Matrix Completion.m

(Difference between revisions)
 Revision as of 02:03, 16 February 2009 (edit)← Previous diff Revision as of 02:06, 16 February 2009 (edit) (undo)Next diff → Line 45: Line 45: disp(sprintf('The relative recovery in the spectral norm is: %d ', norm(M-X)/norm(M))) disp(sprintf('The relative recovery in the spectral norm is: %d ', norm(M-X)/norm(M))) - === SVT() === === SVT() === Line 135: Line 134: numiter = k; numiter = k; - === bdsqr() === === bdsqr() === Line 168: Line 166: sigma = diag(S); sigma = diag(S); bnd = U(end,1:k)'; bnd = U(end,1:k)'; + + + === compute_int() === +
+                                                                function int = compute_int(mu,j,delta,eta,LL,strategy,extra)
+                                                                %COMPUTE_INT:  Determine which Lanczos vectors to reorthogonalize against.
+                                                                %
+                                                                %      int = compute_int(mu,eta,LL,strategy,extra))
+                                                                %
+                                                                %   Strategy 0: Orthogonalize vectors v_{i-r-extra},...,v_{i},...v_{i+s+extra}
+                                                                %               with nu>eta, where v_{i} are the vectors with  mu>delta.
+                                                                %   Strategy 1: Orthogonalize all vectors v_{r-extra},...,v_{s+extra} where
+                                                                %               v_{r} is the first and v_{s} the last Lanczos vector with
+                                                                %               mu > eta.
+                                                                %   Strategy 2: Orthogonalize all vectors with mu > eta.
+                                                                %
+                                                                %   Notice: The first LL vectors are excluded since the new Lanczos
+                                                                %   vector is already orthogonalized against them in the main iteration.
+
+                                                                %  Rasmus Munk Larsen, DAIMI, 1998.
+
+                                                                if (delta= ETA.')
+                                                                end
+                                                                switch strategy
+                                                                case 0
+                                                                I0 = find(abs(mu(1:j))>=delta);
+                                                                if length(I0)==0
+                                                                [mm,I0] = max(abs(mu(1:j)));
+                                                                end
+                                                                int = zeros(j,1);
+                                                                for i = 1:length(I0)
+                                                                for r=I0(i):-1:1
+                                                                if abs(mu(r))0
+                                                                int(1:LL) = 0;
+                                                                end
+                                                                int = find(int);
+                                                                case 1
+                                                                int=find(abs(mu(1:j))>eta);
+                                                                int = max(LL+1,min(int)-extra):min(max(int)+extra,j);
+                                                                case 2
+                                                                int=find(abs(mu(1:j))>=eta);
+                                                                end
+                                                                int = int(:);
+
+ + === lanbpro() === +
+                                                                function [U,B_k,V,p,ierr,work] = lanbpro(varargin)
+
+                                                                %LANBPRO Lanczos bidiagonalization with partial reorthogonalization.
+                                                                %   LANBPRO computes the Lanczos bidiagonalization of a real
+                                                                %   matrix using the  with partial reorthogonalization.
+                                                                %
+                                                                %   [U_k,B_k,V_k,R,ierr,work] = LANBPRO(A,K,R0,OPTIONS,U_old,B_old,V_old)
+                                                                %   [U_k,B_k,V_k,R,ierr,work] = LANBPRO('Afun','Atransfun',M,N,K,R0, ...
+                                                                %                                       OPTIONS,U_old,B_old,V_old)
+                                                                %
+                                                                %   Computes K steps of the Lanczos bidiagonalization algorithm with partial
+                                                                %   reorthogonalization (BPRO) with M-by-1 starting vector R0, producing a
+                                                                %   lower bidiagonal K-by-K matrix B_k, an N-by-K matrix V_k, an M-by-K
+                                                                %   matrix U_k and an M-by-1 vector R such that
+                                                                %        A*V_k = U_k*B_k + R
+                                                                %   Partial reorthogonalization is used to keep the columns of V_K and U_k
+                                                                %   semiorthogonal:
+                                                                %         MAX(DIAG((EYE(K) - V_K'*V_K))) <= OPTIONS.delta
+                                                                %   and
+                                                                %         MAX(DIAG((EYE(K) - U_K'*U_K))) <= OPTIONS.delta.
+                                                                %
+                                                                %   B_k = LANBPRO(...) returns the bidiagonal matrix only.
+                                                                %
+                                                                %   The first input argument is either a real 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 must be the name of an M-file which applies the transpose of
+                                                                %   the same linear operator to the columns of a given matrix,
+                                                                %   and the third and fourth arguments must be M and N, the dimensions
+                                                                %   of then problem.
+                                                                %
+                                                                %   The OPTIONS structure is used to control the reorthogonalization:
+                                                                %     OPTIONS.delta:  Desired level of orthogonality
+                                                                %                     (default = sqrt(eps/K)).
+                                                                %     OPTIONS.eta  :  Level of orthogonality after reorthogonalization
+                                                                %                     (default = eps^(3/4)/sqrt(K)).
+                                                                %     OPTIONS.cgs  :  Flag for switching between different reorthogonalization
+                                                                %                     algorithms:
+                                                                %                      0 = iterated modified Gram-Schmidt  (default)
+                                                                %                      1 = iterated classical Gram-Schmidt
+                                                                %     OPTIONS.elr  :  If OPTIONS.elr = 1 (default) then extended local
+                                                                %                     reorthogonalization is enforced.
+                                                                %     OPTIONS.onesided
+                                                                %                  :  If OPTIONS.onesided = 0 (default) then both the left
+                                                                %                     (U) and right (V) Lanczos vectors are kept
+                                                                %                     semiorthogonal.
+                                                                %                     OPTIONS.onesided = 1 then only the columns of U are
+                                                                %                     are reorthogonalized.
+                                                                %                     OPTIONS.onesided = -1 then only the columns of V are
+                                                                %                     are reorthogonalized.
+                                                                %     OPTIONS.waitbar
+                                                                %                  :  The progress of the algorithm is display graphically.
+                                                                %
+                                                                %   If both R0, U_old, B_old, and V_old are provided, they must
+                                                                %   contain a partial Lanczos bidiagonalization of A on the form
+                                                                %
+                                                                %        A V_old = U_old B_old + R0 .
+                                                                %
+                                                                %   In this case the factorization is extended to dimension K x K by
+                                                                %   continuing the Lanczos bidiagonalization algorithm with R0 as a
+                                                                %   starting vector.
+                                                                %
+                                                                %   The output array work contains information about the work used in
+                                                                %   reorthogonalizing the u- and v-vectors.
+                                                                %      work = [ RU  PU ]
+                                                                %             [ RV  PV ]
+                                                                %   where
+                                                                %      RU = Number of reorthogonalizations of U.
+                                                                %      PU = Number of inner products used in reorthogonalizing U.
+                                                                %      RV = Number of reorthogonalizations of V.
+                                                                %      PV = Number of inner products used in reorthogonalizing V.
+
+                                                                % References:
+                                                                % R.M. Larsen, Ph.D. Thesis, Aarhus University, 1998.
+                                                                %
+                                                                % G. H. Golub & C. F. Van Loan, "Matrix Computations",
+                                                                % 3. Ed., Johns Hopkins, 1996.  Section 9.3.4.
