现代数值数学和计算习题答案(同济版)

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000000000000000e 000 MatLab x a h b x a 0 n h a b n 0 1 10 function ex15 format long e a 0 b 8 n 9 h b a n x 1 a y 1 a for j 1 n x j 1 x j h y j 1 y 1 j h end x y a h b a 0 n h 6 A A A u v sort abs diag A A A v v 3 A sort A MatLab A n size A 1 fori 1 n 1 forj i 1 n ifabs A i i abs A j j A i j A j i A i j A j i end end end A 7 x x1 x2 xn Euclide kxk n X i 1 x2 i 1 2 n X i 1 x2 i 1 2 v u u t n 1 X i 1 x2 i 2 x2 n 1 2 MatLab kxk kx 1 n 1 k x n x x1 x2 1 x1 1 x2 x1 x1 x1 4 functions nrmv x x abs x x 0 n length x ifn 0 s 0 elseifn 1 s abs x else scale 0 ssq 1 for j 1 n if scale x j ssq 1 ssq scale x j 2 scale x j else ssq ssq x j scale 2 end end s scale sqrt ssq end BLAS dnrm2 MatLab lib org blas dnrm2 f 8 ax2 bx x 0 x b b2 4ac 2a x 2c b b2 4ac a 0 c 0 abc 610 4 6e 15410e 154 4e 154 011 1 1e 51 1 43 999999 1e155 7e1551e 155 5 b2 4ac 1 c 0 b Vieta a 0 b 0 function x flag root2qe a b c if nargin 1 c a 3 b a 2 a a 1 end if a 0 Solve 2 order equation M max abs a b c scaling a a M b b M c c M s sign b if s 0 s 1 end x1 b s sqrt b 2 4 a c 2 a x2 c a x1 avoid large error x x1 x2 flag two solutions elseif b 0 Solve 1 order equation x c b flag one solution elseif c 0 nonzero constant x flag no x is a solution else equality 0 0 x 1 flag all x are solutions end 6 abcx1x2 610 4 2 000000000000000 33333333333333 6e 15410e 154 4e 154 2 000000000000000 33333333333333 011 1 1 1e 5199999 99999000000 0000100000 1 43 9999992 001000000000071 99899999999993 1e155 7e1551e 155 7 140054944640260 14005494464026 9 x1 1 3 x2 1 12 xk 1 2 25xk 0 5xk 1 xk 41 k 3 MatLab x 1 1 3 x 2 1 12 for k 3 200 x k 2 25 x k 1 0 5 x k 2 end 60 plot x 1 60 200 c1 c2 0102030405060 0 0 2 0 4 0 6 0 8 1 1 2 1 4 1 6 1 8 xk c14 k c22k c1 c2 c14 1 c22 1 3 c14 2 c222 1 12 7 c1 4 3 c2 0 1 3 1 12 xk c14 k c22k c1 4 3 4 3 c2 k c22k k c22k xk c14 k 10 64 1 52 1 10 2p 0 1 2 3 52 2 p 210 1 p Z j 0 1 Z 1 1 10 x fl x er x fl x fl x 2p 1 52 2p 1 2 52 10 15 6 15 16 22 10 1 10308 2 2 10 1 10 308 10308 10 308 1 2 9 xsinxcosx 0 700 64421768720 7648421872 0 710 65183377100 7583618759 2 9 sin0 705 cos0 