Numerical Methods Supervision 3

Exercises

Questions beginning with S are from the course workbook. Your answers to non-exam questions should be reasonably short: one or two paragraphs each.

1. S2.2
   
   1. What is the formula for finding the square root of a number using Newton Raphson? (This is given in the slide pack but derive it from first principles please.)
   2. When do we say that a numerical method is a second order method and when do we say an iteration has order of convergence 2?
   3. What is the order of convergence of Newton Raphson?
   4. Suppose the derivative of the target function is expensive to compute or not available in computable form: suggest a cheaper iteration based on Newton’s method (makes the same graphical construction) and estimate/find/state its order of convergence.


2. S2.7, part a) only
   Show all the Chebyshev polynomials pass through (1,1). What other point do half of them pass through?


3. Outline the first five iterations of the CORDIC algorithm when computing cos(30). Which angle's cosine has actually been computed? (You do not need to compute the cosine, but you may do so if you like.)


4. S2.10


5. S3.1, parts a) to d) only
   
   1. What condition is needed for a pair of matrices to be multipliable?
   2. What is the complexity (in floating point operations (flops)) of converting to upper triangular form for a general set of coefficients?
   3. What is the complexity of converting to the triangular form generated using Cholesky’s method?
   4. What is the complexity of forwards and backwards substitution?


6. 2015 Paper 1 Question 6


7. S3.10
   
   1. Does an adaptive time step help when modelling non-linear components?
   2. Does an adaptive time step help when modelling time-varying components?