+                                                                %
+                                                                % 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.
+
+                                                                % Check input arguments.
+
+                                                                global LANBPRO_TRUTH
+                                                                LANBPRO_TRUTH=0;
+
+                                                                if LANBPRO_TRUTH==1
+                                                                global MU NU MUTRUE NUTRUE
+                                                                global MU_AFTER NU_AFTER MUTRUE_AFTER NUTRUE_AFTER
+                                                                end
+
+                                                                if nargin<1 | length(varargin)<2
+                                                                error('Not enough input arguments.');
+                                                                end
+                                                                narg=length(varargin);
+
+                                                                A = varargin{1};
+                                                                if isnumeric(A) | isstruct(A)
+                                                                if isnumeric(A)
+                                                                if ~isreal(A)
+                                                                error('A must be real')
+                                                                end
+                                                                [m n] = size(A);
+                                                                elseif isstruct(A)
+                                                                [m n] = size(A.R);
+                                                                end
+                                                                k=varargin{2};
+                                                                if narg >= 3 & ~isempty(varargin{3});
+                                                                p = varargin{3};
+                                                                else
+                                                                p = rand(m,1)-0.5;
+                                                                end
+                                                                if narg < 4, options = []; else options=varargin{4}; end
+                                                                if narg > 4
+                                                                if narg<7
+                                                                error('All or none of U_old, B_old and V_old must be provided.')
+                                                                else
+                                                                U = varargin{5}; B_k = varargin{6}; V = varargin{7};
+                                                                end
+                                                                else
+                                                                U = []; B_k = []; V = [];
+                                                                end
+                                                                if narg > 7, anorm=varargin{8}; else anorm = []; end
+                                                                else
+                                                                if narg<5
+                                                                error('Not enough input arguments.');
+                                                                end
+                                                                Atrans = varargin{2};
+                                                                if ~isstr(Atrans)
+                                                                error('Afunc and Atransfunc must be names of m-files')
+                                                                end
+                                                                m = varargin{3};
+                                                                n = varargin{4};
+                                                                if ~isreal(n) | abs(fix(n)) ~= n | ~isreal(m) | abs(fix(m)) ~= m
+                                                                error('M and N must be positive integers.')
+                                                                end
+                                                                k=varargin{5};
+                                                                if narg < 6, p = rand(m,1)-0.5; else p=varargin{6}; end
+                                                                if narg < 7, options = []; else options=varargin{7}; end
+                                                                if narg > 7
+                                                                if  narg < 10
+                                                                error('All or none of U_old, B_old and V_old must be provided.')
+                                                                else
+                                                                U = varargin{8}; B_k = varargin{9}; V = varargin{10};
+                                                                end
+                                                                else
+                                                                U = []; B_k = []; V=[];
+                                                                end
+                                                                if narg > 10, anorm=varargin{11}; else anorm = [];  end
+                                                                end
+
+                                                                % Quick return for min(m,n) equal to 0 or 1.
+                                                                if min(m,n) == 0
+                                                                U = [];  B_k = [];  V = [];  p = [];  ierr = 0;  work = zeros(2,2);
+                                                                return
+                                                                elseif  min(m,n) == 1
+                                                                if isnumeric(A)
+                                                                U = 1;  B_k = A;  V = 1;  p = 0; ierr = 0; work = zeros(2,2);
+                                                                else
+                                                                U = 1;  B_k = feval(A,1); V = 1; p = 0; ierr = 0; work = zeros(2,2);
+                                                                end
+                                                                if nargout<3
+                                                                U = B_k;
+                                                                end
+                                                                return
+                                                                end
+
+                                                                % Set options.
+                                                                %m2 = 3/2*(sqrt(m)+1);
+                                                                %n2 = 3/2*(sqrt(n)+1);
+                                                                m2 = 3/2;
+                                                                n2 = 3/2;
+                                                                delta = sqrt(eps/k); % Desired level of orthogonality.
+                                                                eta = eps^(3/4)/sqrt(k);    % Level of orth. after reorthogonalization.
+                                                                cgs = 0;             % Flag for switching between iterated MGS and CGS.
+                                                                elr = 2;             % Flag for switching extended local
+                                                                % reorthogonalization on and off.
+                                                                gamma = 1/sqrt(2);   % Tolerance for iterated Gram-Schmidt.
+                                                                onesided = 0; t = 0; waitb = 0;
+
+                                                                % Parse options struct
+                                                                if ~isempty(options) & isstruct(options)
+                                                                c = fieldnames(options);
+                                                                for i=1:length(c)
+                                                                if strmatch(c(i),'delta'), delta = getfield(options,'delta');  end
+                                                                if strmatch(c(i),'eta'), eta = getfield(options,'eta'); end
+                                                                if strmatch(c(i),'cgs'), cgs = getfield(options,'cgs'); end
+                                                                if strmatch(c(i),'elr'), elr = getfield(options,'elr'); end
+                                                                if strmatch(c(i),'gamma'), gamma = getfield(options,'gamma'); end
+                                                                if strmatch(c(i),'onesided'), onesided = getfield(options,'onesided'); end
+                                                                if strmatch(c(i),'waitbar'), waitb=1; end
+                                                                end
+                                                                end
+
+                                                                if waitb
+                                                                waitbarh = waitbar(0,'Lanczos bidiagonalization in progress...');
+                                                                end
+
+                                                                if isempty(anorm)
+                                                                anorm = []; est_anorm=1;
+                                                                else
+                                                                est_anorm=0;
+                                                                end
+
+                                                                % Conservative statistical estimate on the size of round-off terms.
+                                                                % Notice that {\bf u} == eps/2.
+                                                                FUDGE = 1.01; % Fudge factor for ||A||_2 estimate.
+
+                                                                npu = 0; npv = 0; ierr = 0;
+                                                                p = p(:);
+                                                                % Prepare for Lanczos iteration.
+                                                                if isempty(U)
+                                                                V = zeros(n,k); U = zeros(m,k);
+                                                                beta = zeros(k+1,1); alpha = zeros(k,1);
+                                                                beta(1) = norm(p);
+                                                                % Initialize MU/NU-recurrences for monitoring loss of orthogonality.
+                                                                nu = zeros(k,1); mu = zeros(k+1,1);
+                                                                mu(1)=1; nu(1)=1;
+
+                                                                numax = zeros(k,1); mumax = zeros(k,1);
+                                                                force_reorth = 0;  nreorthu = 0; nreorthv = 0;
+                                                                j0 = 1;
+                                                                else
+                                                                j = size(U,2); % Size of existing factorization
