◆ CscSsorSolve()

Sub CscSsorSolve	(	Trans As	String,
		N As	Long,
		Omega As	Double,
		Val() As	Double,
		Colptr() As	Long,
		Rowind() As	Long,
		D() As	Double,
		B() As	Double,
		X() As	Double,
		Optional Info As	Long,
		Optional Base As	Long = `-1`
	)

対称逐次的過剰緩和(SSOR)前処理 (CSC)

目的: 連立一次方程式の疎な係数行列 A に対する対称逐次的過剰緩和(SSOR)前処理を行う. すなわち, 連立一次方程式 M*x = b または M^T*x = b を解く. ここで, M は前処理行列である.

引数

[in]	Trans	= "N": Mx = b を解く. = "T", "C": M^Tx = b を解く.
[in]	N	前処理行列の次数. (N >= 0) (N = 0 の場合, 処理を行わずに戻る)
[in]	Omega	緩和パラメータω. (0 < ω < 2)
[in]	Val()	配列 Val(LVal - 1) (LVal >= Nnz) 行列 A の非ゼロ要素の値. (Nnz は非ゼロ要素数)
[in]	Colptr()	配列 Colptr(LColptr - 1) (LColptr >= N + 1) 行列 A の列ポインタ.
[in]	Rowind()	配列 Rowind(LRowind - 1) (LRowind >= Nnz) 行列 A の行インデクス. (Nnz は非ゼロ要素数)
[in]	D()	配列 D(LD) (LD >= N) CSC_SSOR()で求めた前処理行列 M の対角要素.
[in]	B()	配列 B(LB - 1) (LB >= N) 右辺ベクトル b.
[out]	X()	配列 X(LX - 1) (LX >= N) 解ベクトル x.
[out]	Info	(省略可) = 0: 正常終了. = i < 0: (-i)番目の入力パラメータの誤り. = j > 0: 行列が特異である(j番目の対角要素が0).
[in]	Base	(省略可) Colptr() および Rowind() のインデクス形式. = 0: 0-ベース(C形式): 開始インデクス値が 0. = 1: 1-ベース(Fortran形式): 開始インデクス値が 1. (省略時: Colptr(0) = 1 であれば 1, そうでなければ 0 とみなす)

使用例: 連立一次方程式 Ax = B を SSOR 前処理付き FGMRES 法で解く. ただし,
( 0.2 -0.11 -0.93 ) ( -0.3727 )

A = ( -0.32 0.81 0.37 ), B = ( 0.4319 )

( -0.8 -0.92 -0.29 ) ( -1.4247 )

とする.
Sub Ex_Fgmres_Ssor_Csc()

Const N = 3, Nnz = N * N, Omega = 0.5, Tol = 0.0000000001 '1.0e-10

Dim A(Nnz - 1) As Double, Ia(N) As Long, Ja(Nnz - 1) As Long

Dim B(N - 1) As Double, X(N - 1) As Double

Dim XX(N - 1) As Double, YY(N - 1) As Double

Dim Iter As Long, Res As Double, IRev As Long, Info As Long

A(0) = 0.2: A(1) = -0.32: A(2) = -0.8: A(3) = -0.11: A(4) = 0.81: A(5) = -0.92: A(6) = -0.93: A(7) = 0.37: A(8) = -0.29

Ia(0) = 0: Ia(1) = 3: Ia(2) = 6: Ia(3) = 9

Ja(0) = 0: Ja(1) = 1: Ja(2) = 2: Ja(3) = 0: Ja(4) = 1: Ja(5) = 2: Ja(6) = 0: Ja(7) = 1: Ja(8) = 2

B(0) = -0.3727: B(1) = 0.4319: B(2) = -1.4247

Dim D(N - 1) As Double

Call CsxSsor(N, Omega, A(), Ia(), Ja(), D(), Info)

If Info <> 0 Then Debug.Print "Ssor Info =" + Str(Info)

IRev = 0

Do

Call Fgmres_r(N, B(), X(), Info, XX(), YY(), IRev, Iter, Res)

If IRev = 1 Then '- Matvec

Call CscDusmv("N", N, N, 1, A(), Ia(), Ja(), XX(), 0, YY())

ElseIf IRev = 3 Then '- Psolve

Call CscSsorSolve("N", N, Omega, A(), Ia(), Ja(), D(), YY(), XX(), Info)

If Info <> 0 Then Debug.Print "SsorSolve Info =" + Str(Info)

ElseIf IRev = 10 Then '- Check convergence

If Res < Tol Then IRev = 11

End If

Loop While IRev <> 0

Debug.Print "X =", X(0), X(1), X(2)

Debug.Print "Iter = " + CStr(Iter) + ", Res = " + CStr(Res) + ", Info = " + CStr(Info)

End Sub

CscDusmv
Sub CscDusmv(Trans As String, M As Long, N As Long, Alpha As Double, Val() As Double, Colptr() As Long, Rowind() As Long, X() As Double, Beta As Double, Y() As Double, Optional Info As Long, Optional Base As Long=-1, Optional IncX As Long=1, Optional IncY As Long=1)
y <- αAx + βy または y <- αATx + βy (CSC)

Fgmres_r
Sub Fgmres_r(N As Long, B() As Double, X() As Double, Info As Long, XX() As Double, YY() As Double, IRev As Long, Optional Iter As Long, Optional Res As Double, Optional M As Long=0, Optional MaxIter As Long=500)
最小残差(FGMRES)法による連立一次方程式 Ax = b の解 (リバースコミュニケーション版)

CscSsorSolve
Sub CscSsorSolve(Trans As String, N As Long, Omega As Double, Val() As Double, Colptr() As Long, Rowind() As Long, D() As Double, B() As Double, X() As Double, Optional Info As Long, Optional Base As Long=-1)
対称逐次的過剰緩和(SSOR)前処理 (CSC)

CsxSsor
Sub CsxSsor(N As Long, Omega As Double, Val() As Double, Ptr() As Long, Ind() As Long, D() As Double, Optional Info As Long, Optional Base As Long=-1)
対称逐次的過剰緩和(SSOR)前処理のための初期化 (CSC/CSR)

実行結果: X = 0.86 0.640000000000001 0.509999999999999

Iter = 3, Res = 8.33576197324036E-16, Info = 0