# NAG CL Interfacef11dgc (real_​gen_​solve_​bdilu)

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## 1Purpose

f11dgc 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 block Jacobi or additive Schwarz preconditioning.

## 2Specification

 #include
 void f11dgc (Nag_SparseNsym_Method method, Integer n, Integer nnz, const double a[], Integer la, const Integer irow[], const Integer icol[], Integer nb, const Integer istb[], const Integer indb[], Integer lindb, const Integer ipivp[], const Integer ipivq[], const Integer istr[], const Integer idiag[], const double b[], Integer m, double tol, Integer maxitn, double x[], double *rnorm, Integer *itn, NagError *fail)
The function may be called by the names: f11dgc, nag_sparse_real_gen_solve_bdilu or nag_sparse_nsym_precon_bdilu_solve.

## 3Description

f11dgc solves a real sparse nonsymmetric linear system of equations:
 $Ax=b,$
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.
f11dgc uses the incomplete (possibly overlapping) block $LU$ factorization determined by f11dfc as the preconditioning matrix. A call to f11dgc must always be preceded by a call to f11dfc. Alternative preconditioners for the same storage scheme are available by calling f11dcc or f11dec.
The matrix $A$, and the preconditioning matrix $M$, are represented in coordinate storage (CS) format (see Section 2.1.1 in the F11 Chapter Introduction) in the arrays a, irow and icol, as returned from f11dfc. The array a holds the nonzero entries in these matrices, while irow and icol hold the corresponding row and column indices.
f11dgc is a Black Box function which calls f11bdc, f11bec and f11bfc. 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.

