nag_dptcon (f07jgc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_dptcon (f07jgc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dptcon (f07jgc) computes the reciprocal condition number of a real n  by n  symmetric positive definite tridiagonal matrix A , using the LDLT  factorization returned by nag_dpttrf (f07jdc).

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_dptcon (Integer n, const double d[], const double e[], double anorm, double *rcond, NagError *fail)

3  Description

nag_dptcon (f07jgc) should be preceded by a call to nag_dpttrf (f07jdc), which computes a modified Cholesky factorization of the matrix A  as
A=LDLT ,
where L  is a unit lower bidiagonal matrix and D  is a diagonal matrix, with positive diagonal elements. nag_dptcon (f07jgc) then utilizes the factorization to compute A-11  by a direct method, from which the reciprocal of the condition number of A , 1/κA  is computed as
1/κ1A=1 / A1 A-11 .
1/κA  is returned, rather than κA , since when A  is singular κA  is infinite.

4  References

Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

5  Arguments

1:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
2:     d[dim]const doubleInput
Note: the dimension, dim, of the array d must be at least max1,n.
On entry: must contain the n diagonal elements of the diagonal matrix D from the LDLT factorization of A.
3:     e[dim]const doubleInput
Note: the dimension, dim, of the array e must be at least max1,n-1.
On entry: must contain the n-1 subdiagonal elements of the unit lower bidiagonal matrix L. (e can also be regarded as the superdiagonal of the unit upper bidiagonal matrix U from the UTDU factorization of A.)
4:     anormdoubleInput
On entry: the 1-norm of the original matrix A, which may be computed as shown in Section 9. anorm must be computed either before calling nag_dpttrf (f07jdc) or else from a copy of the original matrix A.
Constraint: anorm0.0.
5:     rconddouble *Output
On exit: the reciprocal condition number, 1/κ1A=1/A1A-11.
6:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
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.
NE_REAL
On entry, anorm=value.
Constraint: anorm0.0.

7  Accuracy

The computed condition number will be the exact condition number for a closely neighbouring matrix.

8  Further Comments

The condition number estimation requires On  floating point operations.
See Section 15.6 of Higham (2002) for further details on computing the condition number of tridiagonal matrices.
The complex analogue of this function is nag_zptcon (f07juc).

9  Example

This example computes the condition number of the symmetric positive definite tridiagonal matrix A  given by
A = 4.0 -2.0 0 0 0 -2.0 10.0 -6.0 0 0 0 -6.0 29.0 15.0 0 0 0 15.0 25.0 8.0 0 0 0 8.0 5.0 .

9.1  Program Text

Program Text (f07jgce.c)

9.2  Program Data

Program Data (f07jgce.d)

9.3  Program Results

Program Results (f07jgce.r)


nag_dptcon (f07jgc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012