NAG Library Routine Document

f07wsf (zpftrs)

1
Purpose

f07wsf (zpftrs) solves a complex Hermitian positive definite system of linear equations with multiple right-hand sides,
AX=B ,  
using the Cholesky factorization computed by f07wrf (zpftrf) stored in Rectangular Full Packed (RFP) format.

2
Specification

Fortran Interface
Subroutine f07wsf ( transr, uplo, n, nrhs, ar, b, ldb, info)
Integer, Intent (In):: n, nrhs, ldb
Integer, Intent (Out):: info
Complex (Kind=nag_wp), Intent (In):: ar(n*(n+1)/2)
Complex (Kind=nag_wp), Intent (Inout):: b(ldb,*)
Character (1), Intent (In):: transr, uplo
C Header Interface
#include <nagmk26.h>
void  f07wsf_ (const char *transr, const char *uplo, const Integer *n, const Integer *nrhs, const Complex ar[], Complex b[], const Integer *ldb, Integer *info, const Charlen length_transr, const Charlen length_uplo)
The routine may be called by its LAPACK name zpftrs.

3
Description

f07wsf (zpftrs) is used to solve a complex Hermitian positive definite system of linear equations AX=B, the routine must be preceded by a call to f07wrf (zpftrf) which computes the Cholesky factorization of A, stored in RFP format. The RFP storage format is described in Section 3.3.3 in the F07 Chapter Introduction. The solution X is computed by forward and backward substitution.
If uplo='U', A=UHU, where U is upper triangular; the solution X is computed by solving UHY=B and then UX=Y.
If uplo='L', A=LLH, where L is lower triangular; the solution X is computed by solving LY=B and then LHX=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
Arguments

1:     transr – Character(1)Input
On entry: specifies whether the normal RFP representation of A or its conjugate transpose is stored.
transr='N'
The matrix A is stored in normal RFP format.
transr='C'
The conjugate transpose of the RFP representation of the matrix A is stored.
Constraint: transr='N' or 'C'.
2:     uplo – Character(1)Input
On entry: specifies how A has been factorized.
uplo='U'
A=UHU, where U is upper triangular.
uplo='L'
A=LLH, 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:     arn×n+1/2 – Complex (Kind=nag_wp) arrayInput
On entry: the Cholesky factorization of A stored in RFP format, as returned by f07wrf (zpftrf).
6:     bldb* – Complex (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 f07wsf (zpftrs) 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

info<0
If info=-i, argument i 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
Parallelism and Performance

f07wsf (zpftrs) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9
Further Comments

The total number of real floating-point operations is approximately 8n2r.
The real analogue of this routine is f07wef (dpftrs).

10
Example

This example solves the system of equations AX=B, where
A= 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i  
and
B= 3.93-06.14i 1.48+06.58i 6.17+09.42i 4.65-04.75i -7.17-21.83i -4.91+02.29i 1.99-14.38i 7.64-10.79i .  
Here A is Hermitian positive definite, stored in RFP format, and must first be factorized by f07wrf (zpftrf).

10.1
Program Text

Program Text (f07wsfe.f90)

10.2
Program Data

Program Data (f07wsfe.d)

10.3
Program Results

Program Results (f07wsfe.r)