NAG FL Interface
f16rbf (dgb_​norm)

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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×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 <nag.h>
double  f16rbf_ (const Integer *inorm, const Integer *m, const Integer *n, const Integer *kl, const Integer *ku, const double ab[], const Integer *ldab)
The routine may be called by the names f16rbf or nagf_blast_dgb_norm.

3 Description

Given a real m×n banded matrix, A, f16rbf calculates one of the values given by
A1=maxji=1m|aij| (the 1-norm of A),
A=maxij= 1n|aij| (the -norm of A),
AF=(i=1mj=1n|aij|2)1/2 (the Frobenius norm of A),   or
maxi,j|aij| (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 https://www.netlib.org/blas/blast-forum/blas-report.pdf

5 Arguments

1: inorm Integer Input
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,j|aij| (not a norm).
Constraints:
  • inorm=171, 173, 174, 175 or 177;
  • if inorm=173, m=1 or n=1.
2: m Integer Input
On entry: m, the number of rows of the matrix A. If m0 on input, f16rbf returns 0.
3: n Integer Input
On entry: n, the number of columns of the matrix A. If n0 on input, f16rbf returns 0.
4: kl Integer Input
On entry: kl, the number of subdiagonals within the band of A. If kl<0 on input, f16rbf returns 0.
5: ku Integer Input
On entry: ku, the number of superdiagonals within the band of A. If ku<0 on input, f16rbf returns 0.
6: ab(ldab,*) Real (Kind=nag_wp) array Input
Note: the second dimension of the array ab must be at least max(1,n).
On entry: the m×n band matrix A.
The matrix is stored in rows 1 to kl+ku+1, more precisely, the element Aij must be stored in
ab(ku+1+i-j,j)  for ​max(1,j-ku)imin(m,j+kl).  
7: ldab Integer Input
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 6×4 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)