F07APF (ZGESVX) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07APF (ZGESVX)

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

F07APF (ZGESVX) uses the $LU$ factorization to compute the solution to a complex system of linear equations
 $AX=B or ATX=B or AHX=B ,$
where $A$ is an $n$ by $n$ 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 F07APF ( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO)
 INTEGER N, NRHS, LDA, LDAF, IPIV(*), LDB, LDX, INFO REAL (KIND=nag_wp) R(*), C(*), RCOND, FERR(NRHS), BERR(NRHS), RWORK(max(1,2*N)) COMPLEX (KIND=nag_wp) A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), WORK(2*N) CHARACTER(1) FACT, TRANS, EQUED
The routine may be called by its LAPACK name zgesvx.

## 3  Description

F07APF (ZGESVX) performs the following steps:
1. Equilibration
The linear system to be solved may be badly scaled. However, the system can be equilibrated as a first stage by setting ${\mathbf{FACT}}=\text{'E'}$. In this case, real scaling factors are computed and these factors then determine whether the system is to be equilibrated. Equilibrated forms of the systems $AX=B$, ${A}^{\mathrm{T}}X=B$ and ${A}^{\mathrm{H}}X=B$ are
 $DR A DC DC-1X = DR B ,$
 $DR A DC T DR-1 X = DC B ,$
and
 $DR A DC H DR-1 X = DC B ,$
respectively, where ${D}_{R}$ and ${D}_{C}$ are diagonal matrices, with positive diagonal elements, formed from the computed scaling factors.
When equilibration is used, $A$ will be overwritten by ${D}_{R}A{D}_{C}$ and $B$ will be overwritten by ${D}_{R}B$ (or ${D}_{C}B$ when the solution of ${A}^{\mathrm{T}}X=B$ or ${A}^{\mathrm{H}}X=B$ is sought).
2. Factorization
The matrix $A$, or its scaled form, is copied and factored using the $LU$ decomposition
 $A=PLU ,$
where $P$ is a permutation matrix, $L$ is a unit lower triangular matrix, and $U$ is upper triangular.
This stage can be by-passed when a factored matrix (with scaled matrices and scaling factors) are supplied; for example, as provided by a previous call to F07APF (ZGESVX) with the same matrix $A$.
3. Condition Number Estimation
The $LU$ factorization of $A$ determines whether a solution to the linear system exists. If some diagonal element of $U$ is zero, then $U$ is exactly singular, no solution exists and the routine returns with a failure. Otherwise the factorized 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 then a warning code is returned on final exit.
4. Solution
The (equilibrated) system is solved for $X$ (${D}_{C}^{-1}X$ or ${D}_{R}^{-1}X$) using the factored form of $A$ (${D}_{R}A{D}_{C}$).
5. Iterative Refinement
Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for the computed solution.
6. Construct Solution Matrix $X$
If equilibration was used, the matrix $X$ is premultiplied by ${D}_{C}$ (if ${\mathbf{TRANS}}=\text{'N'}$) or ${D}_{R}$ (if ${\mathbf{TRANS}}=\text{'T'}$ or $\text{'C'}$) so that it solves the original system before equilibration.

