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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_dsptrf (f07pd)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_lapack_dsptrf (f07pd) computes the Bunch–Kaufman factorization of a real symmetric indefinite matrix, using packed storage.

Syntax

[ap, ipiv, info] = f07pd(uplo, n, ap)

[ap, ipiv, info] = nag_lapack_dsptrf(uplo, n, ap)

Description

nag_lapack_dsptrf (f07pd) factorizes a real symmetric matrix

A

, using the Bunch–Kaufman diagonal pivoting method and packed storage.

A

is factorized as either

A = P U D U^{T} P^{T}

uplo ='U'

A = P L D L^{T} P^{T}

uplo ='L'

, where

P

is a permutation matrix,

U

(or

L

) is a unit upper (or lower) triangular matrix and

D

is a symmetric block diagonal matrix with

1

1

and

2

2

diagonal blocks;

U

(or

L

) has

2

2

unit diagonal blocks corresponding to the

2

2

blocks of

D

. Row and column interchanges are performed to ensure numerical stability while preserving symmetry.

This method is suitable for symmetric matrices which are not known to be positive definite. If

A

is in fact positive definite, no interchanges are performed and no

2

2

blocks occur in

D

References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

1: $uplo$ – string (length ≥ 1)

Specifies whether the upper or lower triangular part of

A

is stored and how

A

is to be factorized.

$uplo ='U'$: The upper triangular part of $A$ is stored and $A$ is factorized as $P U D U^{T} P^{T}$ , where $U$ is upper triangular.
$uplo ='L'$: The lower triangular part of $A$ is stored and $A$ is factorized as $P L D L^{T} P^{T}$ , where $L$ is lower triangular.

Constraint:

uplo ='U'

'L'

2: $n$ – int64int32nag_int scalar

n

, the order of the matrix

A

Constraint:

n \geq 0

3: $ap (:)$ – double array

The dimension of the array ap must be at least

\max (1, n \times (n + 1) / 2)

The

n

n

symmetric matrix

A

, packed by columns.

More precisely,

if $uplo ='U'$ , the upper triangle of $A$ must be stored with element $A_{i j}$ in $ap (i + j (j - 1) / 2)$ for $i \leq j$ ;
if $uplo ='L'$ , the lower triangle of $A$ must be stored with element $A_{i j}$ in $ap (i + (2 n - j) (j - 1) / 2)$ for $i \geq j$ .

Optional Input Parameters

None.

Output Parameters

1: $ap (:)$ – double array

The dimension of the array ap will be

\max (1, n \times (n + 1) / 2)

A

stores details of the block diagonal matrix

D

and the multipliers used to obtain the factor

U

L

as specified by uplo.

2: $ipiv (n)$ – int64int32nag_int array

Details of the interchanges and the block structure of

D

. More precisely,

if $ipiv (i) = k > 0$ , $d_{i i}$ is a $1$ by $1$ pivot block and the $i$ th row and column of $A$ were interchanged with the $k$ th row and column;
if $uplo ='U'$ and $ipiv (i - 1) = ipiv (i) = - l < 0$ , $(\begin{matrix} d_{i - 1, i - 1} & {\bar{d}}_{i, i - 1} \\ {\bar{d}}_{i, i - 1} & d_{i i} \end{matrix})$ is a $2$ by $2$ pivot block and the $(i - 1)$ th row and column of $A$ were interchanged with the $l$ th row and column;
if $uplo ='L'$ and $ipiv (i) = ipiv (i + 1) = - m < 0$ , $(\begin{matrix} d_{i i} & d_{i + 1, i} \\ d_{i + 1, i} & d_{i + 1, i + 1} \end{matrix})$ is a $2$ by $2$ pivot block and the $(i + 1)$ th row and column of $A$ were interchanged with the $m$ th row and column.

3: $info$ – int64int32nag_int scalar

info = 0

unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

W $info > 0$: Element $_$ of the diagonal is exactly zero. The factorization has been completed, but the block diagonal matrix $D$ is exactly singular, and division by zero will occur if it is used to solve a system of equations.

Accuracy

uplo ='U'

, the computed factors

U

and

D

are the exact factors of a perturbed matrix

A + E

, where

|E| \leq c (n) ε P |U| |D| |U^{T}| P^{T},

c (n)

is a modest linear function of

n

, and

ε

is the machine precision.

uplo ='L'

, a similar statement holds for the computed factors

L

and

D

Further Comments

The elements of

D

overwrite the corresponding elements of

A

; if

D

has

2

2

blocks, only the upper or lower triangle is stored, as specified by uplo.

The unit diagonal elements of

U

L

and the

2

2

unit diagonal blocks are not stored. The remaining elements of

U

L

overwrite elements in the corresponding columns of

A

, but additional row interchanges must be applied to recover

U

L

explicitly (this is seldom necessary). If

ipiv (i) = i

, for

i = 1, 2, \dots, n

(as is the case when

A

is positive definite), then

U

L

are stored explicitly in packed form (except for their unit diagonal elements which are equal to

1

The total number of floating-point operations is approximately

\frac{1}{3} n^{3}

A call to nag_lapack_dsptrf (f07pd) may be followed by calls to the functions:

nag_lapack_dsptrs (f07pe) to solve $A X = B$ ;
nag_lapack_dspcon (f07pg) to estimate the condition number of $A$ ;
nag_lapack_dsptri (f07pj) to compute the inverse of $A$ .

The complex analogues of this function are nag_lapack_zhptrf (f07pr) for Hermitian matrices and nag_lapack_zsptrf (f07qr) for symmetric matrices.

Example

This example computes the Bunch–Kaufman factorization of the matrix

A

, where

A = (\begin{matrix} 2.07 & 3.87 & 4.20 & - 1.15 \\ 3.87 & - 0.21 & 1.87 & 0.63 \\ 4.20 & 1.87 & 1.15 & 2.06 \\ - 1.15 & 0.63 & 2.06 & - 1.81 \end{matrix}),

using packed storage.

Open in the MATLAB editor: f07pd_example

function f07pd_example


fprintf('f07pd example results\n\n');

% Indefinite matrix A (lower triangular part stored in packed format)
uplo = 'L';
n = int64(4);
ap = [2.07;   3.87;   4.20;   -1.15;
             -0.21;   1.87;    0.63;
                      1.15;    2.06;
                              -1.81];

% Factorize
[apf, ipiv, info] = f07pd( ...
                           uplo, n, ap);

[ifail] = x04cc( ...
                 uplo, 'Non-unit', n, apf, 'Details of factorization');

fprintf('\nPivot indices\n   ');
fprintf('%11d', ipiv);
fprintf('\n');

f07pd example results

 Details of factorization
             1          2          3          4
 1      2.0700
 2      4.2000     1.1500
 3      0.2230     0.8115    -2.5907
 4      0.6537    -0.5960     0.3031     0.4074

Pivot indices
            -3         -3          3          4

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Chapter Contents

Chapter Introduction

NAG Toolbox