# NAG FL Interfacef07fsf (zpotrs)

## 1Purpose

f07fsf solves a complex Hermitian positive definite system of linear equations with multiple right-hand sides,
 $AX=B ,$
where $A$ has been factorized by f07frf.

## 2Specification

Fortran Interface
 Subroutine f07fsf ( uplo, n, nrhs, a, lda, b, ldb, info)
 Integer, Intent (In) :: n, nrhs, lda, ldb Integer, Intent (Out) :: info Complex (Kind=nag_wp), Intent (In) :: a(lda,*) Complex (Kind=nag_wp), Intent (Inout) :: b(ldb,*) Character (1), Intent (In) :: uplo
#include <nag.h>
 void f07fsf_ (const char *uplo, const Integer *n, const Integer *nrhs, const Complex a[], const Integer *lda, Complex b[], const Integer *ldb, Integer *info, const Charlen length_uplo)
The routine may be called by the names f07fsf, nagf_lapacklin_zpotrs or its LAPACK name zpotrs.

## 3Description

f07fsf is used to solve a complex Hermitian positive definite system of linear equations $AX=B$, this routine must be preceded by a call to f07frf which computes the Cholesky factorization of $A$. 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: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{nrhs}$Integer Input
On entry: $r$, the number of right-hand sides.
Constraint: ${\mathbf{nrhs}}\ge 0$.
4: $\mathbf{a}\left({\mathbf{lda}},*\right)$Complex (Kind=nag_wp) array Input
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the Cholesky factor of $A$, as returned by f07frf.
5: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f07fsf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
6: $\mathbf{b}\left({\mathbf{ldb}},*\right)$Complex (Kind=nag_wp) array Input/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 $X$.
7: $\mathbf{ldb}$Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f07fsf is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
8: $\mathbf{info}$Integer Output
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 f07fvf, and an estimate for ${\kappa }_{\infty }\left(A\right)$ ($\text{}={\kappa }_{1}\left(A\right)$) can be obtained by calling f07fuf.

## 8Parallelism and Performance

f07fsf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07fsf 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.

The total number of real floating-point operations is approximately $8{n}^{2}r$.
This routine may be followed by a call to f07fvf to refine the solution and return an error estimate.
The real analogue of this routine is f07fef.

## 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 and must first be factorized by f07frf.

### 10.1Program Text

Program Text (f07fsfe.f90)

### 10.2Program Data

Program Data (f07fsfe.d)

### 10.3Program Results

Program Results (f07fsfe.r)