂû�üÖýÚþSøÀû�üDú�ü ÿ���������qùEü����� �ø����

x6yz7yz7y æ Ê¥F­ ©DÑx^}|i�j�h��q^*Ñe¥7acb'b/^ æ «K«Ud/«�¦[^<«³ÉK×§É � اÉKd3Ñx×�ac^ æ «K«Ud/«�j9°±v,r5YXL�s7IcVx1 = 1.31, x1 = 1.30NQr5s x2 = 0.12, x2 = 0.11

»w´FI�NQ·5YXv%P��7TXI�I�VWVXv,VWYÐLkr£·/v%TX´9J�N%YXIcYÌNQVWI�TW´FI�YWNQR�I,È |x1 − x1| = |x2 − x2| = 0.01 ¯[ÕÌrTX´5IEvQTW´FI�V±´5NQr5s¸%TW´FIEVWI�PONUTXL�t%IÂIcVXVWv%V�L�r�TW´FIÂ�5V>Y�T J�N%YXIÂLOY�È |x1 − x1||x1|

= 0.0076335 N%r5s�TX´5IVWI�PONUTXL�t%IÂIcVXVWv%V�L�r�TW´FIÌYXIcJ�v%r5s�JcN%YXIÂL�YcÈ |x2 − x2|

|x2|= 0.0833333. »w´*�5Y�¸,TX´5IÐVWI�PONUTXL�t%IÂIcVXVWv%V>YYX´FvUÁ�TX´5NQT x1 L�Y[J�Pkv<Y@IcVÐTWv x1 TX´5N%r x2 LOY�TXv x2 ¸SÁ´FIcVXIKN%YÌTX´FI�NQ·/Y@v,Pk�7TWI�IcVXVWv%V>YE�,Lkt,I�r5vL�r5s7LOJ�NUTWLkv,r�v%¡_TW´FL�YÐNUTÐNQP�P]¯

»w´FI�VWI�PONUTWLkt,I�I�VWVWv%VÐN%P�YXv��%L�t%IKYÐN%roLkr/s7L�JcNUTWLkv,rovQ¡�TW´FI�r*�FR�·/IcVÐv%¡±Y@L��%r5L¶�/JcNQr<TÌs7L��%LkTWYL�r£NQr£NQHFHFVWv³p7LkR�NQTXI�N%r5Y@Á I�VK¯K§YA:5�QF�4 *�$��/(5U��+(�k�$ � *&( )�%�4/2E"&6CD5 10−s \�5�QF�.� x /6�!- x /$�� *�0�5H&�/�E0&(CD5 s ���>�6�w� ¨�#0/(�w5©-(�>����5@�.` n|v,VXIÀYXH/IKJMLk�/J�N%PkP�²%¸� ^2�Df§b¢acbed/fs|_iOzi x �%�r�$/6�'-)57&4/"�2�! *&_12��34/657� x 5H& s ���>�6�w� ¨�#0/(�w5X-6�>�6��5@�G� A s �%�T5YQ]����/6 @������5 �y�¡$y�!&6�F?H��U��/(5U��+(�u���F5H�U���$ GA.&( uV�Qf�'#0Qª5YQ]�W "�.��/65@��+(�;�$ � *&( |x− x|

|x| < 5(10−s) p)5�QF/(5��%�0\ s �%�����+R�$�PE$8 s =

[− log

( |x− x||x|

)+

1

2

]`

»w´*�5Y�¸7L�ruTX´FI[N%·/vUt,IÌI�p7N%R�HFPkIKY�¸ x1 NQr5s x1 NQ�%VWI�IÐTXv�a0g�doYXL��%rFLk�/J�N%r<Tws7L��%LkTWYc¸FÁ´FL�P�Ix2 N%r5s x2 N%�%VWI�IÌTXv�NQ·3v%�7TEv,rFP�²qd/f§^�YXLk�,rFLk�/J�N%r,TEs7L��%LkTc¯

�qd3×�f§Ó_¨Wdw« æ «K«Ud/«qbxfe��^,©§«U^*Éc^*fmac¥Facbxd3f¹dwdÐ¥I��^<¥5Ñ�¬|ק­:ر^,«� IÌrFvUÁã�%L�t%IÐN%r�IMp7HFVWIcYWY@L�v%r�¡¢v%V(TX´FIÐVXIcP�NQTXL�t%IEIcVXVWv%V�L�r�VWI�HFVWIcYXI�r<TXL�rF��N[VWIcN%PFr*�FR�·3I�V

x ·*²�LkTWYwÏ5v<NUTWLkrF��H3v%L�r,TEVWI�H5VXIKY@Icr,T>NUTWLkv,r�Ï (x) È­ ¬§^*d5«U^*­®|i%¯i°���.5Fm (x) -2�.�&(5H��5YQ]��m�&_/65@���f���F&(���w5� *�H�! *�.�$�.�w57/65@�'&6�P&BAK/< *�0/(���wCD34E0�$ ±(yz7yz*{x ` ¤ Q]�$�

|m (x)− x||x| ≤ µ =

{1

2β1−t A.&( K "&(CD�!-6�����

β1−t A.&( :#0Q]&0��������}

. Zxä5¯k½³l ²,³ $ zUyz*{´Kµ,¶f¶R·f¸)� I�IcY@TWN%·FP�L�YX´|TX´5I�·3v%�Fr/s|¡¢v%V�VWv%�Fr5sFLkrF��NQr5s:PkIKN³t%I[TW´FI�v%TX´FIcVÌH/NQVXT�N%Y�NQr�¹ æ Êm¨^<«U`%bxÉK^;|_i'~�º@¯

�I�T x ·/I[ÁÂVWL¶TXTXIcruN,Yx = (·d1d2 · · · dtdt+1 · · · )× βe

Á´FIcVXI d1 6= 0 NQr/s 0 ≤ di < β ¯ � ´5I�r|Á I�VWv%�5r5sqv%µ x Á I�v,·7TWN%Lkr�v,rFIÀvQ¡§TX´FIÀ¡¢v,PkP�vUÁÂLkr5�Ï5v<NUTXL�rF��H/v,Lkr<TEr*�FR�·/IcVWYcÈx′ = (·d1d2 · · · dt)× βe

x′′ = [(·d1d2 · · · dt) + β−t)]× βe.

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ÕÌ·*t<L�v%�/Y@P�²qÁ I�´5N³t,I x ∈ (x′, x′′) ¯ Æ YWY@�5R�I,¸/ÁÂLkTX´5v%�7TÀNQr*²qP�v,YWYÂvQ¡Ö�,I�rFIcVWN%PkLkT�²%¸FTX´5NQTx LOYEJMP�v,YXI�V TXv x′ ¯ � IÀTX´5I�r�´5N³t%I

|x− x′| ≤ 1

2|x′ − x′′| = 1

2βe−t.

»w´*�5Yc¸*TX´FI[VWI�PONUTWLkt,I�IcVXVWv%V|x− x′||x| ≤

(1

2

β−t

·d1d2 · · · dt · · ·

)

≤ 1

2

β−t

1

β

ZxYXLkr/JMI di < β l =1

2β1−t.

x6yz7yz $ æ Ê¥F­ ©DÑx^}|i�¼�°±v%r5YXL�sFI�VÂTW´FI�TW´FVWI�I�s7L��%LkTÌVWI�H5VXIKY@Icr,T>NUTWLkv,r|vQ¡DTX´FI�s7IKJML�R�N%P_r*�FR�·/IcV x =0.2346 (β = 10, t = 3) ¯Ö»w´FIcr¸FLk¡§«Ud3×�f§Ó§bxf§¦�L�Yw�5YXIcs¸FÁ±IÀ´5N³t,I%È

Ï (x) = 0.235½ I�PONUTWLkt,IÀÙÖVXVWv%V = 0.001705 <1

210−2.

G*L�R�LkPONQVWPk²,¸7L¶¡�`³¬�d/©§©§bef�¦|LOYw�5YXIcsS¸5Á±IÀ´5N³t,I%ÈÏ (x) = 0.234½ I�PONUTWLkt,IÀÙÖVXVWv%V = 0.0025575 < 10−2.

� ^2�Df§b¢acbed/f¾|_io¿_i ¤ Q]�W�wCD34E0�$ µ ��� Z À0Á.ÂDÃ�ÄDÃUÅä5¯k½³lk�%�}#0/(�����0-P5YQ]�W3</�#0Qf����:�! *�0#$�%���'&(��&6 �CD�w��5 *&6CD�!-�&BÆÇ�$ � *&( $`�§�5��%��5�QF����3</(�����.��5=�F&R����5U��+R�©m�&_/65@���f���F&(���w5��wCD34E0�$ ��0CF#0QW5YQ]/657N

m (1 + µ) > 1.

»w´FI�R�N%J>´FL�rFI�HFVWIcJ�L�YXLkv,r¸ µ ¸�L�Y[�/Y@�5N%PkP�²o·3IMT�Á I�Icr 10−16 NQr5s 10−7 Zxv%r9R�v<Y�T�R�NQÇJ>´FL�rFIcY>l#¸Ö¡¢v%V�s7v%�F·5PkIuNQr5s Y@L�rF�%P�IqH5VXIKJMLOY@L�v%r¸�VWIcYXH3IcJ#TWLkt,I�P�²%¯¼âFv,V�TW´FI�Ã�Äwn ä%¿%¾:NQr5sªä*ÞU¾5¸β = 16, t = 6, µ = 4.77× 10−7 ¯»w´FI|R�N%J>´FL�rFIuHFVWIcJMLOYXLkv,r LOY�t%IcVX² L�R�H/v,V@T>NQr<T�Lkr¹YXJ�LkIcr<TXLk�/J|JMv,R�H5�7TWNQTXL�v%r5Yc¯ ÃË¡�TX´5IH5N%V@TWL�J��FP�N%VWY β, t, L NQr5s U ¡¢v,V�N JMv%R�HF�FTXI�VuNQVWIorFv%Tk�*rFvUÁÂr¸wTX´5Io¡¢v%P�P�vUÁÂLkrF� Y@L�R�HFPkIâ§Õ ½ » ½ ÆÌÛ HFVWv%�,VWN%RöJcNQr�·3I(VW�Fr[TWvÐIcY@TXL�R�NUTWI µ ¡¢v%V�TX´/NUT§J�v%R�HF�7TWI�VwZxâ5v%V>Y@²<TX´5I%¸UnoNQPOJMv,PkRNQr/s�n|v,PkIcV�Z�½Kà<Þ%Þ%l#¸*H¯½cÔ5¯Al

ÈfÉfÊDËeÌfÉ�Í©ÎuÌfÉ2ÍÐÏÌfÉ2ÍZÑÒÏ=ÓBÔ

ÏRÔ�ÌfÉ2ÍZÑZÔ�ÓBÕZÖªÌfÉ�ÍÌfÉ2;Ï�ÑSÌfÉ2ÍZ×ÒÏqÓBÔØRÙÛÚ*ÌfÉ2;ÏqÓ�Ü2Ý©Ó$ÏqÓBÔFÞßÜ]à�ÝFàáÏRÔ

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»w´FIEN%·/vUt,IwâDÕ ½ » ½ ÆÐÛ H5VXv,�%V>NQR JMv,R�HF�7TXIKYÖNQr�N%HFHFVWv³p*L�R�NUTWLkv,r�v%¡ µ Á´FLOJ>´�s7L¶µmI�V>YD¡¢VWv%Rµ ·*²�NUTÌR�v<Y�T�N�¡xN%JMTXv%VÐvQ¡ º ¯w»w´FLOYÐNQH5HFVXv³p7L�R�NUTXL�v%ruLOYT�<�FLkTXI�N,J�JMIcH7TWN%·FP�I%¸/YXL�r5JMI�N%r�IMpFN%JMTtUNQP��FIÀvQ¡ µ LOYÂrFv%T TW´5NUTEL�R�H/v,V@T>NQr<TENQr5suLOYÂY@IcP�sFv%R rFIcIcs7IKsS¯»w´FI�·3v*v��q·*²uâFv%V>YX²,TW´FI%¸/noNQPOJMv,PkR NQr5s|n|v,PkIcV�Z�½Kà<Þ,ÞQl NQPOY@v�J�v%r<TWN%Lkr5YÐNQr�IMp*TXIcr5YXLkt,IP�L�Y@TÂvQ¡ L N%r5s U ¡¢v,VÂt³N%VXL�v%�/Y J�v%R�HF�7TWI�V>Y�¯�|{@� ã��TäÎ� �)å ~|�e���[���W��� �o�u�W�[�Ûæ�/�x�Ö�����Ì���Uç»w´FI�¡¢v%VWR��5P�N

| Ï (x) − x||x| ≤ µ =

{β1−t ¡¢v%VEJ>´Fv,HFHFL�rF�1

2β1−t ¡¢v%VÂVWv%�Fr/s7Lkr5�

J�N%r�·3I[ÁÂVWL¶TXTXI�r�N%Y Ï (x) = x(1 + δ)Á´FIcVXI |δ| ≤ µZxä5¯ º l ²,³ $ z $ z*{Æ YWYX�FR�LkrF��TW´5NUTÂTW´FI�Ã1ÙÖÙ(ÙhY�T>NQr5s5NQV>sq´Fv,P�s5Y�¸/Á±I�JcNQr�IcN,Y@L�Pk²�s7I�VWLkt,IÀTX´FIÀ¡¢v,PkP�vUÁÂL�rF��Y@L�R�HFP�IÑx¥2g�É�dwd ¤ d3¥7acbef�¦9©(d3bxfma�¥7«Ub¢aK¬�­®^,aKbx`5¯

­ ¬§^*d5«U^*­ |_ioè_ié�q�$5 x /(�!- y E"�P5UV�&Wm�&_/65@���f�ê�]&6���F5<�FCD3<E0�. 0�0\�/6�!-ª���$5rm (x + y) \±(yz $ z*{m (x− y), m (xy) \�/(�!-rm (x/y) -��$�!&657�K5�QF�L#0&(3T�!CD5H�0-:��CD3L\�-(� ÆG�. *�$�!#0�.\!�! *&,-(CF#$5©/6�!-�^$CF&(?5U�'�.�w57` ¤ Q]�$�ë `�m (x± y) = (x± y)(1 + δ) \�V�Q]�$ *� |δ| ≤ µ `ì `�m (xy) = (xy)(1 + δ) \�V�QF�. "� |δ| ≤ µ `í `�� A y 6= 0 \�5�QF�.�Lm (x/y) = (x/y)(1 + δ) \�V�Q]�$ *� |δ| ≤ µ `

l �s#"&63r�CD57�$ 0�L5YQ]/65G-�&u�&(5rCf�$�)5YQ]�K§"¢�¢�¢î��57/6�!-�/6 *-R\r5YQ]�rA.&(����&6VX�����:m�&,/(5U�����)�]&6���F5r��/6V&�AK/2-�-6��5@�'&6�P3)�>�RQf5�Q]&(�o-�Nï `�m (x + y) = x(1 + δ1) + y(1 + δ2) V�Q]�$ *� |δ1| ≤ µ \�/(�- |δ2| ≤ µ `

x6yz $ z*{ æ Ê¥F­ ©DÑx^}|i@~fð�\_bx­ ©DÑx^ ç Ñxd3¥7acbef�¦¼èÌd/befma���©(^<«U¥7acbed/f§É)g�bxac¬e�qd3קf�Ó§bxf§¦�I�T

β = 10, t = 3L�r£½ÀTW´FVXv,�F�%´uäF¯½,¯ x = 0.999× 102, y = 0.111× 100 ¯

x + y = 100.0110 = .100011× 103

Ï (x + y) = 0.100× 103

»w´*�5Yc¸*Ï (x + y) = (x + y)(1 + δ) ¸FÁ´FIcVXIδ = −1.0999× 10−4, |δ| < 1

2(10−2).

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ðùò�ð#íôñ�ð� õ

éö

º ¯ x = 0.999× 102, y = 0.111× 100 ¯xy = 11.0889Ï (xy) = 0.111× 102

»w´<�/Y�¸*Ï (xy) = xy(1 + δ) ¸FÁ´FI�VWIδ = 1.00100× 10−3, |δ| ≤ 1

2(101−3).

ä5¯ x = 0.999× 102, y = 0.111× 100 ¯x

y= 900

Ï (x

y

)= 0.900× 103

δ = 0.

Ô/¯r�IMTβ = 10, t = 4

x = 0.1112, y = 0.2245× 105

xy = 0.24964× 104,Ï (xy) = 0.2496× 104

»w´<�/Y�¸ | Ï (xy)− xy| = .44 NQr5s|δ| = 1.7625× 10−4 <

1

2× 10−3

÷ d/­ ©D×�acbxf§¦Iø bxac¬§d/×�a�¥[ùuק¥7«UÓ¨ � be¦/bxa»w´FI�v,VXIcR ú ÄDÃ�ÂDÃUÅä5¯ àqNQr5suTW´FI�I�p7N%R�HFPkIKY¡¢v,PkP�vUÁÂLkr5��TX´FLOYÐTX´FIcv%VWI�R Y@´FvUÁöTX´/NUTÐTW´FI�VWI�PONUTWLkt,II�VWVWv%V>Y�LkrhJ�v%R�HF�7TWLkr5� TX´FI£Y@�FR�¸ÐsFL¶µmI�VWI�r5J�I%¸ÂHFVWv7s7�5JMT�NQr5s��<�Fv%TXL�I�r<TqL�ryÏ5v<NUTXL�rF� H3v%L�r<TNQVWLkTX´FR�IMTWL�J�NQVWI�Y@R�NQP�P]¯r�EvUÁ I�t%IcVc¸FTX´FIcVXI�N%VXI�J�v%R�HF�7TWI�V>YEÁÂLkTX´Fv,�7TÐ�,�5NQV>s|s7L��%LkTWYEL�r|Á´FLOJ>´N%s5s7L¶TWLkv,r5YÂNQr/suYX�F·7TWVWN,J#TWLkv,r5Y R�N³²qrFvQT·3I�N%J�J��FV>NUTXI,¯Æ �%�5N%VWs s7L��%LkT�LOY�NQrãIMp*TWVWN¼s7L��%LkT�v%r TX´FI|P�vUÁ±IcV�Icr5sãvQ¡ÌTX´FIoN%VXLkTX´FR�I�TXLOJ�VWI��,L�Y@TXIcVÁ´Fv<Y@IÌHF�FVWH/v<Y@IÌL�Y(TXv�J�NQTWJ>´�TW´FIÌP�vUÁ¹v%V>s7I�Vws7L��%LkT±Á´FLOJ>´qÁ v%�5P�s�v%TX´FIcVXÁÂLOYXIE·/I�HF�5YX´FIKs�v%�FTvQ¡DIMp7LOY�TWI�r5J�IÀÁ´FI�r�TX´FI�sFIcJML�R�NQPH3v%L�r<TWYÂN%VXI[NQP�L��%rFIKsS¯âFv,VEJMv%R�HF�FTXI�V>YwÁÂLkTX´|N��%�5N%VWsus7L��%LkT

Ï (x ± y) = (x + y)(1 + δ) |δ| ≤ µ

�EvUÁ I�t,I�VK¸<¡¢v%VwTX´5v,YXI�ÁÂLkTX´5v%�7TÐN��%�/NQV>sqs7L��%LkTÏ (x± y) = x(1 + δ1)± y(1 + δ2),

|δ1| ≤ µ, |δ2| ≤ µ.

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½ I�R�N%V*�mÈE»w´FVWv%�F�,´Fv%�FTÐTX´5L�YÌ·/v*v2�3¸SÁ I�ÁÂL�P�PDN,YXYX�FR�I�TW´5NUT�TX´5I�J�v%R�HF�7T>NUTXL�v%r/Yд5N³t,I·3I�I�r�H3I�VX¡¢v%VWR�Ics�ÁÂL¶TW´�N��,�5NQV>sqs7L��%LkTc¸/N%Y TX´5I�²�N%VXI�v%r�NQP�R�v,Y@TENQP�P_N³tUNQL�P�N%·FPkI�R�N%J>´FL�rFIKY�¯� I�YX´5N%PkPJcNQP�PSVXIKY@�FPkTWYÀ½�TW´FVXv,�F�%´uä�vQ¡�»w´FI�v,VXIcR ú ÄDÃ�ÂDÃUÅäF¯ à�N%Pkv,rF��ÁÂL¶TW´¼Z À0Á.ÂDÃ�ÂDÃUÅäF¯ º l(TX´5IKd]קf§Ó�¥*¨­ª^*fmac¥5ÑÌÑe¥�g�Éod!d ¤ d/¥Facbxf§¦y©(d3bxfma|¥7«Ubxac¬§­ª^<acbx`5¯�»w´FIcYXI�¡¢�Fr5s5NQR�I�r<TWN%PÖPON³ÁEY�¡¢v%VWR TX´FI·5N,Y@LOY¡¢v,V�IKY�T>NQ·FP�LOY@´FL�rF�Ì·/v,�Fr5sFY�¡¢v%VDVWI�PONUTWLkt,IÖI�VWVXv,VWY_Lkr�vQTW´FI�V�Ï5v,NQTXL�rF�ÐH3v%L�r<TDJ�v%R�HF�7T>NUTWLkv,r5Y�¯x $ z $ z7y æ Ê¥F­ ©DÑx^}|i@~w~Ú°±v%r/Y@LOs7I�V TW´FIÀÏ5v,NQTXL�rF��H/v,Lkr<TÐJ�v%R�HF�7T>NUTXL�v%ruvQ¡ x(y + z) È

Ï (x(y + z)) = [x · Ï (y + z)](1 + δ1)= x(y + z)(1 + δ2)(1 + δ1)= x(y + z)(1 + δ1δ2 + δ1 + δ2)≈ x(y + z)(1 + δ3),Á´FIcVXI δ3 = δ1 + δ2 a5YXLkr5J�I δ1 NQr/s δ2 NQVWIÀYXR�NQP�Pe¸*TW´FI�L�VEHFVWv7s7�5JMTwLOYÂrFIc�%P�IcJMTXIcs¯� I£J�NQryr5vUÁ IKN%YXLkP�²®IKY�T>NQ·FP�L�YX´ãTX´FI£·/v,�Fr5s¹v%¡ δ3 ¯ G*�FH5H/v<Y@I β = 10 ¸ÂNQr5s TW´5NUTVWv%�Fr5sFLkrF��LOYÂ�5YXIcsS¯Ö»w´FIcr

|δ3| = |δ1 + δ2| ≤ |δ1|+ |δ2|≤ 1

2× 101−t +

1

2× 101−t

= 101−t.

»w´*�5Yc¸%ac¬§^�«U^*Ñx¥7aKbYb3^q^<«³«³d/«�Ó§×�^qacd�«Ud3×�f§Ó¨>dF«¹bxf `,d/­ª©§×�acbef�¦ ¤ (x(y +z)) beÉ[¥7Ø(d3×_a~Dð 1−t bef aK¬�^;g�d5«UÉca�`%¥FÉK^3i�|{7ü æZ�����e���e��� �)å

n~��]�������X�����:�u�X�[������������,�

°±v%r/Y@LOs7I�VuN%sFsFLkrF� n Ï5v,NQTXL�rF�ªH3v%L�r<Tur*�FR�·/IcVWY x1, x2, . . . , xn ÁÂLkTX´öVXv,�Fr5s7L�rF�/¯ý�ÌIM�5rFIs2 = Ï (x1 + x2) ¯Ö»w´FL�Yw�,Lkt,IcY

s2 = Ï (x1 + x2) = (x1 + x2)(1 + δ2), Zxä5¯ ä<l ²,³ $ z'þ=z*{Á´FI�VWI |δ2| ≤ µ ¯�»w´5NUT[LOY�¸ s2 − (x1 + x2) = δ2(x1 + x2) ¯<�ÌIM�5rFI s3, s4, . . . , snVWIcJM�5VWYXLkt,I�P�²�·<²

si+1 = Ï (si + xi+1), i = 2, 3, . . . , n− 1. ZeäF¯ Ô<l ²_³ $ z@þ=zUy»w´FI�r s3 = Ï (s2 +x3) = (s2 +x3)(1+ δ3) = x1(1+ δ2)(1+ δ3)+x2(1+ δ2)(1+δ3) + x3(1 + δ3) ¯Ö»w´FI�r

s3 − (x1 + x2 + x3) = (x1 + x2)δ2 + (x1 + x2)(1 + δ2)δ3 + x3δ3

≈ (x1 + x2)δ2 + (x1 + x2 + x3)δ3Zxä5¯ ß,l ²,³ $ z'þ=z $

Z¢rFIc�%P�IcJMTXL�rF��TW´FI�TWI�VWR δ2δ3 Á´5L�J>´oLOY�Y@R�NQP�P]¸mN%r5soYXv�v%r3l#¯[»w´*�5Y�¸3·*²�Lkr5sF�5J#TWLkv,r:Á IJ�N%ruYX´FvUÁyTW´5NUTsn − (x1 + x2 + · · ·+ xn) ≈ (x1 + x2)δ2 + (x1 + x2 + x3)δ3

+ · · ·+ (x1 + x2 + · · ·+ xn)δnZxä5¯ ¿<l ²,³ $ z'þ=z@þ

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ð� �]���6ò n õFó �

îKð�>�R�uþF�f�>�

ðü�� ôKÿ

ñ%ò�

é�

ZxN%�,N%LkrurFI��,PkIKJ#TWLkrF��TW´FIÀTXIcVXR�Y δiδj ¸FÁ´5L�J>´�NQVWI[Y@R�NQP�P l#¯»w´FI[Ù��<�5NQTXL�v%r¼Z À0Á.ÂDà � à �äF¯ ¿,lwJcNQru·/I[ÁÂVWLkT@TXIcr|N%Ysn − (x1 + x2 + · · ·+ xn) ≈ x1(δ2 + δ3 + · · ·+ δn)

+x2(δ2 + · · ·+ δn) + x3(δ3 + · · ·+ δn)+ · · ·+ xnδn

Zxä5¯AÞ%l ²,³ $ z'þ=z��Á´FI�VWIÀIcN,J>´ |δi| ≤

1

2β1−t = µ ¯��ÐI��5rFL�rF� δ1 = 0 ¸7Á I[J�N%r�ÁÂVWL¶TWI%È

­ ¬§^*d5«U^*­ |_i'~FgkhB�qd3×�f§Ó�bef�¦ æ «K«Ud/«Ebxf ç Ñxd3¥7aKbxf§¦�èÌd/befma � Ó§Ó�bxacbed/fXj³iG���.5 x1, x2, . . . , xn±(yz@þ=z*{

E0� n m�&,/(5U�������]&6���F5©�FCD3<E"�$ 0�$` ¤ QF�.�m (x1 + x2 + · · ·+ xn)− (x1 + x2 + · · ·+ xn)

≈ x1(δ1 + δ2 + · · ·+ δn) + x2(δ2 + · · ·+ δn) + · · ·+ xnδn, Zxä5¯ á<l ²,³ $ z'þ=z��V�Q]�$ *�)�0/�#0Q |δi| ≤ µ, i = 1, 2, . . . , n `

� = � ',.@+ � â5VXv,R TX´5IÐNQ·3vUt%I ¡¢v%VWR��5P�NÀÁ IEY@IcIwTX´5NQT(Á IEY@´Fv,�FPOs�IMp7H/IKJ#T±YXR�NQP�PkIcV�I�VWVXv,VL�ru�,I�rFIcVWN%P/Á´5I�r|N%sFsFLkrF� n Ï5v,NQTXL�rF��H3v%L�r<TEr*�FR�·3I�V>YwLkruL�r5JMVWIcN,Y@L�rF��v%V>s7I�VwvQ¡§R�NQ�%r5L¶TW�5s7I%È|x1| ≤ |x2| ≤ |x3| ≤ · · · ≤ |xn|.

