# NAG Library Function Document

## 1Purpose

nag_sparse_herm_sol (f11jsc) solves a complex sparse Hermitian system of linear equations, represented in symmetric coordinate storage format, using a conjugate gradient or Lanczos method, without preconditioning, with Jacobi or with SSOR preconditioning.

## 2Specification

 #include #include
 void nag_sparse_herm_sol (Nag_SparseSym_Method method, Nag_SparseSym_PrecType precon, Integer n, Integer nnz, const Complex a[], const Integer irow[], const Integer icol[], double omega, const Complex b[], double tol, Integer maxitn, Complex x[], double *rnorm, Integer *itn, double rdiag[], NagError *fail)

## 3Description

nag_sparse_herm_sol (f11jsc) solves a complex sparse Hermitian linear system of equations
 $Ax=b,$
using a preconditioned conjugate gradient method (see Barrett et al. (1994)), or a preconditioned Lanczos method based on the algorithm SYMMLQ (see Paige and Saunders (1975)). The conjugate gradient method is more efficient if $A$ is positive definite, but may fail to converge for indefinite matrices. In this case the Lanczos method should be used instead. For further details see Barrett et al. (1994).
nag_sparse_herm_sol (f11jsc) 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 Cholesky (IC) preconditioning see nag_sparse_herm_chol_sol (f11jqc).
The matrix $A$ is represented in symmetric coordinate storage (SCS) format (see Section 2.1.2 in the f11 Chapter Introduction) in the arrays a, irow and icol. The array a holds the nonzero entries in the lower triangular part of the matrix, while irow and icol hold the corresponding row and column indices.

## 4References

Barrett R, Berry M, Chan T F, Demmel J, Donato J, Dongarra J, Eijkhout V, Pozo R, Romine C and Van der Vorst H (1994) Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods SIAM, Philadelphia
Paige C C and Saunders M A (1975) Solution of sparse indefinite systems of linear equations SIAM J. Numer. Anal. 12 617–629
Young D (1971) Iterative Solution of Large Linear Systems Academic Press, New York

