F07VUF (ZTBCON) (PDF version)
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F07 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F07VUF (ZTBCON)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

F07VUF (ZTBCON) estimates the condition number of a complex triangular band matrix.

2  Specification

SUBROUTINE F07VUF ( NORM, UPLO, DIAG, N, KD, AB, LDAB, RCOND, WORK, RWORK, INFO)
INTEGER  N, KD, LDAB, INFO
REAL (KIND=nag_wp)  RCOND, RWORK(N)
COMPLEX (KIND=nag_wp)  AB(LDAB,*), WORK(2*N)
CHARACTER(1)  NORM, UPLO, DIAG
The routine may be called by its LAPACK name ztbcon.

3  Description

F07VUF (ZTBCON) estimates the condition number of a complex 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  Parameters

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:     AB(LDAB,*) – COMPLEX (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 F07VUF (ZTBCON) 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:     WORK(2×N) – COMPLEX (KIND=nag_wp) arrayWorkspace
10:   RWORK(N) – REAL (KIND=nag_wp) arrayWorkspace
11:   INFO – INTEGEROutput
On exit: INFO=0 unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

Errors or warnings detected by the routine:
INFO<0
If INFO=-i, the ith parameter 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  Further Comments

A call to F07VUF (ZTBCON) involves solving a number of systems of linear equations of the form Ax=b or AHx=b; the number is usually 5 and never more than 11. Each solution involves approximately 8nk real floating point operations (assuming nk) but takes considerably longer than a call to F07VSF (ZTBTRS) with one right-hand side, because extra care is taken to avoid overflow when A is approximately singular.
The real analogue of this routine is F07VGF (DTBCON).

9  Example

This example estimates the condition number in the 1-norm of the matrix A, where
A= -1.94+4.43i 0.00+0.00i 0.00+0.00i 0.00+0.00i -3.39+3.44i 4.12-4.27i 0.00+0.00i 0.00+0.00i 1.62+3.68i -1.84+5.53i 0.43-2.66i 0.00+0.00i 0.00+0.00i -2.77-1.93i 1.74-0.04i 0.44+0.10i .
Here A is treated as a lower triangular band matrix with two subdiagonals. The true condition number in the 1-norm is 71.51.

9.1  Program Text

Program Text (f07vufe.f90)

9.2  Program Data

Program Data (f07vufe.d)

9.3  Program Results

Program Results (f07vufe.r)


F07VUF (ZTBCON) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012