§YAr5�QF�T�wCD34E0�$ 0�G/( *��/6 � */(�f�2�0-:�������#. "�"/(�������4&( *-2�. �&�AT34/_�6�w��5@CF-2��\J5�QF�.�u5�QF�G��/( @���$ ��$ � *&( 0�VX�����JE0�)/R�0�$&_#$�'/(5H�0-�VX��5YQW5YQ]�K��34/6�����$ K�FCD3<E0�. 0��Z æ Ê^<«³`,bxÉK^;|i%¯/lM¯­ ¬§^*d5«U^*­é|_i'~D|i:�@¨����5YQ]�L�FCD3<E0�. 0� ni1 E.8± $ z*{ $

1 + η1 = (1 + δ1)(1 + δ2) · · · (1 + δn)

1 + η2 = (1 + δ2)(1 + δ3) · · · (1 + δn)

1 + η3 = (1 + δ3) · · · (1 + δn)

```1 + ηn−1 = (1 + δn−1)(1 + δn)

1 + ηn = (1 + δn)

OG� �$&k-2�'¨��!� µ′ =µ

0.9/(�!-u/R�0��CD3<��5YQ]/(5 nµ ≤ 0.1 \�5YQ]�$�

fl(x1 + x2 + · · ·+ xn) =

x1(1 + η1) + x2(1 + η2) + · · ·+ xn−1(1 + ηn−1) + xn(1 + ηn), Zxä5¯ à<l ²,³ $ z'þ=z��V�Q]�$ *� |η1| ≤ (n− 1)µ′ /(�- |ηi| ≤ (n− i + 1)µ′, i = 2, . . . n `´Kµ,¶f¶R·f¸ G*IcI[G*TXI�ÁwNQVXT�Z�½Kà%à%á<l#¸7HFH_¯Â½cä%¾QÇW½Kä º ¯

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» �ìDíUî<ïcð�ñ%ò_ó

ôöõ5÷,ø(ùmúÂû�üÖýÚþSøÀû�üDú�ü ÿ���������qùEü����� �ø�����|{�� � ���x���X���@�7çÂ�[���e��� �:å

n~|�e���[���X��� �:�u�X�������������,�

�ÖVWv7JMI�IKs7L�rF��N%YDL�r�TX´FIÌJ�N%YXIÂvQ¡SN%s5s7L¶TWLkv,r�vQ¡ n Ï5v,NQTXL�rF�ÀH/v,Lkr<T(r*�FR�·/IcVWY�L�r�TX´5IEP�N,Y�T(YXIcJMTXL�v%r¸LkTÐJ�NQru·3I�Y@´FvUÁÂr�TX´/NUT­ ¬§^*d5«U^*­é|_i'~*j_i±(yz��z*{

m (x1 × x2 × · · · × xn) ≈ (1 + ε)

n∏

i=1

xi

V�Q]�$ *� ε = |(1 + δ2)(1 + δ3) · · · (1 + δn)− 1| /(�- |δi| ≤ µ, i = 1, 2, . . . , n `� Ø(d/קf§ÓªdËd/« ε È Æ YWYX�FR�LkrF��TW´5NUT (n− 1)µ < .01 ¸7L¶TÐJcNQru·/I�YX´FvUÁÂr�TX´5NQT

ε < 1.06(n− 1)µ ZxäF¯�½c¾<l ²,³ yz��!z*{Zx»w´5L�YÂN,YXYX�FR�H7TXL�v%ruLOY��,�5L¶TWI�VWIcN%PkLOY@TXLOJ6a*v%ruR�v,Y@TwR�N,J>´FLkr5IcY TX´5L�YÂN,YXYX�FR�H7TXL�v%ruÁÂL�PkPS´Fv,P�sq¡¢v%V¡xNQL�VWPk²�PONQVW�%I�tUNQP��FIcYwv%¡ n lM¯»w´*�5Yc¸

ε ≤ (1 + µ)n−1 − 1 < (n− 1)µ

[1 +

0.05

1− .05

]< 1.06(n− 1)µ. Zxä5¯k½,½Kl y!z��z*{

»w´*�5Yc¸FJMv%R�·FL�rFLkr5��»w´5I�v%VWI�R ú ÄDÃ��DÃUÅä5¯k½cÔ�NQr5s9Z À0Á.ÄDÃ��DÃUÅä5¯k½K¾,l#¸7Á I[J�N%r�ÁÂVWL¶TWI­ ¬§^*d5«U^*­ |_i'~FzáhB�qd3×�f§Ó�bef�¦ æ «³«Ud5« bef ç Ñxd3¥7aKbxf§¦ÚèÐd3bxfma�� ×�Ñxacbx©§Ñebx`%¥Facbxd3fXj³i ¤ Q]�±(yz��z7y *�$��/(5U��+(� �$ � *&( ß���Ò#"&63r�CD5@���f�S5YQ]�W�! *&,-(CF#$5}&�A n m�&_/65@���f�[�F&(���w5k�FCD3<E0�. 0�[�%�s/(5u34&(�051.06(n− 1)µ \�/(�0�0CD3)������5�QF/(5 (n− 1)µ < .01 `�|{�� �M��������y,�W�|�LçÂ� �¹�����q�[��[���]���Æ ¡¢VWI_�<�FIcr<TXP�²�N%VXLOYXLkrF��JMv%R�HF�FTWNUTWLkv,r5NQP�TWN,YB�uL�r:r*�FR�I�VWL�JcNQP�PkL�rFIcN%V�N%Pk�,I�·FV>N�LOYETW´FI�J�v%R�HF�7ÇTWNQTXL�v%ruvQ¡§TX´FI[bef�f§^<«�©�«Udӧק`Qa[vQ¡�T�Á±v n ÇËt%IKJ#TXv,VWY x N%r5s y È

xT y = x1y1 + x2y2 + · · ·+ xnyn ZeäF¯�½ º l yz���zB{Á´FIcVXI xi N%r5s yi ¸ i = 1, . . . , n ¸/NQVWIÐTW´FI�JMv,R�H3v%r5I�r<TWYwv%¡ x N%r5s y ¯

�ÐI��5rFIS1 = Ï (x1y1), ZeäF¯�½cä<l yz���zUyS2 = Ï (S1 + Ï (x2y2)), ZeäF¯�½�Ô*l yz���z $¯¯¯Sk = Ï (Sk−1 + Ï (xkyk)), ZeäF¯�½Kß,l yz���z'þ

k = 3, 4, . . . , n.

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ñQòþò�D�F�Rõ

ðEì� ôï�ð@îKð� �]� »

ê� IÀTX´FIcru´/N³t%I%¸*�5YXL�rF��»w´FI�v,VXIcR ú ÄDÃ��DÃUÅäF¯�½�Ô/¸

S1 = x1y1(1 + δ1), ZxäF¯�½c¿<l yz���z��S2 = [S1 + x2y2(1 + δ2)] (1 + η2) ZxäF¯�½³Þ%l yz���z��

¯¯¯Sn = [Sn−1 + xnyn(1 + δn)] (1 + ηn), ZxäF¯�½cá<l yz���z��

Á´FIcVXI�IcN,J>´ |δi| ≤ µ ¸_NQr/s |ηi| ≤ µ ¯�G7�F·5Y@TXLkTX�7TWLkrF��TW´FI�tUNQP��FIcY�vQ¡ S1 TX´5VXv,�F�%´ Sn−1 L�rSn NQr5s�R�N6�*L�rF��Y@v,R�IÀVWIcN%VXV>NQrF�,I�R�I�r<T>Y�¸,Á±I�JcNQruÁÂVXLkTXI

Sn =n∑

i=1

xiyi(1 + εi) ZxäF¯�½cà<l yz���z��Á´FIcVXI

1 + εi = (1 + δi)(1 + ηi)(1 + ηi+1) · · · (1 + ηn)

≈ 1 + δi + ηi + ηi+1 + · · ·+ ηn (η1 = 0) ZeäF¯ º ¾<l yz��z��Z¢L��%r5v%VWLkrF��TX´5I[HFVXv7s7�/J#TWY δiηj N%r5s ηjηk ¸7Á´FLOJ>´|NQVWI�YXR�NQP�POlM¯âFv,VÂIMpFNQR�HFP�I%¸7Á´FIcr n = 2 ¸7LkTEL�YwIKN%YX²�TXv�J>´FIKJ0��TX´5NQT

S2 = x1y1(1 + ε1) + x2y2(1 + ε2), ZxäF¯ º ½³l yz���zB{��Á´FIcVXI 1+ ε1 ≈ 1+ δ1 + η2, 1+ ε2 ≈ 1+ δ2 + η2 Z¢rFIc�%P�IcJ#TWLkr5��TX´5IÐHFVWv7s7�5J#T>Y(v%¡ δ1η2 N%r5sδ2η2 ¸7Á´FL�J>´|N%VXIÀY@R�N%PkP l#¯âFVWv%R Z ÄDÃ��DÃ��äF¯�½cà<l#¸FNQr/s9Z ÄDÃ��DÃ��äF¯ º ¾<l#¸*Á±I�JcNQruÁÂVXLkTXI,È­ ¬§^*d5«U^*­é|_i'~D¯i$ z���z*{fl(x1y1 + x2y2 + · · ·+ xnyn) = x1y1(1 + ε1) + x2y2(1 + ε2) + · · ·+ xnyn(1 + εn)

V�Q]�$ *� εi /( "�G����+R�$�ßE.8! ÄDÃ��DÃ��í ` ì#"%$ `âFVWv%R »w´FI�v,VXIcR ÂDÃ��DÃUÅäF¯�½c¿�Á±IÀ´5N³t,I

| Ï (xT y)− xT y| ≤n∑

i=1

|xiyi||εi| ZxäF¯ º%º l $ z���zB{.y� Ø(d3×�f§Óed]d/« |εi| bxfãac^<«U­ªÉ�dwd µ ¯�tÐr5s7IcVwTX´FI�N,YXYX�FR�H7TXL�v%r�TX´/NUT nµ < 0.01 ¸7N�·3v%�Fr5s¡¢v%V εi L�ruTXI�VWR�YwvQ¡ µ JcNQr�·/IÀIKY�T>NQ·FP�L�YX´FIKs9ZeY@IcI��EL��%´5N%R Z�½Kà%à%¿<l#¸7HFH_¯±Þ³ÔQÇ@ÞQß,l#¯

tÐYXLkr5��TX´FLOYw·/v,�Fr5sS¸5Á±I�JcNQruÁÂVXLkTXI,È­ ¬§^*d5«U^*­é|_i'~]¼ hB�qd3×�f§Ó§bxf§¦ æ «³«³d/«�bef å f�f§^<«qèÀ«UdÓ§×�`%a ÷ d3­ ©D×_aK¥7acbed/fXj³i±(yz���z*{

|m (xT y)− xT y| ≤ n1.01µ|x|T |y|,

V�Q]�$ *� |x| �05H/(�-R�XA.&( K5YQ]�L+(�"#$5H&( �VX��5YQ;#0&(3T�]&6�!�$�F5@� |xi| `�q^*­ª¥7«'&É2�

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»ë ìDíUî<ïcð�ñ%ò_ó

ôöõ5÷,ø(ùmúÂû�üÖýÚþSøÀû�üDú�ü ÿ���������qùEü����� �ø����Z¢L l Û vQTWIÀTX´5NQTE´FLk�,´�VWI�PONUTWLkt,I�N,J�J��FVWN,JM²�JcNQrurFvQTE·3I[�%�5N%VWN%r<TXI�IKs�Lk¡ |xT y| � |x|T |y| ¯Z¢L�L l�Ä v,�Fr5sFY§�%L�t%Icr�L�r�»w´FIcv%VWI�R�Y ÂDÃ��DÃUÅäF¯�½c¿ÐN%r5s ú ÄDÃ��DÃUÅä5¯k½UÞÂJcNQr�·3IwVXIKs7�5J�Ics�·*²�NE¡xN%J#TWv%V n ·*²[�5YXLkr5�I�p*TXI�r/s7Ics9HFVWIcJ�L�YXLkv,r£v,V�YXv%R�I�H5NQVXTXLOJM�FPONQV[L�R�HFP�I�R�I�r<TWNQTXL�v%r5Y�ZeY@IcI��EL��%´5N%R Z@½cà%à,¿,lM¸H5H¯Ö¿%àQÇ Þ*½Ì¡¢v%VEs7I�TWNQL�POYWlM¯

�|{�( �95,��*)¹�����:���Ðå,��Ý~|�e���[���X��� �:�u�X���+� �[��5�-,�¹�����q�[��[���]�����­ ¬§^*d5«U^*­ |_i'~D¿ie�q�$5 |M | = (|mij |) `}���.5 A /(�- B E0�k5@V�&Lm�&,/(5U�����4�]&6���F5�34/65@ ��'#0�.�±(yz���z*{/(�- c /Gm�&_/65@���f���F&(���w5��wCD34E0�$ $` ¤ Q]�$�ë `�m (cA) = cA + E, |E| ≤ µ|cA|ì `�m (A + B) = (A + B) + E, |E| ≤ µ|A + B|

§YA A /(�- B /6 *�K5UV�&<3</(5U ��'#0���L#0&(3T�F/(5U�'E.���©A.&6 �34/65@ ��>12?H3)CD�95@� ��9�'#0/(5U�'&(�F\�5�QF�.�í `�m (AB) = AB + E, |E| ≤ nµ|A| |B|+ O(µ2) `´Kµ,¶f¶R·f¸ G*IcI � L�Po�<L�r5YXv%r Z�½Kà%¿<ß%l#¸*H¯Â½,½KßF¯

� ^*¥Ff§bxf§¦¼dwd O(µ2)

� = � '%.@+ � Ã1r®TX´FI|N%·/vUt,IqIMp7HFVWIcYWYXLkv,r¸DTX´FIurFv%TWNUTWLkv,r O(µ2) Y@TWNQr/sFY�¡¢v%V�N£J�v%R�HFP�L�JcNUTXIKsIMp7HFVWIcYWYXLkv,r TW´5NUT�LOYq·3v%�Fr/s7Icsy·*² cµ2 ¸ÐÁ´FIcVXI c L�YuN®J�v%r5Y@TWN%r<Tc¸Es7IcH/Icr5s7L�rF� �5H/v,ryTX´FIHFVWv%·FP�I�R�¯�»w´FI[I�p*H5VXIKYXYXLkv,r O(µ2) ÁÂL�PkPS·3I[�5YXIcs�¡¢VWI_�<�FI�r<TWPk²�L�ruTX´FLOY·3v*v��m¯� = � '%.@+ � »w´5IuPON%Y@T�VXIKY@�FPkT�Y@´5vUÁEY�TW´5NUT�TW´FI|R�NUTXVWLkpªR��FPkTXL�HFP�L�JcNUTXL�v%rãL�r®Ï5v<NUTWLkrF�¼H3v%L�r<TNQVWLkTX´FR�IMTWL�J:J�NQr¹·3I:t,I�VW²®L�r5N%JcJM�FV>NUTWI%¸wY@L�r5JMI |A| |B| R�N³² ·/I£R��5J>´yPONQVW�%IcV�TX´5N%r |AB|LkTWYXI�Pk¡�¹ æ Ê^<«U`%beÉc^;|_ioèFº@¯ æ «³«Ud5«/.�d3×�f§Ó§É�bef ­ ^<«U­ªÉ�dwdT¬|d5«U­ªÉ»_V>N%sFL¶TWLkv,r5NQP�Pk²,¸³¡¢v%V�R�NUTWVXLkp�J�v%R�HF�7T>NUTXL�v%r/Y§TX´FI·3v%�Fr/sFY§¡¢v%V�IcVXVWv%VDR�NUTWVXLOJMIKY§N%VXIw�,Lkt,I�rL�r�TXI�VWR�Y�vQ¡mTW´FIErFv,VXR�Y�v%¡3TW´FIER�NUTWVXLOJMIKY�¸,VWNQTX´FIcV�TX´/NQr�L�r�TXI�VWR�Y�vQ¡N%·5Y@v,Pk�FTXIÂtUNQP��FIcY(vQ¡3TX´FIR�NUTWVXLOJMIKYÌN,YÐ�,Lkt,I�r:N%·/vUt,I%¯��EI�VWI�Á±I�VXIcÁÂVXLkTXI�TX´FI�·/v,�Fr5s:¡¢v%V�I�VWVXv,VÐR�NUTWVXLOJMIKYE¡¢v,VÌR�NQTXVWL¶pR��5P¶TWLkHFP�LOJ�NUTWLkv,r5Y[�5YXL�rF�|rFv%VWR�Y�¸�¡¢v%V�IcN%YX²:VWIM¡¢IcVXIcr5JMI�PONUTWI�V�L�r¼TX´5I�·/v*v2�3¯10)�k3)Cf�05G�!&657�.\Q]&6V��.+(�$ 0\�5YQ]/65T�.�w5@ �8(?HVX�%�$�4�$ � *&( <E0&(CD�!-(�:/( *�)3<&( "�)34�0/(�w�����*A0CD��5YQ]/6�[�!&( �3)?HVX�%�$�<�. � "&( 0�ZxYXI�IÀVWI�R�NQV"�7Y Lkr:\_^<`%acbed/f¹j�io|3l#¯Ã1r�TXI�VWR�YÂvQ¡DN�rFv%VWR�¸7Á±I�JcNQruÁÂVXLkTXI Ï (AB) = AB + E,Á´FIcVXI

||E||p ≤ ηµ||A||p||B||p + O(µ2), p = 1, 2, F.Ã1ruH5NQVXTXLOJM�FPONQVK¸7L�rqTWI�VWR�YEvQ¡ ‖ ‖1 rFv%VWR�¸7Á±IÀ´5N³t,I­ ¬§^*d5«U^*­é|_i'~Dèi ||fl(AB)−AB||1 ≤ nµ‖A‖1‖B‖1 + O(µ2) `±(yz���z7y

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ñõ(�ò� � 4ñ�ì� ôï�ð@îKð� �F� � »

ó­ g�d å ­ª©(d5«³aK¥Ffmaq\S©(^*`,bx¥FÑ ÷ ¥FÉc^*É

576 8£',;M.�?98 :<; =<)U;#0<. � 2FB ;M?>=5B ?A)Q'<;M?A0/4 ÃË¡ b LOYEN�t%IcJMTXv,Vc¸<TX´5I�ru¡¢VXv,R N%·/vUt,IÌÁ IÀ´5N³t%I||fl(Ab)−Ab||1 ≤ nµ||A||1||b||1& 6 8:'<;M.�? 8 � 2FB ;M?>=5B ?A)Q'<;M?A0/4/? @�'74u0*.Ë;BA70F-50/47'7B � '<;M.�? 8 ½ IcJcNQP�P3TX´5NQTwN�VXIKNQP3R�NUTXVWLkp O L�YJ�N%PkP�IcsuNQrß&( �5YQ]&"��&(�!/6��3</(5U ��>1�Lk¡ OT O = OOT = I ¯

÷ d5«Ud3ÑxÑe¥7«Rcª|_iOgwði��q�$5 A ∈ Rn×n /6�!- Q ∈ Rn×n &6 �5�QF&*��&(�/(��` ¤ QF�.�C yz���z*{||m (QA)−QA||2 ≤ nµ||A||2

å ­ ©DÑxbe`%¥7aKbxd3f d!dÂaK¬�^o¥FØ(dfb/^|«U^*ÉK×�Ñxa¤ Q]�< "����CD�95T&BA/D�&6 *&(����/6 �8 E�ÄDÃ�FDÃUÅí ` ì#" �$/(8R�45�QF/(5@\r/6�95YQ]&(C��RQP3</(5U ��>1W3)CD�95@� ��9�'#0/(5U�'&(� #0/6��E"�

���!/2#0#.CD */657�����s���.��. */6� \�� A�&(��}&�Ak5�QF�k3</(5U ��'#0�����%��&( �5YQ]&"�2&6�!/(��5YQ]�$� 5YQ]��m�&,/(5U�����k�F&(���F53</(5U ��>1ß3)CD�95@� ��9�'#0/(5U�'&(� ����+(���;&(�w�98s/I�03</(���T/6�!-ß/2#"#0�H�!5H/�E$���P�. � *&6 $`SO��kV��;VX�������$�"�}�����/657�$ ê#"QF/"�57�$ 0�0\)5YQf�%�} *�.��CD�95�A.&( �3:�;5YQ]�êE0/(�0�%�ê&BAW34/6�F8ß�FCD3<�$ ��'#"/6���98ß+,�'/�E$���[/(� ��&( ���5YQf3:�-(�%�$#$Cf�0�.�0-����ê5YQf�%�LE0&_&,¥�`�|{�G � �����L�IH3�IJ �95,��<�LKÝ��� �� �¹���LçÂ���@�]�[���e��� ���L�

� ��ç[��<�(��M�� �ã�����q����[���e�����Ã1r<TX�FLkTXL�t%IcPk²,¸*L¶TÂLOY J�PkIKNQV±TX´5NQT Lk¡§N�PONQVW�%IEr*�FR�·/IcV v%¡Ï5v<NUTXL�rF��H/v,Lkr<TÂJMv,R�H5�7TWNQTXL�v%r5Y±L�Yws7v,rFI%¸TX´5I�r£TX´FI�N%JcJM�FR��FPONUTXIKsoIcVXVWv%VÀJ�N%r:·3I��,�5L¶TWI�PONQVW�%I,¯ON|dfg�^�b/^<«#P«Ud/קf�Ó_¨Wdw« ^<«³«Ud5«£`%¥FfØ(^¼Ó§bxÉK¥FÉ�ac«Ud/קÉ�^�b3^<f�¥7a�¥®ÉKbxf§¦3Ñx^9É�aK^,©�d!d[`%d3­ ©§×�ac¥Facbxd3f±i�âFv,VÌI�pFNQR�HFP�I%¸SJMv,r5Y@LOs7IcVTX´5I�Y@�F·FTXV>N%J#TWLkv,r�v%¡_T�Á v�r<�5R�·3I�V>Y�È

x = 0.54617 N%r5s y = 0.54601

»w´FI[I�p7N,J#TÂtUNQP��FIÀL�Yd = x− y = 0.00016.G*�FH5H/v<Y@I[rFvUÁhÁ±I[�5YXI�¡¢v%�FVEsFLk�,L¶TÐN%VXLkTX´FR�I�TXLOJÌÁÂLkTX´�VWv%�Fr/s7Lkr5�5¯�»w´FI�r�Á IÀ´5N³t%I

x = 0.5462 Z]°±v,VXVWIcJMT±TWv�¡¢v%�FVEYXLk�,rFLk�/J�N%r,TEs7L��%LkTWY>ly = 0.5460 Z]°±v%VWVWIcJ#T TXv�¡¢v,�FVEYXLk�,rFL¶�3J�NQr<TEs7L��%LkTWY>ld = x− y = 0.0002.

�EvUÁö�%v*v7sqLOY TX´5I�NQHFHFVWv³p7LkR�NQTXL�v%rqv%¡ d TWv d Q|»w´5I[VXIcP�NQTXL�t%I�I�VWVXv,V LOY|d− d||d| = 0.25 Z'�<�FLkTXIÀPONQVW�%I RAl

� ´5NUT�´5N%HFH3I�rFIKsãN%·/vUt,IqLOY�TX´FIu¡¢v,PkP�vUÁÂLkr5�5¯®Ã1r ¡¢v%�5V�sFLk�,L¶T�NQVWLkTX´FR�IMTWL�J%¸ÖTW´FI�r<�5R�Ç·3I�V>Y�¾5¯ ßQÔ,¿ º NQr5s ¾F¯AßUÔ<¿%¾oN%VXI�vQ¡ÌN%PkR�v,Y@T�TX´FI�YXN%R�IuYXLkÍcI%¯®G*v/¸ÖÁ´FIcr®TW´FIq�/VWY@T�v,rFIuÁ N,Y

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�q^*­ª¥7«'&X�DÃËTDL�Y_TXvÌ·/I rFvQTWIcs�TX´5NQT�L�r�R�NQr*²[J�N,Y@IKY�¸UYX�F·7TXV>N%JMTXL�v%r[LOY§H/IcV@¡¢v,VXR�Ics�VWNQTX´FIcVN%JcJM�FV>NUTWI�P�²%¯�ÃËTÖL�Y(rFvQT(N[JcNQ�5YXIÂvQ¡3TX´FIÐI�VWVXv,V�Ç� */(5YQ]�$ r��5 *�$+R�0/(�9�T5�QF���. � *&6 0�r34/2-��G���}�"/6 ��9�'�$ #0&(3T�!CD5H/(5U�'&(�F�)&( <�.+(�$�ß5�QF&R�$�4���ß5YQ]��-2/(5H/}/R�0�$&_#$�'/(5H�0-WVX��5�QI5YQ]�4�0CFE$5U */�#$5@�'&6�!`�Ã1r5s7I�IKsS¸mL¶T´FL��%´FP�L��%´<TWY±TX´5I[IcNQVWP�LkIcVÂI�VWVXv,VWYc¯

� b/d3bxÓ§bxf§¦ ÷ ¥Ff§`%^*ÑxÑe¥7acbed/f��âFv,V@TW�Fr5NUTWI�P�²%¸ÌLkr�R�NQr*²yJ�N%YXIcYuJ�NQTWN,Y�TWVXv,HF´FLOJ:J�N%r5JMIcPkPONUTWLkv,rhJ�N%rh·3I N³t%v%LOs7IKsS¯�âFv%VIMpFN%R�H5PkI,¸FJMv%r/Y@LOs7I�V TW´FI�J�N,Y@IÀvQ¡DY@v,Pkt*L�rF��TX´FIK�<�5N,s7V>NUTXLOJÌI_�<�5NUTWLkv,rÈ

