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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_zhptrf (f07pr)

▸▿ 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_zhptrf (f07pr) computes the Bunch–Kaufman factorization of a complex Hermitian indefinite matrix, using packed storage.

Syntax

[ap, ipiv, info] = f07pr(uplo, n, ap)

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

Description

nag_lapack_zhptrf (f07pr) factorizes a complex Hermitian matrix

A

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

A

is factorized as either

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

uplo ='U'

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

uplo ='L'

, where

P

is a permutation matrix,

U

(or

L

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

D

is an Hermitian 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 keeping the matrix Hermitian.

This method is suitable for Hermitian 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 $P U D U^{H} P^{T}$ , where $U$ is upper triangular.
$uplo ='L'$: The lower triangular part of $A$ is stored and $A$ is factorized as $P L D L^{H} P^{T}$ , 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: $ap (:)$ – complex array

The dimension of the array ap must be at least

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

The

n

n

Hermitian 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: $ap (:)$ – complex 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^{H}| 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

and

L

are stored in the corresponding columns of the array ap, 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 real floating-point operations is approximately

\frac{4}{3} n^{3}

A call to nag_lapack_zhptrf (f07pr) may be followed by calls to the functions:

nag_lapack_zhptrs (f07ps) to solve $A X = B$ ;
nag_lapack_zhpcon (f07pu) to estimate the condition number of $A$ ;
nag_lapack_zhptri (f07pw) to compute the inverse of $A$ .

The real analogue of this function is nag_lapack_dsptrf (f07pd).

Example

This example computes the Bunch–Kaufman factorization of the matrix

A

, where

A = (\begin{array}{r} - 1.36 + 0.00 i & 1.58 + 0.90 i & 2.21 - 0.21 i & 3.91 + 1.50 i \\ 1.58 - 0.90 i & - 8.87 + 0.00 i & - 1.84 - 0.03 i & - 1.78 + 1.18 i \\ 2.21 + 0.21 i & - 1.84 + 0.03 i & - 4.63 + 0.00 i & 0.11 + 0.11 i \\ 3.91 - 1.50 i & - 1.78 - 1.18 i & 0.11 - 0.11 i & - 1.84 + 0.00 i \end{array}),

using packed storage.

Open in the MATLAB editor: f07pr_example

function f07pr_example


fprintf('f07pr example results\n\n');

% Hermitian indefinite matrix A (Lower triangular part stored in packed form)
uplo = 'L';
n = int64(4);
ap = [-1.36 + 0i;  1.58 - 0.9i;   2.21 + 0.21i;  3.91 - 1.5i;
                  -8.87 + 0i;    -1.84 + 0.03i; -1.78 - 1.18i;
                                 -4.63 + 0i;     0.11 - 0.11i;
                                                -1.84 + 0i];

% Factorize
[apf, ipiv, info] = f07pr( ...
                           uplo, n, ap);

[ifail] = x04dc( ...
                 uplo, 'Non-unit', n, apf, 'Details of factorization');

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

f07pr example results

 Details of factorization
             1          2          3          4
 1     -1.3600
        0.0000

 2      3.9100    -1.8400
       -1.5000     0.0000

 3      0.3100     0.5637    -5.4176
        0.0433     0.2850     0.0000

 4     -0.1518     0.3397     0.2997    -7.1028
        0.3743     0.0303     0.1578     0.0000

Pivot indices
            -4         -4          3          4

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox