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Numerical Analysis r School of Mathematics and Statistics Xi an Jiaotong University r Numerical Analysis SK1 1 1 2 1 414213562373 O O 3 5 Cq 9k i 2 x 1 1 414 2 x 2 1 4142 2 1 414 1 2 10 3 2 1 4142 1 2 10 4 1 2 1 414 2 1 2 10 3 2 0 35355339 10 3 2 2 1 4142 2 1 2 10 4 2 0 35355339 10 4 r Numerical Analysis SK1 1 6 x 1 X O e J O 1 1 1 2x 1 x 1 x 2 1 x2 1 x2 x 3 1 cos2x x 4 ln 1 1 x2 x 1 1 1 2x 1 x 1 x 1 x 1 x 1 2x 1 2x 1 x 2x2 1 2x 1 x 2 1 x2 1 x2 x 2x2 x 1 x2 1 x2 3 1 cos2x x 2sin2x x 4 ln 1 1 x2 x ln x 1 1 x2 r Numerical Analysis SK1 1 6 x 1 X O e J O 1 1 1 2x 1 x 1 x 2 1 x2 1 x2 x 3 1 cos2x x 4 ln 1 1 x2 x 1 1 1 2x 1 x 1 x 1 x 1 x 1 2x 1 2x 1 x 2x2 1 2x 1 x 2 1 x2 1 x2 x 2x2 x 1 x2 1 x2 3 1 cos2x x 2sin2x x 4 ln 1 1 x2 x ln x 1 1 x2 r Numerical Analysis SK1 1 6 x 1 X O e J O 1 1 1 2x 1 x 1 x 2 1 x2 1 x2 x 3 1 cos2x x 4 ln 1 1 x2 x 1 1 1 2x 1 x 1 x 1 x 1 x 1 2x 1 2x 1 x 2x2 1 2x 1 x 2 1 x2 1 x2 x 2x2 x 1 x2 1 x2 3 1 cos2x x 2sin2x x 4 ln 1 1 x2 x ln x 1 1 x2 r Numerical Analysis SK1 1 6 x 1 X O e J O 1 1 1 2x 1 x 1 x 2 1 x2 1 x2 x 3 1 cos2x x 4 ln 1 1 x2 x 1 1 1 2x 1 x 1 x 1 x 1 x 1 2x 1 2x 1 x 2x2 1 2x 1 x 2 1 x2 1 x2 x 2x2 x 1 x2 1 x2 3 1 cos2x x 2sin2x x 4 ln 1 1 x2 x ln x 1 1 x2 r Numerical Analysis SK1 1 8 z U e g 1 x 5 4 9 x 5 3 7 x 5 2 6 x 5 4 2 1 x x2 2 x3 3 xn n 1 x 5 4 9 x 5 3 7 x 5 2 6 x 5 4 x 5 9 x 5 7 x 5 6 x 5 4 2 n 4 1 x x2 2 x3 3 x4 4 h x 4 1 x 3 1 ix 2 1x 1 r Numerical Analysis SK1 1 9 3O N O y 2971 71 U y 2971 71 29 71 29 70 29 69 29 2 29 1 79 54162637 r Numerical Analysis SK2 2 3 e A LU 1 147 258 3610 LU 100 210 321 147 0 3 6 001 2 6 e 4 2 4 21710 4109 x1 x2 x3 10 3 7 A GGT 200 140 221 2 1 2 042 001 Ax b A GGT GGTx b Gy b y 5 2 1 T GTx y x 2 1 1 T r Numerical Analysis SK2 2 3 e A LU 1 147 258 3610 LU 100 210 321 147 0 3 6 001 2 6 e 4 2 4 21710 4109 x1 x2 x3 10 3 7 A GGT 200 140 221 2 1 2 042 001 Ax b A GGT GGTx b Gy b y 5 2 1 T GTx y x 2 1 1 T r Numerical Analysis SK2 2 8 O A 5 4 7 42 4 7 45 1 2 A kAk1 kAk max 16 10 16 16 ATA 90054 0360 54090 I ATA 36 2 144 0 1 2 36 3 144 kAk2 q max ATA 144 12 r Numerical Analysis SK2 A 1 1 72 1 8 5 84 8 5 81 kAk 1 1 kA 1k max n14 72 20 72 14 72 o 5 18 Cond1 A kAk1kA 1k1 16 5 18 40 9 Cond A kAk kA 1k 40 9 kA 1k2 q