www.pudn.com > RobustSF.zip > laneig.m, change:2001-08-16,size:9695b

```function [V,D,bnd,j,work] = laneig(A,nin,k,sigma,options)

%LANEIG  Compute a few eigenvalues and eigenvectors.
%   LANEIG solves the eigenvalue problem A*v=lambda*v, when A is
%   real and symmetric using the Lanczos algorithm with partial
%   reorthogonalization (PRO).
%
%   [V,D] = LANEIG(A)
%   [V,D] = LANEIG('Afun',N)
%
%   The first input argument is either a real symmetric matrix, or a
%   string containing the name of an M-file which applies a linear
%   operator to the columns of a given matrix.  In the latter case,
%   the second input argument must be N, the order of the problem.
%
%   The full calling sequence is
%
%   [V,D,ERR] = LANEIG(A,K,SIGMA,OPTIONS)
%   [V,D,ERR] = LANEIG('Afun',N,K,SIGMA,OPTIONS)
%
%   On exit ERR contains the computed error bounds.  K is the number of
%   eigenvalues desired and SIGMA is numerical shift or a two letter string
%   which specifies which part of the spectrum should be computed:
%
%   SIGMA            Specified eigenvalues
%
%   'AL'            Algebraically Largest
%   'AS'            Algebraically Smallest
%   'LM'            Largest Magnitude   (default)
%   'SM'            Smallest Magnitude  (does not work when A is an m-file)
%   'BE'            Both Ends.  Computes k/2 eigenvalues
%                   from each end of the spectrum (one more
%                   from the high end if k is odd.)
%
%   The OPTIONS structure specifies certain parameters in the algorithm.
%
%    Field name      Parameter                              Default
%
%    OPTIONS.tol     Convergence tolerance                  16*eps
%    OPTIONS.lanmax  Dimension of the Lanczos basis.
%    OPTIONS.v0      Starting vector for the Lanczos        rand(n,1)-0.5
%                    iteration.
%    OPTIONS.delta   Level of orthogonality among the       sqrt(eps/K)
%                    Lanczos vectors.
%    OPTIONS.eta     Level of orthogonality after           10*eps^(3/4)
%                    reorthogonalization.
%    OPTIONS.cgs     reorthogonalization method used        0
%                    '0' : iterated modified Gram-Schmidt
%                    '1' : iterated classical Gram-Schmidt
%    OPTIONS.elr     If equal to 1 then extended local      1
%                    reorthogonalization is enforced.
%

% References:
% R.M. Larsen, Ph.D. Thesis, Aarhus University, 1998.
%
% B. N. Parlett, ``The Symmetric Eigenvalue Problem'',
% Prentice-Hall, Englewood Cliffs, NJ, 1980.
%
% H. D. Simon, ``The Lanczos algorithm with partial reorthogonalization'',
% Math. Comp. 42 (1984), no. 165, 115--142.

% Rasmus Munk Larsen, DAIMI, 1998

%%%%%%%%%%%%%%%%%%%%% Parse and check input arguments. %%%%%%%%%%%%%%%%%%%%%%

if ~isstr(A)
if nargin<1
error('Not enough input arguments.');
end
[m n] = size(A);
Aisfunc = 0;
if m~=n | ~isequal(A,A') | ~isreal(A)
error('A must be real symmetric')
end
if nargin < 4 | isempty(sigma)
options = [];
else
options = sigma;
end
if nargin < 3 | isempty(k), sigma = 'LM'; else, sigma = k; end
if nargin < 2 | isempty(nin), k = min(n,5); else, k = nin; end
else
if nargin<2
error('Not enough input arguments.');
end
Aisfunc = 1;
n = nin;
if nargin < 5 | isempty(options)
options.tol = 16*eps;
options.lanmax = n;
options.v0 = rand(n,1)-0.5;
end
if nargin < 4 | isempty(sigma), sigma = 'LM'; end
if nargin < 3 | isempty(k), k = min(n,5);  end
end

if ~isnumeric(k) | real(abs(fix(k)))~=k | ~isnumeric(n) | real(abs(fix(n)))~=n
error('Input arguments N and K must be positive integers.')