ax2 + bx + c = 0, a 6= 0.

»w´FIÀ�5YX�5NQPSÁwN³²�TW´FIÀT�Á±v�VWv<v%TWY x1 NQr5s x2 N%VXI[J�v%R�HF�7TWIcs�L�YcÈx1 =

−b +√

b2 − 4ac

2a

x2 =−b−

√b2 − 4ac

2aÃËTÖL�Y(JMP�IcNQVD¡¢VWv%R NQ·3vUt%I TX´5NQTÖL¶¡ a, b ¸,N%r5s c N%VXI r*�FR�·3I�V>YÖYX�5J>´�TX´5NQT −b LOYÖNQ·3v%�FT§TW´FIYWNQR�I�YXLkÍcI�N,Y √b2 − 4ac ZxÁÂL¶TW´£VWIcYXH/IKJ#T�TXv�TX´FI�NQVWLkTX´FR�IMTWL�J��5YXIcs5lM¸mTX´FIcr9N�J�NQTWN%Y@TXVWv%H5´FL�JJ�N%r5JMIcPkPONUTWLkv,r|ÁÂLkP�P§v*JcJM�FV�L�r:JMv,R�H5�7TXL�rF� x2 N%r5soN,YÌN�VWIcYX�FPkTc¸3TX´5I�JMv,R�HF�7TXIKsotUNQP��FI�vQ¡ x2J�N%r�·3I�JMv,R�HFPkI�TXIcPk²�IcVXVWv%r5I�v%�/Y�¯x6yz���z*{ æ Ê¥F­ ©DÑx^}|i g�~ßh ÷ ¥Ff§`%^*ÑeÑx¥7aKbxd3fybxfe�qdd/aM¨ ç bxf§Ó�bef�¦£dwdEac¬§^TS�ק¥FÓ�«³¥Facbx`fj¼°±v%r5YXL�sFI�V_YXv%P�t<ÇL�rF� ax2 + bx + c = 0 ¸3ÁÂLkTX´ a = 1 ¸ b = −105 ¸ c = 1 Zxâ5v%V>Y@²<TX´5I%¸mnoNQPOJMv,PkR NQr/son|v%P�I�VZ�½Kà<Þ,ÞQl#¸*HFH_¯ º ¾UÇ º%º l#¯D»w´FI�r��5YXL�rF� β = 10, t = 8, L = −U = −50 ¸7Á IÀYXI�I�TW´5NUT

x1 =105 +

√1010 − 4

2= 105 Z¢TXVW�FI[NQr5YXÁ I�V#l

x2 =105 − 105

2= 0 ZeJMv%R�HFP�IMTWI�P�²qÁÂVWv%rF�*l#¯

»w´FI�TXVW�FI x2 = 0.000010000000001 ZxJ�v%VWVXIKJ#TWPk²�VWv%�Fr/s7IcsqTXv|½,½ÀYXL��%rFLk�/J�N%r<TÂsFLk�,L¶T>YWlM¯(»w´FIJ�NQTWN,Y�TWVXv,HF´FLOJÂJ�NQr/JMI�P�PONUTXL�v%r�TWv<v2��H5P�N,JMIEL�rqJMv,R�HF�7TXL�rF� x2 ¸7YXLkr/JMI −b NQr/s (√

b2 − 4ac) N%VXIvQ¡�TX´FI�YWNQR�IÀv%V>s7IcVc¯ Û v%TXI�TX´/NUTEL�r�áQÇËsFLk�,L¶TENQVWLkTX´FR�IMTWL�J%¸ √1010 − 4 = 105 ¯N|dfg ÷ ¥Ff�`,^*ÑxÑx¥Facbxd3f ÷ ¥Ff¹Ø(^ � b3d/beÓ�^*Óybef ç bef�Ó§bxf§¦[�qdd/acÉ�d!dÂac¬§^TS�ק¥FÓ�«U¥7acbe`VU

° NQr5J�I�P�P�NQTXL�v%ruJ�NQr�·3I�N³t%v,L�s7IKs�Lk¡DN%r�I,�<�FLktUN%PkIcr,TÂH5N%LkVÂv%¡�¡¢v,VXR��FPON%Y L�Yw�/Y@IKsSÈx1 = −b + Y@L��%r (b)√b2 − 4ac

2a

x2 =c

ax1

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ñõ(�ò� � 4ñ�ì� ôï�ð@îKð� �F� � »2»

Á´FIcVXI[YXLk�,r (b) LOYwTX´FI�YXL��%ruvQ¡ b ¯�tÐYXLkrF��TW´FIcYXIÀ¡¢v%VWR��FPON%Yc¸*Á±IÀIKN%YXLkP�²qYXI�I�TX´5NQTcÈx1 = 100000.00

x2 =1.0000000

100000.00= 0.000010000

�q^*­ª¥F«W&X�|° NQr5J�I�P�P�NQTXL�v%r R�N³²¼Y�TWLkP�P(T>N6�,IqHFPON%J�Iqs7�FVWLkr5�|TX´FI�YX�F·7TXV>N%JMTXL�v%r b2 − 4ac NQr5sYXLk�,rFL¶�3J�NQr<TEs7L��%LkTWYÐÁÂLkP�P·3I�Pkv<Y�TELk¡ b2 ≈ 4ac ¯ÖÃ1r�TW´5NUTÐJcN%YXI%¸FLkTÐLOYEN%s7t*LOYXN%·FP�IÌTWv��5YXI�s7v%�F·5PkIHFVWIcJ�L�YXLkv,r�L�r�JMv%R�HF�FTXL�rF� b2 − 4ac ¯x6yz���z7y æ Ê¥F­ ©DÑx^}|i g/ghâFv,V²%I�TÐNQrFv%TX´FIcV I�pFNQR�HFP�IÀJ�v%r5YXLOs7I�VwTW´FI[HFVWv%·FP�I�R v%¡�Ict³N%Pk�/NUTXL�rF�

f(x) = ex − x− 1 NQT x = .01 ¯tÐYXL�rF���5t,I�s7Lk�,L¶T�NQVWL¶TW´FR�IMTXLOJQ¸/TX´5I�JMv,VXVWIcJMTÐNQr5YXÁ±IcVÂL�Y̾F¯ ¾%¾,¾%¾,ß%¾F½K¿<Þ*¯DÃË¡ f(x) LOYÂI�tUN%Pk�5NQTXIKss7L�VXIKJ#TWPk²�¡¢VWv%R TX´FI[I�p*H5VXIKYXYXLkv,r¸*Á±I[´/N³t%I

f(.01) = 1.0101 − (0.01)− 1 = 0.0001½ IcP�NQTXL�t%IÀÙÖVWVXv,V =0.0001− 0.000050167

0.00005016= 0.99× 100,

L�r5s7LOJ�NUTWLkr5��TW´5NUTEÁ I[J�N%rFrFvQTwTWVX�5Y@TEI�t,I�r�TX´FIÀ�/VWY@TEY@L��%rFLk�/JcNQr<TEs7Lk�,L¶TK¯ç d/«³acקf�¥7aK^*ÑYcXP�`%¥Ff§`%^*ÑxÑe¥7acbed/f®`%¥Ff ¥F¦/¥5bxf Ø(^q¥�b3d/beÓ�^*ӮקÉKbxf§¦�ac¬§^q`%d3fb/^<«U¦3^<fmaÉK^,«Ube^<É)dËd5« ex �ex = 1 + x +

x2

2+

x3

3!+ · · ·

Ã1ruTX´FLOYEJcN%YXIÌÁ IÀ´5N³t%Iex − x− 1 = (1 + x +

x2

2+

x3

3!+ · · · )− x− 1

=x2

2+

x3

3!+

x4

4!· · ·

âFv,V x = 0.01 ¸<TX´FLOYw¡¢v%VWR��5P�N��%L�t%IKY(.01)2

2+

(.01)3

3!+

(.01)4

4!+ · · ·

= .00005 + .000000166666+ .00000000004166+ · · ·= .000050167 h�`,d5«³«U^*`Qa�×�©yacdP��b3^:ÉKbx¦3f�b'�§`,¥Ffma)�§¦3×_«U^*É(j

� = � ',.@+ � Û vQTXI�TX´5NQTÐLk¡ x Á I�VWI�rFI��<NUTWLkt,I%¸FTW´FI�r:�5Y@I�vQ¡DTX´FI�JMv,r<t,I�VW�%Icr,TEYXI�VWLkIKY¡¢v,Vex Á v%�5P�s:rFvQTÀ´5N³t%I�´FIcPkH3IcsS¯�âFv,V�I�pFNQR�HFP�I%¸mTXv�JMv,R�HF�7TXI ex ¡¢v,V[NqrFIc�,NQTXL�t%I�t³N%Pk�5I�v%¡ x ¸J�N%r5JMIcPkPONUTWLkv,ruJcNQru·/I�N³t,v%LOs7Icsq·*²��5YXLkrF�

e−x =1

ex=

1

1 + x +x2

2!+

x3

3!+ · · ·

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En =

∫ 1

0

xnex−1dx

¡¢v%VEsFL¶µmI�VWI�r<TÂtUNQP��FIcYwv%¡ n ¯�Ã1r<TXIc�%V>NUTWLkrF��·*²�H5N%V@T>Yw�%L�t%IcYEn =

∫ 1

0

xnex−1 dx = (xnex−1)10 −∫ 1

0

nxn−1ex−1 dx

v%VEn = 1− nEn−1, n = 2, 3, . . .»w´*�5Yc¸,Lk¡ E1 LOY��*rFvUÁÂr_¸%TW´FI�rq¡¢v%Vws7L¶µmI�VWI�r<TwtUNQP��FIcY±vQ¡ n ¸ En J�NQr�·/IÀJMv,R�HF�7TXIKsS¸*�5YXLkrF��TX´FINQ·3vUt%I�VWIcJM�5VWYXLkt,IС¢v,VXR��FPONF¯Ã1r5s7IcIcsS¸*ÁÂLkTX´ β = 10 N%r5s t = 6 ¸7N%r5s�Y�T>NQVXTXL�rF��ÁÂLkTX´ E1 = 0.367879 N,Y(N�YXLkp<Ç1s7L��%LkTNQH5HFVXv³p7L�R�NUTXL�v%r�TXv E1 = 1/e ¸*Á±IÀ´5N³t,I�¡¢VXv,RÝNQ·3vUt%I,ÈE1 = 0.367879E2 = 0.264242E3 = 0.207274E4 = 0.170904¯¯E9 = −0.068480 Z'g�«³d3f§¦ml

� Ñxac¬§d/ק¦3¬|ac¬§^�bxfmac^*¦/«U¥Ff�Ó£bxÉЩ(d/ÉKb¢aKbYb3^�ac¬�«Ud/ק¦3¬�d3×_aÐac¬§^�befmac^<«RbF¥FÑ [0, 1] P/ac¬§^�`,d/­ ¨©D×_aK^<ÓªbF¥FÑe×�^:dwd E9 beÉ�f�^*¦3¥7acb'b/^3i »w´5L�YÂH5´FI�rFv,R�I�rFv,ruJcNQru·/I[I�p*H5P�N%LkrFIKs�N,Y ¡¢v%P�PkvUÁEYc¯»w´FI:IcVXVWv%V�L�rhJ�v%R�HF�7TWLkrF� E2 ÁwN%Y −2 TXL�R�IKY�TW´FI:IcVXVWv%VqLkrhJ�v%R�HF�7TWLkr5� E1 ¸wTX´FII�VWVWv%VÀLkrªJ�v%R�HF�7TWLkr5� E3 ÁwN%Y −3 TXL�R�IcY�TX´FI�IcVXVWv%V[L�r E2 Z TX´5I�VWIM¡¢v%VWI%¸_TX´FIqI�VWVXv,V�NUT[TX´5L�YY@TXI�H:Á N,YÐI�p7N,J#TWPk²|YXL¶p�TXL�R�IcYÐTX´FI�I�VWVWv%VÐL�r E1 l#¯[»w´*�5Yc¸3TX´FI�I�VWVXv,VÐL�r:JMv,R�HF�7TXL�rF� E9 Á N,Y(−2)(−3)(−4) · · · (−9) = 9! TWLkR�IKYETW´FI�IcVXVWv%VÂL�r E1 ¯Ð»w´FI�I�VWVXv,VELkr E1 ÁwN%YEsF�FI�TWv�TX´FIVWv%�Fr5sFLkrF��v%¡ 1/e �5YXLkrF��YXLkpuYXLk�,rFL¶�3J�NQr<TEs7L��%LkTWYc¸FÁ´FLOJ>´�L�YÂN%·/v,�7T 4.412× 10−7 ¯��EvUÁ I�t,I�VK¸TX´5L�Y�Y@R�NQP�P(IcVXVWv%V[R��FPkTXL�HFPkL�Ics¼·*² 9! �,N³t%I 9! × 4.412 × 10−7 = 0.1601 ¸_Á´FLOJ>´ LOYK�<�FL¶TWIPONQVW�%I%¯

��^*¥7«³«³¥5f�¦3bxf§¦¼ac¬§^;�q^*`,×_«³«U^*f�`,^

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Æ �<NQL�r¸¡¢v,VÀTX´FLOY[I�p7N%R�HFPkI,¸_LkT[TX�FVWrFIKs£v,�7T[TX´/NUT�Á I�J�v%�FPOs9�%I�T�N�R��5J>´¼·/I�T@TWI�V�VWIcYX�FPkTÀ·*²YXLkR�HFP�²:VWIcN%VXV>NQrF�,Lkr5��TX´FI�VWIcJ��FV>Y@L�v%r9Y@vuTX´/NUT[TX´5I�I�VWVXv,VÀNQT[I�t%IcVX²£Y�TWI�H¸�Lkr5Y@TXIKN%s£vQ¡w·/IcLkr5�R�NQ�,rFL¶�/IcsS¸7LOYÂVWIcs7�5J�IcsS¯DÃ1r5sFI�Ics¸FL¶¡DÁ I�VWI�ÁÂVWLkTXI�TX´FI[VWIcJ��FV>Y@L�v%ruN%YEn−1 =

1−En

n, n = . . . , 3, 2

TX´5I�ruTX´FI�I�VWVXv,VwNQTEIcN,J>´uY@TXIcHuÁÂL�P�P·/I�VXIKs7�5JMIKsq·*²uN�¡xN%J#TWv%VÂv%¡ 1/n ¯Ö»w´*�5Y�¸/Y�T>NQVXTXL�rF��ÁÂL¶TW´N�PONQVW�%IÀtUNQP��FI[vQ¡ n ZxYWN³²%¸ n = 20 lwNQr/s�Á v%V"�*Lkr5��·/N%J0�*Á N%VWs¸<Á I[ÁÂL�PkP§Y@IcIÀTX´5NQT E9 ÁÂL�P�P_·3IN%JcJM�FV>NUTWIÐTWv�¡¢�FP�P_Y@Lkp*ÇËsFLk�,L¶TÂHFVWIcJ�L�YXLkv,r¯»_v�v,·7TWN%Lkr|N�Y�T>NQVXTXL�rF��t³N%Pk�5I%¸FÁ IÀrFvQTWIÀTX´5NQTEn =

∫ 1

0

xnen−1dx ≤∫ 1

0

xndx =1

n + 1.

� LkTX´ n = 20 ¸ E20 ≤1

21¯��_IMT[Z YÐTWN6�,I E20 = 0 ¯À»w´FIcr¸SY�T>NQVXTXL�rF��ÁÂLkTX´ E20 = 0 ¸mL¶TÀJ�N%r·3I�YX´FvUÁÂroZxâ5v%V>Y@²<TW´FI%¸7noN%P�J�v%P�Ru¸*NQr/s�n|v%P�I�VÀZ�½Kà<Þ,ÞQl#¸,H¯�½³ÞQl�TX´5NQT E9 = 0.0916123 ¸UÁ´FLOJ>´LOYÂJMv%VWVWIcJ#TwTWv�¡¢�5PkP_YXLkp<Ç1s7L��%LkTÂHFVXIKJMLOY@L�v%r_¯»w´FIÀVXIKN%YXv%r�¡¢v,VÂv%·7T>NQL�rFL�rF��TW´FLOYÂN%J�J��FV>N%JM²�ÁwN%Y±TX´/NUTÂTX´5I�IcVXVWv%V L�r E20 ÁwN%YwNUTwR�v,Y@T

1

21aSTX´FLOY�I�VWVXv,VÐÁwN%YÌR��FPkTXL�HFP�LkIKs:·*² 1

20L�r£JMv,R�HF�7TXL�rF� E19 ¸�,Lkt*L�rF��N%roI�VWVXv,VÐv%¡±NUTÀR�v,Y@T

1

20· 1

21= 0.0024 L�ruTX´FI�J�v%R�HF�7T>NUTWLkv,r�v%¡ E19 ¸5N%r5s�YXv�v%r¯

�|{�\ � �]M��]�Tä ���:� �9��� ����%�»w´FI�JMv%r/JMI�HFTWYEv%¡DÏ5v,NQTXL�rF��H3v%L�r,T�r*�FR�·3I�V>YÐN%r5s|VWv%�Fr5sFLkrF��IcVXVWv%V>Y´5N³t,I[·/IcI�roL�r<TXVWv7s7�5JMIKsNQr/s�s7LOYWJM�5YWY@IKsqL�ruTX´FLOYEJ>´5N%H7TXIcVc¯½,¯ ç Ñxd3¥7acbef�¦ èÐd3bxfma ¬|ק­£Ø(^<«³É2� Æ rFv,VXR�N%PkL�Í�IKs®Ï/v,NUTWLkr5� H/v,Lkr<Tqr*�FR�·/IcVq´5N,Y�TX´5I¡¢v%VWR

x = ±rβe,

Á´FI�VWI e LOY�J�N%PkP�Ics9^<ÊS©(d/f§^*fma%¸ r LOY�TX´FI�Écbe¦/f§bY�D`%¥Ffma,¸�N%r5s β LOYÀTW´FI�ØD¥FÉc^:vQ¡ TX´FIr<�5R�·3I�V�YX²7Y�TWI�R�¯q»w´FI�Ï5v<NUTWLkrF�|H3v%L�r<T�r*�FR�·3I�V�YX²7Y�TWI�RÅL�Y�J>´/NQV>N%J#TWI�VWLkÍcIcs£·*²o¡¢v%�5VH5NQV>NQR�IMTWI�V>Y�È β ^ TW´FIEØD¥FÉK^wa t ^ TX´FI©§«U^<`,bxÉKbxd3f�a L, U ^ TW´FIEÑxdfg�^<«�NQr5s�×_©§©(^<«Ñxbx­®b¢acÉ�dwdÂaK¬�^:^,ÊS©±d/f§^*fma%¯º ¯ æ «³«Ud5«UÉ2�[»w´FI�IcVXVWv%V³ZxY>lwL�r£NuJMv,R�H5�7TWNQTXL�v%roLOY�R�IKN%YX�FVWIcs�I�LkTX´5I�VÀ·*²�/�E.�.&6�9CD5H���. � "&( v%V� *�$��/(5U��+R�4�$ � *&( #¯¤ QF�K "�.��/65@��+(�4�. � "&( 0��34/,¥��L3<&( *�K�$�.�F�.�L5YQ]/(�[/�E.�$&(�9CD5H�:�$ � *&6 0�.`»w´FIÐVWI�PONUTWLkt,IÐI�VWVWv%V(�,Lkt,IcY±NQrqL�r5s7LOJ�NQTXL�v%r�v%¡TX´5IÌr*�FR�·/IcV v%¡�YXLk�,rFL¶�3J�NQr<Tws7Lk�,L¶T>Y(L�r�N%rNQHFHFVWv³p7LkR�NQTXIÌN%r5YXÁ±IcVc¯»w´FI�VXIcP�NQTXL�t%I�IcVXVWv%VÀL�r£VWI�H5VXIKY@Icr,TWLkr5��NuVXIKNQPDr*�FR�·3I�V x ·*²oLkTWY�Ï5v<NUTWLkrF��H3v%L�r<TÀVWI�HFÇVXIKY@Icr,T>NUTWLkv,r Ï (x) LOYq·3v%�Fr/s7Icsh·<²yN r*�FR�·/IcV µ ¸�J�N%PkP�Ics¹TW´FI�3</�#0Qf������! "�"#$�%���'&(�h ­ ¬§^*d5«U^*­ ú ÄDÃ�ÄDÃUÅ|_io¯j³¯

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�|{#}ih �9���������_���]�����Ðå,��Ý~q��%�Ö����Ò�ª�Ì�:���X���âFv,VÖs7IMT>NQL�P�YÖvQ¡mÃ1ÙÖÙÖÙ®Y@TWN%r5sFNQV>sS¸%YXI�I TW´FIER�v%rFv,�%V>NQHF´ij Æ r Æ R�I�VWL�JcNQr Û NQTXL�v%r5N%P5G<TWN%r5sFN%VWsSÈÃ1ÙÖÙÖÙ G<TWN%r5sFN%VWs¹¡¢v,V�Ä L�r5NQVW²ãâ§P�v,NQTXL�rF�QÇ7�Dv%L�r,T Æ VWL¶TW´FR�IMTXLOJQ¸ kªÃ1ÙÖÙÖÙ HF�F·FP�L�JcNUTWLkv,r¸�½Kà%á,ßF¸Ã1ÙÖÙÖÙ Y@TWN%r5sFNQV>s�¡¢v%V ½ N,s7L¶p*ÇËLkr/s7I�H3I�r5sFI�r<TÖâ�P�v,NQTXL�rF���Dv%L�r,T Æ VXLkTX´FR�I�TXLOJQ¸%Ã1Ù(ÙÖÙE¸ Û I�ÁmlÖv%V"�3¸½cà,á<Þ7¸ N%r5snj Û �FR�I�VWL�JcNQP[°±v%R�HF�7TWLkrF�®ÁÂL¶TW´ÚÃ1ÙÖÙÖÙ â�P�v,NQTXL�rF�s�Dv%L�r<T Æ VWLkTX´FR�IMTWL�J�ko·<²¹n ¯ÕÌt%IcV@TWv%r¸FG*Ã Æ n ¸ º ¾%¾F½,¯

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âFv,V�VXIKY@�5P¶T>Y�v%rãI�VWVXv,V�·3v%�5r5sFY�¡¢v,V�·5N%YXLOJqÏ/v,NUTWLkr5�9H3v%L�r<T�R�NUTXVWLkp v%H3I�V>NUTWLkv,r5Yc¸ÖTX´5IJMPON%YWYXL�J�·3v<v2�7Yw·<²/o,N%R�IKY���¯ � L�Po�<L�r5YXv%r Z¢L lqp7rts�uwv#xys{z}|�~y��������xys{��� ~yv��tsT��|���z�vVs{�Z Æ Ù��±l§NQr/sqZ¢L�L ly�7�y�t�t�����yx!��|c|���|t�e���!uwv#x�sXz}|�~�������|��y��sX�W�[sX�EZ'�ÖVWI�r<TXLOJMI�Ç7�ÌNQP�Pe¸ Û I�Áo,I�V>Y@Ic²%¸_½Kà%¿,ä,lwNQVWI�I�p<TWVXIcR�I�P�²��5YXIM¡¢�5P§N%r5s|tUNQP��5NQ·5PkI�VWIcYXv%�5VWJ�IcYc¯ ¤ QF�43<&R��5� *�0#"�$�F5r/6CD5�QF&(? */657/65@��+(��E0&,&_¥I&(� �$ � *&6 k/(�!/6�98R���%���%�)5YQ]�u&(��kE$8ê�EL��%´/NQR Z@½cà%à,¿,lM¯�¢�+R�$ �8} *�.�.�0/6 *#"QF�. k&BA�FCD3<�$ ��'#"/6��/(�/(�98R���%�K3)Cf��5 Q]/6+R�4/�#"&0�!8k&BA�5�QD�%�LE"&,&_¥2`

�ÐLOYWJM�5YWY@L�v%ruv%r�·5N%YXLOJEÏ5v<NUTWLkrF��H3v%L�r,TÂv,H/IcVWNQTXL�v%r5YwNQr/s�VWv%�Fr5sFLkrF��I�VWVWv%V>Y±s7�5IÌTWv�J�NQrFÇJMIcPkPONUTWLkv,r5YÂNQr/sqVWIcJ��FVWYXL�t%IÌJ�v%R�HF�7T>NUTWLkv,r5YÂNQVWIÌ�,Lkt,I�rurFvUÁwN%sFN³²7YÖL�r�R�NQr*²�r*�FR�I�VWL�JcNQPSNQr/NQPkDz7Y@LOY TXI�p*TE·/v*v��7Yc¯��,[��fçÀ�e�§��� ��� ����������� ��� Â�|i@~ ZxN,lk�ÖVWvUt%IÌTX´FIÀ¡¢v,PkP�vUÁÂL�rF��IMp7HFVWIcYWY@L�v%r

| Ï (x)− x||x| ≤ µ =

{ 1

2β1−t ¡¢v%VÂVWv%�5r5s7L�rF�

β1−t ¡¢v%VEJ>´5v%HFHFL�rF�Z¢·/l�G*´FvUÁhTX´5NQT�ZxN<lwJ�NQru·3I[ÁÂVXLkT@TWI�r�LkruTW´FI�¡¢v%VWR

Ï (x) = x(1 + δ), |δ| ≤ µ.

|i g��IMT x ·3I�N�Ï5v,NQTXL�rF��H/v,Lkr<TEr*�FR�·3I�VEN%r5s k ·3I�N�H/v<Y@LkTXL�t%IÀL�r,TWI��,I�VK¸*TX´FIcrÏ (xk

k!) =

xk

k!(1 + ek),

Á´FI�VWI|ek| ≤ 2k µ + O(µ2).

