HP 71B and HP Prime Python: Gaussian Quadrature
A Very Brief Introduction
This is a top-level overview of the Gaussian Quadrature method. For more details, please check out the Sources section.
The design of GQ is to calculate:
∫( f(x) dx, -1, 1) ≈ Σ( wi * f(xi), i = 1, n)
Where do xi and wi come from?
* The xi values come from find the roots of the polynomials P’(x)= 0
* Pn’(x) are the derivatives of the Legendre polynomials Pn(x)
* After getting the roots, xi, the weights, w are calculated by the formula:
wi = 2 / ((1 -xi)^2 * Pn’(xi)^2)
Weights and Points of orders 3 and 5:
Among 3 points:
Roots (x) come from the derivative of the Legendre polynomial of the 3rd order.
Pn’(x) = 3/2 * (5*x^2 – 1)
Point xi |
Weight wi |
-√(3/5) ≈ -0.7745 9669 2 = -√0.6 |
5/9 |
0 |
8/9 |
√(3/5) ≈ 0.7745 9669 2 = √0.6 |
5/9 |
Among 5 points:
Roots (x) come from the derivative of the Legendre polynomial of the 5th order.
Pn’(x) = 15/8 * (21*x^4 – 14*x^2 + 1)
Point xi |
Weight wi |
-0.90617 98459 |
0.23692 68851 |
-0.53846 93101 |
0.47862 86705 |
0 |
128/225 ≈ 0.56888 88889 |
0.53846 93101 |
0.47862 86705 |
0.90617 98459 |
0.23692 68851 |
We can fit any interval [a, b] by this conversion:
∫( f(x) dx, a, b) = (b – a) / 2 * ∫( f( (b – a) / 2* x + (b + a) / 2 dx, -1, 1)
= (b – a) / 2 * Σ( wi * f( (b – a) / 2 * xi + (b + a) / 2 ), i = 1, n)
However, the programs on this blog entry will focus on the basic version of the Gaussian Quadrature.
The Code
The following code works with the following integrals:
∫( f(x) dx, -1, 1) ≈ Σ( wi * f(xi), i = 1, n)
HP 71B Basic
HP 71B: GQ3 (Gaussian Quadrature – 3 Point: Integrals from -1 to 1)
10 DEF FNF(X) = insert function of X here
15 S = 0
20 RADIANS
25 FOR I = 1 TO 3
30 READ X, W
31 DATA -SQR(0.6),5/9
32 DATA 0,8/9
33 DATA SQR(0.6),5/9
40 S = S + W * FNF(X)
55 NEXT I
50 DISP “ INTEGRAL= ”; S
HP 71B: GQ5 (Gaussian Quadrature – 5 Point: Integrals from -1 to 1)
10 DEF FNF(X) = insert function of X here
15 S = 0
20 RADIANS
25 FOR I = 1 TO 5
30 READ X, W
31 DATA -.9061798459, .2369268851
32 DATA -.5384693101, .4786286705
33 DATA 0, 128/255
34 DATA .5384693101, .4786286705
35 DATA .9061798459, .2369268851
40 S = S + W * FNF(X)
45 NEXT I
50 DISP “ INTEGRAL= ”; S
Integral Test: ∫( f(x) dx, -1, 1) ≈ Σ( wi * f(xi), i = 1, n)
Function f(x) |
Actual (approx) |
Gaussian 3 point (GQ3) |
Gaussian 5 point (GQ5) |
(x-3)*(x+4)*(x-5) |
352/3 ≈ 117.333333333 |
117.333333333 |
117.333333334 |
-x^2 + 9*x + 10 |
58/3 ≈ 19.33333333 |
19.3333333334 |
19.3333333333 |
1.5 * cos(x) |
2.524412954 |
2.52450532159 |
2.52441295561 |
exp(-x) |
2.350402387 |
2.35033692868 |
2.35040238646 |
exp(sin(x)) |
2.283194521 |
2.28303911433 |
2.28319665353 |
sin(x) |
0 |
0 |
0 |
1/(x - 2) |
-1.09861228867 |
-1.09803921569 |
-1.09860924181 |
HP Prime Python App
HP Prime Python: gq3.py
# Gaussian Quadrature 3 point
from math import *
def f(x):
# insert Function here
return 1/(x-2)
s=0
x=[-sqrt(0.6),0,sqrt(0.6)]
w=[5/9,8/9,5/9]
for i in range(3):
s=s+w[i]*f(x[i])
print("Integral: ",s)
HP Prime Python: gq5.py
# Gaussian Quadrature 5 point
from math import *
def f(x):
# insert Function here
return sin(x)
s=0
x=[0,-.5384693101056831,.5384693101056831,-.9061798459386640, .9061798459386640]
w=[.5688888888889,.4786286704993665,.4786286704993665,.2369268850561891,.2369268850561891]
for i in range(5):
s=s+w[i]*f(x[i])
print("Integral: ",s)
Function f(x) |
Actual (approx) |
Gaussian 3 point (GQ3) |
Gaussian 5 point (GQ5) |
(x-3)*(x+4)*(x-5) |
352/3 ≈ 117.333333333 |
117.333333333 |
117.333333333 |
-x^2 + 9*x + 10 |
58/3 ≈ 19.33333333 |
19.3333333333 |
19.3333333334 |
1.5 * cos(x) |
2.524412954 |
2.52450532159038 |
2.52441295561081 |
exp(-x) |
2.350402387 |
2.35033692868001 |
2.35040238646284 |
exp(sin(x)) |
2.283194521 |
2.2830391143268 |
2.28319665352905 |
sin(x) |
0 |
0.0 |
0.0 |
1/(x - 2) |
-1.09861228867 |
-1.09803921568627 |
-1.09860924181248 |
Sources
Kamermans, Mike “Pomax”. “Gaussian Quadrature Weights and Abscissae” https://pomax.github.io/bezierinfo/legendre-gauss.html 2011. Retrieved January 19, 2025.
Wikipedia. “Gauss-Legendre quadrature” https://en.wikipedia.org/wiki/Gauss%E2%80%93Legendre_quadrature Last edited January 19, 2025. Retrieved January 19, 2025.
Wikipedia. “Legendre Polynomials” https://en.wikipedia.org/wiki/Legendre_polynomials Last edited December 4, 2024. Retrieved January 19, 2025.
When
it comes to evaluating integrals numerically, is Gaussian Quadrature
better than Simpson’s Method?
Eddie
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