NAG Library Function Document

nag_dpotrs (f07fec)

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

3 Description

nag_dpotrs (f07fec) is used to solve a real symmetric positive definite system of linear equations

A X = B

, this function must be preceded by a call to nag_dpotrf (f07fdc) which computes the Cholesky factorization of

A

. The solution

X

is computed by forward and backward substitution.

uplo = Nag_Upper

A = U^{T} U

, where

U

is upper triangular; the solution

X

is computed by solving

U^{T} Y = B

and then

U X = Y

uplo = Nag_Lower

A = L L^{T}

, where

L

is lower triangular; the solution

X

is computed by solving

L Y = B

and then

L^{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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

or Nag_ColMajor.

2: uplo – Nag_UploTypeInput

On entry: specifies how

A

has been factorized.

$uplo = Nag_Upper$: $A = U^{T} U$ , where $U$ is upper triangular.
$uplo = Nag_Lower$: $A = L L^{T}$ , 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: a[ $\dim$ ] – const doubleInput

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

\max (1, pda \times n)

On entry: the Cholesky factor of

A

, as returned by nag_dpotrf (f07fdc).

6: pda – IntegerInput

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

Constraint:

pda \geq \max (1, n)

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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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.

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

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_dporfs (f07fhc), and an estimate for

κ_{\infty} (A)

(

= κ_{1} (A)

) can be obtained by calling nag_dpocon (f07fgc).

8 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_dporfs (f07fhc) to refine the solution and return an error estimate.

The complex analogue of this function is nag_zpotrs (f07fsc).

9 Example

This example solves the system of equations

A X = B

, where

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 & 8.30 \\ - 13.35 & 2.13 \\ 1.89 & 1.61 \\ - 4.14 & 5.00 \end{matrix}) .

Here

A

is symmetric positive definite and must first be factorized by nag_dpotrf (f07fdc).

NAG Library Function Documentnag_dpotrs (f07fec)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_dpotrs (f07fec)