naginterfaces.library.sparse.complex_​gen_​solve_​jacssor

naginterfaces.library.sparse.complex_gen_solve_jacssor(method, precon, a, irow, icol, omega, b, m, tol, maxitn, x)

complex_gen_solve_jacssor solves a complex sparse non-Hermitian system of linear equations, represented in coordinate storage format, using a restarted generalized minimal residual (RGMRES), conjugate gradient squared (CGS), stabilized bi-conjugate gradient (BI-CGSTAB), or transpose-free quasi-minimal residual (TFQMR) method, without preconditioning, with Jacobi, or with SSOR preconditioning.

For full information please refer to the NAG Library document for f11ds

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f11/f11dsf.html

Parameters

methodstr

Specifies the iterative method to be used.

$method=\text{'RGMRES'}$

Restarted generalized minimum residual method.

$method=\text{'CGS'}$

Conjugate gradient squared method.

$method=\text{'BICGSTAB'}$

Bi-conjugate gradient stabilized ($\ell$) method.

$method=\text{'TFQMR'}$

Transpose-free quasi-minimal residual method.

preconstr, length 1

Specifies the type of preconditioning to be used.

$precon=\text{'N'}$

No preconditioning.

$precon=\text{'J'}$

Jacobi.

$precon=\text{'S'}$

Symmetric successive-over-relaxation (SSOR).

acomplex, array-like, shape $\left(\text{nnz}\right)$

The nonzero elements of the matrix $A$, ordered by increasing row index, and by increasing column index within each row. Multiple entries for the same row and column indices are not permitted. The function complex_gen_sort() may be used to order the elements in this way.

irowint, array-like, shape $\left(\text{nnz}\right)$

The row indices of the nonzero elements supplied in $a$.

icolint, array-like, shape $\left(\text{nnz}\right)$

The column indices of the nonzero elements supplied in $a$.

omegafloat

If $precon=\text{'S'}$, $omega$ is the relaxation parameter $\omega$ to be used in the SSOR method. Otherwise $omega$ need not be initialized and is not referenced.

bcomplex, array-like, shape $\left(n\right)$

The right-hand side vector $b$.

mint

If $method=\text{'RGMRES'}$, $m$ is the dimension of the restart subspace.

If $method=\text{'BICGSTAB'}$, $m$ is the order $\ell$ of the polynomial BI-CGSTAB method.

Otherwise, $m$ is not referenced.

tolfloat

The required tolerance. Let $x_k$ denote the approximate solution at iteration $k$, and $r_k$ the corresponding residual. The algorithm is considered to have converged at iteration $k$ if

$${\parallel r_k\parallel }_{\infty }\le \tau \times ({\parallel b\parallel }_{\infty }+{\parallel A\parallel }_{\infty }{\parallel x_k\parallel }_{\infty })\text{.}$$

If $tol\le 0.0$, $\tau =max\left(\sqrt{ϵ},10ϵ,\sqrt{n}ϵ\right)$ is used, where $ϵ$ is the machine precision. Otherwise $\tau =max(tol,10ϵ,\sqrt{n}ϵ)$ is used.

maxitnint

The maximum number of iterations allowed.

xcomplex, array-like, shape $\left(n\right)$

An initial approximation to the solution vector $x$.

Returns

xcomplex, ndarray, shape $\left(n\right)$

An improved approximation to the solution vector $x$.

rnormfloat

The final value of the residual norm ${\parallel r_k\parallel }_{\infty }$, where $k$ is the output value of $itn$.

itnint

The number of iterations carried out.

Raises

NagValueError

(errno $1$)

On entry, $maxitn=⟨value⟩$.

Constraint: $maxitn\ge 1$

(errno $1$)

On entry, $tol=⟨value⟩$.

Constraint: $tol<1.0$.

(errno $1$)

On entry, $m=⟨value⟩$ and $n=⟨value⟩$.

Constraint: $0<m\le min(n,⟨value⟩)$.

(errno $1$)

On entry, $method=⟨value⟩$.

