F07JRF (ZPTTRF) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F07JRF (ZPTTRF)

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

F07JRF (ZPTTRF) computes the modified Cholesky factorization of a complex n  by n  Hermitian positive definite tridiagonal matrix A .

2  Specification

SUBROUTINE F07JRF ( N, D, E, INFO)
INTEGER  N, INFO
REAL (KIND=nag_wp)  D(*)
COMPLEX (KIND=nag_wp)  E(*)
The routine may be called by its LAPACK name zpttrf.

3  Description

F07JRF (ZPTTRF) factorizes the matrix A  as
A=LDLH ,
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 UHDU , where U  is a unit upper bidiagonal matrix.

4  References

None.

5  Parameters

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 LDLH factorization of A.
3:     E(*) – COMPLEX (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

Errors or warnings detected by the routine:
INFO<0
If INFO=-i, the ith argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
INFO>0
If INFO=i, the leading minor of order i is not positive definite. If i<N, the factorization could not be completed, while if i=N, the factorization was completed, but DN0.

7  Accuracy

The computed factorization satisfies an equation of the form
A+E=LDLH ,
where
E=OεA
and ε  is the machine precision.
Following the use of this routine, F07JSF (ZPTTRS) can be used to solve systems of equations AX=B , and F07JUF (ZPTCON) can be used to estimate the condition number of A .

8  Further Comments

The total number of floating point operations required to factorize the matrix A  is proportional to n .
The real analogue of this routine is F07JDF (DPTTRF).

9  Example

This example factorizes the Hermitian positive definite tridiagonal matrix A  given by
A = 16.0i+00.0 16.0-16.0i 0.0i+0.0 0.0i+0.0 16.0+16.0i 41.0i+00.0 18.0+9.0i 0.0i+0.0 0.0i+00.0 18.0-09.0i 46.0i+0.0 1.0+4.0i 0.0i+00.0 0.0i+00.0 1.0-4.0i 21.0i+0.0 .

9.1  Program Text

Program Text (f07jrfe.f90)

9.2  Program Data

Program Data (f07jrfe.d)

9.3  Program Results

Program Results (f07jrfe.r)


F07JRF (ZPTTRF) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

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