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)

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

Simpson’s Rule: Suppose f(x)=x Let a = 2 and b = 8 ½ (64-4) = 30 1/6(6)(2 + 4*5 +8) = 30

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

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.

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..

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