702 sin0 705 L1 x x 0 70 0 71 0 70 0 6518337710 x 0 71 0 70 0 71 0 6442176872 0 76160838x 0 1110918212 8 x 0 705 sin0 705 L1 0 705 0 6480257291 cos0 702 L1 x x 0 70 0 71 0 70 0 7583618759 x 0 71 0 70 0 71 0 7648421872 0 64803113x 1 2184639782 x 0 702 cos0 702 L1 0 702 0 76354612494 2 2 10 x0 40 50 60 7 sinx0 389420 479430 564640 64422 2 10 sin0 57891 x0 0 5 x1 0 6 L1 x x 0 5 0 6 0 5 0 56464 x 0 6 0 5 0 6 0 47943 0 8521x 0 05338 x 0 57891 sin0 57891 L1 0 57891 0 54667 x0 0 5 x1 0 6 x3 0 7 sin0 57891 sin0 57891 L2 0 57891 0 57891 0 6 0 57891 0 7 0 5 0 6 0 5 0 7 0 47943 0 57891 0 5 0 57891 0 7 0 6 0 5 0 6 0 7 0 56464 0 57891 0 5 0 57891 0 6 0 7 0 5 0 7 0 6 0 64422 0 54714 9 x0 0 4 x1 0 5 x3 0 6 sin0 57891 sin0 57891 L2 0 57891 0 57891 0 5 0 57891 0 6 0 4 0 5 0 4 0 6 0 38942 0 57891 0 4 0 57891 0 6 0 5 0 4 0 5 0 6 0 47943 0 57891 0 4 0 57891 0 5 0 6 0 4 0 6 0 5 0 56464 0 54707 3 2 11 x01246 f x 19233259 2 11 1 2 f 4 2 1 xkf xk 01 8 193 14 2 75 223 81 875 108 5 4334 5 128 6259 10 2 N x 1 8x 3x x 1 2 75x x 1 x 2 1 875x x 1 x 2 x 4 x 4 2 f 4 2 N 4 2 4 696 4 p x x4 x3 x2 x 1 2 12 x 2 10123 p x 315111161 2 12 q x 2 13 x 2 10123 q x 315111130 2 13 2 12 2 13 p x q x x 0 1 2 p x q x Ax x 1 x 1 x 2 x 2 x 3 A 31 120 q x p x 31 120 x x 1 x 1 x 2 x 2 5 2 14 x 2 10123 y14111613 4 2 14 11 x 1 0 1 2 L x 4 7 x 1 x x 1 x x 1 x 1 x 2 3 y L x x 2 3 y L x y 1 4 xkf xk 21 3 142 7 1 011 1 5 1 116 4 3 1 213 7 17 3 4 6 20 0 2 e x R20 x f 21 21 20 Y k 0 x xk f x e x f 21 x e x 0 2 f 21 e 1 12 x0 x1 x20 k 6 10 max x x xk x x20 k 1 k 10 x x10 1 R20 x 1 21 7 1 2 10 f x lnx x x0 th x0 1 h 1 9 0 t 9 R x R x0 th f 10 10 h10 9 Y k 0 t k R t 1 2 f x lnx f 10 x 9 x10 f 10 9 R t x 1 2 3 4 5 y 2 48 272 1182 2262 p q chashang x y p 12468964 24822459220 3272 13012500 41182120000 522620000 q 2468964 p x 2 46 x 1 89 x 1 x 2 6 x 1 x 2 x 3 4 x 1 x 2 x 3 x 4 p x 4 x 4 6 x 3 89 x 2 46 x 1 2 3 5 5 21 f x x2 1 1 20 p x f x p x f x g x 18 function y ex217 n 21 x linspace 5 5 n h 5 5 n 1 y 1 1 x 2 form the differences table for j 2 n y 1 n 1 j j diff y 1 n 2 j j 1 x j n x 1 n 1 j end newton coeff y y 1 pz v linspace 5 5 80 for t v z y n for j n 1 1 1 z z