f11 Chapter Contents
f11 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_sparse_herm_matvec (f11xsc)

## 1  Purpose

nag_sparse_herm_matvec (f11xsc) computes a matrix-vector product involving a complex sparse Hermitian matrix stored in symmetric coordinate storage format.

## 2  Specification

 #include #include
 void nag_sparse_herm_matvec (Integer n, Integer nnz, const Complex a[], const Integer irow[], const Integer icol[], Nag_SparseSym_CheckData check, const Complex x[], Complex y[], NagError *fail)

## 3  Description

nag_sparse_herm_matvec (f11xsc) computes the matrix-vector product
 $y=Ax$
where $A$ is an $n$ by $n$ complex Hermitian sparse matrix, of arbitrary sparsity pattern, stored in symmetric coordinate storage (SCS) format (see Section 2.1.2 in the f11 Chapter Introduction). The array a stores all the nonzero elements in the lower triangular part of $A$, while arrays irow and icol store the corresponding row and column indices respectively.

None.

## 5  Arguments

1:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 1$.
2:    $\mathbf{nnz}$IntegerInput
On entry: the number of nonzero elements in the lower triangular part of the matrix $A$.
Constraint: $1\le {\mathbf{nnz}}\le {\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$.
3:    $\mathbf{a}\left[{\mathbf{nnz}}\right]$const ComplexInput
On entry: the nonzero elements in 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.
4:    $\mathbf{irow}\left[{\mathbf{nnz}}\right]$const IntegerInput
5:    $\mathbf{icol}\left[{\mathbf{nnz}}\right]$const IntegerInput
On entry: the row and column indices of the nonzero elements supplied in array a.
Constraints:
irow and icol must satisfy the following 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 $\mathit{i}=1,2,\dots ,{\mathbf{nnz}}-1$.
6:    $\mathbf{check}$Nag_SparseSym_CheckDataInput
On entry: specifies whether or not the SCS representation of the matrix $A$, values of n, nnz, irow and icol should be checked.
${\mathbf{check}}=\mathrm{Nag_SparseSym_Check}$
Checks are carried out on the values of n, nnz, irow and icol.
${\mathbf{check}}=\mathrm{Nag_SparseSym_NoCheck}$
None of these checks are carried out.
Constraint: ${\mathbf{check}}=\mathrm{Nag_SparseSym_Check}$ or $\mathrm{Nag_SparseSym_NoCheck}$.
7:    $\mathbf{x}\left[{\mathbf{n}}\right]$const ComplexInput
On entry: the vector $x$.
8:    $\mathbf{y}\left[{\mathbf{n}}\right]$ComplexOutput
On exit: the vector $y$.
9:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
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.
An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
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 3.6.5 in the Essential Introduction 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.

## 7  Accuracy

The computed vector $y$ satisfies the error bound
 $y-Ax∞≤cnεA∞x∞,$
where $c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.

## 8  Parallelism and Performance

nag_sparse_herm_matvec (f11xsc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_sparse_herm_matvec (f11xsc) 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.

## 9  Further Comments

### 9.1  Timing

The time taken for a call to nag_sparse_herm_matvec (f11xsc) is proportional to nnz.

## 10  Example

This example reads in a complex sparse Hermitian positive definite matrix $A$ and a vector $x$. It then calls nag_sparse_herm_matvec (f11xsc) to compute the matrix-vector product $y=Ax$.

### 10.1  Program Text

Program Text (f11xsce.c)

### 10.2  Program Data

Program Data (f11xsce.d)

### 10.3  Program Results

Program Results (f11xsce.r)