NAG C Library Function Document

nag_zgbmv (f16sbc)

1
Purpose

nag_zgbmv (f16sbc) performs matrix-vector multiplication for a complex band matrix.

2
Specification

#include <nag.h>
#include <nagf16.h>
void  nag_zgbmv (Nag_OrderType order, Nag_TransType trans, Integer m, Integer n, Integer kl, Integer ku, Complex alpha, const Complex ab[], Integer pdab, const Complex x[], Integer incx, Complex beta, Complex y[], Integer incy, NagError *fail)

3
Description

nag_zgbmv (f16sbc) performs one of the matrix-vector operations
yαAx+βy,  yαATx+βy  or  yαAHx+βy  
where A is an m by n complex band matrix with kl subdiagonals and ku superdiagonals, x and y are complex vectors, and α and β are complex 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 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:     trans Nag_TransTypeInput
On entry: specifies the operation to be performed.
trans=Nag_NoTrans
yαAx+βy.
trans=Nag_Trans
yαATx+βy.
trans=Nag_ConjTrans
yαAHx+βy.
Constraint: trans=Nag_NoTrans, Nag_Trans or Nag_ConjTrans.
3:     m IntegerInput
On entry: m, the number of rows of the matrix A.
Constraint: m0.
4:     n IntegerInput
On entry: n, the number of columns of the matrix A.
Constraint: n0.
5:     kl IntegerInput
On entry: kl, the number of subdiagonals within the band of A.
Constraint: kl0.
6:     ku IntegerInput
On entry: ku, the number of superdiagonals within the band of A.
Constraint: ku0.
7:     alpha ComplexInput
On entry: the scalar α.
8:     ab[dim] const ComplexInput
Note: the dimension, dim, of the array ab must be at least
  • max1,pdab×n when order=Nag_ColMajor;
  • max1,m×pdab when order=Nag_RowMajor.
On entry: the m by 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=max1,i-kl,,minn,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 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: pdabkl+ku+1.
10:   x[dim] const ComplexInput
Note: the dimension, dim, of the array x must be at least
  • max1,1+n-1incx when trans=Nag_NoTrans;
  • max1,1+m-1incx 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 IntegerInput
On entry: the increment in the subscripts of x between successive elements of x.
Constraint: incx0.
12:   beta ComplexInput
On entry: the scalar β.
13:   y[dim] ComplexInput/Output
Note: the dimension, dim, of the array y must be at least
  • max1,1+m-1incy when trans=Nag_NoTrans;
  • max1,1+n-1incy 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 IntegerInput
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 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, 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 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_zgbmv (f16sbc) 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+1.0i 1.0+2.0i 0.0+0.0i 0.0+0.0i 2.0+1.0i 2.0+2.0i 2.0+3.0i 0.0+0.0i 3.0+1.0i 3.0+2.0i 3.0+3.0i 3.0+4.0i 0.0+0.0i 4.0+2.0i 4.0+3.0i 4.0+4.0i 0.0+0.0i 0.0+0.0i 5.0+3.0i 5.0+4.0i 0.0+0.0i 0.0+0.0i 0.0+0.0i 6.0+4.0i ,  
x = 1.0-1.0i 2.0-2.0i 3.0-3.0i 4.0-4.0i ,  
y = -3.5+00.0i -11.5+01.0i -27.5+03.0i -29.0+07.5i -25.5+10.0i -14.5+10.0i ,  
α=1.0+0.0i   and   β=2.0+0.0i .  

10.1
Program Text

Program Text (f16sbce.c)

10.2
Program Data

Program Data (f16sbce.d)

10.3
Program Results

Program Results (f16sbce.r)