NAG Library Routine Document

f07auf (zgecon)


    1  Purpose
    7  Accuracy


f07auf (zgecon) estimates the condition number of a complex matrix A, where A has been factorized by f07arf (zgetrf).


Fortran Interface
Subroutine f07auf ( norm, n, a, lda, anorm, rcond, work, rwork, info)
Integer, Intent (In):: n, lda
Integer, Intent (Out):: info
Real (Kind=nag_wp), Intent (In):: anorm
Real (Kind=nag_wp), Intent (Out):: rcond, rwork(2*n)
Complex (Kind=nag_wp), Intent (In):: a(lda,*)
Complex (Kind=nag_wp), Intent (Out):: work(2*n)
Character (1), Intent (In):: norm
C Header Interface
#include nagmk26.h
void  f07auf_ (const char *norm, const Integer *n, const Complex a[], const Integer *lda, const double *anorm, double *rcond, Complex work[], double rwork[], Integer *info, const Charlen length_norm)
The routine may be called by its LAPACK name zgecon.


f07auf (zgecon) estimates the condition number of a complex matrix A, in either the 1-norm or the -norm:
κ1 A = A1 A-11   or   κ A = A A-1 .  
Note that κA=κ1AH.
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 should be preceded by a call to f06uaf to compute A1 or A, and a call to f07arf (zgetrf) to compute the LU factorization of A. The routine then 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:     n – IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
3:     alda* – Complex (Kind=nag_wp) arrayInput
Note: the second dimension of the array a must be at least max1,n.
On entry: the LU factorization of A, as returned by f07arf (zgetrf).
4:     lda – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f07auf (zgecon) is called.
Constraint: ldamax1,n.
5:     anorm – Real (Kind=nag_wp)Input
On entry: if norm='1' or 'O', the 1-norm of the original matrix A.
If norm='I', the -norm of the original matrix A.
anorm may be computed by calling f06uaf with the same value for the argument norm.
anorm must be computed either before calling f07arf (zgetrf) or else from a copy of the original matrix A (see Section 10).
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:     rwork2×n – Real (Kind=nag_wp) arrayWorkspace
9:     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

f07auf (zgecon) 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 f07auf (zgecon) 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 8n2 real floating-point operations but takes considerably longer than a call to f07asf (zgetrs) 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 f07agf (dgecon).


This example estimates the condition number in the 1-norm of the matrix A, where
A= -1.34+2.55i 0.28+3.17i -6.39-2.20i 0.72-0.92i -0.17-1.41i 3.31-0.15i -0.15+1.34i 1.29+1.38i -3.29-2.39i -1.91+4.42i -0.14-1.35i 1.72+1.35i 2.41+0.39i -0.56+1.47i -0.83-0.69i -1.96+0.67i .  
Here A is nonsymmetric and must first be factorized by f07arf (zgetrf). The true condition number in the 1-norm is 231.86.

Program Text

Program Text (f07aufe.f90)

Program Data

Program Data (f07aufe.d)

Program Results

Program Results (f07aufe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017