NAG Library Routine Document

1Purpose

f07gsf (zpptrs) solves a complex Hermitian positive definite system of linear equations with multiple right-hand sides,
 $AX=B ,$
where $A$ has been factorized by f07grf (zpptrf), using packed storage.

2Specification

Fortran Interface
 Subroutine f07gsf ( uplo, n, nrhs, ap, b, ldb, info)
 Integer, Intent (In) :: n, nrhs, ldb Integer, Intent (Out) :: info Complex (Kind=nag_wp), Intent (In) :: ap(*) Complex (Kind=nag_wp), Intent (Inout) :: b(ldb,*) Character (1), Intent (In) :: uplo
C Header Interface
#include nagmk26.h
 void f07gsf_ (const char *uplo, const Integer *n, const Integer *nrhs, const Complex ap[], Complex b[], const Integer *ldb, Integer *info, const Charlen length_uplo)
The routine may be called by its LAPACK name zpptrs.

3Description

f07gsf (zpptrs) is used to solve a complex Hermitian positive definite system of linear equations $AX=B$, the routine must be preceded by a call to f07grf (zpptrf) which computes the Cholesky factorization of $A$, using packed storage. The solution $X$ is computed by forward and backward substitution.
If ${\mathbf{uplo}}=\text{'U'}$, $A={U}^{\mathrm{H}}U$, where $U$ is upper triangular; the solution $X$ is computed by solving ${U}^{\mathrm{H}}Y=B$ and then $UX=Y$.
If ${\mathbf{uplo}}=\text{'L'}$, $A=L{L}^{\mathrm{H}}$, where $L$ is lower triangular; the solution $X$ is computed by solving $LY=B$ and then ${L}^{\mathrm{H}}X=Y$.

4References

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

5Arguments

1:     $\mathbf{uplo}$ – Character(1)Input
On entry: specifies how $A$ has been factorized.
${\mathbf{uplo}}=\text{'U'}$
$A={U}^{\mathrm{H}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A=L{L}^{\mathrm{H}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{nrhs}$ – IntegerInput
On entry: $r$, the number of right-hand sides.
Constraint: ${\mathbf{nrhs}}\ge 0$.
4:     $\mathbf{ap}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput
Note: the dimension of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
On entry: the Cholesky factor of $A$ stored in packed form, as returned by f07grf (zpptrf).
5:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On entry: the $n$ by $r$ right-hand side matrix $B$.
On exit: the $n$ by $r$ solution matrix $X$.
6:     $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07gsf (zpptrs) is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
7:     $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

7Accuracy

For each right-hand side vector $b$, the computed solution $x$ is the exact solution of a perturbed system of equations $\left(A+E\right)x=b$, where
• if ${\mathbf{uplo}}=\text{'U'}$, $\left|E\right|\le c\left(n\right)\epsilon \left|{U}^{\mathrm{H}}\right|\left|U\right|$;
• if ${\mathbf{uplo}}=\text{'L'}$, $\left|E\right|\le c\left(n\right)\epsilon \left|L\right|\left|{L}^{\mathrm{H}}\right|$,
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.
If $\stackrel{^}{x}$ is the true solution, then the computed solution $x$ satisfies a forward error bound of the form
 $x-x^∞ x∞ ≤cncondA,xε$
where $\mathrm{cond}\left(A,x\right)={‖\left|{A}^{-1}\right|\left|A\right|\left|x\right|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖\left|{A}^{-1}\right|\left|A\right|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$.
Note that $\mathrm{cond}\left(A,x\right)$ can be much smaller than $\mathrm{cond}\left(A\right)$.
Forward and backward error bounds can be computed by calling f07gvf (zpprfs), and an estimate for ${\kappa }_{\infty }\left(A\right)$ ($\text{}={\kappa }_{1}\left(A\right)$) can be obtained by calling f07guf (zppcon).

8Parallelism and Performance

f07gsf (zpptrs) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07gsf (zpptrs) 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.

9Further Comments

The total number of real floating-point operations is approximately $8{n}^{2}r$.
This routine may be followed by a call to f07gvf (zpprfs) to refine the solution and return an error estimate.
The real analogue of this routine is f07gef (dpptrs).

10Example

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 packed form, and must first be factorized by f07grf (zpptrf).

10.1Program Text

Program Text (f07gsfe.f90)

10.2Program Data

Program Data (f07gsfe.d)

10.3Program Results

Program Results (f07gsfe.r)

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