|i%| °±v%r5Y@TXVW�5J#TwIMpFN%R�H5PkIKY TXv�Y@´5vUÁyTX´5NQTwTX´FI�s7LOY@TXVWLk·F�FTXL�t%IÀP�N³ÁÚ¡¢v%VwÏ5v<NUTXL�rF��H/v,Lkr<TÐN,sFs7LkÇTXL�v%roNQr/s�R��FPkTXL�HFP�L�JcNUTXL�v%rosFv<IKYErFv%TдFv,P�sS¯ � ´/NUTÌJcNQr|²%v,�|YXN³²uNQ·3v%�7TETW´FI�J�v%R�R��FÇTWNUTWLkt*LkT�²�N%r5s�N%YWY@v7JMLONUTWLkt*LkT�²À¡¢v%V�TW´FIcYXIEv,H/IcVWNQTXL�v%r5Y%Q/��L�t%IÂVWIcN,Y@v,r5Y�¡¢v%VÖ²,v%�FVÖN%r5YXÁ±IcVWYc¯|i�j �IMT x1, x2, . . . , xn ·3IÀTX´5I n Ï5v<NUTWLkrF��H3v%L�r<Twr*�FR�·/IcVWYc¯��ÌIM�5rFI

s2 = Ï (x1 + x2), sk = Ï (sk−1 + xk), k = 3, . . . , n.

»w´FI�r�¡¢VXv,RÝ»w´FIcv%VWI�R ú ÄDÃ�ÃUÅä5¯k½ º YX´FvUÁÚTW´5NUT

Ï (x1 + x2 + · · ·+ xn) = x1(1 + η1) + x2(1 + η2) + · · ·+ xn(1 + ηn).

|i z ZxN,l���L�t%I[N�HFVWv<v%¡�v%¡§»w´FIcv%VWI�R ú ÄDÃ��DÃUÅä5¯k½cÔ5¯Z¢·/l��ÖVWvUt%IÀ»w´FI�v,VXIcR ú ÄDÃ��DÃ�ÄäF¯�½Kß�·*²��5V>Y@TwIKY�T>NQ·FP�LOY@´FL�rF��TX´5I[VXIKY@�FPkT�Z ÄDÃ��DÃUÅä5¯k½,½Kl#¯|i%¯ ZxN,l�°±v%r/Y�TWVX�5JMTENQrqI�pFNQR�HFP�IÐTWv�YX´FvUÁyTW´5NUTK¸7Á´FI�r�N%s5s7Lkr5��N�P�L�Y@TÂvQ¡_Ï/v,NUTWLkr5��H3v%L�r<Tr*�FR�·3I�V>Yc¸Lkr¼�,I�rFIcVWN%Pe¸mTW´FI�VXv,�Fr5s7L�rF��I�VWVWv%V�ÁÂLkP�PÖ·3I�PkIKYXY�Lk¡±TW´FI�r*�FR�·3I�V>Y[NQVWIN%s5s7IcsuL�r�v,VWs7IcVÂvQ¡§Lkr5J�VXIKN%YXLkr5��R�NQ�,rFLkTX�5s7I,¯

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ôöõ5÷,ø(ùmúÂû�üÖýÚþSøÀû�üDú�ü ÿ���������qùEü����� �ø����Z¢·/luâ§Lkr/suN%rFvQTW´FI�VÂI�pFNQR�HFP�IÌTWv�Y@´FvUÁhTX´5NQTÂTX´FLOYwLOYÂrFvQTEN%PkÁwN³²7Y±r5IcJMIKYXYWNQVWL�Pk²�TWVX�FI,¯

|i�¼StÌY@L�rF��»w´FI�v,VXIcR ú ÄDÃ��DÃUÅä5¯k½UÞ*¸*Y@´5vUÁ¹TX´5NQT ´FL��%´�VXIcP�NQTXL�t%IÐN,J�J��FVWN,JM²�L�Y v,·7TWN%LkrFIKs�L�r�JMv,R�HF�7T@ÇL�rF� xT x ¯|i%¿öG7´FvUÁÚTW´5NUT

ZeN,lqÏ (cA) = cA + E, |E| ≤ µ|cA|Z¢·/lqÏ (A + B) = (A + B) + E, |E| ≤ µ(|A|+ |B|)ZeJclqÏ (AB) = AB + E, |E| ≤ nµ|A| |B|+ O(µ2)

ZË°±v%r5YX�FPkTrø bxÑ�&bef�ÉKd3fZh.~fè!¯3z!j³¸7H_¯S½%½KßF¯ l|i%è�°±v,r5Y@TXVW�5J#T(NÀYXLkR�HFP�IÂIMpFNQR�HFP�IwTXv�YX´FvUÁ®TX´5NQT�TX´FIER�NQTXVWL¶p�R��FPkTXL�HFPkLOJ�NQTXL�v%r�L�r�Ï5v,NQTXL�rF�H3v%L�r<TÐNQVWL¶TW´FR�IMTWL�J�rFIcIcsurFvQTE·3I[N%JcJM�FV>NUTWI%¯|_i'~DðS�(VXvUt,IÐTW´5NUTELk¡ Q LOYwv%VXTX´Fv,�%v%r/NQP/TX´FIcr

Ï (QA) = Q(A + E), Á´5I�VWI ‖E‖2 ≤ n2µ‖A‖2 + O(µ2).

|_i'~!~ª�_IMT y1, . . . , yn ·3I n J�v%P��FR�r�t%IcJMTXv,VWYws7I��5rFIcsuVWIcJ��FVWYXL�t%I�P�²mÈyi+1 = Ayi, i = 1, 2, . . . , n− 1.

�_IMT yi = Ï (yi) ¯Öâ�L�r5suN�·/v,�Fr5sq¡¢v%V TX´FI[VWI�PONUTWLkt,IÐIcVXVWv%V Lkr�JMv,R�HF�7TXL�rF��IcN,J>´ yi, i =1, . . . , n ¯

|_i'~FgS�_IMT β = 10, t = 4 ¯±°±v%R�HF�FTXI Ï (AT A),Á´5I�VWIA =

1 110−4 00 10−4

.

½ I�H3IcNUTE²,v%�FVÂJ�v%R�HF�7T>NUTXL�v%ruÁÂLkTX´ t = 9 ¯±°±v,R�H/NQVWIÐTW´FI[VXIKY@�5P¶T>Y�¯|_i'~D|öG7´FvUÁy´FvUÁ¹TXv�NQVWVWN%rF�%IEJ�v%R�HF�7T>NUTWLkv,r�Lkr�IcN,J>´�vQ¡STW´FIС¢v,PkP�vUÁÂL�rF�5¸*Y@v�TX´5NQT(TW´FIÌP�v,YWY±vQ¡YXL��%rFLk�/J�N%r<T�s7L��%LkTWYÀJ�N%r:·3I�N³t,v%LOs7Ics¯��Ìvuv%rFI�r*�FR�I�VWL�JcNQP§IMpFNQR�HFP�I�L�r:IKN%J>´£JcN%YXI�TWvYX�FHFH3v%VXT²%v,�FVENQr5YXÁ I�VK¯

ZeN,l ex − x− 1 ¸ ¡¢v,VÂrFI��<NUTXL�t%I�tUNQP��FIcYwv%¡ x ¯Z¢·/l √x4 + 1− x2 ¸q¡¢v,VÂP�N%VX�,IÌtUNQP��FIKYÂvQ¡ x ¯ZeJcl 1

x− 1

x + 1¸ ¡¢v,VÂP�N%VX�,IÌtUNQP��FIKYwvQ¡ x ¯

Zxs5l x− sin x ¸ ¡¢v%VÂtUN%Pk�FIKYÂvQ¡ x rFIcN%VÂÍ�IcVXv/¯ZxIKl 1− cos x ¸ ¡¢v%VÂtUNQP��FIcYwv%¡ x rFIKNQVÂÍcI�VWv5¯Z¢¡Wl ex − 1

x¡¢v,V |x| << 1 ¯

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�ñKî�f�>�R� â

ê

Z¢�<l (1− cosx)

x2¡¢v%VEYXR�N%PkP x

|_i'~*j � ´5NUTENQVWI�TX´FI[VWI�PONUTWLkt,I�N%r5s�NQ·5YXv%P��7TWI�IcVXVWv%V>Y±Lkr|NQH5HFVXv³p7L�R�NUTXL�rF�ZxN,l π ·<² 22

7Q

Z¢·/l 1

3·*²:¯ ä%ä%äVQ

ZeJcl 1

6·*²:¯�½c¿%¿VQ

�EvUÁhR�N%r<²�Y@L��%rFLk�/JcNQr<TwsFLk�,L¶T>YENQVWIÐTW´FI�VWI[L�r�IKN%J>´�JMv%R�HF�FTWNUTWLkv,r}Q|_i'~Fz��IMT β = 10, t = 4 ¯±°±v%r5YXL�sFI�VÂJMv,R�H5�7TXL�rF�

a = (1

6− .1666)/.1666.

�EvUÁhR�N%r<²�JMv%VWVWIcJ#TÂs7L��%LkTWYwv%¡�TW´FI[IMpFN%JMTENQr5YXÁ±IcVwÁÂLkP�PS²%v,���,IMT%Q|_i'~D¯ °±v%r5YXL�sFI�VwI�tUNQP��5NQTXL�rF�

e =√

a2 + b2.

�EvUÁÚJ�N%r�TX´5I�J�v%R�HF�7T>NUTXL�v%rq·/I�v%VW�,N%rFLkÍcIcs�YXv[TX´/NUTwvUt%I�VXÏ5vUÁãLkruJMv,R�H5�7TXL�rF� a2 + b2¡¢v%VÂPONQVW�%I�t³N%Pk�5IcYwvQ¡ a v%V b J�N%r�·3I�N³t%v,L�sFIcsXQ|_i'~]¼ � ´5NUTqN%r5Y@Á I�V>Y�ÁÂLkP�PE²,v%�ã�%IMT�Lk¡[²%v,�ãJ�v%R�HF�7TWI�TX´FI�¡¢v,PkP�vUÁÂLkr5�9r*�FR�·/IcVWY�v%rã²%v%�5VJ�NQPOJM�5P�NQTXv%Vwv,VwJ�v%R�HF�7TWI�V�Q

ZxN,l √108 − 1Z¢·/l √10−20 − 1ZeJcl 1016 − 50

°±v%R�HF�7TWIÌTW´FI�NQ·/Y@v,Pk�7TWI[NQr5s�VXIcP�NQTXL�t%IÀI�VWVWv%V>Y(L�r�IcN%J>´�J�N,Y@I,¯|_i'~D¿ � ´5NUTÂHFVWv%·FP�I�R s7v�²%v,��¡¢v%VWIcYXI�IÀL�r|Y@v,Pkt*L�rF��TW´FIK�<�5N%sFVWNQTXLOJÌI,�<�5NUTWLkv,r5Y

ZxN,l x2 − 106x + 1 = 0,Z¢·/l 10−10x2 − 1010x + 1010 = 0

�5Y@L�rF��TX´5I[Á±IcPkPkÇU�*rFvUÁÂrq¡¢v%VWR��FPONFÈx =

−b±√

b2 − 4ac

2a.

� ´5NUTÂVWI�R�Ics7²�sFv�²,v%�uY@�F�,�%IKY�T�Q Û vUÁöYXv%P�t%IÐTX´FI[I,�<�5NUTWLkv,r5Y �5Y@L�rF��²%v,�FVÂY@�F�,�%IKY�TWIcsVXIcR�IKs7²%¸7ÁÂLkTX´ t = 4 ¯

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ôöõ5÷,ø(ùmúÂû�üÖýÚþSøÀû�üDú�ü ÿ���������qùEü����� �ø����|_i'~DèöG7´FvUÁÚTW´5NUTÂTW´FI[Lkr<TWI��%V>NQP

yi =

∫ 1

0

xi

x + 5dx

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Ty = b

Á´FIcVXI T = (tij) LOYÂN�rFv,r5YXLkrF�,�FPONQV �FHFH3I�VwTXVWLONQrF�,�FP�N%VwR�NQTXVWL¶pqN%r5s y = (y1, y2, . . . , yn)T ¯G*H3IcJ�L¶�/JcNQP�Pk²,¸t11y1 + t12y2 + · · ·+ t1nyn = b1

t22y2 + · · ·+ t2nyn = b2

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tnnyn = bn

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(A + E)x = b + δb����������������� �"!$#��"%&�(')%+*-,�.0/21 341 57698:,;376=<(8:,;341 >(1?,A@�*-1 5CBD<(E EGFH@=I�B�@�J"IK/L6=1 5NM=1 IK@=<(<(/71 @PO�B0IK/7FQIK@R ,A>SFTBU,A/2VW<(/7/2IK/X,A@Y,AE Z�541 5NJ"IK/L8:,;37/71 [Q>(IK8XM=\P34,;341 IK@P5^]Q_�<`BD,A5`,;aXE 1?,;37<(V9Bb1 3469376=<dce,;371 IK@Y,AEUf�6HZ�541 >�,AEg , R IK/7,;34IK/hZ�1 @i.0/71 34,A1 @�*Y,A@PV�6=<(E Vdj�1 541 371 @=OX,AMPM�IK1 @T378X<�@T375e,;3bke/2OKIK@=@=<Lc�,;341 IK@Y,AE g , R IK/4,;37IK/2ZT*�lH34,A@PJ$IK/7Vm @P1 jT<�/2541 3nZT*�<o34>K]dpq1 E FH1 @=57IK@9V=1 <(Vr,A@�\=@T341 8X<(E Z�V=<�,;376r1 @ts�uKvKwP]xkyJ$<(E E I^Bb546P1 M�1 @Q6=1 5N@Y,A8X<X6Y,A5L541 @=>(<R <(<�@L<(5734, R E 1 546P<�VL,;3-k�/7OKIK@=@P<Dce,;371 IK@Y,AE g , R IK/4,;37IK/2ZT]�pq1 E FH1 @=57IK@�z 5 R I�IKFL{N|�}�~��K��}P�P���)�?�`�e�?�)}P�Y�K���K��}� �P���=�K}P��1 5b,A@i<o[�37/2<�8X<(E Zd1 8XM�IK/234,A@T3e,A@PVxjT<(/2Zx\=57<(J$\=E R IHIKFXJ"IK/b,A@TZx@H\=8X<(/21 >^,AE�,A@Y,AE Z�573�]

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(T + E)x = b,

Á´FIcVXI�TX´FI�Icr,TWVXL�IcYÌvQ¡±TX´FI�IcVXVWv%VÌR�NQTXVWL¶p E NQVWI4�,�5L¶TWI�YXR�NQP�Pe¯�Ã1r:¡xN,J#Tc¸SLk¡ E = (eij) N%r5sT = (tij) ¸7TX´FIcr

|eij | ≤ nµ|tij |+ O(µ2)

YX´FvUÁÂLkr5��TX´5NQT�TX´FIqI�VWVXv,V�J�N%r9·3IqIct%IcrªY@R�NQP�P�I�V[TW´5NQr¼TW´FIqIcVXVWv%V[R�N%sFI�Lkr VWv%�5r5s7L�rF��TX´FII�r<TWVXL�IcY�vQ¡ T ¯®»w´*�5Yc¸T5YQ]�;E0/�#0¥P��CFE.��5@��5UCD5@�'&6�I�! *&,#0���0��A.&( ��.&6�9+,�����ª/(��C,���F�$ �5U ��'/(�f�6CD��/6 ��8R��57�$3 �%����57/2E.�o�M¯æ Ê¥F­ ©DÑx^�j_i@~7g � fê��f§É�aK¥7ØDÑx^ � Ñx¦3d5«Ubxac¬§­�� ùu¥FקÉcÉKbx¥Ff æ Ñebx­ªbef�¥7aKbxd3f�ø b¢aK¬�d3×�ax(þ=z*{.y è[bYb3d5aKbxf§¦°±v%r5YXLOs7I�V�YXv%P�t*LkrF�9TW´FI 2 × 2 Y@²7Y@TXIcR �5YXLkr5�£TW´FI:Y@TWN%r5sFNQV>sªIcPkL�R�Lkr/NUTXL�v%r¹R�IMTW´Fv7sS¸J�N%PkP�Icsê�ÀN%�5YXYXLONQr�I�P�LkR�L�r5NUTWLkv,rÈ Ax = b ¸,Á´FI�VWI A =

(10−10 11 2

), b =

(13

) ¯�»w´5NQTLOY�¸10−10x1 + x2 = 1

x1 + 2x2 = 3.ÙÖP�LkR�L�r5NUTWLkr5� x2 ¡¢VWv%R�TW´FI�Y@IKJMv,r5sqI,�<�5NUTWLkv,r¸7Á±IÀv,·7TWN%Lkr10−10x1 + x2 = 1

(2− 1010)x2 = 3− 1010

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�%L�t*Lkr5� x2 = 1 ¸ x1 = 0 ¸7Á´FIcVXIKN%Y±TX´5I[IMpFN%JMTÂYXv%P��7TWLkv,r�LOY x1 = x2 = 1 ¯»w´*�5Y�¸§ac¬§^®¥FØ(dfb/^㩧«³d_`%^*ÉcÉ|beÉo`%Ñe^*¥7«UÑYc ×�f§Écac¥7ØDÑx^3io»w´FI�VWIcN,s7I�V>Y[NQVWI�N%Y*�%IKsoTWvt%IcVXLk¡¢²oTW´FI�R�YXI�P�t%IKYÀTX´5NQT[TX´FI�JMv%R�HF�FTXIcs YXv%P��7TWLkv,r x = (1, 0)T LOYÀTX´FIqIMpFN%JMT�Y@v,Pk�7TWLkv,r¼vQ¡TX´5I�Y@²7Y@TXI�R (A + E)x = b ¸7Á´FI�VWI E L�YwPONQVW�%I,¯ÃË¡§N%r�NQP��%v,VXLkTX´5R LOYÂY@TWNQ·5PkIÌ¡¢v%VwN��,Lkt,I�r�R�NQTXVWL¶p A ¸*TX´FIcr�v,rFI�Á±v,�FP�sqPkLo�%IÌTXv�YXI�IÌTX´5NQTTX´5I�NQP��%v%VWLkTX´FR LOY�Y@TWNQ·5PkI�¡¢v%V[Ict%IcVX²|R�NUTWVXLkp A L�r N��,Lkt,I�r¼JMPON%YWY�¯�»w´*�5Yc¸_Á I�R�N³²:�%L�t%I�N¡¢v%VWR�NQPs7I��5rFLkTXL�v%r�vQ¡DY�T>NQ·FL�P�L¶T�²qN,Y ¡¢v%P�PkvUÁEYcÈ

� ^2�Df§b¢acbed/f:j_i@~f|_i Æ r�NQP��%v%VWLkTX´FR�LOY§Y@TWN%·FP�IÖ¡¢v%VDN�JMPON%YWY_v%¡FR�NQTXVWL�J�IcY C Lk¡F¡¢v%V§I�t,I�VW²ÀR�NQTXVWL¶p  $ ¡ $A Lkr C ¸mTX´FI�JMv,R�HF�7TXIKs:YXv%P��7TXL�v%r:·<²�TW´FI�NQP��%v%VWLkTX´FR LOYÐTW´FI�IMpFN%JMT�YXv%P��7TXL�v%r£vQ¡±NqrFIcN%VX·*²HFVWv%·FP�I�R�¯

»w´*�5Y�¸F¡¢v%VwTX´5I[PkL�rFIcN%VÂY@²7Y@TXI�R HFVWv%·FP�I�RAx = b,

NQruNQP��%v,VXLkTX´FRÎLOYwY�T>NQ·FP�IС¢v,VÂN�JMPON%YWY vQ¡_R�NQTXVWL�J�IcY C Lk¡¡¢v%VwI�t,I�VW² A ∈ C N%r5s�¡¢v,VwIcN%J>´ b ¸7L¶THFVWv7s7�5JMIKYÂN�JMv%R�HF�FTXIcs�YXv%P��7TXL�v%r x TX´5NQTÐYXNQTXLOY��/IcY(A + E)x = δ = b + δb

¡¢v%VEYXv%R�I E NQr5s δb ¸FÁ´FIcVXI (A + E) LOYÂJMP�v,YXI�TXv A NQr5s b + δb L�YÂJ�Pkv<Y@I�TWv b ¯ü£{7ü �ã���L���e���e�����W��� �)åö�Ö��� �y,�u���]��� ���:���o��%�Ö��5���[���]���

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}.

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c(x) =

∣∣∣∣xf ′(x)

f(x)

∣∣∣∣ . Z¢Ô/¯ º l ²,³ þ=z'þ=z7yÆ YÂN%rqIMpFN%R�H5PkI,¸<P�IMT f(x) = ex ¯Ö»w´FI�r c(x) = |x| ¸7Á´FLOJ>´�LOY P�N%VX�,IE¡¢v,V PONQVW�%I�tUNQP��FIcYvQ¡ x ¯Ì»w´5NQTÌR�IcN%r5Yc¸ /k��34/6���� "�.��/65@��+(���$ � *&( :��� x #0/6�k� *&,-(CF#0�</u��/( U�2�) *�$��/65@��+(���$ � *&( :���ex \��$&<5YQf�%�©� *&�E$���$3Û�%�K�����9?B#0&6�!-(��5U�'&(��0-R\�V�QF�.� x �%����/6 @�2��¯ÃË¡ f v%V x L�Y[N�t%IKJ#TXv,Vc¸mTW´FI�r£TX´5I�JMv%r/s7L¶TWLkv,r:r*�FR�·3I�V�JcNQr£·/I�s7I��5rFIcs£L�r£TW´FI�YXN%R�IÁwN³²��5Y@L�rF��rFv,VXR�YwL�r5Y�TWIcN,s�v%¡_TW´FI�NQ·5YXv%P��7TWI�tUN%Pk�FIKY�¯»w´*�5Y�¸%TX´FIÌJMv%r/s7L¶TWLkv,r�r*�FR�·3I�VÖv%¡mN�¡¢�5r5J#TWLkv,r�vQ¡Y@Ict%IcVWN%P7t³N%VXLONQ·5PkIKYÂZ¢v%V±NÀt%IcJMTXv,V>l§J�N%r·3I�s7I��5rFIKso·*²|VXIcHFPON%JML�rF� f ′(x) ·<²�LkTWY��%V>N%s7L�I�r<TK¯ÌÃ1r:TX´FLOYÀJ�N%YXI c(x) =

‖x5 f‖‖f(x)‖ ¸SÁ´FI�VWI

5f LOYwTX´FIÀ�,VWN,s7LkIcr<Tc¯âFv,V±I�p7N%R�HFPkI,¸,Lk¡ f(x) = x1−x2 ¸%TW´FI�r 5f = (1,−1) ¸<N%r5s�TX´5I�J�v%r5s7LkTXL�v%r�r*�FR�·/IcVc(x) vQ¡ f ZxÁÂL¶TW´|VXIKY@H3IcJMT TWv�TX´FI[L�r7�5rFLkTXI�rFv%VWR�l(LOYw�%L�t%Icr�·*²mÈ

c(x) =‖x5 f‖∞‖f(x)‖∞ =

2 max{|x1|, |x2|}|x1 − x2|

Z¢Ô/¯ ä<l ²,³ þ=z'þ=z $Á´FLOJ>´�YX´FvUÁEY±TX´/NUT f(x) L�YEbeÑxÑ¢¨W`%d3f§Ó�bxacbxd3f§^<Ó b'd x1 ' x2 ¯æ Ê¥F­ ©DÑx^�j_i@~D¼ ÷ d3f§Ó�bxacbxd3fª¬|×�­£Ø(^<«�dwdwaK¬�^½� ¥7a�«UbxÊm¨*¦[^*`Qacd/«�èÀ«Udӧק`QaG*�FHFH3v,YXI A ∈ R NQr/s x L�YEN%r�r7Ç]t,IcJMTXv%VK¯(»w´FIcr|L¶T�J�NQr�·3I�Y@´5vUÁÂrP¹ æ Ê^<«U`%beÉc^:j_i@~7g]ºTX´/NUTwTX´FI[J�v%r5s7LkTXL�v%r�r*�FR�·3I�V κ vQ¡ Ax Z¢ÁÂLkTX´uVWIcYXH/IKJ#T TXv�H/IcV@TW�FVW·5NUTWLkv,r5Y vQ¡ x l L�Yw�,Lkt,I�r�·*²

κ = ‖A‖ ‖x‖‖Ax‖ , Z¢Ô/¯ Ô*l ²,³ þ=z'þ=z@þÁ´FIcVXI�TX´5I[R�NQTXVWL¶p�rFv%VWR LOY TX´FI�YX�F·3v%V>s7Lkr/NUTXIÀR�NUTWVXLkpqrFv,VXR�¯ÃË¡ A LOYEY"�<�5NQVWI�N%r5s�rFv%r5YXL�rF�%�FPONQVK¸<TX´FIcr

κ ≤ ‖A‖‖A−1‖. Z¢Ô/¯ ß,l ²,³ þ=z'þ=z��æ Ê¥F­ ©DÑx^�j_i@~f¿ ÷ d3f§Ó�bxacbxd3fª¬|×�­£Ø(^<«�dwdwaK¬�^:èÐd3ÑYcf§d/­ªbe¥FÑ���dd/acÉ»w´FIqH5VXv,·FPkIcRÅv%¡w�5r5s7L�rF�:VWv*vQT>Y�vQ¡EN|H3v%P�²*rFv%R�LONQP J�NQr ·3Iq´5Lk�,´FPk²9L�PkPkÇËJ�v%r5sFL¶TWLkv,rFIcsS¯� I��5V>Y�T�L�P�Pk�5Y@TXV>NUTWI�TX´FLOY�ÁÂLkTX´ N|Y@L�R�HFPkI��,�/N%s7V>NUTWL�J�H3v%P�²*rFv%R�LONQP]¯|°±v,r5Y@LOs7IcV�Y@v,Pkt*L�rF��TX´5I�<�5N%sFVWNQTXLOJÐI,�,�/NUTXL�v%r_È