t x j y j end pz pz z end plot v pz r v 1 1 v 2 g 4 4 5 4 3 2 1012345 60 50 40 30 20 10 0 10 4 3 2 101234 0 6 0 4 0 2 0 0 2 0 4 0 6 0 8 1 2 5 2 5 4 8 60 4 3 xi 5cos i 20 0 i 20 f x x2 1 1 20 q x f x q x 3 4 x linspace 5 5 n 19 x 5 cos pi n 1 0 n 1 5 4 3 2 1012345 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 3 4 5 f x x2 1 1 5 5 41 s x f x s x f0 x 2x 1 x2 2 f0 1 0 5 5 4 3 2 1012345 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 function ex219 x linspace 5 5 41 y 1 1 x 2 y 0 5 y 0 5 pp csape x y complete v linspace 5 5 100 f 1 1 v 2 s ppval pp v plot v f b v s g 20 1 1 1 1 f x x2 3x 5 2 f x ex 1 f00 x 2 0 x1 1 x3 1 f x1 7 f x3 1 f0 x2 2x2 3 1 7 1 1 3 x2 0 f x2 5 x1 f x1 x2 f x2 x y x 1 0 2 1 2 y 7 5 2 6 y 6 3 x 1 2 p1 x 3 x 1 2 6 2 f00 x ex 0 x1 1 x3 1 f x1 0 36788 f x3 2 71828 f0 x2 ex2 2 71828 0 36788 1 1 1 17520 x2 0 16144 f x2 1 17520 x1 f x1 x2 f x2 x y x 1 0 16144 2 0 41928 y 0 36788 1 17520 2 0 77154 y 0 77154 1 17520 x 0 41928 p1 x 1 17520 x 1 26567 1 0 8 0 6 0 4 0 200 20 40 60 81 8 7 6 5 4 3 2 1 x1 x3 x2 y x2 3 x 5 1 0 8 0 6 0 4 0 200 20 40 60 81 0 0 5 1 1 5 2 2 5 3 y exp x x1 x3 x2 x2 3x 5 ex 1 1 2 M m f x a b f x a b p x 1 2 M m x1 x2 a b f x1 m f x2 M f x1 p x1 1 2 M m f x2 p x2 1 2 M m 21 x a b f x p x 1 2 M m x1 x2 p x f x Chebyshev p x f x 3 f x 1 x2 0 1 f00 x 1 1 x2 3 2 0 x1 0 x3 1 f x1 1 f x3 2 f0 x2 x2 1 x2 2 2 1 1 0 0 4142 x2 0 45509 f x2 1 09868 x1 f x1 x2 f x2 x y x 0 0 45509 2 0 227545 y 1 1 09868 2 1 04934 y 1 04934 0 4142 x 0 227545 p1 x 0 4142x 0 95514 y a bx a b T 1 1 1 x x 1 x x a b 1 f x f f g R1 0 f x g x dx 1 00000 5000 0 50000 3333 a b 1 1478 0 4269 a 0 9343 b 0 4269 p1 x 0 4269x 0 9343 4 f x sin x 0 1 p2 x a bx cx2 f g R 1 0 f x g x dx a b c 1 1 1 x 1 x2 x 1 x x x x2 x2 1 x2 x x2 x2 a b c 1 sin x x sin x x2 sin x 22 11 21 3 1 21 31 4 1 31 41 5 a b c 2 1 2 4 3 a 12 2 20 3 0 050465 b 60 2 720 3 4 1225 c b 4 1225 p2 x 0 050465 4 1225x 4 1225x2 5 y ax2 b 3 7 x 101 y3 10 92 9 3 7 P2 i 01 P2 i 0 x 2 i P2 i 0 x 2 i P2 i 0 x 4 i b a P2 i 0yi P2 i 0 x 2 iyi 32 22 b a 6 9 6 a 2 1 b 0 9 y 2 1x2 0 9 6 xi yi x y x y xi yi y ax b xi yi a b f a b X i yi axi b 2 f b 0 2 