# NAG FL Interfacef04aef (withdraw_​real_​square_​solve_​ref)

Note: this routine is deprecated and will be withdrawn at Mark 28. Replaced by f07abf.

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## 1Purpose

f04aef calculates the accurate solution of a set of real linear equations with multiple right-hand sides using an $LU$ factorization with partial pivoting, and iterative refinement.

## 2Specification

Fortran Interface
 Subroutine f04aef ( a, lda, b, ldb, n, m, c, ldc, aa, ldaa, bb, ldbb,
 Integer, Intent (In) :: lda, ldb, n, m, ldc, ldaa, ldbb Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: a(lda,*), b(ldb,*) Real (Kind=nag_wp), Intent (Inout) :: c(ldc,*), aa(ldaa,*), bb(ldbb,*) Real (Kind=nag_wp), Intent (Out) :: wkspce(max(1,n))
#include <nag.h>
 void f04aef_ (const double a[], const Integer *lda, const double b[], const Integer *ldb, const Integer *n, const Integer *m, double c[], const Integer *ldc, double wkspce[], double aa[], const Integer *ldaa, double bb[], const Integer *ldbb, Integer *ifail)
The routine may be called by the names f04aef or nagf_linsys_withdraw_real_square_solve_ref.

## 3Description

Given a set of real linear equations $AX=B$, the routine first computes an $LU$ factorization of $A$ with partial pivoting, $PA=LU$, where $P$ is a permutation matrix, $L$ is lower triangular and $U$ is unit upper triangular. An approximation to $X$ is found by forward and backward substitution. The residual matrix $R=B-AX$ is then calculated using additional precision, and a correction $D$ to $X$ is found by solving $LUD=PR$. $X$ is replaced by $X+D$ and this iterative refinement of the solution is repeated until full machine accuracy has been obtained.

## 4References

Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

## 5Arguments

1: $\mathbf{a}\left({\mathbf{lda}},*\right)$Real (Kind=nag_wp) array Input
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 $n×n$ matrix $A$.
2: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f04aef is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
3: $\mathbf{b}\left({\mathbf{ldb}},*\right)$Real (Kind=nag_wp) array Input
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On entry: the $n×m$ right-hand side matrix $B$.
4: $\mathbf{ldb}$Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f04aef is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
5: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
6: $\mathbf{m}$Integer Input
On entry: $m$, the number of right-hand sides.
Constraint: ${\mathbf{m}}\ge 0$.
7: $\mathbf{c}\left({\mathbf{ldc}},*\right)$Real (Kind=nag_wp) array Output
Note: the second dimension of the array c must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On exit: the $n×m$ solution matrix $X$.
8: $\mathbf{ldc}$Integer Input
On entry: the first dimension of the array c as declared in the (sub)program from which f04aef is called.
Constraint: ${\mathbf{ldc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9: $\mathbf{wkspce}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)\right)$Real (Kind=nag_wp) array Workspace
10: $\mathbf{aa}\left({\mathbf{ldaa}},*\right)$Real (Kind=nag_wp) array Output
Note: the second dimension of the array aa must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: the triangular factors $L$ and $U$, except that the unit diagonal elements of $U$ are not stored.
11: $\mathbf{ldaa}$Integer Input
On entry: the first dimension of the array aa as declared in the (sub)program from which f04aef is called.
Constraint: ${\mathbf{ldaa}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
12: $\mathbf{bb}\left({\mathbf{ldbb}},*\right)$Real (Kind=nag_wp) array Output
Note: the second dimension of the array bb must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On exit: the final $n×m$ residual matrix $R=B-AX$.
13: $\mathbf{ldbb}$Integer Input
On entry: the first dimension of the array bb as declared in the (sub)program from which f04aef is called.
Constraint: ${\mathbf{ldbb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
14: $\mathbf{ifail}$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).

## 6Error 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:
${\mathbf{ifail}}=1$
Matrix $A$ is approximately singular.
${\mathbf{ifail}}=2$
The matrix $A$ is too ill-conditioned.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{ldaa}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldaa}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{lda}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{ldb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{ldbb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldbb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{ldc}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-99$
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.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

The computed solutions should be correct to full machine accuracy. For a detailed error analysis see page 107 of Wilkinson and Reinsch (1971).

## 8Parallelism and Performance

f04aef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f04aef 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.

The time taken by f04aef is approximately proportional to ${n}^{3}$.
If there is only one right-hand side, it is simpler to use f04atf.

## 10Example

This example solves the set of linear equations $AX=B$ where
 $A=( 33 16 72 -24 -10 -57 -8 -4 -17 ) and B=( -359 281 85 ) .$

### 10.1Program Text

Program Text (f04aefe.f90)

### 10.2Program Data

Program Data (f04aefe.d)

### 10.3Program Results

Program Results (f04aefe.r)