## 5Arguments

1:    $\mathbf{method}$Nag_SparseSym_MethodInput
On entry: specifies the iterative method to be used.
${\mathbf{method}}=\mathrm{Nag_SparseSym_CG}$
${\mathbf{method}}=\mathrm{Nag_SparseSym_SYMMLQ}$
Lanczos method (SYMMLQ).
Constraint: ${\mathbf{method}}=\mathrm{Nag_SparseSym_CG}$ or $\mathrm{Nag_SparseSym_SYMMLQ}$.
2:    $\mathbf{precon}$Nag_SparseSym_PrecTypeInput
On entry: specifies the type of preconditioning to be used.
${\mathbf{precon}}=\mathrm{Nag_SparseSym_NoPrec}$
No preconditioning.
${\mathbf{precon}}=\mathrm{Nag_SparseSym_JacPrec}$
Jacobi.
${\mathbf{precon}}=\mathrm{Nag_SparseSym_SSORPrec}$
Symmetric successive-over-relaxation (SSOR).
Constraint: ${\mathbf{precon}}=\mathrm{Nag_SparseSym_NoPrec}$, $\mathrm{Nag_SparseSym_JacPrec}$ or $\mathrm{Nag_SparseSym_SSORPrec}$.
3:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 1$.
4:    $\mathbf{nnz}$IntegerInput
On entry: the number of nonzero elements in the lower triangular part of the matrix $A$.
Constraint: $1\le {\mathbf{nnz}}\le {\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$.
5:    $\mathbf{a}\left[{\mathbf{nnz}}\right]$const ComplexInput
On entry: the nonzero elements of the lower triangular part 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 nag_sparse_herm_sort (f11zpc) may be used to order the elements in this way.
6:    $\mathbf{irow}\left[{\mathbf{nnz}}\right]$const IntegerInput
7:    $\mathbf{icol}\left[{\mathbf{nnz}}\right]$const IntegerInput
On entry: the row and column indices of the nonzero elements supplied in array a.
Constraints:
irow and icol must satisfy these constraints (which may be imposed by a call to nag_sparse_herm_sort (f11zpc)):
• $1\le {\mathbf{irow}}\left[\mathit{i}\right]\le {\mathbf{n}}$ and $1\le {\mathbf{icol}}\left[\mathit{i}\right]\le {\mathbf{irow}}\left[\mathit{i}\right]$, for $\mathit{i}=0,1,\dots ,{\mathbf{nnz}}-1$;
• ${\mathbf{irow}}\left[\mathit{i}-1\right]<{\mathbf{irow}}\left[\mathit{i}\right]$ or ${\mathbf{irow}}\left[\mathit{i}-1\right]={\mathbf{irow}}\left[\mathit{i}\right]$ and ${\mathbf{icol}}\left[\mathit{i}-1\right]<{\mathbf{icol}}\left[\mathit{i}\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{nnz}}-1$.
8:    $\mathbf{omega}$doubleInput
On entry: if ${\mathbf{precon}}=\mathrm{Nag_SparseSym_SSORPrec}$, omega is the relaxation parameter $\omega$ to be used in the SSOR method. Otherwise omega need not be initialized.
Constraint: $0.0<{\mathbf{omega}}<2.0$.
9:    $\mathbf{b}\left[{\mathbf{n}}\right]$const ComplexInput
On entry: the right-hand side vector $b$.
10:  $\mathbf{tol}$doubleInput
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 ${\mathbf{tol}}\le 0.0$, $\tau =\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(\sqrt{\epsilon },10\epsilon ,\sqrt{n}\epsilon \right)$ 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: ${\mathbf{tol}}<1.0$.
11:  $\mathbf{maxitn}$IntegerInput
On entry: the maximum number of iterations allowed.
Constraint: ${\mathbf{maxitn}}\ge 1$.
12:  $\mathbf{x}\left[{\mathbf{n}}\right]$ComplexInput/Output
On entry: an initial approximation to the solution vector $x$.
On exit: an improved approximation to the solution vector $x$.
13:  $\mathbf{rnorm}$double *Output
On exit: the final value of the residual norm $‖{r}_{k}‖$, where $k$ is the output value of itn.
14:  $\mathbf{itn}$Integer *Output
On exit: the number of iterations carried out.
15:  $\mathbf{rdiag}\left[{\mathbf{n}}\right]$doubleOutput
On exit: the elements of the diagonal matrix ${D}^{-1}$, where $D$ is the diagonal part of $A$. Note that since $A$ is Hermitian the elements of ${D}^{-1}$ are necessarily real.
16:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ACCURACY
The required accuracy could not be obtained. However, a reasonable accuracy has been achieved and further iterations could not improve the result.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_COEFF_NOT_POS_DEF
The matrix of the coefficients a appears not to be positive definite. The computation cannot continue.
NE_CONVERGENCE
The solution has not converged after $〈\mathit{\text{value}}〉$ iterations.
NE_INT
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, ${\mathbf{nnz}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nnz}}\le {\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
A serious error, code $〈\mathit{\text{value}}〉$, has occurred in an internal call. Check all function calls and array sizes. Seek expert help.
NE_INVALID_SCS
On entry, $\mathit{I}=〈\mathit{\text{value}}〉$, ${\mathbf{icol}}\left[\mathit{I}-1\right]=〈\mathit{\text{value}}〉$ and ${\mathbf{irow}}\left[\mathit{I}-1\right]=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{icol}}\left[\mathit{I}-1\right]\ge 1$ and ${\mathbf{icol}}\left[\mathit{I}-1\right]\le {\mathbf{irow}}\left[\mathit{I}-1\right]$.
On entry, $i=〈\mathit{\text{value}}〉$, ${\mathbf{irow}}\left[i-1\right]=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{irow}}\left[i-1\right]\ge 1$ and ${\mathbf{irow}}\left[i-1\right]\le {\mathbf{n}}$.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_NOT_STRICTLY_INCREASING
On entry, ${\mathbf{a}}\left[i-1\right]$ is out of order: $i=〈\mathit{\text{value}}〉$.
On entry, the location (${\mathbf{irow}}\left[\mathit{I}-1\right],{\mathbf{icol}}\left[\mathit{I}-1\right]$) is a duplicate: $\mathit{I}=〈\mathit{\text{value}}〉$. Consider calling nag_sparse_herm_sort (f11zpc) to reorder and sum or remove duplicates.
NE_PRECOND_NOT_POS_DEF
The preconditioner appears not to be positive definite. The computation cannot continue.
NE_REAL
On entry, ${\mathbf{omega}}=〈\mathit{\text{value}}〉$.
Constraint: $0.0<{\mathbf{omega}}<2.0$.
On entry, ${\mathbf{tol}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{tol}}<1.0$.
NE_ZERO_DIAG_ELEM
The matrix $A$ has a non-real diagonal entry in row $〈\mathit{\text{value}}〉$.
The matrix $A$ has a zero diagonal entry in row $〈\mathit{\text{value}}〉$.
The matrix $A$ has no diagonal entry in row $〈\mathit{\text{value}}〉$.

## 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

nag_sparse_herm_sol (f11jsc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_sparse_herm_sol (f11jsc) 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 nag_sparse_herm_sol (f11jsc) for each iteration is roughly proportional to nnz. One iteration with the Lanczos method (SYMMLQ) requires a slightly larger number of operations than one iteration with the conjugate gradient method.
The number of iterations required to achieve a prescribed accuracy cannot easily be determined a priori, as it can depend dramatically on the conditioning and spectrum of the preconditioned matrix of the coefficients $\stackrel{-}{A}={M}^{-1}A$.

## 10Example

This example solves a complex sparse Hermitian positive definite system of equations using the conjugate gradient method, with SSOR preconditioning.

### 10.1Program Text

Program Text (f11jsce.c)

### 10.2Program Data

Program Data (f11jsce.d)

### 10.3Program Results

Program Results (f11jsce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017