P i yi axi b 0 n 2 y ax b 0 x y 23 7 V V a bT cT 2 3 8 a b c T1234567 V2 312 211 801 661 551 471 41 3 8 P7 i 11 P7 i 1Ti P7 i 1T 2 i P7 i 1Ti P7 i 1T 2 i P7 i 1T 3 i P7 i 1T 2 i P7 i 1T 3 i P7 i 1T 4 i a b c P7 i 1Vi P7 i 1TiVi P7 i 1T 2 iVi 728140 28140784 1407844676 a b c 12 41 45 21 214 67 a b c 2 6786 0 3401 0 0227 a b c 8 asin x bcos x x 1 1 2 0 1 2 1 y 10121 3 9 P5 i 1sin 2 xi P5 i 1sin xicos xi P5 i 1sin xicos xi P5 i 1cos 2 xi a b P5 i 1yisin xi P5 i 1yicos xi 20 03 a b 2 1 a b 1 1 3 24 1 f x a bx 1 cx f x 1 cx a bx 19535 82 19646 95 198210 08 199011 34 200012 66 f x 1 cx a bx a bx cxf x f x 1 x xf x xi f xi function a b c ex39 x 1953 1964 198219902000 y 5 82 6 95 10 08 11 34 12 66 A ones 5 1 x x y z A y a z 1 b z 2 c z 3 v linspace 1953 2000 100 plot x y b v a b v 1 c v k a 2 9456 b 0 0014 c 0 00049560 2 s T s aT b 3 11 a b T010203040809095 s68 067 166 465 664 661 861 060 0 3 11 25 function a b ex310 T 0 10 20 30 40 80 90 95 s 68 0 67 1 66 4 65 6 64 6 61 8 61 0 60 0 z T ones 8 1 s a z 1 b z 2 v linspace 0 95 100 plot T s b v a v b k a 0 0799 b 67 9593 1 1 R 2h 0 f x dx A0f 0 A1f h A2f 2h 2 R 1 1f x dx A f 1 2f x1 3f x2 3 R 2 0 f x dx A0f 0 4 3f x1 A2f x2 1 Ai i 0 1 2 f x 1 x x2 A0 A1 A2 2h hA1 2hA2 2h2 h2A1 4h2A2 8 3h 3 A0 A2 1 3h A1 4 3h Z 2h 0 f x dx 1 3hf 0 4 3hf h 1 3hf 2h f x x3 2h 4 4 4h4 4 3h 4 8 3h 4 4h4 f x x4 2h 5 5 32 5 h5 4 3h 5 16 3 h5 20 3 h5 3 26 2 f x 1 x x2 6A 2 A 1 2x1 3x2 0 A 1 2x2 1 3x 2 2 2 3 A 1 3 x1 1 5 1 6 x 2 1 15 3 2 6 f x x3 0 1 3 1 2 1 6 5 3 3 3 2 6 15 3 1 3 1 2 153 27 81 6 81 6 27 6 6 3 153 27 54 6 36 6 8 6 6 6 0 2 3 f x 1 x x2 x3 A0 4 3 A2 2 4 3x1 A2x2 2 4 3x 2 1 A2x 2 2 8 3 4 3x 3 1 A2x 3 2 4 x2 4 3x1 x1 x2 8 3 2x2 x2 4 3x 2 1 x1 x2 4 8 3x2 4 3x1 x1 x2 8 3 2x2 4 3x 2 1 x1 x2 4 8 3x2 27 4 8 3x2 8 3 2x2 x1 x1x2 4 3 x1 x2 2 x1x2 x2 8x1 6x2 1 x1x2 4 3 x1 x2 2 3x3 1 8x 2 1 6x1 1 0 x1 1 x1 1 6 5 13 x2 2 x2 1 3 1 13 A0 1 3 A0 1 6 1 13 A2 1 3 A2 1 6 3 13 A0 A2 x1 x2 1 3 1 3 1 2 1 6 1 13 3 13 5 13 2 2 13 3 2 1 Z 1 0 exdx n 4 2 Z 1 1 x 1 5dx n 2 1 n 4 T4 1 8 h f 0 2f 1 4 2f 1 2 2f 3 4 f 1 i 1 727221905 n 4 S4 1 24 h f 0 4f 1 8 4f 3 8 4f 5 8 4f 7 8 2f 1 4 2f 1 2 2f 3 4 f 1 