NAG C Library Function Document

nag_dsbmv (f16pdc)

1
Purpose

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

2
Specification

#include <nag.h>
#include <nagf16.h>
void  nag_dsbmv (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)

3
Description

nag_dsbmv (f16pdc) performs the matrix-vector operation
yαAx+βy,  
where A is an n by 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 http://www.netlib.org/blas/blast-forum/blas-report.pdf

5
Arguments

1:     order Nag_OrderTypeInput
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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     uplo Nag_UploTypeInput
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 IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     k IntegerInput
On entry: k, the number of subdiagonals or superdiagonals of the matrix A.
Constraint: k0.
5:     alpha doubleInput
On entry: the scalar α.
6:     ab[dim] const doubleInput
Note: the dimension, dim, of the array ab must be at least max1,pdab×n.
On entry: the n by 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=max1,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,,minn,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,,minn,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=max1,i-k,,i.
7:     pdab IntegerInput
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 doubleInput
Note: the dimension, dim, of the array x must be at least max1,1+n-1incx.
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 IntegerInput
On entry: the increment in the subscripts of x between successive elements of x.
Constraint: incx0.
10:   beta doubleInput
On entry: the scalar β.
11:   y[dim] doubleInput/Output
Note: the dimension, dim, of the array y must be at least max1,1+n-1incy.
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 IntegerInput
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 3.7 in How to Use the NAG Library and its Documentation).

6
Error 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.
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 2.7.5 in How to Use the NAG Library and its Documentation 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

nag_dsbmv (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)