max A 1 TA 1 q max ATA 1 r 1 36 1 6 Cond2 A kAk2kA 1k2 12 6 2 r Numerical Analysis SK2 2 15 A B n R n c y 1 Cond cA Cond A 2 Cond AB Cond A Cond B 3 Cond2 RA Cond2 A Cond2 AR 4 Cond2 A max ATA min ATA 5 A Cond2 A max min yyy 1 Cond cA kcAk k cA 1k c kAk c 1 kAk kAk kA 1k Cond A r Numerical Analysis SK2 2 Cond AB kABk k AB 1k kAk kBk kA 1k kB 1k Cond A Cond B 3 du RA T RA ATRTRA ATA 1o kRAk2 q max RA T RA q max ATA kAk2 RA 1 T RA 1 A 1R 1 T RA 1 R A 1 TA 1R 1 RA 1 T RA 1 qu A 1 TA 1 d q k A r Numerical Analysis SK2 kA 1k2 q max A 1 TA 1 r max n RA 1 T RA 1 o kRA 1k2 Cond2 RA kRAk2k RA 1k2 kAk2kA 1k2 Cond A 2o kARk2 q max AR T AR q max RTATAR q max R 1ATAR q max ATA kAk2 AR 1 T AR 1 R 1A 1 T AR 1 A 1 TRR 1A 1 A 1 TA 1 k AR 1k2 r max n AR 1 T AR 1 o q max A 1 TA 1 kA 1k2 r Numerical Analysis SK2 Cond2 AR kARk2k AR 1k2 kAk2kA 1k2 Cond A 4 kyAB BA A AB A k AB x x B AB x Bx BA Bx Bx kA 1k2 q max A 1 TA 1 q max AT 1A 1 q max AAT 1 s 1 min AAT s 1 min ATA Cond2 A kAk2kA 1k2 s max ATA min ATA r Numerical Analysis SK2 5 Cond2 A kAk2kA 1k2 q max ATA q max A 1 TA 1 q max A2 q max A2 1 s max A2 min A2 max min r Numerical Analysis SK2 2 16 O 3 dC d C A QR 1 A 111 2 1 1 2 45 1 3 dC 11 A 11 a21 2 c 1 5 s 2 5 E3 dC P12 1 5 2 5 0 2 5 1 5 0 001 r Numerical Analysis SK2 K P12A 5 1 5 1 5 0 3 5 3 5 2 45 12 P12A 11 a31 2 c 5 3 s 2 3 E3 dC P13 5 3 0 2 3 010 2 3 0 5 3 K A 1 P13P12A 3 33 0 3 5 3 5 0 6 5 9 5 r Numerical Analysis SK2 13 A 1 12 a32 6 5 c 1 5 s 2 5 E P23 100 0 1 5 2 5 0 2 5 1 5 K A 2 P23A 1 3 33 03 3 003 R P P23P13P12 1 3 2 3 2 3 2 3 1 3 2 3 2 3 2 3 1 3 r Numerical Analysis SK2 Q PT 1 3 2 3 2 3 2 3 1 3 2 3 2 3 2 3 1 3 d C 11 H1 I3 2u1uT 1 I3 1 1 w1wT 1 1 sign a11 ka11k2 3 w1 a 0 1 1e1 1 2 2 T 3 1 0 0 T 4 2 2 T 1 1 1 a 0 11 12 r Numerical Analysis SK2 H1 I3 1w1wT 1 100 010 001 1 3 422 211 211 1 3 1 2 2 22 1 2 12 A 1 H1A 33 3 00 3 0 33 12 f H2 I2 2u2uT 2 I2 1 2 w2wT 2 r Numerical Analysis SK2 2 sign a 1 22 k g a 1 2 k2 kga 1 2 k2 3 2 2 2 a 1 22 9 w2 g a 1 2 2e 1 1 0 3 T 3 0 T 3 3 T w2wT 2 3 3 3 3 9 9 99 f H2 I2 1 2 w2wT 2 10 01 1 9 9 9 99 01 10 H2 10T 0 f H2 100 001 010 r Numerical Analysis SK2 A 2 H2A 1 33 3 0 33 00 3 R H H2H1 1 3 1 2 2 2 12 22 1 Q HT 1 3 1 2 2 2 12 22 1 A QR r Numerical Analysis SK2 2 18 e 1 2x1 4x2 11 3x1 5x2 3 