F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07NPF (ZSYSVX)

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

F07NPF (ZSYSVX) uses the diagonal pivoting factorization to compute the solution to a complex system of linear equations
 $AX=B ,$
where $A$ is an $n$ by $n$ symmetric matrix and $X$ and $B$ are $n$ by $r$ matrices. Error bounds on the solution and a condition estimate are also provided.

## 2  Specification

 SUBROUTINE F07NPF ( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK, RWORK, INFO)
 INTEGER N, NRHS, LDA, LDAF, IPIV(*), LDB, LDX, LWORK, INFO REAL (KIND=nag_wp) RCOND, FERR(*), BERR(*), RWORK(*) COMPLEX (KIND=nag_wp) A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), WORK(max(1,LWORK)) CHARACTER(1) FACT, UPLO
The routine may be called by its LAPACK name zsysvx.

## 3  Description

F07NPF (ZSYSVX) performs the following steps:
1. If ${\mathbf{FACT}}=\text{'N'}$, the diagonal pivoting method is used to factor $A$. The form of the factorization is $A=UD{U}^{\mathrm{T}}$ 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.
2. If some ${d}_{ii}=0$, so that $D$ is exactly singular, then the routine returns with ${\mathbf{INFO}}=i$. Otherwise, the factored form of $A$ is used to estimate the condition number of the matrix $A$. If the reciprocal of the condition number is less than machine precision, ${\mathbf{INFO}}=\mathbf{N}+{\mathbf{1}}$ is returned as a warning, but the routine still goes on to solve for $X$ and compute error bounds as described below.
3. The system of equations is solved for $X$ using the factored form of $A$.
4. Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.

## 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 http://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  Parameters

1:     FACT – CHARACTER(1)Input
On entry: specifies whether or not the factorized form of the matrix $A$ has been supplied.
${\mathbf{FACT}}=\text{'F'}$
AF and IPIV contain the factorized form of the matrix $A$. AF and IPIV will not be modified.
${\mathbf{FACT}}=\text{'N'}$
The matrix $A$ will be copied to AF and factorized.
Constraint: ${\mathbf{FACT}}=\text{'F'}$ or $\text{'N'}$.
2:     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 $\text{'L'}$.
3:     N – INTEGERInput
On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
4:     NRHS – INTEGERInput
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{NRHS}}\ge 0$.
5:     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: 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.
6:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F07NPF (ZSYSVX) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
7:     AF(LDAF,$*$) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array AF must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: if ${\mathbf{FACT}}=\text{'F'}$, AF contains the block diagonal matrix $D$ and the multipliers used to obtain the factor $U$ or $L$ from the factorization ${\mathbf{A}}=UD{U}^{\mathrm{T}}$ or ${\mathbf{A}}=LD{L}^{\mathrm{T}}$ as computed by F07NRF (ZSYTRF).
On exit: if ${\mathbf{FACT}}=\text{'N'}$, AF returns the block diagonal matrix $D$ and the multipliers used to obtain the factor $U$ or $L$ from the factorization ${\mathbf{A}}=UD{U}^{\mathrm{T}}$ or ${\mathbf{A}}=LD{L}^{\mathrm{T}}$.
8:     LDAF – INTEGERInput
On entry: the first dimension of the array AF as declared in the (sub)program from which F07NPF (ZSYSVX) is called.
Constraint: ${\mathbf{LDAF}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
9:     IPIV($*$) – INTEGER arrayInput/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 entry: if ${\mathbf{FACT}}=\text{'F'}$, IPIV contains details of the interchanges and the block structure of $D$, as determined by F07NRF (ZSYTRF).
• 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 ${\mathbf{IPIV}}\left(i-1\right)={\mathbf{IPIV}}\left(i\right)=-l<0$, $\left(\begin{array}{cc}{d}_{i-1,i-1}& {\stackrel{-}{d}}_{i,i-1}\\ {\stackrel{-}{d}}_{i,i-1}& {d}_{ii}\end{array}\right)$ is a $2$ by $2$ pivot block and the $\left(i-1\right)$th row and column of $A$ were interchanged with the $l$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.
On exit: if ${\mathbf{FACT}}=\text{'N'}$, IPIV contains details of the interchanges and the block structure of $D$, as determined by F07NRF (ZSYTRF), as described above.
10:   B(LDB,$*$) – COMPLEX (KIND=nag_wp) arrayInput
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 $B$.
11:   LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07NPF (ZSYSVX) is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
12:   X(LDX,$*$) – COMPLEX (KIND=nag_wp) arrayOutput
Note: the second dimension of the array X must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{NRHS}}\right)$.
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$ or $\mathbf{N}+{\mathbf{1}}$, the $n$ by $r$ solution matrix $X$.
13:   LDX – INTEGERInput
On entry: the first dimension of the array X as declared in the (sub)program from which F07NPF (ZSYSVX) is called.
Constraint: ${\mathbf{LDX}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
14:   RCOND – REAL (KIND=nag_wp)Output
On exit: the estimate of the reciprocal condition number of the matrix $A$. If ${\mathbf{RCOND}}=0.0$, the matrix may be exactly singular. This condition is indicated by . Otherwise, if RCOND is less than the machine precision, the matrix is singular to working precision. This condition is indicated by ${\mathbf{INFO}}=\mathbf{N}+{\mathbf{1}}$.
15:   FERR($*$) – REAL (KIND=nag_wp) arrayOutput
Note: the dimension of the array FERR must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{NRHS}}\right)$.
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$ or $\mathbf{N}+{\mathbf{1}}$, an estimate of the forward error bound for each computed solution vector, such that ${‖{\stackrel{^}{x}}_{j}-{x}_{j}‖}_{\infty }/{‖{x}_{j}‖}_{\infty }\le {\mathbf{FERR}}\left(j\right)$ where ${\stackrel{^}{x}}_{j}$ is the $j$th column of the computed solution returned in the array X and ${x}_{j}$ is the corresponding column of the exact solution $X$. The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error.
16:   BERR($*$) – REAL (KIND=nag_wp) arrayOutput
Note: the dimension of the array BERR must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{NRHS}}\right)$.
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$ or $\mathbf{N}+{\mathbf{1}}$, an estimate of the component-wise relative backward error of each computed solution vector ${\stackrel{^}{x}}_{j}$ (i.e., the smallest relative change in any element of $A$ or $B$ that makes ${\stackrel{^}{x}}_{j}$ an exact solution).
17:   WORK($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LWORK}}\right)$) – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, ${\mathbf{WORK}}\left(1\right)$ returns the optimal LWORK.
18:   LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F07NPF (ZSYSVX) is called.
${\mathbf{LWORK}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2×{\mathbf{N}}\right)$, and for best performance, when ${\mathbf{FACT}}=\text{'N'}$, ${\mathbf{LWORK}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2×{\mathbf{N}},{\mathbf{N}}×\mathit{nb}\right)$, where $\mathit{nb}$ is the optimal block size for F07NRF (ZSYTRF).
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.
19:   RWORK($*$) – REAL (KIND=nag_wp) arrayWorkspace
Note: the dimension of the array RWORK must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
20:   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 argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
If ${\mathbf{INFO}}\le {\mathbf{N}}$, $d\left(i,i\right)$ is exactly zero. The factorization has been completed, but the factor $D$ is exactly singular, so the solution and error bounds could not be computed. ${\mathbf{RCOND}}=0.0$ is returned.
${\mathbf{INFO}}={\mathbf{N}}+1$
$D$ is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of RCOND would suggest.

