NAG FL Interface
f04baf (real_square_solve)
1
Purpose
f04baf computes the solution to a real system of linear equations $AX=B$, where $A$ is an $n$ by $n$ matrix and $X$ and $B$ are $n$ by $r$ matrices. An estimate of the condition number of $A$ and an error bound for the computed solution are also returned.
2
Specification
Fortran Interface
Integer, Intent (In) 
:: 
n, nrhs, lda, ldb 
Integer, Intent (Inout) 
:: 
ifail 
Integer, Intent (Out) 
:: 
ipiv(n) 
Real (Kind=nag_wp), Intent (Inout) 
:: 
a(lda,*), b(ldb,*) 
Real (Kind=nag_wp), Intent (Out) 
:: 
rcond, errbnd 

C Header Interface
#include <nag.h>
void 
f04baf_ (const Integer *n, const Integer *nrhs, double a[], const Integer *lda, Integer ipiv[], double b[], const Integer *ldb, double *rcond, double *errbnd, Integer *ifail) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
f04baf_ (const Integer &n, const Integer &nrhs, double a[], const Integer &lda, Integer ipiv[], double b[], const Integer &ldb, double &rcond, double &errbnd, Integer &ifail) 
}

The routine may be called by the names f04baf or nagf_linsys_real_square_solve.
3
Description
The $LU$ decomposition with partial pivoting and row interchanges is used to factor $A$ as $A=PLU$, where $P$ is a permutation matrix, $L$ is unit lower triangular, and $U$ is upper triangular. 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
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
5
Arguments

1:
$\mathbf{n}$ – Integer
Input

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

2:
$\mathbf{nrhs}$ – Integer
Input

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

3:
$\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (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 $n$ coefficient matrix $A$.
On exit: if ${\mathbf{ifail}}\ge {\mathbf{0}}$, the factors $L$ and $U$ from the factorization $A=PLU$. The unit diagonal elements of $L$ are not stored.

4:
$\mathbf{lda}$ – Integer
Input

On entry: the first dimension of the array
a as declared in the (sub)program from which
f04baf is called.
Constraint:
${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

5:
$\mathbf{ipiv}\left({\mathbf{n}}\right)$ – Integer array
Output

On exit: if ${\mathbf{ifail}}\ge {\mathbf{0}}$, the pivot indices that define the permutation matrix $P$; 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.

6:
$\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Real (Kind=nag_wp) array
Input/Output

Note: the second dimension of the array
b
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On entry: the $n$ by $r$ matrix of righthand sides $B$.
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$ or ${\mathbf{n}+{\mathbf{1}}}$, the $n$ by $r$ solution matrix $X$.

7:
$\mathbf{ldb}$ – Integer
Input

On entry: the first dimension of the array
b as declared in the (sub)program from which
f04baf is called.
Constraint:
${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

8:
$\mathbf{rcond}$ – Real (Kind=nag_wp)
Output

On exit: if no constraints are violated, an estimate of the reciprocal of the condition number of the matrix $A$, computed as ${\mathbf{rcond}}=1/\left({\Vert A\Vert}_{1}{\Vert {A}^{1}\Vert}_{1}\right)$.

9:
$\mathbf{errbnd}$ – Real (Kind=nag_wp)
Output

On exit: if
${\mathbf{ifail}}={\mathbf{0}}$ or
${\mathbf{n}+{\mathbf{1}}}$, an estimate of the forward error bound for a computed solution
$\hat{x}$, such that
${\Vert \hat{x}x\Vert}_{1}/{\Vert x\Vert}_{1}\le {\mathbf{errbnd}}$, where
$\hat{x}$ is a column of the computed solution returned in the array
b and
$x$ is the corresponding column of the exact solution
$X$. If
rcond is less than
machine precision,
errbnd is returned as unity.

10:
$\mathbf{ifail}$ – Integer
Input/Output

On entry:
ifail must be set to
$0$,
$1$ or
$1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value
$1$ or
$1$ is recommended. If message printing is undesirable, then the value
$1$ is recommended. Otherwise, the value
$0$ is recommended.
When the value $\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit:
${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6
Error Indicators and Warnings
If on entry
${\mathbf{ifail}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
 ${\mathbf{ifail}}>0\hspace{0.17em}\text{and}\hspace{0.17em}{\mathbf{ifail}}\le {\mathbf{n}}$

Diagonal element $\u2329\mathit{\text{value}}\u232a$ of the upper triangular factor is zero. The factorization has been completed, but the solution could not be computed.
 ${\mathbf{ifail}}={\mathbf{n}}+1$

A solution has been computed, but
rcond is less than
machine precision so that the matrix
$A$ is numerically singular.
 ${\mathbf{ifail}}=1$

On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 0$.
 ${\mathbf{ifail}}=2$

On entry, ${\mathbf{nrhs}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
 ${\mathbf{ifail}}=4$

On entry, ${\mathbf{lda}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
 ${\mathbf{ifail}}=7$

On entry, ${\mathbf{ldb}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 7 in the Introduction to the NAG Library FL Interface for further information.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library FL Interface for further information.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
The integer allocatable memory required is
n, and the real allocatable memory required is
$4\times {\mathbf{n}}$. In this case the factorization and the solution
$X$ have been computed, but
rcond and
errbnd have not been computed.
See
Section 9 in the Introduction to the NAG Library FL Interface for further information.
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.
f04baf uses the approximation
${\Vert E\Vert}_{1}=\epsilon {\Vert A\Vert}_{1}$ to estimate
errbnd. See Section 4.4 of
Anderson et al. (1999) for further details.
8
Parallelism and Performance
f04baf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f04baf 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 required to solve the equations $AX=B$ is proportional to $\left(\frac{2}{3}{n}^{3}+{n}^{2}r\right)$. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
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 complex analogue of
f04baf is
f04caf.
10
Example
This example solves the equations
where
An estimate of the condition number of $A$ and an approximate error bound for the computed solutions are also printed.
10.1
Program Text
10.2
Program Data
10.3
Program Results