d01tcc (dim1_gauss_wgen) : NAG Library CL Interface, Mark 27

3 Description

d01tcc returns the weights

w_{i}

and abscissae

x_{i}

for use in the summation

S = \sum_{i = 1}^{n} w_{i} f (x_{i}),

which approximates a definite integral (see Davis and Rabinowitz (1975) or Stroud and Secrest (1966)). The following types are provided:

(a)Gauss–Legendre

$S ≃ \int_{a}^{b} f (x) d x, exact for f (x) = P_{2 n - 1} (x) .$

Constraint: $b > a$ .
(b)Gauss–Jacobi
normal weights:

$S ≃ \int_{a}^{b} {(b - x)}^{c} {(x - a)}^{d} f (x) d x, exact for f (x) = P_{2 n - 1} (x),$

adjusted weights:

$S ≃ \int_{a}^{b} f (x) d x, exact for f (x) = {(b - x)}^{c} {(x - a)}^{d} P_{2 n - 1} (x) .$

Constraint: $c > −1$ , $d > −1$ , $b > a$ .
(c)Exponential Gauss
normal weights:

$S ≃ \int_{a}^{b} {| x - \frac{a + b}{2} |}^{c} f (x) d x, exact for f (x) = P_{2 n - 1} (x),$

adjusted weights:

$S ≃ \int_{a}^{b} f (x) d x, exact for f (x) = {| x - \frac{a + b}{2} |}^{c} P_{2 n - 1} (x) .$

Constraint: $c > −1$ , $b > a$ .
(d)Gauss–Laguerre
normal weights:

$\begin{array}{l} S & ≃ \int_{a}^{\infty} {| x - a |}^{c} e^{- b x} f (x) d x (b > 0), \\ ≃ \int_{- \infty}^{a} {| x - a |}^{c} e^{- b x} f (x) d x (b < 0), exact for f (x) = P_{2 n - 1} (x), \end{array}$

adjusted weights:

$\begin{array}{l} S & ≃ \int_{a}^{\infty} f (x) d x (b > 0), \\ ≃ \int_{- \infty}^{a} f (x) d x (b < 0), exact for f (x) = {| x - a |}^{c} e^{- b x} P_{2 n - 1} (x) . \end{array}$

Constraint: $c > −1$ , $b \neq 0$ .
(e)Gauss–Hermite
normal weights:

$S ≃ \int_{- \infty}^{+ \infty} {| x - a |}^{c} e^{- b {(x - a)}^{2}} f (x) d x, exact for f (x) = P_{2 n - 1} (x),$

adjusted weights:

$S ≃ \int_{- \infty}^{+ \infty} f (x) d x, exact for f (x) = {| x - a |}^{c} e^{- b {(x - a)}^{2}} P_{2 n - 1} (x) .$

Constraint: $c > −1$ , $b > 0$ .
(f)Rational Gauss
normal weights:

$\begin{array}{l} S & ≃ \int_{a}^{\infty} \frac{{| x - a |}^{c}}{{| x + b |}^{d}} f (x) d x (a + b > 0), \\ ≃ \int_{- \infty}^{a} \frac{{| x - a |}^{c}}{{| x + b |}^{d}} f (x) d x (a + b < 0), exact for f (x) = P_{2 n - 1} (\frac{1}{x + b}), \end{array}$

adjusted weights:

$\begin{array}{l} S & ≃ \int_{a}^{\infty} f (x) d x (a + b > 0), \\ ≃ \int_{- \infty}^{a} f (x) d x (a + b < 0), exact for f (x) = \frac{{| x - a |}^{c}}{{| x + b |}^{d}} P_{2 n - 1} (\frac{1}{x + b}) . \end{array}$

Constraint: $c > −1$ , $d > c + 1$ , $a + b \neq 0$ .

In the above formulae,

P_{2 n - 1} (x)

stands for any polynomial of degree

2 n - 1

or less in

x

The method used to calculate the abscissae involves finding the eigenvalues of the appropriate tridiagonal matrix (see Golub and Welsch (1969)). The weights are then determined by the formula

w_{i} = {\sum_{j = 0}^{n - 1} P_{j}^{*} {(x_{i})}^{2}}^{−1},

where

P_{j}^{*} (x)

is the

j

th orthogonal polynomial with respect to the weight function over the appropriate interval.

The weights and abscissae produced by d01tcc may be passed to d01fbc, which will evaluate the summations in one or more dimensions.

5 Arguments

quad_type

– Nag_QuadType Input

On entry: indicates the type of quadrature rule.

