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Program 4 Numerical Linear Algebra 1 Fall 2010
Due date: via email by 11:59PM on Friday, 11/5/10
General Task
Implement the Implicit QR algorithm for symmetric tridiagonal matrices. Explore the con
vergence behavior for various sizes of
n
and eigevalue distributions of
T
. Section 8.3.5
discusses this algorithm and has an example. Pay attention to the behavior of the subdi
agonal elements in positions other than
n,n

1, i.e., watch for deﬂation of the problem.
You do not have to exploit this in your algorithm but if you do discuss how it helps your
computational complexity.
There are forms of tridiagonal matrices that have known eigenvalues in addition to ex
amples you can ﬁnd in various text books. For example, if
T
∈
R
n
and the diagonals are
constant, i.e., a typical row has only
β,α,β
as the three nonzero elements then
λ
k
=
α
+ 2
β
cos
±
kπ
n
+ 1
²
(verify this identity if you use it). You can also use Gershgorin’s three theorems do generate
a matrix with various distributions on eigenvalues. (One is in the text on p. 320). See
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 Fall '06
 gallivan

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