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UD Factorization & Kalman Filtering

UD Factorization & Kalman Filtering

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15 Aug 2011 (Updated )

UD and LD factorization of nonnegative matrices and associated Kalman filter implementations.

[ldi,r]=ldinv(ld,tolp,toln)
% LDINV : inverse of unit lower triangular l and diagonal d
%
%         function [ldi,r]=ldinv(ld,tolp,toln)
%
%         ld  : LDL factorization
%         tolp: tolerance for positiveness default 1e-12, see also toln
%         toln: error generation if diagonal part < toln, default: tolp
%               if toln < 0 all the diagonal parts < max(0,tolp) are set to zero
%
%         ldi : inv(ld), d on diagonal
%         r   : rank(d)
%
%         Used to compute : inv(p)=inv(l)'*inv(d)*inv(l)
%         [ldi]=udinv(ld) computes inv(l),inv(d)
%         Then: inv(p)=udt2sym(ldinv(sym2ud(p)))
%
%         See also ldt2sym, ld2sym, sym2ld
%
%         References: Factorization methods for discrete sequential estimation
%                     1977, Gerald J. Bierman
%
% L.G. van Willigenburg, W.L. de Koning, Update August 2011

  function [ldi,r]=ldinv(ld,tolp,toln)

  if nargin > 3; error('  one, two or three input arguments required'); end;
  if nargin==1; tolp=1e-12; toln=tolp;
  elseif nargin==2; toln=tolp; end; tolp=max(0,tolp); 
  if (toln>tolp); error('  toln > tolp'); end;

  [n,m]=size(ld);
  if n~=m; error('  ld must be square'); end;
  if n==0; error('  Compatible but empty inputs'); end;

  r=n; ldi=zeros(n);
  for i=1:n
    if ld(i,i)<toln; error('  toln violated');
    elseif ld(i,i)<=tolp; ldi(i,i)=0; r=r-1;
    else ldi(i,i)=1/ld(i,i); end
  end

  for j=n-1:-1:1
    jp1=j+1;
    for k=n:-1:jp1
      sum=-ld(k,j);
      for i=k-1:-1:jp1
        sum=sum-ldi(k,i)*ld(i,j);
      end
      ldi(k,j)=sum;
    end
  end

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