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

3

Description

nag_dsptrs (f07pec) is used to solve a real symmetric indefinite system of linear equations

A X = B

, the function must be preceded by a call to nag_dsptrf (f07pdc) which computes the Bunch–Kaufman factorization of

A

, using packed storage.

uplo = Nag_Upper

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

, where

P

is a permutation matrix,

U

is an upper triangular matrix and

D

is a symmetric block diagonal matrix with

1

1

and

2

2

blocks; the solution

X

is computed by solving

P U D Y = B

and then

U^{T} P^{T} X = Y

uplo = Nag_Lower

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

, where

L

is a lower triangular matrix; the solution

X

is computed by solving

P L D Y = B

and then

L^{T} P^{T} X = Y

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 3.3.1.3 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: specifies how

A

has been factorized.

$uplo = Nag_Upper$: $A = P U D U^{T} P^{T}$ , where $U$ is upper triangular.
$uplo = Nag_Lower$: $A = P L D L^{T} P^{T}$ , 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: $nrhs$ – IntegerInput

On entry:

r

, the number of right-hand sides.

Constraint:

nrhs \geq 0

5: $ap [\dim]$ – const doubleInput

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

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

On entry: the factorization of

A

stored in packed form, as returned by nag_dsptrf (f07pdc).

6: $ipiv [\dim]$ – const IntegerInput

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

\max (1, n)

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

D

, as returned by nag_dsptrf (f07pdc).

7: $b [\dim]$ – 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: the

n

r

solution matrix

X

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

9: $fail$ – 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_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

For each right-hand side vector

b

, the computed solution

x

is the exact solution of a perturbed system of equations

(A + E) x = b

, where

if $uplo = Nag_Upper$ , $|E| \leq c (n) ε P |U| |D| |U^{T}| P^{T}$ ;
if $uplo = Nag_Lower$ , $|E| \leq c (n) ε P |L| |D| |L^{T}| P^{T}$ ,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision.

\hat{x}

is the true solution, then the computed solution

x

satisfies a forward error bound of the form

\frac{{‖x - \hat{x}‖}_{\infty}}{{‖x‖}_{\infty}} \leq c (n) cond (A, x) ε

where

cond (A, x) = {‖|A^{- 1}| |A| |x|‖}_{\infty} / {‖x‖}_{\infty} \leq cond (A) = {‖|A^{- 1}| |A|‖}_{\infty} \leq κ_{\infty} (A)

Note that

cond (A, x)

can be much smaller than

cond (A)

Forward and backward error bounds can be computed by calling nag_dsprfs (f07phc), and an estimate for

κ_{\infty} (A)

(

= κ_{1} (A)

) can be obtained by calling nag_dspcon (f07pgc).

8

Parallelism and Performance

nag_dsptrs (f07pec) 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

2 n^{2} r

This function may be followed by a call to nag_dsprfs (f07phc) to refine the solution and return an error estimate.

The complex analogues of this function are nag_zhptrs (f07psc) for Hermitian matrices and nag_zsptrs (f07qsc) for symmetric matrices.

10

Example

This example solves the system of equations

A X = B

, 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}) and B = (\begin{matrix} - 9.50 & 27.85 \\ - 8.38 & 9.90 \\ - 6.07 & 19.25 \\ - 0.96 & 3.93 \end{matrix}) .

Here

A

is symmetric indefinite, stored in packed form, and must first be factorized by nag_dsptrf (f07pdc).

NAG Library Function Document

nag_dsptrs (f07pec)

▸▿ 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