end

% Quick return for n<2  or k<1
if n < 1 | k<1
if nargout < 2
V = zeros(k,1);
else
V = eye(n,k);
D = zeros(k,k);
bnd =zeros(k,1);
end
return
end
if n == 1
if ~Aisfunc
D = A;
V = 1;
bnd = 0;
else
D = feval(A,1);
V = 1;
dnb = 0;
end
if nargout<2
V=D;
end
return
end

% A is the matrix of all zeros (not detectable if A is a string)
if ~Aisfunc
if nnz(A)==0
if nargout < 2
V = zeros(k,1);
else
V = eye(n,k);
D = zeros(k,k);
bnd =zeros(k,1);
end
return
end
end

lanmax = n;
tol = 16*eps;
r = rand(n,1)-0.5;
part = sigma;
% Parse options struct
if ~isempty(options) & isstruct(options)
c = fieldnames(options);
for i=1:length(c)
if strmatch(c(i),'v0'), r = getfield(options,'v0'); r=r(:); end
if strmatch(c(i),'tol'), tol = getfield(options,'tol'); end
if strmatch(c(i),'lanmax'), lanmax = getfield(options,'lanmax'); end
end
end

% Protect against absurd arguments.
tol = max(tol,eps);
lanmax = min(lanmax,n);
if size(r,1)~=n
error('v0 must be a vector of length n')
end

lanmax = min(lanmax,n);
if k>lanmax
error('K must satisfy  K <= LANMAX <= N.');
end
ksave = k;

if strcmp(sigma,'SM') & ~isstr(A)
sigma = 0;
end

% Prepare for shift-and-invert if sigma is numeric.
if  isnumeric(sigma)
part = 'LM';
if isstr(A)
error('Shift-and-invert works only when the matrix A is given explicitly.');
else
pmmd = symmmd(A);
A = A(pmmd,pmmd);
[S.L,S.U] = lu(A - sigma*speye(n));
condU = condest(S.U);
dsigma = n * full(max(max(abs(A)))) * eps;
if sigma < 0
sgnsig = -1;
else
sgnsig = 1;
end
sigitr = 1;
while condU > 1/eps & ((dsigma <= 1 & sigitr <= 10) | ~isfinite(condU))
disps1 = sprintf(['sigma = %10e is near an exact eigenvalue of A,\n' ...
'so we cannot use the LU factorization of (A-sigma*I): ' ...
' condest(U) = %10e.\n'],sigma,condU);
if abs(sigma) < 1
sigma = sigma + sgnsig * dsigma;
disps2 = sprintf('We are trying sigma + %10e = %10e instead.\n', ...
sgnsig*dsigma,sigma);
else
sigma = sigma * (1 + dsigma);
disps2 = sprintf('We are trying sigma * (1 + %10e) = %10e instead.\n', ...
dsigma,sigma);
end
%     if nargout < 3 & dispn ~= 0
disp([disps1 disps2])
%     end
[S.L,S.U] = lu(A - sigma*speye(n));
condU = condest(S.U);
dsigma = 10 * dsigma;
sigitr = sigitr + 1;
end
end
A = S;
end

neig = 0; nrestart=-1;
if ~strcmp(part,'BE')
j = min(2*k+2,lanmax);
else
j = min(k+1,lanmax);
end

%%%%%%%%%%%%%%%%%%%%% Here begins the computation  %%%%%%%%%%%%%%%%%%%%%%

V = []; T = []; anorm = []; work = zeros(1,2); rnorm=-1;

while neig < k
%%%%%%%%%%%%%%%%%%%%% Compute Lanczos tridiagonalization %%%%%%%%%%%%%%%%%
j = min(lanmax,j+1-mod(j,2));
% "Trick" to avoid unwanted zero eigenvalues when laneig is used for
% SVD calculations. (Nothing to if lanmax is odd, though.)

