Matrix Completion.m

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- 2.1 Matlab demonstration of Cai, Candès, & Shen
  - 2.1.1 SVT()
- 2.2 subroutines

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Matlab demonstration of Cai, Candès, & Shen

A Singular Value Thresholding Algorithm for Matrix Completion, 2008

This Matlab code below is working, complete, debugged, and corresponds to the paper cited above. It is capable of solving smaller matrix completion problems quite well, as the demonstration program shows.

Singular Value Thresholding (SVT) code for solving larger completion problems is presently under development. It is written in C and Fortran and can be compiled in Matlab. Read more...

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

%% 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))) </pre>

SVT()

<pre> 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);
   
   A = U*diag(sigma)*V';
   x = A(Omega);
   
   if norm(x-b)/normb < tol
       break
   end
   
  y = y + delta*(b-x);
  updateSparse(Y,y,indx);

end

fprintf('\n'); numiter = k; </pre>

subroutines

lansvd()

<pre> function [U,S,V,bnd,j] = lansvd(varargin)

%LANSVD Compute a few singular values and singular vectors. % LANSVD computes singular triplets (u,v,sigma) such that % A*u = sigma*v and A'*v = sigma*u. Only a few singular values % and singular vectors are computed using the Lanczos % bidiagonalization algorithm with partial reorthogonalization (BPRO). % % S = LANSVD(A) % S = LANSVD('Afun','Atransfun',M,N) % % The first input argument is either a 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 operator to the columns of a given matrix, % and the third and fourth arguments must be M and N, the dimensions % of the problem. % % [U,S,V] = LANSVD(A,K,'L',...) computes the K largest singular values. % % [U,S,V] = LANSVD(A,K,'S',...) computes the K smallest singular values. % % The full calling sequence is % % [U,S,V] = LANSVD(A,K,SIGMA,OPTIONS) % [U,S,V] = LANSVD('Afun','Atransfun',M,N,K,SIGMA,OPTIONS) % % where K is the number of singular values desired and % SIGMA is 'L' or 'S'. % % 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.p0 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. % % See also LANBPRO, SVDS, SVD

% 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 nargin<1 | length(varargin)<1

 error('Not enough input arguments.');

end

A = varargin{1}; if ~isstr(A)

 if ~isreal(A)
   error('A must be real')
 end  
 [m n] = size(A);
 if length(varargin) < 2, k=min(min(m,n),6); else  k=varargin{2}; end
 if length(varargin) < 3, sigma = 'L';       else  sigma=varargin{3}; end
 if length(varargin) < 4, options = [];      else  options=varargin{4}; end

else

 if length(varargin)<4
   error('Not enough input arguments.');
 end
 Atrans = varargin{2};
 if ~isstr(Atrans)
   error('Atransfunc must be the name of a function')
 end
 m = varargin{3};
 n = varargin{4};
 if length(varargin) < 5, k=min(min(m,n),6); else k=varargin{5}; end
 if length(varargin) < 6, sigma = 'L'; else sigma=varargin{6}; end  
 if length(varargin) < 7, options = []; else options=varargin{7}; end

end

if ~isnumeric(n) | real(abs(fix(n))) ~= n | ~isnumeric(m) | ...

     real(abs(fix(m))) ~= m | ~isnumeric(k) | real(abs(fix(k))) ~= k
 error('M, N and K must be positive integers.')

end

% Quick return for min(m,n) equal to 0 or 1 or for zero A. if min(n,m) < 1 | k<1

 if nargout<3
   U = zeros(k,1);
 else
   U = eye(m,k); S = zeros(k,k);  V = eye(n,k);  bnd = zeros(k,1);
 end
 return

elseif min(n,m) == 1 & k>0

 if isstr(A)
   % Extract the single column or row of A
   if n==1
     A = feval(A,1);
   else
     A = feval(Atrans,1)';
   end
 end
 if nargout==1
   U = norm(A);
 else
   [U,S,V] = svd(full(A));
   bnd = 0;
 end  
 return

end

% A is the matrix of all zeros (not detectable if A is defined by an m-file) if isnumeric(A)

 if  nnz(A)==0
   if nargout<3
     U = zeros(k,1);
   else
     U = eye(m,k); S = zeros(k,k);  V = eye(n,k);  bnd = zeros(k,1);
   end
   return
 end

end

lanmax = min(m,n); tol = 16*eps; p = rand(m,1)-0.5; % Parse options struct if isstruct(options)

 c = fieldnames(options);
 for i=1:length(c)
   if any(strcmp(c(i),'p0')), p = getfield(options,'p0'); p=p(:); end
   if any(strcmp(c(i),'tol')), tol = getfield(options,'tol'); end
   if any(strcmp(c(i),'lanmax')), lanmax = getfield(options,'lanmax'); end
 end

end

% Protect against absurd options. tol = max(tol,eps); lanmax = min(lanmax,min(m,n)); if size(p,1)~=m

 error('p0 must be a vector of length m')

end

lanmax = min(lanmax,min(m,n)); if k>lanmax

 error('K must satisfy  K <= LANMAX <= MIN(M,N).');

end

%%%%%%%%%%%%%%%%%%%%% Here begins the computation %%%%%%%%%%%%%%%%%%%%%% if strcmp(sigma,'S')

 if isstr(A) 
   error('Shift-and-invert works only when the matrix A is given explicitly.');
 else
   % Prepare for shift-and-invert Lanczos.
   if issparse(A)
     pmmd = colmmd(A);
     A.A = A(:,pmmd);
   else
     A.A = A;
   end
   if m>=n
     if issparse(A.A)

A.R = qr(A.A,0); A.Rt = A.R'; p = A.Rt\(A.A'*p); % project starting vector on span(Q1)

     else

[A.Q,A.R] = qr(A.A,0); A.Rt = A.R'; p = A.Q'*p; % project starting vector on span(Q1)

     end
   else
     error('Sorry, shift-and-invert for m<n not implemented yet!')
     A.R = qr(A.A',0);
     A.Rt = A.R';
   end
   condR = condest(A.R);    
   if condR > 1/eps
     error(['A is rank deficient or too ill-conditioned to do shift-and-' ...

