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
Character (1), Intent (In)	::	uplo

C Header Interface

#include <nag.h>

void	f04bhf_ (const char uplo, const Integer n, const Integer nrhs, double a[], const Integer lda, Integer ipiv[], double b[], const Integer ldb, double rcond, double errbnd, Integer ifail, const Charlen length_uplo)

The routine may be called by the names f04bhf or nagf_linsys_real_symm_solve.

3 Description

The diagonal pivoting method is used to factor

A

A = U D U^{T}

, if

uplo ='U'

, or

A = L D L^{T}

, if

uplo ='L'

, where

U

(or

L

) is a product of permutation and unit upper (lower) triangular matrices, and

D

is symmetric and block diagonal with

1

1

and

2

2

diagonal blocks. 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.

uplo ='L'

, the lower triangle of the matrix

A

is stored.

Constraint:

uplo ='U'

'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

n

symmetric matrix

A

uplo ='U'

, the leading n by n upper triangular part of the array a contains the upper triangular part of the matrix

A

, and the strictly lower triangular part of a is not referenced.

uplo ='L'

, the leading n by n lower triangular part of the array 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 \geq 0

, the block diagonal matrix

D

and the multipliers used to obtain the factor

U

L

from the factorization

A = U D U^{T}

A = L D L^{T}

as computed by f07mdf.

5: $lda$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f04bhf is called.

Constraint:

lda \geq \max (1, n)

6: $ipiv (n)$ – Integer array Output

On exit: if

ifail \geq 0

, details of the interchanges and the block structure of

D

, as determined by f07mdf.

$ipiv (k) > 0$: Rows and columns $k$ and $ipiv (k)$ were interchanged, and $d_{k k}$ is a $1$ by $1$ diagonal block.
$uplo ='U'$ and $ipiv (k) = ipiv (k - 1) < 0$: Rows and columns $k - 1$ and $- ipiv (k)$ were interchanged and $d_{k - 1 : k, k - 1 : k}$ is a $2$ by $2$ diagonal block.
$uplo ='L'$ and $ipiv (k) = ipiv (k + 1) < 0$: Rows and columns $k + 1$ and $- ipiv (k)$ were interchanged and $d_{k : k + 1, k : k + 1}$ is a $2$ by $2$ diagonal block.

7: $b (ldb *)$ – 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

r

matrix of right-hand sides

B

On exit: if

ifail = 0

n + 1

, the

n

r

solution matrix

X

8: $ldb$ – Integer Input

On entry: the first dimension of the array b as declared in the (sub)program from which f04bhf is called.

Constraint:

ldb \geq \max (1, n)

9: $rcond$ – 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

rcond = 1 / ({‖A‖}_{1} {‖A^{- 1}‖}_{1})

10: $errbnd$ – Real (Kind=nag_wp) Output

On exit: if

ifail = 0

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.

11: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

- 1 or 1

. If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 or 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this argument, the recommended value is

0

. 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:

$ifail > 0 and ifail \leq n$: Diagonal block $〈value〉$ of the block diagonal matrix is zero. The factorization has been completed, but 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 not one of 'U' or 'u' or 'L' 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: $lda \geq \max (1, n)$ .

$ifail = - 8$: 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 $\max (2 \times n, lwork)$ , where lwork is the optimum workspace required by f07maf. If this failure occurs it may be possible to solve the equations by calling the packed storage version of f04bhf, f04bjf, or by calling f07maf directly with less than the optimum workspace (see Chapter F07).
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. f04bhf 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

f04bhf 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} + 2 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 analogues of f04bhf are f04chf for complex Hermitian matrices, and f04dhf for complex symmetric matrices.

10 Example

This example solves the equations

A X = B,

where

A

is the symmetric indefinite matrix

A = (\begin{matrix} - 1.81 & 2.06 & 0.63 & - 1.15 \\ 2.06 & 1.15 & 1.87 & 4.20 \\ 0.63 & 1.87 & - 0.21 & 3.87 \\ - 1.15 & 4.20 & 3.87 & 2.07 \end{matrix}) and B = (\begin{matrix} 0.96 & 3.93 \\ 6.07 & 19.25 \\ 8.38 & 9.90 \\ 9.50 & 27.85 \end{matrix}) .

An estimate of the condition number of

A

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

Interfaces: FL CL AD

NAG FL Interface Introduction

F04 (Linsys) Chapter Contents

F04 (Linsys) Chapter Introduction

f04bh: FL CL AD

NAG FL Interfacef04bhf (real_​symm_​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
f04bhf (real_symm_solve)