NAG CL Interface
f16pdc (dsbmv)

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1 Purpose

f16pdc performs matrix-vector multiplication for a real symmetric band matrix.

2 Specification

#include <nag.h>
void  f16pdc (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer k, double alpha, const double ab[], Integer pdab, const double x[], Integer incx, double beta, double y[], Integer incy, NagError *fail)
The function may be called by the names: f16pdc, nag_blast_dsbmv or nag_dsbmv.

3 Description

f16pdc performs the matrix-vector operation
yαAx+βy,  
where A is an n×n real symmetric band matrix with k subdiagonals and k superdiagonals, x and y are n-element real vectors, and α and β are real scalars.

4 References

Basic Linear Algebra Subprograms Technical (BLAST) Forum (2001) Basic Linear Algebra Subprograms Technical (BLAST) Forum Standard University of Tennessee, Knoxville, Tennessee https://www.netlib.org/blas/blast-forum/blas-report.pdf

5 Arguments

1: order Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: uplo Nag_UploType Input
On entry: specifies whether the upper or lower triangular part of A is stored.
uplo=Nag_Upper
The upper triangular part of A is stored.
uplo=Nag_Lower
The lower triangular part of A is stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
3: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
4: k Integer Input
On entry: k, the number of subdiagonals or superdiagonals of the matrix A.
Constraint: k0.
5: alpha double Input
On entry: the scalar α.
6: ab[dim] const double Input
Note: the dimension, dim, of the array ab must be at least max(1,pdab×n).
On entry: the n×n symmetric band matrix A.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of Aij, depends on the order and uplo arguments as follows:
if order=Nag_ColMajor and uplo=Nag_Upper,
Aij is stored in ab[k+i-j+(j-1)×pdab], for j=1,,n and i=max(1,j-k),,j;
if order=Nag_ColMajor and uplo=Nag_Lower,
Aij is stored in ab[i-j+(j-1)×pdab], for j=1,,n and i=j,,min(n,j+k);
if order=Nag_RowMajor and uplo=Nag_Upper,
Aij is stored in ab[j-i+(i-1)×pdab], for i=1,,n and j=i,,min(n,i+k);
if order=Nag_RowMajor and uplo=Nag_Lower,
Aij is stored in ab[k+j-i+(i-1)×pdab], for i=1,,n and j=max(1,i-k),,i.
7: pdab Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array ab.
Constraint: pdabk+1.
8: x[dim] const double Input
Note: the dimension, dim, of the array x must be at least max(1,1+(n-1)|incx|).
On entry: the n-element vector x.
If incx>0, xi must be stored in x[(i-1)×incx], for i=1,2,,n.
If incx<0, xi must be stored in x[(n-i)×|incx|], for i=1,2,,n.
Intermediate elements of x are not referenced. If n=0, x is not referenced and may be NULL.
9: incx Integer Input
On entry: the increment in the subscripts of x between successive elements of x.
Constraint: incx0.
10: beta double Input
On entry: the scalar β.
11: y[dim] double Input/Output
Note: the dimension, dim, of the array y must be at least max(1,1+(n-1)|incy|).
On entry: the vector y. See x for details of storage.
If beta=0, y need not be set.
On exit: the updated vector y.
12: incy Integer Input
On entry: the increment in the subscripts of y between successive elements of y.
Constraint: incy0.
13: 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

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, incx=value.
Constraint: incx0.
On entry, incy=value.
Constraint: incy0.
On entry, k=value.
Constraint: k0.
On entry, n=value.
Constraint: n0.
NE_INT_2
On entry, pdab=value, k=value.
Constraint: pdabk+1.
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.

7 Accuracy

The BLAS standard requires accurate implementations which avoid unnecessary over/underflow (see Section 2.7 of Basic Linear Algebra Subprograms Technical (BLAST) Forum (2001)).

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f16pdc is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example computes the matrix-vector product
y=αAx+βy  
where
A = ( 1.0 2.0 3.0 0.0 0.0 2.0 2.0 3.0 4.0 0.0 3.0 3.0 3.0 4.0 5.0 0.0 4.0 4.0 4.0 5.0 0.0 0.0 5.0 5.0 5.0 ) ,  
x = ( -1.0 2.0 -3.0 2.0 -1.0 ) ,  
y = ( 10.0 1.5 9.5 8.5 24.0 ) ,  
α= 1.5   and   β= 1.0 .  

10.1 Program Text

Program Text (f16pdce.c)

10.2 Program Data

Program Data (f16pdce.d)

10.3 Program Results

Program Results (f16pdce.r)