d01tbf (dim1_gauss_wres) : NAG Library, Mark 27

d01tbf 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.

The routine may be called by the names d01tbf or nagf_quad_dim1_gauss_wres.

d01tbf 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.

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

On entry: 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

−5

On entry: the parameters

a

and

b

which occur in the quadrature formulae described in Section 3.

Constraints:

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

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.

On exit: the n weights.

On exit: the n abscissae.

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

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

n = ⟨ value ⟩

.
n-rule used:

n = ⟨ value ⟩

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

n = ⟨ value ⟩

No nonzero weights were generated for the provided parameters.

On entry,

key = ⟨ value ⟩

.
Constraint:

key = 0

3

−3

4

−4

−5

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

The value of a and/or b is invalid for Gauss-Hermite quadrature.
On entry, $key = ⟨ value ⟩$ .
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, $key = ⟨ value ⟩$ .
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, $key = ⟨ value ⟩$ .
On entry, $a = ⟨ value ⟩$ and $b = ⟨ value ⟩$ .
Constraint: $| a + b | > 0.0$ .

On entry,

n = ⟨ value ⟩

.
Constraint:

n > 0

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 7 in the Introduction to the NAG Library FL Interface for further information.

Your licence key may have expired or may not have been installed correctly.

See Section 8 in the Introduction to the NAG Library FL Interface for further information.

Dynamic memory allocation failed.

See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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

d01tbf is not threaded in any implementation.

Timing is negligible.

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

NAG FL Interface
d01tbf (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 FL Interfaced01tbf (dim1_​gauss_​wres)