nag_quad_1d_gauss_wset (d01tbc) (PDF version)

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nag_quad_1d_gauss_wset (d01tbc)

▸▿ 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

nag_quad_1d_gauss_wset (d01tbc) returns the weights and abscissae appropriate to a Gaussian quadrature formula with a specified number of abscissae. The formulae provided are for Gauss–Legendre, rational Gauss, Gauss–Laguerre and Gauss–Hermite.

2

Specification

#include <nag.h>

#include <nagd01.h>

void	nag_quad_1d_gauss_wset (Nag_QuadType quad_type, double a, double b, Integer n, double weight[], double abscis[], NagError *fail)

3

Description

nag_quad_1d_gauss_wset (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

1: $quad_type$ – Nag_QuadTypeInput

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

or

Nag_Quad_Gauss_Rational_Adjusted

.

2: $a$ – doubleInput

3: $b$ – doubleInput

On entry: the quantities

a

and

b

as described in the appropriate sub-section of 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: $b > 0$ .

4: $n$ – IntegerInput

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

or

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.

5: $weight [n]$ – doubleOutput

On exit: the n weights.

6: $abscis [n]$ – doubleOutput

On exit: the n abscissae.

7: $fail$ – 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

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 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.

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.

7

Accuracy

The weights and abscissae are stored for standard values of a and b to full machine accuracy.

8

Parallelism and Performance

nag_quad_1d_gauss_wset (d01tbc) is not threaded in any implementation.

9

Further Comments

Timing is negligible.

10

Example

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

10.1

Program Text

Program Text (d01tbce.c)

10.2

Program Data

None.

10.3

Program Results

Program Results (d01tbce.r)

nag_quad_1d_gauss_wset (d01tbc) (PDF version)

d01 Chapter Contents

d01 Chapter Introduction

NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017