f(x) = x2 − 2x + 1 = 0.»w´FI�VWv*vQTWYÀN%VXI x = 1, 1.Û vUÁ H/IcV@TW�FVW·:TW´FI�JMv*Iòñ�J�LkIcr,T º ·*²:¾5¯ ¾,¾%¾%¾5½%¯�»w´FI�JMv,R�H5�7TXIKsVWv<v%TWYÂv%¡DTX´FI�H/IcV@TW�FVX·3Ics�H3v%P�²*rFv%R�LONQP f(x) = x2 − 2.00001x + 1 NQVWI%È x1 = 1.0032 NQr5s

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x2 = 0.9968. »w´5I�VXIcP�NQTXL�t%I�I�VWVXv,VWY�L�r x1 NQr5s x2 N%VXI�¾5¯ ¾,¾%ä º a_v,r:TW´FI�vQTW´FI�V[´/NQr5sS¸_TX´FIVWI�PONUTXL�t%I I�VWVWv%V§Lkr�TX´5IEsFNUT>NÌLOY 5×10−6 ¯�»w´*�5Y�¸,NL��3</(���6�]�$ �5@CD $E0/65@�'&6�k���k5YQ]��-�/657/)#0Q]/(�f�2�0-5YQ]�� "&_&65'����CFE���5H/(�w5@�'/6���98U¯ ­ ¬�^;ø beÑ�&bef�ÉKd/fãèÐd3ÑYcf§d3­ªbx¥FÑ»w´FI�NQ·3vUt%I�IMpFNQR�HFP�I�Lkr*t%v,Pkt,Ics:R��FPkTXL�HFP�I�VWv*vQT>Y�¯�n|�FPkTXL�HFPkI�VWv*vQTWY�v,V�VWv*vQT>Y�J�Pkv<Y@I�TXv�IKN%J>´vQTW´FI�V_Lkr*tUNQVWL�N%·FP�²ÐR�N6�,IDTX´FIÖVWv*vQTXÇe�5r/s7Lkr5�ÂHFVXv,·FP�I�R�LkP�P¶Ç1JMv,r5s7LkTXL�v%rFIKs�a³´FvUÁ I�t%IcVc¸�TX´FI��! *&2E.�o�.3#0/(�nE0�W�����9?B#0&6�!-(��5U�'&(��0-��$+R�$��V�Q]�$��5�QF�W *&,&(5@�W/6 *�W+(�. �8[V��$�����$�H�]/6 */(5H�0-U¯Ú°±v%r5YXLOs7I�V�TX´FI¡¢v%P�P�vUÁÂLkrF��Á I�P�P¶Ç7�*rFvUÁÂrqI�p7N%R�HFPkIÀ·*² � L�Po�<L�r5YXv%rÈ

p(x) = (x − 1)(x− 2) · · · (x− 20)

= x20 − 210x19 + · · ·»w´FI�VXv*v%TWY�vQ¡ p(x) N%VXI 1, 2, . . . , 20 ¯ Û vUÁ H3I�VXTX�FVW·ªTW´FI|JMv*Iòñ�JML�I�r<T�vQ¡ x19 ¡¢VWv%R−210 TXv −210− 2−23 ¸SPkIKN³t<L�rF��vQTX´5I�VÀJMv*Iòñ�JML�I�r<T>YÌ�5r5J>´5NQr5�%Ics¯À»w´5L�YÀJ>´5N%rF�%I�N%R�v%�Fr<TWYTXv�N%HFHFVWv³p*L�R�NUTWI�P�² 1.12× 10−7 ¸FÁ´FLOJ>´uL�YEYXR�NQP�Pe¯(G7I�t%IcVWN%PmVXv*vQT>YwvQ¡§TX´FI[H3I�VXTX�5VX·3IcsuH3v%P�²,ÇrFv,R�LONQP]¸7J�N%VXI�¡¢�FPkP�²qJ�v%R�HF�7TWIcs�·<² � LkPo�*Lkr5YXv%r_¸7Á±IcVXIÌ¡¢v%�Fr/s�TWv�·3I[t%IcVX²�s7LkµmI�VWI�r<T¡¢VXv,RÎTX´FIv%VWL��%L�r5NQP§VXv*v%TWYc¯�â5v%VÀIMpFN%R�H5PkI,¸TX´5I�VWv*vQT>Y x = 16 N%r5s x = 17 ·3IcJ�N%R�I�NQHFHFVWv³p7LkR�NQTXI�P�²I_�<�5N%PmTXv 16.73± 2.81i ¯�»w´FLOYEJ>´5NQr5�%IÀJ�NQr�·3I[IcN,Y@L�Pk²qIMp7HFPONQL�rFIcs�·*²�JMv,R�HF�7TXL�rF��TX´5I�JMv%rFÇs7LkTXL�v%r£r<�5R�·3I�V>YÌv%¡ÖTX´5I�L�r5s7L�t*L�sF�5NQP�VWv*vQTWYc¯[ÃËT[J�NQr£·/I�YX´FvUÁÂry¢ æ Ê^,«U`,bxÉK^ j_i gF|Jh�¥wjA£�TW´5NUTTX´5I�JMv%r/s7L¶TWLkv,r¼r<�5R�·3I�V�v%¡±TW´FI�VWv<v%T x = xj ÁÂLkTX´ VWIcYXH/IKJ#T[TWv�TX´FI�H3I�VXTX�5VX·5NQTXL�v%r9vQ¡ TX´FIYXLkrF�,PkI[J�v<IBñ�JML�I�r<T ai LOY JMv%r/s j =

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• »w´5I�NQP��%v%VWLkTX´FR�R�L��%´<TE·/I[�5r5Y�T>NQ·FP�I• »w´5IÀTXIcY@TEIMpFN%R�H5PkIKY±R�N³²�·3IÀTXv*v�YXH/IKJMLONQP• »w´5IÀN%Pk�,v%VWL¶TW´FR�¸<TX´Fv,�F�%´uY@�5JcJMIKYXY@¡¢�FP]¸<R�L��%´<T´5N³t%IСxNQL�PkIKsqL�r�TX´FIÀH5N%V@TWL�J��FP�N%V J�LkV>JM�5R�ÇY@TWN%r5JMIKY

ü£{�� �:��%�Ö��/���[���]�u� æÝ�����x�À�±�e� �)åh�Ö��� ã��X������ �����������y,�u���]���°±v%r/Y@LOs7I�V TW´FIÀ¡¢v%P�PkvUÁÂL�rF��P�LkrFIKNQVÂYX²*Y@TXIcR�È

x1 + 2x2 = 3

2x1 + 3.999x2 = 5.999

»w´FI�I�pFN%J#T�Y@v,Pk�7TWLkv,r�L�YcÈ x1 = x2 = 1 ¯ Û vUÁ�R�N��%I�N�YXR�NQP�P_H3I�VXTX�FVW·5NQTXL�v%r|L�r�TX´FI�VXL��%´<T@Ç´5N%r5s�YXLOs7I[v%·7T>NQL�rFL�rF��TW´FI�YX²*Y@TXIcR�Èx1 + 2x2 = 3

2x1 + 3.999x2 = 6

»w´FI(YXv%P��7TWLkv,r�vQ¡*TX´5I±H3I�VXTX�FVW·3Ics�YX²7Y�TWI�R�¸³v%·7T>NQL�rFIcsÀ·*²Ó�ÀNQ�5YWYXL�N%r[I�P�LkR�L�r5NUTWLkv,r[ÁÂL¶TW´�H5N%V@TWL�N%PHFL�t%v%TXL�rF�uZeJMv,r5Y@LOs7IcVXIKs�TWv�·/I�N�Y@TWNQ·5PkIÀR�IMTW´Fv7s�L�r�HFV>N%J#TWL�J�IKl±LOY�È x1 = 3, x2 = 0.¤ QfCf�0\�/ê+R�$ �8}��3</(���©#0Q]/6�����u����5YQ]�� ��>�RQf5TQ]/6�!-;�0�'-2�}#"QF/(�f�2�0-ê5�QF���$&6�9CD5@�'&6�ª/6�957&6?���.5YQ]�$ $`

Ã1ruTX´FLOYEYXIcJ#TWLkv,r�Á I�Y@TX�5s7²qTX´FI[I�µ3IKJ#TEv%¡DYXR�NQP�PH/IcV@TW�FVW·5NUTWLkv,r5Y vQ¡§TX´FI[L�rFHF�7T�sFNUT>N ANQr/s b v%ruTX´5I�JMv%R�HF�FTXIcsuYXv%P��7TXL�v%r x vQ¡§TX´FI�YX²7Y�TWI�R Ax = b ¯G*L�r5JMIÀL�r�TW´FIÀPkL�rFIKNQVÂY@²7Y@TXIcR HFVWv%·FP�I�R Ax = b ¸*TW´FI[L�rFHF�7TÐsFNQTWN�NQVWI A N%r5s b ¸*TX´FIcVXIJMv,�FPOso·3I�L�R�HF�FVWL¶TWLkIKYÀI�LkTX´FIcV[Lkr b v,VÀLkr A v,VÀLkr9·/v%TX´¯ � I�ÁÂL�P�PDTX´FIcVXI�¡¢v%VWI�J�v%r5YXL�sFI�VÌTX´FIIMµmIcJMTÐvQ¡ÖH3I�VXTX�5VX·5NQTXL�v%r5YEv%r�TX´5I�YXv%P��7TXL�v%r x L�roIKN%J>´�vQ¡�TW´FIcYXI�J�N,Y@IKYEYXI�H5N%VWNQTXI�P�²%¯ � I�ÁÂL�PkPYXI�I�TW´5NUTÀL�r:NQP�P§TX´FIKY@I�J�N%YXIcYc¸mN�r*�FR�·3I�V[JcNQP�PkIKs�TX´FI}#0&(�-(��5U�'&(�ß�FCD3<E0�. �v%¡�TX´FI�R�NUTXVWLkp AHFPON³²7Y N%r�L�R�H/v,V@T>NQr<TEVWv%P�I%¯�]�h§���� ¨:©Ó®�·�¥æ�³²TÏ]®³° ¥­¤á°Pª¦´¦¥#§à�e¨ §�¨d¥­¬ª®¬«ê§à©q¬¦¥=­&®/´á¨ªÐ Þi§ãÐa® µb®y·�¥¸�É°

b¯=¯ þ=z���z*{ � I�N%YWY@�FR�I�´FI�VWI�TX´5NQTÂTX´FIcVXI�N%VXI�L�R�H5�FVXLkTXL�IcYELkr b ·F�7TÂTX´5I[R�NQTXVWL¶p A LOYwIMpFN,J#Tc¯­ ¬§^*d5«U^*­�j_i@~fè_i:hB�qbx¦3¬maÀèÐ^<«³ac×�«KØD¥7aKbxd3f ­ ¬�^*d/«U^<­ªj ÃË¡ δb NQr5s δx ¸,NQVWI%¸%VXIKY@H3IcJMTXL�t%I�P�²%¸±'��z���z*{ TX´5I±H3I�VXTX�FVW·5NQTXL�v%r5Y_vQ¡ b N%r5s x L�r�TX´5I±P�Lkr5IcNQV§Y@²7Y@TXIcR Ax = b ¸ A LOY_rFv,r5YXLkrF�,�FPONQV�N%r5s b 6= 0,TX´5I�r

‖δb‖Cond(A)‖b‖ ≤

‖δx‖‖x‖ ≤ Cond(A)

‖δb‖‖b‖ .

´Kµ,¶f¶R·f¸ G*L�r5JMIAx = b,

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NQr/sA(x + δx) = b + δb,� I[´5N³t%I

Aδx = δb.»w´5NQTEL�Yc¸δx = A−1δb.»§N6�*Lkr5��N�YX�F·3v%V>s7Lkr/NUTXIÀR�NUTWVXLkp*Ç]t,IcJMTXv%V rFv,VXR Á IÀ�%I�T

‖δx‖ ≤ ‖A−1‖ ‖δb‖. ZxÔ5¯ ¿,l ��z���z*{Æ �<NQL�r¸*TWN��*LkrF��TW´FI�YXN%R�I�r5v%VWR v%ru·/v%TX´|YXL�s7IKYwvQ¡ Ax = b, Á I��,IMT ‖Ax‖ = ‖b‖ v,V

‖b‖ = ‖Ax‖ ≤ ‖A‖ ‖x‖ Z¢Ô5¯ ÞQl ��z��z7y°±v%R�·FL�rFLkr5��Z �DÃ��DÃUÅÔ5¯ ¿,l NQr/s9Z �DÃ��DÃ�ÄÔ5¯ ÞQlM¸7Á±IÀ´5N³t,I

‖δx‖‖x‖ ≤ ‖A‖ ‖A

−1‖‖δb‖‖b‖ . Z¢Ô/¯ á<l ��z���z $ÕÌruTX´FI[v%TX´FIcV ´/NQr5sS¸ Aδx = δb �%L�t%IKY

‖δx‖ ≥ ‖δb‖‖A‖ Z¢Ô/¯ à<l ��z���z'þÆ POYXv5¸*¡¢VWv%R Ax = b ¸7Á IÀ´5N³t%I

1

‖x‖ ≥1

‖A−1‖‖b‖ . Z¢Ô5¯�½c¾<l ��z���z��°±v%R�·FLkr5LkrF�|Z �DÃ��Dà �Ô5¯ à,l NQr5s¼Z �DÃ��DÃ��Ô5¯�½c¾,lM¸7Á±IÀ´5N³t,I

‖δx‖x‖ ≥

‖δb‖‖A‖‖A−1‖‖b‖ .

• »w´FIÀvQTX´5I�VÂH5N%V@TEJ�N%r�·3I�YXLkR�L�P�N%VXL�Pk²qHFVXvUt,IcsS¯

� ^2�Df§b¢acbed/fãj�iOgwði ¤ QF�K�wCD34E0�$ ‖A‖ ‖A−1‖ �%�K#"/6�����0-<5YQ]�:#"&6�!-6��5@�'&6�ê�FCD3<E0�. �&�A A /6�!-�%�L-��$�!&657�0-kE$8 Cond(A) ` å fmac^<«³©�«U^<ac¥7aKbxd3fydwd ­ ¬�^*d/«U^<­ ú �DÃ��DÃUÅj_i@~fè

»w´FI�v,VXIcR ú �DÃ��DÃUÅÔ/¯k½KàuYWN³²7Y�TX´5NQT�N�VWI�PONUTXL�t%I�J>´5N%rF�%I�L�r¼TX´FI�YXv%P��7TWLkv,r¼JcNQr¼·3IqN%Y[PONQVW�%I�N,YCond(A) R��5P¶TWLkHFP�L�Ics£·<²oTX´FI�VWI�PONUTXL�t%I�J>´5NQr5�%I�L�r£TX´FI�t%IKJ#TWv%V b ¯�»w´*�5Yc¸X� A)5�QF�k#0&(�-(��5U�'&(��FCD3<E0�. k�%���!&65�57&,&I��/( @���.\K5�QF�.�S/I��34/6�����]�$ �5@CD $E0/65@�'&6�ª���e5YQ]�u+(�0#.5H&( b VX������Q]/6+R�}+R�$ �8�9��5U5@�o�u�@ÆT�0#$5�&6� 5YQ]�k�$&(�>CD5@�'&6�!` l � 5�QF�W&(5YQ]�$ 4Q]/6�!-(\�� Ak5�QF�u#0&6�!-(��5U�'&(�e�wCD34E0�$ <�%�k��/( U�2�.\5YQ]�$�P�$+R�$�I/��03</(���D�]�$ �5@CD $E0/65@�'&6�ê��� b 3)�>�RQf5�#"QF/(�����:5�QF���$&6�9CD5@�'&6�P-6 */R��5U�'#"/6���98�`

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ô �Fú/ù���û�÷7û úª¹�ø�õ�ùÂ÷,ý�ø��§û úeº��)��ùEü�ìøÀü�wû úÂûAø[ü±ûOüÖý

� = � ',.@+ � Ã1r|t*L�I�Á�vQ¡(»w´FIcv%VWI�R ú �DÃ��DÃUÅÔ5¯�½càF¸5Á´5NUTд/NQHFH3I�rFIKs|ÁÂLkTX´|TW´FI�NQ·3vUt%I[I�pFNQR�HFP�IJ�N%r�·3I[IcN,Y@L�P�²�IMp7HFPONQL�rFIcsS¯ Û vQTWIÀTX´5NQT ¡¢v,VwTX´FLOYÂI�p7N%R�HFPkI,¸ Cond(A) = O(104) ¯æ Ê¥F­ ©DÑx^�j_i g�~ � fybxÑeÑ ¨>`%d3f�Ó§b¢aKbxd3f�^*Ó®Ñebxf§^*¥7«�É$cÉcac^*­ ©§«Ud5ØDÑx^*­x(þ=z7yW�A =

1 2 12 4.0001 2.0021 2.002 2.004

, b =

48.00215.006

»w´FIÀIMpFN,J#TEY@v,Pk�FTXL�v%r x =

111

¯±°±´5NQrF�,I b TWv b′ =

48.00205.0061

¯

�q^*Ñe¥7acb'b/^|èÐ^<«³ac×�«³Ø§¥Facbxd3f��‖b′ − b‖‖b‖ =

‖δb‖‖b‖ = 1.879× 10−5 ZH��3</(���xl .

ÃË¡DÁ±I[YXv%P�t%IÌTX´FI�YX²7Y�TWI�R Ax′ = b′ ¸FÁ I��,IMT x′ = x + δx =

3.0850−0.04361.0022

.

Z x′ LOYEJMv,R�H5PkI�TXI�P�²qs7LkµmI�VWI�r<T¡¢VXv,R x l�q^*Ñe¥7acb'b/^ æ «³«³d/«¹bxfÎac¬§^hÉKd3Ñx×�acbxd3f�� ‖δx‖

‖x‖ = 1.3461. ÃËT�L�Y�IKN%YXLkP�²ªt,I�VWL¶�5IKsªTW´5NUT�TW´FIL�rFI_�<�5N%PkLkT�²�L�rq»w´FIcv%VWI�R ú �DÃ��DÃUÅÔ5¯�½càÀLOY(YWNUTXLOY@�5IcsSÈ Cond(A) · ‖δb‖

‖b‖ = 96.5920 > 1.3461 =||δx||||x|| ¯¤ Q]�r�! *�0-6�'#.5H�0-�#0Q]/6�����LV�/(�0\JQ]&(V��$+(�. 0\�&(+(�. ��98u����5U��34/657�0-�`æ Ê¥F­ ©DÑx^�j_i g/g � g�^<ÑeÑ ¨>`%d3f�Ó§b¢aKbxd3f�^*Óª©§«Ud5ØDÑx^*­

A =

(1 23 4

), b =

(37

).

»w´FIÀIMpFN,J#TEY@v,Pk�FTXL�v%r x =

(11

) ¯��I�T b′ = b + δb =

(3.00017.0001

) ¯¤ Q]�I *�$��/(5U��+R�P#0Q]/(�f�2�I��� b N ‖b′ − b‖

‖b‖ = 1.875 × 10−5 ZeÉc­®¥FÑxÑ]lW¯ Û vQTXI�TW´5NUTCond(A) = 14.9330 ZeÉc­ª¥5ÑxÑelW¯ ¤ QDCf�I/ -6 */R��5U�'#P#0Q]/6�����P����5YQ]�;�$&(�9CD5U�'&(� x �%�;�!&(5�"1��F�0#.5H�0-�`ÖÃ1r�¡xN,J#T x′ YXNQTXLOY�¡¢²*L�rF� Ax′ = b′ L�Y

x′ =

(0.99991.0001

)≈ x =

(11

). ¬|d5aK^!� ‖δx‖‖x‖ = 10−5 ZxÉK­ª¥FÑeÑel .

�]�h§���Ì ¨X©Ó®�·�¥æ�³²æÏ�®³°Ö¥­¤á°�ªª´¦¥#§à�q¨Î§�¨d¥­¬ª®è¡ ´¦¥­°[§$°A

�EIcVXIÂÁ IÐN,YXYX�FR�IwTX´5NQT(TW´FI�VWIÐNQVWIÂL�R�H5�FVXLkTXL�IcY(v%rFP�²�L�r A N%r5s�N%Y(N[VXIKY@�5P¶T(Á IE´5N³t,I A+∆AL�r�´5N%r5sS¸F·5�7T b LOYÂI�p7N,J#TK¯

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ñQò�ð�òÿîKð� �]�[ù��

îó Y � � � �6ò

ð#í³ñ÷2�>�ñcî%ò

� Y �ð�ñô þ

ò� ÿ óñô ö �

­ ¬§^*d5«U^*­ j�iOgw|i1 �_(^fd]aoèÐ^<«³ac×�«KØD¥7aKbxd3f ­ ¬�^*d/«U^<­ $ Æ YWYX�FR�I A LOYÀrFv%r/Y@L�rF�%�5P�N%V�N%r5s±'��z���z7yb 6= 0 ¯DG*�FHFH3v,YXI�TW´5NUT ∆A NQr5s δx NQVWI%¸KVXIKY@H3IcJMTXL�t%I�P�²%¸cTX´5I(H3I�VXTX�FVW·5NUTWLkv,r5YSv%¡ A NQr5s x Lkr[TW´FIP�LkrFIKNQVEYX²7Y�TWI�R Ax = b. âF�FVXTX´FIcVXR�v,VXI,¸FN%YWY@�FR�I�TX´/NUT ∆A L�YEYX�5J>´uTX´/NUT ‖∆A‖ <

1

‖A−1‖ ¯»w´FIcr‖δx‖‖x‖ ≤ Cond(A)

‖∆A‖‖A‖ /

(1− Cond(A)

‖∆A‖‖A‖

).

´Kµ,¶f¶R·f¸�� I[´5N³t%I(A + ∆A)(x + δx) = b,v%V

(A + ∆A)x + (A + ∆A)δx = b. Z¢Ô5¯�½%½³l ��z���z��G*L�r5JMI Ax = b, Á I[´5N³t,IС¢VWv%R Z �DÃ��DÃ��Ô5¯�½%½³l

(A + ∆A)δx = −∆Ax Z¢Ô/¯k½ º l ��z���z��v%V

δx = −A−1∆A(x + δx). ZxÔ5¯�½cä,l ��z���z��»§N6�*Lkr5��TW´FI[rFv,VXR v,r�·3vQTW´|Y@LOs7IcYc¸7Á I�´/N³t%I

‖δx‖ ≤ ‖A−1‖ ‖∆A‖ · (‖x‖+ ‖δx‖) Z¢Ô5¯�½�Ô*l ��z���z��=‖A−1‖ ‖A‖ ‖∆A‖

‖A‖ (‖x‖+ ‖δx‖)

TX´/NUTELOY�¸ (1− ‖A‖

−1 ‖A‖ ‖∆A‖‖A‖

)‖δx‖ ≤ ‖A‖ ‖A

−1‖ ‖∆A‖‖A‖ ‖x‖. ZxÔ5¯�½Kß,l ��z��z*{��

G*L�r5JMI ‖A−1‖ ‖∆A‖ < 1, TX´5I�IMp7HFVWIcYWY@L�v%r¼�Fr/s7I�V�H5NQVWI�r<TX´5IcYXL�Y[v%¡wTX´FI�P�IM¡ T�´5NQr/s Y@LOs7I�LOYH3v,YXL¶TWLkt,I%¯ � I�J�NQr:TX´*�5YÀs7L�t*L�s7I�·3vQTW´9YXL�sFIcY�vQ¡ÖTW´FI�Lkr5I_�<�5NQP�LkT�²|·<²�TW´FLOYÀr<�5R�·3I�VÀÁÂL¶TW´Fv%�FTJ>´5N%rF�%L�rF��TX´FI[L�rFI_�<�5N%PkLkT�²%¯ Æ ¡ TXI�VETX´FLOY�¸5L¶¡DÁ±I�N%P�YXv�s7Lkt*LOs7I[·*² ‖x‖ ¸7Á I�v,·7TWN%Lkr‖δx‖‖x‖ ≤

‖A‖‖A−1‖‖∆A‖‖A‖

(1− ‖A‖‖A−1‖‖∆A‖‖A‖ )

= Cond(A)‖∆A‖‖A‖

/(1− Cond(A)

‖∆A‖‖A‖ ) Z¢Ô5¯�½c¿<l ��z���zB{R{

Á´FLOJ>´�H5VXvUt,IcY±TX´FIÀTW´FI�v,VXIcR�¯� = � ',.@+�± � Ä IcJcNQ�5YXI�vQ¡ TX´FI�N%YWYX�FR�H7TXL�v%r£TX´5NQT ‖∆A‖ <

1

‖A−1‖ Z¢Á´5L�J>´¼LOY��,�5L¶TWIVWIcN%YXv%r/NQ·FP�IETWv�N%YWY@�FR�I³l#¸*TX´5IÀsFI�rFv,R�L�r5NQTXv%VÂv,rqTW´FI[VWLk�,´,Tw´5N%r5s�Y@LOs7IÀvQ¡�TX´FI[L�rFI_�<�5N%PkLkT�²�L�r»w´FIcv%VWI�R ú �DÃ��DÃ�ÄÔ/¯ º ä�L�YÐPkIKYXYETX´5N%r|v%rFI,¯ ¤ QfCf�:�$+R�$�[� A ‖∆A‖‖A‖ �%����3</(��� \�5�QF�.�P5YQ]�$ *�4#"&6CD��-WE0�</

-( "/R��5@�'#)#0Q]/(�f�2�:���I5�QF���$&(�>CD5@�'&6�ê� A Cond(A) �%����/( @���,`

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û �ì�íUî<ïcð�ñ%òé

ô �Fú/ù���û�÷7û úª¹�ø�õ�ùÂ÷,ý�ø��§û úeº��)��ùEü�ìøÀü�wû úÂûAø[ü±ûOüÖý