## 4References

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

## 5Arguments

1: $\mathbf{method}$Nag_SparseNsym_Method Input
On entry: specifies the iterative method to be used.
${\mathbf{method}}=\mathrm{Nag_SparseNsym_RGMRES}$
Restarted generalized minimum residual method.
${\mathbf{method}}=\mathrm{Nag_SparseNsym_CGS}$
Conjugate gradient squared method.
${\mathbf{method}}=\mathrm{Nag_SparseNsym_BiCGSTAB}$
Bi-conjugate gradient stabilized ($\ell$) method.
${\mathbf{method}}=\mathrm{Nag_SparseNsym_TFQMR}$
Transpose-free quasi-minimal residual method.
Constraint: ${\mathbf{method}}=\mathrm{Nag_SparseNsym_RGMRES}$, $\mathrm{Nag_SparseNsym_CGS}$, $\mathrm{Nag_SparseNsym_BiCGSTAB}$ or $\mathrm{Nag_SparseNsym_TFQMR}$.
2: $\mathbf{n}$Integer Input
3: $\mathbf{nnz}$Integer Input
4: $\mathbf{a}\left[{\mathbf{la}}\right]$const double Input
5: $\mathbf{la}$Integer Input
6: $\mathbf{irow}\left[{\mathbf{la}}\right]$const Integer Input
7: $\mathbf{icol}\left[{\mathbf{la}}\right]$const Integer Input
8: $\mathbf{nb}$Integer Input
9: $\mathbf{istb}\left[{\mathbf{nb}}+1\right]$const Integer Input
10: $\mathbf{indb}\left[{\mathbf{lindb}}\right]$const Integer Input
11: $\mathbf{lindb}$Integer Input
12: $\mathbf{ipivp}\left[{\mathbf{lindb}}\right]$const Integer Input
13: $\mathbf{ipivq}\left[{\mathbf{lindb}}\right]$const Integer Input
14: $\mathbf{istr}\left[{\mathbf{lindb}}+1\right]$const Integer Input
15: $\mathbf{idiag}\left[{\mathbf{lindb}}\right]$const Integer Input
On entry: the values returned in arrays irow, icol, ipivp, ipivq, istr and idiag by a previous call to f11dfc.
The arrays istb, indb and a together with the scalars n, nnz, la, nb and lindb must be the same values that were supplied in the preceding call to f11dfc.
16: $\mathbf{b}\left[{\mathbf{n}}\right]$const double Input
On entry: the right-hand side vector $b$.
17: $\mathbf{m}$Integer Input
On entry: if ${\mathbf{method}}=\mathrm{Nag_SparseNsym_RGMRES}$, m is the dimension of the restart subspace.
If ${\mathbf{method}}=\mathrm{Nag_SparseNsym_BiCGSTAB}$, m is the order $\ell$ of the polynomial Bi-CGSTAB method. Otherwise, m is not referenced.
Constraints:
• if ${\mathbf{method}}=\mathrm{Nag_SparseNsym_RGMRES}$, $0<{\mathbf{m}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}},50\right)$;
• if ${\mathbf{method}}=\mathrm{Nag_SparseNsym_BiCGSTAB}$, $0<{\mathbf{m}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}},10\right)$.
18: $\mathbf{tol}$double Input
On entry: 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
 $‖rk‖∞≤τ×(‖b‖∞+‖A‖∞‖xk‖∞).$
If ${\mathbf{tol}}\le 0.0$, $\tau =\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(\sqrt{\epsilon },\sqrt{n}\epsilon \right)$ is used, where $\epsilon$ is the machine precision. Otherwise $\tau =\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{tol}},10\epsilon ,\sqrt{n}\epsilon \right)$ is used.
Constraint: ${\mathbf{tol}}<1.0$.
19: $\mathbf{maxitn}$Integer Input
On entry: the maximum number of iterations allowed.
Constraint: ${\mathbf{maxitn}}\ge 1$.
20: $\mathbf{x}\left[{\mathbf{n}}\right]$double Input/Output
On entry: an initial approximation to the solution vector $x$.
On exit: an improved approximation to the solution vector $x$.
21: $\mathbf{rnorm}$double * Output
On exit: the final value of the residual norm ${‖{r}_{k}‖}_{\infty }$, where $k$ is the output value of itn.
22: $\mathbf{itn}$Integer * Output
On exit: the number of iterations carried out.
23: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ACCURACY
The required accuracy could not be obtained. However a reasonable accuracy may have been achieved. You should check the output value of rnorm for acceptability. This error code usually implies that your problem has been fully and satisfactorily solved to within or close to the accuracy available on your system. Further iterations are unlikely to improve on this situation.
NE_ALG_FAIL
Algorithmic breakdown. A solution is returned, although it is possible that it is completely inaccurate.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_CONVERGENCE
The solution has not converged after $⟨\mathit{\text{value}}⟩$ iterations.
NE_INT
On entry, ${\mathbf{istb}}\left[0\right]=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{istb}}\left[0\right]\ge 1$.
On entry, ${\mathbf{maxitn}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{maxitn}}\ge 1$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 1$.
On entry, ${\mathbf{nnz}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nnz}}\ge 1$.
NE_INT_2
On entry, ${\mathbf{la}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{nnz}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{la}}\ge 2×{\mathbf{nnz}}$.
On entry, ${\mathbf{nb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: $1\le {\mathbf{nb}}\le {\mathbf{n}}$.
On entry, ${\mathbf{nnz}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nnz}}\le {{\mathbf{n}}}^{2}$.
NE_INT_3
On entry, ${\mathbf{lindb}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{istb}}\left[{\mathbf{nb}}\right]-1=⟨\mathit{\text{value}}⟩$ and ${\mathbf{nb}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lindb}}\ge {\mathbf{istb}}\left[{\mathbf{nb}}\right]-1$.
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{method}}=\mathrm{Nag_SparseNsym_RGMRES}$, $1\le {\mathbf{m}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}},⟨\mathit{\text{value}}⟩\right)$.
If ${\mathbf{method}}=\mathrm{Nag_SparseNsym_BiCGSTAB}$, $1\le {\mathbf{m}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}},⟨\mathit{\text{value}}⟩\right)$.
NE_INT_ARRAY
On entry, ${\mathbf{indb}}\left[⟨\mathit{\text{value}}⟩\right]=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: $1\le {\mathbf{indb}}\left[m-1\right]\le {\mathbf{n}}$, for $m={\mathbf{istb}}\left[0\right],{\mathbf{istb}}\left[0\right]+1,\dots ,{\mathbf{istb}}\left[{\mathbf{nb}}\right]-1$.
NE_INTERNAL_ERROR
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_INVALID_CS
On entry, ${\mathbf{icol}}\left[⟨\mathit{\text{value}}⟩\right]=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: $1\le {\mathbf{icol}}\left[\mathit{i}-1\right]\le {\mathbf{n}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{nnz}}$.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to f11dfc and f11dgc.
On entry, ${\mathbf{irow}}\left[⟨\mathit{\text{value}}⟩\right]=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: $1\le {\mathbf{irow}}\left[\mathit{i}-1\right]\le {\mathbf{n}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{nnz}}$.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to f11dfc and f11dgc.
NE_INVALID_CS_PRECOND
The CS representation of the preconditioner is invalid.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to f11dfc and f11dgc.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_NOT_STRICTLY_INCREASING
On entry, element $⟨\mathit{\text{value}}⟩$ of a was out of order.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to f11dfc and f11dgc.
On entry, for $b=⟨\mathit{\text{value}}⟩$, ${\mathbf{istb}}\left[b\right]=⟨\mathit{\text{value}}⟩$ and ${\mathbf{istb}}\left[b-1\right]=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{istb}}\left[\mathit{b}\right]>{\mathbf{istb}}\left[\mathit{b}-1\right]$, for $\mathit{b}=1,2,\dots ,{\mathbf{nb}}$.
On entry, location $⟨\mathit{\text{value}}⟩$ of $\left({\mathbf{irow}},{\mathbf{icol}}\right)$ was a duplicate.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to f11dfc and f11dgc.
NE_REAL
On entry, ${\mathbf{tol}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{tol}}<1.0$.

## 7Accuracy

On successful termination, the final residual ${r}_{k}=b-A{x}_{k}$, where $k={\mathbf{itn}}$, satisfies the termination criterion
 $‖rk‖∞≤τ×(‖b‖∞+‖A‖∞‖xk‖∞).$
The value of the final residual norm is returned in rnorm.

## 8Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f11dgc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f11dgc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The time taken by f11dgc for each iteration is roughly proportional to the value of nnzc returned from the preceding call to f11dfc.
The number of iterations required to achieve a prescribed accuracy cannot be easily determined a priori, as it can depend dramatically on the conditioning and spectrum of the preconditioned coefficient matrix $\overline{A}={M}^{-1}A$.
Some illustrations of the application of f11dgc to linear systems arising from the discretization of two-dimensional elliptic partial differential equations, and to random-valued randomly structured linear systems, can be found in Salvini and Shaw (1996).

## 10Example

This example program reads in a sparse matrix $A$ and a vector $b$. It calls f11dfc, with the array ${\mathbf{lfill}}=0$ and the array ${\mathbf{dtol}}=0.0$, to compute an overlapping incomplete $LU$ factorization. This is then used as an additive Schwarz preconditioner on a call to f11dgc which uses the Bi-CGSTAB method to solve $Ax=b$.

### 10.1Program Text

Program Text (f11dgce.c)

### 10.2Program Data

Program Data (f11dgce.d)

### 10.3Program Results

Program Results (f11dgce.r)