MATH 417: Numerical Analysis
Instructor: Prof. Wolfgang Bangerth
Teaching Assistants: Dukjin Nam
Homework assignment 8 – due 4/5/2007
Problem 1 (Steepest descent iteration). Repeat what you did for Prob-
lem 1 of Homework 6 (Jacobi iteration) and Problem 2 of Homework 7 (Gauss-
Seidel iteration), but use the steepest descent algorithm instead to compute the
vectors x(k). Generate the same plots as before. Compare your results with
the previous results, in particular compare how quickly the iterations appear to
converge. (5 points)
Problem 2 (Conjugate gradient iteration). Do the same as in Problem 1
one last time, but use the Conjugate Gradient algorithm this time to compute
the vectors x(k). Generate the same plots as before. Compare your results with
the previous results, in particular compare how quickly the iterations appear to
converge. (5 points)
Problem 3 (Lagrange interpolation).
(a) Compute the Lagrange interpolation polynomials L4,k, k = 0 . . . 3, for the
points x0= 1, x1= 2, x2= 1.5 and x3= 1.6.
(b) Calculate the interpolating polynomial for the data set where yk= log xk
at the four points xk. Write the polynomial in the form p4(x) = a3x3+
a2x2+a1x+a0.(4 points)
Problem 4 (Lagrange interpolation). The polynomial p4(x) calculated in
Problem 3 by construction interpolates the function f(x) = log x. Compute an
upper bound for the error on the interval [1,2], using the theorem that states
how large |f(x)−p4(x)|can at most be. (3 points)
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