æ Ê¥F­ ©DÑx^�j_i gFj °±v%r/Y@LOs7I�V(Ù�pFNQR�HFP�I ² � Ã�Ä ³Ô5¯ º ½Âv,r5JMIÐR�v,VXI,¯�°±´/NQrF�,I a23 = 2.002 TWv º ¯ ¾%¾ º ½�a�%IcI�H b �5p*IKsS¯Ö»w´*�5Y

∆A = −10−4

0 0 00 0 10 0 0

ZeÉc­ª¥5ÑxÑel .

Û vUÁ�Y@v,Pkt,IÐTW´FI�YX²*Y@TXIcR�È (A + ∆A)x′ = b :

x′ =

3.0852−0.04371.0011

, δx = x′ − x =

2.0852−1.04370.0021

½ I�PONUTWLkt,IÀÙÖVXVWv%V =‖δx‖‖x‖ = 1.3463 Z'�Sקb¢ac^:Ñx¥7«U¦3^*lM¯

Û v%TXI�TX´/NUT Cond(A) = O(105) ¯�]�h§���Ý ¨:©Ó®�·�¥æ�³²TÏ]®³° ¥­¤á°Pª¦´¦¥#§à�e¨¦Õ §�¨´ªÉ�³¥­¬d¥­¬ª®¶¡ ´¦¥­°B§"°

A´a¨³Ð襭¬¦®çy®�·�¥¸�É°

bâ§Lkr/NQP�Pk²,¸_Á I�N%YWY@�FR�I�rFvUÁ TX´5NQT�·/v%TX´9TX´5I�Lkr5HF�7T�sFNQTWN A N%r5s b ´5N³t,I�L�R�HF�FVWL¶TWLkIKY�¯ Æ Y�NVWIcYX�FP¶TwÁ I�´/N³t%IETW´FI[Y@²7Y@TXIcR ÁÂL¶TW´ A + ∆A N%Y TW´FIÀR�NUTXVWLkpqNQr5s b + δb N%Y±TX´FI[VWL��%´<T ´/NQr5sYXL�s7IÀt,IcJ#TWv%VK¯­ ¬§^*d5«U^*­ j_i g/zi9Z7ùu^*f�^<«U¥FÑ�èÐ^<«³ac×�«KØD¥7aKbxd3f ­ ¬�^*d/«³^*­®l Æ YWY@�FR�I�TX´5NQT A LOYÐrFv,r5YXLkr7DZ'��z���z $ �%�5P�N%Vc¸ b 6= 0 ¸/NQr5s ‖∆A‖ <

1

‖A−1‖ ¯Ö»w´FI�r‖δx‖‖x‖ ≤

Cond(A)

1− Cond(A) · ‖∆A‖‖A‖

(‖∆A‖‖A‖ +

‖δb‖‖b‖

).

´Kµ,¶f¶R·f¸ G*�F·FTXV>N%J#TWLkr5�Ax = b¡¢VWv%R

(A + ∆A)(x + δx) = b + δbÁ I�´/N³t%I(A + ∆A)(x + δx)−Ax = δbv%V

(A + ∆A)(x + δx)− (A + ∆A)x + (A + ∆A)x−Ax = δbv%V(A + ∆A)(δx) −∆Ax = δbv%V

A(I −A−1(−∆A))δx = δb + ∆Ax. Z¢Ô5¯�½³Þ%l ��z���z*{�y

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îó Y � � � �6ò

ð#í³ñ÷2�>�ñcî%ò

� Y �ð�ñô þ

ò� ÿ óñô û

ê

�I�T A−1(−∆A) = F ¯Ö»w´FI�r‖F‖ = ‖A−1(−∆A)‖ ≤ ‖A−1‖ ‖∆A‖ < 1 Z¢·*²�N,YXYX�FR�H7TXL�v%r3l .

G*L�r5JMI ‖F‖ < 1 ¸ I−F LOYÖL�r*t%I�VXTXL�·FP�I�ZxYXI�IÌ»w´FI�v,VXIcR ú Å�Ã�FDÃ�Fº ¯ º Þ%lDNQr5s�TW´FI�r�¡¢VWv%R Z �DÃ��DÃUÅBÄÔ5¯�½³ÞQl§Á IE´5N³t,Iδx = (I − F )−1A−1(δb + ∆Ax).Æ �<NQL�r¸7�5YXLkr5��»w´5I�v%VWI�R ú Å�Ã�FDÃ�Fº ¯ º Þ*¸*Á I[J�NQr�ÁÂVWL¶TWI‖(I − F )−1‖ ≤ 1

1− ‖F‖ Z¢Ô5¯�½cá<l ��z���zB{ $»w´*�5Yc¸

‖δx‖ ≤ ‖A−1‖1− ‖F‖(‖δb‖+ ‖∆A‖ ‖x‖)

v%V‖δx‖‖x‖ ≤

‖A−1‖(1− ‖F‖) ·

(‖δb‖‖x‖ + ‖∆A‖

) Z¢Ô5¯�½cà<l ��z���zB{0þ≤ ‖A−1‖

(1− ‖F‖)

(‖δb‖ ‖A‖‖b‖ + ‖∆A‖

) Z Û vQTWIÀTX´5NQT 1

‖x‖ ≤‖A‖‖b‖ l#¯

»w´5NQTEL�Yc¸‖δx‖‖x‖ ≤

‖A−1‖ ‖A‖(1− ‖F‖)

(‖δb‖‖b‖ +

‖∆A‖‖A‖

). Z¢Ô5¯ º ¾<l ��z���zB{ò�

Æ �<NQL�r‖F‖ = ‖A−1(−∆A)‖ ≤ ‖A−1‖ ‖∆A‖ =

‖A−1‖ ‖A‖‖A‖ · ‖∆A‖. ZxÔ5¯ º ½³l ��z��z*{��

G*L�r5JMI ‖F‖ ≤ 1 ¸FÁ I[J�NQruÁÂVWL¶TWIÌ¡¢VWv%R Z �DÃ��DÃUÅ��Ô5¯ º ¾,l±NQr/s9Z �DÃ��DÃUÅ��Ô5¯ º ½³l‖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‖

).

Z¢Ô5¯ º%º l ��z���zB{��

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ûë ì�íUî<ïcð�ñ%òé

ô �Fú/ù���û�÷7û úª¹�ø�õ�ùÂ÷,ý�ø��§û úeº��)��ùEü�ìøÀü�wû úÂûAø[ü±ûOüÖý

� = � ',.@+�± � � IqNQ�<NQL�r9YXI�I�¡¢VWv%R Z �DÃ��DÃUÅ�FÔ/¯ º,º lÌTX´/NUT�Ict%I�r¼Lk¡±TW´FI�VWI�PONUTWLkt,I�H/IcV@TW�FVX·/NUTXL�v%r/Y‖δb‖‖b‖ NQr5s ‖∆A‖

‖A‖ NQVWI£YXR�NQP�Pe¸ÂTW´FI�VWI9R�L��%´<T�·/I¼Nãs7VWN,Y�TWL�J£J>´5NQr5�%IoL�rÚTW´FI Y@v,Pk�7TWLkv,r¸ELk¡Cond(A) L�Y�PONQVW�%I%¯¼»w´*�5Yc¸ Cond(A) HFP�N³²7Y�TX´FI�JMVW�5J�L�N%P(VWv%P�IqL�rªTX´5IuYXI�r5YXLkTXL�t<LkT�²¼vQ¡ÂTW´FIYXv%P��7TXL�v%r¯

¬|d/ac¥7aKbxd3fed]d/« ÷ d3f�Ó§b¢aKbxd3fª¬|ק­£Ø(^<«³ÉtÐrFPkIKYXYÖvQTX´5I�VWÁÂL�YXIÂY�T>NUTXIKsS¸<Á´FI�r�Á IÂÁÂVXLkTXI Cond(A) ¸*Á±IÂÁÂL�PkP/R�IcNQrq°±v,r5s 2(A) ¸TW´5NUTÖLOY�¸%TX´FIÌJMv%r/s7L¶TWLkv,r�r<�5R�·3I�VÖÁÂLkTX´�VWIcYXH/IKJ#T�TWv º ÇËrFv%VWR�¯�»w´FIÐJ�v%r5sFL¶TWLkv,r�r*�FR�Ç·3I�VÂv%¡DN�R�NUTWVXLkp A ÁÂLkTX´�VXIKY@H3IcJMTwTXv�N�Y@�5·/v,VWs7L�r5NQTXI p r5v%VWR (p = 1, 2,∞) ÁÂL�PkP·3I�s7I�rFv%TXIKs£·*² Condp(A) ¸TW´5NUT�LOY�¸ Cond1(A) ÁÂLkP�PÖY@TWNQr/so¡¢v,VÀTX´FI�J�v%r5s7LkTXL�v%rr*�FR�·/IcVÂvQ¡ A ÁÂLkTX´�VWIcYXH/IKJ#TÂTWvu½�Ç]r5v%VWRu¸*I�TWJQ¯

ü£{�( ���u��� �y,�u����%���]�Ì�Î�)åy�Ö�����¹�u�:���x���]�u� ����������¹�)åÚ�� �[��5�-,»w´FI�¡¢v%P�PkvUÁÂL�rF��NQVWI�YXv%R�I�LkR�H3v%VXTWN%r,T�Zx·F�7T[IcN,Y@²�TWvuH5VXvUt,IKlÂH5VXv,H/IcV@TWLkIKYÌv%¡(TW´FI�JMv%r/s7L¶TWLkv,rr*�FR�·3I�VÂv%¡§N�R�NUTWVXLkpS¯ZxÃ@l Cond(A) ≥ 1 ¯Z¢ÃXÃ@l Cond2(A) = 1 Lk¡±NQr/s|v%rFP�²�L¶¡ A LOYÀN�rFv,rFÍ�IcVXvqYWJ�N%P�N%VÐR��FP¶TWLkH5PkI�vQ¡±N%rov%VXTX´5v%�%v,r5NQPR�NQTXVWL¶pS¸7L]¯ I,¯k¸ AT A = αI ¸FÁ´FIcVXI α 6= 0 ¯ M &(5H��5YQ]/65T5YQf�%���! *&0�]�$ �5U8I&�A�/6�ª&( �5YQ]&"��&(�!/6��3</(5U ��>1 A 34/,¥��.��5YQ]�<3</(5U ��>1ê�$&ê/(5U?

5U */2#.5U��+R��A.&( L��5'��Cf�$�K���I�FCD3<�. ��'#0/(�J#"&63r�CD57/65@�'&6�]�$` $ZxÃ@ÃXÃ@l Cond(αA) = Cond(A) ¸FÁ´FI�VWI α LOYEN�rFv%rFÍcI�VWv�YWJ�N%P�N%Vc¯Z¢Ã��Àl Cond2(A) =

σ µC¶^·σ µC¸ ¹ ¸%Á´FI�VWI σ µC¶^·�NQr/s σ µC¸ ¹ÀNQVWI±TX´FIwPONQVW�%IcY@T�NQr5s�Y@R�N%PkP�IcY@TÖY@L�rF�%�5P�N%VtUN%Pk�FIKYwvQ¡ A ¯

Z��Àl Cond2(AT A) = (Cond2(A))2 ¯

Z��ÌÃ@l Cond2(A) = Cond2(AT ); Cond1(A) = Cond∞(AT ) ¯

Z��ÌÃ@Ã@l Cond(AB) ≤ Cond(A) Cond(B) ¸/L¶¡ A N%r5s B NQVWI�JMv,R�H/NUTXL�·FP�I[¡¢v%VÐR�NQTXVWL¶puR��FPkÇTWLkH5PkLOJ�NQTXL�v%r¯� I[r5vUÁÚ¡¢v%VWR�NQP�Pk²�s7IM�5r5IÀTX´FI�L�P�P¶Ç1JMv,r5s7LkTXL�v%rFL�rF��NQr5s�Á I�P�P¶Ç1JMv,r5s7LkTXL�v%rFL�rF��LkruTWI�VWR�YEvQ¡TX´5I�JMv%r/s7L¶TWLkv,r�r*�FR�·/IcVc¯

� ^2�Df§b¢acbed/fhj_i gF¯_i ¤ Q]�)��8R��5H�.3 Ax = b �%�L�����>?B#"&6�!-6��5@�'&6�!�0-W� A Cond(A) �%�4^$CD��57�:��/6 @�2�,` ¸�2¡$y l 5�QF�. �VX�%�$�.\���5��%��V��$���9?B#0&(�-(��5U�'&(�!�0-2`�¾÷ d3fb/^*fmacbed/f��WtErFP�IcYWYSvQTW´FI�VWÁÂLOY@I�Y@TWNUTWIcsS¸K·*² Cond(A) Á±I�ÁÂL�P�PQR�IcNQr Cond2(A) ¯

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ñþò�ïQñQò�ð

�ñ� ��ò

ð#í³ñ�ì�F�R�D�

ð� �]��ü�� ôLÿ

ñQò��òî�îKð#ò� º û

ó� = � ',.@+�± � »w´Fv,�F�%´�TW´FI(JMv,r5s7LkTXL�v%rÀr*�FR�·3I�VK¸³N%YSs7I��5rFIcs[N%·/vUt,I%¸cL�YSr5v%VWR�Ç1s7IcH/Icr5s7I�r<TK¸TX´5I�J�v%r5sFL¶TWLkv,r:r*�FR�·3I�V>YÌÁÂL¶TW´£VWIcYXH/IKJ#T�TXvuT�Á v�s7Lkµ3IcVXIcr<T�rFv,VXR�YÀNQVWI�VWI�PONUTWIcs ZxYXI�Iê��v%P��F·NQr/sÔ�ÖNQr;�v<NQr Z�½Kà%à,¿,l#¸5H¯ º ¿,lM¯�ZeâFv%VÌIMpFNQR�HFP�I%¸/L¶T�J�NQr|·3I�YX´FvUÁÂruTX´/NUTÐLk¡ A LOYÐNQr n× nR�NUTWVXLkpS¸/TW´FI�r 1

n≤ Cond2(A)°±v,r5s ∞(A)

≤ n ¯Al:Ã1r9�%IcrFI�V>NQP]¸J� A4/u3</(5U ��>1}�%�4V��$���9?B#0&(�!-6��5@�'&6�!�0-ê&6 �����9?B#0&(�-(��5U�'&(��"-LVX��5YQ) *�.�@�F�0#.5�57&L&(�!�r�!&6 �3L\=��5��%��/6� �$&������9?B#0&(�-(��5U�'&(��"-)&( �V��$���9?B#0&(�-(��5U�'&(�!�0-VX��5YQW *�.�U�]�0#$5�5H&)�$&(3<�4&(5YQ]�$ ��!&( �3:�$`æ Ê¥F­ ©DÑx^�j_i gw¼

ZxN<l�°±v,r5Y@LOs7IcVA =

(1 0.9999

0.9999 1

); TX´FIcr A−1 = 103

(5.0003 −4.99997−4.9997 5.0003

).

½,¯E»w´FI[JMv%r/s7L¶TWLkv,r�r*�FR�·/IcVWYwÁÂLkTX´|VWIcYXH3IcJ#TwTWv�TW´FI[L�r7�5rFLkT�²qrFv,VXR NQr5s£½MÇËrFv%VWRÝN%VXI‖A‖∞ = ‖A‖1 = 1.9999; ‖A−1‖∞ = ‖A−1‖1 = 104

°±v%r5s ∞(A) = °±v%r5s 1(A) = 1.9999× 104º ¯E»w´FI[JMv%r/s7L¶TWLkv,r�r*�FR�·/IcVEÁÂLkTX´�VXIKY@H3IcJMTwTXv�TX´5I º Ç]r5v%VWR L�Y‖A‖2 =

√ρ(A) = 1.9999, ‖A−1‖2 =

√ρ(A−1) = 104

°±v,r5s 2(A) = 1.9999× 104.

� = � ',.@+ � â5v%V�TW´FI�N%·/vUt,I�IMpFNQR�HFP�I%¸DLkT�TX�5VXrFIKs¼v%�7T�TX´5NQT�TX´5IqJMv,r5s7LkTXL�v%r r*�FR�·3I�VÁÂLkTX´ VWIcYXH/IKJ#TÀTXvoN%r<²£rFv%VWR LOYÀTX´5I�YXN%R�I,¯q»w´FLOY[LOY�¸§´FvUÁ I�t%IcVc¸Sr5vQT�NQP�ÁwN³²*YÌTW´FI�J�N,Y@I,¯�Ã1r�%IcrFI�V>NQP]¸<´FvUÁ±Ict%IcVc¸,TX´FIc²qNQVWIÌJ�Pkv<Y@IcPk²�VWI�PONUTWIcsS¯wZ�G*I�IÀ·3I�P�vUÁyTX´5IÀJ�v%r5sFL¶TWLkv,rqr*�FR�·3I�Vwv%¡_TX´5I�EL�P�·/IcV@TER�NUTWVXLkpqÁÂLkTX´�VWIcYXH/IKJ#TwTXv�s7LkµmI�VWI�r<TErFv%VWR�Y�¯Al�]�h»���� Þe�q¡ ®½¼ ®³ß�ß?­�¾ª¨ª�yÒn¨ÎÍ�ß�ß?­ ·��q¨ªÐe§g¥#§ã�q¨ª®�ÐÀ¿«´¦¥­°B§�·�®tÕ½,¯ ­ ¬�^½N|bxѢر^,«³aI� ¥7a�«UbxÊ

A =

1 12

13 · · · 1

n

12

13

14 · · · 1

n+1¯¯¯ ¯¯¯1n

1n+1 · · · · · · 1

2n−1

âFv%V n = 10, Cond2(A) = 1.6025×1013; Cond∞(A) = 3.5353×1013; Cond1(A) =3.5353× 1013.º ¯ ­ ¬�^ èÐ^*b7� ¥7ac«Ub¢Ê A = (aij) ÁÂL¶TW´ aii Ø α, aij = 1 ¡¢v%V i 6= j ¯�»w´FI�R�NUTWVXLkp·/IKJMv%R�IKY�L�PkPkÇ1JMv%r/s7L¶TWLkv,rFIcsªÁ´FI�r α LOY�JMP�v,YXI�TWv®½uv%V n − 1 ¯ âFv,V�I�pFNQR�HFP�I%¸�Á´FIcrα = 0.9999 N%r5s n = 5 ¸ Cond(A) = 5× 104 ¯

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ô �Fú/ù���û�÷7û úª¹�ø�õ�ùÂ÷,ý�ø��§û úeº��)��ùEü�ìøÀü�wû úÂûAø[ü±ûOüÖý

ä5¯©¦[¥Ff§Ó§^,«U­ªd3f§Ó�^b� ¥7a�«Ub¢Ê A = (aij) ¸*Á´FIcVXI aij = vn−ji ; vi = i Ç]TX´qJ�v%R�H/v,rFI�r<T(v%¡N%r n ÇËt%IcJMTXv,V v ¯�âFv%V n = 5 ¸ v = (1, 2, 3, 4, 5)T , Cond(A) = 2.6170× 104 ¯�»w´FL�YR�NQTXVWL¶poNQVWLOY@IKYEL�r£YXI�t%IcVWN%P_HFV>N%JMTXLOJ�N%P�NQH5HFPkLOJ�NQTXL�v%r5Yc¸mLkr/JMP��5s7L�rF��H3v%P�²*rFv%R�LONQP�Lkr<TXIcVXH3vQÇPONUTWLkv,r¯

�]�h»���Ì ®/�yÒ éÁ´³°Ö©É®Á¿¤¦Õ�¥½¥­¬ª®«� �q¨³Ðq§à¥#§à�q¨*¯ ¤a¡ÂªÉ®³°Ãªa®ô²��a°Í�ß�ß?­��I�q¨³Ðq§à¥#§à�q¨a§�¨ª©`ÄÆ ¡¢VXI,�,�5I�r<TXP�²ªN%Y*�%IKss�<�FIcY@TXL�v%rªLOYcÈq´FvUÁ�PONQVW�%I Cond(A) ´5N%Y�TXv9·/I�·3IM¡¢v,VXI�TX´FI�YX²*Y@TXIcRAx = b L�Y±JMv,r5Y@LOs7IcVXIKs�TWv�·3IÐL�PkPkÇËJ�v%r5sFL¶TWLkv,rFIcsXQ � IÐÁÂL�PkP/�5YXIлw´FI�v,VXIcR ú �DÃ��DÃ�ÂÔ5¯ º ßÌTXv�NQr5YXÁ±IcVDTX´FI�<�FIcY@TXL�v%r_¯G*�FHFH3v,YXI�¡¢v%VEYXLkR�HFP�LOJMLkT�²

‖∆A‖‖A‖ =

‖δb‖‖b‖ = 10−d.

»w´FIcr¸7¡¢VWv%RÝ»w´FIcv%VWI�R ú �DÃ��DÃ�ÂÔ/¯ º ß7¸*LkT¡¢v%P�PkvUÁEY TW´5NUT ‖δx‖‖x‖ L�YK/0��� *&$12��3</(5H�.�98uP�IcYWY TX´5N%r�v,VÂI_�<�5NQPTXv 2× Cond(A)× 10−d ¯»w´FLOY YWN³²7YÖTW´5NUTÂLk¡_TW´FI[sFNUT>N�´5N,YwN�VWI�PONUTWLkt,IÌIcVXVWv%V±vQ¡ 10−d N%r5s�Lk¡_TW´FIÀVXIcP�NQTXL�t%I�I�VWVXv,VL�r�TX´FI�Y@v,Pk�7TWLkv,r�´5N%YÖTXv�·3IÐ�,�5NQV>NQr<TWI�Ics�TXv�·/IÐPkIKYXYÖTX´5N%r�v%VÖI,�<�5NQP5TXv 10−t, TX´FIcr Cond(A)´5N,Y�TXv�·/I�P�IcYWYÖTW´5NQr�v,V±I,�<�5NQP5TXv 1

2× 10d−t ¯�»w´*�5Y�¸V�QF�.5YQ]�$ �/:��8R��5H�.3Ð�%�������9?B#0&(�!-6��5@�'&6�!�0-

&( �V��$���9?B#0&(�-(��5U�'&(�!�0-4-��H�F�.�-R�T&(� 7� $ 5YQ]�T/2#0#.CD */2#.8L&BA�5YQ]�T-2/(5H/L/6�!-R\á 7��� $ QF&(Vß3)CF#"Q��. � *&6 ���;5YQ]�K�$&(�9CD5U�'&(�[#"/6�PE0�L57&6���. "/(5H�"-2`âFv,V�I�pFNQR�HFP�I%¸YX�FHFH3v,YXI�TW´5NUT�TX´5I�sFNUT>Nq´5N³t,I�NuVWI�PONUTXL�t%I�I�VWVXv,VÌv%¡±NQ·3v%�FT 10−5 N%r5sNQr�N%JcJM�FV>N%J�²�vQ¡DNQ·3v%�7T 10−3 L�YEYXv%�5�%´<Tc¸*TW´FI�r Cond(A) ≤ 1

2× 102 = 50 ¯±ÕÌruTW´FI[vQTW´FI�V

´5N%r5sS¸�Lk¡ÂTX´FI�N%JcJM�FV>N%J�²:vQ¡ÐN%·/v,�7T 10−2 LOY�YXv%�F�,´<Tc¸§TX´FIcr Cond(A) ≤ 1

2× 103 = 500 ¯»w´*�5Yc¸%L�r�TX´FIÐ�5VWY@T±JcN%YXIwTX´FIÐYX²7Y�TWI�R ÁÂLkP�P5·3IÐÁ I�P�PkÇËJ�v%r5s7LkTXL�v%r5Ics�L¶¡ Cond(A) L�Y(PkIKYXY�TX´5N%r�v%VI_�<�5N%P<TXv�ßQ¾5¸UÁ´FLkP�IwLkr�TX´5IwYXIcJ�v%r5s�J�N,Y@I,¸UTX´FIÂYX²7Y�TWI�R ÁÂL�PkPF·3IwÁ±IcPkPkÇ1JMv%r/s7L¶TWLkv,rFIcs�L¶¡ Cond(A)LOYwPkIKYXY TW´5NQr�v%VÂI,�<�5NQPmTWv�ß%¾%¾5¯

æ Écacbx­®¥7acbef�¦ � `,`%×�«U¥F`6c�d]«Ud/­ ac¬§^ ÷ d/f§Ó�bxacbed/fª¬|×�­£Ø(^<«Ã1r|�,I�rFIcVWN%Pe¸FLk¡DTX´FI�sFNUT>N�NQVWI�NQHFH5VXv³p7L�R�NQTXIcPk²qN,J�J��FVWNQTXI�N%r5s�L¶¡ Cond(A) = 10s ¸TW´FI�r|TW´FI�VWI�ÁÂL�P�P�·3I�v,rFPk²|N%·/v,�7T t − s Y@L��%rFLk�/JcNQr<T�s7L��%LkT�N,J�J��FVWN,JM²�Lkr|TW´FI�J�v%R�ÇHF�FTXIcs�YXv%P��7TXL�v%ruÁ´FIcr�TW´FI�YXv%P��7TXL�v%ruL�YÂJ�v%R�HF�7TWIcsuL�r t Ç1s7L��%LkTENQVWL¶TW´FR�IMTXLOJQ¯âFv,V�·/I�T@TWI�Vq�5r5s7I�V>Y@TWNQr/s7Lkr5� vQ¡�JMv,r5s7LkTXL�v%rFL�rF�5¸ÂY@TWN%·FLkP�LkT�²ãN%r5s¹N%J�J��FV>N%JM²,¸±Á I:N%�,NQL�rVWIM¡¢I�V TW´FI[VXIKN%s7IcVWY±TWv�TX´FI[H5N%H/IcVÂvQ¡Dı�5r5J>´ Z�½cà,á<Þ%l#¯

�]�h»���Ý Ñn¬¦®«�I�q¨³Ðe§g¥#§à�e¨ ¯ ¤á¡½ªa®³°b´á¨ªÐÛ¯�®t´³°W¨¦®tÕVÕ楸� Þ §�¨¦©q¤áßg´Á°B§à¥[ä»w´FI�JMv%r/s7L¶TWLkv,r�r*�FR�·3I�VwNQPOYXv��%L�t%IKY(NQrqLkr5sFL�JcNUTXL�v%r�Á´FI�r�N�R�NUTWVXLkp A LOY J�v%R�HF�7T>NUTWLkv,r5NQP�Pk²JMP�v,YXIÌTXv�N�Y@L�rF�,�FP�N%VwR�NQTXVWL¶pSÈ�� A Cond(A) �%����/( @���.\ A �%�L#$��&R�$�:57&<��������CD��/( $`

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ñþò�ïQñQò�ð

�ñ� ��ò

ð#í³ñ�ì�F�R�D�

ð� �]��ü�� ôLÿ

ñQò��òî�îKð#ò� º û »

