% Regularization Tools. % Version 3.1  13-September-01. % Copyright (c) 1993 and 1998 by Per Christian Hansen and IMM. % % Demonstration. %   regudemo  - Tutorial introduction to Regularization Tools. % % Test examples. %   baart     - Fredholm integral equation of the first kind. %   blur      - Image deblurring test problem. %   deriv2    - Computation of the second derivative. %   foxgood   - Severely ill-posed problem. %   heat      - Inverse heat equation. %   ilaplace  - Inverse Laplace transformation. %   parallax  - Stellar parallax problem with 28 fixed observations. %   phillips  - Philips' "famous" test problem. %   shaw      - One-dimensional image restoration problem. %   spikes    - Test problem with a "spiky" solution. %   ursell    - Integral equation with no square integrable solution. %   wing      - Test problem with a discontinuous solution. % % Regularization routines. %   cgls      - Computes the least squares solution based on k steps %               of the conjugate gradient algorithm. %   discrep   - Minimizes the solution (semi-)norm subject to an upper %               bound on the residual norm (discrepancy principle). %   dsvd      - Computes the damped SVD/GSVD solution. %   lsqi      - Minimizes the residual norm subject to an upper bound %               on the (semi-)norm of the solution. %   lsqr_b    - Computes the least squares solution based on k steps %               of the LSQR algorithm (Lanczos bidiagonalization). %   maxent    - Computes the maximum entropy regularized solution. %   mtsvd     - Computes the modified TSVD solution. %   nu        - Computes the solution based on k steps of Brakhage's %               iterative nu-method. %   pcgls     - Same as cgls, but for general-form regularization. %   plsqr_b   - Same as lsqr, but for general-form regularization. %   pnu       - Same as nu, but for general-form regularization. %   tgsvd     - Computes the truncated GSVD solution. %   tikhonov  - Computes the Tikhonov regularized solution. %   tsvd      - Computes the truncated SVD solution. %   ttls      - Computes the truncated TLS solution. % % Analysis routines. %   fil_fac   - Computes filter factors for some regularization methods. %   gcv       - Plots the GCV function and computes its minimum. %   l_corner  - Locates the L-shaped corner of the L-curve. %   l_curve   - Computes the L-curve, plots it, and computes its corner. %   lagrange  - Plots the Lagrange function ||Ax-b||^2 + lambda^2*||Lx||^2, %               and its derivative. %   picard    - Plots the (generalized) singular values, the Fourier %               coefficient for the right-hand side, and a (smoothed curve of) %               the solution's Fourier-coefficients. %   plot_lc   - Plots an L-curve. %   quasiopt  - Plots the quasi-optimality function and computes its minimum. % % Routines for transforming a problem in general form into one in % standard form, and back again. %   gen_form  - Transforms a standard-form solution back into the %               general-form setting. %   std_form  - Transforms a general-form problem into one in %               standard form. % % Utility routines. %   bidiag    - Bidiagonalization of a matrix by Householder transformations. %   cgsvd     - Computes the compact generalized SVD of a matrix pair. %   csvd      - Computes the compact SVD of an m-by-n matrix. %   get_l     - Produces a p-by-n matrix which is the discrete %               approximation to the d'th order derivative operator. %   lanc_b    - Performs k steps of the Lanczos bidiagonalization %               process with/without reorthogonalization. %   regutm    - Generates random test matrices for regularization methods. % % Auxiliary routines required by some of the above routines. %   app_hh_l  - Applies a Householder transformation from the left. %   gen_hh    - Generates a Householder transformation. %   heb_new   - Newton-Raphson iteration with Hebden's rational %               approximation, used in lsqi. %   lsolve    - Inversion with A-weighted generalized inverse of L. %   ltsolve   - Inversion with transposed A-weighted inverse of L. %   newton    - Newton iteration, used in discrep. %   pinit     - Initialization for treating general-form problems. %   pythag    - Computes sqrt(a^2 + b^2). %   spleval   - Computes points on a spline or spline curve.  % The following three routines are not documented, since they are only used% internally by gcv, l_corner, and quasiopt, respectively.%   gcvfun    - Computes the GCV function%   lcfun     - Computes the curvature of the L-curve%   quasifun  - Computes the quasi-optimality function.%% In certain occations (cf. the manual), the function l_corner requires the % following nine functions from the Spline Toolbox, Version 2.0: %   fnder, ppbrk, ppmak, ppual, sp2pp, sorted, spbrk, spmak, sprpp. % If the Spline Toolbox is not available, then the package can still be used % with almost full functionality, the only exception being that one is not % able to compute the "corner" of discrete L-curves.