NAG Library Routine Document

f07puf (zhpcon)


f07puf (zhpcon) estimates the condition number of a complex Hermitian indefinite matrix A, where A has been factorized by f07prf (zhptrf), using packed storage.


Fortran Interface
Subroutine f07puf ( uplo, n, ap, ipiv, anorm, rcond, work, info)
Integer, Intent (In):: n, ipiv(*)
Integer, Intent (Out):: info
Real (Kind=nag_wp), Intent (In):: anorm
Real (Kind=nag_wp), Intent (Out):: rcond
Complex (Kind=nag_wp), Intent (In):: ap(*)
Complex (Kind=nag_wp), Intent (Out):: work(2*n)
Character (1), Intent (In):: uplo
C Header Interface
#include <nagmk26.h>
void  f07puf_ (const char *uplo, const Integer *n, const Complex ap[], const Integer ipiv[], const double *anorm, double *rcond, Complex work[], Integer *info, const Charlen length_uplo)
The routine may be called by its LAPACK name zhpcon.


f07puf (zhpcon) estimates the condition number (in the 1-norm) of a complex Hermitian indefinite matrix A:
κ1A=A1A-11 .  
Since A is Hermitian, 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 f06udf to compute A1 and a call to f07prf (zhptrf) to compute the Bunch–Kaufman factorization of A. The routine then uses Higham's implementation of Hager's method (see Higham (1988)) to estimate A-11.


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:     uplo – Character(1)Input
On entry: specifies how A has been factorized.
A=PUDUHPT, where U is upper triangular.
A=PLDLHPT, 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:     ap* – Complex (Kind=nag_wp) arrayInput
Note: the dimension of the array ap must be at least max1,n×n+1/2.
On entry: the factorization of A stored in packed form, as returned by f07prf (zhptrf).
4:     ipiv* – Integer arrayInput
Note: the dimension of the array ipiv must be at least max1,n.
On entry: details of the interchanges and the block structure of D, as returned by f07prf (zhptrf).
5:     anorm – Real (Kind=nag_wp)Input
On entry: the 1-norm of the original matrix A, which may be computed by calling f06udf with its argument norm='1'. anorm must be computed either before calling f07prf (zhptrf) or else from a copy of the original matrix A.
Constraint: anorm0.0.
6:     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.
7:     work2×n – Complex (Kind=nag_wp) arrayWorkspace
8:     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

f07puf (zhpcon) 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 f07puf (zhpcon) involves solving a number of systems of linear equations of the form Ax=b; the number is usually 5 and never more than 11. Each solution involves approximately 8n2 real floating-point operations but takes considerably longer than a call to f07psf (zhptrs) 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 f07pgf (dspcon).


This example estimates the condition number in the 1-norm (or -norm) of the matrix A, where
A= -1.36+0.00i 1.58+0.90i 2.21-0.21i 3.91+1.50i 1.58-0.90i -8.87+0.00i -1.84-0.03i -1.78+1.18i 2.21+0.21i -1.84+0.03i -4.63+0.00i 0.11+0.11i 3.91-1.50i -1.78-1.18i 0.11-0.11i -1.84+0.00i .  
Here A is Hermitian indefinite, stored in packed form, and must first be factorized by f07prf (zhptrf). The true condition number in the 1-norm is 9.10.

Program Text

Program Text (f07pufe.f90)

Program Data

Program Data (f07pufe.d)

Program Results

Program Results (f07pufe.r)