¤ Qf�%�:3<�0/R��CD *�<&BAK�!�0/6 ��!�.�"�:57&���������CD��/( ���5U8��%�4/k3<&( "�</�#0#.CD "/(5H�)34�0/R��CD *�:5YQ]/6�P5YQ]�-��$5H�. �3)���!/6�F5K&�A A `:â5v%V�I�pFNQR�HFP�I%¸�JMv,r5Y@LOs7IcV[TX´FI�Á±IcPkPkÇ7�<r5vUÁÂr n × n �FHFH3I�V�TXVWL�N%rF�%�5P�N%VR�NUTWVXLkp A = (aij) ÁÂL¶TW´ aii = 1 ¸ÌN%r5s aij = −1 Lk¡ j > i ¯�»w´FI£R�NQTXVWL¶pÚ´/N%YqTX´FIs7I�TXI�VWR�Lkr/NQr<TÂI_�<�5NQPmTWv|½%¸7´FvUÁ I�t,I�VK¸,LkTELOYÂrFIcN%VXP�²�YXLkrF�,�FPONQVw¡¢v%VÂPONQVW�%I n ¯ Û v%TXI�TX´/NUT

°±v,r5s ∞(A) = n2n−1.

G*L�R�LkPONQVWPk²,¸X5�QF�4��3</(���9����0��&�A)5�QF�k-2�.5H�. �3)���!/6�F5G&BA</W34/65@ ��>1I-�&,���<�&(5r��"#0�.�0�./6 ����98W34�0/6�5YQ]/(5 A �%��#$��&R�$�©5H&�/r��������CD��/( �34/65@ ��>1f`§âFv,VIMpFN%R�H5PkI,¸cJMv,r5YXL�s7IcV A = diag(0.1, 0.1, . . . , 0.1)vQ¡/v%V>s7I�V ½c¾,¾%¾F¯ det(A) = 10−1000 ¸%Á´FL�J>´�L�YDN�Y@R�N%PkP*r*�FR�·3I�VK¯ �EvUÁ I�t%IcVc¸ A L�YDJ�v%r5YXL�sFI�VWIcsTXv�·3I[H/IcV@¡¢IKJ#TWPk²�rFv%r5YXL�rF�%�FPONQVK¸*·/IKJ�N%�5Y@I�°±v,r5s 2(A) = 1 ¯�]�h»��ã� ¨:°�´á¡£¢aßà®tÕ��³²½Í�ß�ß?­'·��e¨ªÐe§g¥#§ã�q¨¦®yÐŨ7§à©Á®³¨¦ç�´áß-¤ª®èÏê°Ö�Xªáßg®Á¡ Õ�DI�VXTX�FVW·5NQTXL�v%r�NQr5N%Pk²7YXL�Y§vQ¡/TX´FIÂIcLk�,I�r*tUNQP��FIwHFVWv%·FP�I�R ÁÂL�P�PF·/IÂs7LOYWJM�5YWY@IKs�L�rq°±´5NQHFTXI�V ��� �à5¯§»w´5IJMv,r5s7LkTXL�v%rFL�rF��v%¡�TW´FI�I�L��%Icr*t³N%Pk�5IcYEN%r5suI�L��%I�r*t,IcJ#TWv%V>Y ÁÂLkP�P_·3I�L�r,TWVXv7s7�/JMIcs�TW´FI�VWI%¯©�EI�VWI[Á I���5Y@TEHFVWIcYXI�r<TEN�¡¢IcÁöI�pFNQR�HFP�IcYwvQ¡�TX´FI[Á I�P�PkÇU�*rFvUÁÂruLkP�PkÇËJ�v%r5s7LkTXL�v%r5Ics�I�L��%I�r*tUNQP��FI�HFVWv%·FP�I�R�Yc¯æ Ê¥F­ ©DÑx^�j_i gF¿ °±v%r5YXL�sFI�VwTX´5I 10× 10 R�NUTWVXLkpSÈxRþ=z���z7y

A =

1 1

1 1 Ư ¯ ¯ ¯ ¯ ¯Æ ¯ ¯ ¯ 1

1

.

»w´FI IcLk�,I�r*tUNQP��FIcYuvQ¡ A NQVWI NQP�P 1 ¯ Û vUÁÅH/IcV@TW�FVW·öTW´FIÚZ�½K¾F¸�½KluJMv*IBñ�J�LkIcr<T�v%¡ A ·*²hNYXR�N%PkP��<�5NQr<TXLkT�² ε = 10−10 ¯ »w´FI�r¹TW´FI:IcLk�,I�r*tUNQP��FIcY�v%¡ÀTX´FI£H/IcV@TW�FVX·3IcsÚR�NQTXVWL¶pyJMv,R�ÇHF�7TWIcs��5Y@L�rF�ÐTX´FI n Æ »�� Æ Ä9¡¢�Fr5J#TWLkv,r�^*bx¦�Z TW´5NUTD�5YXIcY�NEr*�FR�I�VWL�JcNQP�Pk²�IMµmIcJMTXL�t%I±I�L��%I�r*tUNQP��FI�ÇJMv,R�HF�7TWNQTXL�v%r�NQP��%v,VXLkTX´FR�l±Á I�VWIС¢v,�Fr5s�TXv�·3I%È0

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

Z'¬|d/ac^�aK¬�^:`K¬§¥Ff§¦/^£bxfãac¬§^o^*bx¦3^*fbF¥FÑxק^*É%iOlæ Ê¥F­ ©DÑx^�j_i gFè ­ ¬�^;ø beÑ�&bef�ÉKd/f_¨Ö.�bxÓ§bx¥5¦/d3f�¥5ѳ� ¥7a�«UbxÊ

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Æ �<NQL�r¸cLkT_YX´Fv%�FPOs�rFvQT�·/I�TW´Fv%�5�%´<TSTX´5NQT_NQrÀI�L��%Icr*t³N%Pk�5I�HFVXv,·FP�I�RhJcNQrÀ·/I(LkP�P¶Ç1JMv,r5s7LkTXL�v%rFIKsv%r5Pk²oÁ´5I�r¼TX´5I�I�L��%I�r*tUNQP��FIKY[NQVWI�R��FPkTXL�HFPkIqv%V�NQVWI�JMP�v,YXI�TXv|IKN%J>´9vQTX´5I�VK¯ Æ r IcLk�,I�r*tUNQP��FIHFVWv%·FP�I�R ÁÂLkTX´:Á I�P�PkÇËYXI�H5N%VWNQTXIKs�IcLk�,I�r*tUNQP��FIcYÌJ�N%r|·/I�t%IcVX²uLkP�PkÇËJ�v%r5s7LkTXL�v%r5Ics�TWv<v/¯À°±v,r5Y@LOs7IcVTX´5I 20× 20 TWVXLONQrF�,�FPONQVwR�NUTXVWLkp£ZY�*rFvUÁÂr�N%Y TW´FI � LkPo�*Lkr5YXv%rFÇ]·FLOs7LONQ�,v%r5N%P3R�NQTXVWL¶p5l#È

A =

20 20

19 20 Ư ¯ ¯ ¯ ¯ ¯Æ ¯ ¯ ¯ 20

1

.

»w´FI�IcLk�,I�r*tUNQP��FIcY�vQ¡ A N%VXI 1, 2, . . . , 20 ¯ Û vUÁ H3I�VXTX�5VX·¼TW´FI£Z º ¾F¸�½Kl�Icr<TXVW²£v%¡ A ·*² ε =10−10) ¯öÃË¡ÌTW´FIoI�L��%Icr*t³N%Pk�5IcY�vQ¡�TX´5L�Y�YXP�Lk�,´,TWPk² H/IcV@TW�FVW·/IKs R�NQTXVWL¶p¹NQVWI�J�v%R�HF�7TWIcsã�5YXLkr5�n Æ »�� Æ Ä�¡¢�Fr/J#TXL�v%r¼^*bx¦�¸SL¶T[ÁÂL�PkP�·3I�YXI�I�r:TX´5NQT�Y@v,R�I�vQ¡±TX´FIcR ÁÂL�PkPÖJ>´5N%rF�%I�s7V>N%Y@TXLOJ�N%PkP�²aTX´5I�²�ÁÂL�P�P5I�t,I�r�·3IcJMv,R�IEJMv%R�HFP�IMpS¸<N%Y�YX´FvUÁÂr�Lkr�TW´FIw¡¢v%P�PkvUÁÂL�rF�[�,VWN%HF´¯ Æ �,N%Lkr¸%TX´FLOYÖJcNQr�·/IIMp7HFPONQL�rFIKs:·*²o�5YXL�rF�uTX´FI�s7I��5rFLkTXL�v%r¼vQ¡ÂJMv,r5s7LkTXL�v%r9r*�FR�·3I�V[v%¡wN%r£L�r5s7L�t*L�s7�/NQPÖIcLk�,I�r*tUNQP��FI�%L�t%IcrqL�r:°±´5NQH7TWI�V ��� �à5¯

0 5 10 15 20 25−6

−4

−2

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6

Re

Im

perturbedoriginal

ç bx¦3×�«³^:j_i%|i ­ ¬�^ æ be¦/^*fbF¥FÑe×�^*É�dwdÐ¥¼\_Ñxbe¦/¬maKÑYc®èÌ^,«³aK×_«³Ø(^*Óeø bxÑ-&bxf§Écd3fì� ¥7a�«Ub¢Ê�i

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A =

n (n− 1) (n− 2) · · · 3 2 1(n− 1) (n− 1) (n− 2) · · · 3 2 1

0 (n− 2) (n− 2)¯ ¯ ¯ ¯¯¯ ¯¯¯¯¯¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯¯¯ ¯¯¯¯¯¯ ¯ ¯ ¯ ¯ ¯ ¯ 2

¯¯¯¯¯¯ 2 2 10 · · · · · · · · · 0 1 1

O�� n ���#. *�0/(�.�.�0\r5�QF�)��34/6�����.�05��.�>���$�F+(/(�9CF�.�<E0�0#0&(3<��� *&*�� *�.�0�0��+(�$�98W�����9?B#0&(�!-6��5@�'&6�!�0-�`âFv,VÌI�pFNQR�HFP�I%¸mÁ´FI�r n = 12 ¸/TW´FI�JMv%r/s7L¶TWLkv,r:r*�FR�·/IcVWYÌvQ¡ÖTW´FI��/VWY@TÌ¡¢IcÁ I�L��%Icr*t³N%Pk�5IcYÐN%VXIvQ¡Dv%V>s7IcV �5rFL¶T�²qÁ´FLkP�IÀTX´5v,YXI�v%¡�TW´FI[PON%Y@T TW´FVWI�I[NQVWIÀvQ¡Dv%V>s7I�V 107 ¯ü£{�G �u�����ÉÈ ���U�����@�W���Ì�Ûå,�� K �Ì�±�e�����W��� ��������]�

æ��e����/�x�Ö�����âFv,PkP�vUÁÂL�rF�<�EL��%´5N%RÅZ�½Kà%à,¿F¸7HFH¯ ä%¾QÇËä5½Kl#¸FN%r5su·5N%YXIcs�v%r�v,�FVÐs7LOYXJ��5YWY@L�v%r5YwL�ruTX´FLOYÐJ>´/NQH7TWI�VK¸Á IÌY@TWNUTWI�N�¡¢I�ÁÚ´5I�P�H7¡¢�FPS�%�5L�s7IcPkL�rFIKYÖ¡¢v,V s7IKY@L��%r5LkrF��N�Y@TWN%·FP�IÌNQP��%v,VXLkTX´5R�YcÈ �ÐvUÁ±Ict%I�V(rFvQTWIj ¤ Q]�$ *�L�%���&4����3T�!���: *�0#$� �]��A.&( :-2�����>���F���f����5H/�E$���)/(� ��&( ���5YQf3:��k�Z'�EL��%´/NQRÅZ@½cà,à%¿,lM¸*H¯Öä%¾<l• Æ t,v%LOsqJ�NQTWN,Y�TWVXv,HF´FLOJÌJcNQr5J�I�P�P�NQTXL�v%r5Yc¸7L¶¡DH3v,YWY@L�·FP�I%¯• Æ t,v%LOs��FrFr5IcJMIKYXYWNQVW²�vUt%IcV@Ï/vUÁÚNQr5su�Fr5sFI�VXÏ5vUÁ[¯• Ã1r|TXV>NQr5Y@¡¢v%VWR�Lkr5��TX´FI�HFVWv%·FP�I�R TWvuN%rFvQTW´FI�VÌR�NQTX´FIcR�NUTXLOJ�N%PkP�²�I_�<�FL�tUNQP�I�r<T�H5VXv,·FPkIcR�¸�5Y@IÐv%rFP�²�Á I�P�PkÇËJ�v%r5s7LkTXL�v%r5Ics�TXV>NQr/Y�¡¢v,VXR�NUTWLkv,r¸%YX�5J>´qN,Y�v%VXTX´Fv,�%v,r5NQPFR�NQTXVWL¶p�R��FPkTXL�HFP�L¶ÇJ�NQ·5LkP�L¶T�²,¯• ÃË¡�N�r<�5R�IcVXLOJ�N%P3YWJ>´FIcR�IÀNQH5H/IKNQV>YÖTWv�·3IÀ�Fr5Y@TWN%·FPkI,¸*Pkv*v���¡¢v,VÂs7L¶µmI�VWI�r<T ¡¢v%VWR��FPONUTWLkv,r5Yc¸Á´FL�J>´�NQVWI(R�NUTW´FI�R�NQTXLOJ�NQP�P�²�I,�<�FLktUN%PkIcr,TK¸U·F�7T�rFv%T§r*�FR�I�VWLOJ�NQP�P�²�ZeY@IcI(TW´FI �5Y@I vQ¡/R�v7s*ÇL¶�5IKsw��V>NQR G7J>´5R�LOs*T�HFVWv7JMIcYWYXIcY�t%I�V>YX�5Y_J�P�N,YXYXL�JcNQP­��VWN%R G7J>´FR�LOs*T�HFVXv7J�IcYWY@IKY_L�r�YXv%P�t<ÇLkrF��P�IcN,Y�TXÇËY"�<�5NQVWIcY±HFVWv%·FP�I�R�YwL�ro°±´/NQH7TWI�V ��� �á,lM¯• Æ VWVWN%rF�%I�²%v%�5VÀJ�v%R�HF�7T>NUTWLkv,r5NQP�YWJ>´FI�R�I|Z¢Lk¡ H3v,YWYXLk·FP�IKlÌL�r Y@�5J>´9NuÁ N³²�TW´5NUT[TW´FI�Lkr7ÇTXI�VWR�Ics7LONUTWI��,�/NQr<TXLkTXL�IcYÂN%VXIÀR��/J>´uYXR�NQP�PkIcV±TW´5NQruTW´FIÀ�5r5N%P_NQr5YXÁ I�VK¯• tEHmsFNUTWI[TX´FI�Y@v,Pk�7TWLkv,r|v%rFP�²q·*²u�5YXLkrF��NqY@R�NQP�P_J�v%VWVXIKJ#TXL�v%rqa7TX´5NQTÐLOY�¸/�FH3s5NUTXI�N%YÂr5I�ÁY@v,Pk�7TWLkv,r½Ø v,P�s|YXv%P��7TWLkv,r Ü YXR�NQP�P�JMv,VXVWIcJMTXL�v%r¸5L¶¡DTX´5Iu#0&( � *�0#$5@�'&6�[#0/6�ßE0�4#0&(3T�!CD5H�0-VX��5�QW�0Cò�e#$�'�.�w5w¨���CD *�.�

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Cond(A) = ‖A‖ ‖A−1‖

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§������.��. */6� \X� AT5YQ]�L-�/(5H/</( "�K/"�2�! *&_1���3</(5H�$�98k/�#0#$CD */(5H��/(�-)� A Cond(A) = 10s \X5YQ]�$�5YQ]�$ *��/( *��/2E"&6CD5 (t− s) ���>�6�w� ¨�#0/(�w5�-(�>����5'��&�A�/�#0#.CD "/�#$8L���W5�QF�G�$&(�9CD5U�'&(�F\�� AT��5X�%�G#0&63r�CD57�0-��� t ?B-6�>�6��5r/( ���5YQf3<�$5@�'#,`�]�hÊ���Ì Þq¥¸´�ªÉ§�ß�§g¥Wä �Á²�´á¨Í̽ßà©Á�a°B§à¥­¬a¡Æ r�N%Pk�,v%VWL¶TW´FR LOY(YWNQLOs�TXv�·/IÐN�ا¥F`W&�g�¥F«³ÓãÉcac¥7ØDÑx^�¥FÑx¦3d5«Ubxac¬§­ Lk¡SL¶TwJMv%R�HF�FTXIcYÖTX´FIÐIMpFN%JMTYXv%P��7TXL�v%rov%¡(N�rFIKNQVW·*²uH5VXv,·FPkIcR�¯ÀG*v,R�I�I�p7N%R�HFPkIKYEv%¡±Y�T>NQ·FP�I�N%Pk�,v%VWL¶TW´FR�YÐN%VXIuZxN%YÌÁ I�ÁÂL�PkPYXI�IÀP�NQTXIcVÂLkruTX´5I�·3v*v��FlwNQVWI%È

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uvT

uT v¸ÂÁ´FIcVXI u N%r5s v N%VXI m J�v%P��FR�rt%IKJ#TWv%V>Y�¯Zxt*LkL l|°±v%R�HF�7T>NUTXL�v%rqvQ¡TW´FIÌR�NQTXVWL¶p B = A−uvT ¸*Á´FIcVXI A N%r5s B NQVWIÂT�Á v n×nR�NUTWVXLOJMIKYÂNQr5s u NQr5s v N%VXI�T�Á v�J�v%P��FR�r�t,IcJMTXv%V>Y�¯

j_i zS�ÌI�t,I�P�v%H�N%r�NQP��%v,VXLkTX´5RhTWvÌJMv,R�HF�7TXI(TX´FI HFVWv7s7�5J#T C = AB Lkr�IKN%J>´�v%¡7TX´FI±¡¢v%P�P�vUÁÂLkrF�JcN%YXIcYc¯OlÖv,�FV�NQP��%v,VXLkTX´5R Y@´Fv,�FPOsoT>N6�%I�N,s7tUNQr<TWN%�%I�v%¡±TW´FI�Y@H3IcJ�L�N%PÖY�TWVX�/J#TX�5VXI�vQ¡±TX´FIR�NQTXVWL�J�IcYÂL�r|IcN,J>´uJcN%YXI%¯á��L�t%I�Ï5v%H7Ç1JMv,�Fr<TENQr5s|YX´FvUÁöY@TXv,VWN%�%IÀVWI_�<�FL�VXIcR�Icr<TELkr|IKN%J>´JcN%YXI%¯ZeN,l A NQr/s B NQVWIÀ·/v%TX´uPkvUÁ I�V TWVXLONQrF�,�FPONQVwR�NUTXVWLOJMIcYc¯Z¢·/l A LOYÂNQVW·FL¶TWVWN%VX²�N%r5s B LOYwP�vUÁ±IcV±TWVXLONQr5�%�FPONQVK¯ZeJcl A NQr/s B NQVWIÀ·/v%TX´�TXVWL�sFL�N%�%v%r/NQP]¯Zxs5l A LOYÂNQVW·FL¶TWVWN%VX²�N%r5s B LOYw�FHFH3I�VG�EIcYWYXI�r*·/IcVX�/¯ZxIKl A LOYw�FHFH3I�VT�ÐIcYWY@Icr<·3I�VW��NQr5s B L�YwTWVXLOs7LONQ�,v%r5N%Pe¯Z¢¡Wl A LOYw�FHFH3I�VT�ÐIcYWY@Icr<·3I�VW��NQr5s B L�YÂ�5HFH/IcVÂTXVWLONQrF�,�FP�N%Vc¯

j_i%¯ Æ Y"�<�5NQVWIÐR�NUTWVXLkp A = (aij) LOYwYXN%L�s�TXv�·/I[N�ا¥5f�Ó¹­ª¥7ac«Ub¢Êov%¡�·5N%r5s7ÁÂLOs*TX´ 2k + 1Lk¡aij = 0 Á´FI�rFIct%IcV |i− j| > k.

�ÌI�t,I�P�v%H�NQruNQP��%v%VWLkTX´FRÎTWv�J�v%R�HF�7TWI�TX´FIÀHFVWv7s7�5J#T C = AB ¸FÁ´FIcVXI A L�YÂN%VX·5L¶TWVWN%VX²N%r5s B L�YÂN�·5N%r5sqR�NQTXVWL¶pqv%¡�·5N%r5s7ÁÂLOs*TX´ º ¸*TWN6�*L�rF��N%s7tUNQr<T>NQ�%I�vQ¡�TX´FI�Y@TXVW�5JMTX�FVWIÀvQ¡TW´FI[R�NUTWVXLkp B ¯±ÕÌt,I�VWÁÂVXLkTXI A ÁÂL¶TW´ AB NQr5s��%L�t%I�Ï5v,H7Ç1JMv%�5r,TK¯

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j_i�¼��IMT A N%r5s B ·3I£T�Á±vãY@²*R�R�IMTXVWLOJoR�NUTWVXLOJMIKY�vQ¡[TW´FI¼YXN%R�I£v%V>s7I�VK¯ý�ÌI�t,I�P�v%HÚN%rNQP��%v%VWLkTX´FR TXv£J�v%R�HF�7TWI C = A + B ¸DTWN��*LkrF�£N%s7tUNQr<T>NQ�%I�v%¡ÐY@²*R�R�IMTWVX²9¡¢v%V�IcN,J>´R�NQTXVWL¶pS¯¦lÖv%�5VENQP��%v%VWLkTX´FR YX´Fv%�5P�suvUt%IcVXÁÂVWLkTXI B ÁÂLkTX´ C ¯ � ´5NQTEL�Y TW´FIÀÏ5v%HFÇËJ�v%�Fr<T%Qj_i%¿Z�IMT ar NQr/s br sFI�rFv%TXI%¸5VXIKY@H3IcJMTXL�t%I�P�²%¸<TW´FI r Ò2Ó�JMv,Pk�FR�r5Ywv%¡�TW´FI[R�NUTWVXLOJMIKY A NQr5s B ¯»w´FI�r�s7I�t,I�P�v%H�NQr�NQP��%v,VXLkTX´FR TXv�JMv,R�H5�7TXIÀTW´FI[HFVWv*sF�5J#T AB ¡¢VWv%R�TW´FIÀ¡¢v%VWR��5P�N

AB =

n∑

i=1

aibTi .

��Lkt,IÌÏ/v%H7Ç1JMv,�Fr<TwN%r5s�Y�TWv%V>NQ�%IÌVXI,�<�FLkVWI�R�Icr,TÂv%¡�TW´FI[NQP��%v%VWLkTX´FR�¯j_i%è °±v%r5YXL�sFI�V TX´FI[R�NQTXVWL¶p

A =

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

0 10 10¯ ¯ ¯ ¯¯¯ ¯¯¯¯¯¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯¯¯ ¯¯¯¯¯¯ ¯ ¯ ¯ ¯ ¯ ¯ 2

¯¯¯¯¯¯ 2 2 10 · · · · · · · · · 0 1 1

â�L�r5sqTX´FI[IcLk�,I�r*tUNQP��FIcY v%¡_TX´5L�YÂR�NQTXVWL¶pq�/Y@L�rF��n Æ »�� Æ ÄhJMv,R�R�N%r5su^<be¦_¯ Û vUÁhH3I�VXÇTX�FVW·�TW´FI�Z�½,¸k½ º l_I�P�I�R�I�r<T§TXv 10−9 NQr5s�JMv,R�H5�7TXI±TX´FIÂIcLk�,I�r*tUNQP��FIcY�vQ¡/TX´FLOYDH3I�VXTX�FVW·/IKsR�NQTXVWL¶pS¯ � ´5NUT�J�v%r5J�Pk�/Y@L�v%r9s7v�²%v%�9R�N6�,I�N%·/v,�7TÀTX´FI�J�v%r5s7LkTXL�v%r5LkrF��v%¡(TX´5I�I�L��%I�rFÇt³N%Pk�5IcY%Qj�i'~DðöÃË¡ f(x) L�YÐN�VWIcN%P¶ÇËtUNQP��FIcs�s7Lkµ3IcVXIcr,TWL�N%·FP�IÀ¡¢�Fr5J#TWLkv,r�vQ¡�N�VWIcN%PStUNQVWL�N%·FPkI x ¸5TX´FIcr�HFVXvUt,ITX´5NQT |f ′(x)||x|

|f(x)| LOY TX´FI�J�v%r5s7LkTXL�v%rur*�FR�·3I�VÐvQ¡ f(x) NUT x ¯j�i'~!~ ZxN,luG*´FvUÁhTX´5NQTEL¶¡ f(x) = log x ¸7TX´5I�ruTX´FI�J�v%r5sFL¶TWLkv,r�r*�FR�·/IcVc¸ c(x) =

∣∣∣∣1

log x

∣∣∣∣ ¯Z¢·/lutÐYXL�rF��TW´FI�N%·/vUt,I[VXIKY@�5P¶T�Z¢v%V�vQTW´FI�VWÁÂLOY@I³l#¸mY@´FvUÁ�TW´5NUT log x LOY�LkP�P¶Ç1JMv,r5s7LkTXL�v%rFIKsrFIKNQV x = 1 ¯

j�i'~Fg�G*´FvUÁ TW´5NUT[TW´FI�JMv,r5s7LkTXL�v%r9r*�FR�·3I�V κ ¡¢v%VÀTX´5I�HFVXv7s7�/J#T Ax ZxÁÂL¶TW´ VXIKY@H3IcJMT�TWv�TX´FIH/IcV@TW�FVX·/NUTXL�v%r�vQ¡ x l LOY κ = ||A|| ||x||||Ax|| ¯j�i'~D|�G*´FvUÁ[¸§·<²¼JMv,R�HF�7TXL�rF�|TW´FIqJ�v%r5s7LkTXL�v%r r*�FR�·/IcVc¸§TW´5NUT�TW´FIqH5VXv,·FPkIcR v%¡ÂJ�v%R�HF�7TWLkr5�√

