NAG Library Routine Document

f16rbf (dgb_norm)

1
Purpose

f16rbf calculates the value of the 1-norm, the -norm, the Frobenius norm or the maximum absolute value of the elements of a real m by n band matrix stored in banded form.
It can also be used to compute the value of the 2-norm of a row n-vector or a column m-vector.

2
Specification

Fortran Interface
Function f16rbf ( inorm, m, n, kl, ku, ab, ldab)
Real (Kind=nag_wp):: f16rbf
Integer, Intent (In):: inorm, m, n, kl, ku, ldab
Real (Kind=nag_wp), Intent (In):: ab(ldab,*)
C Header Interface
#include <nagmk26.h>
double  f16rbf_ (const Integer *inorm, const Integer *m, const Integer *n, const Integer *kl, const Integer *ku, const double ab[], const Integer *ldab)

3
Description

Given a real m by n banded matrix, A, f16rbf calculates one of the values given by
A1=maxji=1maij (the 1-norm of A),
A=maxij= 1naij (the -norm of A),
AF=i=1mj=1naij21/2 (the Frobenius norm of A),   or
maxi,jaij (the maximum absolute element value of A).
If m or n is 1 then additionally f16rbf can calculate the value A2=ai2 (the 2-norm of A).

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:     inorm – IntegerInput
On entry: specifies the value to be returned. The integer codes shown below can be replaced by the equivalent named constants of the form NAG_?_NORM. These named constants are available via the nag_library module and are also used in the example program for clarity.
inorm=171 (NAG_ONE_NORM)
The 1-norm.
inorm=173 (NAG_TWO_NORM)
The 2-norm of a row or column vector.
inorm=174 (NAG_FROBENIUS_NORM)
The Frobenius (or Euclidean) norm.
inorm=175 (NAG_INF_NORM)
The -norm.
inorm=177 (NAG_MAX_NORM)
The value maxi,jaij (not a norm).
Constraints:
  • inorm=171, 173, 174, 175 or 177;
  • if inorm=173, m=1 or n=1.
2:     m – IntegerInput
On entry: m, the number of rows of the matrix A. If m0 on input, f16rbf returns 0.
3:     n – IntegerInput
On entry: n, the number of columns of the matrix A. If n0 on input, f16rbf returns 0.
4:     kl – IntegerInput
On entry: kl, the number of subdiagonals within the band of A. If kl0 on input, f16rbf returns 0.
5:     ku – IntegerInput
On entry: ku, the number of superdiagonals within the band of A. If ku0 on input, f16rbf returns 0.
6:     abldab* – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array ab must be at least max1,n.
On entry: the m by n band matrix A.
The matrix is stored in rows 1 to kl+ku+1, more precisely, the element Aij must be stored in
abku+1+i-jj  for ​max1,j-kuiminm,j+kl. 
7:     ldab – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f16rbf is called.
Constraint: ldabkl+ku+1.

6
Error Indicators and Warnings

If any constraint on an input parameter is violated, an error message is printed and program execution is terminated.

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

f16rbf is not threaded in any implementation.

9
Further Comments

None.

10
Example

Calculates the various norms of a 6 by 4 banded matrix with two subdiagonals and one superdiagonal.

10.1
Program Text

Program Text (f16rbfe.f90)

10.2
Program Data

Program Data (f16rbfe.d)

10.3
Program Results

Program Results (f16rbfe.r)