$quad_type = Nag_Quad_Gauss_Legendre$: Gauss–Legendre, with normal weights.
$quad_type = Nag_Quad_Gauss_Jacobi$: Gauss–Jacobi, with normal weights.
$quad_type = Nag_Quad_Gauss_Jacobi_Adjusted$: Gauss–Jacobi, with adjusted weights.
$quad_type = Nag_Quad_Gauss_Exponential$: Exponential Gauss, with normal weights.
$quad_type = Nag_Quad_Gauss_Exponential_Adjusted$: Exponential Gauss, with adjusted weights.
$quad_type = Nag_Quad_Gauss_Laguerre$: Gauss–Laguerre, with normal weights.
$quad_type = Nag_Quad_Gauss_Laguerre_Adjusted$: Gauss–Laguerre, with adjusted weights.
$quad_type = Nag_Quad_Gauss_Hermite$: Gauss–Hermite, with normal weights.
$quad_type = Nag_Quad_Gauss_Hermite_Adjusted$: Gauss–Hermite, with adjusted weights.
$quad_type = Nag_Quad_Gauss_Rational$: Rational Gauss, with normal weights.
$quad_type = Nag_Quad_Gauss_Rational_Adjusted$: Rational Gauss, with adjusted weights.

Constraint:

quad_type = Nag_Quad_Gauss_Legendre

Nag_Quad_Gauss_Jacobi

Nag_Quad_Gauss_Jacobi_Adjusted

Nag_Quad_Gauss_Exponential

Nag_Quad_Gauss_Exponential_Adjusted

Nag_Quad_Gauss_Laguerre

Nag_Quad_Gauss_Laguerre_Adjusted

Nag_Quad_Gauss_Hermite

Nag_Quad_Gauss_Hermite_Adjusted

Nag_Quad_Gauss_Rational

Nag_Quad_Gauss_Rational_Adjusted

a

– double Input

b

– double Input

c

– double Input

d

– double Input

On entry: the parameters

a

b

c

and

d

which occur in the quadrature formulae described in Section 3. c is not used if

quad_type = Nag_Quad_Gauss_Legendre

; d is not used unless

quad_type = Nag_Quad_Gauss_Jacobi

Nag_Quad_Gauss_Jacobi_Adjusted

Nag_Quad_Gauss_Rational

Nag_Quad_Gauss_Rational_Adjusted

. For some rules c and d must not be too large (see Section 6).

Constraints:

if $quad_type = Nag_Quad_Gauss_Legendre$ , $a < b$ ;
if $quad_type = Nag_Quad_Gauss_Jacobi$ or $Nag_Quad_Gauss_Jacobi_Adjusted$ , $a < b$ and $c > - 1.0$ and $d > - 1.0$ ;
if $quad_type = Nag_Quad_Gauss_Exponential$ or $Nag_Quad_Gauss_Exponential_Adjusted$ , $a < b$ and $c > - 1.0$ ;
if $quad_type = Nag_Quad_Gauss_Laguerre$ or $Nag_Quad_Gauss_Laguerre_Adjusted$ , $b \neq 0.0$ and $c > - 1.0$ ;
if $quad_type = Nag_Quad_Gauss_Hermite$ or $Nag_Quad_Gauss_Hermite_Adjusted$ , $b > 0.0$ and $c > - 1.0$ ;
if $quad_type = Nag_Quad_Gauss_Rational$ or $Nag_Quad_Gauss_Rational_Adjusted$ , $a + b \neq 0.0$ and $c > - 1.0$ and $d > c + 1.0$ .

n

– Integer Input

On entry:

n

, the number of weights and abscissae to be returned. If

quad_type = Nag_Quad_Gauss_Exponential_Adjusted

Nag_Quad_Gauss_Hermite_Adjusted

and

c \neq 0.0

, an odd value of n may raise problems (see

fail . code =

NE_INDETERMINATE).

Constraint:

n > 0

weight [n]

– double Output

On exit: the n weights.

abscis [n]

– double Output

On exit: the n abscissae.

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_CONSTRAINT

On entry, a, b, c, or d is not in the allowed range:

a = ⟨ value ⟩

b = ⟨ value ⟩

c = ⟨ value ⟩

d = ⟨ value ⟩

and

quad_type = ⟨ value ⟩

NE_CONVERGENCE

The algorithm for computing eigenvalues of a tridiagonal matrix has failed to converge.

NE_INDETERMINATE

The contribution of the central abscissa to the summation is indeterminate.

NE_INT

On entry,

n = ⟨ value ⟩

.
Constraint:

n > 0

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_TOO_BIG

One or more of the weights are larger than

rmax

, the largest floating point number on this computer (see X02ALC):

rmax = ⟨ value ⟩

.
Possible solutions are to use a smaller value of

n

; or, if using adjusted weights to change to normal weights.

NE_TOO_SMALL

One or more of the weights are too small to be distinguished from zero on this machine.
The underflowing weights are returned as zero, which may be a usable approximation.
Possible solutions are to use a smaller value of

n

; or, if using normal weights, to change to adjusted weights.

8 Parallelism and Performance

d01tcc 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.

10 Example

This example returns the abscissae and (adjusted) weights for the seven-point Gauss–Laguerre formula.

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
d01tcc (dim1_gauss_wgen)

▸▿ 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 Interfaced01tcc (dim1_​gauss_​wgen)

▸▿ 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
d01tcc (dim1_gauss_wgen)