NAG Library Routine Document

f07jdf  (dpttrf)

 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

Fortran Interface
Subroutine f07jdf ( n, d, e, info)
Integer, Intent (In):: n
Integer, Intent (Out):: info
Real (Kind=nag_wp), Intent (Inout):: d(*), e(*)
C Header Interface
#include nagmk26.h
void  f07jdf_ ( const Integer *n, double d[], double e[], Integer *info)
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)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017