+                                                                % Allocate space for Lanczos vectors
+                                                                U = [U, zeros(m,k-j)];
+                                                                V = [V, zeros(n,k-j)];
+                                                                alpha = zeros(k+1,1);  beta = zeros(k+1,1);
+                                                                alpha(1:j) = diag(B_k); if j>1 beta(2:j) = diag(B_k,-1); end
+                                                                beta(j+1) = norm(p);
+                                                                % Reorthogonalize p.
+                                                                if j=gamma*normold
+                                                                stop = 1;
+                                                                else
+                                                                normold = alpha(j);
+                                                                end
+                                                                end
+                                                                end
+
+                                                                if est_anorm
+                                                                if j==2
+                                                                anorm = max(anorm,FUDGE*sqrt(alpha(1)^2+beta(2)^2+alpha(2)*beta(2)));
+                                                                else
+                                                                anorm = max(anorm,FUDGE*sqrt(alpha(j-1)^2+beta(j)^2+alpha(j-1)* ...
+                                                                beta(j-1) + alpha(j)*beta(j)));
+                                                                end
+                                                                end
+
+                                                                if ~fro & alpha(j) ~= 0
+                                                                % Update estimates of the level of orthogonality for the
+                                                                %  columns 1 through j-1 in V.
+                                                                nu = update_nu(nu,mu,j,alpha,beta,anorm);
+                                                                numax(j) = max(abs(nu(1:j-1)));
+                                                                end
+
+                                                                if j>1 & LANBPRO_TRUTH
+                                                                NU(1:j-1,j-1) = nu(1:j-1);
+                                                                NUTRUE(1:j-1,j-1) = V(:,1:j-1)'*r/alpha(j);
+                                                                end
+
+                                                                if elr>0
+                                                                nu(j-1) = n2*eps;
+                                                                end
+
+                                                                % IF level of orthogonality is worse than delta THEN
+                                                                %    Reorthogonalize v_j against some previous  v_i's, 0<=i delta | force_reorth ) & alpha(j)~=0
+                                                                % Decide which vectors to orthogonalize against:
+                                                                if fro | eta==0
+                                                                int = [1:j-1]';
+                                                                elseif force_reorth==0
+                                                                int = compute_int(nu,j-1,delta,eta,0,0,0);
+                                                                end
+                                                                % Else use int from last reorth. to avoid spillover from mu_{j-1}
+                                                                % to nu_j.
+
+                                                                % Reorthogonalize v_j
+                                                                [r,alpha(j),rr] = reorth(V,r,alpha(j),int,gamma,cgs);
+                                                                npv = npv + rr*length(int); % number of inner products.
+                                                                nu(int) = n2*eps;  % Reset nu for orthogonalized vectors.
+
+                                                                % If necessary force reorthogonalization of u_{j+1}
+                                                                % to avoid spillover
+                                                                if force_reorth==0
+                                                                force_reorth = 1;
+                                                                else
+                                                                force_reorth = 0;
+                                                                end
+                                                                nreorthv = nreorthv + 1;
+                                                                end
+                                                                end
+
+
+                                                                % Check for convergence or failure to maintain semiorthogonality
+                                                                if alpha(j) < max(n,m)*anorm*eps & j 0
+                                                                % A vector numerically orthogonal to span(Q_k(:,1:j)) was found.
+                                                                % Continue iteration.
+                                                                bailout=0;
+                                                                break;
+                                                                end
+                                                                end
+                                                                if bailout
+                                                                j = j-1;
+                                                                ierr = -j;
+                                                                break;
+                                                                else
+                                                                r=r/nrmnew; % Continue with new normalized r as starting vector.
+                                                                force_reorth = 1;
+                                                                if delta>0
+                                                                fro = 0;    % Turn off full reorthogonalization.
+                                                                end
+                                                                end
+                                                                elseif  j delta*alpha(j)
+                                                                %    fro = 1;
+                                                                ierr = j;
+                                                                end
+
+                                                                if j>1 & LANBPRO_TRUTH
+                                                                NU_AFTER(1:j-1,j-1) = nu(1:j-1);
+                                                                NUTRUE_AFTER(1:j-1,j-1) = V(:,1:j-1)'*r/alpha(j);
+                                                                end
+
+
+                                                                if alpha(j) ~= 0
+                                                                V(:,j) = r/alpha(j);
+                                                                else
+                                                                V(:,j) = r;
+                                                                end
+
+                                                                %%%%%%%%%% Lanczos step to generate u_{j+1}. %%%%%%%%%%%%%
+                                                                if waitb
+                                                                waitbar((2*j+1)/(2*k),waitbarh)
+                                                                end
+
+                                                                if isnumeric(A)
+                                                                p = A*V(:,j) - alpha(j)*U(:,j);
+                                                                elseif isstruct(A)
+                                                                p = A.Rt\V(:,j) - alpha(j)*U(:,j);
+                                                                else
+                                                                p = feval(A,V(:,j)) - alpha(j)*U(:,j);
+                                                                end
+                                                                beta(j+1) = norm(p);
+                                                                % Extended local reorthogonalization
+                                                                if beta(j+1)= gamma*normold
+                                                                stop = 1;
+                                                                else
+                                                                normold = beta(j+1);
+                                                                end
+                                                                end
+                                                                end
+
+                                                                if est_anorm
+                                                                % We should update estimate of ||A||  before updating mu - especially
+                                                                % important in the first step for problems with large norm since alpha(1)
+                                                                % may be a severe underestimate!
+                                                                if j==1
+                                                                anorm = max(anorm,FUDGE*pythag(alpha(1),beta(2)));
+                                                                else
+                                                                anorm = max(anorm,FUDGE*sqrt(alpha(j)^2+beta(j+1)^2 + alpha(j)*beta(j)));
+                                                                end
+                                                                end
+
+
+                                                                if ~fro & beta(j+1) ~= 0
+                                                                % Update estimates of the level of orthogonality for the columns of V.
+                                                                mu = update_mu(mu,nu,j,alpha,beta,anorm);
+                                                                mumax(j) = max(abs(mu(1:j)));
+                                                                end
+
+                                                                if LANBPRO_TRUTH==1
+                                                                MU(1:j,j) = mu(1:j);
+                                                                MUTRUE(1:j,j) = U(:,1:j)'*p/beta(j+1);
+                                                                end
+
+                                                                if elr>0
+                                                                mu(j) = m2*eps;
+                                                                end
+
+                                                                % IF level of orthogonality is worse than delta THEN
+                                                                %    Reorthogonalize u_{j+1} against some previous  u_i's, 0<=i<=j.
+                                                                if onesided~=1 & (fro | mumax(j) > delta | force_reorth) & beta(j+1)~=0
