F07WEF (DPFTRS) (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

F07WEF (DPFTRS) solves a real symmetric positive definite system of linear equations with multiple right-hand sides,
AX=B ,
where A has been factorized by F07WDF (DPFTRF), stored in Rectangular Full Packed (RFP) format. The RFP storage format is described in Section 3.3.3 in the F07 Chapter Introduction.

2  Specification

REAL (KIND=nag_wp)  A(N*(N+1)/2), B(LDB,*)
The routine may be called by its LAPACK name dpftrs.

3  Description

F07WEF (DPFTRS) is used to solve a real symmetric positive definite system of linear equations AX=B, the routine must be preceded by a call to F07WDF (DPFTRF) which computes the Cholesky factorization of A, where A is stored in RFP format. 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

Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software 37, 2

5  Parameters

1:     TRANSR – CHARACTER(1)Input
On entry: specifies whether the RFP representation of A is normal or transposed.
The matrix A is stored in normal RFP format.
The matrix A is stored in transposed RFP format.
Constraint: TRANSR='N' or 'T'.
2:     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'.
3:     N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint: N0.
4:     NRHS – INTEGERInput
On entry: r, the number of right-hand sides.
Constraint: NRHS0.
5:     A(N×N+1/2) – REAL (KIND=nag_wp) arrayInput
On entry: the Cholesky factorization of A stored in RFP format, as returned by F07WDF (DPFTRF).
6:     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.
7:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07WEF (DPFTRS) is called.
Constraint: LDBmax1,N.
8:     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 and κA is the condition number when using the -norm.
Note that condA,x can be much smaller than condA.

8  Further Comments

The total number of floating point operations is approximately 2n2r.
The complex analogue of this routine is F07WSF (ZPFTRS).

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 RFP format, and must first be factorized by F07WDF (DPFTRF).

9.1  Program Text

Program Text (f07wefe.f90)

9.2  Program Data

Program Data (f07wefe.d)

9.3  Program Results

Program Results (f07wefe.r)

F07WEF (DPFTRS) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

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