Midterm 1
Problem 1
Show that if r is a zero of multiplicity 2 of the function f (x) (i.e., f (r) = f 0 (r) = 0 6=
f 00 (r)), then the iterative procedure
xn+1 = xn − 2
f (xn )
f 0 (xn )
is quadratically convergent, under suitable assumptions.
P.S. Note that if r is a zero of multiplicity of k, then the quadratic convergence in Newton’s
method will be restored by making modification xn+1 = xn − kf (xn )/f 0 (xn )
Problem 2
Consider the steepest descent method for solving Ax = b: x(k+1) = x(k) + tk v (k) , v (k) =
b − Ax(k) , tk = (v (k) , b − Ax(k) )/(v (k) , Av (k) ).
1) For what values of tk does the method of steepest descent reduce to the Richardson
method.
2) What values on tk and A must be fulfilled so that the method of steepest descent is
equivalent to the Jacobi method?
3) Denote x∗ = A−1 b, q(x) = (x.Ax) − 2(x, b) and show that
q(x(k+1) ) − q(x∗ )
1
=1− ,
(k)
∗
q(x ) − q(x )
β
where
(1)
1
(v (k) , v (k) )2
= (k)
.
β
(v , Av (k) )(v (k) , A−1 v (k) )
Next using the inequality
(λmax + λmin )2
β≤
4λmax λmin
derive the convergence rate for (1).
Problem 3
Write a program for finding natural cubic spline interpolation of f (x) = 1/(1 + x2 ) on
the interval [−5, 5] by using equally spaced knots with n = 5, 10, 20, 40, 80. Use conjugate
gradient algorithm to solve linear system of equations. For each n plot both cubic spline
interpolation and f (x) on the same graph. Print the values of zi for n = 20. Compute
max[−5,5] f (x) − S(x) for each n.
Problem 4
1
Characterize the family of n×n nonsingular matrices A for which one step of Gauss-Seidel
algorithm solves Ax = b, starting at the vector x = 0.
Problem 5
What condition will have to be placed on the nodes x0 and x1 if the interpolation problem
p(xi ) = ci0 ,
p00 (xi ) = ci2 , (i = 0, 1)
is to be solvable by a cubic polynomial (for arbitrary cij )?
2