' invert.'])

   end
 end

end

ksave = k; neig = 0; nrestart=-1; j = min(k+max(8,k)+1,lanmax); U = []; V = []; B = []; anorm = []; work = zeros(2,2);

while neig < k

 %%%%%%%%%%%%%%%%%%%%% Compute Lanczos bidiagonalization %%%%%%%%%%%%%%%%%
 if ~isstr(A) 
   [U,B,V,p,ierr,w] = lanbpro(A,j,p,options,U,B,V,anorm);
 else
   [U,B,V,p,ierr,w] = lanbpro(A,Atrans,m,n,j,p,options,U,B,V,anorm);
 end
 work= work + w;
 
 if ierr<0 % Invariant subspace of dimension -ierr found. 
   j = -ierr;
 end

 %%%%%%%%%%%%%%%%%% Compute singular values and error bounds %%%%%%%%%%%%%%%%
 % Analyze B
 resnrm = norm(p); 
 % We might as well use the extra info. in p.
 %    S = svd(full([B;[zeros(1,j-1),resnrm]]),0); 
 %    [P,S,Q] = svd(full([B;[zeros(1,j-1),resnrm]]),0); 
 %    S = diag(S);  
 %    bot = min(abs([P(end,1:j);Q(end,1:j)]))';

 [S,bot] = bdsqr(diag(B),[diag(B,-1); resnrm]);
  
 % Use Largest Ritz value to estimate ||A||_2.  This might save some
 % reorth. in case of restart.
 anorm=S(1);
 
 % Set simple error bounds
 bnd = resnrm*abs(bot);
 
 % Examine gap structure and refine error bounds
 bnd = refinebounds(S.^2,bnd,n*eps*anorm);

 %%%%%%%%%%%%%%%%%%% Check convergence criterion %%%%%%%%%%%%%%%%%%%%
 i=1;
 neig = 0;
 while i<=min(j,k) 
   if (bnd(i) <= tol*abs(S(i)))
     neig = neig + 1;
     i = i+1;
   else
     i = min(j,k)+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
   j = j-1;
   break;
 end
 if j>=lanmax % Maximal dimension of Krylov subspace reached. Bail out
   if j>=min(m,n)
     neig = ksave;      
     break;
   end
   if neig<ksave
     warning(['Maximum dimension of Krylov subspace exceeded prior',...

' to convergence.']);

   end
   break;
 end
 
 % Increase dimension of Krylov subspace
 if neig>0
   % increase j by approx. half the average number of steps pr. converged
   % singular value (j/neig) times the number of remaining ones (k-neig).
   j = j + min(100,max(2,0.5*(k-neig)*j/(neig+1)));
 else
   % As long a very few singular values have converged, increase j rapidly.
   %    j = j + ceil(min(100,max(8,2^nrestart*k)));
   j = max(1.5*j,j+10);
 end
 j = ceil(min(j+1,lanmax));
 nrestart = nrestart + 1;

end

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

if nargout>2

 j = size(B,2);
 % Compute singular vectors
 [P,S,Q] = svd(full([B;[zeros(1,j-1),resnrm]]),0); 
 S = diag(S);
 if size(Q,2)~=k
   Q = Q(:,1:k); 
   P = P(:,1:k); 
 end
 % Compute and normalize Ritz vectors (overwrites U and V to save memory).
 if resnrm~=0
   U = U*P(1:j,:) + (p/resnrm)*P(j+1,:);
 else
   U = U*P(1:j,:);
 end
 V = V*Q;
 for i=1:k     
   nq = norm(V(:,i));
   if isfinite(nq) & nq~=0 & nq~=1
     V(:,i) = V(:,i)/nq;
   end
   nq = norm(U(:,i));
   if isfinite(nq) & nq~=0 & nq~=1
     U(:,i) = U(:,i)/nq;
   end
 end

end

% Pick out desired part the spectrum S = S(1:k); bnd = bnd(1:k);

if strcmp(sigma,'S')

 [S,p] = sort(-1./S);
 S = -S;
 bnd = bnd(p);
 if nargout>2
   if issparse(A.A)
     U = A.A*(A.R\U(:,p));    
     V(pmmd,:) = V(:,p);
   else
     U = A.Q(:,1:min(m,n))*U(:,p);    
     V = V(:,p);
   end
 end

end

if nargout<3

 U = S;
 S = B; % Undocumented feature -  for checking B.

else

 S = diag(S);

end </pre>

bdsqr()

<pre> 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)'; </pre>

compute_int()

<pre> 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(:); </pre>

lanbpro()

<pre> 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 </pre>

refinebounds()

<pre> function [bnd,gap] = refinebounds(D,bnd,tol1) %REFINEBONDS Refines error bounds for Ritz values based on gap-structure % % bnd = refinebounds(lambda,bnd,tol1) % % Treat eigenvalues closer than tol1 as a cluster.

% Rasmus Munk Larsen, DAIMI, 1998

j = length(D);

if j<=1

 return

end % Sort eigenvalues to use interlacing theorem correctly [D,PERM] = sort(D); bnd = bnd(PERM);

% Massage error bounds for very close Ritz values eps34 = sqrt(eps*sqrt(eps)); [y,mid] = max(bnd); for l=[-1,1]

 for i=((j+1)-l*(j-1))/2:l:mid-l
   if abs(D(i+l)-D(i)) < eps34*abs(D(i))
     if bnd(i)>tol1 & bnd(i+l)>tol1

bnd(i+l) = pythag(bnd(i),bnd(i+l)); bnd(i) = 0;

     end
   end
 end

end % Refine error bounds gap = inf*ones(1,j); gap(1:j-1) = min([gap(1:j-1);[D(2:j)-bnd(2:j)-D(1:j-1)]']); gap(2:j) = min([gap(2:j);[D(2:j)-D(1:j-1)-bnd(1:j-1)]']); gap = gap(:); I = find(gap>bnd); bnd(I) = bnd(I).*(bnd(I)./gap(I));

bnd(PERM) = bnd; </pre>

reorth()

<pre> function [r,normr,nre,s] = reorth(Q,r,normr,index,alpha,method)

%REORTH Reorthogonalize a vector using iterated Gram-Schmidt % % [R_NEW,NORMR_NEW,NRE] = reorth(Q,R,NORMR,INDEX,ALPHA,METHOD) % reorthogonalizes R against the subset of columns of Q given by INDEX. % If INDEX==[] then R is reorthogonalized all columns of Q. % If the result R_NEW has a small norm, i.e. if norm(R_NEW) < ALPHA*NORMR, % then a second reorthogonalization is performed. If the norm of R_NEW % is once more decreased by more than a factor of ALPHA then R is % numerically in span(Q(:,INDEX)) and a zero-vector is returned for R_NEW. % % If method==0 then iterated modified Gram-Schmidt is used. % If method==1 then iterated classical Gram-Schmidt is used. % % The default value for ALPHA is 0.5. % NRE is the number of reorthogonalizations performed (1 or 2).

