# NAG FL Interfacef07qsf (zsptrs)

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## 1Purpose

f07qsf solves a complex symmetric system of linear equations with multiple right-hand sides,
 $AX=B ,$
where $A$ has been factorized by f07qrf, using packed storage.

## 2Specification

Fortran Interface
 Subroutine f07qsf ( uplo, n, nrhs, ap, ipiv, b, ldb, info)
 Integer, Intent (In) :: n, nrhs, ipiv(*), 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
#include <nag.h>
 void f07qsf_ (const char *uplo, const Integer *n, const Integer *nrhs, const Complex ap[], const Integer ipiv[], Complex b[], const Integer *ldb, Integer *info, const Charlen length_uplo)
The routine may be called by the names f07qsf, nagf_lapacklin_zsptrs or its LAPACK name zsptrs.

## 3Description

f07qsf is used to solve a complex symmetric system of linear equations $AX=B$, the routine must be preceded by a call to f07qrf which computes the Bunch–Kaufman factorization of $A$, using packed storage.
If ${\mathbf{uplo}}=\text{'U'}$, $A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $P$ is a permutation matrix, $U$ is an upper triangular matrix and $D$ is a symmetric block diagonal matrix with $1×1$ and $2×2$ blocks; the solution $X$ is computed by solving $PUDY=B$ and then ${U}^{\mathrm{T}}{P}^{\mathrm{T}}X=Y$.
If ${\mathbf{uplo}}=\text{'L'}$, $A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $L$ is a lower triangular matrix; the solution $X$ is computed by solving $PLDY=B$ and then ${L}^{\mathrm{T}}{P}^{\mathrm{T}}X=Y$.

## 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.
${\mathbf{uplo}}=\text{'U'}$
$A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, 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: $\mathbf{ap}\left(*\right)$Complex (Kind=nag_wp) array Input
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 factorization of $A$ stored in packed form, as returned by f07qrf.
5: $\mathbf{ipiv}\left(*\right)$Integer array Input
Note: the dimension of the array ipiv must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: details of the interchanges and the block structure of $D$, as returned by f07qrf.
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 $n×r$ right-hand side matrix $B$.
On exit: the $n×r$ 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 f07qsf 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 ${\mathbf{uplo}}=\text{'U'}$, $|E|\le c\left(n\right)\epsilon P|U||D||{U}^{\mathrm{T}}|{P}^{\mathrm{T}}$;
• if ${\mathbf{uplo}}=\text{'L'}$, $|E|\le c\left(n\right)\epsilon P|L||D||{L}^{\mathrm{T}}|{P}^{\mathrm{T}}$,
$c\left(n\right)$ 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
 $‖x-x^‖∞ ‖x‖∞ ≤c(n)cond(A,x)ε$
where $\mathrm{cond}\left(A,x\right)={‖|{A}^{-1}||A||x|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖|{A}^{-1}||A|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$.
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 f07qvf, and an estimate for ${\kappa }_{\infty }\left(A\right)$ ($\text{}={\kappa }_{1}\left(A\right)$) can be obtained by calling f07quf.

## 8Parallelism and Performance

f07qsf 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 f07qvf to refine the solution and return an error estimate.
The real analogue of this routine is f07pef.

## 10Example

This example solves the system of equations $AX=B$, where
 $A= ( -0.39-0.71i 5.14-0.64i -7.86-2.96i 3.80+0.92i 5.14-0.64i 8.86+1.81i -3.52+0.58i 5.32-1.59i -7.86-2.96i -3.52+0.58i -2.83-0.03i -1.54-2.86i 3.80+0.92i 5.32-1.59i -1.54-2.86i -0.56+0.12i )$
and
 $B= ( -55.64+41.22i -19.09-35.97i -48.18+66.00i -12.08-27.02i -0.49-01.47i 6.95+20.49i -6.43+19.24i -4.59-35.53i ) .$
Here $A$ is symmetric, stored in packed form, and must first be factorized by f07qrf.

### 10.1Program Text

Program Text (f07qsfe.f90)

### 10.2Program Data

Program Data (f07qsfe.d)

### 10.3Program Results

Program Results (f07qsfe.r)