NAG CL Interface
f07pgc (dspcon)

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1 Purpose

f07pgc estimates the condition number of a real symmetric indefinite matrix A, where A has been factorized by f07pdc, using packed storage.

2 Specification

#include <nag.h>
void  f07pgc (Nag_OrderType order, Nag_UploType uplo, Integer n, const double ap[], const Integer ipiv[], double anorm, double *rcond, NagError *fail)
The function may be called by the names: f07pgc, nag_lapacklin_dspcon or nag_dspcon.

3 Description

f07pgc estimates the condition number (in the 1-norm) of a real symmetric indefinite 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 function actually returns an estimate of the reciprocal of κ1(A).
The function should be preceded by a call to f16rdc to compute A1 and a call to f07pdc 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.

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: order Nag_OrderType Input
On entry: the order argument 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=Nag_RowMajor. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: uplo Nag_UploType Input
On entry: specifies how A has been factorized.
uplo=Nag_Upper
A=PUDUTPT, where U is upper triangular.
uplo=Nag_Lower
A=PLDLTPT, where L is lower triangular.
Constraint: uplo=Nag_Upper or Nag_Lower.
3: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
4: ap[dim] const double Input
Note: the dimension, dim, of the array ap must be at least max(1,n×(n+1)/2).
On entry: the factorization of A stored in packed form, as returned by f07pdc.
5: ipiv[dim] const Integer Input
Note: the dimension, dim, of the array ipiv must be at least max(1,n).
On entry: details of the interchanges and the block structure of D, as returned by f07pdc.
6: anorm double Input
On entry: the 1-norm of the original matrix A, which may be computed by calling f16rdc with its argument norm=Nag_OneNorm. anorm must be computed either before calling f07pdc or else from a copy of the original matrix A.
Constraint: anorm0.0.
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 * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
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 contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL
On entry, anorm=value.
Constraint: anorm0.0.

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.
f07pgc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

A call to f07pgc 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 2n2 floating-point operations but takes considerably longer than a call to f07pec with one right-hand side, because extra care is taken to avoid overflow when A is approximately singular.
The complex analogues of this function are f07puc for Hermitian matrices and f07quc for symmetric matrices.

10 Example

This example estimates the condition number in the 1-norm (or -norm) of the matrix A, where
A= ( 2.07 3.87 4.20 -1.15 3.87 -0.21 1.87 0.63 4.20 1.87 1.15 2.06 -1.15 0.63 2.06 -1.81 ) .  
Here A is symmetric indefinite, stored in packed form, and must first be factorized by f07pdc. The true condition number in the 1-norm is 75.68.

10.1 Program Text

Program Text (f07pgce.c)

10.2 Program Data

Program Data (f07pgce.d)

10.3 Program Results

Program Results (f07pgce.r)