VBScriptで固有値・固有ベクトルを求める (反復法)
n × n の正方行列 A と
n次元のベクトル x について
Ax = λx (ただし x ≠ 0) が成り立つとき
λを固有値, x を固有ベクトルという.
最初に適当なベクトルx0から始めて xk+1 = Axk を反復すると
xk は行列 A の最大固有値に対応する固有ベクトルに収束する.
固有値はレイリー(Rayleigh)商
λ = (xkTxk+1) / (xkTxk)
により求める.
べき乗法、累乗法とも言う.
Option Explicit Private Const N = 3 'ベキ乗法で最大固有値を求める Call Main Private Sub Main() Dim a: a = Array(_ Array(5.0, 4.0, 1.0, 1.0), _ Array(4.0, 5.0, 1.0, 1.0), _ Array(1.0, 1.0, 4.0, 2.0), _ Array(1.0, 1.0, 2.0, 4.0)) Dim x: x = Array(1.0 ,0.0 ,0.0 ,0.0) 'ベキ乗法 Dim lambda: lambda = power(a, x) WScript.StdOut.WriteLine WScript.StdOut.WriteLine "eigenvalue" WScript.StdOut.WriteLine Right(Space(14) & FormatNumber(lambda, 10, -1, 0, 0), 14) WScript.StdOut.WriteLine "eigenvector" Call disp_vector(x) End Sub 'ベキ乗法 Private Function power(ByVal a, ByRef x0) Dim i, j, k Dim lambda: lambda = 0.0 '正規化 (ベクトル x0 の長さを1にする) Call normarize(x0) Dim e0: e0 = 0.0 For i = 0 To N e0 = e0 + x0(i) Next For k = 1 To 200 '1次元配列を表示 WScript.StdOut.Write Right(Space(3) & k, 3) & vbTab Call disp_vector(x0) '行列の積 x1 = A × x0 Dim x1: x1 = Array(0.0, 0.0, 0.0, 0.0) For i = 0 To N For j = 0 To N x1(i) = x1(i) + a(i)(j) * x0(j) Next Next '内積 Dim p0: p0 = 0.0 Dim p1: p1 = 0.0 For i = 0 To N p0 = p0 + x1(i) * x1(i) p1 = p1 + x1(i) * x0(i) Next '固有値 lambda = p0 / p1 '正規化 (ベクトル x1 の長さを1にする) Call normarize(x1) '収束判定 Dim e1: e1 = 0.0 For i = 0 To N e1 = e1 + x1(i) Next If Abs(e1 - e0) < 0.00000000001 Then Exit For x0 = x1 e0 = e1 Next power = lambda End Function '1次元配列を表示 Private Sub disp_vector(ByVal row) Dim col For Each col In row WScript.StdOut.Write Right(Space(14) & FormatNumber(col, 10, -1, 0, 0), 14) & vbTab Next WScript.StdOut.WriteLine End Sub '正規化 (ベクトルの長さを1にする) Private Sub normarize(ByRef x()) Dim s: s = 0.0 Dim i For i = 0 To N s = s + x(i) * x(i) Next s = Sqr(s) For i = 0 To N x(i) = x(i) / s Next End Sub
z:\vbs>cscript //nologo 1101.vbs 1 1.0000000000 0.0000000000 0.0000000000 0.0000000000 2 0.7624928517 0.6099942813 0.1524985703 0.1524985703 3 0.6745979924 0.6589096670 0.2353248811 0.2353248811 4 0.6517382447 0.6501601860 0.2761602732 0.2761602732 5 0.6421032522 0.6419452154 0.2963188771 0.2963188771 6 0.6373267026 0.6373108932 0.3063082594 0.3063082594 7 0.6349074765 0.6349058954 0.3112772079 0.3112772079 8 0.6336860412 0.6336858831 0.3137548428 0.3137548428 9 0.6330719648 0.6330719490 0.3149919005 0.3149919005 10 0.6327640473 0.6327640457 0.3156099832 0.3156099832 11 0.6326098648 0.6326098646 0.3159189122 0.3159189122 12 0.6325327172 0.6325327172 0.3160733485 0.3160733485 13 0.6324941293 0.6324941293 0.3161505596 0.3161505596 14 0.6324748319 0.6324748319 0.3161891634 0.3161891634 15 0.6324651822 0.6324651822 0.3162084649 0.3162084649 16 0.6324603572 0.6324603572 0.3162181155 0.3162181155 17 0.6324579446 0.6324579446 0.3162229408 0.3162229408 18 0.6324567383 0.6324567383 0.3162253534 0.3162253534 19 0.6324561352 0.6324561352 0.3162265597 0.3162265597 20 0.6324558336 0.6324558336 0.3162271629 0.3162271629 21 0.6324556828 0.6324556828 0.3162274644 0.3162274644 22 0.6324556074 0.6324556074 0.3162276152 0.3162276152 23 0.6324555697 0.6324555697 0.3162276906 0.3162276906 24 0.6324555509 0.6324555509 0.3162277283 0.3162277283 25 0.6324555415 0.6324555415 0.3162277472 0.3162277472 26 0.6324555367 0.6324555367 0.3162277566 0.3162277566 27 0.6324555344 0.6324555344 0.3162277613 0.3162277613 28 0.6324555332 0.6324555332 0.3162277637 0.3162277637 29 0.6324555326 0.6324555326 0.3162277648 0.3162277648 30 0.6324555323 0.6324555323 0.3162277654 0.3162277654 31 0.6324555322 0.6324555322 0.3162277657 0.3162277657 32 0.6324555321 0.6324555321 0.3162277659 0.3162277659 33 0.6324555321 0.6324555321 0.3162277659 0.3162277659 34 0.6324555321 0.6324555321 0.3162277660 0.3162277660 35 0.6324555320 0.6324555320 0.3162277660 0.3162277660 eigenvalue 10.0000000000 eigenvector 0.6324555320 0.6324555320 0.3162277660 0.3162277660