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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_sparse_complex_herm_matvec (f11xs)

## Purpose

nag_sparse_complex_herm_matvec (f11xs) computes a matrix-vector product involving a complex sparse Hermitian matrix stored in symmetric coordinate storage format.

## Syntax

[y, ifail] = f11xs(a, irow, icol, check, x, 'n', n, 'nz', nz)
[y, ifail] = nag_sparse_complex_herm_matvec(a, irow, icol, check, x, 'n', n, 'nz', nz)

## Description

nag_sparse_complex_herm_matvec (f11xs) 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 Symmetric coordinate storage (SCS) format 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.

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{a}\left({\mathbf{nz}}\right)$ – complex array
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_complex_herm_sort (f11zp) may be used to order the elements in this way.
2:     $\mathrm{irow}\left({\mathbf{nz}}\right)$int64int32nag_int array
3:     $\mathrm{icol}\left({\mathbf{nz}}\right)$int64int32nag_int array
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_complex_herm_sort (f11zp)):
• $1\le {\mathbf{irow}}\left(\mathit{i}\right)\le {\mathbf{n}}$ and $1\le {\mathbf{icol}}\left(\mathit{i}\right)\le {\mathbf{irow}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nz}}$;
• ${\mathbf{irow}}\left(\mathit{i}-1\right)<{\mathbf{irow}}\left(\mathit{i}\right)$ 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 $\mathit{i}=2,3,\dots ,{\mathbf{nz}}$.
4:     $\mathrm{check}$ – string (length ≥ 1)
Specifies whether or not the SCS representation of the matrix $A$, values of n, nz, irow and icol should be checked.
${\mathbf{check}}=\text{'C'}$
Checks are carried out on the values of n, nz, irow and icol.
${\mathbf{check}}=\text{'N'}$
None of these checks are carried out.
Constraint: ${\mathbf{check}}=\text{'C'}$ or $\text{'N'}$.
5:     $\mathrm{x}\left({\mathbf{n}}\right)$ – complex array
The vector $x$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the array x.
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 1$.
2:     $\mathrm{nz}$int64int32nag_int scalar
Default: the dimension of the arrays a, irow, icol. (An error is raised if these dimensions are not equal.)
The number of nonzero elements in the lower triangular part of the matrix $A$.
Constraint: $1\le {\mathbf{nz}}\le {\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$.

### Output Parameters

1:     $\mathrm{y}\left({\mathbf{n}}\right)$ – complex array
The vector $y$.
2:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{check}}\ne \text{'C'}$ or $\text{'N'}$.
${\mathbf{ifail}}=2$
 On entry, ${\mathbf{n}}<1$, or ${\mathbf{nz}}<1$, or ${\mathbf{nz}}>{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$.
${\mathbf{ifail}}=3$
On entry, the arrays irow and icol fail to satisfy the following constraints:
• $1\le {\mathbf{irow}}\left(i\right)\le {\mathbf{n}}$ and $1\le {\mathbf{icol}}\left(i\right)\le {\mathbf{irow}}\left(i\right)$, for $i=1,2,\dots ,{\mathbf{nz}}$;
• ${\mathbf{irow}}\left(i-1\right)<{\mathbf{irow}}\left(i\right)$ or ${\mathbf{irow}}\left(i-1\right)={\mathbf{irow}}\left(i\right)$ and ${\mathbf{icol}}\left(i-1\right)<{\mathbf{icol}}\left(i\right)$, for $i=2,3,\dots ,{\mathbf{nz}}$.
Therefore a nonzero element has been supplied which does not lie in the lower triangular part of $A$, is out of order, or has duplicate row and column indices. Call nag_sparse_complex_herm_sort (f11zp) to reorder and sum or remove duplicates.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## 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.

### Timing

The time taken for a call to nag_sparse_complex_herm_matvec (f11xs) is proportional to nz.

## Example

This example reads in a complex sparse Hermitian positive definite matrix $A$ and a vector $x$. It then calls nag_sparse_complex_herm_matvec (f11xs) to compute the matrix-vector product $y=Ax$.
```function f11xs_example

fprintf('f11xs example results\n\n');

% Sparse Hermitian matrix product y = Ax
a = [6 ...
-1 + 1i  6 + 0i ...
0 + 1i 5 + 0i  ...
5 + 0i ...
2 - 2i                        4 + 0i  ...
1 + 1i  2 + 0i         6 + 0i ...
-4 + 3i                0 + 1i -1 + 0i 6 + 0i ...
-1 - 1i         0 - 1i        9 + 0i ...
1 + 3i                        1 + 2i -1 + 0i        1 + 4i 9 + 0i];
irow = [int64(1)  2 2  3 3  4  5 5  6 6 6  7 7 7 7  8 8 8  9 9 9 9 9];
icol = [int64(1)  1 2  2 3  4  1 5  3 4 6  2 5 6 7  4 6 8  1 5 6 8 9];

x = [ 1 + 9i;      2 - 8i;      3 + 7i;
4 - 6i;      5 + 5i;      6 - 4i;
7 + 3i;      8 - 2i;      9 + 1i];

check = 'C';
[y, ifail] = f11xs( ...
a, irow, icol, check, x);

disp('Matrix-vector product');
disp(y);

```
```f11xs example results

Matrix-vector product
8.0000 +54.0000i
-10.0000 -92.0000i
25.0000 +27.0000i
26.0000 -28.0000i
54.0000 +12.0000i
26.0000 -22.0000i
47.0000 +65.0000i
71.0000 -57.0000i
60.0000 +70.0000i

```