Symmetric Matrix Eigenvalue Problems - Numerical Methods in Engineering with MATLAB - page 368

sturmSeq

Given the diagonals c and d of A = [ c \ d \ c ], and the valueof λ ,thisfunctionreturns

the Sturm sequence P 0 ( λ ) , P 1 ( λ ) ,..., P n ( λ ). Note that P n ( λ ) = | A − λ I | .

functionp=sturmSeq(c,d,lambda)
% Returns Sturm sequence p associated with
%thetridiagonalmatrixA=[c\d\c]andlambda.
%USAGE:p=sturmSeq(c,d,lambda).
%Notethat

|

A - lambda*I

|

= p(n).

n = length(d) + 1;
p = ones(n,1);
p(2) = d(1) - lambda;
fori=2:n-1
p(i+1) = (d(i) - lambda)*p(i) - (c(i-1)ˆ2 )*p(i-1);

end

count eVals

Thisfunction counts the number of sign changes in the Sturm sequence and returns
the number of eigenvalues of the matrix A = [ c \ d \ c ]that aresmaller than λ .

function num eVals(c,d,lambda)

% Counts eigenvalues smaller than lambda of matrix
%A=[c\d\c].UsestheSturmsequence.
% USAGE: num

_

eVals = count

_

_

eVals = count

_

eVals(c,d,lambda).

p = sturmSeq(c,d,lambda);
n = length(p);
oldSign = 1; num

_

eVals = 0;

fori=2:n
pSign = sign(p(i));
if pSign == 0; pSign = -oldSign; end
if pSign*oldSign < 0
num

_

eVals = num

_

eVals + 1;

end
oldSign = pSign;

end