16 Apr 1998
18 Mar 2008)
Analysis and Solution of Discrete Ill-Posed Problems.
function x = gen_form(L_p,x_s,A,b,K,M)
%GEN_FORM Transform a standard-form problem back to the general-form setting.
% x = gen_form(L_p,x_s,A,b,K,M) (method 1)
% x = gen_form(L_p,x_s,x_0) (method 2)
% Transforms the standard-form solution x_s back to the required
% solution to the general-form problem:
% x = L_p*x_s + d ,
% where L_p and d depend on the method as follows:
% method = 1: L_p = pseudoinverse of L, d = K*(b - A*L_p*x_s)
% method = 2: L_p = A-weighted pseudoinverse of L, d = x_0.
% Usually, the standard-form problem is generated by means of
% function std_form.
% Note that x_s may have more that one column.
% References: L. Elden, "Algorithms for regularization of ill-
% conditioned least-squares problems", BIT 17 (1977), 134-145.
% L. Elden, "A weighted pseudoinverse, generalized singular values,
% and constrained lest squares problems", BIT 22 (1982), 487-502.
% M. Hanke, "Regularization with differential operators. An itera-
% tive approach", J. Numer. Funct. Anal. Optim. 13 (1992), 523-540.
% Per Christian Hansen, IMM, 06/12/93.
% Nargin determines which method.
[p,q] = size(x_s); Km = size(K,1);
x = L_p*x_s;
x = L_p*x_s + K*(M*(b*ones(1,q) - A*(L_p*x_s)));
x_0 = A; [p,q] = size(x_s);
x = L_p*x_s + x_0*ones(1,q);