x1 2x2 6 2x1 x2 7 1 A 24 3 5 12 21 b 11 3 6 7 T r Numerical Analysis SK2 ATAx ATb 18 3 346 x1 x2 51 48 x1 3 0403 x2 1 2418 r Numerical Analysis SK3 3 2 k 10 x1 4x2 4x3 13 4x1 10 x2 8x3 11 4x1 8x2 10 x3 25 n S 1 2 n S o S x k 1 1 13 4x k 2 4x k 3 10 x k 1 2 11 4x k 1 8x k 3 10 x k 1 3 25 4x k 1 8x k 2 10 pd m S x k 1 1 13 4x k 2 4x k 3 10 x k 1 2 11 4x k 1 1 8x k 3 10 x k 1 3 25 4x k 1 1 8x k 1 2 10 r Numerical Analysis SK3 3 2 k 10 x1 4x2 4x3 13 4x1 10 x2 8x3 11 4x1 8x2 10 x3 25 n S 1 2 n S o S x k 1 1 13 4x k 2 4x k 3 10 x k 1 2 11 4x k 1 8x k 3 10 x k 1 3 25 4x k 1 8x k 2 10 pd m S x k 1 1 13 4x k 2 4x k 3 10 x k 1 2 11 4x k 1 1 8x k 3 10 x k 1 3 25 4x k 1 1 8x k 1 2 10 r Numerical Analysis SK3 t S x k 1 SOR 1 w x k wx k 1 G S x k 1 1 0 2x k 1 1 2 13 4x k 2 4x k 3 10 x k 1 2 0 2x k 2 1 2 11 4x k 1 1 8x k 3 10 x k 1 3 0 2x k 3 1 2 25 4x k 1 1 8x k 1 2 10 X A 1044 4108 4810 A S f O 41 10 42 104 410 84 43 A 296 A r Numerical Analysis SK3 2D A 10 4 4 410 8 4 810 2D A 216 1 2D A S 2 A pd m S 3 A w 1 2 0 2 t S 3 3 k 2x1 3x2 10 x3 3 5x1 2x2 x3 12 x1 4x2 2x3 20 n S nd r Numerical Analysis SK3 2D A 10 4 4 410 8 4 810 2D A 216 1 2D A S 2 A pd m S 3 A w 1 2 0 2 t S 3 3 k 2x1 3x2 10 x3 3 5x1 2x2 x3 12 x1 4x2 2x3 20 n S nd r Numerical Analysis SK3 U 5x1 2x2 x3 12 x1 4x2 2x3 20 2x1 3x2 10 x3 3 KT X l S pd m S 0 1 S pd m S S B D E 1F 200 110 11 2 1 01 1 00 1 000 1 2 00 1 2 10 0 1 2 1 2 01 1 00 1 000 0 1 2 1 2 0 1 2 1 2 00 1 2 I B 1 2 1 2 0 1 2 1 2 00 1 2 1 2 2 0 1 0 2 3 1 2 B max i i 1 2 1 pd m S r Numerical Analysis SK3 3 10 x 0 0 0 0 T F e 2x1 x2 x3 2 x1 2x2 x3 0 x1 x2 2x3 2 11 S r 0 b Ax 0 b d 0 r 0 b 2 0 2 T kr 0 k2 2 8 d 0 T Ad 0 4 1 0 1 2 1 1 121 112 1 0 1 24 0 kr 0 k2 2 d 0 TAd 0 1 3 x 1 x 0 0d 0 2 3 1 0 1 T r Numerical Analysis SK3 12 S r 1 b Ax 1 b 2 3A 1 0 1 4 3 0 1 0 T kr 1 k2 2 16 9 0 kr 1 k2 2 kr 0 k2 2 16 9 1 8 2 9 d 1 r 1 0d 0 4 3 0 1 0 T 2 9 2 0 2 T 4 9 1 3 1 T d 1 T Ad 1 16 81 1 3 1 2 1 1 121 112 1 3 1 64 27 1 kr 1 k2 2 d 1 TAd 1 16 9 27 64 3 4 r Numerical Analysis SK3 x 2 x 1 1d 1 2 3 0 2 3 T 3 4 4 9 1 3 1 T 1 1 1 T r 2 b Ax 2 2 0 2 2 1 1 121 112 1 1 1 2 0 2 2 0 2 0 0 0 x x 2 1 1 1 T r Numerical Analysis SK4 4 1 y f x 3eZ XeL 1 xi0125 yi2312147 KF L 1 1 1 KF L3 x x 1 x 2 x 5 0 1 0 2 0 5 2 x 0 x 2 x 5 1 0 1 2 1 5 