to COMP 205 Homework 4 Feb 11, 1998 Due: Feb 18, 1998
Problems
matrix,
Derive expressions for the two singular values of 
as functions of w,x,y and z.
(10 points)
complex system 
, where 
. Moreover, the
matrices 
and 
are
real. Compare the operation counts of Gaussian elimination applied to
the real system and the original complex system, assuming that complex
multiplication takes four real multiplications and two additions.
How many real f.p. operations does complex division require?
is called Hessenberg if 
when
. How many operations are required to solve 
by Gaussian elimination? If 
is normalized so that 
for all i,j, and partial pivoting is used in Gaussian elimination, show that
the elements of 
(in 
decomposition, are less
than n in magnitude.
tri-diagonal matrix:
Show that it has a 
factorization 
,
where
Give an efficient algorithm to compute 
and 
. What is
its operation count?