Integer, Intent (In)	::	n, nrhs, lda, ldb
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (Inout)	::	a(lda,), b(ldb,)
Real (Kind=nag_wp), Intent (Out)	::	rcond, errbnd
Character (1), Intent (In)	::	uplo

C Header Interface

#include <nag.h>

void	f04bdf_ (const char uplo, const Integer n, const Integer nrhs, double a[], const Integer lda, double b[], const Integer ldb, double rcond, double errbnd, Integer ifail, const Charlen length_uplo)

The routine may be called by the names f04bdf or nagf_linsys_real_posdef_solve.

3 Description

The Cholesky factorization is used to factor

A

A = U^{T} U

, if

uplo ='U'

, or

A = L L^{T}

, if

uplo ='L'

, where

U

is an upper triangular matrix and

L

is a lower triangular matrix. The factored form of

A

is then used to solve the system of equations

A X = 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: $uplo$ – Character(1) Input: On entry: if $uplo ='U'$ , the upper triangle of the matrix $A$ is stored.
If $uplo ='L'$ , the lower triangle of the matrix $A$ is stored.

Constraint: $uplo ='U'$ or $'L'$ .
2: $n$ – Integer Input: On entry: the number of linear equations $n$ , i.e., the order of the matrix $A$ .

Constraint: $n \geq 0$ .
3: $nrhs$ – Integer Input: On entry: the number of right-hand sides $r$ , i.e., the number of columns of the matrix $B$ .

Constraint: $nrhs \geq 0$ .
4: $a (lda, *)$ – 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 $n \times n$ symmetric matrix $A$ .
If $uplo ='U'$ , the leading n by n upper triangular part of a contains the upper triangular part of the matrix $A$ , and the strictly lower triangular part of a is not referenced.

If $uplo ='L'$ , the leading n by n lower triangular part of a contains the lower triangular part of the matrix $A$ , and the strictly upper triangular part of a is not referenced.

On exit: if $ifail = 0$ or $n + 1$ , the factor $U$ or $L$ from the Cholesky factorization $A = U^{T} U$ or $A = L L^{T}$ .
5: $lda$ – Integer Input: On entry: the first dimension of the array a as declared in the (sub)program from which f04bdf is called.

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

On entry: the $n \times r$ matrix of right-hand sides $B$ .

On exit: if $ifail = 0$ or $n + 1$ , the $n \times r$ solution matrix $X$ .
7: $ldb$ – Integer Input: On entry: the first dimension of the array b as declared in the (sub)program from which f04bdf is called.

Constraint: $ldb \geq \max (1, n)$ .
8: $rcond$ – Real (Kind=nag_wp) Output: On exit: if $ifail = 0$ or $n + 1$ , an estimate of the reciprocal of the condition number of the matrix $A$ , computed as $rcond = 1 / ({‖ A ‖}_{1} {‖ A^{- 1} ‖}_{1})$ .
9: $errbnd$ – Real (Kind=nag_wp) Output: On exit: if $ifail = 0$ or $n + 1$ , an estimate of the forward error bound for a computed solution $\hat{x}$ , such that ${‖ \hat{x} - x ‖}_{1} / {‖ x ‖}_{1} \leq errbnd$ , 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: $ifail$ – 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:

$ifail > 0 and ifail \leq n$: The principal minor of order $⟨ value ⟩$ of the matrix $A$ is not positive definite. The factorization has not been completed and the solution could not be computed.

$ifail = n + 1$: A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.

$ifail = - 1$: On entry, $uplo \neq'U'$ or $'L'$ : $uplo = ⟨ value ⟩$ .

$ifail = - 2$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

$ifail = - 3$: On entry, $nrhs = ⟨ value ⟩$ .
Constraint: $nrhs \geq 0$ .

$ifail = - 5$: On entry, $lda = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $lda \geq \max (1, n)$ .

$ifail = - 7$: On entry, $ldb = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $ldb \geq \max (1, n)$ .

$ifail = - 99$: 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.
The integer allocatable memory required is n, and the real allocatable memory required is $3 \times n$ . Allocation failed before the solution could be computed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The computed solution for a single right-hand side,

\hat{x}

, satisfies an equation of the form

(A + E) \hat{x} = b,

where

{‖ E ‖}_{1} = O (ε) {‖ A ‖}_{1}

and

ε

is the machine precision. An approximate error bound for the computed solution is given by

\frac{{‖ \hat{x} - x ‖}_{1}}{{‖ x ‖}_{1}} \leq κ (A) \frac{{‖ E ‖}_{1}}{{‖ A ‖}_{1}},

where

κ (A) = {‖ A^{- 1} ‖}_{1} {‖ A ‖}_{1}

, the condition number of

A

with respect to the solution of the linear equations. f04bdf uses the approximation

{‖ E ‖}_{1} = ε {‖ A ‖}_{1}

to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f04bdf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f04bdf 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 total number of floating-point operations required to solve the equations

A X = B

is proportional to

(\frac{1}{3} n^{3} + n^{2} r)

. 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 f04bdf is f04cdf.

10 Example

This example solves the equations

A X = B,

where

A

is the symmetric positive definite matrix

A = (\begin{matrix} 4.16 & - 3.12 & 0.56 & - 0.10 \\ - 3.12 & 5.03 & - 0.83 & 1.18 \\ 0.56 & - 0.83 & 0.76 & 0.34 \\ - 0.10 & 1.18 & 0.34 & 1.18 \end{matrix}) and B = (\begin{matrix} 8.70 & 8.30 \\ - 13.35 & 2.13 \\ 1.89 & 1.61 \\ - 4.14 & 5.00 \end{matrix}) .

An estimate of the condition number of

A

and an approximate error bound for the computed solutions are also printed.

f04bd: FL CL CPP AD PY MB

NAG FL Interfacef04bdf (real_​posdef_​solve)

▸▿ 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
f04bdf (real_posdef_solve)