NAG Library Function Document

nag_dgetrs (f07aec)

void	nag_dgetrs (Nag_OrderType order, Nag_TransType trans, Integer n, Integer nrhs, const double a[], Integer pda, const Integer ipiv[], double b[], Integer pdb, NagError *fail)

3 Description

nag_dgetrs (f07aec) is used to solve a real system of linear equations

A X = B

A^{T} X = B

, the function must be preceded by a call to nag_dgetrf (f07adc) which computes the

L U

factorization of

A

A = P L U

. The solution is computed by forward and backward substitution.

trans = Nag_NoTrans

, the solution is computed by solving

P L Y = B

and then

U X = Y

trans = Nag_Trans

Nag_ConjTrans

, the solution is computed by solving

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

Constraint:

order = Nag_RowMajor

or Nag_ColMajor.

2: trans – Nag_TransTypeInput

On entry: indicates the form of the equations.

$trans = Nag_NoTrans$: $A X = B$ is solved for $X$ .
$trans = Nag_Trans$ or $Nag_ConjTrans$: $A^{T} X = B$ is solved for $X$ .

Constraint:

trans = Nag_NoTrans

Nag_Trans

Nag_ConjTrans

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)

The

(i, j)

th element of the matrix

A

is stored in

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

On entry: the

L U

factorization of

A

, as returned by nag_dgetrf (f07adc).

6: pda – IntegerInput

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

Constraint:

pda \geq \max (1, n)

7: ipiv[ $\dim$ ] – const IntegerInput

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

\max (1, n)

On entry: the pivot indices, as returned by nag_dgetrf (f07adc).

8: 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

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

10: 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

|E| \leq c (n) ε P |L| |U|,

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)

, and

cond (A^{T})

can be much larger (or smaller) than

cond (A)

Forward and backward error bounds can be computed by calling nag_dgerfs (f07ahc), and an estimate for

κ_{\infty} (A)

can be obtained by calling nag_dgecon (f07agc) with

norm = Nag_InfNorm

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_dgerfs (f07ahc) to refine the solution and return an error estimate.

The complex analogue of this function is nag_zgetrs (f07asc).

9 Example

This example solves the system of equations

A X = B

, where

A = (\begin{matrix} 1.80 & 2.88 & 2.05 & - 0.89 \\ 5.25 & - 2.95 & - 0.95 & - 3.80 \\ 1.58 & - 2.69 & - 2.90 & - 1.04 \\ - 1.11 & - 0.66 & - 0.59 & 0.80 \end{matrix}) and B = (\begin{matrix} 9.52 & 18.47 \\ 24.35 & 2.25 \\ 0.77 & - 13.28 \\ - 6.22 & - 6.21 \end{matrix}) .

Here

A

is nonsymmetric and must first be factorized by nag_dgetrf (f07adc).

NAG Library Function Documentnag_dgetrs (f07aec)

+− 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_dgetrs (f07aec)