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NAG Library Function Document

nag_dppsv (f07gac)

▸▿ 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_dppsv (f07gac) computes the solution to a real system of linear equations

A X = B,

where

A

is an

n

n

symmetric positive definite matrix stored in packed format and

X

and

B

are

n

r

matrices.

2

Specification

#include <nag.h>

#include <nagf07.h>

void	nag_dppsv (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer nrhs, double ap[], double b[], Integer pdb, NagError *fail)

3

Description

nag_dppsv (f07gac) uses the Cholesky decomposition to factor

A

A = U^{T} U

uplo = Nag_Upper

A = L L^{T}

uplo = Nag_Lower

, 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 http://www.netlib.org/lapack/lug

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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $uplo$ – Nag_UploTypeInput

On entry: if

uplo = Nag_Upper

, the upper triangle of

A

is stored.

uplo = Nag_Lower

, the lower triangle of

A

is stored.

Constraint:

uplo = Nag_Upper

Nag_Lower

3: $n$ – IntegerInput

On entry:

n

, the number of linear equations, i.e., the order of the matrix

A

Constraint:

n \geq 0

4: $nrhs$ – IntegerInput

On entry:

r

, the number of right-hand sides, i.e., the number of columns of the matrix

B

Constraint:

nrhs \geq 0

5: $ap [\dim]$ – 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: if

fail . code =

NE_NOERROR, the factor

U

L

from the Cholesky factorization

A = U^{T} U

A = L L^{T}

, in the same storage format as

A

6: $b [\dim]$ – doubleInput/Output

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

$\max (1, pdb \times nrhs)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdb)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

B

is stored in

$b [(j - 1) \times pdb + i - 1]$ when $order = Nag_ColMajor$ ;
$b [(i - 1) \times pdb + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

n

r

right-hand side matrix

B

On exit: if

fail . code =

NE_NOERROR, the

n

r

solution matrix

X

7: $pdb$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array b.

Constraints:

if $order = Nag_ColMajor$ , $pdb \geq \max (1, n)$ ;
if $order = Nag_RowMajor$ , $pdb \geq \max (1, nrhs)$ .

8: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .

On entry, $nrhs = 〈value〉$ .
Constraint: $nrhs \geq 0$ .

On entry, $pdb = 〈value〉$ .
Constraint: $pdb > 0$ .
NE_INT_2: On entry, $pdb = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdb \geq \max (1, n)$ .

On entry, $pdb = 〈value〉$ and $nrhs = 〈value〉$ .
Constraint: $pdb \geq \max (1, nrhs)$ .
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_MAT_NOT_POS_DEF: The leading minor of order $〈value〉$ of $A$ is not positive definite, so the factorization could not be completed, and the solution has not been computed.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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. See Section 4.4 of Anderson et al. (1999) for further details.

nag_dppsvx (f07gbc) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_real_sym_posdef_packed_lin_solve (f04bec) solves

A x = b

and returns a forward error bound and condition estimate. nag_real_sym_posdef_packed_lin_solve (f04bec) calls nag_dppsv (f07gac) to solve the equations.

8

Parallelism and Performance

nag_dppsv (f07gac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_dppsv (f07gac) 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 function. 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 is approximately

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

, where

r

is the number of right-hand sides.

The complex analogue of this function is nag_zppsv (f07gnc).

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 \\ - 13.35 \\ 1.89 \\ - 4.14 \end{matrix}) .

Details of the Cholesky factorization of

A

are also output.

NAG Library Function Document

nag_dppsv (f07gac)

▸▿ 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

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