d01tbc (dim1_gauss_wres) : NAG Library CL Interface, Mark 27

3 Description

d01tbc returns the weights and abscissae for use in the Gaussian quadrature of a function

f (x)

. The quadrature takes the form

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

where

w_{i}

are the weights and

x_{i}

are the abscissae (see Davis and Rabinowitz (1975), Fröberg (1970), Ralston (1965) or Stroud and Secrest (1966)).

Weights and abscissae are available for Gauss–Legendre, rational Gauss, Gauss–Laguerre and Gauss–Hermite quadrature, and for a selection of values of

n

(see Section 5).

(a)Gauss–Legendre Quadrature:

$S ≃ \int_{a}^{b} f (x) d x$

where $a$ and $b$ are finite and it will be exact for any function of the form

$f (x) = \sum_{i = 0}^{2 n - 1} c_{i} x^{i} .$
(b)Rational Gauss quadrature, adjusted weights:

$S ≃ \int_{a}^{\infty} f (x) d x (a + b > 0) or S ≃ \int_{- \infty}^{a} f (x) d x (a + b < 0)$

and will be exact for any function of the form

$f (x) = \sum_{i = 2}^{2 n + 1} \frac{c_{i}}{{(x + b)}^{i}} = \frac{\sum_{i = 0}^{2 n - 1} c_{2 n + 1 - i} {(x + b)}^{i}}{{(x + b)}^{2 n + 1}} .$
(c)Gauss–Laguerre quadrature, adjusted weights:

$S ≃ \int_{a}^{\infty} f (x) d x (b > 0) or S ≃ \int_{- \infty}^{a} f (x) d x (b < 0)$

and will be exact for any function of the form

$f (x) = e^{- b x} \sum_{i = 0}^{2 n - 1} c_{i} x^{i} .$
(d)Gauss–Hermite quadrature, adjusted weights:

$S ≃ \int_{- \infty}^{+ \infty} f (x) d x$

and will be exact for any function of the form

$f (x) = e^{- b {(x - a)}^{2}} \sum_{i = 0}^{2 n - 1} c_{i} x^{i} (b > 0) .$
(e)Gauss–Laguerre quadrature, normal weights:

$S ≃ \int_{a}^{\infty} e^{- b x} f (x) d x (b > 0) or S ≃ \int_{- \infty}^{a} e^{- b x} f (x) d x (b < 0)$

and will be exact for any function of the form

$f (x) = \sum_{i = 0}^{2 n - 1} c_{i} x^{i} .$
(f)Gauss–Hermite quadrature, normal weights:

$S ≃ \int_{- \infty}^{+ \infty} e^{- b {(x - a)}^{2}} f (x) d x$

and will be exact for any function of the form

$f (x) = \sum_{i = 0}^{2 n - 1} c_{i} x^{i} .$

Note: the Gauss–Legendre abscissae, with

a = −1

b = + 1

, are the zeros of the Legendre polynomials; the Gauss–Laguerre abscissae, with

a = 0

b = 1

, are the zeros of the Laguerre polynomials; and the Gauss–Hermite abscissae, with

a = 0

b = 1

, are the zeros of the Hermite polynomials.

4 References

Davis P J and Rabinowitz P (1975) Methods of Numerical Integration Academic Press

Fröberg C E (1970) Introduction to Numerical Analysis Addison–Wesley

Ralston A (1965) A First Course in Numerical Analysis pp. 87–90 McGraw–Hill

Stroud A H and Secrest D (1966) Gaussian Quadrature Formulas Prentice–Hall

5 Arguments

quad_type

– Nag_QuadType Input

On entry: indicates the quadrature formula.

$quad_type = Nag_Quad_Gauss_Legendre$: Gauss–Legendre quadrature on a finite interval, using normal weights.
$quad_type = Nag_Quad_Gauss_Laguerre$: Gauss–Laguerre quadrature on a semi-infinite interval, using normal weights.
$quad_type = Nag_Quad_Gauss_Laguerre_Adjusted$: Gauss–Laguerre quadrature on a semi-infinite interval, using adjusted weights.
$quad_type = Nag_Quad_Gauss_Hermite$: Gauss–Hermite quadrature on an infinite interval, using normal weights.
$quad_type = Nag_Quad_Gauss_Hermite_Adjusted$: Gauss–Hermite quadrature on an infinite interval, using adjusted weights.
$quad_type = Nag_Quad_Gauss_Rational_Adjusted$: Rational Gauss quadrature on a semi-infinite interval, using adjusted weights.

Constraint:

quad_type = Nag_Quad_Gauss_Legendre

Nag_Quad_Gauss_Laguerre

Nag_Quad_Gauss_Laguerre_Adjusted

Nag_Quad_Gauss_Hermite

Nag_Quad_Gauss_Hermite_Adjusted

Nag_Quad_Gauss_Rational_Adjusted

a

– double Input

b

– double Input

On entry: the parameters

a

and

b

which occur in the quadrature formulae described in Section 3.

Constraints:

if $quad_type = Nag_Quad_Gauss_Rational_Adjusted$ , $a + b \neq 0.0$ ;
if $quad_type = Nag_Quad_Gauss_Laguerre$ or $Nag_Quad_Gauss_Laguerre_Adjusted$ , $b \neq 0.0$ ;
if $quad_type = Nag_Quad_Gauss_Hermite$ or $Nag_Quad_Gauss_Hermite_Adjusted$ , $b > 0.0$ .

Constraints:

Rational Gauss: $a + b \neq 0.0$ ;
Gauss–Laguerre: $b \neq 0.0$ ;
Gauss–Hermite: $b > 0$ .

n

– Integer Input

On entry:

n

, the number of weights and abscissae to be returned.

Constraint:

n = 1

2

3

4

5

6

8

10

12

14

16

20

24

32

48

64

Note: if

n > 0

and is not a member of the above list, the maxmium value of

n

stored below

n

will be used, and all subsequent elements of abscis and weight will be returned as zero.

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

The value of a and/or b is invalid for the chosen quad_type. Either:

On entry, argument $⟨ value ⟩$ had an illegal value.
The value of a and/or b is invalid for Gauss-Hermite quadrature.
On entry, $quad_type = ⟨ value ⟩$ .
On entry, $a = ⟨ value ⟩$ and $b = ⟨ value ⟩$ .
Constraint: $b > 0.0$ .
The value of a and/or b is invalid for Gauss-Laguerre quadrature.
On entry, $quad_type = ⟨ value ⟩$ .
On entry, $a = ⟨ value ⟩$ and $b = ⟨ value ⟩$ .
Constraint: $| b | > 0.0$ .
The value of a and/or b is invalid for rational Gauss quadrature.
On entry, $quad_type = ⟨ value ⟩$ .
On entry, $a = ⟨ value ⟩$ and $b = ⟨ value ⟩$ .
Constraint: $| a + b | > 0.0$ .

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_QUAD_GAUSS_NPTS_RULE

The n-point rule is not among those stored.
On entry:

n = ⟨ value ⟩

.
n-rule used:

n = ⟨ value ⟩

NE_TOO_SMALL

Underflow occurred in calculation of normal weights.
Reduce n or use adjusted weights:

n = ⟨ value ⟩

NE_WEIGHT_ZERO

No nonzero weights were generated for the provided parameters.

NAG CL Interface
d01tbc (dim1_gauss_wres)

▸▿ 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 Interfaced01tbc (dim1_​gauss_​wres)

▸▿ 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
d01tbc (dim1_gauss_wres)