+                                                                % Decide which vectors to orthogonalize against.
+                                                                if fro | eta==0
+                                                                int = [1:j]';
+                                                                elseif force_reorth==0
+                                                                int = compute_int(mu,j,delta,eta,0,0,0);
+                                                                else
+                                                                int = [int; max(int)+1];
+                                                                end
+                                                                % Else use int from last reorth. to avoid spillover from nu to mu.
+
+                                                                %    if onesided~=0
+                                                                %      fprintf('i = %i, nr = %i, fro = %i\n',j,size(int(:),1),fro)
+                                                                %    end
+                                                                % Reorthogonalize u_{j+1}
+                                                                [p,beta(j+1),rr] = reorth(U,p,beta(j+1),int,gamma,cgs);
+                                                                npu = npu + rr*length(int);  nreorthu = nreorthu + 1;
+
+                                                                % Reset mu to epsilon.
+                                                                mu(int) = m2*eps;
+
+                                                                if force_reorth==0
+                                                                force_reorth = 1; % Force reorthogonalization of v_{j+1}.
+                                                                else
+                                                                force_reorth = 0;
+                                                                end
+                                                                end
+
+                                                                % Check for convergence or failure to maintain semiorthogonality
+                                                                if beta(j+1) < max(m,n)*anorm*eps  & j 0
+                                                                % A vector numerically orthogonal to span(Q_k(:,1:j)) was found.
+                                                                % Continue iteration.
+                                                                bailout=0;
+                                                                break;
+                                                                end
+                                                                end
+                                                                if bailout
+                                                                ierr = -j;
+                                                                break;
+                                                                else
+                                                                p=p/nrmnew; % Continue with new normalized p as starting vector.
+                                                                force_reorth = 1;
+                                                                if delta>0
+                                                                fro = 0;    % Turn off full reorthogonalization.
+                                                                end
+                                                                end
+                                                                elseif  j delta*beta(j+1)
+                                                                %    fro = 1;
+                                                                ierr = j;
+                                                                end
+
+                                                                if LANBPRO_TRUTH==1
+                                                                MU_AFTER(1:j,j) = mu(1:j);
+                                                                MUTRUE_AFTER(1:j,j) = U(:,1:j)'*p/beta(j+1);
+                                                                end
+                                                                end
+                                                                if waitb
+                                                                close(waitbarh)
+                                                                end
+
+                                                                if j5
+                                                                work = [[nreorthu,npu];[nreorthv,npv]];
+                                                                end
+
+
+
+                                                                function mu = update_mu(muold,nu,j,alpha,beta,anorm)
+
+                                                                % UPDATE_MU:  Update the mu-recurrence for the u-vectors.
+                                                                %
+                                                                %   mu_new = update_mu(mu,nu,j,alpha,beta,anorm)
+
+                                                                %  Rasmus Munk Larsen, DAIMI, 1998.
+
+                                                                binv = 1/beta(j+1);
+                                                                mu = muold;
+                                                                eps1 = 100*eps/2;
+                                                                if j==1
+                                                                T = eps1*(pythag(alpha(1),beta(2)) + pythag(alpha(1),beta(1)));
+                                                                T = T + eps1*anorm;
+                                                                mu(1) = T / beta(2);
+                                                                else
+                                                                mu(1) = alpha(1)*nu(1) - alpha(j)*mu(1);
+                                                                %  T = eps1*(pythag(alpha(j),beta(j+1)) + pythag(alpha(1),beta(1)));
+                                                                T = eps1*(sqrt(alpha(j).^2+beta(j+1).^2) + sqrt(alpha(1).^2+beta(1).^2));
+                                                                T = T + eps1*anorm;
+                                                                mu(1) = (mu(1) + sign(mu(1))*T) / beta(j+1);
+                                                                % Vectorized version of loop:
+                                                                if j>2
+                                                                k=2:j-1;
+                                                                mu(k) = alpha(k).*nu(k) + beta(k).*nu(k-1) - alpha(j)*mu(k);
+                                                                %T = eps1*(pythag(alpha(j),beta(j+1)) + pythag(alpha(k),beta(k)));
+                                                                T = eps1*(sqrt(alpha(j).^2+beta(j+1).^2) + sqrt(alpha(k).^2+beta(k).^2));
+                                                                T = T + eps1*anorm;
+                                                                mu(k) = binv*(mu(k) + sign(mu(k)).*T);
+                                                                end
+                                                                %  T = eps1*(pythag(alpha(j),beta(j+1)) + pythag(alpha(j),beta(j)));
+                                                                T = eps1*(sqrt(alpha(j).^2+beta(j+1).^2) + sqrt(alpha(j).^2+beta(j).^2));
+                                                                T = T + eps1*anorm;
+                                                                mu(j) = beta(j)*nu(j-1);
+                                                                mu(j) = (mu(j) + sign(mu(j))*T) / beta(j+1);
+                                                                end
+                                                                mu(j+1) = 1;
+
+
+                                                                function nu = update_nu(nuold,mu,j,alpha,beta,anorm)
+
+                                                                % UPDATE_MU:  Update the nu-recurrence for the v-vectors.
+                                                                %
+                                                                %  nu_new = update_nu(nu,mu,j,alpha,beta,anorm)
+
+                                                                %  Rasmus Munk Larsen, DAIMI, 1998.
+
+                                                                nu = nuold;
+                                                                ainv = 1/alpha(j);
+                                                                eps1 = 100*eps/2;
+                                                                if j>1
+                                                                k = 1:(j-1);
+                                                                %  T = eps1*(pythag(alpha(k),beta(k+1)) + pythag(alpha(j),beta(j)));
+                                                                T = eps1*(sqrt(alpha(k).^2+beta(k+1).^2) + sqrt(alpha(j).^2+beta(j).^2));
+                                                                T = T + eps1*anorm;
+                                                                nu(k) = beta(k+1).*mu(k+1) + alpha(k).*mu(k) - beta(j)*nu(k);
+                                                                nu(k) = ainv*(nu(k) + sign(nu(k)).*T);
+                                                                end
+                                                                nu(j) = 1;
+
+                                                                function x = pythag(y,z)
+                                                                %PYTHAG Computes sqrt( y^2 + z^2 ).
+                                                                %
+                                                                % x = pythag(y,z)
+                                                                %
+                                                                % Returns sqrt(y^2 + z^2) but is careful to scale to avoid overflow.
+
+                                                                % Christian H. Bischof, Argonne National Laboratory, 03/31/89.
+
+                                                                [m n] = size(y);
+                                                                if m>1 | n>1
+                                                                y = y(:); z=z(:);
+                                                                rmax = max(abs([y z]'))';
+                                                                id=find(rmax==0);
+                                                                if length(id)>0
+                                                                rmax(id) = 1;
+                                                                x = rmax.*sqrt((y./rmax).^2 + (z./rmax).^2);
+                                                                x(id)=0;
+                                                                else
+                                                                x = rmax.*sqrt((y./rmax).^2 + (z./rmax).^2);
+                                                                end
+                                                                x = reshape(x,m,n);
+                                                                else
+                                                                rmax = max(abs([y;z]));
+                                                                if (rmax==0)
+                                                                x = 0;
+                                                                else
+                                                                x = rmax*sqrt((y/rmax)^2 + (z/rmax)^2);
+                                                                end
+                                                                end

