# ?tbcon

Estimates the reciprocal of the condition number of a triangular band matrix.

## Syntax

FORTRAN 77:

call stbcon( norm, uplo, diag, n, kd, ab, ldab, rcond, work, iwork, info )

call dtbcon( norm, uplo, diag, n, kd, ab, ldab, rcond, work, iwork, info )

call ctbcon( norm, uplo, diag, n, kd, ab, ldab, rcond, work, rwork, info )

call ztbcon( norm, uplo, diag, n, kd, ab, ldab, rcond, work, rwork, info )

Fortran 95:

call tbcon( a, rcond [,uplo] [,diag] [,norm] [,info] )

## Description

This routine estimates the reciprocal of the condition number of a triangular band matrix A in either the 1-norm or infinity-norm:

κ1(A) =||A||1 ||A-1||1 = κ∞(AT) = κ∞(AH)

κ∞(A) =||A||∞ ||A-1||∞ =κ1 (AT) = κ1(AH) .

## Input Parameters

norm

  CHARACTER*1. Must be '1' or 'O' or 'I'.

  If norm = '1' or 'O', then the routine estimates the condition number of matrix A in 1-norm.

  If `norm = 'I'`, then the routine estimates the condition number of matrix A in infinity-norm.

uplo

  CHARACTER*1. Must be 'U' or 'L'. Indicates whether A is upper or lower triangular:

  If `uplo = 'U'`, the array ap stores the upper triangle of A in packed form.

  If `uplo = 'L'`, the array ap stores the lower triangle of A in packed form.

diag

  CHARACTER*1. Must be 'N' or 'U'.

  If `diag = 'N'`, then A is not a unit triangular matrix.

  If `diag = 'U'`, then A is unit triangular: diagonal elements are assumed to be 1 and not referenced in the array ab.

n

  INTEGER. The order of the matrix A; n ≥ 0.

kd

  INTEGER. The number of superdiagonals or subdiagonals in the matrix A; kd ≥ 0.

ab, work

  REAL for stbcon

  DOUBLE PRECISION for dtbcon

  COMPLEX for ctbcon

  DOUBLE COMPLEX for ztbcon.

  Arrays: `ab(ldab,*)`, `work(*)`.

  The array ab contains the band matrix A. The second dimension of ab must be at least `max(1,n)`. The array work is a workspace for the routine.

  The dimension of work must be at least `max(1, 3*n)` for real flavors and `max(1, 2*n)` for complex flavors.

ldab

  INTEGER. The first dimension of the array ab. (ldab ≥ kd +1).

iwork

  INTEGER. Workspace array, DIMENSION at least `max(1, n)`.

rwork

  REAL for ctbcon

  DOUBLE PRECISION for ztbcon.

  Workspace array, DIMENSION at least `max(1, n)`.

## Output Parameters

rcond

  REAL for single precision flavors.

  DOUBLE PRECISION for double precision flavors.

  An estimate of the reciprocal of the condition number. The routine sets rcond =0 if the estimate underflows; in this case the matrix is singular (to working precision). However, anytime rcond is small compared to 1.0, for the working precision, the matrix may be poorly conditioned or even singular.

info

  INTEGER. If `info = 0`, the execution is successful.

  If info = -i, the i-th parameter had an illegal value.

## Fortran 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their Fortran 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see Fortran 95 Interface Conventions.

Specific details for the routine tbcon interface are as follows:

a

  Stands for argument ab in Fortan 77 interface. Holds the array A of size `(kd+1,n)`.

norm

  Must be '1', 'O', or 'I'. The default value is '1'.

uplo

  Must be 'U' or 'L'. The default value is 'U'.

diag

  Must be 'N' or 'U'. The default value is 'N'.

## Application Notes

The computed rcond is never less than r (the reciprocal of the true condition number) and in practice is nearly always less than 10r. A call to this routine involves solving a number of systems of linear equations `A*x = b`; the number is usually 4 or 5 and never more than 11. Each solution requires approximately `2*n(kd + 1)` floating-point operations for real flavors and `8*n(kd + 1)` operations for complex flavors.

Parent topic: Routines for Estimating the Condition Number