(Jacobi Gauss-Seidel SOR ) 1. (Theory of Iteration Method) Jacobi Gauss-Seidel SOR 2. Jacobi (Jacobi s Iteration Method) Jacobi 3. Gauss-Seide
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1 03 9 (Jacobi Gauss-Seidel SOR (Theory of Iteration Method Jacobi Gauss-Seidel SOR Jacobi (Jacobi s Iteration Method Jacobi 3 Gauss-Seidel (Gauss-Seidel Method Gauss-Seidel 4 SOR (SOR Method SOR 9 Ax = b (9 x k+ x k+ := Mx k + c M M n (R, c R n (9 (iteration method (9 x = Mx + c = Ax = b (93
2 04 9 (9 x k+ := x k + (M Ix k + c (94 (9 {x 0, x, } 9 x 0 R n (9 {x i } 0 M ( ( lim n x k = x x = Mx + c x k = Mx k + c } x x k = M(x x k = = M k (x x 0 x x k M k x x 0 M < ( M < x k = Mx k + c = M(Mx k + c + c = k = M k x 0 + ( M i c lim k x k = (I M c i= ( Banach x x k M x x k (95 M (9 M c A
3 9 Jacobi 05 9 Jacobi Jacobi Jacobi Jacobi 3 (Jacobi x 0 R n for k =,, x (k+ := x (k + a (b n = a x (k x (k+ := x (k + a (b n = a x (k x n (k+ := x n (k + (b n n = a n x (k (9 x k+ := Jx k + c (96 J = 0 a a a n a a a 0 a n,n a n,n a n a n,n 0, c = b a b a b n J M n (R p J (λ p J (λ = J λi λ a a a n a = a a λ a n a a n a n λ a λ a a n = ( n a a a a a λ a n nn a n a n λ (97 a ii 0 (i =,,, n a λ a a n p J (λ = a a λ a n a n a n λ (98
4 06 9 Jacobi Gerschgorin 9 (Gerschgorin A = [a i ] (i, =,,, n C a ii, n =, i a i S i (i =,,, n A λ n i= S i A λ a ii λ n =, i i {,,, n} J λ λ max i max n i=, i n =, i a i a ii a i a i a ii (99 (90 M M n max a i i (9 =, i a ii [ Varga ] n(n + n(n + = n(n + k kn(n + kn(n + 9 LU Jacobi n Jacobi
5 93 Gauss-Seidel Gauss-Seidel Jacobi Gauss-Seidel 3 x k+ i x (k+ i x (k+ x (k+ x (k+ i x (k x(k k + x(k Gauss-Seidel 4 (Gauss-Seidel x 0 R n k =,, x (k+ := x (k + a (b n = a x (k x (k+ := x (k + a (b a x (k+ n = a x (k x n (k+ := x n (k + (b n n = a n x (k+ x n (k i Jacobi x k Gauss-Seidel Jacobi a a a D =, L = a n a n,n, U = a a n a n,n (9 x k+ := D (b Lx k+ Ux k (93 x k+ x k+ := (I + D L D Ux k + (I + D L D b x k+ := (D + L Ux k + (D + L b (94 G = (D + L U 93 n i=,i a i a ii < ( =,,, n Gauss-Seidel G J <
6 08 9 ( J < G J J = ( a i M n (R e T = [ ] J e J e < e J := L J + U J L J, U J J U J e ( J I L J e L J 0 (L J n = L J n = 0 0 (I L J = I + L J + (L J + + (L J n I + L J + L J + + L J n = (I L J G = (D + L = (I + D L D U = (I L J U J (I L I (I L J, J < G J G e (I L J U J e (I L J (I L J + ( J Ie = {I + (I L J ( J }e (I + ( J Ie = J e ( 93 (Stein-Rosenberg J = L J + U J M n (R G = (I L J U J ( λ n (G = λ n (J = 0 ( 0 < λ n (G < λ n (J < (3 λ n (G = λ n (J = (4 λ n (G > λ n (J > λ n (A A M n (R A M n (R a i 0 Jacobi Gauss-Seidel Jacobi Gauss-Seidel
7 94 SOR SOR Gauss-Seidel ω SOR(Successive Over-Relaxation 5 (SOR x 0 k = 0,,, x (k+ := x (k + ω a (b n = a i x (k x (k+ := x (k + ω a (b a x (k+ n = a x (k x n (k+ := x n (k + ω (b n n = a n x (k+ x n (k (95 ω (9 x k+ := x k + ωd {b Lx k+ (D + Ux k } (96 ω D(I + ωd Lx k+ := ω D { I ω(i + D U } x k + b (97 ω = Gauss-Seidel B(ω = ω D(I + ωd L H(ω = I B(ω A H(ω M n (R SOR x k+ := H(ωx k + B(ω b (98 94 (Kahan A M n (R max λ(h(ω ω for ω ( det(i + ωd L = det =
8 0 9 det(h(ω = det{(i + ωd LH(ω} = det(i ω(i + D U = det{( ωi ωd U} n = ( ω n = λ i (H(ω i= max i λ i (H(ω ω ( 0 < ω < SOR ω A 94 (Ostrowski-Reich Hermite A 0 < ω < ( A (9 max i λ i (H(ω < A := L + D + U A Hermite B(ω = ω D(I + ωd L L = U B(ω + B(ω A = = = ω D(I + ωd L + ω (I + ωl D D A ω D + L + ω D + U A ( ω D λ A (B(ω A C x C n x A A Hermite A A (B(ω Ax = λx x (B(ω Ax = λx Ax x (B (ω Ax = λx Ax x (B(ω + B (ω Ax = Re(λx Ax Re(λ > 0
9 94 SOR Q := A (B(ω A = A B(ω I (Q I(Q + I = I B (ωa = H(ω µ, x H(ω y := (Q + I x y 0 µ (Q I(Q + I x = H(ωx = µx (Q Iy = µ(q + Iy (µ Qy = ( + µy Qy = + µ µ y λ = ( + µ/( µ A (B(ω A µ = λ + Re(λ λ + + Re(λ µ < max λ(h(ω ω 943 (property A A M n (C property A P PAP = D M M D ( D, D 944 (Young-Varga A property A ω b = + max λ(j max λ(j max λ(h(ω = ω b = + max λ(j ( ω b ω 94 SOR (95 (96 (96 B(ω H(ω (97 (98
10 9 95 A x A =, x = 0 n A 3 (Tridiagonal b - Axi IEEE 倍精度計算, 次元数 : 0 00 ω= 9 ω= 05 Jacobi 00 Gauss-Seidel 0000 e-06 e-08 ω= 5 e 反復回数 9: Jacobi Gauss-Seidel SOR n = 0 Jacobi Gauss-Seidel SOR (ω = 05, 5, 9 b Ax k 9 SOR (ω = 5, Gauss-Seidel Jacobi SOR ω SOR 9 n = 00 IEEE SOR SOR 0 Jacobi Gauss-Seidel
11 95 3 b - Axi ω=9, 次元数 : gmp 56bit 計算 IEEE 倍精度 000 e-06 e 反復回数 9: (SOR, 00, ω = 9 998
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