Matlab demonstration of Cai, Candès, & Shen

test_SVT.m

% Written by: Emmanuel Candes
% Email: emmanuel@acm.caltech.edu
% Created: October 2008

%% Set path and global variables
global SRB
SRB = true;

%% Setup a matrix
randn('state',2008);
rand('state',2008);

n = 1000; r = 10;
M = randn(n,r)*randn(r,n);

df = r*(2*n-r);
oversampling = 5;  m = 5*df;

Omega = randsample(n^2,m);
data = M(Omega);

%% Set parameters and solve

p  = m/n^2;  delta = 1.2/p;
maxiter = 500;
tol = 1e-4;

%% Approximate minimum nuclear norm solution by SVT algorithm

tic
[U,S,V,numiter] = SVT(n,Omega,data,delta,maxiter,tol);
toc

%% Show results

X = U*S*V';

disp(sprintf('The relative error on Omega is: %d ', norm(data-X(Omega))/norm(data)))
disp(sprintf('The relative recovery error is: %d ', norm(M-X,'fro')/norm(M,'fro')))
disp(sprintf('The relative recovery in the spectral norm is: %d ', norm(M-X)/norm(M)))

SVT()

function [U,Sigma,V,numiter]  = SVT(n,Omega,b,delta,maxiter,tol)
%
% Finds the minimum of tau ||X||_* + .5 || X ||_F^2
%
% subject to P_Omega(X) = P_Omega(M)
%
% using linear Bregman iterations
%
% Usage:  [U,S,V,numiter]  = SVT(n,Omega,b,delta,maxiter,opts)
%
% Inputs:
%
% n - size of the matrix X assumed n by n
%
% Omega - set of observed entries
%
% b - data vector of the form M(Omega)
%
% delta - step size
%
% maxiter - maximum number of iterations
%
% Outputs: matrix X stored in SVD format X = U*diag(S)*V'
%
% U - nxr left singular vectors
%
% S - rx1 singular values
%
% V - nxr right singular vectors
%
% numiter - number of iterations to achieve convergence

% Description:
% Reference:
%
%    Cai, Candes and Shen
%    A singular value thresholding algorithm for matrix completion
%    Submitted for publication, October 2008.
%
% Written by: Emmanuel Candes
% Email: emmanuel@acm.caltech.edu
% Created: October 2008

m = length(Omega); [temp,indx] = sort(Omega);
tau = 5*n; incre = 5;

[i, j] = ind2sub([n,n], Omega);
ProjM = sparse(i,j,b,n,n,m);

normProjM = normest(ProjM);
k0 = ceil(tau/(delta*normProjM));

normb = norm(b);

y = k0*delta*b;
Y = sparse(i,j,y,n,n,m);
r = 0;

fprintf('\nIteration:   ');
for k = 1:maxiter,
fprintf('\b\b\b%3d',k);
s = r + 1;

OK = 0;
while ~OK
[U,Sigma,V] = lansvd(Y,s,'L');
OK = (Sigma(s,s) <= tau);
s = s + incre;
end

sigma = diag(Sigma); r = sum(sigma > tau);
U = U(:,1:r); V = V(:,1:r); sigma = sigma(1:r) - tau; Sigma = diag(sigma);

x = XonOmega(U*diag(sigma),V,Omega);

if (norm(x-b)/normb < tol)
break
end

y = y + delta*(b-x);
end

fprintf('\n');
numiter = k;

bdsqr()

function [sigma,bnd] = bdsqr(alpha,beta)

% BDSQR: Compute the singular values and bottom element of
%        the left singular vectors of a (k+1) x k lower bidiagonal
%        matrix with diagonal alpha(1:k) and lower bidiagonal beta(1:k),
%        where length(alpha) = length(beta) = k.
%
% [sigma,bnd] = bdsqr(alpha,beta)
%
% Input parameters:
%   alpha(1:k)   : Diagonal elements.
%   beta(1:k)    : Sub-diagonal elements.
% Output parameters:
%   sigma(1:k)  : Computed eigenvalues.
%   bnd(1:k)    : Bottom elements in left singular vectors.

% Below is a very slow replacement for the BDSQR MEX-file.

%warning('PROPACK:NotUsingMex','Using slow matlab code for bdsqr.')
k = length(alpha);
if min(size(alpha)') ~= 1  | min(size(beta)') ~= 1
error('alpha and beta must be vectors')
elseif length(beta) ~= k
error('alpha and beta must have the same lenght')
end
B = spdiags([alpha(:),beta(:)],[0,-1],k+1,k);
[U,S,V] = svd(full(B),0);
sigma = diag(S);
bnd = U(end,1:k)';

compute_int()

function int = compute_int(mu,j,delta,eta,LL,strategy,extra)
%COMPUTE_INT:  Determine which Lanczos vectors to reorthogonalize against.
%
%      int = compute_int(mu,eta,LL,strategy,extra))
%
%   Strategy 0: Orthogonalize vectors v_{i-r-extra},...,v_{i},...v_{i+s+extra}
%               with nu>eta, where v_{i} are the vectors with  mu>delta.
%   Strategy 1: Orthogonalize all vectors v_{r-extra},...,v_{s+extra} where
%               v_{r} is the first and v_{s} the last Lanczos vector with
%               mu > eta.
%   Strategy 2: Orthogonalize all vectors with mu > eta.
%
%   Notice: The first LL vectors are excluded since the new Lanczos
%   vector is already orthogonalized against them in the main iteration.

%  Rasmus Munk Larsen, DAIMI, 1998.

if (delta<eta)
error('DELTA should satisfy DELTA >= ETA.')
end
switch strategy
case 0
I0 = find(abs(mu(1:j))>=delta);
if length(I0)==0
[mm,I0] = max(abs(mu(1:j)));
end
int = zeros(j,1);
for i = 1:length(I0)
for r=I0(i):-1:1
if abs(mu(r))<eta | int(r)==1
break;
else
int(r) = 1;
end
end
int(max(1,r-extra+1):r) = 1;
for s=I0(i)+1:j
if abs(mu(s))<eta | int(s)==1
break;
else
int(s) = 1;
end
end
int(s:min(j,s+extra-1)) = 1;
end
if LL>0
int(1:LL) = 0;
end
int = find(int);
case 1
int=find(abs(mu(1:j))>eta);
int = max(LL+1,min(int)-extra):min(max(int)+extra,j);
case 2
int=find(abs(mu(1:j))>=eta);
end
int = int(:);

lanbpro()

function [U,B_k,V,p,ierr,work] = lanbpro(varargin)

%LANBPRO Lanczos bidiagonalization with partial reorthogonalization.
%   LANBPRO computes the Lanczos bidiagonalization of a real
%   matrix using the  with partial reorthogonalization.
%
%   [U_k,B_k,V_k,R,ierr,work] = LANBPRO(A,K,R0,OPTIONS,U_old,B_old,V_old)
%   [U_k,B_k,V_k,R,ierr,work] = LANBPRO('Afun','Atransfun',M,N,K,R0, ...
%                                       OPTIONS,U_old,B_old,V_old)
%
%   Computes K steps of the Lanczos bidiagonalization algorithm with partial
%   reorthogonalization (BPRO) with M-by-1 starting vector R0, producing a
%   lower bidiagonal K-by-K matrix B_k, an N-by-K matrix V_k, an M-by-K
%   matrix U_k and an M-by-1 vector R such that
%        A*V_k = U_k*B_k + R
%   Partial reorthogonalization is used to keep the columns of V_K and U_k
%   semiorthogonal:
%         MAX(DIAG((EYE(K) - V_K'*V_K))) <= OPTIONS.delta
%   and
%         MAX(DIAG((EYE(K) - U_K'*U_K))) <= OPTIONS.delta.
%
%   B_k = LANBPRO(...) returns the bidiagonal matrix only.
%
%   The first input argument is either a real 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 must be the name of an M-file which applies the transpose of
%   the same linear operator to the columns of a given matrix,
%   and the third and fourth arguments must be M and N, the dimensions
%   of then problem.
%
%   The OPTIONS structure is used to control the reorthogonalization:
%     OPTIONS.delta:  Desired level of orthogonality
%                     (default = sqrt(eps/K)).
%     OPTIONS.eta  :  Level of orthogonality after reorthogonalization
%                     (default = eps^(3/4)/sqrt(K)).
%     OPTIONS.cgs  :  Flag for switching between different reorthogonalization
%                     algorithms:
%                      0 = iterated modified Gram-Schmidt  (default)
%                      1 = iterated classical Gram-Schmidt
%     OPTIONS.elr  :  If OPTIONS.elr = 1 (default) then extended local
%                     reorthogonalization is enforced.
%     OPTIONS.onesided
%                  :  If OPTIONS.onesided = 0 (default) then both the left
%                     (U) and right (V) Lanczos vectors are kept
%                     semiorthogonal.
%                     OPTIONS.onesided = 1 then only the columns of U are
%                     are reorthogonalized.
%                     OPTIONS.onesided = -1 then only the columns of V are
%                     are reorthogonalized.
%     OPTIONS.waitbar
%                  :  The progress of the algorithm is display graphically.
%
%   If both R0, U_old, B_old, and V_old are provided, they must
%   contain a partial Lanczos bidiagonalization of A on the form
%
%        A V_old = U_old B_old + R0 .
%
%   In this case the factorization is extended to dimension K x K by
%   continuing the Lanczos bidiagonalization algorithm with R0 as a
%   starting vector.
%
%   The output array work contains information about the work used in
%   reorthogonalizing the u- and v-vectors.
%      work = [ RU  PU ]
%             [ RV  PV ]
%   where
%      RU = Number of reorthogonalizations of U.
%      PU = Number of inner products used in reorthogonalizing U.
%      RV = Number of reorthogonalizations of V.
%      PV = Number of inner products used in reorthogonalizing V.

