NAG Library Routine Document

f07vuf (ztbcon)


f07vuf (ztbcon) estimates the condition number of a complex triangular band matrix.


Fortran Interface
Subroutine f07vuf ( norm, uplo, diag, n, kd, ab, ldab, rcond, work, rwork, info)
Integer, Intent (In):: n, kd, ldab
Integer, Intent (Out):: info
Real (Kind=nag_wp), Intent (Out):: rcond, rwork(n)
Complex (Kind=nag_wp), Intent (In):: ab(ldab,*)
Complex (Kind=nag_wp), Intent (Out):: work(2*n)
Character (1), Intent (In):: norm, uplo, diag
C Header Interface
#include <nagmk26.h>
void  f07vuf_ (const char *norm, const char *uplo, const char *diag, const Integer *n, const Integer *kd, const Complex ab[], const Integer *ldab, double *rcond, Complex work[], double rwork[], Integer *info, const Charlen length_norm, const Charlen length_uplo, const Charlen length_diag)
The routine may be called by its LAPACK name ztbcon.


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.


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


1:     norm – Character(1)Input
On entry: indicates whether κ1A or κA is estimated.
norm='1' or 'O'
κ1A is estimated.
κA is estimated.
Constraint: norm='1', 'O' or 'I'.
2:     uplo – Character(1)Input
On entry: specifies whether A is upper or lower triangular.
A is upper triangular.
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.
A is a nonunit triangular matrix.
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* – 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:     work2×n – Complex (Kind=nag_wp) arrayWorkspace
10:   rworkn – Real (Kind=nag_wp) arrayWorkspace
11:   info – IntegerOutput
On exit: info=0 unless the routine detects an error (see Section 6).

Error Indicators and Warnings

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


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.

Parallelism and Performance

f07vuf (ztbcon) 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.

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).


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.

Program Text

Program Text (f07vufe.f90)

Program Data

Program Data (f07vufe.d)

Program Results

Program Results (f07vufe.r)