3 x 0 x 1 x 5 2 0 2 1 2 5 12 x 0 x 1 x 2 5 0 5 1 5 2 147 2 x x2 x3 R3 x f 4 4 x x 1 x 2 x 5 r Numerical Analysis SK4 02 131 21294 51474591 N3 x 2 x 4x x 1 x x 1 x 2 2 x x2 x3 R3 x f 0 1 2 5 x x x 1 x 2 x 5 r Numerical Analysis SK4 4 7 f x x7 x4 3x 1 f 20 21 27 f 20 21 28 d m X f xi xi 1 xi k f 7 k 0uxi xi 1 xi k m f 20 21 27 f 7 7 1 f 20 21 28 f 8 8 0 8 0 r Numerical Analysis SK4 4 7 f x x7 x4 3x 1 f 20 21 27 f 20 21 28 d m X f xi xi 1 xi k f 7 k 0uxi xi 1 xi k m f 20 21 27 f 7 7 1 f 20 21 28 f 8 8 0 8 0 r Numerical Analysis SK4 4 9 y f x 3eZ XeL 1 xi 101 yi 101 y0 i 000 2 xi012 yi1 10 y0 i 00 y00 i 2 D A L 1 L 1 1 1 10 0011 000 1 2 11111 3 2 110 1 2 3 2 3 2 r Numerical Analysis SK4 H5 x 1 0 x 1 x 1 2 2 x 1 2x 3 2 x 1 2x2 3 2 x 1 2x2 x 1 R5 x f 1 1 0 0 1 1 x x 1 2x2 x 1 2 R5 x f 6 6 x 1 2x2 x 1 2 2 L 01 010 0101 1 1 2 2 3 1 10247 2011 1 2 9 4 37 8 r Numerical Analysis SK4 4 10 xi i 0 1 n p li x i 0 1 n KF y 1 n P i 0 li x 1 2 n P i 0 li x xk i xk k 1 2 n 3 n P i 0 li x xi x k 0 k 1 2 n 4 n X i 0 li 0 xk i 1 k 0 0 k 1 2 n 1 nx0 x1 xn k n 1 r Numerical Analysis SK4 f x Ln x Rn x n X i 0 li x f xi f n 1 n 1 x x0 x xn 1 f x 1 Kf n 1 x 0 n X i 1 li x 1 2 f x xk k 1 2 n Kf n 1 x 0 n X i 1 li xk i xk k 1 2 n r Numerical Analysis SK4 3 xi x k k X j 0 cj kx j i x k j n X i 0 li x xi x k n X i 0 li x k X j 0 cj kx j i 1 k jxk j k X j 0 cj k 1 k jxk j n X i 0 li x xj i k X j 0 cj k 1 k jxk jxj 2 xk k X j 0 cj k 1 k j xk 1 1 k 0 k 1 2 n r Numerical Analysis SK4 4 i k 0 3 1 x 0 K n X i 0 li 0 1 ii k 1 2 n 3 2 x 0 K n X i 0 li 0 xk i 0 iii f x xn 1 K xn 1 n X i 0 li x xn 1 i xn 1 n 1 n 1 x x0 x xn r Numerical Analysis SK4 xn 1 n X i 0 li x xn 1 i x x0 x xn x 0 0 n X i 0 li 0 x n 1 i 1 n 1x0 x1 xn n X i 0 li 0 xn 1 i 1 nx0 x1 xn r Numerical Analysis SK4 4 11 f x a0 a1x a2x2 an 1xn 1 anxn f x 0 kn x1 x2 xn y 1 n X i 1 xk i f 0 xi 0 0 k n 2 1 an k n 1 K f x an x x0 x x1 x xn an x f 0 x an 0 x f0 xi an 0 xi n X i 1 xk i f 0 xi 1 an n X i 1 xk i 0 xi Pg x x k 1 an n X i 1 g xi 0 xi 1 an g x1 x2 xn 1 an g n 1 n 1 r Numerical Analysis SK4 k 0 1 n 2 g n 1 x xk n 1 0 k n 1 g n 1 x xn 1 n 1 n 1 3 y n X i 1 ik i 1 i i 1 i i 1 i n 0 k 0 1 n 2 P x x x1 x x2 x xn 0 xi xi x1 xi xi 1 xi xi 1 xi xn r Numerical Analysis SK4 k 0 1 n 2 g n 1 x xk n 