NAG FL Interface
f04aef (withdraw_​real_​square_​solve_​ref)

Note: this routine is deprecated and will be withdrawn at Mark 28.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

1 Purpose

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.

2 Specification

Fortran Interface
Subroutine f04aef ( a, lda, b, ldb, n, m, c, ldc, wkspce, aa, ldaa, bb, ldbb, ifail)
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))
C Header Interface
#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.

3 Description

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.

4 References

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

5 Arguments

1: alda* Real (Kind=nag_wp) array Input
Note: the second dimension of the array a must be at least max1,n.
On entry: the n by n matrix A.
2: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f04aef is called.
Constraint: ldamax1,n.
3: bldb* Real (Kind=nag_wp) array Input
Note: the second dimension of the array b must be at least max1,m.
On entry: the n by m right-hand side matrix B.
4: ldb Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f04aef is called.
Constraint: ldbmax1,n.
5: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
6: m Integer Input
On entry: m, the number of right-hand sides.
Constraint: m0.
7: cldc* Real (Kind=nag_wp) array Output
Note: the second dimension of the array c must be at least max1,m.
On exit: the n by m solution matrix X.
8: ldc Integer Input
On entry: the first dimension of the array c as declared in the (sub)program from which f04aef is called.
Constraint: ldcmax1,n.
9: wkspcemax1,n Real (Kind=nag_wp) array Workspace
10: aaldaa* Real (Kind=nag_wp) array Output
Note: the second dimension of the array aa must be at least max1,n.
On exit: the triangular factors L and U, except that the unit diagonal elements of U are not stored.
11: ldaa Integer Input
On entry: the first dimension of the array aa as declared in the (sub)program from which f04aef is called.
Constraint: ldaamax1,n.
12: bbldbb* Real (Kind=nag_wp) array Output
Note: the second dimension of the array bb must be at least max1,m.
On exit: the final n by m residual matrix R=B-AX.
13: ldbb Integer Input
On entry: the first dimension of the array bb as declared in the (sub)program from which f04aef is called.
Constraint: ldbbmax1,n.
14: 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 -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry 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:
ifail=1
Matrix A is approximately singular.
ifail=2
The matrix A is too ill-conditioned.
ifail=3
On entry, ldaa=value and n=value.
Constraint: ldaamax1,n.
On entry, lda=value and n=value.
Constraint: ldamax1,n.
On entry, ldb=value and n=value.
Constraint: ldbmax1,n.
On entry, ldbb=value and n=value.
Constraint: ldbbmax1,n.
On entry, ldc=value and n=value.
Constraint: ldcmax1,n.
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
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.
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.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

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

8 Parallelism 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.

9 Further Comments

The time taken by f04aef is approximately proportional to n3.
If there is only one right-hand side, it is simpler to use f04atf.

10 Example

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 .  

10.1 Program Text

Program Text (f04aefe.f90)

10.2 Program Data

Program Data (f04aefe.d)

10.3 Program Results

Program Results (f04aefe.r)