NAG Library Function Document

nag_dsptrf (f07pdc)

+− 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

1 Purpose

nag_dsptrf (f07pdc) computes the Bunch–Kaufman factorization of a real symmetric indefinite matrix, using packed storage.

2 Specification

#include <nag.h>

#include <nagf07.h>

void	nag_dsptrf (Nag_OrderType order, Nag_UploType uplo, Integer n, double ap[], Integer ipiv[], NagError *fail)

3 Description

nag_dsptrf (f07pdc) 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 = Nag_Upper

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

uplo = Nag_Lower

, 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

4 References

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

5 Arguments

1: order – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: uplo – Nag_UploTypeInput

On entry: specifies whether the upper or lower triangular part of

A

is stored and how

A

is to be factorized.

$uplo = Nag_Upper$: 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.
$uplo = Nag_Lower$: 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 = Nag_Upper

Nag_Lower

3: n – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

4: ap[ $\dim$ ] – doubleInput/Output

Note: the dimension, dim, of the array ap must be at least

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

On entry: the

n

n

symmetric matrix

A

, packed by rows or columns.

The storage of elements

A_{i j}

depends on the order and uplo arguments as follows:

if $order = Nag_ColMajor$ and $uplo = Nag_Upper$ ,
$A_{i j}$ is stored in $ap [(j - 1) \times j / 2 + i - 1]$ , for $i \leq j$ ;
if $order = Nag_ColMajor$ and $uplo = Nag_Lower$ ,
$A_{i j}$ is stored in $ap [(2 n - j) \times (j - 1) / 2 + i - 1]$ , for $i \geq j$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Upper$ ,
$A_{i j}$ is stored in $ap [(2 n - i) \times (i - 1) / 2 + j - 1]$ , for $i \leq j$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Lower$ ,
$A_{i j}$ is stored in $ap [(i - 1) \times i / 2 + j - 1]$ , for $i \geq j$ .

On exit:

A

is overwritten by details of the block diagonal matrix

D

and the multipliers used to obtain the factor

U

L

as specified by uplo.

5: ipiv[n] – IntegerOutput

On exit: details of the interchanges and the block structure of

D

. More precisely,

if $ipiv [i - 1] = k > 0$ , $d_{i i}$ 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 $uplo = Nag_Upper$ and $ipiv [i - 2] = ipiv [i - 1] = - l < 0$ , $(\begin{matrix} d_{i - 1, i - 1} & {\bar{d}}_{i, i - 1} \\ {\bar{d}}_{i, i - 1} & d_{i i} \end{matrix})$ 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 $uplo = Nag_Lower$ and $ipiv [i - 1] = ipiv [i] = - m < 0$ , $(\begin{matrix} d_{i i} & d_{i + 1, i} \\ d_{i + 1, i} & d_{i + 1, i + 1} \end{matrix})$ 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.

6: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_SINGULAR: $D (⟨value⟩, ⟨value⟩)$ 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.

7 Accuracy

uplo = Nag_Upper

, 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 = Nag_Lower

, a similar statement holds for the computed factors

L

and

D

8 Parallelism and Performance

nag_dsptrf (f07pdc) is not threaded by NAG in any implementation.

nag_dsptrf (f07pdc) 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 Users' Note for your implementation for any additional implementation-specific information.

9 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 - 1] = 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_dsptrf (f07pdc) may be followed by calls to the functions:

nag_dsptrs (f07pec) to solve $A X = B$ ;
nag_dspcon (f07pgc) to estimate the condition number of $A$ ;
nag_dsptri (f07pjc) to compute the inverse of $A$ .

The complex analogues of this function are nag_zhptrf (f07prc) for Hermitian matrices and nag_zsptrf (f07qrc) for symmetric matrices.

10 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.

NAG Library Function Documentnag_dsptrf (f07pdc)

+− 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 Library Function Document

nag_dsptrf (f07pdc)