NAG FL Interface
f07nnf (zsysv)
1
Purpose
f07nnf computes the solution to a complex system of linear equations
where
$A$ is an
$n$ by
$n$ symmetric matrix and
$X$ and
$B$ are
$n$ by
$r$ matrices.
2
Specification
Fortran Interface
Subroutine f07nnf ( 
uplo, n, nrhs, a, lda, ipiv, b, ldb, work, lwork, info) 
Integer, Intent (In) 
:: 
n, nrhs, lda, ldb, lwork 
Integer, Intent (Inout) 
:: 
ipiv(*) 
Integer, Intent (Out) 
:: 
info 
Complex (Kind=nag_wp), Intent (Inout) 
:: 
a(lda,*), b(ldb,*) 
Complex (Kind=nag_wp), Intent (Out) 
:: 
work(max(1,lwork)) 
Character (1), Intent (In) 
:: 
uplo 

C Header Interface
#include <nag.h>
void 
f07nnf_ (const char *uplo, const Integer *n, const Integer *nrhs, Complex a[], const Integer *lda, Integer ipiv[], Complex b[], const Integer *ldb, Complex work[], const Integer *lwork, Integer *info, const Charlen length_uplo) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
f07nnf_ (const char *uplo, const Integer &n, const Integer &nrhs, Complex a[], const Integer &lda, Integer ipiv[], Complex b[], const Integer &ldb, Complex work[], const Integer &lwork, Integer &info, const Charlen length_uplo) 
}

The routine may be called by the names f07nnf, nagf_lapacklin_zsysv or its LAPACK name zsysv.
3
Description
f07nnf uses the diagonal pivoting method to factor $A$ as
$A=UD{U}^{\mathrm{T}}$ if ${\mathbf{uplo}}=\text{'U'}$ or $A=LD{L}^{\mathrm{T}}$ if ${\mathbf{uplo}}=\text{'L'}$,
where $U$ (or $L$) is a product of permutation and unit upper (lower) triangular matrices, and $D$ is symmetric and block diagonal with $1$ by $1$ and $2$ by $2$ diagonal blocks. The factored form of $A$ is then used to solve the system of equations $AX=B$.
4
References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
5
Arguments

1:
$\mathbf{uplo}$ – Character(1)
Input

On entry: if
${\mathbf{uplo}}=\text{'U'}$, the upper triangle of
$A$ is stored.
If ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of $A$ is stored.
Constraint:
${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.

2:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.

3:
$\mathbf{nrhs}$ – Integer
Input

On entry: $r$, the number of righthand sides, i.e., the number of columns of the matrix $B$.
Constraint:
${\mathbf{nrhs}}\ge 0$.

4:
$\mathbf{a}\left({\mathbf{lda}},*\right)$ – Complex (Kind=nag_wp) array
Input/Output

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
$n$ by
$n$ symmetric matrix
$A$.
 If ${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
 If ${\mathbf{uplo}}=\text{'L'}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.
On exit: if
${\mathbf{info}}={\mathbf{0}}$, the block diagonal matrix
$D$ and the multipliers used to obtain the factor
$U$ or
$L$ from the factorization
$A=UD{U}^{\mathrm{T}}$ or
$A=LD{L}^{\mathrm{T}}$ as computed by
f07nrf.

5:
$\mathbf{lda}$ – Integer
Input

On entry: the first dimension of the array
a as declared in the (sub)program from which
f07nnf is called.
Constraint:
${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