1 0 k n 1 g n 1 x xn 1 n 1 n 1 3 y n X i 1 ik i 1 i i 1 i i 1 i n 0 k 0 1 n 2 P x x x1 x x2 x xn 0 xi xi x1 xi xi 1 xi xi 1 xi xn r Numerical Analysis SK4 d 5 n X i 1 f xi 0 xi f x1 x2 xn f n 1 n 1 3 f x xk K n X i 1 xk i 0 xi xk n 1 n 1 0 k 0 1 2 n 2 AO xi i i 1 2 n k n X i 1 ik i 1 i 2 i i 1 i i 1 i n 0 k 0 1 2 n 2 r Numerical Analysis SK4 4 13 y f x 3eZ XeL xi01234 yi 8 701956 v e ng 1 S0 0 0 S0 4 48 1 w hi 1 i 1 2 3 4 i hi hi hi 1 1 2 i 1 i 1 2 i 1 2 3 d0 6 h1 y1 y0 h1 y0 0 6 7 8 0 6 d4 6 h4 y0 4 y4 y3 h4 6 48 56 19 66 di 6 hi hi 1 yi 1 yi hi 1 yi yi 1 hi 3 yi 1 2yi yi 1 i 1 2 3 r Numerical Analysis SK4 n 21000 1 2 2 1 2 00 0 1 2 2 1 2 0 00 1 2 2 1 2 00012 M0 M1 M2 M3 M4 6 18 36 54 66 M0 0 M1 6 M2 12 M3 18 M4 24 S x x3 8 1 x 8x x3 8 0 x 1 2 x 3 2 x 1 3 8 2 x 2 x 1 x3 8 1 x 2 2 3 x 3 3 x 2 3 2 3 x 16 x 2 x3 8 2 x 3 3 4 x 3 4 x 3 3 16 4 x 52 x 3 x3 8 3 x 4 r Numerical Analysis SK5 5 2 XL p x p x c0 c1x c2x2 xi1345678910 yi2781011111098 9 P i 1 1 9 P i 1 x 9 P i 1 x2 9 P i 1 x 9 P i 1 x2 9 P i 1 x3 9 P i 1 x2 9 P i 1 x3 9 P i 1 x4 c0 c1 c2 9 P i 1 yi 9 P i 1 xiyi 9 P i 1 x2 i yi 953381 533813017 381301725317 c0 c1 c2 76 489 3168 r Numerical Analysis SK5 c0 1 459663866 c1 3 605309396 c2 0 267570665 p x 1 459663866 3 605309396x 0 267570665x2 5 3 y x3 m 0 1 C g C g p1 x c0 c1x K 0 x 1 1 x x k j 1 0 xk jdx 1 k j 1 0 f 1 0 xdx 2 3 1 f 1 0 x 3 2dx 2 5 r Numerical Analysis SK5 c0 1 459663866 c1 3 605309396 c2 0 267570665 p x 1 459663866 3 605309396x 0 267570665x2 5 3 y x3 m 0 1 C g C g p1 x c0 c1x K 0 x 1 1 x x k j 1 0 xk jdx 1 k j 1 0 f 1 0 xdx 2 3 1 f 1 0 x 3 2dx 2 5 r Numerical Analysis SK5 5 0 0 0 1 0 1 1 1 c0 c1 0 f 1 f 1 1 2 1 2 1 3 c0 c1 2 3 2 5 c0 4 15 c1 4 5 p1 x 4 15 4 5x r Numerical Analysis SK5 5 4 y arcsinx3 0 1 C g p2 x c0g0 x c1g1 x c2g2 x k E g0 x 1 0 xg0 g0 x 1 1 0 xdx 1 2 r0 g0 g0 1 1 1 0 dx 1 g1 x x 0 r0 x 1 2 1 xg1 g1 1 0 x x 1 2 2dx 1 24 r1 g1 g1 1 0 x 1 2 2dx 1 12 r Numerical Analysis SK5 b1 1 r1 1 2 c1 r1 r0 1 12 g2 x x b1 g1 x c1g0 x x 1 2 2 1 12 x2 x 1 6 g0 f 1 f 1 0 arcsinxdx 2 1 g1 f 1 0 x 1 2 arcsinxdx 1 2 8 g2 g2 1 0 x2 x 1 6 2dx 1 180 g2 f 1 0 x2 x 1 6 arcsinxdx 8 7 18 r Numerical Analysis SK5 p2 x c0g0 x c1g1 x c2g2 x g0 f g0 g0 g0 x g1 f g1 g1 g1 x g2 f g2 g2 g2 x 2 1 6 3 2 x 1 2 45 2 70 x2 x 