NAG CL Interface
f16pbc (dgbmv)

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1 Purpose

f16pbc performs matrix-vector multiplication for a real band matrix.

2 Specification

#include <nag.h>
void  f16pbc (Nag_OrderType order, Nag_TransType trans, Integer m, Integer n, Integer kl, Integer ku, 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: f16pbc, nag_blast_dgbmv or nag_dgbmv.

3 Description

f16pbc performs one of the matrix-vector operations
yαAx+βy,   or  yαATx+βy,  
where A is an m×n real band matrix with kl subdiagonals and ku superdiagonals, x and y are real vectors, and α and β are real scalars.
If m=0 or n=0, no operation is performed.

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: trans Nag_TransType Input
On entry: specifies the operation to be performed.
trans=Nag_NoTrans
yαAx+βy.
trans=Nag_Trans or Nag_ConjTrans
yαATx+βy.
Constraint: trans=Nag_NoTrans, Nag_Trans or Nag_ConjTrans.
3: m Integer Input
On entry: m, the number of rows of the matrix A.
Constraint: m0.
4: n Integer Input
On entry: n, the number of columns of the matrix A.
Constraint: n0.
5: kl Integer Input
On entry: kl, the number of subdiagonals within the band of A.
Constraint: kl0.
6: ku Integer Input
On entry: ku, the number of superdiagonals within the band of A.
Constraint: ku0.
7: alpha double Input
On entry: the scalar α.
8: ab[dim] const double Input
Note: the dimension, dim, of the array ab must be at least
  • max(1,pdab×n) when order=Nag_ColMajor;
  • max(1,m×pdab) when order=Nag_RowMajor.
On entry: the m×n band matrix A.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements Aij, for row i=1,,m and column j=max(1,i-kl),,min(n,i+ku), depends on the order argument as follows:
  • if order=Nag_ColMajor, Aij is stored as ab[(j-1)×pdab+ku+i-j];
  • if order=Nag_RowMajor, Aij is stored as ab[(i-1)×pdab+kl+j-i].
9: 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: pdabkl+ku+1.
10: x[dim] const double Input
Note: the dimension, dim, of the array x must be at least
  • max(1,1+(n-1)|incx|) when trans=Nag_NoTrans;
  • max(1,1+(m-1)|incx|) when trans=Nag_Trans or Nag_ConjTrans.
On entry: the vector x.
If trans=Nag_NoTrans, then x is an n-element vector.
  • 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.
Otherwise, x is an m-element vector.
  • If incx>0, xi must be stored in x[(i-1)×incx], for i=1,2,,m.
  • If incx<0, xi must be stored in x[(m-i)×|incx|], for i=1,2,,m.
  • Intermediate elements of x are not referenced. If m=0, x is not referenced and may be NULL.
11: incx Integer Input
On entry: the increment in the subscripts of x between successive elements of x.
Constraint: incx0.
12: beta double Input
On entry: the scalar β.
13: y[dim] double Input/Output
Note: the dimension, dim, of the array y must be at least
  • max(1,1+(m-1)|incy|) when trans=Nag_NoTrans;
  • max(1,1+(n-1)|incy|) when trans=Nag_Trans or Nag_ConjTrans.
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.
14: incy Integer Input
On entry: the increment in the subscripts of y between successive elements of y.
Constraint: incy0.
15: 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, kl=value.
Constraint: kl0.
On entry, ku=value.
Constraint: ku0.
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
NE_INT_3
On entry, pdab=value, kl=value, ku=value.
Constraint: pdabkl+ku+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.
f16pbc is not threaded in any implementation.

9 Further Comments

None.

10 Example

A vector y, of length 6, is updated using y2y+Ax , where A is a 6×4 banded matrix with two subdiagonals and one superdiagonal, and x is a vector of length 4.

10.1 Program Text

Program Text (f16pbce.c)

10.2 Program Data

Program Data (f16pbce.d)

10.3 Program Results

Program Results (f16pbce.r)