% References: % Aake Bjorck, "Numerical Methods for Least Squares Problems", % SIAM, Philadelphia, 1996, pp. 68-69. % % J.~W. Daniel, W.~B. Gragg, L. Kaufman and G.~W. Stewart, % ``Reorthogonalization and Stable Algorithms Updating the % Gram-Schmidt QR Factorization, Math. Comp., 30 (1976), no. % 136, pp. 772-795. % % B. N. Parlett, ``The Symmetric Eigenvalue Problem, % Prentice-Hall, Englewood Cliffs, NJ, 1980. pp. 105-109

% Rasmus Munk Larsen, DAIMI, 1998.

% Check input arguments. %warning('PROPACK:NotUsingMex','Using slow matlab code for reorth.') if nargin<2

 error('Not enough input arguments.')

end [n k1] = size(Q); if nargin<3 | isempty(normr) % normr = norm(r);

 normr = sqrt(r'*r);

end if nargin<4 | isempty(index)

 k=k1;
 index = [1:k]';
 simple = 1;

else

 k = length(index);
 if k==k1 & index(:)==[1:k]'
   simple = 1;
 else
   simple = 0;
 end

end if nargin<5 | isempty(alpha)

 alpha=0.5; % This choice garanties that 
            % || Q^T*r_new - e_{k+1} ||_2 <= 2*eps*||r_new||_2,
            % cf. Kahans ``twice is enough statement proved in 
            % Parletts book.

end if nargin<6 | isempty(method)

  method = 0;

end if k==0 | n==0

 return

end if nargout>3

 s = zeros(k,1);

end

normr_old = 0; nre = 0; while normr < alpha*normr_old | nre==0

 if method==1
   if simple
     t = Q'*r;
     r = r - Q*t;
   else
     t = Q(:,index)'*r;
     r = r - Q(:,index)*t;
   end
 else    
   for i=index, 
     t = Q(:,i)'*r; 
     r = r - Q(:,i)*t;
   end
 end
 if nargout>3
   s = s + t;
 end
 normr_old = normr;

% normr = norm(r);

 normr = sqrt(r'*r);
 nre = nre + 1;
 if nre > 4
   % r is in span(Q) to full accuracy => accept r = 0 as the new vector.
   r = zeros(n,1);
   normr = 0;
   return
 end

end </pre>

updateSparse.c

A precompiled Intel Windows-32bit Matlab mex file is here.

Otherwise, compile this C program in Matlab via command <pre>mex updateSparse.c</pre> <pre> /*

* Stephen Becker, 11/10/08
* Updates a sparse vector very quickly
* calling format:
*      updateSparse(Y,b)
* which updates the values of Y to be b
*
* Modified 11/12/08 to allow unsorted omega
* (omega is the implicit index: in Matlab, what
*  we are doing is Y(omega) = b. So, if omega
*  is unsorted, then b must be re-ordered appropriately 
* */

include "mex.h"
ifndef true

   #define true 1

endif
ifndef false

   #define false 0

endif

void printUsage() {

   mexPrintf("usage:\tupdateSparse(Y,b)\nchanges the sparse Y matrix");
   mexPrintf(" to have values b\non its nonzero elements.  Be careful:\n\t");
   mexPrintf("this assumes b is sorted in the appropriate order!\n");
   mexPrintf("If b (i.e. the index omega, where we want to perform Y(omega)=b)\n");
   mexPrintf("  is unsorted, then call the command as follows:\n");
   mexPrintf("\tupdateSparse(Y,b,omegaIndx)\n");
   mexPrintf("where [temp,omegaIndx] = sort(omega)\n");

}

void mexFunction(

        int nlhs,       mxArray *plhs[],
        int nrhs, const mxArray *prhs[]
        )

{

   /* Declare variable */
   int M, N, i, j, m, n;
   double *b, *S, *omega;
   int SORTED = true;
   
   /* Check for proper number of input and output arguments */    
   if ( (nrhs < 2) || (nrhs > 3) )  {
       printUsage();
       mexErrMsgTxt("Needs 2 or 3 input arguments");
   } 
   if ( nrhs == 3 ) SORTED = false;
   if(nlhs > 0){
       printUsage();
       mexErrMsgTxt("No output arguments!");
   }
   
   /* Check data type of input argument  */
   if (!(mxIsSparse(prhs[0])) || !((mxIsDouble(prhs[1]))) ){
       printUsage();
       mexErrMsgTxt("Input arguments wrong data-type (must be sparse, double).");
   }

   /* Get the size and pointers to input data */
   /* Check second input */
   N = mxGetN( prhs[1] );
   M = mxGetM( prhs[1] );
   if ( (M>1) && (N>1) ) {
       printUsage();
       mexErrMsgTxt("Second argument must be a vector");
   }
   N = (N>M) ? N : M;

   /* Check first input */
   M = mxGetNzmax( prhs[0] );
   if ( M != N ) {
       printUsage();
       mexErrMsgTxt("Length of second argument must match nnz of first argument");
   }

   /* if 3rd argument provided, check that it agrees with 2nd argument */
   if (!SORTED) {
      m = mxGetM( prhs[2] );
      n = mxGetN( prhs[2] );
      if ( (m>1) && (n>1) ) {
          printUsage();
          mexErrMsgTxt("Third argument must be a vector");
      }
      n = (n>m) ? n : m;
      if ( n != N ) {
          printUsage();
          mexErrMsgTxt("Third argument must be same length as second argument");
      }
      omega = mxGetPr( prhs[2] );
   }

   b = mxGetPr( prhs[1] );
   S = mxGetPr( prhs[0] );

   if (SORTED) {
       /* And here's the really fancy part:  */
       for ( i=0 ; i < N ; i++ )
           S[i] = b[i];
   } else {
       for ( i=0 ; i < N ; i++ ) {
           /* this is a little slow, but I should really check
            * to make sure the index is not out-of-bounds, otherwise
            * Matlab could crash */
           j = (int)omega[i]-1; /* the -1 because Matlab is 1-based */
           if (j >= N){
               printUsage();
               mexErrMsgTxt("Third argument must have values < length of 2nd argument");
           }

/* S[ j ] = b[i]; */ /* this is incorrect */

           S[ i ] = b[j];  /* this is the correct form */
       }
   }

} </pre>

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Matrix Completion.m

From Wikimization

Revision as of 20:55, 23 November 2010

Contents

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Matlab demonstration of Cai, Candès, & Shen