1 6 0 685834706x2 0 601776314x 5 245574288 5 5 V4 f x sin 2x 3 m 1 1 Cng cn V4 p0 x 1 p1 x x p2 x 1 2 3x 2 1 p3 x 1 2 5x 3 3x r Numerical Analysis SK5 p2 x c0g0 x c1g1 x c2g2 x g0 f g0 g0 g0 x g1 f g1 g1 g1 x g2 f g2 g2 g2 x 2 1 6 3 2 x 1 2 45 2 70 x2 x 1 6 0 685834706x2 0 601776314x 5 245574288 5 5 V4 f x sin 2x 3 m 1 1 Cng cn V4 p0 x 1 p1 x x p2 x 1 2 3x 2 1 p3 x 1 2 5x 3 3x r Numerical Analysis SK5 d pk pk 2 2k 1 p0 p0 2 p1 p1 2 3 p2 p2 2 5 p3 p3 2 7 p0 f 1 1 sin 2 xdx 0 p1 f 1 1 x sin 2 xdx 8 2 p2 f 1 2 1 1 3x2 1 sin 2 xdx 0 p3 f 1 2 1 1 5x3 3x sin 2 xdx 48 4 2 10 r Numerical Analysis SK5 p x c0p0 x c1p1 x c2p2 x c3p3 x p0 f p0 p0 p0 x p1 f p1 p1 p1 x p2 f p2 p2 p2 x p3 f p3 p3 p3 x 2520 240 2 4 x 420 2 10 4 x3 1 553191208x 0 562228340 x3 5 6 y x3 m 1 4 1 C g p1 x c0 c1x y x y0 1 2x 1 2 y00 1 4x 3 2 0 1 4 x 1 x0 1 4 x2 1 r Numerical Analysis SK5 p x c0p0 x c1p1 x c2p2 x c3p3 x p0 f p0 p0 p0 x p1 f p1 p1 p1 x p2 f p2 p2 p2 x p3 f p3 p3 p3 x 2520 240 2 4 x 420 2 10 4 x3 1 553191208x 0 562228340 x3 5 6 y x3 m 1 4 1 C g p1 x c0 c1x y x y0 1 2x 1 2 y00 1 4x 3 2 0 1 4 x 1 x0 1 4 x2 1 r Numerical Analysis SK5 d 5 28 k x i c0 c1 xi 1 i i 0 1 2 1 2 xi c1 0 1 2 c0 1 4c1 x 1 c0 c1 x1 1 c0 c1 1 2 x 1 2 1 c1 0 c0 17 48 c2 2 3 x1 9 16 E 1 48 1 48 p1 x 17 48 2 3x r Numerical Analysis SK5 5 10 y e x3 m 1 1 Cq C g 1 T2 x 2x2 1 x0 2 2 x1 2 2 E p1 x e 2 2 e 2 2 e 2 2 2 2 2 2 x 2 2 e 2 2 1 2 e 2 2 e 2 2 x 2 2 1 2 e 2 2 e 2 2 1 2 e 2 2 e 2 2 x max 1 x 1 R1 x M2 2 1 2 e 4 0 679570457 r Numerical Analysis SK5 2 e x 1 x 1 2x 2 e 1 p2 x 1 x 1 2x 2 p1 x p2 x a2 2 T2 x 1 x 1 2x 2 1 2 2 2x 2 1 5 4 x r Numerical Analysis SK5 5 12 T k x Tk 2x 1 1 T k x k 0 1 2 3 4 2 y T k x 3 m 0 1 u x 1 x x2 3 y T k x 2 T2k x 1 T 0 x T0 2x 1 1 T 1 x T1 2x 1 2x 1 T 2 x T2 2x 1 2 2x 1 2 1 T 3 x T3 2x 1 4 2x 1 3 3 2x 1 T 4 x T4 2x 1 8 2x 1 4 8 2x 1 2 1 r Numerical Analysis SK5 2 T k T j 1 0 1 x x2Tk 2x 1 Tj 2x 1 dx 2x 1 t dx 1 2dt 2x 1 t dx 1 2dt 1 1 1 1 t2Tk t Tj t dt Tk Tj 0 k 6 j k j 0 2 k j 6 0 3 T n x 2 Tn 2x2 1 cos narccos 2x2 1 cosn arccos 2x2 1 2x2 cos 1 2cos2 2 x cos 2 2arccosx T n x2 cos 2narccosx T2n x r Numerical Analysis SK5 2 T k T j 1 0 1 x x2Tk 2x 1 Tj 2x 1 dx 2x 1 t dx 1 2dt 2x 1 t dx 1 2dt 1 1 1 1 t2Tk t Tj t dt Tk Tj 0 k 6 j k j 0 2 k j 6 0 