# NAG CL Interfacef11xnc (complex_​gen_​matvec)

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## 1Purpose

f11xnc computes a matrix-vector or conjugate transposed matrix-vector product involving a complex sparse non-Hermitian matrix stored in coordinate storage format.

## 2Specification

 #include
 void f11xnc (Nag_TransType trans, Integer n, Integer nnz, const Complex a[], const Integer irow[], const Integer icol[], Nag_SparseNsym_CheckData check, const Complex x[], Complex y[], NagError *fail)
The function may be called by the names: f11xnc, nag_sparse_complex_gen_matvec or nag_sparse_nherm_matvec.

## 3Description

f11xnc computes either the matrix-vector product $y=Ax$, or the conjugate transposed matrix-vector product $y={A}^{\mathrm{H}}x$, according to the value of the argument trans, where $A$ is a complex $n×n$ sparse non-Hermitian 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 the 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 f11xnc will be to compute the matrix-vector product required in the application of f11bsc to sparse complex linear systems. This is illustrated in Section 10 in f11drc.

None.

## 5Arguments

1: $\mathbf{trans}$Nag_TransType Input
On entry: specifies whether or not the matrix $A$ is conjugate transposed.
${\mathbf{trans}}=\mathrm{Nag_NoTrans}$
$y=Ax$ is computed.
${\mathbf{trans}}=\mathrm{Nag_ConjTrans}$
$y={A}^{\mathrm{H}}x$ is computed.
Constraint: ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$ or $\mathrm{Nag_ConjTrans}$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 1$.
3: $\mathbf{nnz}$Integer Input
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 Complex Input
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 f11znc may be used to order the elements in this way.
5: $\mathbf{irow}\left[{\mathbf{nnz}}\right]$const Integer Input
6: $\mathbf{icol}\left[{\mathbf{nnz}}\right]$const Integer Input
On entry: the row and column indices of the nonzero elements supplied in array a.
Constraints:
• $1\le {\mathbf{irow}}\left[\mathit{i}\right]\le {\mathbf{n}}$ 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_CheckData Input
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 Complex Input
On entry: the vector $x$.
9: $\mathbf{y}\left[{\mathbf{n}}\right]$Complex Output
On exit: the vector $y$.
10: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

A nonzero element has been supplied which does not lie within the matrix $A$, is out of order, or has duplicate row and column indices. Consider calling f11znc to reorder and sum or remove duplicates.
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_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 7.5 in the Introduction to the NAG Library CL Interface 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 8 in the Introduction to the NAG Library CL Interface 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: $\mathit{I}=⟨\mathit{\text{value}}⟩$.

## 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
• ${‖y-{A}^{\mathrm{H}}x‖}_{\infty }\le c\left(n\right)\epsilon {‖{A}^{\mathrm{H}}‖}_{\infty }{‖x‖}_{\infty }$, if ${\mathbf{trans}}=\mathrm{Nag_ConjTrans}$,
where $c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.

## 8Parallelism and Performance

f11xnc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f11xnc 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 f11xnc is proportional to nnz.

### 9.2Use of check

It is expected that a common use of f11xnc will be to compute the matrix-vector product required in the application of f11bsc to sparse complex linear systems. In this situation f11xnc 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 complex sparse matrix $A$ and a vector $x$. It then calls f11xnc to compute the matrix-vector product $y=Ax$ and the conjugate transposed matrix-vector product $y={A}^{\mathrm{H}}x$.

### 10.1Program Text

Program Text (f11xnce.c)

### 10.2Program Data

Program Data (f11xnce.d)

### 10.3Program Results

Program Results (f11xnce.r)