## 7  Accuracy

For each right-hand side vector $b$, the computed solution $\stackrel{^}{x}$ is the exact solution of a perturbed system of equations $\left(A+E\right)\stackrel{^}{x}=b$, where
 $E1 = Oε A1 ,$
where $\epsilon$ is the machine precision. See Chapter 11 of Higham (2002) for further details.
If $\stackrel{^}{x}$ is the true solution, then the computed solution $x$ satisfies a forward error bound of the form
 $x-x^∞ x^∞ ≤ wc condA,x^,b$
where $\mathrm{cond}\left(A,\stackrel{^}{x},b\right)={‖\left|{A}^{-1}\right|\left(\left|A\right|\left|\stackrel{^}{x}\right|+\left|b\right|\right)‖}_{\infty }/{‖\stackrel{^}{x}‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖\left|{A}^{-1}\right|\left|A\right|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$. If $\stackrel{^}{x}$ is the $j$th column of $X$, then ${w}_{c}$ is returned in ${\mathbf{BERR}}\left(j\right)$ and a bound on ${‖x-\stackrel{^}{x}‖}_{\infty }/{‖\stackrel{^}{x}‖}_{\infty }$ is returned in ${\mathbf{FERR}}\left(j\right)$. See Section 4.4 of Anderson et al. (1999) for further details.

## 8  Further Comments

The factorization of $A$ requires approximately $\frac{4}{3}{n}^{3}$ floating point operations.
For each right-hand side, computation of the backward error involves a minimum of $16{n}^{2}$ floating point operations. Each step of iterative refinement involves an additional $24{n}^{2}$ operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required. Estimating the forward error involves solving a number of systems of equations of the form $Ax=b$; the number is usually $4$ or $5$ and never more than $11$. Each solution involves approximately $8{n}^{2}$ operations.
The real analogue of this routine is F07MBF (DSYSVX). The complex Hermitian analogue of this routine is F07MPF (ZHESVX).

## 9  Example

This example solves the equations
 $AX=B ,$
where $A$ is the complex symmetric matrix
 $A = -0.56+0.12i -1.54-2.86i 5.32-1.59i 3.80+0.92i -1.54-2.86i -2.83-0.03i -3.52+0.58i -7.86-2.96i 5.32-1.59i -3.52+0.58i 8.86+1.81i 5.14-0.64i 3.80+0.92i -7.86-2.96i 5.14-0.64i -0.39-0.71i$
and
 $B = -6.43+19.24i -4.59-35.53i -0.49-01.47i 6.95+20.49i -48.18+66.00i -12.08-27.02i -55.64+41.22i -19.09-35.97i .$
Error estimates for the solutions, and an estimate of the reciprocal of the condition number of the matrix $A$ are also output.

### 9.1  Program Text

Program Text (f07npfe.f90)

### 9.2  Program Data

Program Data (f07npfe.d)

### 9.3  Program Results

Program Results (f07npfe.r)