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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zspcon (f07qu)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_lapack_zspcon (f07qu) estimates the condition number of a complex symmetric matrix A, where A has been factorized by nag_lapack_zsptrf (f07qr), using packed storage.


[rcond, info] = f07qu(uplo, ap, ipiv, anorm, 'n', n)
[rcond, info] = nag_lapack_zspcon(uplo, ap, ipiv, anorm, 'n', n)


nag_lapack_zspcon (f07qu) estimates the condition number (in the 1-norm) of a complex symmetric matrix A:
κ1A=A1A-11 .  
Since A is symmetric, κ1A=κA=AA-1.
Because κ1A is infinite if A is singular, the function actually returns an estimate of the reciprocal of κ1A.
The function should be preceded by a computation of A1 and a call to nag_lapack_zsptrf (f07qr) to compute the Bunch–Kaufman factorization of A. The function 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


Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Specifies how A has been factorized.
A=PUDUTPT, where U is upper triangular.
A=PLDLTPT, where L is lower triangular.
Constraint: uplo='U' or 'L'.
2:     ap: – complex array
The dimension of the array ap must be at least max1,n×n+1/2
The factorization of A stored in packed form, as returned by nag_lapack_zsptrf (f07qr).
3:     ipiv: int64int32nag_int array
The dimension of the array ipiv must be at least max1,n
Details of the interchanges and the block structure of D, as returned by nag_lapack_zsptrf (f07qr).
4:     anorm – double scalar
The 1-norm of the original matrix A. anorm must be computed either before calling nag_lapack_zsptrf (f07qr) or else from a copy of the original matrix A.
Constraint: anorm0.0.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the dimension of the array ipiv.
n, the order of the matrix A.
Constraint: n0.

Output Parameters

1:     rcond – double scalar
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.
2:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

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.

Further Comments

A call to nag_lapack_zspcon (f07qu) 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 nag_lapack_zsptrs (f07qs) with one right-hand side, because extra care is taken to avoid overflow when A is approximately singular.
The real analogue of this function is nag_lapack_dspcon (f07pg).


This example estimates the condition number in the 1-norm (or -norm) of the matrix A, where
A= -0.39-0.71i 5.14-0.64i -7.86-2.96i 3.80+0.92i 5.14-0.64i 8.86+1.81i -3.52+0.58i 5.32-1.59i -7.86-2.96i -3.52+0.58i -2.83-0.03i -1.54-2.86i 3.80+0.92i 5.32-1.59i -1.54-2.86i -0.56+0.12i .  
Here A is symmetric, stored in packed form, and must first be factorized by nag_lapack_zsptrf (f07qr). The true condition number in the 1-norm is 32.92.
function f07qu_example

fprintf('f07qu example results\n\n');

% Estimate condition number of A, where A is complex symmetric matrix
% such that the lower triangular part is stored in packed format

% A in full format first to get 1-norm of A
n = int64(4);
a =  [ -0.39 - 0.71i   5.14 - 0.64i  -7.86 - 2.96i   3.80 + 0.92i;
        5.14 - 0.64i   8.86 + 1.81i  -3.52 + 0.58i   5.32 - 1.59i;
       -7.86 - 2.96i  -3.52 + 0.58i  -2.83 - 0.03i  -1.54 - 2.86i;
        3.80 + 0.92i   5.32 - 1.59i  -1.54 - 2.86i  -0.56 + 0.12i];
anorm = norm(a,1);

% pack A in array ap
uplo = 'L';
ap = [];
for j = 1:n
  ap = [ap; a(j:n,j)];

% Factorize packed form
[apf, ipiv, info] = f07qr( ...
                           uplo, n, ap);

% Get reciprocal condition number
[rcond, info] = f07qu( ...
                       uplo, apf, ipiv, anorm);

fprintf('Estimate of condition number = %9.2e\n', 1/rcond);

f07qu example results

Estimate of condition number =  2.06e+01

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