NAG Library Routine Document

f07bgf  (dgbcon)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

f07bgf (dgbcon) estimates the condition number of a real band matrix A, where A has been factorized by f07bdf (dgbtrf).

2
Specification

Fortran Interface
Subroutine f07bgf ( norm, n, kl, ku, ab, ldab, ipiv, anorm, rcond, work, iwork, info)
Integer, Intent (In):: n, kl, ku, ldab, ipiv(*)
Integer, Intent (Out):: iwork(n), info
Real (Kind=nag_wp), Intent (In):: ab(ldab,*), anorm
Real (Kind=nag_wp), Intent (Out):: rcond, work(3*n)
Character (1), Intent (In):: norm
C Header Interface
#include nagmk26.h
void  f07bgf_ ( const char *norm, const Integer *n, const Integer *kl, const Integer *ku, const double ab[], const Integer *ldab, const Integer ipiv[], const double *anorm, double *rcond, double work[], Integer iwork[], Integer *info, const Charlen length_norm)
The routine may be called by its LAPACK name dgbcon.

3
Description

f07bgf (dgbcon) estimates the condition number of a real band matrix A, in either the 1-norm or the -norm:
κ1A=A1A-11   or   κA=AA-1 .  
Note that κA=κ1AT.
Because the condition number is infinite if A is singular, the routine actually returns an estimate of the reciprocal of the condition number.
The routine should be preceded by a call to f06rbf to compute A1 or A, and a call to f07bdf (dgbtrf) to compute the LU factorization of A. The routine then uses Higham's implementation of Hager's method (see Higham (1988)) to estimate A-11 or A-1.

4
References

Higham N J (1988) FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation ACM Trans. Math. Software 14 381–396

5
Arguments

1:     norm – Character(1)Input
On entry: indicates whether κ1A or κA is estimated.
norm='1' or 'O'
κ1A is estimated.
norm='I'
κA is estimated.
Constraint: norm='1', 'O' or 'I'.
2:     n – IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
3:     kl – IntegerInput
On entry: kl, the number of subdiagonals within the band of the matrix A.
Constraint: kl0.
4:     ku – IntegerInput
On entry: ku, the number of superdiagonals within the band of the matrix A.
Constraint: ku0.
5:     abldab* – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array ab must be at least max1,n.
On entry: the LU factorization of A, as returned by f07bdf (dgbtrf).
6:     ldab – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07bgf (dgbcon) is called.
Constraint: ldab2×kl+ku+1.
7:     ipiv* – Integer arrayInput
Note: the dimension of the array ipiv must be at least max1,n.
On entry: the pivot indices, as returned by f07bdf (dgbtrf).
8:     anorm – Real (Kind=nag_wp)Input
On entry: if norm='1' or 'O', the 1-norm of the original matrix A.
If norm='I', the -norm of the original matrix A.
anorm may be computed by calling f06rbf with the same value for the argument norm.
anorm must be computed either before calling f07bdf (dgbtrf) or else from a copy of the original matrix A (see Section 10).
Constraint: anorm0.0.
9:     rcond – Real (Kind=nag_wp)Output
On exit: an estimate of the reciprocal of the condition number of A. rcond is set to zero if exact singularity is detected or the estimate underflows. If rcond is less than machine precision, A is singular to working precision.
10:   work3×n – Real (Kind=nag_wp) arrayWorkspace
11:   iworkn – Integer arrayWorkspace
12:   info – IntegerOutput
On exit: info=0 unless the routine detects an error (see Section 6).

6
Error Indicators and Warnings

info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.

7
Accuracy

The computed estimate rcond is never less than the true value ρ, and in practice is nearly always less than 10ρ, although examples can be constructed where rcond is much larger.

8
Parallelism and Performance

f07bgf (dgbcon) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9
Further Comments

A call to f07bgf (dgbcon) involves solving a number of systems of linear equations of the form Ax=b or ATx=b; the number is usually 4 or 5 and never more than 11. Each solution involves approximately 2n2kl+ku floating-point operations (assuming nkl and nku) but takes considerably longer than a call to f07bef (dgbtrs) with one right-hand side, because extra care is taken to avoid overflow when A is approximately singular.
The complex analogue of this routine is f07buf (zgbcon).

10
Example

This example estimates the condition number in the 1-norm of the matrix A, where
A= -0.23 2.54 -3.66 0.00 -6.98 2.46 -2.73 -2.13 0.00 2.56 2.46 4.07 0.00 0.00 -4.78 -3.82 .  
Here A is nonsymmetric and is treated as a band matrix, which must first be factorized by f07bdf (dgbtrf). The true condition number in the 1-norm is 56.40.

10.1
Program Text

Program Text (f07bgfe.f90)

10.2
Program Data

Program Data (f07bgfe.d)

10.3
Program Results

Program Results (f07bgfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017