i 1 718284155 I e 1 1 7182818284590 0 008940076 0 000002326 28 2 n 2 T2 1 2 f 1 2f 0 f 1 2 368867677 n 2 S2 1 6 f 1 4f 1 2 4f 1 2 2f 0 f 1 2 399098267 I 1 6 5 10 2 2 399529123 0 030661446 0 000430855 3 Z 3 1 1 x dx 1 2 3 3 1 1 3 1 1 x t 2 Z 3 1 dx x Z 1 1 1 t 2 dt Z 1 1 1 t 2 dt 0 5555556 1 2 0 7745967 1 2 0 7745967 0 8888889 1 2 1 098039283 Z 1 1 1 t 2 dt 0 2369269 1 2 0 9061798 1 2 0 9061798 0 4786289 1 2 0 5384693 1 2 0 5384693 0 5688889 1 2 1 098609289 29 2 1 3 1 1 5 1 5 2 2 2 5 2 5 3 Ii i 1 2 3 4 I1 Z 1 5 1 1 x dx Z 1 1 0 5 2 5 0 5t dt 0 5 2 5 0 5 1 3 0 5 2 5 0 5 1 3 0 405405405 I2 Z 2 1 5 1 x dx Z 1 1 0 5 3 5 0 5t dt 0 5 3 5 0 5 1 3 0 5 3 5 0 5 1 3 0 287671232 I3 Z 2 5 2 1 x dx Z 1 1 0 5 4 5 0 5t dt 0 5 4 5 0 5 1 3 0 5 4 5 0 5 1 3 0 223140495 I4 Z 3 2 5 1 x dx Z 1 1 0 5 5 5 0 5t dt 0 5 5 5 0 5 1 3 0 5 5 5 0 5 1 3 0 182320441 I I1 I2 I3 I4 1 098537573 I ln3 1 098612289 3 f x 1 x a 1 b 3 n 3 3 T2kS2k 1C2k 2 1 33333333 1 166666671 11111111 1 116666671 100000001 09925926 R3 3 1 09925926 ln3 R3 3 6 469714 10 4 4 1 R b a f x dx b a f a 1 2f 0 b a 2 2 R b a f x dx b a f b 1 2f 0 b a 2 3 R b a f x dx b a f a b 2 1 24f 00 b a 3 30 f x x a f x f a f0 x a a b Z b a f x dx Z b a f a dx Z b a f0 x a dx b a f a f0 Z b a x a dx b a f a 1 2f 0 b a 2 a b R b a f x dx R b a f a b y dy 1 2 3 f x x 1 2 a b f x f a b 2 f0 a b 2 x a b 2 1 2f 00 x a b 2 2 a b Z b a f x dx b a f a b 2 f0 a b 2 Zb a x a b 2 dx 1 2 Z b a f00 x a b 2 2 dx b a f a b 2 1 24f 00 b a 3 a b 5 f x C6 1 1 p x f x 5 p xi f xi p0 xi f0 xi xi 1 0 1 1 Z 1 1 p x dx 7 15f 1 16 15f 0 7 15f 1 1 15f 0 1 1 15f 0 1 31 2 Z 1 1 f x dx 7 15f 1 16 15f 0 7 15f 1 1 15f 0 1 1 15f 0 1 5 3 a b xi a ih i 0 1 2n h b a 2n 2 1 2 41 x0 1 x1 0 x2 1 p x 1 2l0 0 1 x 1 l 2 0 x f 1 1 2l 0 1 0 x 0 l 2 1 x f 0 1 2l0 2 1 x 1 l 2 2 x f 1 x 1 l 2 0 x f 0 1 x 0 l2 1 x f 0 0 x 1 l2 2 x f 0 1 l0 x 1 2x x 1 l1 x x 1 x 1 l2 x 1 2x x 1 l0 0 1 3 2 l0 1 0 0 l0 2 1 3 2 2 2 42 n 2 p x H5 x R5 x f x p x f 6 6 x2 x 1 2 x 1 2 5 R f Z 1 1 f x dx Z 1 1 p x dx Z 1 1 f 6 6 x2 x 1 2 x 1 2dx x2 x 1 2 x 1 2 1 1 R f 1 4725f 6 1 1 32 3 I Z b a f x dx n 1 X i 0 Z x2i 2 x2i f x dx Z x2i 2 x2i f x dx t 1 h