Graphics Programs Reference
In-Depth Information
Solution
Referring to Eqs. (9.50), we get
a
1
=
a
2
=
a
3
=
4
4
5
r
1
=
2
r
2
=
4
r
3
=
2
Hence
λ
min
≥
min(
a
i
−
r
i
)
=
4
−
4
=
0
λ
max
≤
max(
a
i
+
r
i
)
=
4
+
4
=
8
Bracketing Eigenvalues
The Sturm sequence property togetherwith Gerschgorin's theoremprovides uscon-
venient tools forbracketing each eigenvalueofa symmetrictridiagonal matrix.
eValBrackets
The function
eValBrackets
brackets the
m
smallest eigenvalues of a symmetric
tridiagonal matrix
A
=
[
c
\
d
\
c
]. It returns the sequence
r
1
,
r
2
,...,
r
m
+
1
, whereeach
interval (
r
i
,
r
i
+
1
) containsexactly oneeigenvalue. The algorithmfirst finds the global
boundson the eigenvalues by Gerschgorin's theorem. The method of bisectionin
conjunctionwith the Sturm sequence propertyis thenused to determine the upper
boundson
λ
m
,λ
m
−
1
,...,λ
1
in thatorder.
functionr=eValBrackets(c,d,m)
%BracketseachofthemlowesteigenvaluesofA=[c\d\c]
% so that there is one eivenvalue in [r(i), r(i+1)].
%USAGE:r=eValBrackets(c,d,m).
[eValMin,eValMax]= gerschgorin(c,d); % Find global limits
r = ones(m+1,1); r(1) = eValMin;
% Search for eigenvalues in descending order
fork=m:-1:1
% First bisection of interval (eValMin,eValMax)
eVal = (eValMax + eValMin)/2;
h = (eValMax - eValMin)/2;
fori=1:100
% Find number of eigenvalues less than eVal
num
eVals(c,d,eVal);
% Bisect again & find the half containing eVal
_
eVals = count
_
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