void	f07mac (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer nrhs, double a[], Integer pda, Integer ipiv[], double b[], Integer pdb, NagError *fail)

The function may be called by the names: f07mac, nag_lapacklin_dsysv or nag_dsysv.

3 Description

f07mac uses the diagonal pivoting method to factor

A

order	uplo	$A$
Nag_ColMajor	Nag_Upper	$U D U^{T}$
Nag_ColMajor	Nag_Lower	$L D L^{T}$
Nag_RowMajor	Nag_Upper	$U^{T} D U$
Nag_RowMajor	Nag_Lower	$L^{T} D L$

where

U

(or

L

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

D

is symmetric and block diagonal with

1 \times 1

and

2 \times 2

diagonal blocks. The factored form of

A

is then used to solve the system of equations

A X = B

Note that, in general, different permutations (pivot sequences) and diagonal block structures are obtained for

uplo = Nag_Upper

Nag_Lower

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

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

5 Arguments

1: $order$ – Nag_OrderType Input

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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $uplo$ – Nag_UploType Input

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: $n$ – Integer Input

On entry:

n

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

A

Constraint:

n \geq 0

4: $nrhs$ – Integer Input

On entry:

r

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

B

Constraint:

nrhs \geq 0

5: $a [\dim]$ – double Input/Output

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

\max (1, pda \times n)

On entry: the

n \times n

symmetric matrix

A

order = Nag_ColMajor

A_{i j}

is stored in

a [(j - 1) \times pda + i - 1]

order = Nag_RowMajor

A_{i j}

is stored in

a [(i - 1) \times pda + j - 1]

uplo = Nag_Upper

, the upper triangular part of

A

must be stored and the elements of the array below the diagonal are not referenced.

uplo = Nag_Lower

, the lower triangular part of

A

must be stored and the elements of the array above the diagonal are not referenced.

On exit: if

fail . code =

NE_NOERROR, 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}

A = U^{T} D U

A = L^{T} D L

as computed by f07mdc.

6: $pda$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) of the matrix

A

in the array a.

Constraint:

pda \geq \max (1, n)

7: $ipiv [\dim]$ – Integer Output

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

\max (1, n)

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 $1 \times 1$ 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 $2 \times 2$ 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 $2 \times 2$ pivot block and the $(i + 1)$ th row and column of $A$ were interchanged with the $m$ th row and column.

8: $b [\dim]$ – double Input/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 \times r

right-hand side matrix

B

On exit: if

fail . code =

NE_NOERROR, the

n \times r

solution matrix

X

9: $pdb$ – Integer Input

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 $order = Nag_RowMajor$ , $pdb \geq \max (1, nrhs)$ .

10: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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, $pda = ⟨ value ⟩$ .
Constraint: $pda > 0$ .

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

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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SINGULAR: Element $⟨ value ⟩$ of the diagonal is exactly zero. The factorization has been completed, but the block diagonal matrix $D$ is exactly singular, so the solution could not be computed.

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.

f07mbc is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, f04bhc solves

A x = b

and returns a forward error bound and condition estimate. f04bhc calls f07mac to solve the equations.

8 Parallelism and Performance

f07mac 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 analogues of f07mac are f07mnc for Hermitian matrices, and f07nnc for symmetric matrices.

10 Example

This example solves the equations

A x = b,

where

A

is the symmetric 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 \\ 6.07 \\ 8.38 \\ 9.50 \end{matrix}) .

Details of the factorization of

A

are also output.

f07ma: FL CL CPP AD

NAG CL Interfacef07mac (dsysv)

▸▿ 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 CL Interface
f07mac (dsysv)