AMSC/CMSC 460 Homework 5 Due 5/1
1. Assume we have a Matlab function
[p, mult, U] = hessreduce(H),
that reduces the upper Hessenberg matrix to triangular form by Gaussian elimination
with partial pivoting. Specifically, p(i) is 1if there was pivoting at the i-th stage of
the reduction of Hand and p(i) is 0if there was not pivoting. The quantity mult(i) is
the i-th multiplier in the reduction; i.e., H(i+1,i)/H(i,i). The matrix Uis the upper
triangular matrix produced by the reduction. Write a Matlab function
x = hesssolve(b, p, mult, U)
to solve the system Hx=b. Here p,mult, and Uare the output from hessreduce.
Answer:
n = size(H,1);
for i = 1:n-1
if p(i)
temp = b(i); b(i) = b(i+1); b(i+1) = temp;
end
b(i+1) = b(i+1) - mult(i)*b(i);
end
for j=n:-1:1
x(j) = b(j)/L(j,j);
b(1:j-1) = b(1:j-1) - L(1:j-1,j)*x(j);
end
2. A symmetric matrix Apositive definite if xTAx > 0 whenever x6= 0. Establish
the following facts. [Hint: You may use the facts in Page 77 of the Afternotes. You
may also use the fact that a matrix Uhas linearly independent columns if and only if
x6= 0 implies Ux 6= 0.]
a. Ais nonsingular.
b. A−1is positive definite.
c. If Bis the submatrix of Asubtended by rows and columns i1< i2<· · · < ik,
then Bis positive definite. What does this say about the diagonal elements of
A.
d. The element of largest magnitude in Aoccurs on the diagonal. [Hint: Prove
the result for the 2×2 case and then use part c.]
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