( ) 5 Reduction ( ) A M n (C) Av = λv (v 0) (11.1) λ C A (eigenvalue) v C n A λ (eigenvector) M n (R) A λ(a) A M n (R) n A λ

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1 ( ) 5 Reduction ( ) A M n (C) Av = λv (v 0) (11.1) λ C A (eigenvalue) v C n A λ (eigenvector) M n (R) A λ(a) A M n (R) n A λi = 0 A C n 5

2 A n λ 1 (A) λ 2 (A) λ n (A) A M n (R) A T = A A λ(a) (Jacobi ) A, B M 2 (R) 1 2 A = 2 1, B = ( ) P M n (R) A A λi = 0 P 1 AP λi = P 1 A λi P = 0 P 1 AP A P 1 v 5 ( ) ( Jacobi ) (QR Bisection ) Reduction(Householder, Lanczos ) A (A λi)v = 0 (11.2) λ v rank(a λi) < n 1 Leverrier-Faddeev ( 5) ( 13 )

3 相似変換 行列の Reduction Householder 法, Lanczos 法 固有値のみを求める方法 固有多項式の解を求める方法 Bisection 法 ( 実対称行列用 ), Danilevski 法 相似変換を重ねることで対角もしくは ( 上 ) 三角行列に収束させる方法 Jacobi 法 ( 実対称行列用 ), QR 法,LR 法 固有値 固有ベクトルを同時に求める方法 Jacobi 法 ( 実対称行列用 ), べき乗法, 逆べき乗法 11.1: 11.3 λ 1 = λ 1 (A) v 1 (i < j λ i λ j, λ i > λ j ) λ i = λ i (A) v i n x 0 x k := A k x 0 x 0 = c 1 v 1 + c 2 v c n v n ( ) k ( ) k x k = (λ 1 ) k c λ2 λn 1v 1 + c 2 v c n v n λ 1 λ 1 ( ) k x k = (λ 1 ) k λ2 c 1 v 1 + O λ 1 v 1 λ 1 Reiley λ 1 (Ax k+1, x k ) (x k, x k ) overflow x k = 1

4 ( ) 1. x 0 ( x 0 = 1) 2. for k = 0, 1, 2,... (a) y k+1 := Ax k (b) γ k+1 := (y k+1, x k )/(x k, x k ) (c) (d) x k+1 := y k+1 / y k+1 γ k λ 1 (A) x k γ k x k 5 A A 1 A A LU 20 ( ) 1. x 0 ( x 0 = 1) 2. A LU L A U A = A 3. for k = 0, 1, 2,... (a) y k+1 := U 1 A L 1 A x k Ay k+1 = x k y k+1 (b) δ k+1 := (y k+1, x k )/(x k, x k ) (c) (d) x k+1 := y k+1 / y k+1 δ k λ n (A) x k A κ(a)

5 11.4. LR ( ) A A = IEEE754 λ 1 (A) Maximum Eigenvalue: e+01 x Ax x i eigenvector[i] A * eivenvector[i] / eigenvector[i] e e e e e e e e e e+01 Minimum Eigenvalue: e-01 i eigenvector[i] A * eivenvector[i] / eigenvector[i] e e e e e e e e e e A = A T (Hint: ) 11.4 LR LU 1 1 U(upper triangular matrix) R Wilkinson

6 (LR ) 1. A 0 := A 2. for k = 0, 1, 2,... (a) L k R k = A k LU (b) A k+1 := R k L k LU λ 1 A i λ 2... λ n A LR Iterative Times: e e e e e e e e e e e e e e e e e e e e e e e e e+00 Iterative Times: e e e e e e e e e e e e e e e e e e e e e e e e e-01 Iterative Times: e e e e e e e e e e e e e e e e e e e e e e e e e-01 Iterative Times: e e e e e e e e e e-02

