F07JGF (DPTCON) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.


    1  Purpose
    7  Accuracy

1  Purpose

F07JGF (DPTCON) computes the reciprocal condition number of a real n  by n  symmetric positive definite tridiagonal matrix A , using the LDLT  factorization returned by F07JDF (DPTTRF).

2  Specification

REAL (KIND=nag_wp)  D(*), E(*), ANORM, RCOND, WORK(N)
The routine may be called by its LAPACK name dptcon.

3  Description

F07JGF (DPTCON) should be preceded by a call to F07JDF (DPTTRF), 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. F07JGF (DPTCON) 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  Parameters

1:     N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint: N0.
2:     D* – REAL (KIND=nag_wp) arrayInput
Note: the dimension 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* – REAL (KIND=nag_wp) arrayInput
Note: the dimension 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:     ANORM – REAL (KIND=nag_wp)Input
On entry: the 1-norm of the original matrix A, which may be computed by calling F06RPF with its parameter NORM='1'. ANORM must be computed either before calling F07JDF (DPTTRF) or else from a copy of the original matrix A.
Constraint: ANORM0.0.
5:     RCOND – REAL (KIND=nag_wp)Output
On exit: the reciprocal condition number, 1/κ1A=1/A1A-11.
6:     WORKN – REAL (KIND=nag_wp) arrayWorkspace
7:     INFO – INTEGEROutput
On exit: INFO=0 unless the routine detects an error (see Section 6).

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.

7  Accuracy

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

8  Parallelism and Performance

Not applicable.

9  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 routine is F07JUF (ZPTCON).

10  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 .  

10.1  Program Text

Program Text (f07jgfe.f90)

10.2  Program Data

Program Data (f07jgfe.d)

10.3  Program Results

Program Results (f07jgfe.r)

F07JGF (DPTCON) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2015