x ¡¢v,V x > 0 LOYEN�Á±IcPkPkÇËJ�v%r5sFL¶TWLkv,rFIcsqHFVXv,·FP�I�R�¯j�i'~*j � v,V*��v,�7TEN�·3v%�5r5s�¡¢v%V TX´FI[VWI�PONUTWLkt,IÌIcVXVWv%V Á´FIcruN�·5N,J0�<ÁwNQV>s�Y@TWN%·FPkI[N%Pk�,v%VWL¶TW´FR�L�YNQHFHFP�L�IcsqTXv�N�HFVWv%·FP�I�R ÁÂLkTX´uTW´FI�JMv,r5s7LkTXL�v%rur<�5R�·3I�V κ ¯

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ô �Fú/ù���û�÷7û úª¹�ø�õ�ùÂ÷,ý�ø��§û úeº��)��ùEü�ìøÀü�wû úÂûAø[ü±ûOüÖý

j�i'~FzyP�IMT A ·3I�rFv%r5YXL�rF�%�FPONQV[N%r5s ∆A ·/I�YX�5J>´£TX´5NQT ||∆A||||A|| < Cond(A) ¯�»w´FIcr£HFVWvUt%ITW´5NUT A + ∆A L�Ywr5v%r5YXLkr5�%�FPONQVK¯

j�i'~D¯ ZeN,lu�EvUÁ�NQVWI Cond2(A) N%r5s Cond2(A−1) VXIcP�NQTXIKsXQZ¢·/l�G*´FvUÁhTX´5NQTL]¯ Cond2(A) ≥ 1LkL]¯ Cond2(A

T A) = (Cond2(A))2 ¯j�i'~]¼ ZeN,lu�I�T A ·/I�N%ruv,V@TW´Fv%�,v%r5N%P/R�NQTXVWL¶pS¯�»w´FI�r|YX´FvUÁÚTW´5NUT Cond2(A) = 1 ¯Z¢·/l�G*´FvUÁ�TW´5NUTÌTX´FI Cond2(A) = 1 L¶¡±N%r5s|v,rFPk²�Lk¡ A L�Y�NqYWJ�N%P�N%VÐR��5P¶TWLkHFP�I�vQ¡(NQrv%VXTX´5v%�%v,r5NQP3R�NUTWVXLkpS¯j�i'~D¿n�_IMT U = (uij) ·3I�N�rFv%r5YXL�rF�%�FPONQVw�FH5H/IcVwTXVWL�N%rF�%�FPONQVwR�NUTWVXLkpS¯�»w´FI�r|YX´FvUÁÚTW´5NUT

Cond2(U) ≥ max(uii)

min(uii).

�ÐI�r5J�IÐJMv,r5Y@TXVW�5J#T±N�Y@L�R�HFPkIÐIMpFNQR�HFP�IÂvQ¡NQr�L�P�P¶Ç1JMv,r5s7LkTXL�v%rFIKs�rFv%r7Ç1s7LONQ�,v%r5N%PFY@²*R�R�IMT@ÇVWLOJÌH3v,YXLkTXL�t%I[s7IM�/rFL¶TWI�R�NUTWVXLkpS¯j�i'~Dèn�_IMT A = LDLT ·3I�N�YX²<R�R�IMTWVXLOJ[H3v,YXL¶TWLkt,I�s7IM�/rFL¶TWI�R�NUTWVXLkpS¯©�_IMT D = diag(Dii) ¯»w´5I�r|Y@´5vUÁ

Cond2(A) ≥ max(dii)

min(dii).

�ÐI�r5J�IoJMv,r5Y�TWVX�/J#TqNQr¹I�p7N%R�HFPkIu¡¢v,VqNQr¹L�PkPkÇËJ�v%r5sFL¶TWLkv,rFIcs rFv%rFÇËs7LONQ�,v%r5N%PÂY@²*R�R�IMTXVWLOJH3v,YXLkTXL�t%I[s7IM�/rFL¶TWI[R�NUTXVWLkpm¯j�iOgwðS�(VXvUt,IÐTW´5NUT¡¢v,VEN��,Lkt,I�rurFv%VWR�¸ Cond(AB) ≤ Cond(A) · Cond(B) ¯j�iOg=~ ZeN,luâ§Lkr/sq¡¢v,VÂÁ´5NUTÂtUNQP��FIKYÂvQ¡ a TX´FI[R�NQTXVWL¶p A =

(1 aa 1

) LOYÂL�PkPkÇËJ�v%r5sFL¶TWLkv,rFIcsXQZ¢·/l � ´5NQTEL�Y TW´FI�JMv,r5s7LkTXL�v%rur<�5R�·3I�VEv%¡ A Q

j�iOg3gô��L�t%Icr�NQr�IMpFNQR�HFP�IÖTWvÌYX´FvUÁ¼TX´5NQT§NÌY�T>NQ·FP�I±N%Pk�,v%VWL¶TW´FR NQHFHFP�L�IcsÀTWvÌNQr�LkP�P¶Ç1JMv,r5s7LkTXL�v%rFIKsH5VXv,·FPkIcRÝJ�N%r�H5VXv7s7�5J�I[NQr�Lkr/N%J�J��FV>NUTXIÀYXv%P��7TXL�v%r¯j�iOgw| ZeN,lu�I�T ai ·3I�TW´FI�LkTX´®J�v<IBñ�JML�I�r<T�vQ¡ÐN�H3v%P�²*rFv%R�LONQP p(x) NQr/s¼PkI�T ∂ai NQr/s ∂xjs7IcrFvQTWI�Y@R�NQP�PÖH3I�VXTX�FVW·5NQTXL�v%r5Y[v%¡ ai N%r5s£TX´FIK�1TX´ VWv<v%T xj ¯u»w´FI�rªYX´FvUÁ TW´5NUTTX´5IÌJMv,r5s7LkTXL�v%r�r*�FR�·/IcV±v%¡STX´FI�VWv<v%T xj ÁÂL¶TW´�VXIKY@H3IcJMT(TWv�H3I�VXTX�FVW·5NUTWLkv,r5YÖv%¡STX´FIJMv*IBñ�J�LkIcr<T ai LOY

|aixi−1j |

|p′(xi)|.

Z¢·/l v �����f�ÕÔ�O ¤ �qO�ùîA0CD�!#$5@�'&6� õ=¶GÖØ×�Ù-ÚGÖ ¸±JMv,R�HF�7TXIuTW´FIoJMv,r5s7LkTXL�v%rãr<�5R�·3I�V>Y�vQ¡TX´5I[VXv*vQT>Y x = i, i = 1, 2, . . . , 20 v%¡�TW´FI � LkPo�*Lkr/Y@v,r�H3v%P�²*rFv%R�LONQPp(x) = (x− 1)(x− 2) . . . (x− 20) = x20 − 210x19 + . . .

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ÁÂLkTX´�VXIKY@H3IcJMT±TWv�H/IcV@TW�FVX·/NUTXL�v%r�vQ¡�TX´FI[JMv*IBñ�J�LkIcr<T x19 ¡¢VWv%R −210 TWv −210 +2−23 ¯X�ÖVWIcYXI�r<T²%v%�5V VWIcYX�FPkTWYwL�rqT>NQ·F�5P�N%Vw¡¢v%VWRÝNQr5s�ÁÂVWL¶TWI�²,v%�FVÂJ�v%r5J�Pk�5YXL�v%rqv,rTX´5IqL�PkPkÇËJ�v%r5sFL¶TWLkv,rFLkr5�ovQ¡wTX´5IqVWv<v%TWY[v%¡ÂTX´FI � LkPo�*Lkr/Y@v,r H/v,Pk²*rFv,R�LONQP]¯|Ù�p7HFPONQL�rÁ´*²qJMIcV@T>NQL�r�VWv*vQT>YÂNQVWIÌR�v%VWIÀLkP�PkÇËJ�v%r5s7LkTXL�v%r5IcsqTX´5N%r�TW´FI[vQTW´FI�V>Yc¯

� � ­ _ � . � ¬ � � � ­ ÷ �ê� è��<��ùu� � � \ � ¬ �è����ê.I_ æ � \e�<¬ ÷ N � è ­ æ � j� ikj�i'~etÐYXLkrF��TX´5I�n Æ »�� Æ Ä¹¡¢�5r5J#TWLkv,rÜÛ�«³¥5f�Ó�Ýx¸7JMVWIcNQTXI�N 5× 5 V>NQr5sFv%RÎR�NUTXVWLkpqNQr5s�TX´FIcrHFVXL�r<TÂv%�7TÂTW´FI�¡¢v%P�PkvUÁÂL�rF��v%�7TWHF�7T>Y�ÈÆ Z º ¸kÈAl#¸ Æ Z�È�¸�½Kl#¸ Æ Z�È�¸ ß,l#¸Æ Z@½%¸½,È º È(ß%l#¸ Æ Z(¢�½%¸/ßY£ lM¸ Æ Z¢Ô/ÈDÇW½,È ½,¸5ß7ÈDÇ>½%Èw½KlM¯� ikj�iOgntÐYXLkrF��TX´FIÌ¡¢�Fr5JMTXL�v%rÞÛodËd/«�Ýe¸,ÁÂVWLkTXIÀN�n Æ »�� Æ Ä¹HFVWv%�,VWN%R TWv��5r/s�TW´FI�bef�f§^<«�©§«UdÓ_¨×�`%a�NQr5sud/×�ac^<«�©§«UdÓ§×�`%a[vQ¡_T�Á v n Ç]t,IcJMTXv%V>Y u NQr5s v ¯

[s] = bxf�©§«³d§Z¢�_¸ tFl[A] = d/×�a�©§«Ud�Zx�¸ tFl

»_IKY�T ²%v%�5V±HFVWv%�,VWN%R ·*²�JMVWIcNQTXL�rF�[T�Á v�s7LkµmI�VWI�r<T t,IcJMTXv%V>Y u NQr/s v �5Y@L�rF��VWN%r5soZ¢Ô/¸k½³l#¯� ikj�io|��IcN%VXr�´FvUÁÚTWv��5Y@IÀTW´FIÀ¡¢v%P�P�vUÁÂLkrF��n Æ »�� Æ ÄhJ�v%R�R�NQr5s5Y TXv�JMVWIcNQTXI[Y@H3IcJ�L�N%PSR�NUTXVWLkÇJMIcYcÈ`,d/­ª©§¥Ff °±v,R�H5NQrFL�v%ruR�NUTWVXLkpÓ§bx¥5¦ �ÌLONQ�%v,r5NQPSR�NQTXVWL�J�IcY v%VwTW´FI�s7LONQ�%v,r5NQPOY vQ¡DN�R�NQTXVWL¶pd3f§^<É n:NUTXVWLkpqÁÂL¶TW´|NQP�PI�r<TXVWL�IcY I_�<�5N%PmTXv|½Ò,^<«³d3É ßmI�VWv�R�NQTXVWL¶p«U¥Ff§Ó ½ NQr5s7v,R R�NUTWVXLkp

g�beÑ�&bxf§ÉKd/f � L�Po�<L�r5YXv%rtZ YwI�L��%Icr*t³N%Pk�5IÌTWIcY@TER�NUTXVWLkp¬§¥Ffy&/^*Ñ �ÌNQr]�,I�PSR�NUTWVXLkpaKd^<©§Ñeb¢aKÒ »�v*I�HFP�LkTXÍ[R�NUTWVXLkp¬§bxÑxØ �ÐLkP�·3I�VXTER�NQTXVWL¶pac«Ubx× ÙÖp<TWVWN,J#TÂTW´FI[�FHFH3I�VwTWVXLONQrF�,�FPONQVwH5NQVXTÂvQ¡�N�R�NQTXVWL¶pac«UbxÑ ÙÖp<TWVWN,J#TÂTW´FI[PkvUÁ I�V TWVXLONQrF�,�FPONQV H5NQVXTÂvQ¡�N�R�NQTXVWL¶pbF¥5f�Ó§^<« �ÖN%r5s7IcVXR�v%r/s7IÀR�NQTXVWL¶p«U¥Ff§Ó�h�fXj n:NUTXVWLkp¹ÁÂLkTX´hVWN%r5s7v%R I�r<TWVXL�IcYc¸ÐJ>´Fv<Y@Icr¹¡¢VXv,R�NªrFv%VWR�NQPÌsFL�Y@TXVWLk·5�7TXL�v%rÁÂLkTX´�R�IcN%r�Í�I�VWv5¸*tUNQVWLONQr5J�I�v,rFI[NQr5suY@TWNQr/sFNQV>s�sFI�t*L�NQTXL�v%ruv%rFI,¯

� ikj�ikjZ�IcN%VXr:´FvUÁ�TXvu�5YXI�TX´5I�¡¢v,PkP�vUÁÂL�rF��n Æ »�� Æ Ä�¡¢�Fr/J#TXL�v%r/YС¢v,V�E"/(���'#434/65@ ��>1I#0&(3T�!CD?57/65@�'&6�]�:Z¢²,v%�¼ÁÂL�PkP±PkIKNQVWr¼N%·/v,�7T�TW´FI�N%Pk�,v%VWL¶TW´FR�YÀvQ¡wTX´FIKY@I�¡¢�Fr5J#TWLkv,r5Y�PONUTWI�V�Lkr¼TW´FL�Y·/v*v��FlMÈ

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a\b �L�rFIcN%VwI_�<�5NUTWLkv,ruYXv%P��7TWLkv,r�v%¡ Ax = b ¯befb noNQTXVWL¶pqL�r*t%IcVWYXIÓ§^<a �ÌIMTXIcVXR�L�r5NQr<T`,d/f§Ó °±v%r/s7L¶TWLkv,r�r*�FR�·/IcV^*be¦ ÙÖL��%Icr<tUN%Pk�FIKYÂNQr5s�I�L��%Icr*t%IcJMTXv,VWYf§d/«³­ �ÖNQVWL�v%�5Y R�NUTWVXLkp�NQr/s�t,IcJMTXv%VwrFv,VXR�Y©±d/Ñ'c °±´5N%VWN,J#TXIcVXLOY@TXLOJÐH3v%P�²*rFv%R�LONQP©±d/Ñ'c�bF¥FÑ »w´FI[tUNQP��FIÀvQ¡DN�H/v,Pk²*rFv,R�L�N%PmNQTÐN��%L�t%I�rur*�FR�·3I�V©DÑxd/a �ÖP�vQTXTXL�rF��tUNQVWLkv,�5Y±¡¢�Fr5J#TWLkv,r5Yc¯«U¥Ffy& ½ N%r]��vQ¡�N�R�NUTXVWLkpm¯Ñe× �qt�¡xN,J#TWv%VWLkÍKNUTXL�v%r_¯�m« Î ½ ¡xN%J#TWv%VWLkÍKNUTWLkv,r¯É_bÓ G*L�rF�%�5P�N%Vq�ÖNQP��FI��ÐIKJMv%R�H3v,YXL¶TWLkv,r¯

� ikj�iOz � VWL¶TWI�n Æ »�� Æ ÄÚHFVWv%�,VWN%R�Y(TXv�J�VXIKNUTWIÐTW´FIÀ¡¢v%P�PkvUÁÂL�rF��Á I�P�P¶Ç7�*rFvUÁÂrqR�NUTWVXLOJMIKY�ÈZeN,l [A] = g�beÑ�&ÖZxr/l�TXvoJ�VXIKNUTWI�TX´FI � L�Po�<L�r5YXv%r¼·FLOs7LONQ�%v,r5NQP�R�NQTXVWL¶p A = (aij) vQ¡v%V>s7IcV n È

aii = n− i + 1, i = 1, 2, · · · , 20

ai−1,i = n, i = 2, 3, · · · , naij = 0, vQTW´FI�VWÁÂL�YXI .

Z¢·/l [A] = èÌ^*b*h�fXjÐTXv�J�VXIKNUTWIÐTW´FI��DI�LR�NUTWVXLkp A = (aij) vQ¡§v%V>s7I�V n Èaij = α, α is a parameter near 1 v%V n− 1.

aii = 1 ¡¢v%V i 6= j.

ZeJcl��ÖVWLkr<T�TW´FI£J�v%r5sFL¶TWLkv,ryr<�5R�·3I�V>Y�vQ¡�IKN%J>´¹vQ¡ÀTX´5IcYXI|T�Á±v®R�NQTXVWL�J�IcY�ÁÂLkTX´ n =10, 20, 50 ¸5NQr5sª½c¾%¾5¸5�5YXL�rF��TW´FI�n Æ »�� Æ Äö¡¢�Fr/J#TXL�v%r9`%d3f�Ó±¸mÁÂL¶TW´:VWIcYXH/IKJ#TÌTXvTX´5I[R�NQTXVWL¶p�rFv%VWR�Y || · ||2, ||||F , N%r5s || · ||∞ ¯

� ikj�io¯StÌY@L�rF�7j�´5I�P�Hy÷EJ�v%R�R�NQr5sFY(¡¢v%VÁj³`,Ñxd`W&É÷EN%r5swj³^<aKbx­ª^c÷U¸%PkIKNQVWr�´FvUÁyTWv�R�IcN,Y@�5VXIÂTWLkR�ÇL�rF��¡¢v%VÌNQr�NQP��%v%VWLkTX´FR�¯� ikj�i%¼�tÌY@L�rF�:n Æ »�� Æ Ä ¡¢�Fr5JMTXL�v%r5YàÛodËd/«�Ý�P�ÛOÉcbeÒ%^GÝ-PáÛOÒ%^<«UdbÝe¸�ÁÂVWLkTXIqN|n Æ »�� Æ Ä HFVWv%�,VWN%RTWv��/r5s�TW´FIÀHFVWv*sF�5J#TwvQ¡�T�Á±v��FHFH3I�VwTXVWLONQrF�,�FP�N%V±R�NQTXVWL�J�IcY A N%r5s B vQ¡�v%V>s7I�V m× nN%r5s n× p ¸7VWIcYXH/IKJ#TWLkt,I�P�²%¯�»_IKY�TE²,v%�FVwHFVWv%�,VWN%R��5YXLkrF�

A = ac«³be×±ZxVWN%r5s¼Z¢Ô/¸ ä<l@l#¸B = ac«³be×±ZxVWN%r5s9ZxäF¸ ä,lXl#¯

� ikj�io¿ ¤ QF�X�CD U�]&(�$�G&BA�5�QD�%�r�"1f�. "#.�%�$�T�%��5H&�5H����5�5�QF/(5J5�QF�LË������oE"�$ �5J34/65@ ��>1:�%�������9?B#0&(�!-6��5@�'&6�!�0-VX��5YQW "���U�F�"#$5©57&<�.&6�9+,�����k5�QF�K�9����0/( ���8R��57�$3á� *&�E$���$34`ZxLOl|°±VWIcNUTWI A = ´FL�P�· (10) ¯<�DIcV@TW�FVX·£TX´5IoZ�½c¾5¸k½³lÌI�r<TXVW²ovQ¡ A ·<² 10−5 ¯�° NQP�PDTX´FIH3I�VXTX�FVW·/IKs�R�NUTXVWLkp B ¯T�I�T b = V>NQr5s (10, 1) ¯Ð°±v,R�H5�7TXI x = A\b, y = B\b ¯°±v%R�HF�FTXI ||x − y|| N%r5s ||x− y||

||x|| ¯ � ´5NUTÐJ�v%r5J�Pk�5YXL�v%ru²%v%�us7V>N³ÁÚ¡¢VWv%R ´FIcVXI'Q

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Z¢L�LOl|°±v%R�HF�FTXIÀTX´5I�JMv%r/s7L¶TWLkv,r�r*�FR�·/IcVWYwv%¡§·3vQTW´ A NQr5s B È Cond(A), Cond(B) ¯Z¢L�L�LOl|°±v%R�HF�FTXIªTW´FI J�v%r5sFL¶TWLkv,r r<�5R�·3I�V9vQ¡ A �5YXL�rF�ÚTW´FI n Æ »�� Æ Ä�JMv,R�R�N%r5sCond(A) N%r5suTX´5I�r|�5YXI[L¶TÐTXv�JMv,R�HF�7TXI�TX´FI[TW´FI�v,VXI�TXLOJ�N%PS�FHFH3I�Vз3v%�5r5s��%L�t%IcrL�r ­ ¬�^*d/«³^*­ ú �DÃ��DÃ�Äj�iOgw|_¯Ö°±v,R�H5NQVWIÐTW´FLOY·/v,�Fr5suÁÂLkTX´uTX´FI�N,J#TW�5NQPSVWI�PONUTWLkt,IÌIcVXVWv%VK¯

� ikj�ioèn�§IcV@¡¢v,VXR�TW´FIoVWIcYXH/IKJ#TWLkt,I�I�p*H3I�VWL�R�Icr<TWY�Y@TWNQTXIcs L�ryG7IcJ#TWLkv,r¹Ô5¯ Þov,r®TW´FI|Ù�pFNQR�HFP�IcY² � Ã�FDÃ�ÄÔ5¯ º áUÇ ² � Ã�FDà �Ô5¯ ä%¾9TWvãYX´FvUÁ TW´5NUTuTX´5I9IcLk�,I�r*tUNQP��FI:H5VXv,·FPkIcR�Yq¡¢v,VuTX´FIKY@I9R�NUTXVWLOJMIcY�N%VXI£LkP�P¶ÇJMv%r/s7L¶TWLkv,rFIcs¯��ikj�i'~Dð � VWL¶TWI[N�n Æ »�� Æ ÄyHFVWv%�%V>NQRÎTWvZxN,l�°±v%r/Y�TWVX�5JMTwTX´FI n× n PkvUÁ I�V TWVXLONQrF�,�FPONQVwR�NUTXVWLkp A = (aij) N,Y ¡¢v%P�PkvUÁEYcÈ

aij = 1 Lk¡ i = jaij = −1 Lk¡ i > jaij = 0 Lk¡ i < j

Z¢·/l��DI�VX¡¢v%VWR�NQr�I�p*H3I�VWL�R�Icr<T±TWv�YX´FvUÁ¹TX´5NQT YXv%P��7TXL�v%rqvQ¡ Ax = b ÁÂLkTX´ A N,Y(NQ·3vUt%INQr/sªTW´FI�t%IcJMTXv,V b JMVWIcNUTWIcs Y@�5J>´®TX´5NQT b = Ax ¸ Á´FIcVXI x = (1, 1, . . . , 1)T ¸·3IcJMv,R�IcY�R�v%VWIwNQr5s�R�v%VWIwLkr5N,J�J��FVWNQTXIwN%Y n Lkr/JMVWIcN%YXIcYDs7�5IwTXvÀTX´5IÂLkr5J�VXIKN%YXLkr5�%P�²L�PkPkÇËJ�v%r5sFL¶TWLkv,rFLkr5��v%¡ A ¯��I�T x sFI�rFv%T TW´FI�JMv,R�HF�7TXIKs�Y@v,Pk�7TWLkv,r¯�ÖVWIcYXI�r<T²%v,�FVÂVXIKY@�5P¶T>YwLkruTX´5IÀ¡¢v%P�PkvUÁÂL�rF��¡¢v%VWRuÈ

n Cond(A) x = A\b ½ IcP�NQTXL�t%IÀI�VWVWv%V||x− x||2||x||2

½ IKY@LOs7�5N%PSrFv%VWR||b−Ax||2||b||2½c¾º ¾ä%¾Ô,¾ßQ¾

��ikj�i'~!~etÐYXLkrF�Ìn Æ »�� Æ Ä:¡¢�5r5J#TWLkv,rLbF¥Ff§Ó�^<« (v) ¸UÁ´FIcVXI v = VWN%r5s (20, 1) ¸KJ�VXIKNUTXI±N 20×20�ÖNQr5s7IcVXR�v,r5s7I[R�NUTWVXLkp A ¯ Û vUÁ�TWN��%I x = v,rFIcY�Z º ¾5¸k½³lwNQr5s b = A ∗ x ¯Ì°±v%R�HF�FTXIrFvUÁ y = A\b ¯|°±v,R�H5NQVWI y ÁÂLkTX´ x ·*²9J�v%R�HF�7TWLkr5� y − x NQr5s ‖y − x‖ ¯ � ´5NQTJMv%r/JMP��5Y@L�v%r/Y sFv�²,v%��s7V>N³Á�Q��ikj�i'~Fg nËK�>�,QF/(3ãâ �àÏr/(�����$ �8�&�A ¤ �.��5NÔê/(5U ��'#0��� $

�IcN%VXr|´5vUÁ�TXvu�5Y@I)�EL��%´5N%RÔZ Yi�ÀNQP�PkIcVX²�v%¡�TXIcY@TÀR�NQTXVWL�J�IcYÌLkr9n Æ »�� Æ Ä�Z T�²*H3I<QF�.� ��2/6�����. �8�¡¢v%VEN�JMv,R�HFPkI�TXIÀP�L�Y@T>lM¯

��ikj�i'~D|* ÖD�&63r�CD5@���f��5YQ]��ä/63r����Ðå!/( ��'/(�#0�dæhËK�>�,QF/(3� ë�ç�ç&èÖ$(é�$ ¯°±v%r5YXL�sFI�VÂJMv,R�H5�7TXL�rF�qÉK¥F­ ©DÑx^�bF¥7«Ubx¥Ff§`%^�vQ¡ n r*�FR�·/IcVWY x1, . . . , xn s7I��5rFIKs�·*²S2

n =1

n− 1

n∑

i=1

(xi − x)2 ,

Á´FIcVXI x =1

n

n∑

i=1

xi .

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�ÌIcYWJMVWL�·/I�t³N%VXL�v%�/YÂR�NUTX´5I�R�NUTWL�JcNQP�Pk²uI_�<�FL�tUNQP�I�r<TÐÁwN³²*YEvQ¡ÖJMv,R�H5�7TXL�rF��TW´FL�Y��<�5NQr<TWL¶T�²N%r5s�s7LOYXJ��5YXY TW´FI�L�VEs7Lkµ3IcVXIcr<TEr<�5R�IcVXLOJ�N%PSY�T>NQ·FL�PkLkT�²qH5VXv,H/IcV@TWLkIKY�¯