3 T n x 2 Tn 2x2 1 cos narccos 2x2 1 cosn arccos 2x2 1 2x2 cos 1 2cos2 2 x cos 2 2arccosx T n x2 cos 2narccosx T2n x r Numerical Analysis SK6 6 2 n 8 EzF Ez O e 37 O 1 1 0 1 1 x dx h b a 8 xi a ih i 0 1 8 EzF b a f x dx h 2 f a 2 7 X i 1 f xi f b 1 0 1 1 x dx 1 16 1 2 7 X i 1 1 1 0 125i 1 2 0 69412185 r Numerical Analysis SK6 Ez b a f x dx h 6 f a 2 7 X i 1 f xi 4 8 X i 1 f xi 1 xi 2 f b 1 h 1 8 0 125 xi ih 0 125i xi 1 xi 2 2i 1 16 1 0 1 1 x dx 1 48 1 2 7 X i 1 1 1 0 125i 4 8 X i 1 1 1 2i 1 16 1 2 1 48 1 2 7 X i 1 1 1 0 125i 64 8 X i 1 1 15 2i 1 2 0 69314765 r Numerical Analysis SK6 6 4 e EzF O I 1 0 exdx m 0 1 U L1 2 10 5 dEzF R f b a 12 h2f 00 a b Kf x ex Kf 00 x ex R f 1 12h 2e 0 1 R f 1 12h 2e 1 2 10 5 h 0 004698 h 0 4 10 2 v n 1 h 1 0 4 10 2 250 m 0 1 250 K R f 1 2 10 5 r Numerical Analysis SK6 6 7 e 9 O 1 h 0 f x dx A0f 0 B0f 0 0 A1f h B1f0 h 3 1 1 x2f x dx A0f x0 A1f x1 1 f x 1 x x2 x3 A0 A1 h B0 A1h B1 1 2h 2 A1h2 2B1h 1 3h 3 A1h3 3B1h2 1 4h 4 A0 h 2 B0 h2 12 A1 h 2 B1 h2 12 K h 0 f x dx h 2f 0 h2 12f 0 0 h 2f h h2 12f 0 h f x x4 KR x4 h 0 x4dx 1 2h 5 1 3h 5 1 30h 5 6 0 r Numerical Analysis SK6 m 3 U2 n e x f 4 4 x2 x h 2 R f R e I e Q e h 0 f 4 4 x2 x h 2dx h 2 e 0 h2 12e 0 0 h 2 e h h2 12e 0 h f 4 24 h 0 x2 x h 2dx h5 720f 4 0 h 3 1 X f x 1 x x2 x3 A0 A1 2 3 A0 x0 A1x1 0 A0 x2 0 A1x2 1 2 5 A0 x3 0 A1x3 1 0 r Numerical Analysis SK6 A0 A1 1 3 x0 q 3 5 x1 q 3 5 1 1x 2f x dx 1 3f q 3 5 1 3f q 3 5 f x x4 K R x4 1 1x 6dx 1 3 q 3 5 4 q 3 5 4 2 7 6 25 8 175 6 0 m 3 e x 1 4 f 4 x q 3 5 2 x q 3 5 2 K R f R e I e Q e I e 1 1 x2 1 4 f 4 x r 3 5 2 x r 3 5 2 dx f 4 24 1 1 x2 x2 3 5 2 dx 1 12f 4 4 175 1 525f 4 1 1 r Numerical Analysis SK6 2 E u w x x2 g0 x g1 x g2 x g2 x E 5 f x CqL g0 x 1 g1 x x 0 0 0 xg0 g0 x 1 1 1 x2 x dx 0 0 g0 g0 1 1 1 1 x2dx 2 3 g1 x x g2 x x b1 g1 x c1g0 x x b1 x c1 r Numerical Analysis SK6 1 xg1 g1 x2 x 1 1 x2 x3dx 0 1 g1 g1 x x 1 1 x2 x2dx 2 5 b1 1 1 0 c1 1 0 3 5 g2 x x 2 3 5 0 g2 x x2 3 5 x0 q 3 5 x1 q 3 5 K L1 x x x1 x0 x1 f x0 x x0 x1 x0 f x1 x q 3 5 2 q 3 5 f r 3 5 x q 3 5 2 q 3 5 f r 3 5 r Numerical Analysis SK6 K 1 1 x2f x dx 1 1 x2L1 x dx 1 2 q 3 5 1 1 x2 x r 3 5 dx f r 3 5 1 2 q 3 5 1 1 x2 x r 3 5 dx f r 3 5 1 3f r 3 5 1 3f r 3 5 3 pd xi u w x 2g d 2 1 1 x2f x dx A0f r 3 5 A1f r 3 5 X X A0 