Constraint: $method=\text{'RGMRES'}$, $\text{'CGS'}$ or $\text{'BICGSTAB'}$.

(errno $1$)

On entry, $omega=⟨value⟩$.

Constraint: $0.0<omega<2.0$

(errno $1$)

On entry, $precon=⟨value⟩$.

Constraint: $precon=\text{'N'}$, $\text{'J'}$ or $\text{'S'}$.

(errno $1$)

On entry, $\text{nnz}=⟨value⟩$ and $n=⟨value⟩$.

Constraint: $1\le \text{nnz}\le n^2$.

(errno $1$)

On entry, $\text{nnz}=⟨value⟩$.

Constraint: $\text{nnz}\ge 1$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $2$)

On entry, the location ($irow[\text{I}-1],icol[\text{I}-1]$) is a duplicate: $\text{I}=⟨value⟩$.

(errno $2$)

On entry, $a[i-1]$ is out of order: $i=⟨value⟩$.

(errno $2$)

On entry, $i=⟨value⟩$, $icol[i-1]=⟨value⟩$ and $n=⟨value⟩$.

Constraint: $icol[i-1]\ge 1$ and $icol[i-1]\le n$.

(errno $2$)

On entry, $i=⟨value⟩$, $irow[i-1]=⟨value⟩$ and $n=⟨value⟩$.

Constraint: $irow[i-1]\ge 1$ and $irow[i-1]\le n$.

(errno $3$)

The matrix $A$ has a zero diagonal entry in row $⟨value⟩$.

(errno $3$)

The matrix $A$ has no diagonal entry in row $⟨value⟩$.

(errno $6$)

Algorithmic breakdown. A solution is returned, although it is possible that it is completely inaccurate.

(errno $7$)

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

Warns

NagAlgorithmicWarning

(errno $4$)

The required accuracy could not be obtained. However, a reasonable accuracy may have been achieved.

(errno $5$)

The solution has not converged after $⟨value⟩$ iterations.

Notes

complex_gen_solve_jacssor solves a complex sparse non-Hermitian system of linear equations:

$$Ax=b\text{,}$$

using an RGMRES (see Saad and Schultz (1986)), CGS (see Sonneveld (1989)), BI-CGSTAB($\ell$) (see Van der Vorst (1989) and Sleijpen and Fokkema (1993)), or TFQMR (see Freund and Nachtigal (1991) and Freund (1993)) method.

complex_gen_solve_jacssor allows the following choices for the preconditioner:

• no preconditioning;

• Jacobi preconditioning (see Young (1971));

• symmetric successive-over-relaxation (SSOR) preconditioning (see Young (1971)).

For incomplete $LU$ (ILU) preconditioning see complex_gen_solve_ilu().

The matrix $A$ is represented in coordinate storage (CS) format (see the F11 Introduction) in the arrays $a$, $irow$ and $icol$. The array $a$ holds the nonzero entries in the matrix, while $irow$ and $icol$ hold the corresponding row and column indices.

complex_gen_solve_jacssor is a Black Box function which calls complex_gen_basic_setup(), complex_gen_basic_solver() and complex_gen_basic_diag(). If you wish to use an alternative storage scheme, preconditioner, or termination criterion, or require additional diagnostic information, you should call these underlying functions directly.

References

Freund, R W, 1993, A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems, SIAM J. Sci. Comput. (14), 470–482

Freund, R W and Nachtigal, N, 1991, QMR: a Quasi-Minimal Residual Method for Non-Hermitian Linear Systems, Numer. Math. (60), 315–339

Saad, Y and Schultz, M, 1986, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. (7), 856–869

Sleijpen, G L G and Fokkema, D R, 1993, BiCGSTAB $\left(\ell \right)$ for linear equations involving matrices with complex spectrum, ETNA (1), 11–32

Sonneveld, P, 1989, CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. (10), 36–52

Van der Vorst, H, 1989, Bi-CGSTAB, a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. (13), 631–644

Young, D, 1971, Iterative Solution of Large Linear Systems, Academic Press, New York