# NAG CL Interfacef11xsc (complex_​herm_​matvec)

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## 1Purpose

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

## 2Specification

 #include
 void f11xsc (Integer n, Integer nnz, const Complex a[], const Integer irow[], const Integer icol[], Nag_SparseSym_CheckData check, const Complex x[], Complex y[], NagError *fail)
The function may be called by the names: f11xsc, nag_sparse_complex_herm_matvec or nag_sparse_herm_matvec.

## 3Description

f11xsc computes the matrix-vector product
 $y=Ax$
where $A$ is an $n×n$ 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.

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 1$.
2: $\mathbf{nnz}$Integer Input
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 Complex Input
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 f11zpc may be used to order the elements in this way.
4: $\mathbf{irow}\left[{\mathbf{nnz}}\right]$const Integer Input
5: $\mathbf{icol}\left[{\mathbf{nnz}}\right]$const Integer Input
On entry: the row and column indices of the nonzero elements supplied in array a.
Constraints:
irow and icol must satisfy the following constraints (which may be imposed by a call to 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_CheckData Input
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 Complex Input
On entry: the vector $x$.
8: $\mathbf{y}\left[{\mathbf{n}}\right]$Complex Output
On exit: the vector $y$.
9: $\mathbf{fail}$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 in the lower triangular part of $A$, is out of order, or has duplicate row and column indices. Consider calling f11zpc 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: ${\mathbf{nnz}}\le {\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$.
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 7.5 in the Introduction to the NAG Library CL Interface 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 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
 $‖y-Ax‖∞≤c(n)ε‖A‖∞‖x‖∞,$
where $c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.

## 8Parallelism and Performance

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

### 9.1Timing

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

## 10Example

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

### 10.1Program Text

Program Text (f11xsce.c)

### 10.2Program Data

Program Data (f11xsce.d)

### 10.3Program Results

Program Results (f11xsce.r)