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Chapter Contents

Chapter Introduction

NAG Toolbox: nag_quad_1d_gauss_wres (d01tb)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_quad_1d_gauss_wres (d01tb) 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.

Syntax

[weight, abscis, ifail] = d01tb(key, a, b, n)

[weight, abscis, ifail] = nag_quad_1d_gauss_wres(key, a, b, n)

Description

nag_quad_1d_gauss_wres (d01tb) 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 Arguments).

(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.

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

Parameters

Compulsory Input Parameters

1: $key$ – int64int32nag_int scalar

Indicates the quadrature formula.

$key = 0$: Gauss–Legendre quadrature on a finite interval, using normal weights.
$key = 3$: Gauss–Laguerre quadrature on a semi-infinite interval, using normal weights.
$key = - 3$: Gauss–Laguerre quadrature on a semi-infinite interval, using adjusted weights.
$key = 4$: Gauss–Hermite quadrature on an infinite interval, using normal weights.
$key = - 4$: Gauss–Hermite quadrature on an infinite interval, using adjusted weights.
$key = - 5$: Rational Gauss quadrature on a semi-infinite interval, using adjusted weights.

Constraint:

key = 0

,

3

,

- 3

,

4

,

- 4

or

- 5

.

2: $a$ – double scalar

3: $b$ – double scalar

The quantities

a

and

b

as described in the appropriate sub-section of Description.

Constraints:

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

4: $n$ – int64int32nag_int scalar

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.

Optional Input Parameters

None.

Output Parameters

1: $weight (n)$ – double array: The n weights.
2: $abscis (n)$ – double array: The n abscissae.
3: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W $ifail = 1$: The n-point rule is not among those stored.

W $ifail = 2$: Underflow occurred in calculation of normal weights.

W $ifail = 3$: No nonzero weights were generated for the provided parameters.

$ifail = 11$: Constraint: $key = 0$ , $3$ , $- 3$ , $4$ , $- 4$ or $- 5$ .

$ifail = 12$

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

Constraint: $|a + b| > 0.0$ .
Constraint: $|b| > 0.0$ .
Constraint: $b > 0.0$ .

$ifail = 14$: Constraint: $n > 0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

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

Further Comments

Timing is negligible.

Example

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

Open in the MATLAB editor: d01tb_example

function d01tb_example


fprintf('d01tb example results\n\n');

key = int64(-3);
a = 0;
b = 1;
n = int64(6);

[weight, abscis, ifail] = d01tb( ...
				 key, a, b, n);

fprintf('  Weights    Abscissae\n');
fprintf('%9.4f%12.4f\n',[weight abscis]');



function [fv, iflag, user] = f(x, nx, iflag, user)
  fv = sin(x)./x.*log(10*(1-x));

d01tb example results

  Weights    Abscissae
   0.5735      0.2228
   1.3693      1.1889
   2.2607      2.9927
   3.3505      5.7751
   4.8868      9.8375
   7.8490     15.9829

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

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