6760, Math 21a, Fall 2009

Week 8 Problems F, Math 21a, Multivariable Calculus

Monte Carlo and Riemann Integration

Course head: Oliver Knill

Office: SciCtr 434

Email: knill@math.harvard.edu

Problem F:
In this exercise, we compute the double integral of the function
f(x,y) = ( 1+ x e^{-y3})^{1/2}
over the rectangle [0,1] x [0,1]. There is no expression using known functions
and numbers to get the answer, but we do it numerically: 1) Find a Riemann Sum with dx=1/M, dy =1/M, where M is an integer like 100. Call this sum "RiemannSum". 2) Find a Monte Carlo Sum by choosing two random numbers x,y in [0,1] and compute the mean f(x,y) over 10'000 experiments. Call this average of random variables "MonteCarlo". 3) Find a numerical integral on the square [0,1] x [0,1]. Call this integral "Numerical" 4) Find the fraction |RiemannSum-Numerical|/|MonteCarlo-Numerical| This tells us by which factor the Riemann Sum error is larger than the Monte Carlo Sum error. To get full credit, we need to have the Mathematica lines and the numerical answers. Below is Mathematica sample code for f(x,y) = exp(x*y) + sqrt(x): Note that all built in functions like Exp[x]=E^x are capitalized.
1) Take M=6 and add the sum of M ^{2}=36 numbers
RiemannSum = [ f(1/M,1/M) + f(2/M,1/M) + ....+ f(M/M,M/M) ]/MThis number "RiemannSum" is a Riemann approximation of the integral. 2) Now use a random number generator to generate 36 random numbers x(i),y(i) in the interval [0,1]. Then produce the average: MonteCarlo = [f(x(1),y(1)) + ... + f(x(36),y(36))]/36You can do this with a friend. One generates random numbers, the other punches in the numbers. 3) Finally, use the integrator in the calculator to find the integral Numerical = integrate f(x,y) , (x,0,1), (y,0,1)on the interval [0,1] x [0,1]. 4) Find |RiemannSum-Numerical|/|MonteCarlo-Numerical| as before |

Questions and comments to knill@math.harvard.edu

Math21b | Math 21a | Fall 2009 |
Department of Mathematics |
Faculty of Art and Sciences |
Harvard University