F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07NSF (ZSYTRS)

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.

## 1  Purpose

F07NSF (ZSYTRS) solves a complex symmetric system of linear equations with multiple right-hand sides,
 $AX=B ,$
where $A$ has been factorized by F07NRF (ZSYTRF).

## 2  Specification

 SUBROUTINE F07NSF ( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
 INTEGER N, NRHS, LDA, IPIV(*), LDB, INFO COMPLEX (KIND=nag_wp) A(LDA,*), B(LDB,*) CHARACTER(1) UPLO
The routine may be called by its LAPACK name zsytrs.

## 3  Description

F07NSF (ZSYTRS) is used to solve a complex symmetric system of linear equations $AX=B$, this routine must be preceded by a call to F07NRF (ZSYTRF) which computes the Bunch–Kaufman factorization of $A$.
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 $1$ by $1$ and $2$ by $2$ 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 ${L}^{\mathrm{T}}{P}^{\mathrm{T}}X=Y$.

## 4  References

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

## 5  Parameters

1:     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:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     NRHS – INTEGERInput
On entry: $r$, the number of right-hand sides.
Constraint: ${\mathbf{NRHS}}\ge 0$.
4:     A(LDA,$*$) – COMPLEX (KIND=nag_wp) arrayInput
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: details of the factorization of $A$, as returned by F07NRF (ZSYTRF).
5:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F07NSF (ZSYTRS) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
6:     IPIV($*$) – INTEGER arrayInput
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 F07NRF (ZSYTRF).
7:     B(LDB,$*$) – 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 $X$.
8:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07NSF (ZSYTRS) is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
9:     INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, the $i$th 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 $\left(A+E\right)x=b$, where
• if ${\mathbf{UPLO}}=\text{'U'}$, $\left|E\right|\le c\left(n\right)\epsilon P\left|U\right|\left|D\right|\left|{U}^{\mathrm{T}}\right|{P}^{\mathrm{T}}$;
• if ${\mathbf{UPLO}}=\text{'L'}$, $\left|E\right|\le c\left(n\right)\epsilon P\left|L\right|\left|D\right|\left|{L}^{\mathrm{T}}\right|{P}^{\mathrm{T}}$,
$c\left(n\right)$ 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
 $x-x^∞ x∞ ≤cncondA,xε$
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 F07NVF (ZSYRFS), and an estimate for ${\kappa }_{\infty }\left(A\right)$ ($\text{}={\kappa }_{1}\left(A\right)$) can be obtained by calling F07NUF (ZSYCON).

The total number of real floating point operations is approximately $8{n}^{2}r$.
This routine may be followed by a call to F07NVF (ZSYRFS) to refine the solution and return an error estimate.
The real analogue of this routine is F07MEF (DSYTRS).

## 9  Example

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 and must first be factorized by F07NRF (ZSYTRF).

### 9.1  Program Text

Program Text (f07nsfe.f90)

### 9.2  Program Data

Program Data (f07nsfe.d)

### 9.3  Program Results

Program Results (f07nsfe.r)