6:
$\mathbf{ipiv}\left(*\right)$ – Integer array
Output

Note: the dimension of the array
ipiv
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: details of the interchanges and the block structure of
$D$. More precisely,
 if ${\mathbf{ipiv}}\left(i\right)=k>0$, ${d}_{ii}$ is a $1$ by $1$ pivot block and the $i$th row and column of $A$ were interchanged with the $k$th row and column;
 if ${\mathbf{uplo}}=\text{'U'}$ and ${\mathbf{ipiv}}\left(i1\right)={\mathbf{ipiv}}\left(i\right)=l<0$, $\left(\begin{array}{cc}{d}_{i1,i1}& {\overline{d}}_{i,i1}\\ {\overline{d}}_{i,i1}& {d}_{ii}\end{array}\right)$ is a $2$ by $2$ pivot block and the $\left(i1\right)$th row and column of $A$ were interchanged with the $l$th row and column;
 if ${\mathbf{uplo}}=\text{'L'}$ and ${\mathbf{ipiv}}\left(i\right)={\mathbf{ipiv}}\left(i+1\right)=m<0$, $\left(\begin{array}{cc}{d}_{ii}& {d}_{i+1,i}\\ {d}_{i+1,i}& {d}_{i+1,i+1}\end{array}\right)$ is a $2$ by $2$ pivot block and the $\left(i+1\right)$th row and column of $A$ were interchanged with the $m$th row and column.

7:
$\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Complex (Kind=nag_wp) array
Input/Output

Note: the second dimension of the array
b
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
to solve the equations
$Ax=b$, where
$b$ is a single righthand side,
b may be supplied as a onedimensional array with length
${\mathbf{ldb}}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n$ by $r$ righthand side matrix $B$.
On exit: if ${\mathbf{info}}={\mathbf{0}}$, the $n$ by $r$ solution matrix $X$.

8:
$\mathbf{ldb}$ – Integer
Input

On entry: the first dimension of the array
b as declared in the (sub)program from which
f07nnf is called.
Constraint:
${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

9:
$\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – Complex (Kind=nag_wp) array
Workspace

On exit: if
${\mathbf{info}}={\mathbf{0}}$,
${\mathbf{work}}\left(1\right)$ returns the optimal
lwork.

10:
$\mathbf{lwork}$ – Integer
Input

On entry: the dimension of the array
work as declared in the (sub)program from which
f07nnf is called.
${\mathbf{lwork}}\ge 1$, and for best performance
${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\times \mathit{nb}\right)$, where
$\mathit{nb}$ is the optimal block size for
f07nrf.
If
${\mathbf{lwork}}=1$, a workspace query is assumed; the routine only calculates the optimal size of the
work array, returns this value as the first entry of the
work array, and no error message related to
lwork is issued.

11:
$\mathbf{info}$ – Integer
Output
On exit:
${\mathbf{info}}=0$ unless the routine detects an error (see
Section 6).
6
Error 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.
 ${\mathbf{info}}>0$

Element $\u2329\mathit{\text{value}}\u232a$ of the diagonal is exactly zero.
The factorization has been completed, but the block diagonal matrix $D$
is exactly singular, so the solution could not be computed.
7
Accuracy
The computed solution for a single righthand side,
$\hat{x}$, satisfies an equation of the form
where
and
$\epsilon $ is the
machine precision. An approximate error bound for the computed solution is given by
where
$\kappa \left(A\right)={\Vert {A}^{1}\Vert}_{1}{\Vert A\Vert}_{1}$, the condition number of
$A$ with respect to the solution of the linear equations. See Section 4.4 of
Anderson et al. (1999) and Chapter 11 of
Higham (2002) for further details.
f07npf is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively,
f04dhf solves
$Ax=b$ and returns a forward error bound and condition estimate.
f04dhf calls
f07nnf to solve the equations.
8
Parallelism and Performance
f07nnf 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 implementationspecific information.
The total number of floatingpoint operations is approximately $\frac{4}{3}{n}^{3}+8{n}^{2}r$, where $r$ is the number of righthand sides.
The real analogue of this routine is
f07maf. The complex Hermitian analogue of this routine is
f07mnf.
10
Example
This example solves the equations
where
$A$ is the complex symmetric matrix
and
Details of the factorization of $A$ are also output.
10.1
Program Text
10.2
Program Data
10.3
Program Results