% References:
% R.M. Larsen, Ph.D. Thesis, Aarhus University, 1998.
%
% G. H. Golub & C. F. Van Loan, "Matrix Computations",
% 3. Ed., Johns Hopkins, 1996.  Section 9.3.4.
%
% 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.

% Check input arguments.

global LANBPRO_TRUTH
LANBPRO_TRUTH=0;

if LANBPRO_TRUTH==1
global MU NU MUTRUE NUTRUE
global MU_AFTER NU_AFTER MUTRUE_AFTER NUTRUE_AFTER
end

if nargin<1 | length(varargin)<2
error('Not enough input arguments.');
end
narg=length(varargin);

A = varargin{1};
if isnumeric(A) | isstruct(A)
if isnumeric(A)
if ~isreal(A)
error('A must be real')
end
[m n] = size(A);
elseif isstruct(A)
[m n] = size(A.R);
end
k=varargin{2};
if narg >= 3 & ~isempty(varargin{3});
p = varargin{3};
else
p = rand(m,1)-0.5;
end
if narg < 4, options = []; else options=varargin{4}; end
if narg > 4
if narg<7
error('All or none of U_old, B_old and V_old must be provided.')
else
U = varargin{5}; B_k = varargin{6}; V = varargin{7};
end
else
U = []; B_k = []; V = [];
end
if narg > 7, anorm=varargin{8}; else anorm = []; end
else
if narg<5
error('Not enough input arguments.');
end
Atrans = varargin{2};
if ~isstr(Atrans)
error('Afunc and Atransfunc must be names of m-files')
end
m = varargin{3};
n = varargin{4};
if ~isreal(n) | abs(fix(n)) ~= n | ~isreal(m) | abs(fix(m)) ~= m
error('M and N must be positive integers.')
end
k=varargin{5};
if narg < 6, p = rand(m,1)-0.5; else p=varargin{6}; end
if narg < 7, options = []; else options=varargin{7}; end
if narg > 7
if  narg < 10
error('All or none of U_old, B_old and V_old must be provided.')
else
U = varargin{8}; B_k = varargin{9}; V = varargin{10};
end
else
U = []; B_k = []; V=[];
end
if narg > 10, anorm=varargin{11}; else anorm = [];  end
end

% Quick return for min(m,n) equal to 0 or 1.
if min(m,n) == 0
U = [];  B_k = [];  V = [];  p = [];  ierr = 0;  work = zeros(2,2);
return
elseif  min(m,n) == 1
if isnumeric(A)
U = 1;  B_k = A;  V = 1;  p = 0; ierr = 0; work = zeros(2,2);
else
U = 1;  B_k = feval(A,1); V = 1; p = 0; ierr = 0; work = zeros(2,2);
end
if nargout<3
U = B_k;
end
return
end

% Set options.
%m2 = 3/2*(sqrt(m)+1);
%n2 = 3/2*(sqrt(n)+1);
m2 = 3/2;
n2 = 3/2;
delta = sqrt(eps/k); % Desired level of orthogonality.
eta = eps^(3/4)/sqrt(k);    % Level of orth. after reorthogonalization.
cgs = 0;             % Flag for switching between iterated MGS and CGS.
elr = 2;             % Flag for switching extended local
% reorthogonalization on and off.
gamma = 1/sqrt(2);   % Tolerance for iterated Gram-Schmidt.
onesided = 0; t = 0; waitb = 0;

% Parse options struct
if ~isempty(options) & isstruct(options)
c = fieldnames(options);
for i=1:length(c)
if strmatch(c(i),'delta'), delta = getfield(options,'delta');  end
if strmatch(c(i),'eta'), eta = getfield(options,'eta'); end
if strmatch(c(i),'cgs'), cgs = getfield(options,'cgs'); end
if strmatch(c(i),'elr'), elr = getfield(options,'elr'); end
if strmatch(c(i),'gamma'), gamma = getfield(options,'gamma'); end
if strmatch(c(i),'onesided'), onesided = getfield(options,'onesided'); end
if strmatch(c(i),'waitbar'), waitb=1; end
end
end

if waitb
waitbarh = waitbar(0,'Lanczos bidiagonalization in progress...');
end

if isempty(anorm)
anorm = []; est_anorm=1;
else
est_anorm=0;
end

% Conservative statistical estimate on the size of round-off terms.
% Notice that {\bf u} == eps/2.
FUDGE = 1.01; % Fudge factor for ||A||_2 estimate.

npu = 0; npv = 0; ierr = 0;
p = p(:);
% Prepare for Lanczos iteration.
if isempty(U)
V = zeros(n,k); U = zeros(m,k);
beta = zeros(k+1,1); alpha = zeros(k,1);
beta(1) = norm(p);
% Initialize MU/NU-recurrences for monitoring loss of orthogonality.
nu = zeros(k,1); mu = zeros(k+1,1);
mu(1)=1; nu(1)=1;

numax = zeros(k,1); mumax = zeros(k,1);
force_reorth = 0;  nreorthu = 0; nreorthv = 0;
j0 = 1;
else
j = size(U,2); % Size of existing factorization
% Allocate space for Lanczos vectors
U = [U, zeros(m,k-j)];
V = [V, zeros(n,k-j)];
alpha = zeros(k+1,1);  beta = zeros(k+1,1);
alpha(1:j) = diag(B_k); if j>1 beta(2:j) = diag(B_k,-1); end
beta(j+1) = norm(p);
% Reorthogonalize p.
if j<k & beta(j+1)*delta < anorm*eps,
fro = 1;
ierr = j;
end
int = [1:j]';
[p,beta(j+1),rr] = reorth(U,p,beta(j+1),int,gamma,cgs);
npu =  rr*j;  nreorthu = 1;  force_reorth= 1;

