function lam=vereigsym(A)
%    VEREIGSYM     Verified eigenvalues of a symmetric interval (or real) matrix.
%
%    This is an INTLAB file. It requires to have INTLAB installed under
%    MATLAB to function properly.
%
%    CONVENTION. A real symmetric n-by-n matrix Ao has n real eigenvalues
%    lambda(Ao,i), i=1,...,n. We order them in nonincreasing sequence, i.e., to satisfy
%        lambda(Ao,1) >= lambda(Ao,2) >= ... >= lambda(Ao,n).
%
%    For a symmetric interval matrix A (that is, satisfying A'=A),
%        lam=vereigsym(A)
%    produces an interval vector lam verified to satisfy
%        lambda(Ao,i) in lam(i), i=1,...,n
%    for each SYMMETRIC Ao in A. Multiplicity of eigenvalues is taken into
%    account. Nothing is said of eigenvalues of nonsymmetric matrices in A.
%    If no verified result if given, then lam is an interval vector of NaN's.
%
%    The interval vector lam has an additional property: for each i and j,
%    the intervals lam(i) and lam(j) are either identical, or disjoint.
%
%    Accordingly, for a real symmetric matrix A,
%        lam=vereigsym(A)
%    produces an interval vector lam verified to satisfy
%        lambda(A,i) in lam(i), i=1,...,n,
%    i.e., enclosing the vector of eigenvalues of A.

%
%    Based partially on the section "Eigenvalues of symmetric matrices" in
%    J. Rohn, A Handbook of Results on Interval Linear Problems,
%    posted at http://www.cs.cas.cz/~rohn
%
%    WARRANTY
%
%    Because the program is licensed free of charge, there is
%    no warranty for the program, to the extent permitted by applicable
%    law. Except when otherwise stated in writing the copyright holder
%    and/or other parties provide the program "as is" without warranty
%    of any kind, either expressed or implied, including, but not
%    limited to, the implied warranties of merchantability and fitness
%    for a particular purpose. The entire risk as to the quality and
%    performance of the program is with you. Should the program prove
%    defective, you assume the cost of all necessary servicing, repair
%    or correction.
%
%    History
%
%    2008-02-02   first version
%    2008-02-08   version for posting (p-coded)
%    2008-04-01   version for posting (decoded)
%
gr=getround;
setround(0);
% setting default output
[m,n]=size(A);
lam=repmat(infsup(NaN,NaN),n,1);
if (m~=n)||~isreal(A)||~isequal(A,A') % wrong data
setround(gr); return
end
% creating full matrix
if issparse(A)
A=full(A); % sparse matrices not implemented yet
end
% case of real matrix
if ~isintval(A)
L=ol(A); % diagonal matrix
lam=diag(L); % vector of verified eigenvalues of A
lam=lam(n:-1:1); % reorder from largest to smallest
setround(gr); return
end
% case of interval matrix
setround(gr); return
end
L=ol(Ac); % diagonal matrix
if isnan(L.inf(1,1)) % verified eigenvalues of Ac not computed
setround(gr); return
end
lambda=diag(L); % vector of verified eigenvalues of Ac