levrec.m

地球物理反演程序

function x=levrec(aa,b);
% LEVREC: solve Tx=b using Levinson's recursion
%
% x=levrec(aa,b) 
%
% This function solves the matrix equation Tx=b for the vector
% x using Levinson recursion. The symmetric Toeplitz matrix T is
% assumed specified completely by the autocorrelation vector
% aa.
%        ( aa(1) aa(2) aa(3) aa(4) ...
%        ( aa(2) aa(1) aa(2) aa(3) aa(4) ... 
%     T= ( aa(3) aa(2) aa(1) aa(2) aa(3) aa(4) ...
%        ( aa(4) aa(3) aa(2) aa(1) aa(2) aa(3) aa(4) ...  
%        ( .............................................
% aa= input autocorrelation vector. If not normalized to aa(1)=1.0
%      then it will be
% b= input rhs vector (b will be converted to a column vector if
%     it is not already) 
% x= solution vector (column vector)
%
% The algorithm is taken from "Matrix Computations" by Golub and
% Van Loan, Second Edition, Johns Hopkins University Press, 1989
% (see page 187)
%
% Implemented by G.F. Margrave, May 1991  
%
% NOTE: It is illegal for you to use this software for a purpose other
% than non-profit education or research UNLESS you are employed by a CREWES
% Project sponsor. By using this software, you are agreeing to the terms
% detailed in this software's Matlab source file.
 
% BEGIN TERMS OF USE LICENSE
%
% This SOFTWARE is maintained by the CREWES Project at the Department
% of Geology and Geophysics of the University of Calgary, Calgary,
% Alberta, Canada.  The copyright and ownership is jointly held by 
% its author (identified above) and the CREWES Project.  The CREWES 
% project may be contacted via email at:  [e-mail]
% 
% The term 'SOFTWARE' refers to the Matlab source code, translations to
% any other computer language, or object code
%
% Terms of use of this SOFTWARE
%
% 1) Use of this SOFTWARE by any for-profit commercial organization is
%    expressly forbidden unless said organization is a CREWES Project
%    Sponsor.
%
% 2) A CREWES Project sponsor may use this SOFTWARE under the terms of the 
%    CREWES Project Sponsorship agreement.
%
% 3) A student or employee of a non-profit educational institution may 
%    use this SOFTWARE subject to the following terms and conditions:
%    - this SOFTWARE is for teaching or research purposes only.
%    - this SOFTWARE may be distributed to other students or researchers 
%      provided that these license terms are included.
%    - reselling the SOFTWARE, or including it or any portion of it, in any
%      software that will be resold is expressly forbidden.
%    - transfering the SOFTWARE in any form to a commercial firm or any 
%      other for-profit organization is expressly forbidden.
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% END TERMS OF USE LICENSE

% convert aa and b to column vectors if needed
 [l,m]=size(b);
  if l<m
    b=b';
  end
 [l,m]=size(aa);
  if l<m
    aa=aa';
  end
% normalize aa
 if aa(1)~= 1.0
   aa=aa/max(aa);
 end
% check for valid autocorr
 if aa(1)~= max(aa)
   error(' Invalid autocorrelation: zero lag not maximum')
 end
% initialize some stuff
 a=aa(2:length(aa));
 n=length(b);
 y=zeros(size(a));
 x=zeros(size(b));
 z=zeros(size(a)); 
 y(1)=-a(1);
 x(1)=b(1);
 beta=1;
 alpha=-a(1);
% main recursion loop
 for k=1:n-1
	beta=(1-alpha^2)*beta;
	mu=(b(k+1)-a(1:k)'*x(k:-1:1))/beta;
	nu(1:k)=x(1:k)+mu*y(k:-1:1);
	x(1:k)=nu(1:k);
   	x(k+1)=mu;
   	if k<(n-1)
     	alpha=-(a(k+1)+a(1:k)'*y(k:-1:1))/beta;
     	z(1:k)=y(1:k)+alpha*y(k:-1:1);
     	y(1:k)=z(1:k);
     	y(k+1)=alpha;
    end
 end