% Compute Gerscgorin bound on ||B_k||_2
if est_anorm
anorm = FUDGE*sqrt(norm(B_k'*B_k,1));
end
mu = m2*eps*ones(k+1,1); nu = zeros(k,1);
numax = zeros(k,1); mumax = zeros(k,1);
force_reorth = 1;  nreorthu = 0; nreorthv = 0;
j0 = j+1;
end

if isnumeric(A)
At = A';
end

if delta==0
fro = 1; % The user has requested full reorthogonalization.
else
fro = 0;
end

if LANBPRO_TRUTH==1
MUTRUE = zeros(k,k); NUTRUE = zeros(k-1,k-1);
MU = zeros(k,k); NU = zeros(k-1,k-1);

MUTRUE_AFTER = zeros(k,k); NUTRUE_AFTER = zeros(k-1,k-1);
MU_AFTER = zeros(k,k); NU_AFTER = zeros(k-1,k-1);
end

% Perform Lanczos bidiagonalization with partial reorthogonalization.
for j=j0:k
if waitb
waitbar(j/k,waitbarh)
end

if beta(j) ~= 0
U(:,j) = p/beta(j);
else
U(:,j) = p;
end

% Replace norm estimate with largest Ritz value.
if j==6
B = [[diag(alpha(1:j-1))+diag(beta(2:j-1),-1)]; ...
[zeros(1,j-2),beta(j)]];
anorm = FUDGE*norm(B);
est_anorm = 0;
end

%%%%%%%%%% Lanczos step to generate v_j. %%%%%%%%%%%%%
if j==1
if isnumeric(A)
r = At*U(:,1);
elseif isstruct(A)
r = A.R\U(:,1);
else
r = feval(Atrans,U(:,1));
end
alpha(1) = norm(r);
if est_anorm
anorm = FUDGE*alpha(1);
end
else
if isnumeric(A)
r = At*U(:,j) - beta(j)*V(:,j-1);
elseif isstruct(A)
r = A.R\U(:,j) - beta(j)*V(:,j-1);
else
r = feval(Atrans,U(:,j))  - beta(j)*V(:,j-1);
end
alpha(j) = norm(r);

% Extended local reorthogonalization
if alpha(j)<gamma*beta(j) & elr & ~fro
normold = alpha(j);
stop = 0;
while ~stop
t = V(:,j-1)'*r;
r = r - V(:,j-1)*t;
alpha(j) = norm(r);
if beta(j) ~= 0
beta(j) = beta(j) + t;
end
if alpha(j)>=gamma*normold
stop = 1;
else
normold = alpha(j);
end
end
end

if est_anorm
if j==2
anorm = max(anorm,FUDGE*sqrt(alpha(1)^2+beta(2)^2+alpha(2)*beta(2)));
else
anorm = max(anorm,FUDGE*sqrt(alpha(j-1)^2+beta(j)^2+alpha(j-1)* ...
beta(j-1) + alpha(j)*beta(j)));
end
end

if ~fro & alpha(j) ~= 0
% Update estimates of the level of orthogonality for the
%  columns 1 through j-1 in V.
nu = update_nu(nu,mu,j,alpha,beta,anorm);
numax(j) = max(abs(nu(1:j-1)));
end

if j>1 & LANBPRO_TRUTH
NU(1:j-1,j-1) = nu(1:j-1);
NUTRUE(1:j-1,j-1) = V(:,1:j-1)'*r/alpha(j);
end

if elr>0
nu(j-1) = n2*eps;
end

% IF level of orthogonality is worse than delta THEN
%    Reorthogonalize v_j against some previous  v_i's, 0<=i<j.
if onesided~=-1 & ( fro | numax(j) > delta | force_reorth ) & alpha(j)~=0
% Decide which vectors to orthogonalize against:
if fro | eta==0
int = [1:j-1]';
elseif force_reorth==0
int = compute_int(nu,j-1,delta,eta,0,0,0);
end
% Else use int from last reorth. to avoid spillover from mu_{j-1}
% to nu_j.

% Reorthogonalize v_j
[r,alpha(j),rr] = reorth(V,r,alpha(j),int,gamma,cgs);
npv = npv + rr*length(int); % number of inner products.
nu(int) = n2*eps;  % Reset nu for orthogonalized vectors.

% If necessary force reorthogonalization of u_{j+1}
% to avoid spillover
if force_reorth==0
force_reorth = 1;
else
force_reorth = 0;
end
nreorthv = nreorthv + 1;
end
end

% Check for convergence or failure to maintain semiorthogonality
if alpha(j) < max(n,m)*anorm*eps & j<k,
% If alpha is "small" we deflate by setting it
% to 0 and attempt to restart with a basis for a new
% invariant subspace by replacing r with a random starting vector:
%j
%disp('restarting, alpha = 0')
alpha(j) = 0;
bailout = 1;
for attempt=1:3
r = rand(m,1)-0.5;
if isnumeric(A)
r = At*r;
elseif isstruct(A)
r = A.R\r;
else
r = feval(Atrans,r);
end
nrm=sqrt(r'*r); % not necessary to compute the norm accurately here.
int = [1:j-1]';
[r,nrmnew,rr] = reorth(V,r,nrm,int,gamma,cgs);
npv = npv + rr*length(int(:));        nreorthv = nreorthv + 1;
nu(int) = n2*eps;
if nrmnew > 0
% A vector numerically orthogonal to span(Q_k(:,1:j)) was found.
% Continue iteration.
bailout=0;
break;
end
end
if bailout
j = j-1;
ierr = -j;
break;
else
r=r/nrmnew; % Continue with new normalized r as starting vector.
force_reorth = 1;
if delta>0
fro = 0;    % Turn off full reorthogonalization.
end
end
elseif  j<k & ~fro & anorm*eps > delta*alpha(j)
%    fro = 1;
ierr = j;
end

if j>1 & LANBPRO_TRUTH
NU_AFTER(1:j-1,j-1) = nu(1:j-1);
NUTRUE_AFTER(1:j-1,j-1) = V(:,1:j-1)'*r/alpha(j);
end

if alpha(j) ~= 0
V(:,j) = r/alpha(j);
else
V(:,j) = r;
end

%%%%%%%%%% Lanczos step to generate u_{j+1}. %%%%%%%%%%%%%
if waitb
waitbar((2*j+1)/(2*k),waitbarh)
end

if isnumeric(A)
p = A*V(:,j) - alpha(j)*U(:,j);
elseif isstruct(A)
p = A.Rt\V(:,j) - alpha(j)*U(:,j);
else
p = feval(A,V(:,j)) - alpha(j)*U(:,j);
end
beta(j+1) = norm(p);
% Extended local reorthogonalization
if beta(j+1)<gamma*alpha(j) & elr & ~fro
normold = beta(j+1);
stop = 0;
while ~stop
t = U(:,j)'*p;
p = p - U(:,j)*t;
beta(j+1) = norm(p);
if alpha(j) ~= 0
alpha(j) = alpha(j) + t;
end
if beta(j+1) >= gamma*normold
stop = 1;
else
normold = beta(j+1);
end
end
end

if est_anorm
% We should update estimate of ||A||  before updating mu - especially
% important in the first step for problems with large norm since alpha(1)
% may be a severe underestimate!
if j==1
anorm = max(anorm,FUDGE*pythag(alpha(1),beta(2)));
else
anorm = max(anorm,FUDGE*sqrt(alpha(j)^2+beta(j+1)^2 + alpha(j)*beta(j)));
end
end

if ~fro & beta(j+1) ~= 0
% Update estimates of the level of orthogonality for the columns of V.
mu = update_mu(mu,nu,j,alpha,beta,anorm);
mumax(j) = max(abs(mu(1:j)));
end

if LANBPRO_TRUTH==1
MU(1:j,j) = mu(1:j);
MUTRUE(1:j,j) = U(:,1:j)'*p/beta(j+1);
end

if elr>0
mu(j) = m2*eps;
end

% IF level of orthogonality is worse than delta THEN
%    Reorthogonalize u_{j+1} against some previous  u_i's, 0<=i<=j.
if onesided~=1 & (fro | mumax(j) > delta | force_reorth) & beta(j+1)~=0
% Decide which vectors to orthogonalize against.
if fro | eta==0
int = [1:j]';
elseif force_reorth==0
int = compute_int(mu,j,delta,eta,0,0,0);
else
int = [int; max(int)+1];
end
% Else use int from last reorth. to avoid spillover from nu to mu.

