◆ Tfqmr_r()

Sub Tfqmr_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 MaxIter As	Long = `500`
	)

Solution of linear system Ax = b using transpose free quasi minimum residual (TFQMR) method (Reverse communication version)

Purpose: This routine solves the linear system Ax = b using the transpose free quasi minimum residual (TFQMR) iterative method with preconditioning.

Parameters

[in]	N	Dimension of the matrix A. (N >= 0) (if N = 0, returns without computation)
[in]	B()	Array B(LB - 1) (LB >= N) Right hand side vector b.
[in,out]	X()	Array X(LX - 1) (LX >= N) [in] Initial guess of solution. [out] Obtained approximate solution.
[out]	Info	= 0: Successful exit = i < 0: The (-i)-th argument is invalid. = 11: Maximum number of iterations exceeded. = 12: Breakdown occurred.
[in,out]	XX()	Array XX(LXX - 1) (LXX >= N) Vector XX for Matvec, MatvecTrans, Psolve and PsolveTrans operations.
[in,out]	YY()	Array YY(LYY - 1) (LYY >= N) Vector YY for Matvec, MatvecTrans, Psolve and PsolveTrans operations.
[in,out]	IRev	Control variable for reverse communication. [in] Before first call, IRev should be initialized to zero. On succeeding calls, IRev should not be altered (except if converged). [out] If IRev is not zero, complete the following process and call this routine again. = 0: Computation finished. See return code in info. = 1: Matvec operation. User should set AXX in YY. Do not alter any other variables. = 3: Psolve operation. User should set solution of MXX = YY in XX. Do not alter any other variables. = 10: To be returned for the convergence test on every iteration . Set IRev = 11 if converged. Do not alter IRev otherwise. The latest values in X(), Iter and Res can be used to decide the convergence. Further, these values may be used to output the intermediate results.
[out]	Iter	(Optional) Actual number of iterations performed for convergence.
[out]	Res	(Optional) Final residual norm value norm(b - A*x).
[in]	MaxIter	(Optional) Maximum number of iterations. (MaxIter > 0) (default = 500)

Example Program: Solve the system of linear equations Ax = B, where
( 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 )

Sub Ex_Tfqmr_r()

Const N = 3, Nnz = N * N, 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, I As Long

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

IRev = 0

Do

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

If IRev = 1 Then '- Matvec

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

ElseIf IRev = 3 Then '- Psolve

For I = 0 To N - 1

XX(I) = YY(I)

Next

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

CsrDusmv
Sub CsrDusmv(Trans As String, M As Long, N As Long, Alpha As Double, Val() As Double, Rowptr() As Long, Colind() 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 or y <- αATx + βy (CSR)

Tfqmr_r
Sub Tfqmr_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 MaxIter As Long=500)
Solution of linear system Ax = b using transpose free quasi minimum residual (TFQMR) method (Reverse ...

Example Results: X = 0.859999999999982 0.640000000000015 0.509999999999994

Iter = 3, Res = 1.96978951268998E-14, Info = 0