F07HGF (DPBCON) (PDF version)
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NAG Library Manual

NAG Library Routine Document


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

F07HGF (DPBCON) estimates the condition number of a real symmetric positive definite band matrix A, where A has been factorized by F07HDF (DPBTRF).

2  Specification

The routine may be called by its LAPACK name dpbcon.

3  Description

F07HGF (DPBCON) estimates the condition number (in the 1-norm) of a real symmetric positive definite band matrix A:
κ1A=A1A-11 .
Since A is symmetric, κ1A=κA=AA-1.
Because κ1A is infinite if A is singular, the routine actually returns an estimate of the reciprocal of κ1A.
The routine should be preceded by a call to F06REF to compute A1 and a call to F07HDF (DPBTRF) to compute the Cholesky factorization of A. The routine then uses Higham's implementation of Hager's method (see Higham (1988)) to estimate A-11.

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:     UPLO – CHARACTER(1)Input
On entry: specifies how A has been factorized.
A=UTU, where U is upper triangular.
A=LLT, where L is lower triangular.
Constraint: UPLO='U' or 'L'.
2:     N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint: N0.
3:     KD – INTEGERInput
On entry: kd, the number of superdiagonals or subdiagonals of the matrix A.
Constraint: KD0.
4:     AB(LDAB,*) – REAL (KIND=nag_wp) arrayInput
Note: the second dimension of the array AB must be at least max1,N.
On entry: the Cholesky factor of A, as returned by F07HDF (DPBTRF).
5:     LDAB – INTEGERInput
On entry: the first dimension of the array AB as declared in the (sub)program from which F07HGF (DPBCON) is called.
Constraint: LDABKD+1.
6:     ANORM – REAL (KIND=nag_wp)Input
On entry: the 1-norm of the original matrix A, which may be computed by calling F06REF with its parameter NORM='1'. ANORM must be computed either before calling F07HDF (DPBTRF) or else from a copy of the original matrix A.
Constraint: ANORM0.0.
7:     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.
8:     WORK(3×N) – REAL (KIND=nag_wp) arrayWorkspace
9:     IWORK(N) – INTEGER arrayWorkspace
10:   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:
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 F07HGF (DPBCON) involves solving a number of systems of linear equations of the form Ax=b; the number is usually 4 or 5 and never more than 11. Each solution involves approximately 4nk floating point operations (assuming nk) but takes considerably longer than a call to F07HEF (DPBTRS) 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 F07HUF (ZPBCON).

9  Example

This example estimates the condition number in the 1-norm (or -norm) of the matrix A, where
A= 5.49 2.68 0.00 0.00 2.68 5.63 -2.39 0.00 0.00 -2.39 2.60 -2.22 0.00 0.00 -2.22 5.17 .
Here A is symmetric and positive definite, and is treated as a band matrix, which must first be factorized by F07HDF (DPBTRF). The true condition number in the 1-norm is 74.15.

9.1  Program Text

Program Text (f07hgfe.f90)

9.2  Program Data

Program Data (f07hgfe.d)

9.3  Program Results

Program Results (f07hgfe.r)

F07HGF (DPBCON) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

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