# NAG Library Function Document

## 1Purpose

nag_sparse_nsym_matvec (f11xac) computes a matrix-vector or transposed matrix-vector product involving a real sparse nonsymmetric matrix stored in coordinate storage format.

## 2Specification

 #include #include
 void nag_sparse_nsym_matvec (Nag_TransType trans, Integer n, Integer nnz, const double a[], const Integer irow[], const Integer icol[], Nag_SparseNsym_CheckData check, const double x[], double y[], NagError *fail)

## 3Description

nag_sparse_nsym_matvec (f11xac) computes either the matrix-vector product $y=Ax$, or the transposed matrix-vector product $y={A}^{\mathrm{T}}x$, according to the value of the argument trans, where $A$ is an $n$ by $n$ sparse nonsymmetric matrix, of arbitrary sparsity pattern. The matrix $A$ is stored in coordinate storage (CS) format (see Section 2.1.1 in the f11 Chapter Introduction). The array a stores all nonzero elements of $A$, while arrays irow and icol store the corresponding row and column indices respectively.
It is envisaged that a common use of nag_sparse_nsym_matvec (f11xac) will be to compute the matrix-vector product required in the application of nag_sparse_nsym_basic_solver (f11bec) to sparse linear systems. An illustration of this usage appears in Section 10 in nag_sparse_nsym_precon_ssor_solve (f11ddc).

None.

## 5Arguments

1:    $\mathbf{trans}$Nag_TransTypeInput
On entry: specifies whether or not the matrix $A$ is transposed.
${\mathbf{trans}}=\mathrm{Nag_NoTrans}$
$y=Ax$ is computed.
${\mathbf{trans}}=\mathrm{Nag_Trans}$
$y={A}^{\mathrm{T}}x$ is computed.
Constraint: ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$ or $\mathrm{Nag_Trans}$.
2:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 1$.
3:    $\mathbf{nnz}$IntegerInput
On entry: the number of nonzero elements in the matrix $A$.
Constraint: $1\le {\mathbf{nnz}}\le {{\mathbf{n}}}^{2}$.
4:    $\mathbf{a}\left[{\mathbf{nnz}}\right]$const doubleInput
On entry: the nonzero elements in 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_nsym_sort (f11zac) may be used to order the elements in this way.
5:    $\mathbf{irow}\left[{\mathbf{nnz}}\right]$const IntegerInput
6:    $\mathbf{icol}\left[{\mathbf{nnz}}\right]$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_nsym_sort (f11zac)):
• $1\le {\mathbf{irow}}\left[\mathit{i}\right]\le {\mathbf{n}}$ and $1\le {\mathbf{icol}}\left[\mathit{i}\right]\le {\mathbf{n}}$, 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 $\mathit{i}=1,2,\dots ,{\mathbf{nnz}}-1$.
7:    $\mathbf{check}$Nag_SparseNsym_CheckDataInput
On entry: specifies whether or not the CS representation of the matrix $A$, values of n, nnz, irow and icol should be checked.
${\mathbf{check}}=\mathrm{Nag_SparseNsym_Check}$
Checks are carried on the values of n, nnz, irow and icol.
${\mathbf{check}}=\mathrm{Nag_SparseNsym_NoCheck}$
None of these checks are carried out.
Constraint: ${\mathbf{check}}=\mathrm{Nag_SparseNsym_Check}$ or $\mathrm{Nag_SparseNsym_NoCheck}$.
8:    $\mathbf{x}\left[{\mathbf{n}}\right]$const doubleInput
On entry: the vector $x$.
9:    $\mathbf{y}\left[{\mathbf{n}}\right]$doubleOutput
On exit: the vector $y$.
10:  $\mathbf{fail}$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_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_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: ${\mathbf{nnz}}\le {{\mathbf{n}}}^{2}$.
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.
NE_INVALID_CS
On entry, $i=〈\mathit{\text{value}}〉$, ${\mathbf{icol}}\left[i-1\right]=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{icol}}\left[i-1\right]\ge 1$ and ${\mathbf{icol}}\left[i-1\right]\le {\mathbf{n}}$.
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_nsym_sort (f11zac) to reorder and sum or remove duplicates.

## 7Accuracy

The computed vector $y$ satisfies the error bound:
• ${‖y-Ax‖}_{\infty }\le c\left(n\right)\epsilon {‖A‖}_{\infty }{‖x‖}_{\infty }$, if ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, or
• ${‖y-{A}^{\mathrm{T}}x‖}_{\infty }\le c\left(n\right)\epsilon {‖{A}^{\mathrm{T}}‖}_{\infty }{‖x‖}_{\infty }$, if ${\mathbf{trans}}=\mathrm{Nag_Trans}$,
where $c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.

## 8Parallelism and Performance

nag_sparse_nsym_matvec (f11xac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_sparse_nsym_matvec (f11xac) 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.

### 9.1Timing

The time taken for a call to nag_sparse_nsym_matvec (f11xac) is proportional to nnz.

### 9.2Use of check

It is expected that a common use of nag_sparse_nsym_matvec (f11xac) will be to compute the matrix-vector product required in the application of nag_sparse_nsym_basic_solver (f11bec) to sparse linear systems. In this situation nag_sparse_nsym_matvec (f11xac) is likely to be called many times with the same matrix $A$. In the interests of both reliability and efficiency you are recommended to set ${\mathbf{check}}=\mathrm{Nag_SparseNsym_Check}$ for the first of such calls, and to set ${\mathbf{check}}=\mathrm{Nag_SparseNsym_NoCheck}$ for all subsequent calls.

## 10Example

This example reads in a sparse matrix $A$ and a vector $x$. It then calls nag_sparse_nsym_matvec (f11xac) to compute the matrix-vector product $y=Ax$ and the transposed matrix-vector product $y={A}^{\mathrm{T}}x$.

### 10.1Program Text

Program Text (f11xace.c)

### 10.2Program Data

Program Data (f11xace.d)

### 10.3Program Results

Program Results (f11xace.r)

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