## 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 $A$ is supplied on entry, and if not, whether the matrix $A$ should be equilibrated before it is factorized.
${\mathbf{FACT}}=\text{'F'}$
AF and IPIV contain the factorized form of $A$. If ${\mathbf{EQUED}}\ne \text{'N'}$, the matrix $A$ has been equilibrated with scaling factors given by R and C. A, AF and IPIV are not modified.
${\mathbf{FACT}}=\text{'N'}$
The matrix $A$ will be copied to AF and factorized.
${\mathbf{FACT}}=\text{'E'}$
The matrix $A$ will be equilibrated if necessary, then copied to AF and factorized.
Constraint: ${\mathbf{FACT}}=\text{'F'}$, $\text{'N'}$ or $\text{'E'}$.
2:     TRANS – CHARACTER(1)Input
On entry: specifies the form of the system of equations.
${\mathbf{TRANS}}=\text{'N'}$
$AX=B$ (No transpose).
${\mathbf{TRANS}}=\text{'T'}$
${A}^{\mathrm{T}}X=B$ (Transpose).
${\mathbf{TRANS}}=\text{'C'}$
${A}^{\mathrm{H}}X=B$ (Conjugate transpose).
Constraint: ${\mathbf{TRANS}}=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.
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/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 $n$ matrix $A$.
If ${\mathbf{FACT}}=\text{'F'}$ and ${\mathbf{EQUED}}\ne \text{'N'}$, A must have been equilibrated by the scaling factors in R and/or C.
On exit: if ${\mathbf{FACT}}=\text{'F'}$ or $\text{'N'}$, or if ${\mathbf{FACT}}=\text{'E'}$ and ${\mathbf{EQUED}}=\text{'N'}$, A is not modified.
If ${\mathbf{FACT}}=\text{'E'}$ or ${\mathbf{EQUED}}\ne \text{'N'}$, $A$ is scaled as follows:
• if ${\mathbf{EQUED}}=\text{'R'}$, $A={D}_{R}A$;
• if ${\mathbf{EQUED}}=\text{'C'}$, $A=A{D}_{C}$;
• if ${\mathbf{EQUED}}=\text{'B'}$, $A={D}_{R}A{D}_{C}$.
6:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F07APF (ZGESVX) 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 factors $L$ and $U$ from the factorization $A=PLU$ as computed by F07ARF (ZGETRF). If ${\mathbf{EQUED}}\ne \text{'N'}$, AF is the factorized form of the equilibrated matrix $A$.
If ${\mathbf{FACT}}=\text{'N'}$ or $\text{'E'}$, AF need not be set.
On exit: if ${\mathbf{FACT}}=\text{'N'}$, AF returns the factors $L$ and $U$ from the factorization $A=PLU$ of the original matrix $A$.
If ${\mathbf{FACT}}=\text{'E'}$, AF returns the factors $L$ and $U$ from the factorization $A=PLU$ of the equilibrated matrix $A$ (see the description of A for the form of the equilibrated matrix).
If ${\mathbf{FACT}}=\text{'F'}$, AF is unchanged from entry.
8:     LDAF – INTEGERInput
On entry: the first dimension of the array AF as declared in the (sub)program from which F07APF (ZGESVX) 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 the pivot indices from the factorization $A=PLU$ as computed by F07ARF (ZGETRF); at the $i$th step row $i$ of the matrix was interchanged with row ${\mathbf{IPIV}}\left(i\right)$. ${\mathbf{IPIV}}\left(i\right)=i$ indicates a row interchange was not required.
If ${\mathbf{FACT}}=\text{'N'}$ or $\text{'E'}$, IPIV need not be set.
On exit: if ${\mathbf{FACT}}=\text{'N'}$, IPIV contains the pivot indices from the factorization $A=PLU$ of the original matrix $A$.
If ${\mathbf{FACT}}=\text{'E'}$, IPIV contains the pivot indices from the factorization $A=PLU$ of the equilibrated matrix $A$.
If ${\mathbf{FACT}}=\text{'F'}$, IPIV is unchanged from entry.
10:   EQUED – CHARACTER(1)Input/Output
On entry: if ${\mathbf{FACT}}=\text{'N'}$ or $\text{'E'}$, EQUED need not be set.
If ${\mathbf{FACT}}=\text{'F'}$, EQUED must specify the form of the equilibration that was performed as follows:
• if ${\mathbf{EQUED}}=\text{'N'}$, no equilibration;
• if ${\mathbf{EQUED}}=\text{'R'}$, row equilibration, i.e., $A$ has been premultiplied by ${D}_{R}$;
• if ${\mathbf{EQUED}}=\text{'C'}$, column equilibration, i.e., $A$ has been postmultiplied by ${D}_{C}$;
• if ${\mathbf{EQUED}}=\text{'B'}$, both row and column equilibration, i.e., $A$ has been replaced by ${D}_{R}A{D}_{C}$.
On exit: if ${\mathbf{FACT}}=\text{'F'}$, EQUED is unchanged from entry.
Otherwise, if no constraints are violated, EQUED specifies the form of equilibration that was performed as specified above.
Constraint: if ${\mathbf{FACT}}=\text{'F'}$, ${\mathbf{EQUED}}=\text{'N'}$, $\text{'R'}$, $\text{'C'}$ or $\text{'B'}$.
11:   R($*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array R must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: if ${\mathbf{FACT}}=\text{'N'}$ or $\text{'E'}$, R need not be set.
If ${\mathbf{FACT}}=\text{'F'}$ and ${\mathbf{EQUED}}=\text{'R'}$ or $\text{'B'}$, R must contain the row scale factors for $A$, ${D}_{R}$; each element of R must be positive.
On exit: if ${\mathbf{FACT}}=\text{'F'}$, R is unchanged from entry.
Otherwise, if no constraints are violated and ${\mathbf{EQUED}}=\text{'R'}$ or $\text{'B'}$, R contains the row scale factors for $A$, ${D}_{R}$, such that $A$ is multiplied on the left by ${D}_{R}$; each element of R is positive.
