NAG FL Interface
f07jgf (dptcon)

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1 Purpose

f07jgf computes the reciprocal condition number of a real n × n symmetric positive definite tridiagonal matrix A , using the LDLT factorization returned by f07jdf.

2 Specification

Fortran Interface
Subroutine f07jgf ( n, d, e, anorm, rcond, work, info)
Integer, Intent (In) :: n
Integer, Intent (Out) :: info
Real (Kind=nag_wp), Intent (In) :: d(*), e(*), anorm
Real (Kind=nag_wp), Intent (Out) :: rcond, work(n)
C Header Interface
#include <nag.h>
void  f07jgf_ (const Integer *n, const double d[], const double e[], const double *anorm, double *rcond, double work[], Integer *info)
The routine may be called by the names f07jgf, nagf_lapacklin_dptcon or its LAPACK name dptcon.

3 Description

f07jgf should be preceded by a call to f07jdf, 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 then utilizes the factorization to compute A-11 by a direct method, from which the reciprocal of the condition number of A , 1/κ(A) is computed as
1/κ1(A)=1 / (A1A-11) .  
1/κ(A) is returned, rather than κ(A) , since when A is singular κ(A) is infinite.

4 References

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

5 Arguments

1: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
2: d(*) Real (Kind=nag_wp) array Input
Note: the dimension of the array d must be at least max(1,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) array Input
Note: the dimension of the array e must be at least max(1,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 argument norm='1'. anorm must be computed either before calling f07jdf 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/κ1(A)=1/(A1A-11).
6: work(n) Real (Kind=nag_wp) array Workspace
7: info Integer Output
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

Background information to multithreading can be found in the Multithreading documentation.
f07jgf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The condition number estimation requires O(n) 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.

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)