A1 r Numerical Analysis SK6 f x 1 x A0 A1 2 3 r 3 5A0 r 3 5A1 0 A0 A1 1 3 1 1x 2f x dx 1 3f q 3 5 1 3f q 3 5 4 X Ai 1 g0 2 xi g1 xi 2 5 2x2 i 1 5x2 i i 0 1 A0 1 5 q 3 5 2 1 3 A1 1 5 q 3 5 2 1 3 r Numerical Analysis SK6 1 1 x2f x dx 1 3f r 3 5 1 3f r 3 5 2 g2 g2 1 1x 2 x2 3 5 2dx 8 175 R f 2 4 f 4 1 525f 4 1 1 5 pd V4 1 1 x2f x dx A0 x2 0f x0 A1x 2 1f x1 r Numerical Analysis SK6 6 9 y y x XeL xi0 60 80 911 11 21 4 yi0 73600 83650 909511 11051 24461 6017 n O y0 1 y00 1 f 0 x1 f x2 f x0 2h h2 6 f 00 Kg0 1 f 1 1 f 0 9 0 2 1 1105 0 9095 0 2 1 005 f 00 x1 f x0 2f x1 f x2 h2 h2 12f 4 Ky00 1 f 0 9 2f 1 f 1 1 0 01 0 9095 2 1 1105 0 01 2 r Numerical Analysis SK6 6 10 e X L 2 f 0 h af0 0 b f 2h f h R f f 0 h af0 0 b f 2h f h O f 1 x x2 R xk 0 k 0 1 2 K 1 a bh 0 2 3bh 0 a 1 3 b 2 3h u f 0 h 1 3f 0 0 2 3h f 2h f h r Numerical Analysis SK6 R x3 3h2 2 3h 8h 3 h3 5 3h 2 6 0 m 2 e x f000 3 x x h 2 R f R e e0 h 1 3e 0 0 2 3h e 2h e h 1 3e 0 0 2 3he 2h 1 3 f000 3 h2 2 3h f000 3 2h3 5 18h 2f000 r Numerical Analysis SK7 7 3 3 m a b 0 m f 0 x M y 0 2 M x0 a b dS xk 1 xk f xk k 0 1 2 S xk u f x 0 x x x f x 0 x 1 f 0 x 0 m f 0 x M 0 2 M 0 f 0 x 2 2 f0 x 0 1 1 f 0 x 1 1 f0 x 1 x a b x a b x 1 f 0 x 1 S xk ux K lim k xk 1 lim k xk f xk x x f x f x 0 r Numerical Analysis SK7 7 4 S g y lim k s 2 r 2 q 2 2 z m ng 2 x0 2 x 1 p 2 2 2 x 0 x2 q 2 p 2 2 2 x 1 xk 1 2 x k x x x 2 x 0 x 1 2 2 x 1 2 1 S x x f x f0 x x x x mg x m x x m 1g x x x mg0 x x x x g x mg x x x g0 x 0 x 1 g x x x g0 x mg x x x g0 x mg0 x g0 x x x g00 x x x g x mg x x x g0 x 2 0 x 1 mg2 x m2g2 x 1 1 m 6 0 m 1 m 1 m 1 r Numerical Analysis SK7 2 xk 1 xk m f xk f0 xk k 0 1 S x 1 m f x f0 x 1 m x x mg x m x x m 1g x x x mg0 x 1 n 0 x 1 m mg2 x m2g2 x 1 1 0 3 xk 1 xk u xk u0 xk k 0 1 S x x u x u0 x x f x f0 x f0 x 2 f x f00 x f0 x 2 x f x f0 x f0 x 2 f x f00 x 0 x 1 f0 x 2 f x f00 x f0 x 2 f x f00 x f x f0 x 2f0 x f00 x f0 x f00 x f x f000 x f0 x 2 f x f00 x 2 0 x 1 f0 x 4 f0 x 4 1 1 0 r Numerical Analysis SK7 7 14 S pd m S 2x cosy 0 2y sinx 0 x 1 y 1 O dC x 1 2cosy y 1 2sinx 1 S xk 1 1 2cosyk yk 1 1 2sinxk k 0 1 x0 y0 0 5 U S 2 pd m S xk 1 1 2cosyk yk 1 1 2sinxk 1 k 0 1 2 r Numerical Analysis SK7 7 15 Xe x1 2x2 3 0 2x2 1 x2 2 5 0 3x1 1 x2 2NC O f x f1 x f2 x T x1 2x2 3 2x2 1 x2 2 5 T K