%    if onesided~=0
%      fprintf('i = %i, nr = %i, fro = %i\n',j,size(int(:),1),fro)
%    end
% Reorthogonalize u_{j+1}
[p,beta(j+1),rr] = reorth(U,p,beta(j+1),int,gamma,cgs);
npu = npu + rr*length(int);  nreorthu = nreorthu + 1;

% Reset mu to epsilon.
mu(int) = m2*eps;

if force_reorth==0
force_reorth = 1; % Force reorthogonalization of v_{j+1}.
else
force_reorth = 0;
end
end

% Check for convergence or failure to maintain semiorthogonality
if beta(j+1) < max(m,n)*anorm*eps  & j<k,
% If beta is "small" we deflate by setting it
% to 0 and attempt to restart with a basis for a new
% invariant subspace by replacing p with a random starting vector:
%j
%disp('restarting, beta = 0')
beta(j+1) = 0;
bailout = 1;
for attempt=1:3
p = rand(n,1)-0.5;
if isnumeric(A)
p = A*p;
elseif isstruct(A)
p = A.Rt\p;
else
p = feval(A,p);
end
nrm=sqrt(p'*p); % not necessary to compute the norm accurately here.
int = [1:j]';
[p,nrmnew,rr] = reorth(U,p,nrm,int,gamma,cgs);
npu = npu + rr*length(int(:));  nreorthu = nreorthu + 1;
mu(int) = m2*eps;
if nrmnew > 0
% A vector numerically orthogonal to span(Q_k(:,1:j)) was found.
% Continue iteration.
bailout=0;
break;
end
end
if bailout
ierr = -j;
break;
else
p=p/nrmnew; % Continue with new normalized p as starting vector.
force_reorth = 1;
if delta>0
fro = 0;    % Turn off full reorthogonalization.
end
end
elseif  j<k & ~fro & anorm*eps > delta*beta(j+1)
%    fro = 1;
ierr = j;
end

if LANBPRO_TRUTH==1
MU_AFTER(1:j,j) = mu(1:j);
MUTRUE_AFTER(1:j,j) = U(:,1:j)'*p/beta(j+1);
end
end
if waitb
close(waitbarh)
end

if j<k
k = j;
end

B_k = spdiags([alpha(1:k) [beta(2:k);0]],[0 -1],k,k);
if nargout==1
U = B_k;
elseif k~=size(U,2) | k~=size(V,2)
U = U(:,1:k);
V = V(:,1:k);
end
if nargout>5
work = [[nreorthu,npu];[nreorthv,npv]];
end

function mu = update_mu(muold,nu,j,alpha,beta,anorm)

% UPDATE_MU:  Update the mu-recurrence for the u-vectors.
%
%   mu_new = update_mu(mu,nu,j,alpha,beta,anorm)

%  Rasmus Munk Larsen, DAIMI, 1998.

binv = 1/beta(j+1);
mu = muold;
eps1 = 100*eps/2;
if j==1
T = eps1*(pythag(alpha(1),beta(2)) + pythag(alpha(1),beta(1)));
T = T + eps1*anorm;
mu(1) = T / beta(2);
else
mu(1) = alpha(1)*nu(1) - alpha(j)*mu(1);
%  T = eps1*(pythag(alpha(j),beta(j+1)) + pythag(alpha(1),beta(1)));
T = eps1*(sqrt(alpha(j).^2+beta(j+1).^2) + sqrt(alpha(1).^2+beta(1).^2));
T = T + eps1*anorm;
mu(1) = (mu(1) + sign(mu(1))*T) / beta(j+1);
% Vectorized version of loop:
if j>2
k=2:j-1;
mu(k) = alpha(k).*nu(k) + beta(k).*nu(k-1) - alpha(j)*mu(k);
%T = eps1*(pythag(alpha(j),beta(j+1)) + pythag(alpha(k),beta(k)));
T = eps1*(sqrt(alpha(j).^2+beta(j+1).^2) + sqrt(alpha(k).^2+beta(k).^2));
T = T + eps1*anorm;
mu(k) = binv*(mu(k) + sign(mu(k)).*T);
end
%  T = eps1*(pythag(alpha(j),beta(j+1)) + pythag(alpha(j),beta(j)));
T = eps1*(sqrt(alpha(j).^2+beta(j+1).^2) + sqrt(alpha(j).^2+beta(j).^2));
T = T + eps1*anorm;
mu(j) = beta(j)*nu(j-1);
mu(j) = (mu(j) + sign(mu(j))*T) / beta(j+1);
end
mu(j+1) = 1;

function nu = update_nu(nuold,mu,j,alpha,beta,anorm)

% UPDATE_MU:  Update the nu-recurrence for the v-vectors.
%
%  nu_new = update_nu(nu,mu,j,alpha,beta,anorm)

%  Rasmus Munk Larsen, DAIMI, 1998.

nu = nuold;
ainv = 1/alpha(j);
eps1 = 100*eps/2;
if j>1
k = 1:(j-1);
%  T = eps1*(pythag(alpha(k),beta(k+1)) + pythag(alpha(j),beta(j)));
T = eps1*(sqrt(alpha(k).^2+beta(k+1).^2) + sqrt(alpha(j).^2+beta(j).^2));
T = T + eps1*anorm;
nu(k) = beta(k+1).*mu(k+1) + alpha(k).*mu(k) - beta(j)*nu(k);
nu(k) = ainv*(nu(k) + sign(nu(k)).*T);
end
nu(j) = 1;

function x = pythag(y,z)
%PYTHAG Computes sqrt( y^2 + z^2 ).
%
% x = pythag(y,z)
%
% Returns sqrt(y^2 + z^2) but is careful to scale to avoid overflow.

% Christian H. Bischof, Argonne National Laboratory, 03/31/89.

[m n] = size(y);
if m>1 | n>1
y = y(:); z=z(:);
rmax = max(abs([y z]'))';
id=find(rmax==0);
if length(id)>0
rmax(id) = 1;
x = rmax.*sqrt((y./rmax).^2 + (z./rmax).^2);
x(id)=0;
else
x = rmax.*sqrt((y./rmax).^2 + (z./rmax).^2);
end
x = reshape(x,m,n);
else
rmax = max(abs([y;z]));
if (rmax==0)
x = 0;
else
x = rmax*sqrt((y/rmax)^2 + (z/rmax)^2);
end
end