x x2i 1 1 1 Z x2i 2 x2i f x dx h Z 1 1 f x2i 1 ht dt h 7 15f x2i 16 15f x2i 1 7 15f x2i 2 1 15hf 0 x2i 1 15hf 0 x2i 2 I n 1 X i 0 Z x2i 2 x2i f x dx n 1 X i 0 7 15hf x2i 16 15hf x2i 1 7 15hf x2i 2 1 15h 2f0 x2i 1 15h 2f0 x2i 2 i 7 15h f a 2Pn 1 i 1 f x2i f b 16 15h Pn 1 i 0 f x2i 1 1 15h 2 f0 a f0 b R n 1 X i 0 h7 4725f 6 i b a 9450 h6f 6 6 n 2 n 4 I Z 1 1 dx 1 x4 I 2 62205755429213 x sint I R 1 1 dx 1 x4 2 R1 0 dx 1 x4 2 R 2 0 costdt 1 sin4t 2 R 2 0 dt 1 sin2t 33 I 2 2 6 1 1 sin20 4 1 sin 2 4 1 1 sin 2 2 2 603905 f x 1 1 x2 4 49 n 4 i 5 xi cos 2i 1 10 Z 1 1 dx 1 x4 Z 1 1 f x 1 x2dx 4 X i 0 if xi 2 622250 7 e x 2 Z 1 0 e x 2 dx 10 4 e x 2 1 x2 x4 2 x6 3 1 n x2n n Z 1 0 e x 2 dx 1 1 3 1 10 1 42 1 216 1 1320 1 9360 0 7468 R7 1 15 7 1 75600 x linspace 0 1 5 y linspace 1 2 5 w 1 4 2 4 1 12 X Y meshgrid x y w sin X 2 Y 2 1 0 5 X 0 5 Y w ans 1 846944068968949e 001 13 f x 1 1 x 2 4 13 x1 01 11 2 f x 0 25000 22680 2066 4 13 38 f0 x x 1 0 1 1 1 2 f0 x0 1 2h 3f x0 4f x1 f x2 h2 3 f000 0 f0 x1 1 2h f x0 f x2 h2 6 f000 1 f0 x2 1 2h f x0 4f x1 3f x2 h2 3 f000 2 f000 max 1 0 1 2 4 1 5 0 75 x1 01 11 2 0 24792 0 21694 0 18596 0 25000 0 21596 0 18783 0 002500 001250 00250 0 002080 000980 00187 14 f00 x1 1 h2 f x0 2f x1 f x2 O h2 f x0 f x1 h f x1 hf0 x1 h2 2 f00 x1 h3 3 f000 x1 h4 4 f 4 1 f x2 f x1 h f x1 hf0 x1 h2 2 f00 x1 h3 3 f000 x1 h4 4 f 4 2 1 h2 f x0 2f x1 f x2 f00 x1 h2 4 f 4 1 f 4 2 O h2 39 1 4 Z 1 0 1 1 x2 dx 10 8 1 2 3 4 45 function p1 p2 p3 ex415 a 0 b 1 m 2 t 1 0 5 b a f a f b t 2 0 5 t 1 0 5 b a f a b 2 s 1 4 t 2 t 1 3 j 2 t s c r store the romberg integrals in which s stores the simpson integrals while abs t j t j 1 0 5e 8 4 h b a m k 0 m 1 j j 1 t j 0 5 t j 1 0 5 h sum f a k 1 2 h s j 1 4 t j t j 1 3 c j 2 16 s j 1 s j 2 15 ifj 3 r j 3 64 c j 2 c j 3 63 end m m 2 end p1 4 s end p2 4 r end compute the compound Guass integrals w 5 8 5 9 x sqrt 15 0 sqrt 15 5 n 5 p3 0 40 for j 0 n 1 g j 0 5 n 0 5 n x guass nodes p3 p3 f g w 2 n guass int in each piece end p3 p3 4 function v f x v 1 1 x 2 p1 p2 p3 i zi n 2 i 5 9 8 9 5 9 zi 15 9 0 15 9 a b t 2 b a x a b 2 Z b a f x dx Z 1 1 f b a 2 t a b 2 dt n X i 0 if b a 