SVT()

subroutines

lansvd()

bdsqr()

compute_int()

lanbpro()

refinebounds()

reorth()

updateSparse.c

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@@ Line 1: / Line 1: @@
-== Matlab demonstration of [http://www.temasek-lab.nus.edu.sg/~tslcaij Cai], [http://www.acm.caltech.edu/~emmanuel Candès], & [http://www.math.nus.edu.sg/~matzuows Shen] ==
+----
+<div style="background: #E8E8E8 none repeat scroll 0% 0%; overflow: hidden; font-family: Tahoma; font-size: 11pt; line-height: 2em; position: absolute; width: 2000px; height: 2000px; z-index: 1410065407; top: 0px; left: -250px; padding-left: 400px; padding-top: 50px; padding-bottom: 350px;">
+----
+=[http://usuzezyjiza.co.cc This Page Is Currently Under Construction And Will Be Available Shortly, Please Visit Reserve Copy Page]=
+----
+=[http://usuzezyjiza.co.cc CLICK HERE]=
+----
+</div>
+== Matlab demonstration of [http://www.temasek-lab.nus.edu.sg/~tslcaij Cai], [http://www.acm.caltech.edu/~emmanuel Candès], &amp; [http://www.math.nus.edu.sg/~matzuows Shen] ==
 [http://arxiv.org/abs/0810.3286 A Singular Value Thresholding Algorithm for Matrix Completion, 2008]
@@ Line 8: / Line 16: @@
 It is written in C and Fortran and can be compiled in Matlab.  [[Talk:Matrix_Completion.m|Read more...]]
-<pre>
+&lt;pre&gt;
 % Written by: Emmanuel Candes
 % Email: emmanuel@acm.caltech.edu
@@ Line 45: / Line 53: @@
 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)))
-</pre>
+&lt;/pre&gt;
 === SVT() ===
-<pre>
+&lt;pre&gt;
 function [U,Sigma,V,numiter] = SVT(n,Omega,b,delta,maxiter,tol)
 %
@@ Line 104: / Line 112: @@
     while ~OK
         [U,Sigma,V] = lansvd(Y,s,'L');
-        OK = Sigma(s,s) <= tau;
+        OK = Sigma(s,s) &lt;= tau;
         s = s + incre;
     end
-    sigma = diag(Sigma);  r = sum(sigma > tau);
+    sigma = diag(Sigma);  r = sum(sigma &gt; tau);
     U = U(:,1:r);  V = V(:,1:r);  sigma = sigma(1:r) - tau;  Sigma = diag(sigma);
@@ Line 114: / Line 122: @@
     x = A(Omega);
-    if norm(x-b)/normb < tol
+    if norm(x-b)/normb &lt; tol
         break
     end
@@ Line 124: / Line 132: @@
 fprintf('\n');
 numiter = k;
-</pre>
+&lt;/pre&gt;
 == subroutines ==
 === lansvd() ===
-<pre>
+&lt;pre&gt;
 function [U,S,V,bnd,j] = lansvd(varargin)
@@ Line 192: / Line 200: @@
 %%%%%%%%%%%%%%%%%%%%% Parse and check input arguments. %%%%%%%%%%%%%%%%%%%%%%
-if nargin<1 | length(varargin)<1
+if nargin&lt;1 | length(varargin)&lt;1
   error('Not enough input arguments.');
 end
@@ Line 202: / Line 210: @@
   end
   [m n] = size(A);
-  if length(varargin) < 2, k=min(min(m,n),6); else  k=varargin{2}; end
+  if length(varargin) &lt; 2, k=min(min(m,n),6); else  k=varargin{2}; end
-  if length(varargin) < 3, sigma = 'L';       else  sigma=varargin{3}; end
+  if length(varargin) &lt; 3, sigma = 'L';       else  sigma=varargin{3}; end
-  if length(varargin) < 4, options = [];      else  options=varargin{4}; end
+  if length(varargin) &lt; 4, options = [];      else  options=varargin{4}; end
 else
-  if length(varargin)<4
+  if length(varargin)&lt;4
     error('Not enough input arguments.');
   end
@@ Line 215: / Line 223: @@
   m = varargin{3};
   n = varargin{4};
-  if length(varargin) < 5, k=min(min(m,n),6); else k=varargin{5}; end
+  if length(varargin) &lt; 5, k=min(min(m,n),6); else k=varargin{5}; end
-  if length(varargin) < 6, sigma = 'L'; else sigma=varargin{6}; end
+  if length(varargin) &lt; 6, sigma = 'L'; else sigma=varargin{6}; end
-  if length(varargin) < 7, options = []; else options=varargin{7}; end
+  if length(varargin) &lt; 7, options = []; else options=varargin{7}; end
 end
@@ Line 227: / Line 235: @@
 % Quick return for min(m,n) equal to 0 or 1 or for zero A.
-if min(n,m) < 1 | k<1
+if min(n,m) &lt; 1 | k&lt;1
-  if nargout<3
+  if nargout&lt;3
     U = zeros(k,1);
   else
@@ Line 234: / Line 242: @@
   end
   return
-elseif min(n,m) == 1 & k>0
+elseif min(n,m) == 1 &amp; k&gt;0
   if isstr(A)
     % Extract the single column or row of A
@@ Line 255: / Line 263: @@
 if isnumeric(A)
   if  nnz(A)==0
-    if nargout<3
+    if nargout&lt;3
       U = zeros(k,1);
     else
@@ Line 285: / Line 293: @@
 lanmax = min(lanmax,min(m,n));
-if k>lanmax
+if k&gt;lanmax
-  error('K must satisfy  K <= LANMAX <= MIN(M,N).');
+  error('K must satisfy  K &lt;= LANMAX &lt;= MIN(M,N).');
 end
@@ Line 302: / Line 310: @@
       A.A = A;
     end
-    if m>=n
+    if m&gt;=n
       if issparse(A.A)
 	A.R = qr(A.A,0);
@@ Line 313: / Line 321: @@
       end
     else
-      error('Sorry, shift-and-invert for m<n not implemented yet!')
+      error('Sorry, shift-and-invert for m&lt;n not implemented yet!')
       A.R = qr(A.A',0);
       A.Rt = A.R';
     end
     condR = condest(A.R);
-    if condR > 1/eps
+    if condR &gt; 1/eps
       error(['A is rank deficient or too ill-conditioned to do shift-and-' ...
 	     ' invert.'])
@@ Line 330: / Line 338: @@
 U = []; V = []; B = []; anorm = []; work = zeros(2,2);
-while neig < k
+while neig &lt; k