if  ~isstr(A)
[V,T,r,anorm,ierr,w] = lanpro(A,j,r,options,V,T,anorm);
else
[V,T,r,anorm,ierr,w] = lanpro(A,n,j,r,options,V,T,anorm);
end
work= work + w;

if ierr<0 % Invariant subspace of dimension -ierr found.
j = -ierr;
end

%%%%%%%%%%%%%%%%%% Compute eigenvalues and error bounds %%%%%%%%%%%%%%%%%%
% Analyze T
[D,top,bot,err] = tqlb([full(diag(T))],full([0;diag(T,1)]));
%  if err>0
%    printf(['TQLB failed. Eigenvalue no. %i did not converge in 30', ...
%	  ' iterations'],err);
%  end
%  full(T)
%  [P,D] = eig(full(T));
%  D = diag(D);
%  bot = P(end,:)';
%  [P(1,:)' P(end,:)']
[D,I] = sort(D);
bot = bot(I);

% Set simple error bounds
rnorm = norm(r);
bnd = rnorm*abs(bot);

% Use Largest Ritz value to estimate ||A||_2. This might save some
% reorth. in case of restart.
anorm = max(abs(D));

% Estimate gap structure and refine error bounds
bnd = refinebounds(D,bnd,n*eps*anorm);

%%%%%%%%%%%%%%%%%%% Check convergence criterion %%%%%%%%%%%%%%%%%%%%
% Reorder eigenvalues according to SIGMA
switch part
case 'AS'
IPART = 1:j;
case 'AL'
IPART = j:-1:1;
case 'LM'
[dummy,IPART] = sort(-abs(D));
case 'BE'
if j<k
IPART=1:j;
else
mid = floor(k/2);
par = rem(k,1);
IPART = [1:mid,(j-mid-par):j]';
end
otherwise
error(['Illegal value for SIGMA: ',part]);
end
D = D(IPART);  bnd = bnd(IPART);
if isnumeric(sigma)
D = sigma + 1./D;
end

% Check if enough have converged.
neig = 0;
for i=1:min(j,k)
if bnd(i) <= tol*abs(D(i))
neig = neig + 1;
end
end

%%%%%%%%%%% Check whether to stop or to extend the Krylov basis? %%%%%%%%%%
if ierr<0 % Invariant subspace found
if j<k
warning(['Invariant subspace of dimension ',num2str(j-1),' found.'])
end
break;
end
if j>=lanmax % Maximal dimension of Krylov subspace reached => Bail out!
if neig<ksave
warning(['Maximum dimension of Krylov subspace exceeded prior',...
' to convergence.']);
end
break;
end

% Increase dimension of Krylov subspace and try again.
if neig>0
%    j = j + ceil(min(20,max(2,((j-1)*(k-neig+1))/(2*(neig+1)))));
j = j + min(100,max(2,0.5*(k-neig)*j/(neig+1)));
elseif neig<k
%    j = j + ceil(min(20,max(8,(k-neig)/2)));
j = max(1.5*j,j+10);
end
j = min(j+1,lanmax);
nrestart = nrestart + 1;
end

%%%%%%%%%%%%%%%% Lanczos converged (or failed). Prepare output %%%%%%%%%%%%%%%
k = min(ksave,j);

if nargout>1
j = size(T,1);
[Q,D] = eig(full(T)); D = diag(D);
[D,I] = sort(D);
% Compute and normalize Ritz vectors (overwrite V to save memory).
V = V*Q(:,I(IPART(1:k)));
for i=1:k
nq = norm(V(:,i));
if isfinite(nq) & nq~=0 & nq~=1
V(:,i) = V(:,i)/nq;
end
end
[D,I] = sort(D);
D = D(IPART(1:k));
if isnumeric(sigma)
D = sigma + 1./D;
V(pmmd,:) = V;
end
end

% Pick out desired part of the spectrum
if length(D)~=k
D = D(1:k);
bnd = bnd(1:k);
end

if nargout<2
V = D;
else
D = diag(D);
end

```