naginterfaces.library.sparse.real_​gen_​solve_​ilu

naginterfaces.library.sparse.real_gen_solve_ilu(method, nnz, a, irow, icol, ipivp, ipivq, istr, idiag, b, m, tol, maxitn, x)

real_gen_solve_ilu solves a real sparse nonsymmetric 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, with incomplete $LU$ preconditioning.

For full information please refer to the NAG Library document for f11dc

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f11/f11dcf.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.

nnzint

The number of nonzero elements in the matrix $A$. This must be the same value as was supplied in the preceding call to real_gen_precon_ilu().

afloat, array-like, shape $\left(\text{la}\right)$

The values returned in the array $a$ by a previous call to real_gen_precon_ilu().

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

The values returned in arrays $irow$, $icol$, $ipivp$, $ipivq$, $istr$ and $idiag$ by a previous call to real_gen_precon_ilu().

$ipivp$ and $ipivq$ are restored on exit.

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

The values returned in arrays $irow$, $icol$, $ipivp$, $ipivq$, $istr$ and $idiag$ by a previous call to real_gen_precon_ilu().

$ipivp$ and $ipivq$ are restored on exit.

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

The values returned in arrays $irow$, $icol$, $ipivp$, $ipivq$, $istr$ and $idiag$ by a previous call to real_gen_precon_ilu().

$ipivp$ and $ipivq$ are restored on exit.

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

The values returned in arrays $irow$, $icol$, $ipivp$, $ipivq$, $istr$ and $idiag$ by a previous call to real_gen_precon_ilu().

$ipivp$ and $ipivq$ are restored on exit.

istrint, array-like, shape $(n+1)$

The values returned in arrays $irow$, $icol$, $ipivp$, $ipivq$, $istr$ and $idiag$ by a previous call to real_gen_precon_ilu().

$ipivp$ and $ipivq$ are restored on exit.

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

The values returned in arrays $irow$, $icol$, $ipivp$, $ipivq$, $istr$ and $idiag$ by a previous call to real_gen_precon_ilu().

$ipivp$ and $ipivq$ are restored on exit.

bfloat, 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.

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

An initial approximation to the solution vector $x$.

Returns

xfloat, 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: $m\ge 1$ and $m\le min(n,⟨value⟩)$.

(errno $1$)

On entry, $method=⟨value⟩$.

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

(errno $1$)

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

Constraint: $\text{la}\ge 2\times nnz$.

(errno $1$)

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

Constraint: $nnz\le n^2$.

(errno $1$)

On entry, $nnz=⟨value⟩$.

Constraint: $nnz\ge 1$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $2$)

On entry, the location ($irow[i-1],icol[i-1]$) is a duplicate: $i=⟨value⟩$.

(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 $CS$ representation of the preconditioner is invalid.

(errno $6$)

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

(errno $7$)

A serious error, code $⟨value⟩$, has occurred in an internal call. Check all function calls and array sizes. Seek expert help.

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

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

real_gen_solve_ilu solves a real sparse nonsymmetric linear system of equations:

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

using a preconditioned 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.

real_gen_solve_ilu uses the incomplete $LU$ factorization determined by real_gen_precon_ilu() as the preconditioning matrix. A call to real_gen_solve_ilu must always be preceded by a call to real_gen_precon_ilu(). Alternative preconditioners for the same storage scheme are available by calling real_gen_solve_jacssor().

The matrix $A$, and the preconditioning matrix $M$, are represented in coordinate storage (CS) format (see the F11 Introduction) in the arrays $a$, $irow$ and $icol$, as returned from real_gen_precon_ilu(). The array $a$ holds the nonzero entries in these matrices, while $irow$ and $icol$ hold the corresponding row and column indices.

real_gen_solve_ilu is a Black Box function which calls real_gen_basic_setup(), real_gen_basic_solver() and real_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

Salvini, S A and Shaw, G J, 1996, An evaluation of new NAG Library solvers for large sparse unsymmetric linear systems, NAG Technical Report TR2/96

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