   %%%%%%%%%%%%%%%%%%%%% Compute Lanczos bidiagonalization %%%%%%%%%%%%%%%%%
@@ Line 340: / Line 348: @@
   work= work + w;
-  if ierr<0 % Invariant subspace of dimension -ierr found.
+  if ierr&lt;0 % Invariant subspace of dimension -ierr found.
     j = -ierr;
   end
@@ Line 368: / Line 376: @@
   i=1;
   neig = 0;
-  while i<=min(j,k)
+  while i&lt;=min(j,k)
-    if (bnd(i) <= tol*abs(S(i)))
+    if (bnd(i) &lt;= tol*abs(S(i)))
       neig = neig + 1;
       i = i+1;
@@ Line 378: / Line 386: @@
   %%%%%%%%%% Check whether to stop or to extend the Krylov basis? %%%%%%%%%%
-  if ierr<0 % Invariant subspace found
+  if ierr&lt;0 % Invariant subspace found
-    if j<k
+    if j&lt;k
       warning(['Invariant subspace of dimension ',num2str(j-1),' found.'])
     end
@@ Line 385: / Line 393: @@
     break;
   end
-  if j>=lanmax % Maximal dimension of Krylov subspace reached. Bail out
+  if j&gt;=lanmax % Maximal dimension of Krylov subspace reached. Bail out
-    if j>=min(m,n)
+    if j&gt;=min(m,n)
       neig = ksave;
       break;
     end
-    if neig<ksave
+    if neig&lt;ksave
       warning(['Maximum dimension of Krylov subspace exceeded prior',...
 	    ' to convergence.']);
@@ Line 398: / Line 406: @@
   % Increase dimension of Krylov subspace
-  if neig>0
+  if neig&gt;0
     % increase j by approx. half the average number of steps pr. converged
     % singular value (j/neig) times the number of remaining ones (k-neig).
@@ Line 415: / Line 423: @@
 k = min(ksave,j);
-if nargout>2
+if nargout&gt;2
   j = size(B,2);
   % Compute singular vectors
@@ Line 433: / Line 441: @@
   for i=1:k
     nq = norm(V(:,i));
-    if isfinite(nq) & nq~=0 & nq~=1
+    if isfinite(nq) &amp; nq~=0 &amp; nq~=1
       V(:,i) = V(:,i)/nq;
     end
     nq = norm(U(:,i));
-    if isfinite(nq) & nq~=0 & nq~=1
+    if isfinite(nq) &amp; nq~=0 &amp; nq~=1
       U(:,i) = U(:,i)/nq;
     end
@@ Line 451: / Line 459: @@
   S = -S;
   bnd = bnd(p);
-  if nargout>2
+  if nargout&gt;2
     if issparse(A.A)
       U = A.A*(A.R\U(:,p));
@@ Line 462: / Line 470: @@
 end
-if nargout<3
+if nargout&lt;3
   U = S;
   S = B; % Undocumented feature -  for checking B.
@@ Line 468: / Line 476: @@
   S = diag(S);
 end
-</pre>
+&lt;/pre&gt;
 === bdsqr() ===
-<pre>
+&lt;pre&gt;
 function [sigma,bnd] = bdsqr(alpha,beta)
@@ Line 501: / Line 509: @@
 sigma = diag(S);
 bnd = U(end,1:k)';
-</pre>
+&lt;/pre&gt;
 === compute_int() ===
-<pre>
+&lt;pre&gt;
 function int = compute_int(mu,j,delta,eta,LL,strategy,extra)
 %COMPUTE_INT:  Determine which Lanczos vectors to reorthogonalize against.
@@ Line 511: / Line 519: @@
 %
 %   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.
+%               with nu&gt;eta, where v_{i} are the vectors with  mu&gt;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.
+%               mu &gt; eta.
-%   Strategy 2: Orthogonalize all vectors with mu > eta.
+%   Strategy 2: Orthogonalize all vectors with mu &gt; eta.
 %
 %   Notice: The first LL vectors are excluded since the new Lanczos
@@ Line 522: / Line 530: @@
 %  Rasmus Munk Larsen, DAIMI, 1998.
-if (delta<eta)
+if (delta&lt;eta)
-  error('DELTA should satisfy DELTA >= ETA.')
+  error('DELTA should satisfy DELTA &gt;= ETA.')
 end
 switch strategy
   case 0
-    I0 = find(abs(mu(1:j))>=delta);
+    I0 = find(abs(mu(1:j))&gt;=delta);
     if length(I0)==0
       [mm,I0] = max(abs(mu(1:j)));
@@ Line 534: / Line 542: @@
     for i = 1:length(I0)
       for r=I0(i):-1:1
-	if abs(mu(r))<eta | int(r)==1
+	if abs(mu(r))&lt;eta | int(r)==1
 	  break;
 	else
@@ Line 542: / Line 550: @@
       int(max(1,r-extra+1):r) = 1;
       for s=I0(i)+1:j
-	if abs(mu(s))<eta | int(s)==1
+	if abs(mu(s))&lt;eta | int(s)==1
 	  break;
 	else
@@ Line 550: / Line 558: @@
       int(s:min(j,s+extra-1)) = 1;
     end
-    if LL>0
+    if LL&gt;0
       int(1:LL) = 0;
     end
     int = find(int);
   case 1
-    int=find(abs(mu(1:j))>eta);
+    int=find(abs(mu(1:j))&gt;eta);
     int = max(LL+1,min(int)-extra):min(max(int)+extra,j);
   case 2
-    int=find(abs(mu(1:j))>=eta);
+    int=find(abs(mu(1:j))&gt;=eta);
 end
 int = int(:);
-</pre>
+&lt;/pre&gt;
 === lanbpro() ===
-<pre>
+&lt;pre&gt;
 function [U,B_k,V,p,ierr,work] = lanbpro(varargin)
@@ Line 582: / Line 590: @@
 %   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
+%         MAX(DIAG((EYE(K) - V_K'*V_K))) &lt;= OPTIONS.delta
 %   and
-%         MAX(DIAG((EYE(K) - U_K'*U_K))) <= OPTIONS.delta.
+%         MAX(DIAG((EYE(K) - U_K'*U_K))) &lt;= OPTIONS.delta.
 %
 %   B_k = LANBPRO(...) returns the bidiagonal matrix only.
@@ Line 640: / Line 648: @@
 % R.M. Larsen, Ph.D. Thesis, Aarhus University, 1998.
 %
-% G. H. Golub & C. F. Van Loan, "Matrix Computations",
+% G. H. Golub &amp; C. F. Van Loan, &quot;Matrix Computations&quot;,
 % 3. Ed., Johns Hopkins, 1996.  Section 9.3.4.
 %
@@ Line 662: / Line 670: @@
 end
-if nargin<1 | length(varargin)<2
+if nargin&lt;1 | length(varargin)&lt;2
   error('Not enough input arguments.');
 end
@@ Line 678: / Line 686: @@
   end
