nag_zhpcon (f07puc) nag_zhpcon (f07puc) estimates the condition number of a complex Hermitian indefinite matrix A, where A has been factorized by nag_zhptrf (f07prc), using packed storage

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NAG C Library Function Document
nag_zhpcon (f07puc)
1 Purpose
nag_zhpcon (f07puc) estimates the condition number of a complex Hermitian indeﬁnite matrix A, where A
has been factorized by nag_zhptrf (f07prc), using packed storage.
2 Speciﬁcation
void nag_zhpcon (Nag_OrderType order, Nag_UploType uplo, Integer n,
const Complex ap[], const Integer ipiv[], double anorm, double *rcond,
NagError *fail)
3 Description
nag_zhpcon (f07puc) estimates the condition number (in the 1-norm) of a complex Hermitian indeﬁnite
matrix A:
 1ðAÞ¼k Ak1kA
 1k1:
Since A is Hermitian,  1ðAÞ¼ 1ðAÞ¼k Ak1kA
 1k1.
Because  1ðAÞ is inﬁnite if A is singular, the function actually returns an estimate of the reciprocal of
 1ðAÞ.
The function should be preceded by a call to nag_zhp_norm (f16udc) to compute kAk1 and a call to
nag_zhptrf (f07prc) to compute the Bunch–Kaufman factorization of A. The function then uses Higham’s
implementation of Hager’s method (see Higham (1988)) to estimate kA
 1k1.
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: order – Nag_OrderType Input
On entry: the order parameter speciﬁes the two-dimensional storage scheme being used, i.e., row-
major ordering or column-major ordering. C language deﬁned storage is speciﬁed by
order ¼ Nag RowMajor. See Section 2.2.1.4 of the Essential Introduction for a more detailed
explanation of the use of this parameter.
Constraint: order ¼ Nag RowMajor or Nag ColMajor.
2: uplo – Nag_UploType Input
On entry: indicates how A has been factorized as follows:
if uplo ¼ Nag Upper, A ¼ PUDU
HP
T, where U is upper triangular;
if uplo ¼ Nag Lower, A ¼ PLDL
HP
T, where L is lower triangular.
Constraint: uplo ¼ Nag Upper or Nag Lower.
f07 – Linear Equations (LAPACK) f07puc
[NP3645/7] f07puc.13: n – Integer Input
On entry: n, the order of the matrix A.
Constraint: n   0.
4: ap½dim  – const Complex Input
Note: the dimension, dim, of the array ap must be at least maxð1;n  ð n þ 1Þ=2Þ.
On entry: details of the factorization of A stored in packed form, as returned by nag_zhptrf (f07prc).
5: ipiv½dim  – const Integer Input
Note: the dimension, dim, of the array ipiv must be at least maxð1;nÞ.
On entry: details of the interchanges and the block structure of D, as returned by nag_zhptrf
(f07prc).
6: anorm – double Input
On entry: the 1-norm of the original matrix A, which may be computed by calling nag_zhp_norm
(f16udc). anorm must be computed either before calling nag_zhptrf (f07prc) or else from a copy of
the original matrix A.
Constraint: anorm   0:0.
7: rcond – double * 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: fail – NagError * Output
The NAG error parameter (see the Essential Introduction).
6 Error Indicators and Warnings
NE_INT
On entry, n = hvaluei.
Constraint: n   0.
NE_REAL
On entry, anorm = hvaluei.
Constraint: anorm   0:0.
NE_ALLOC_FAIL
Memory allocation failed.
NE_BAD_PARAM
On entry, parameter hvaluei had an illegal value.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the
call is correct then please consult NAG for assistance.
f07puc NAG C Library Manual
f07puc.2 [NP3645/7]7 Accuracy
The computed estimate rcond is never less than the true value  , and in practice is nearly always less than
10 , although examples can be constructed where rcond is much larger.
8 Further Comments
A call to nag_zhpcon (f07puc) involves solving a number of systems of linear equations of the form
Ax ¼ b; the number is usually 5 and never more than 11. Each solution involves approximately 8n
2 real