12:   C($*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array C must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: if ${\mathbf{FACT}}=\text{'N'}$ or $\text{'E'}$, C need not be set.
If ${\mathbf{FACT}}=\text{'F'}$ or ${\mathbf{EQUED}}=\text{'C'}$ or $\text{'B'}$, C must contain the column scale factors for $A$, ${D}_{C}$; each element of C must be positive.
On exit: if ${\mathbf{FACT}}=\text{'F'}$, C is unchanged from entry.
Otherwise, if no constraints are violated and ${\mathbf{EQUED}}=\text{'C'}$ or $\text{'B'}$, C contains the row scale factors for $A$, ${D}_{C}$; each element of C is positive.
13:   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: if ${\mathbf{EQUED}}=\text{'N'}$, B is not modified.
If ${\mathbf{TRANS}}=\text{'N'}$ and ${\mathbf{EQUED}}=\text{'R'}$ or $\text{'B'}$, B is overwritten by ${D}_{R}B$.
If ${\mathbf{TRANS}}=\text{'T'}$ or $\text{'C'}$ and ${\mathbf{EQUED}}=\text{'C'}$ or $\text{'B'}$, B is overwritten by ${D}_{C}B$.
14:   LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07APF (ZGESVX) is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
15:   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 $X$ to the original system of equations. Note that the arrays $A$ and $B$ are modified on exit if ${\mathbf{EQUED}}\ne \text{'N'}$, and the solution to the equilibrated system is ${D}_{C}^{-1}X$ if ${\mathbf{TRANS}}=\text{'N'}$ and ${\mathbf{EQUED}}=\text{'C'}$ or $\text{'B'}$, or ${D}_{R}^{-1}X$ if ${\mathbf{TRANS}}=\text{'T'}$ or $\text{'C'}$ and ${\mathbf{EQUED}}=\text{'R'}$ or $\text{'B'}$.
16:   LDX – INTEGERInput
On entry: the first dimension of the array X as declared in the (sub)program from which F07APF (ZGESVX) is called.
Constraint: ${\mathbf{LDX}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
17:   RCOND – REAL (KIND=nag_wp)Output
On exit: if no constraints are violated, an estimate of the reciprocal condition number of the matrix $A$ (after equilibration if that is performed), computed as ${\mathbf{RCOND}}=1.0/\left({‖A‖}_{1}{‖{A}^{-1}‖}_{1}\right)$.
18:   FERR(NRHS) – REAL (KIND=nag_wp) arrayOutput
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.
19:   BERR(NRHS) – REAL (KIND=nag_wp) arrayOutput
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).
20:   WORK($2×{\mathbf{N}}$) – COMPLEX (KIND=nag_wp) arrayWorkspace
21:   RWORK($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2×{\mathbf{N}}\right)$) – REAL (KIND=nag_wp) arrayOutput
On exit: ${\mathbf{RWORK}}\left(1\right)$ contains the reciprocal pivot growth factor $‖A‖/‖U‖$. The ‘max absolute element’ norm is used. If ${\mathbf{RWORK}}\left(1\right)$ is much less than $1$, then the stability of the $LU$ factorization of the (equilibrated) matrix $A$ could be poor. This also means that the solution X, condition estimator RCOND, and forward error bound FERR could be unreliable. If factorization fails with , then ${\mathbf{RWORK}}\left(1\right)$ contains the reciprocal pivot growth factor for the leading INFO columns of $A$.
22:   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 ${\mathbf{INFO}}=i$, ${u}_{ii}$ is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, so the solution and error bounds could not be computed. ${\mathbf{RCOND}}=0.0$ is returned.
${\mathbf{INFO}}={\mathbf{N}}+1$
The triangular matrix $U$ 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
 $E≤cnεPLU ,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision. See Section 9.3 of Higham (2002) for further details.
If $x$ is the true solution, then the computed solution $\stackrel{^}{x}$ 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 $\frac{8}{3}{n}^{3}$ floating point operations.
Estimating the forward error involves solving a number of systems of linear equations of the form $Ax=b$ or ${A}^{\mathrm{T}}x=b$; the number is usually $4$ or $5$ and never more than $11$. Each solution involves approximately $8{n}^{2}$ operations.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.
The real analogue of this routine is F07ABF (DGESVX).

## 9  Example

This example solves the equations
 $AX=B ,$
where $A$ is the general matrix
 $A= -1.34+02.55i 0.28+3.17i -6.39-02.20i 0.72-00.92i -1.70-14.10i 33.10-1.50i -1.50+13.40i 12.90+13.80i -3.29-02.39i -1.91+4.42i -0.14-01.35i 1.72+01.35i 2.41+00.39i -0.56+1.47i -0.83-00.69i -1.96+00.67i$
and
 $B= 26.26+51.78i 31.32-06.70i 64.30-86.80i 158.60-14.20i -5.75+25.31i -2.15+30.19i 1.16+02.57i -2.56+07.55i .$
Error estimates for the solutions, information on scaling, an estimate of the reciprocal of the condition number of the scaled matrix $A$ and an estimate of the reciprocal of the pivot growth factor for the factorization of $A$ are also output.

### 9.1  Program Text

Program Text (f07apfe.f90)

### 9.2  Program Data

Program Data (f07apfe.d)

### 9.3  Program Results

Program Results (f07apfe.r)

F07APF (ZGESVX) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual