nagcpp::quad::dim1_gauss_wres (d01tb) : NAG Library, Mark 27

2 Specification

#include "d01/nagcpp_d01tb.hpp"

template <typename WEIGHT, typename ABSCIS> void function dim1_gauss_wres(const types::f77_integer key, const double a, const double b, const types::f77_integer n, WEIGHT &&weight, ABSCIS &&abscis, OptionalD01TB opt)

3 Description

dim1_gauss_wres 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

key

– types::f77_integer Input

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

a

– double Input

On entry: the parameters

a

and

b

which occur in the quadrature formulae described in Section 3.

Constraints:

if $key = −5$ , $a + b \neq 0.0$ ;
if $key = 3$ or $−3$ , $b \neq 0.0$ ;
if $key = 4$ or $−4$ , $b > 0.0$ .

Constraints:

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

b

– double Input

On entry: the parameters

a

and

b

which occur in the quadrature formulae described in Section 3.

Constraints:

if $key = −5$ , $a + b \neq 0.0$ ;
if $key = 3$ or $−3$ , $b \neq 0.0$ ;
if $key = 4$ or $−4$ , $b > 0.0$ .

Constraints:

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

n

– types::f77_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 array Output

On exit: the n weights.

abscis (n)

– double array Output

On exit: the n abscissae.

opt

– OptionalD01TB Input/Output

Optional parameter container, derived from Optional.

6 Exceptions and Warnings

Errors or warnings detected by the function:

All errors and warnings have an associated numeric error code field, errorid, stored either as a member of the thrown exception object (see errorid), or as a member of opt.ifail, depending on how errors and warnings are being handled (see Error Handling for more details).

Raises: WarningException

$errorid = 1$: The n-point rule is not among those stored.
On entry: $n = ⟨ value ⟩$ .
n-rule used: $n = ⟨ value ⟩$ .

$errorid = 2$: Underflow occurred in calculation of normal weights.
Reduce n or use adjusted weights: $n = ⟨ value ⟩$ .

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

Raises: ErrorException

$errorid = 11$: On entry, $key = ⟨ value ⟩$ .
Constraint: $key = 0, 3, −3, 4, −4 or −5$ .

$errorid = 12$: 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: $| b | > 0.0$ .

$errorid = 12$: 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: $b > 0.0$ .

$errorid = 12$: 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: $| a + b | > 0.0$ .

$errorid = 14$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n > 0$ .

$errorid = 10601$: On entry, argument $⟨ value ⟩$ must be a vector of size $⟨ value ⟩$ array.
Supplied argument has $⟨ value ⟩$ dimensions.

$errorid = 10601$: On entry, argument $⟨ value ⟩$ must be a vector of size $⟨ value ⟩$ array.
Supplied argument was a vector of size $⟨ value ⟩$ .

$errorid = 10601$: On entry, argument $⟨ value ⟩$ must be a vector of size $⟨ value ⟩$ array.
The size for the supplied array could not be ascertained.

$errorid = 10602$: On entry, the raw data component of $⟨ value ⟩$ is null.

$errorid = 10603$: On entry, unable to ascertain a value for $⟨ value ⟩$ .

$errorid = −99$: An unexpected error has been triggered by this routine.

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

$errorid = −999$: Dynamic memory allocation failed.

NAG CPP Interface
nagcpp::quad::dim1_gauss_wres (d01tb)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Exceptions and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

NAG CPP Interfacenagcpp::quad::dim1_gauss_wres (d01tb)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Exceptions and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

NAG CPP Interface
nagcpp::quad::dim1_gauss_wres (d01tb)