Topic: Numerical Computing

1.

a.) An iteration gives these three values:

x0: 4.472135954999580 x1: 4.352501798965644 x2: 4.324984929600801

Assuming linear convergence, use an excel spreadsheet to extrapolate to the true value, using each of the three formulas (on the bottom). Show 15 decimal places in each cell.

b.) Set up an excel spreadsheet to solve x*exp(x) = 1 by using accelerated convergence of iteration of x = exp(-x). The excel spreadsheet should do two regular iterations of x = exp(-x), then an extrapolation to get the next x, then two more regular iterations, then extrapolation, etc. until complete convergence. Use the third extrapolation formula. Also, put in columns for bits of accuracy and digits of accuracy, using the true value 0.567143290409784.

Further clarification: The x column of your spreadsheet should be organized like this:

A6 starting x A7 iteration from A6 A8 iteration from A7 A9 extrapolation from A6,A7,A8 A10 iteration from A9 A11 iteration from A10 A12 extrapolation from A9,A10,A11 and so forth.

Formulas:

T = (x2*x0 - x1*x1)/(x0 - 2*x1 + x2) (the "naive" formula)

T = x0 - (x0 - x1)^2/((x0-x1)-(x1-x2)))

T = x2 - (x2 - x1)^2/((x0-x1)-(x1-x2))