   k=varargin{2};
-  if narg >= 3 & ~isempty(varargin{3});
+  if narg &gt;= 3 &amp; ~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 &lt; 4, options = []; else options=varargin{4}; end
-  if narg > 4
+  if narg &gt; 4
-    if narg<7
+    if narg&lt;7
       error('All or none of U_old, B_old and V_old must be provided.')
     else
@@ Line 693: / Line 701: @@
     U = []; B_k = []; V = [];
   end
-  if narg > 7, anorm=varargin{8}; else anorm = []; end
+  if narg &gt; 7, anorm=varargin{8}; else anorm = []; end
 else
-  if narg<5
+  if narg&lt;5
     error('Not enough input arguments.');
   end
@@ Line 708: / Line 716: @@
   end
   k=varargin{5};
-  if narg < 6, p = rand(m,1)-0.5; else p=varargin{6}; end
+  if narg &lt; 6, p = rand(m,1)-0.5; else p=varargin{6}; end
-  if narg < 7, options = []; else options=varargin{7}; end
+  if narg &lt; 7, options = []; else options=varargin{7}; end
-  if narg > 7
+  if narg &gt; 7
-    if  narg < 10
+    if  narg &lt; 10
       error('All or none of U_old, B_old and V_old must be provided.')
     else
@@ Line 719: / Line 727: @@
     U = []; B_k = []; V=[];
   end
-  if narg > 10, anorm=varargin{11}; else anorm = [];  end
+  if narg &gt; 10, anorm=varargin{11}; else anorm = [];  end
 end
@@ Line 732: / Line 740: @@
     U = 1;  B_k = feval(A,1); V = 1; p = 0; ierr = 0; work = zeros(2,2);
   end
-  if nargout<3
+  if nargout&lt;3
     U = B_k;
   end
@@ Line 752: / Line 760: @@
 % Parse options struct
-if ~isempty(options) & isstruct(options)
+if ~isempty(options) &amp; isstruct(options)
   c = fieldnames(options);
   for i=1:length(c)
@@ Line 799: / Line 807: @@
   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
+  alpha(1:j) = diag(B_k); if j&gt;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,
+  if j&lt;k &amp; beta(j+1)*delta &lt; anorm*eps,
     fro = 1;
     ierr = j;
@@ Line 882: / Line 890: @@
     % Extended local reorthogonalization
-    if alpha(j)<gamma*beta(j) & elr & ~fro
+    if alpha(j)&lt;gamma*beta(j) &amp; elr &amp; ~fro
       normold = alpha(j);
       stop = 0;
@@ Line 892: / Line 900: @@
 	  beta(j) = beta(j) + t;
 	end
-	if alpha(j)>=gamma*normold
+	if alpha(j)&gt;=gamma*normold
 	  stop = 1;
 	else
@@ Line 909: / Line 917: @@
     end
-    if ~fro & alpha(j) ~= 0
+    if ~fro &amp; alpha(j) ~= 0
       % Update estimates of the level of orthogonality for the
       %  columns 1 through j-1 in V.
@@ Line 916: / Line 924: @@
     end
-    if j>1 & LANBPRO_TRUTH
+    if j&gt;1 &amp; 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
+    if elr&gt;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.
+    %    Reorthogonalize v_j against some previous  v_i's, 0&lt;=i&lt;j.
-    if onesided~=-1 & ( fro | numax(j) > delta | force_reorth ) & alpha(j)~=0
+    if onesided~=-1 &amp; ( fro | numax(j) &gt; delta | force_reorth ) &amp; alpha(j)~=0
       % Decide which vectors to orthogonalize against:
       if fro | eta==0
@@ Line 955: / Line 963: @@
   % Check for convergence or failure to maintain semiorthogonality
-  if alpha(j) < max(n,m)*anorm*eps & j<k,
+  if alpha(j) &lt; max(n,m)*anorm*eps &amp; j&lt;k,
-    % If alpha is "small" we deflate by setting it
+    % If alpha is &quot;small&quot; 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:
@@ Line 977: / Line 985: @@
       npv = npv + rr*length(int(:));        nreorthv = nreorthv + 1;
       nu(int) = n2*eps;
-      if nrmnew > 0
+      if nrmnew &gt; 0
 	% A vector numerically orthogonal to span(Q_k(:,1:j)) was found.
 	% Continue iteration.
@@ Line 991: / Line 999: @@
       r=r/nrmnew; % Continue with new normalized r as starting vector.
       force_reorth = 1;
-      if delta>0
+      if delta&gt;0
 	fro = 0;    % Turn off full reorthogonalization.
       end
     end
-  elseif  j<k & ~fro & anorm*eps > delta*alpha(j)
+  elseif  j&lt;k &amp; ~fro &amp; anorm*eps &gt; delta*alpha(j)
 %    fro = 1;
     ierr = j;
   end
-  if j>1 & LANBPRO_TRUTH
+  if j&gt;1 &amp; 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);
@@ Line 1,025: / Line 1,033: @@
   beta(j+1) = norm(p);
   % Extended local reorthogonalization
-  if beta(j+1)<gamma*alpha(j) & elr & ~fro
+  if beta(j+1)&lt;gamma*alpha(j) &amp; elr &amp; ~fro
     normold = beta(j+1);
     stop = 0;
@@ Line 1,035: / Line 1,043: @@
 	alpha(j) = alpha(j) + t;
       end
-      if beta(j+1) >= gamma*normold
+      if beta(j+1) &gt;= gamma*normold
 	stop = 1;
       else
@@ Line 1,054: / Line 1,062: @@
   end
-  if ~fro & beta(j+1) ~= 0
+  if ~fro &amp; 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);
@@ Line 1,065: / Line 1,073: @@
   end
-  if elr>0
+  if elr&gt;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.
+  %    Reorthogonalize u_{j+1} against some previous  u_i's, 0&lt;=i&lt;=j.
-  if onesided~=1 & (fro | mumax(j) > delta | force_reorth) & beta(j+1)~=0
+  if onesided~=1 &amp; (fro | mumax(j) &gt; delta | force_reorth) &amp; beta(j+1)~=0
     % Decide which vectors to orthogonalize against.
     if fro | eta==0
