NAG CL Interface
f11xec (real_​symm_​matvec)

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1 Purpose

f11xec computes a matrix-vector product involving a real sparse symmetric matrix stored in symmetric coordinate storage format.

2 Specification

#include <nag.h>
void  f11xec (Integer n, Integer nnz, const double a[], const Integer irow[], const Integer icol[], Nag_SparseSym_CheckData check, const double x[], double y[], NagError *fail)
The function may be called by the names: f11xec, nag_sparse_real_symm_matvec or nag_sparse_sym_matvec.

3 Description

f11xec computes the matrix-vector product
y=Ax  
where A is an n×n symmetric 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 nonzero elements in the lower triangular part of A, while arrays irow and icol store the corresponding row and column indices respectively.
It is envisaged that a common use of f11xec will be to compute the matrix-vector product required in the application of f11gec to sparse symmetric linear systems. An illustration of this usage appears in f11jdc.

4 References

None.

5 Arguments

1: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n1.
2: nnz Integer Input
On entry: the number of nonzero elements in the lower triangular part of A.
Constraint: 1nnzn×(n+1)/2.
3: a[nnz] const double 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 f11zbc may be used to order the elements in this way.
4: irow[nnz] const Integer Input
5: icol[nnz] const Integer Input
On entry: the row and column indices of the nonzero elements supplied in array a.
Constraints:
irow and icol must satisfy these constraints (which may be imposed by a call to f11zbc):
  • 1irow[i]n and 1icol[i]irow[i], for i=0,1,,nnz-1;
  • irow[i-1]<irow[i] or irow[i-1]=irow[i] and icol[i-1]<icol[i], for i=1,2,,nnz-1.
6: 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.
check=Nag_SparseSym_Check
Checks are carried out on the values of n, nnz, irow and icol.
check=Nag_SparseSym_NoCheck
None of these checks are carried out.
See also Section 9.2.
Constraint: check=Nag_SparseSym_Check or Nag_SparseSym_NoCheck.
7: x[n] const double Input
On entry: the vector x.
8: y[n] double Output
On exit: the vector y.
9: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error 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 f11zbc 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.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n1.
On entry, nnz=value.
Constraint: nnz1.
NE_INT_2
On entry, nnz=value and n=value.
Constraint: nnzn×(n+1)/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, I=value, icol[I-1]=value and irow[I-1]=value.
Constraint: icol[I-1]1 and icol[I-1]irow[I-1].
On entry, i=value, irow[i-1]=value and n=value.
Constraint: irow[i-1]1 and irow[i-1]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, a[i-1] is out of order: i=value.
On entry, the location (irow[I-1],icol[I-1]) is a duplicate: I=value.

7 Accuracy

The computed vector y satisfies the error bound
y-Axc(n)εAx,  
where c(n) is a modest linear function of n, and ε is the machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f11xec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f11xec 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 f11xec is proportional to nnz.

9.2 Use of check

It is expected that a common use of f11xec will be to compute the matrix-vector product required in the application of f11gec to sparse symmetric linear systems. In this situation f11xec 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 check=Nag_SparseSym_Check for the first of such calls, and to set check=Nag_SparseSym_NoCheck for all subsequent calls.

10 Example

This example reads in a symmetric positive definite sparse matrix A and a vector x. It then calls f11xec to compute the matrix-vector product y=Ax.

10.1 Program Text

Program Text (f11xece.c)

10.2 Program Data

Program Data (f11xece.d)

10.3 Program Results

Program Results (f11xece.r)