NAG FL Interface
f07jrf (zpttrf)

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1 Purpose

f07jrf computes the modified Cholesky factorization of a complex n × n Hermitian positive definite tridiagonal matrix A .

2 Specification

Fortran Interface
Subroutine f07jrf ( n, d, e, info)
Integer, Intent (In) :: n
Integer, Intent (Out) :: info
Real (Kind=nag_wp), Intent (Inout) :: d(*)
Complex (Kind=nag_wp), Intent (Inout) :: e(*)
C Header Interface
#include <nag.h>
void  f07jrf_ (const Integer *n, double d[], Complex e[], Integer *info)
The routine may be called by the names f07jrf, nagf_lapacklin_zpttrf or its LAPACK name zpttrf.

3 Description

f07jrf 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 Arguments

1: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
2: d(*) Real (Kind=nag_wp) array Input/Output
Note: the dimension of the array d must be at least max(1,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) array Input/Output
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 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 Integer Output
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 d(n)0.

7 Accuracy

The computed factorization satisfies an equation of the form
A+E=LDLH ,  
where
E=O(ε)A  
and ε is the machine precision.
Following the use of this routine, f07jsf can be used to solve systems of equations AX=B , and f07juf can be used to estimate the condition number of A .

8 Parallelism and Performance

f07jrf 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 real analogue of this routine is f07jdf.

10 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 ) .  

10.1 Program Text

Program Text (f07jrfe.f90)

10.2 Program Data

Program Data (f07jrfe.d)

10.3 Program Results

Program Results (f07jrfe.r)