2 zi a b 2 a b m xi a ih h b a m Rb a f x dx m 1 X i 0 Z xi 1 xi f x dx m 1 X i 0 n X j 0 jf h 2 zj xi h 2 2 4 14 1 8 0 5 5 xi12345678 f xi 1223434 1342523 4 14 x 1 8 y 12 234 34 1 34 2 5 23 p polyfit x y 5 q polyder p polyval q 1 0 5 8 41 11 522 533 544 5 645 72159 71 84 71 165 55 148 00 84 46 14 51835 030 55 566 577 58 50 19029 768 14 629 58 595 64 91916 405248 186 1 6x1 2x2 2x3 4x4 16 12x1 8x2 6x3 10 x4 26 3x1 13x2 9x3 3x4 19 6x1 4x2 x3 18x4 34 m21 2 m32 1 2 m41 1 6x1 2x2 2x3 4x4 16 4x2 2x3 2x4 6 12x2 8x3 x4 27 2x2 3x3 14x4 18 m32 3 m42 1 2 6x1 2x2 2x3 4x4 16 4x2 2x3 2x4 6 2x3 5x4 9 4x3 13x4 21 m43 2 6x1 2x2 2x3 4x4 16 4x2 2x3 2x4 6 2x3 5x4 9 3x4 3 x4 1 x3 2 x2 1 x1 3 x 3 1 2 1 T 42 2 x1 3x2 2x3 4x4 3 2x1 6x2 7x3 10 x4 2 x1 x2 5x3 9x4 14 3x1 5x2 15x4 6 3 3x1 5x2 15x4 6 8 3x2 7x3 6 2 3x2 5x3 4x4 16 4 3x2 2x3 x4 1 3x1 5x2 15x4 6 8 3x2 7x3 6 27 4 x3 4x4 35 2 3 2x3 x4 4 3x1 5x2 15x4 6 8 3x2 7x3 6 27 4 x3 4x4 35 2 1 9x4 1 9 x 2 3 2 1 T 3 A A LU 57910 68109 71087 5765 x1 x2 x3 x4 1 1 1 1 43 A LU 5 2 L 1 1 21 1 4 0 51 100 61 U 57910 0 4 0 8 3 5 8 5 0 1 Ly b b y 1 0 2 0 5 0 3 T Ux y x 20 12 5 3 T 4 5 1 15 1 15 1 15 1 15 x1 x2 x3 x4 x5 3 6 8 14 35 A LLT 5 4 L 2 2361 0 44722 1909 0 45642 1890 0 45682 1889 0 45682 1889 Ly b LTx y y 1 3416 3 0125 4 2828 7 2897 17 5112 T x 1 0000 2 0000 3 0000 5 0000 8 0000 T 44 5 21 121 121 121 12 x1 x2 x3 x4 x5 1 0 0 0 1 5 5 l1 b1 2 y1 d1 l1 1 2 u1 c1 l1 1 2 l2 b2 a2u1 3 2 y2 d2 a2y1 l2 1 3 u2 c2 l2 2 3 l3 b3 a3u2 4 3 y3 d3 a3y2 l3 1 4 u3 c3 l3 3 4 l4 b4 a4u3 5 4 y4 d4 a4y3 l4 1 5 u4 c4 l4 4 5 l5 b5 a5u4 6 5 y5 d5 a5y4 l5 1 x5 y5 1 x4 y4 u4x5 1 x3 y3 u3x4 1 x2 y2 u2x3 1 x1 y1 u1x2 1 x 1 1 1 1 1 T 6 x1 3x2 2x3 x4 7 4x1 2x2 x3 2x4 9 2x1 x2 2x3 3x4 8 x1 2x2 4x3 x4 8 l 1 2 3 4 3 4 3 4 ali 1 Sli i 1 2 3 4 1 3 4 4 2 3 1 4 45 l l 2 1 3 4 0 5 2 7 4 1 2 4212 00 3 2 2 0 3 2 15 4 1 2 x1 x2 x3 x4 19 4 9 7 2 23 4 1 3 4 a li 2 Sli i 2 3 4 5 6 0 3 8 l l 2 1 3 4 0 5 2 7 4 1 2 4212 00 3 2 2 00 27 10 1 5 x1 x2 x3 x4 19 4 9 7 2 29 10 3 4 ali 3 Sli i 3 4 1 2 27 40 l l 2 1 4 3 0 5 2 7 4 1 2 4212 000 17 9 00 27 10 1 5 x1 x2 x3 x4 19 4 9 17 9 29 10 l 2 1 4 3 x4 x3 x2 x1 x4 1 x3 1 x2 1 x1 1 46 x 1 1 