NAG Library Routine Document

f07vgf  (dtbcon)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

f07vgf (dtbcon) estimates the condition number of a real triangular band matrix.

2
Specification

Fortran Interface
Subroutine f07vgf ( norm, uplo, diag, n, kd, ab, ldab, rcond, work, iwork, info)
Integer, Intent (In):: n, kd, ldab
Integer, Intent (Out):: iwork(n), info
Real (Kind=nag_wp), Intent (In):: ab(ldab,*)
Real (Kind=nag_wp), Intent (Out):: rcond, work(3*n)
Character (1), Intent (In):: norm, uplo, diag
C Header Interface
#include nagmk26.h
void  f07vgf_ ( const char *norm, const char *uplo, const char *diag, const Integer *n, const Integer *kd, const double ab[], const Integer *ldab, double *rcond, double work[], Integer iwork[], Integer *info, const Charlen length_norm, const Charlen length_uplo, const Charlen length_diag)
The routine may be called by its LAPACK name dtbcon.

3
Description

f07vgf (dtbcon) estimates the condition number of a real triangular 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 computes A1 or A exactly, and 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:     uplo – Character(1)Input
On entry: specifies whether A is upper or lower triangular.
uplo='U'
A is upper triangular.
uplo='L'
A is lower triangular.
Constraint: uplo='U' or 'L'.
3:     diag – Character(1)Input
On entry: indicates whether A is a nonunit or unit triangular matrix.
diag='N'
A is a nonunit triangular matrix.
diag='U'
A is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 1.
Constraint: diag='N' or 'U'.
4:     n – IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
5:     kd – IntegerInput
On entry: kd, the number of superdiagonals of the matrix A if uplo='U', or the number of subdiagonals if uplo='L'.
Constraint: kd0.
6:     abldab* – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array ab must be at least max1,n.
On entry: the n by n triangular band matrix A.
The matrix is stored in rows 1 to kd+1, more precisely,
  • if uplo='U', the elements of the upper triangle of A within the band must be stored with element Aij in abkd+1+i-jj​ for ​max1,j-kdij;
  • if uplo='L', the elements of the lower triangle of A within the band must be stored with element Aij in ab1+i-jj​ for ​jiminn,j+kd.
If diag='U', the diagonal elements of A are assumed to be 1, and are not referenced.
7:     ldab – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07vgf (dtbcon) is called.
Constraint: ldabkd+1.
8:     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.
9:     work3×n – Real (Kind=nag_wp) arrayWorkspace
10:   iworkn – Integer arrayWorkspace
11:   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

f07vgf (dtbcon) 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 f07vgf (dtbcon) 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 2nk floating-point operations (assuming nk) but takes considerably longer than a call to f07vef (dtbtrs) 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 f07vuf (ztbcon).

10
Example

This example estimates the condition number in the 1-norm of the matrix A, where
A= -4.16 0.00 0.00 0.00 -2.25 4.78 0.00 0.00 0.00 5.86 6.32 0.00 0.00 0.00 -4.82 0.16 .  
Here A is treated as a lower triangular band matrix with one subdiagonal. The true condition number in the 1-norm is 69.62.

10.1
Program Text

Program Text (f07vgfe.f90)

10.2
Program Data

Program Data (f07vgfe.d)

10.3
Program Results

Program Results (f07vgfe.r)

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