7 11.5. QR e e e e e e e e e e e e e e e-01 i eigenvalues e e e e e LR QR 11.5 QR A A = [a 1 a 2 a n ] n Gram-Schmidt a 1, a 2,..., a n q 1, q 2,..., q n 22 (Gram-Schmidt ) 1. for i = 1, 2,..., n (a) u i := a i i 1 j=1 (a i, q j )q j (b) q i := u i / u i 2 u 1 2 (a 2, q 1 ) (a 3, q 1 ) (a n, q 1 ) [a 1 a 2 a n ] = [q 1 q 2 q n ] u 2 2 (a 3, q 2 ) (a n, q 2 ) u n 1 2 (a n, q n 1 ) u n 2 A = QR Q QQ T = I R A QR LR QR QR

8 (QR ) 1. A 0 := A 2. for k = 0, 1, 2,... (a) Q k R k = A k A k QR (b) A k+1 := R k Q k LR QR Iterative Times: e e e e e e e e e e e e e e e e e e e e e e e e e+00 Iterative Times: e e e e e e e e e e e e e e e e e e e e e e e e e-01 Iterative Times: e e e e e e e e e e e e e e e e e e e e e e e e e-01 Iterative Times: e e e e e e e e e e e e e e e e e e e e e e e e e-01 Iterative Times: e e e e e e e e e e-15

9 11.6. Reduction e e e e e e e e e e e e e e e-01 i eigenvalues e e e e e-01 LR QR Reduction n, q 1, q 2,...,q n 11.6 Reduction 0 0 Reduction Householder 24 (Householder ) 1. A 0 := A 2. for i = 1, 2,..., n 1 (a) s 2 := n j=i+1 a 2 ji (b) s := sign(a i+1,i ) s 2 (c) c := 1/(s 2 + a i+1,i s) (d) w i := [ a i+1,i + s a i+2,i... a ni ] T (e) P i := I cw i w T i (f) q i := caw i c 2 w i(caw i ) T w i (g) A i+1 := P i A i P i = A i q i w T i w i q T i

10 A P = P 1 P 2 P n 1 α 1 P 1 AP = β 1 α (11.3) β n 1 α n Hessenberg A P 1 AP = α 1 β 1 β 1 α β n 1 β n 1 α n (11.4) 3 Householder P 1.000e e e e e e e e e e e e e e e e e e e e e e e e e-01 3 P 1 AP 5.000e e e e e e e e e e e e e e e e e e e e e e e e e-01 2 Reduction Lanczos 25 ( Lanczos ) 1. x 1 2 = 1 x 1 2. for i = 1, 2,..., n (a) α i := (x i, Ax i ) (b) y i+1 := Ax i β i 1 x i 1 α i x i ( β 0 = 0) (c) β i := y i Lanczos

11 11.6. Reduction 135 (d) x i+1 := (1/β i )y i+1 Lanczos x 1 = [ ] T P 1.000e e e e e e e e e e e e e e e e e e e e e e e e e-01 3 P 1 AP 5.000e e e e e e e e e e e e e e e e e e e e e e e e e-01 Householder Lanczos 3 1. A M 3 (R) A = (a) A λ 1 v 1 λ (b) A LU (c) A λ 3 v 3 λ A 1 = 2 3, A 2 = , A = 2 3

12 A A Jordan 2 Jordan 3 4. (11.4) 3 H (a) H λi = 0 p 0 (λ) = 1, p 1 (λ) := α 1 λ p i (λ) := (α i λ)p i 1 (λ) β 2 i 1 p i 2(λ) p n (λ) = H λi (b) Hessenberg 5. A = [a i j ] M n (R) monic q n (x) = x n + n 1 i=0 c ix i c n, c n 1,..., c 0 Leverrier-Faddeev [21, 10] trace(a) = n i=1 a ii N 1 := I c n 1 := trace(an 1 ) N 2 := AN 1 + c n 1 I c n 2 := 1 2 trace(an 2) N 3 := AN 2 + c n 2 I c n 3 := 1 3 trace(an 3). c 1 := 1 n 1 trace(an n 1) N n := AN n 1 + c 1 I c 0 := 1 n trace(an n) 3 2 Jordan

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