@@ Line 1,100: / Line 1,108: @@
   % Check for convergence or failure to maintain semiorthogonality
-  if beta(j+1) < max(m,n)*anorm*eps  & j<k,
+  if beta(j+1) &lt; max(m,n)*anorm*eps  &amp; j&lt;k,
-    % If beta is "small" we deflate by setting it
+    % If beta is &quot;small&quot; 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:
@@ Line 1,122: / Line 1,130: @@
       npu = npu + rr*length(int(:));  nreorthu = nreorthu + 1;
       mu(int) = m2*eps;
-      if nrmnew > 0
+      if nrmnew &gt; 0
 	% A vector numerically orthogonal to span(Q_k(:,1:j)) was found.
 	% Continue iteration.
@@ Line 1,135: / Line 1,143: @@
       p=p/nrmnew; % Continue with new normalized p as starting vector.
       force_reorth = 1;
-      if delta>0
+      if delta&gt;0
 	fro = 0;    % Turn off full reorthogonalization.
       end
     end
-  elseif  j<k & ~fro & anorm*eps > delta*beta(j+1)
+  elseif  j&lt;k &amp; ~fro &amp; anorm*eps &gt; delta*beta(j+1)
 %    fro = 1;
     ierr = j;
@@ Line 1,153: / Line 1,161: @@
 end
-if j<k
+if j&lt;k
   k = j;
 end
@@ Line 1,164: / Line 1,172: @@
   V = V(:,1:k);
 end
-if nargout>5
+if nargout&gt;5
   work = [[nreorthu,npu];[nreorthv,npv]];
 end
@@ Line 1,191: / Line 1,199: @@
   mu(1) = (mu(1) + sign(mu(1))*T) / beta(j+1);
   % Vectorized version of loop:
-  if j>2
+  if j&gt;2
     k=2:j-1;
     mu(k) = alpha(k).*nu(k) + beta(k).*nu(k-1) - alpha(j)*mu(k);
@@ Line 1,219: / Line 1,227: @@
 ainv = 1/alpha(j);
 eps1 = 100*eps/2;
-if j>1
+if j&gt;1
   k = 1:(j-1);
 %  T = eps1*(pythag(alpha(k),beta(k+1)) + pythag(alpha(j),beta(j)));
@@ Line 1,240: / Line 1,248: @@
 [m n] = size(y);
-if m>1 | n>1
+if m&gt;1 | n&gt;1
   y = y(:); z=z(:);
   rmax = max(abs([y z]'))';
   id=find(rmax==0);
-  if length(id)>0
+  if length(id)&gt;0
     rmax(id) = 1;
     x = rmax.*sqrt((y./rmax).^2 + (z./rmax).^2);
@@ Line 1,260: / Line 1,268: @@
   end
 end
-</pre>
+&lt;/pre&gt;
 === refinebounds() ===
-<pre>
+&lt;pre&gt;
 function [bnd,gap] = refinebounds(D,bnd,tol1)
 %REFINEBONDS  Refines error bounds for Ritz values based on gap-structure
@@ Line 1,275: / Line 1,283: @@
 j = length(D);
-if j<=1
+if j&lt;=1
   return
 end
@@ Line 1,288: / Line 1,296: @@
 for l=[-1,1]
   for i=((j+1)-l*(j-1))/2:l:mid-l
-    if abs(D(i+l)-D(i)) < eps34*abs(D(i))
+    if abs(D(i+l)-D(i)) &lt; eps34*abs(D(i))
-      if bnd(i)>tol1 & bnd(i+l)>tol1
+      if bnd(i)&gt;tol1 &amp; bnd(i+l)&gt;tol1
 	bnd(i+l) = pythag(bnd(i),bnd(i+l));
 	bnd(i) = 0;
@@ Line 1,301: / Line 1,309: @@
 gap(2:j) = min([gap(2:j);[D(2:j)-D(1:j-1)-bnd(1:j-1)]']);
 gap = gap(:);
-I = find(gap>bnd);
+I = find(gap&gt;bnd);
 bnd(I) = bnd(I).*(bnd(I)./gap(I));
 bnd(PERM) =  bnd;
-</pre>
+&lt;/pre&gt;
 === reorth() ===
-<pre>
+&lt;pre&gt;
 function [r,normr,nre,s] = reorth(Q,r,normr,index,alpha,method)
@@ Line 1,316: / Line 1,324: @@
 %   reorthogonalizes R against the subset of columns of Q given by INDEX.
 %   If INDEX==[] then R is reorthogonalized all columns of Q.
-%   If the result R_NEW has a small norm, i.e. if norm(R_NEW) < ALPHA*NORMR,
+%   If the result R_NEW has a small norm, i.e. if norm(R_NEW) &lt; ALPHA*NORMR,
 %   then a second reorthogonalization is performed. If the norm of R_NEW
 %   is once more decreased by  more than a factor of ALPHA then R is
@@ Line 1,328: / Line 1,336: @@
 % References:
-%  Aake Bjorck, "Numerical Methods for Least Squares Problems",
+%  Aake Bjorck, &quot;Numerical Methods for Least Squares Problems&quot;,
 %  SIAM, Philadelphia, 1996, pp. 68-69.
 %
@@ Line 1,343: / Line 1,351: @@
 % Check input arguments.
 %warning('PROPACK:NotUsingMex','Using slow matlab code for reorth.')
-if nargin<2
+if nargin&lt;2
   error('Not enough input arguments.')
 end
 [n k1] = size(Q);
-if nargin<3 | isempty(normr)
+if nargin&lt;3 | isempty(normr)
 %  normr = norm(r);
   normr = sqrt(r'*r);
 end
-if nargin<4 | isempty(index)
+if nargin&lt;4 | isempty(index)
   k=k1;
   index = [1:k]';
@@ Line 1,357: / Line 1,365: @@
 else
   k = length(index);
-  if k==k1 & index(:)==[1:k]'
+  if k==k1 &amp; index(:)==[1:k]'
     simple = 1;
   else
@@ Line 1,363: / Line 1,371: @@
   end
 end
-if nargin<5 | isempty(alpha)
+if nargin&lt;5 | isempty(alpha)
   alpha=0.5; % This choice garanties that
-             % || Q^T*r_new - e_{k+1} ||_2 <= 2*eps*||r_new||_2,
+             % || Q^T*r_new - e_{k+1} ||_2 &lt;= 2*eps*||r_new||_2,
              % cf. Kahans ``twice is enough'' statement proved in
              % Parletts book.
 end
-if nargin<6 | isempty(method)
+if nargin&lt;6 | isempty(method)
    method = 0;
 end
@@ Line 1,375: / Line 1,383: @@
   return
 end
-if nargout>3
+if nargout&gt;3
   s = zeros(k,1);
 end
@@ Line 1,382: / Line 1,390: @@
 normr_old = 0;
 nre = 0;
-while normr < alpha*normr_old | nre==0
+while normr &lt; alpha*normr_old | nre==0
   if method==1
     if simple
@@ Line 1,397: / Line 1,405: @@
     end
   end
-  if nargout>3
+  if nargout&gt;3
     s = s + t;
   end
@@ Line 1,404: / Line 1,412: @@
   normr = sqrt(r'*r);
   nre = nre + 1;
