d01 Chapter Contents
d01 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_quad_1d_gauss_wset (d01tbc)

## 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 Gauss–Legendre, rational Gauss, Gauss–Laguerre and Gauss–Hermite.

## 2  Specification

 #include #include
 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 $f\left(x\right)$. The quadrature takes the form
 $S=∑i=1nwifxi$
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).
 $S≃∫abfxdx$
where $a$ and $b$ are finite and it will be exact for any function of the form
 $fx=∑i=0 2n-1cixi.$
 $S≃∫a∞fx dx a+b> 0 or S≃∫-∞a fx dx a+b< 0$
and will be exact for any function of the form
 $fx=∑i=2 2n+1cix+bi=∑i=0 2n-1c2n+1-ix+bix+b2n+1.$
 $S≃∫a∞fx dx b> 0 or S≃∫-∞a fx dx b< 0$
and will be exact for any function of the form
 $fx=e-bx∑i=0 2n-1cixi.$
 $S≃∫-∞ +∞ fx dx$
and will be exact for any function of the form
 $fx=e-b x-a 2∑i=0 2n-1cixi b>0.$
(e) Gauss–Laguerre quadrature, normal weights:
 $S≃∫a∞e-bxfx dx b> 0 or S≃∫-∞a e-bxfx dx b< 0$
and will be exact for any function of the form
 $fx=∑i=0 2n-1cixi.$
(f) Gauss–Hermite quadrature, normal weights:
 $S≃∫-∞ +∞ e-b x-a 2fx dx$
and will be exact for any function of the form
 $fx=∑i=0 2n-1cixi.$
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

On entry: indicates the quadrature formula.
${\mathbf{quad_type}}=\mathrm{Nag_Quad_Gauss_Legendre}$
Gauss–Legendre quadrature on a finite interval, using normal weights.
${\mathbf{quad_type}}=\mathrm{Nag_Quad_Gauss_Rational_Adjusted}$
Rational Gauss quadrature on a semi-infinite interval, using adjusted weights.
${\mathbf{quad_type}}=\mathrm{Nag_Quad_Gauss_Laguerre_Adjusted}$
Gauss–Laguerre quadrature on a semi-infinite interval, using adjusted weights.
${\mathbf{quad_type}}=\mathrm{Nag_Quad_Gauss_Hermite_Adjusted}$
Gauss–Hermite quadrature on an infinite interval, using adjusted weights.
${\mathbf{quad_type}}=\mathrm{Nag_Quad_Gauss_Laguerre}$
Gauss–Laguerre quadrature on a semi-infinite interval, using normal weights.
${\mathbf{quad_type}}=\mathrm{Nag_Quad_Gauss_Hermite}$
Gauss–Hermite quadrature on an infinite interval, using normal weights.
Constraint: ${\mathbf{quad_type}}=\mathrm{Nag_Quad_Gauss_Legendre}$, $\mathrm{Nag_Quad_Gauss_Rational_Adjusted}$, $\mathrm{Nag_Quad_Gauss_Laguerre_Adjusted}$, $\mathrm{Nag_Quad_Gauss_Hermite_Adjusted}$, $\mathrm{Nag_Quad_Gauss_Laguerre}$ or $\mathrm{Nag_Quad_Gauss_Hermite}$.
3:     bdoubleInput
On entry: the quantities $a$ and $b$ as described in the appropriate sub-section of Section 3.
Constraints:
• if ${\mathbf{quad_type}}=\mathrm{Nag_Quad_Gauss_Rational_Adjusted}$, ${\mathbf{a}}+{\mathbf{b}}\ne 0.0$;
• if ${\mathbf{quad_type}}=\mathrm{Nag_Quad_Gauss_Laguerre}$ or $\mathrm{Nag_Quad_Gauss_Laguerre_Adjusted}$, ${\mathbf{b}}\ne 0.0$;
• if ${\mathbf{quad_type}}=\mathrm{Nag_Quad_Gauss_Hermite}$ or $\mathrm{Nag_Quad_Gauss_Hermite_Adjusted}$, ${\mathbf{b}}>0.0$.
4:     nIntegerInput
On entry: $n$, the number of weights and abscissae to be returned.
Constraint: ${\mathbf{n}}=1$, $2$, $3$, $4$, $5$, $6$, $8$, $10$, $12$, $14$, $16$, $20$, $24$, $32$, $48$ or $64$.
5:     weight[n]doubleOutput
On exit: the n weights.
6:     abscis[n]doubleOutput
On exit: the n abscissae.
7:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
The value of a and/or b is invalid: ${\mathbf{a}}=〈\mathit{\text{value}}〉$ and ${\mathbf{b}}=〈\mathit{\text{value}}〉$.
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.
The n-point rule is not among those stored.
NE_TOO_SMALL
Underflow occurred in calculation of normal weights. Reduce n or use adjusted weights: ${\mathbf{n}}=〈\mathit{\text{value}}〉$.

## 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 (d01tbce.c)

### 9.2  Program Data

Program Data (d01tbce.d)

### 9.3  Program Results

Program Results (d01tbce.r)