Graphics Programs Reference
In-Depth Information
is
s
rpm, it is imperativetoensurethat there are no naturalfrequencies (which are
related to the eigenvalues) close to thatspeed.
Eigenvalueshifting is also be used to speedupconvergence. Suppose that we are
computing the smallest eigenvalue
λ
1
of the matrix
A
. The ideaistointroduce a shift
λ
1
/λ
2
assmall as possible.Since
λ
1
=
λ
1
−
s
that makes
s
, we should choose
s
≈
λ
1
(
s
=
λ
1
shouldbe avoided to prevent divisionbyzero). Of course, this methodworks
onlyif wehave aprior estimate of
λ
1
.
The inverse powermethodwith eigenvalueshifting is a particularlypowerful tool
for finding eigenvectors if the eigenvalues areknown. By shifting very close to an
eigenvalue, the corresponding eigenvector can becomputedinoneor two iterations.
Power Method
The powermethod converges to the eigenvalue
farthest from zero
and the associated
eigenvector. It is very similar to the inverse powermethod; the onlydifference be-
tween the two methods is the interchangeof
v
and
z
in Eq. (9.28). The outline of the
procedure is:
1. Let
v
be an approximation to
x
n
(a randomvector of unit magnitude will do).
2. Compute the vector
z
=
Av
(9.35)
3. Compute
|
z
|
.
4. Let
v
=
z
/
|
z
|
and repeat steps 2-4 until the change in
v
is negligible.
|
|=±
λ
n
and
v
=
λ
n
is deter-
At the conclusion of the procedure,
z
x
n
(the sign of
minedinthe same way as in the inverse powermethod).
invPower
Given the matrix
A
and the scalar
s
, the function
invPower
returns the eigenvalueof
A
closest to
s
and the corresponding eigenvector. The matrix
A
∗
=
s
I
is decomposed
as soonas it isformed,sothatonly the solutionphase (forward and back substitu-
tion) is neededinthe iterative loop. If
A
is banded, the efficiency of the program can
be improvedbyreplacing
LUdec
and
LUsol
by functionsthatspecialize in banded
matrices—see Example 9.6. The program linethatforms
A
∗
must also be modified to
becompatible with the storage scheme used for
A
.
A
−
function [eVal,eVec] = invPower(A,s,maxIter,tol)
% Inverse power mehod for finding the eigenvalue of A
% closest to s & the correstponding eigenvector.
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