F07GEF (DPPTRS) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


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

F07GEF (DPPTRS) solves a real symmetric positive definite system of linear equations with multiple right-hand sides,
AX=B ,
where A has been factorized by F07GDF (DPPTRF), using packed storage.

2  Specification

REAL (KIND=nag_wp)  AP(*), B(LDB,*)
The routine may be called by its LAPACK name dpptrs.

3  Description

F07GEF (DPPTRS) is used to solve a real symmetric positive definite system of linear equations AX=B, the routine must be preceded by a call to F07GDF (DPPTRF) which computes the Cholesky factorization of A, using packed storage. The solution X is computed by forward and backward substitution.
If UPLO='U', A=UTU, where U is upper triangular; the solution X is computed by solving UTY=B and then UX=Y.
If UPLO='L', A=LLT, where L is lower triangular; the solution X is computed by solving LY=B and then LTX=Y.

4  References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5  Parameters

1:     UPLO – CHARACTER(1)Input
On entry: specifies how A has been factorized.
A=UTU, where U is upper triangular.
A=LLT, where L is lower triangular.
Constraint: UPLO='U' or 'L'.
2:     N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint: N0.
3:     NRHS – INTEGERInput
On entry: r, the number of right-hand sides.
Constraint: NRHS0.
4:     AP(*) – REAL (KIND=nag_wp) arrayInput
Note: the dimension of the array AP must be at least max1,N×N+1/2.
On entry: the Cholesky factor of A stored in packed form, as returned by F07GDF (DPPTRF).
5:     B(LDB,*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array B must be at least max1,NRHS.
On entry: the n by r right-hand side matrix B.
On exit: the n by r solution matrix X.
6:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07GEF (DPPTRS) is called.
Constraint: LDBmax1,N.
7:     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:
If INFO=-i, the ith parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.

7  Accuracy

For each right-hand side vector b, the computed solution x is the exact solution of a perturbed system of equations A+Ex=b, where cn is a modest linear function of n, and ε is the machine precision.
If x^ is the true solution, then the computed solution x satisfies a forward error bound of the form
x-x^ x cncondA,xε
where condA,x=A-1Ax/xcondA=A-1AκA.
Note that condA,x can be much smaller than condA.
Forward and backward error bounds can be computed by calling F07GHF (DPPRFS), and an estimate for κA (=κ1A) can be obtained by calling F07GGF (DPPCON).

8  Further Comments

The total number of floating point operations is approximately 2n2r.
This routine may be followed by a call to F07GHF (DPPRFS) to refine the solution and return an error estimate.
The complex analogue of this routine is F07GSF (ZPPTRS).

9  Example

This example solves the system of equations AX=B, where
A= 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18   and   B= 8.70 8.30 -13.35 2.13 1.89 1.61 -4.14 5.00 .
Here A is symmetric positive definite, stored in packed form, and must first be factorized by F07GDF (DPPTRF).

9.1  Program Text

Program Text (f07gefe.f90)

9.2  Program Data

Program Data (f07gefe.d)

9.3  Program Results

Program Results (f07gefe.r)

F07GEF (DPPTRS) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

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