Numerical Integration UC Berkeley Fall 2004, E77 Copyright 2005, Andy Packard. This work is licensed under the Creative Commons Attribution-ShareAlike License. To view a copy of this license, visit or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

Integration For a function f, The “integral of f from a to b” is the area under the graph of the function. If f is continuous, then the area is well defined, as the common limit of upper and lower sums. The integral is denoted a b x graph of f(x)

Integration: Fundamental theorem of Calculus If a function g is the antiderivative of f, namely for all x, then the fundamental theorem of calculus gives that the integral of f can be computed by evaluating g But finding antiderivatives can be hard… In many cases of engineering and scientific (and economics, etc) interest –the functions do not have known antiderivatives, so… –the integral (area) must be approximated by a finite number of function evaluations.

Numerical Integration: Trapezoid a b x graph of f(x) a b x approximate this region with this trapezoid

Composite Trapezoid a b x graph of f(x) Apply “ trapezoid ” approximation to k subdivisions of [a b] a b x graph of f(x) “ trapezoid ” approximation with 3 subdivisions of [a b]

Composite Trapezoid a b x graph of f(x) “ trapezoid ” approximation with 3 subdivisions of [a b] h Let h=(b-a)/3. The sum of all the approximations is

Derivation of Simpson’s Rule: Suppose f(x)=x 3

Simpson’s Rule: Suppose f(x)=x 2

Simpson’s Rule: Suppose f(x)=x

Simpson’s Rule: Suppose f(x)=1

Simpson’s Rule: Derivation Put all of those together, along with Hence: If f is any cubic polynomial, then This is the basis for Simpson ’ s rule.

Simpson’s Rule For any function f, the Simpson ’ s approximation to is Evaluate the function at the endpoints and in the middle

Composite Simpson’s Rule a b x graph of f(x) Simpson on 3 subdivisions of [a b] h h=(b-a)/3 Add them up. Total of 7 function evaluations.

Composite Simpson’s Rule a b x graph of f(x) Simpson on 2 subdivisions of [a b] h=(b-a)/2 5 function evaluations giving the approximation as I S =I 1 S +I 2 S

Numerical Integration: Ad-Hoc stopping criteria Pick a method (trapezoid, or Simpson’s). Set a stopping tolerance TOL. Pick k, an initial number of subdivisions Iterate as below –Apply composite method using k divisions –Apply composite method using 2k divisions –If answers are within TOL, stop, and return the 2k division answer –If answers are not within TOL, increase k and repeat. With only this much analysis, the computed answer is not specifically known to be accurate to any specific precision.

Adaptive Stepsize Only use small h (the “stepsize”) where the convergence demands it. Recursive implementation is straightforward. function I = adr(fh,a,b,tol) Compute I1 using 1 subdivision Compute I2 using 2 subdivisions If the answers are within tol, I = I2; Else m = (a+b)/2; ILeft = adr(fh,a,m,tol/2); IRight = adr(fh,m,b,tol/2); I = ILeft + IRight; end

Error Analysis More error analysis can be done making assumptions about the magnitude of the derivatives of f –A bound on the magnitude of 3 rd derivative of f yields accuracy bounds for the Trapezoid method –A bound on the magnitude of 5 th derivative of f yields accuracy bounds for the Simpson’s method Learn more about these in Math 128A

Matlab functions trapz –fixed trapezoidal approximation –data is vectors x and y ( y represents f(x) ) –1 division between each data pair quad –Adaptive Simpson’s method –Data is function handle, interval endpoints, stopping tolerance