D01TBF (PDF version)

D01 Chapter Contents

D01 Chapter Introduction

NAG Library Manual

NAG Library Routine Document

D01TBF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

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.

2 Specification

SUBROUTINE D01TBF (

KEY, A, B, N, WEIGHT, ABSCIS, IFAIL)

INTEGER	KEY, N, IFAIL
REAL (KIND=nag_wp)	A, B, WEIGHT(N), ABSCIS(N)

3 Description

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.

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 Parameters

1: KEY – INTEGERInput

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

or

-5

.

2: A – REAL (KIND=nag_wp)Input

3: B – REAL (KIND=nag_wp)Input

On entry: the quantities

a

and

b

as described in the appropriate sub-section of Section 3.

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) – REAL (KIND=nag_wp) arrayOutput

On exit: the N weights.

6: ABSCIS(N) – REAL (KIND=nag_wp) arrayOutput

On exit: the N abscissae.

7: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

,

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. 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).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

or

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

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

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

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

$IFAIL = 11$: On entry, $KEY = ⟨value⟩$ .
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:

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: $|A + B| > 0.0$ .

$IFAIL = 14$: On entry, $N = ⟨value⟩$ .
Constraint: $N > 0$ .

7 Accuracy

The weights and abscissae are stored for standard values of A and B to full machine accuracy.

8 Further Comments

Timing is negligible.

9 Example

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

9.1 Program Text

Program Text (d01tbfe.f90)

9.2 Program Data

None.

9.3 Program Results

Program Results (d01tbfe.r)

D01TBF (PDF version)

D01 Chapter Contents

D01 Chapter Introduction

NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012