ﬂoating-point operations but takes considerably longer than a call to nag_zhptrs (f07psc) with 1 right-hand
side, because extra care is taken to avoid overﬂow when A is approximately singular.
The real analogue of this function is nag_dspcon (f07pgc).
9 Example
To estimate the condition number in the 1-norm (or inﬁnity-norm) of the matrix A, where
A ¼
 1:36 þ 0:00i 1:58 þ 0:90i 2:21   0:21i 3:91 þ 1:50i
1:58   0:90i  8:87 þ 0:00i  1:84   0:03i  1:78 þ 1:18i
2:21 þ 0:21i  1:84 þ 0:03i  4:63 þ 0:00i 0:11 þ 0:11i
3:91   1:50i  1:78   1:18i 0:11   0:11i  1:84 þ 0:00i
1
C C A
0
B B @ :
Here A is Hermitian indeﬁnite, stored in packed form, and must ﬁrst be factorized by nag_zhptrf (f07prc).
The true condition number in the 1-norm is 9.10.
9.1 Program Text
/* nag_zhpcon (f07puc) Example Program.
*
* Copyright 2001 Numerical Algorithms Group.
*
* Mark 7, 2001.
*/
#include <stdio.h>
#include <nag.h>
#include <nag_stdlib.h>
#include <naga02.h>
#include <nagf07.h>
#include <nagf16.h>
#include <nagx02.h>
int main(void)
{
/* Scalars */
double anorm, rcond;
Integer ap_len, i, j, n;
Integer exit_status=0;
NagError fail;
Nag_UploType uplo_enum;
Nag_OrderType order;
/* Arrays */
Integer *ipiv=0;
char uplo[2];
Complex *ap=0;
#ifdef NAG_COLUMN_MAJOR
#define A_UPPER(I,J) ap[J*(J-1)/2 + I - 1]
#define A_LOWER(I,J) ap[(2*n-J)*(J-1)/2 + I - 1]
order = Nag_ColMajor;
#else
#define A_LOWER(I,J) ap[I*(I-1)/2 + J - 1]
#define A_UPPER(I,J) ap[(2*n-I)*(I-1)/2 + J - 1]
order = Nag_RowMajor;
f07 – Linear Equations (LAPACK) f07puc
[NP3645/7] f07puc.3#endif
INIT_FAIL(fail);
Vprintf("f07puc Example Program Results\n\n");
/* Skip heading in data file */
Vscanf("%*[^\n] ");
Vscanf("%ld%*[^\n] ", &n);
ap_len = n * (n + 1)/2;
/* Allocate memory */
if ( !(ipiv = NAG_ALLOC(n, Integer)) ||
!(ap = NAG_ALLOC(ap_len, Complex)) )
{
Vprintf("Allocation failure\n");
exit_status = -1;
goto END;
}
/* Read A from data file */
Vscanf(" ’ %1s ’%*[^\n] ", uplo);
if (*(unsigned char *)uplo == ’L’)
uplo_enum = Nag_Lower;
else if (*(unsigned char *)uplo == ’U’)
uplo_enum = Nag_Upper;
else
{
Vprintf("Unrecognised character for Nag_UploType type\n");
exit_status = -1;
goto END;
}
if (uplo_enum == Nag_Upper)
{
for (i = 1; i <= n; ++i)
{
for (j = i; j <= n; ++j)
Vscanf(" ( %lf , %lf )", &A_UPPER(i,j).re, &A_UPPER(i,j).im);
}
Vscanf("%*[^\n] ");
}
else
{
for (i = 1; i <= n; ++i)
{
for (j = 1; j <= i; ++j)
Vscanf(" ( %lf , %lf )", &A_LOWER(i,j).re, &A_LOWER(i,j).im);
}
Vscanf("%*[^\n] ");
}
/* Compute norm of A */
f16udc(order, Nag_OneNorm, uplo_enum, n, ap, &anorm, &fail);
if (fail.code != NE_NOERROR)
{
Vprintf("Error from f16udc.\n%s\n", fail.message);
exit_status = 1;
goto END;
}
/* Factorize A */
f07prc(order, uplo_enum, n, ap, ipiv, &fail);
if (fail.code != NE_NOERROR)
{
Vprintf("Error from f07prc.\n%s\n", fail.message);
exit_status = 1;
goto END;
}
/* Estimate condition number */
f07puc(order, uplo_enum, n, ap, ipiv, anorm, &rcond, &fail);
if (fail.code != NE_NOERROR)
{
Vprintf("Error from f07puc.\n%s\n", fail.message);
exit_status = 1;
goto END;
f07puc NAG C Library Manual
f07puc.4 [NP3645/7]}
if (rcond >= X02AJC)
Vprintf("Estimate of condition number =%10.2e\n", 1.0/rcond);
else
Vprintf("A is singular to working precision\n");
END:
if (ipiv) NAG_FREE(ipiv);
if (ap) NAG_FREE(ap);
return exit_status;
}
9.2 Program Data
f07puc Example Program Data
4 :Value of N
’L’ :Value of UPLO
(-1.36, 0.00)
( 1.58,-0.90) (-8.87, 0.00)
( 2.21, 0.21) (-1.84, 0.03) (-4.63, 0.00)
( 3.91,-1.50) (-1.78,-1.18) ( 0.11,-0.11) (-1.84, 0.00) :End of matrix A
9.3 Program Results
f07puc Example Program Results
Estimate of condition number = 6.68e+00
f07 – Linear Equations (LAPACK) f07puc
[NP3645/7] f07puc.5 (last)

Because

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