f04abf calculates the accurate solution of a set of real symmetric positive definite linear equations with multiple right-hand sides, using a Cholesky factorization and iterative refinement.

2 Specification

Fortran Interface

Subroutine f04abf (

a, lda, b, ldb, n, m, c, ldc, wkspce, bb, ldbb, ifail)

Integer, Intent (In)	::	lda, ldb, n, m, ldc, ldbb
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	b(ldb,*)
Real (Kind=nag_wp), Intent (Inout)	::	a(lda,*), c(ldc,m), bb(ldbb,m)
Real (Kind=nag_wp), Intent (Out)	::	wkspce(n)

C Header Interface

#include <nag.h>

void	f04abf_ (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 bb[], const Integer ldbb, Integer *ifail)

The routine may be called by the names f04abf or nagf_linsys_withdraw_real_posdef_solve_ref.

3 Description

Given a set of real linear equations

A X = B

, where

A

is symmetric positive definite, f04abf first computes a Cholesky factorization of

A

A = L L^{T}

, where

L

is lower triangular. An approximation to

X

is found by forward and backward substitution. The residual matrix

R = B - A X

is then calculated using additional precision, and a correction

D

X

is found by solving

L L^{T} D = R

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: $a (lda, *)$ – Real (Kind=nag_wp) array Input/Output: Note: the second dimension of the array a must be at least $\max (1, n)$ .

On entry: the upper triangle of the $n \times n$ positive definite symmetric matrix $A$ . The elements of the array below the diagonal need not be set.

On exit: the elements of the array below the diagonal are overwritten; the upper triangle of $A$ is unchanged.
2: $lda$ – Integer Input: On entry: the first dimension of the array a as declared in the (sub)program from which f04abf is called.

Constraint: $lda \geq \max (1, n)$ .
3: $b (ldb, *)$ – Real (Kind=nag_wp) array Input: Note: the second dimension of the array b must be at least $\max (1, m)$ .

On entry: the $n \times m$ 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 f04abf is called.

Constraint: $ldb \geq \max (1, n)$ .
5: $n$ – Integer Input: On entry: $n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .
6: $m$ – Integer Input: On entry: $m$ , the number of right-hand sides.

Constraint: $m \geq 0$ .
7: $c (ldc, m)$ – Real (Kind=nag_wp) array Output: Note: the second dimension of the array c must be at least $\max (1, m)$ .

On exit: the $n \times m$ solution matrix $X$ .
8: $ldc$ – Integer Input: On entry: the first dimension of the array c as declared in the (sub)program from which f04abf is called.

Constraint: $ldc \geq \max (1, n)$ .
9: $wkspce (n)$ – Real (Kind=nag_wp) array Workspace
10: $bb (ldbb, m)$ – Real (Kind=nag_wp) array Output: Note: the second dimension of the array bb must be at least $\max (1, m)$ .

On exit: the final $n \times m$ residual matrix $R = B - A X$ .
11: $ldbb$ – Integer Input: On entry: the first dimension of the array bb as declared in the (sub)program from which f04abf is called.

Constraint: $ldbb \geq \max (1, n)$ .
12: $ifail$ – 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

−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 not positive definite.

$ifail = 2$: Matrix $A$ is too ill-conditioned; iterative refinement fails to improve the solution.

$ifail = 3$: On entry, $lda = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $lda \geq \max (1, n)$ .

On entry, $ldb = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $ldb \geq \max (1, n)$ .

On entry, $ldbb = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $ldbb \geq \max (1, n)$ .

On entry, $ldc = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $ldc \geq \max (1, n)$ .

On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 0$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

$ifail = - 99$: 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 39 of Wilkinson and Reinsch (1971).

8 Parallelism and Performance

f04abf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f04abf 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 f04abf is approximately proportional to

n^{3}

If there is only one right-hand side, it is simpler to use f04asf.

10 Example

This example solves the set of linear equations

A X = B

where

A = (\begin{array}{r} 5 & 7 & 6 & 5 \\ 7 & 10 & 8 & 7 \\ 6 & 8 & 10 & 9 \\ 5 & 7 & 9 & 10 \end{array}) and B = (\begin{array}{r} 23 \\ 32 \\ 33 \\ 31 \end{array}) .

f04ab: FL CL CPP AD

NAG FL Interfacef04abf (withdraw_​real_​posdef_​solve_​ref)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
f04abf (withdraw_real_posdef_solve_ref)