CS 417/517 Computational Methods and Software
Spring 2004
HW 3
Assigned: Thurs Feb 5, 2004; Due: Thurs Feb 12, 2004
Homework problems will be posted in the course directory:
www.cs.odu.edu/˜pothen/Courses/CS417. If any corrections are found to be neces-
sary, they will be posted in this directory as well. You must show your work to receive credit for
your answers. In problems where you are asked to give reasons, an answer without a stated reason
will receive no credit.
1. Alice buys three apples, twelve bananas, and one cantaloupe for $2.36. Bob buys twelve
apples and two cantaloupes for $5.26. Carol buys two bananas and three cantaloupes for
$2.77. How much do single pieces of each fruit cost?
2. Matlab has a collection of ill-conditioned matrices that cause large errors in the computed
solution when floating point arithmetic is used. These matrices include magic,pascal,
hilb, etc. For this problem, let us investigate the n×nHilbert matrix H, whose elements
are given by
hij = 1/(i+j−1),for i= 1,2, . . . , n;j= 1,2, . . . , n.
(a) In Matlab, type type hilb.m to see Matlab’s efficient ‘vectorized’ code for com-
puting the Hilbert matrix. There are five lines of code in this Matlab program. Explain
what each line of code does.
Hint. Use Matlab’s help command as you need. In particular, you might find
help punct to give you useful information.
(b) Create a Hilbert matrix of order nand create a right hand side vector bsuch that the
exact solution to Hx =bis the vector of all ones. Solve the system of equations
Hx =bin Matlab. Compare the solution that you compute with the exact solution. Do
this for three values of matrix sizes: n= 10,n= 20, and n= 30.
(c) The difference between the computed solution and the exact solution (in absolute value)
is known to be less than or equal to the product of two numbers: the condition number
of the matrix and the machine precision. Use the Matlab command cond to calcu-
late the condition numbers of the three Hilbert matrices; then compute the product of
the condition number and machine precision in Matlab. Do the values of the product
correctly predict the errors you see for the three values of n?