NAG FL Interface
f07hgf (dpbcon)

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1 Purpose

f07hgf estimates the condition number of a real symmetric positive definite band matrix A, where A has been factorized by f07hdf.

2 Specification

Fortran Interface
Subroutine f07hgf ( uplo, n, kd, ab, ldab, anorm, rcond, work, iwork, info)
Integer, Intent (In) :: n, kd, ldab
Integer, Intent (Out) :: iwork(n), info
Real (Kind=nag_wp), Intent (In) :: ab(ldab,*), anorm
Real (Kind=nag_wp), Intent (Out) :: rcond, work(3*n)
Character (1), Intent (In) :: uplo
C Header Interface
#include <nag.h>
void  f07hgf_ (const char *uplo, const Integer *n, const Integer *kd, const double ab[], const Integer *ldab, const double *anorm, double *rcond, double work[], Integer iwork[], Integer *info, const Charlen length_uplo)
The routine may be called by the names f07hgf, nagf_lapacklin_dpbcon or its LAPACK name dpbcon.

3 Description

f07hgf estimates the condition number (in the 1-norm) of a real symmetric positive definite band matrix A:
κ1(A)=A1A-11 .  
Since A is symmetric, κ1(A)=κ(A)=AA-1.
Because κ1(A) is infinite if A is singular, the routine actually returns an estimate of the reciprocal of κ1(A).
The routine should be preceded by a call to f06ref to compute A1 and a call to f07hdf 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 Arguments

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 Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
3: kd Integer Input
On entry: kd, the number of superdiagonals or subdiagonals of the matrix A.
Constraint: kd0.
4: ab(ldab,*) Real (Kind=nag_wp) array Input
Note: the second dimension of the array ab must be at least max(1,n).
On entry: the Cholesky factor of A, as returned by f07hdf.
5: ldab Integer Input
On entry: the first dimension of the array ab as declared in the (sub)program from which f07hgf 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 argument norm='1'. anorm must be computed either before calling f07hdf 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) array Workspace
9: iwork(n) Integer array Workspace
10: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

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.

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

Background information to multithreading can be found in the Multithreading documentation.
f07hgf 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 f07hgf 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 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.

10 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. The true condition number in the 1-norm is 74.15.

10.1 Program Text

Program Text (f07hgfe.f90)

10.2 Program Data

Program Data (f07hgfe.d)

10.3 Program Results

Program Results (f07hgfe.r)