Computational Physics
Numerical Integration
Dr. Guy Tel-Zur
Tulips by Anna Cervova, publicdomainpictures.net
MHJ- Chap. 7 – Numerical
Integration
• Agenda
– 1D integration (Trapezoidal and Simpson rules)
– Gaussian quadrature (when the abscissas are not
equally spaced)
– Singular integrals
– Physics case study
– Parallel Computing (part 1)
Newton-Cotes quadrature: equal
step methods
Step size:
The strategy then is to find a reliable Taylor expansion for
f(x) in the smaller sub intervals
7.2
Let’s define x0 = a + h and use x0 as the midpoint.
The general form for the Taylor expansion around x0 goes
like:
Let us now suppose that we split the integral in Eq. (7.2) in two parts, one from x0−h to
x0 and the other from x0 to x0 + h, that is, our integral is rewritten as:
Next we assume that we can use the two-point formula for the derivative, meaning
that we approximate f(x) in these two regions by a straight line, as indicated in the
figure. This means that every small element under the function f(x) looks like a
trapezoid.
we are trying to approximate our
function f(x) with a first order polynomial, that is f(x) = a + bx. The constant b is the
slope given by the first derivative at x = x0:
The Trapezoidal Rule:
Trapezoidal Rule
• A worked out example in “C”.
• We will learn how to Integrate a general form
of a function?
• The code is under:
lecture02/code_ch_07/trapez.c
// Computational Physics
// Guy Tel-Zur, August 2010
// Trapezoidal Rule
// Based on MHJ Page 129, Chapter 7, 2009 Fall Edition
// Usage: standard gcc -o fn fn.c -lm
#include <math.h>
#include<stdio.h>
int main() {
// functions declarations:
double TrapezoidalRule(double a , double b , int n,
double(*func )(double));
double MyFunction(double x);
// variables declarations:
double a = 0.0;
double b = 10.0;
int n = 1000;
double s;
s = TrapezoidalRule(a, b, n, &MyFunction);
printf("Trapezoidal Rule Integral=%Lf\n",s);
return 0;
} // end of main
double TrapezoidalRule (double a , double b , int n,
double(*func )(double)) {
double TrapezSum;
double fa ,fb ,x ,step ;
int j;
step =(b-a)/((double)n );
fa =(*func)(a)/2.;
fb =(*func)(b)/2.;
TrapezSum= 0.;
for(j=1;j<=n-1;j++) {
x=j*step+a;
TrapezSum+=(*func)(x);
}
TrapezSum=(TrapezSum+fb+fa)*step;
return TrapezSum;
} // end TrapezoidalRule
double MyFunction(double x) {
// write below the mathematical expression of the
// function to be integrated
return x*x;
}
Section 7.5 – Rectangle Rule
Another very simple approach is the so-called midpoint or rectangle method. In this case the integration
area is split in a given number of rectangles with length h and height given by the mid-point value
of the function. This gives the following simple rule for approximating an integral
•
Demo: ch7 code folder. Program: rectangle_trapez.c
double RectangleRule (double a , double b , int n, double(*func )(double))
{
double RectangleSum;
double fa ,fb ,x ,step ;
int j;
step =(b-a)/((double)n );
RectangleSum= 0.;
for(j=0;j<=n;j++) {
x=(j+0.5)*step+a;
RectangleSum+=(*func)(x);
}
RectangleSum *= step;
return RectangleSum;
} // end Rectangle Rule
Error analysis
• Trapezoidal & Rectangle rules:
– Local error: α h^3
– Global error: α h^2
– Global error: α (b-a)
Can’t we do better?
• So far: Linear two-points approximations for f
• Now (Simpson’s): three-points formula (a
parabola):
• Recall from Chapter 3, the 1st and the 2nd
derivatives approximations:
• f’=
• f’’=
• With the latter two expressions we can now
approximate the function f as:
I gave up using Equation Editor – Sorry…
Next Slide…
Simpson’s Rule
Note that the improved accuracy in the evaluation of the derivatives gives a
better error approximation, O(h5) vs. O(h3) . - This is the local error.
The global error goes like: O(h4).
The full integral is therefore:
Simpson’s Rule
7.3 Adaptive Integration
• Naïve approach, for example, two parts of
Trapezoidal Integration with a different step
size
Recursive Adaptive Integration
7.4 Gaussian Quadrature (GQ)
We could fit a polynomial of order N with N equally spaced points, but Gauss,
who was a genius, suggested to fit a polynomial of order 2N-1 with N points!!!
Recommended online video:
http://physics.orst.edu/~rubin/COURSES/VideoLecs/Integration/Integration.html
Speaking about references, you
can always check Google Books
http://www.google.com/search?tbs=bks%3A1&tbo=1&q=computational+physics&btn
G=Search+Books
Another recommended reference:
Computational Physics By Steven E. Koonin
So far – equally spaced points:
Our derivation is exact if f is polynomial
4.9
Note: This is a set of N linear equations (for each p) !!!
Legendre Polynomials:
Guy: Scaling of the
weights: ωn’=(b-a)/2* ω n
Sage demo
• Upload: “Gaussian Quadrature Demo.sws”
• Reference:
http://wiki.sagemath.org/interact/calculus
Let’s sum what we did so far
2 points integration: Trapezoidal Rule (+Rectangle Rule),fit straight line
3 points integration: Simpson’s Rule, fit a parabola
Quadrature Formula:
Gauss Quadrature:
More Orthogonal Functions
The program1.cpp is a ready for comparing the 3 integrations methods mentioned.
The Trapezoidal, Simpson’s and GQ function are to be written.
>> Explain this program (open DevC++ IDE)