NAG CL Interface
f16rbc (dgb_​norm)

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1 Purpose

f16rbc 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.

2 Specification

#include <nag.h>
void  f16rbc (Nag_OrderType order, Nag_NormType norm, Integer m, Integer n, Integer kl, Integer ku, const double ab[], Integer pdab, double *r, NagError *fail)
The function may be called by the names: f16rbc, nag_blast_dgb_norm or nag_dgb_norm.

3 Description

Given a real m×n banded matrix, A, f16rbc 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).

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: norm Nag_NormType Input
On entry: specifies the value to be returned.
norm=Nag_OneNorm
The 1-norm.
norm=Nag_FrobeniusNorm
The Frobenius (or Euclidean) norm.
norm=Nag_InfNorm
The -norm.
norm=Nag_MaxNorm
The value maxi,j|aij| (not a norm).
Constraint: norm=Nag_OneNorm, Nag_FrobeniusNorm, Nag_InfNorm or Nag_MaxNorm.
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: 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].
8: 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.
9: r double * Output
On exit: the value of the norm specified by norm.
10: 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, 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

f16rbc 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 (f16rbce.c)

10.2 Program Data

Program Data (f16rbce.d)

10.3 Program Results

Program Results (f16rbce.r)