The routine may be called by the names f07apf, nagf_lapacklin_zgesvx or its LAPACK name zgesvx.
3Description
f07apf 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
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
$$A=PLU\text{,}$$
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 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.
4References
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
5Arguments
1: $\mathbf{fact}$ – 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: $\mathbf{trans}$ – 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: $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.
4: $\mathbf{nrhs}$ – IntegerInput
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Note: the second dimension of the array af
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
On entry: if ${\mathbf{fact}}=\text{'F'}$, af contains the factors $L$ and $U$ from the factorization $A=PLU$ as computed by f07arf. 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: $\mathbf{ldaf}$ – IntegerInput
On entry: the first dimension of the array af as declared in the (sub)program from which f07apf is called.
Note: the dimension of the array ipiv
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
On entry: if ${\mathbf{fact}}=\text{'F'}$, ipiv contains the pivot indices from the factorization $A=PLU$ as computed by f07arf; 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: $\mathbf{equed}$ – 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: $\mathbf{r}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array r
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
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: $\mathbf{c}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array c
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
On entry: if ${\mathbf{fact}}=\text{'N'}$ or $\text{'E'}$, c need not be set.
If ${\mathbf{fact}}=\text{'F'}$ and ${\mathbf{equed}}=\text{'C'}$ 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.
Note: the second dimension of the array x
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{nrhs}})$.
On exit: if ${\mathbf{info}}={\mathbf{0}}$ or ${\mathbf{n}+{\mathbf{1}}}$, the $n\times r$ 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: $\mathbf{ldx}$ – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which f07apf is called.
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 ${\mathbf{rcond}}=1.0/\left({\Vert A\Vert}_{1}{\Vert {A}^{-1}\Vert}_{1}\right)$.
18: $\mathbf{ferr}\left({\mathbf{nrhs}}\right)$ – 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 ${\Vert {\hat{x}}_{j}-{x}_{j}\Vert}_{\infty}/{\Vert {x}_{j}\Vert}_{\infty}\le {\mathbf{ferr}}\left(j\right)$ where ${\hat{x}}_{j}$ 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: $\mathbf{berr}\left({\mathbf{nrhs}}\right)$ – 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 ${\hat{x}}_{j}$ (i.e., the smallest relative change in any element of $A$ or $B$ that makes ${\hat{x}}_{j}$ an exact solution).
21: $\mathbf{rwork}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,2\times {\mathbf{n}})\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{rwork}}\left(1\right)$ contains the reciprocal pivot growth factor $\Vert A\Vert /\Vert U\Vert $. 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 ${\mathbf{info}}>{\mathbf{0}}\hspace{0.17em}\text{and}\hspace{0.17em}{\mathbf{info}}\le \mathbf{n}$, then ${\mathbf{rwork}}\left(1\right)$ contains the reciprocal pivot growth factor for the leading info columns of $A$.
22: $\mathbf{info}$ – IntegerOutput
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.
Element $\u27e8\mathit{\text{value}}\u27e9$ of the diagonal 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.
${\mathbf{info}}={\mathbf{n}}+1$
$U$ 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.
7Accuracy
For each right-hand side vector $b$, the computed solution $\hat{x}$ is the exact solution of a perturbed system of equations $(A+E)\hat{x}=b$, where
where
$\mathrm{cond}(A,\hat{x},b)={\Vert \left|{A}^{-1}\right|(\left|A\right|\left|\hat{x}\right|+\left|b\right|)\Vert}_{\infty}/{\Vert \hat{x}\Vert}_{\infty}\le \mathrm{cond}\left(A\right)={\Vert \left|{A}^{-1}\right|\left|A\right|\Vert}_{\infty}\le {\kappa}_{\infty}\left(A\right)$.
If $\hat{x}$ is the $j$th column of $X$, then ${w}_{c}$ is returned in ${\mathbf{berr}}\left(j\right)$ and a bound on ${\Vert x-\hat{x}\Vert}_{\infty}/{\Vert \hat{x}\Vert}_{\infty}$ is returned in ${\mathbf{ferr}}\left(j\right)$. See Section 4.4 of Anderson et al. (1999) for further details.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07apf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07apf 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.
9Further 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.
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.