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Foundations of Computational Math

Due date: 5PM Wednesday, 11/26/08
Grading: Since this overlaps with the assignment of Program 3, all of these questions
will be graded based on effort.

Problem 8.1

Suppose is a symmetric positive definite tridiagonal matrix of the form

where n = 2k, Dr and Db are diagonal matrices of order n/2, L is a lower triangular matrix
with nonzeros restricted to its main diagonal and its first subdiagonal, and U is upper
triangular matrix with nonzeros restricted to its main diagonal and its first superdiagonal.

Assume that Ax = b can be solved using Jacobi ’s method, i.e., the iteration converges
acceptably fast. Partition each iterate xi into the top half and bottom half, i.e.,

8.1.a. Assume an initial guess x0 is given and identify what information, i.e., the pieces
of xi for 0≤i≤j, determines the values found in the vectors and for any j > 0.

8.1.b. Can the relationships from 8.1.a be used to design an iteration that approximates
the solution essentially as well but only requires half of the work of Jacobi’s
method ?

8.1.c. Relate your new method from 8.1.b to applying Gauss-Seidel to solve Bx = b
starting from the same initial guess x0.

Problem 8.2

8.2.a

Consider the two matrices :

Suppose you solve systems of linear equations involving A1 and A2 using Jacobi’s method.
For which matrix would you expect faster convergence?

8.2.b

Consider the matrix

(i) Will Jacobi’s method converge when solving Ax = b?
(ii) Will Gauss-Seidel converge when solving Ax = b?

Problem 8.3

Consider solving a linear system Ax = b where A is symmetric positive definite using steepest descent.

(8.3.a) Suppose you use steepest descent without preconditioning . Show that the residuals,
rk and rk+1 are orthogonal for all k.

(8.3.b) Suppose you use steepest descent with preconditioning . Are the residuals, rk
and rk+1 are orthogonal for all k? If not is there any vector from step k that is
guaranteed to be orthogonal to rk+1?

Problem 8.4

Let f(x) = x^3 − 3x + 1. This polynomial has three distinct roots

(8.4.a) Consider using the iteration function

Which, if any, of the three roots can you compute with and how would you
choose x(0) for each computable root?

(8.4.b) Consider using the iteration function

Which, if any, of the three roots can you compute with and how would you
choose x(0) for each computable root?

(8.4.c) For each of the roots you identified as computable using either or ,
apply the iteration to find the values of the roots . (You need not turn in any
code, but using a simple program to do this is recommended.)

8.4.d

Let be a continuous function. Show that if is a contraction mapping
on [a, b] then the sequence {x(k)} defined by x(k+1) = is a Cauchy sequence.

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