F07JDF (DPTTRF) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F07JDF (DPTTRF)

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.

 Contents

    1  Purpose
    7  Accuracy

1  Purpose

F07JDF (DPTTRF) computes the modified Cholesky factorization of a real n  by n  symmetric positive definite tridiagonal matrix A .

2  Specification

SUBROUTINE F07JDF ( N, D, E, INFO)
INTEGER  N, INFO
REAL (KIND=nag_wp)  D(*), E(*)
The routine may be called by its LAPACK name dpttrf.

3  Description

F07JDF (DPTTRF) factorizes the matrix A  as
A=LDLT ,  
where L  is a unit lower bidiagonal matrix and D  is a diagonal matrix with positive diagonal elements. The factorization may also be regarded as having the form UTDU , where U  is a unit upper bidiagonal matrix.

4  References

None.

5  Arguments

1:     N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint: N0.
2:     D* – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array D must be at least max1,N.
On entry: must contain the n diagonal elements of the matrix A.
On exit: is overwritten by the n diagonal elements of the diagonal matrix D from the LDLT factorization of A.
3:     E* – REAL (KIND=nag_wp) arrayInput/Output
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 matrix A.
On exit: is overwritten by the n-1 subdiagonal elements of the lower bidiagonal matrix L. (E can also be regarded as containing the n-1 superdiagonal elements of the upper bidiagonal matrix U.)
4:     INFO – INTEGEROutput
On exit: INFO=0 unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

INFO<0
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
INFO>0andINFO<N
The leading minor of order value is not positive definite, the factorization could not be completed.
INFO>0andINFO=N
The leading minor of order n is not positive definite, the factorization was completed, but DN0.

7  Accuracy

The computed factorization satisfies an equation of the form
A+E=LDLT ,  
where
E=OεA  
and ε  is the machine precision.
Following the use of this routine, F07JEF (DPTTRS) can be used to solve systems of equations AX=B , and F07JGF (DPTCON) can be used to estimate the condition number of A .

8  Parallelism and Performance

F07JDF (DPTTRF) is not threaded in any implementation.

9  Further Comments

The total number of floating-point operations required to factorize the matrix A  is proportional to n .
The complex analogue of this routine is F07JRF (ZPTTRF).

10  Example

This example factorizes the symmetric positive definite tridiagonal matrix A  given by
A = 4.0 -2.0 0.0 0.0 0.0 -2.0 10.0 -6.0 0.0 0.0 0.0 -6.0 29.0 15.0 0.0 0.0 0.0 15.0 25.0 8.0 0.0 0.0 0.0 8.0 5.0 .  

10.1  Program Text

Program Text (f07jdfe.f90)

10.2  Program Data

Program Data (f07jdfe.d)

10.3  Program Results

Program Results (f07jdfe.r)


F07JDF (DPTTRF) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016