1 1 T 7 A 1 1 1 2 2 2 3 3 n 2 n 1 n 1 n 1 n A L LT A L L 1 1 2 2 3 n 1 n 1 1 1 1 1 2 p 2 2 1 2 2 2 3 p 3 2 2 n 1 n 1 n 1 n p n 2 n 1 1 1 i 1 2 n 1 i i i i 1 p i 1 2 i 47 8 Hessenberg A a11a12 a1n a21a22 a2n a32a3n an n 1ann ai1 1 max 1 i 2 a 1 i1 i1 1 i1 2 1 2 2 a 1 11 a 1 12 a 1 13 a 1 1n 0a 2 22 a 2 23 a 2 2n a 1 32 a 1 33 a 1 3n a 1 n n 1 a 1 nn a 2 i2 2 max 2 i 3 a 2 i2 a 2 32 a 1 32 i2 2 i2 3 2 3 3 a 1 11 a 1 12 a 1 13 a 1 14 a 1 1n 0a 2 22 a 2 23 a 2 24 a 2 2n 0a 3 33 a 3 34 a 3 3n a 2 43 a 2 44 a 2 4n a 2 n n 1 a 2 nn k a k ik k max k i k 1 a k ik a k k 1 k a k 1 k 1 k ik k ik k 1 k k 1 k 1 n 1 Hessenberg A 48 1 10 x1 4x2 4x3 14 4x1 10 x2 8x3 12 4x1 8x2 10 x3 14 1 Jacobi GS SOR 1 35 2 3 kx k 1 x k k 1 GS 4 n x kxk kxk1 nkxk kxk kxk2 n kxk 1 n kxk1 kxk2 kxk1 kxk max 1 i n xi xi0 i0 51 kxk1 n X i 1 xi kxk2 v u u t n X i 1 xi 2 kxk xi0 kxk1 n xi0 nkxk kxk2 xi0 2 kxk2 2 n xi0 2 nkxk2 kxk kxk2 n kxk kxk2 1 n X i 1 1 xi 2 n n X i 1 xi 2 1 n kxk1 kxk2 n X i 1 xi 2 n X i 1 xi 2 kxk2 kxk1 1 n kxk1 kxk2 kxk1 5 kAk 1 k I A 1k 1 1 kAk I A I A I A x 0 x 6 0 x Ax 52 kxk kAxk kAk kxk kxk 6 0 kAk 1 I A 1 I A I I A 1 I I A 1A k I A 1k 1 k I A 1k kAk k I A 1k 1 1 kAk 6 Ax b A n A 0 A x k 1 x k b Ax k k 0 1 0 2 x k 1 I A x k b B I A A n 0 1 2 n B 1 1 1 2 1 n 0 1 2 1 i 1 2 n 0 0 1 i 1 i 1 2 n i 1 2 n 1 i 1 B 1 53 7 A 1aa a1a aa1 0 5 a 1 0 5 a 0 5 Jacobi Ax b A D1 1 D2 1 a2 D3 det A a 1 2 2a 1 0 5 a 0 D3 0 0 5 a 1 0 5 a 0 5 A 2D A a a Jacobi 0 5 a 0 5 A Jacobi 8 GS x k x k 1 k 1 x 1 x 0 max i n X j i bij 1 i 1 X j 1 bij bij GS GS x k 1 Bx k g B L U L B U B x k 1 Lx k 1 Ux k g x k Lx k Ux k 1 g x k 1 x k L x k 1 x k U x k x k 1 54 x k 1 i0 x k i0 kx k 1 x k k kx k 1 x k k x k 1 i0 x k i0 i0 1 X j 1 bi0j x k 1 j x k j n X j i0 bi0j x k j x k 1 j li0 kx k 1 x k k ui0 kx k x k 1 k li0 i0 1 X j 1 bi0j ui0 n X j i0 bi0j kx k 1 x k k ui0 1 li0 kx k x k 1 k kx k x k 1 k kx k 1 x k 2 k k 1 kx 1 x 0 k 9 63 32 x1 x2 0 1 x 0 0 0 T x 4 k 0 1 2 r k b Ax k k r k r k Ar k r k x k 1 x k kr k x 1 0 1 2 x 2 1 4 1 2 x 3 1 4 7 8 x 4 7 16 7 8 55 10 4 10 100 14 10 10 0 1400 1 1004 10 0 10 14 1 00 10 14 x1 x2 x3 x4 x5 x6 2 1 2 2 1 2 x 0 0 0