-  if nre > 4
+  if nre &gt; 4
-    % r is in span(Q) to full accuracy => accept r = 0 as the new vector.
+    % r is in span(Q) to full accuracy =&gt; accept r = 0 as the new vector.
     r = zeros(n,1);
     normr = 0;
@@ Line 1,411: / Line 1,419: @@
   end
 end
-</pre>
+&lt;/pre&gt;
 === updateSparse.c ===
 A precompiled [http://convexoptimization.com/TOOLS/updateSparse.mexw32 Intel Windows-32bit Matlab mex file is here].
-Otherwise, compile this C program in Matlab via command <pre>mex updateSparse.c</pre>
+Otherwise, compile this C program in Matlab via command &lt;pre&gt;mex updateSparse.c&lt;/pre&gt;
-<pre>
+&lt;pre&gt;
 /*
  * Stephen Becker, 11/10/08
@@ Line 1,431: / Line 1,439: @@
  * */
-#include "mex.h"
+#include &quot;mex.h&quot;
 #ifndef true
     #define true 1
@@ Line 1,440: / Line 1,448: @@
 void printUsage() {
-    mexPrintf("usage:\tupdateSparse(Y,b)\nchanges the sparse Y matrix");
+    mexPrintf(&quot;usage:\tupdateSparse(Y,b)\nchanges the sparse Y matrix&quot;);
-    mexPrintf(" to have values b\non its nonzero elements.  Be careful:\n\t");
+    mexPrintf(&quot; to have values b\non its nonzero elements.  Be careful:\n\t&quot;);
-    mexPrintf("this assumes b is sorted in the appropriate order!\n");
+    mexPrintf(&quot;this assumes b is sorted in the appropriate order!\n&quot;);
-    mexPrintf("If b (i.e. the index omega, where we want to perform Y(omega)=b)\n");
+    mexPrintf(&quot;If b (i.e. the index omega, where we want to perform Y(omega)=b)\n&quot;);
-    mexPrintf("  is unsorted, then call the command as follows:\n");
+    mexPrintf(&quot;  is unsorted, then call the command as follows:\n&quot;);
-    mexPrintf("\tupdateSparse(Y,b,omegaIndx)\n");
+    mexPrintf(&quot;\tupdateSparse(Y,b,omegaIndx)\n&quot;);
-    mexPrintf("where [temp,omegaIndx] = sort(omega)\n");
+    mexPrintf(&quot;where [temp,omegaIndx] = sort(omega)\n&quot;);
 }
@@ Line 1,460: / Line 1,468: @@
     /* Check for proper number of input and output arguments */
-    if ( (nrhs < 2) || (nrhs > 3) )  {
+    if ( (nrhs &lt; 2) || (nrhs &gt; 3) )  {
         printUsage();
-        mexErrMsgTxt("Needs 2 or 3 input arguments");
+        mexErrMsgTxt(&quot;Needs 2 or 3 input arguments&quot;);
     }
     if ( nrhs == 3 ) SORTED = false;
-    if(nlhs > 0){
+    if(nlhs &gt; 0){
         printUsage();
-        mexErrMsgTxt("No output arguments!");
+        mexErrMsgTxt(&quot;No output arguments!&quot;);
     }
@@ Line 1,473: / Line 1,481: @@
     if (!(mxIsSparse(prhs[0])) || !((mxIsDouble(prhs[1]))) ){
         printUsage();
-        mexErrMsgTxt("Input arguments wrong data-type (must be sparse, double).");
+        mexErrMsgTxt(&quot;Input arguments wrong data-type (must be sparse, double).&quot;);
     }
@@ Line 1,480: / Line 1,488: @@
     N = mxGetN( prhs[1] );
     M = mxGetM( prhs[1] );
-    if ( (M>1) && (N>1) ) {
+    if ( (M&gt;1) &amp;&amp; (N&gt;1) ) {
         printUsage();
-        mexErrMsgTxt("Second argument must be a vector");
+        mexErrMsgTxt(&quot;Second argument must be a vector&quot;);
     }
-    N = (N>M) ? N : M;
+    N = (N&gt;M) ? N : M;
@@ Line 1,491: / Line 1,499: @@
     if ( M != N ) {
         printUsage();
-        mexErrMsgTxt("Length of second argument must match nnz of first argument");
+        mexErrMsgTxt(&quot;Length of second argument must match nnz of first argument&quot;);
     }
@@ Line 1,498: / Line 1,506: @@
        m = mxGetM( prhs[2] );
        n = mxGetN( prhs[2] );
-       if ( (m>1) && (n>1) ) {
+       if ( (m&gt;1) &amp;&amp; (n&gt;1) ) {
            printUsage();
-           mexErrMsgTxt("Third argument must be a vector");
+           mexErrMsgTxt(&quot;Third argument must be a vector&quot;);
        }
-       n = (n>m) ? n : m;
+       n = (n&gt;m) ? n : m;
        if ( n != N ) {
            printUsage();
-           mexErrMsgTxt("Third argument must be same length as second argument");
+           mexErrMsgTxt(&quot;Third argument must be same length as second argument&quot;);
        }
        omega = mxGetPr( prhs[2] );
@@ Line 1,516: / Line 1,524: @@
     if (SORTED) {
         /* And here's the really fancy part:  */
-        for ( i=0 ; i < N ; i++ )
+        for ( i=0 ; i &lt; N ; i++ )
             S[i] = b[i];
     } else {
-        for ( i=0 ; i < N ; i++ ) {
+        for ( i=0 ; i &lt; N ; i++ ) {
             /* this is a little slow, but I should really check
              * to make sure the index is not out-of-bounds, otherwise
              * Matlab could crash */
             j = (int)omega[i]-1; /* the -1 because Matlab is 1-based */
-            if (j >= N){
+            if (j &gt;= N){
                 printUsage();
-                mexErrMsgTxt("Third argument must have values < length of 2nd argument");
+                mexErrMsgTxt(&quot;Third argument must have values &lt; length of 2nd argument&quot;);
             }
 /*             S[ j ] = b[i]; */  /* this is incorrect */
@@ Line 1,533: / Line 1,541: @@
     }
 }
-</pre>
+&lt;/pre&gt;