Regularization Tools. Version 2.0 16-March-93. Copyright (c) 1993 by Per Christian Hansen and UNI-C. Demonstration. regudemo - Tutorial introduction to Regularization Tools. Test examples. baart - Fredholm integral equation of the first kind. 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 a damped SVD/GSVD solution. lsqi - Minimizes the residual norm subject to an upper bound on the (semi-)norm of the solution. lsqr - 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 - 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||^, and its derivative. picard - Plots the (generalized) singular values, the Fourier coefficient for the right-hand side, and a (smoothed curve of) the solution 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. bsvd - Computes the singular values, or the compact SVD, of a bidiagonal matrix stored in compact form. csd - Computes the CS decomposition. 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. gsvd - Computes the generalized SVD of a matrix pair. lanc_b - Performs k steps of the Lanczos bidiagonalization process with/without reorthogonalization. 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. heb_new2 - Ditto, used in discrep. lsolve - Inversion with A-weighted generalized inverse of L. ltsolve - Inversion with transposed A-weighted inverse of L. mgs - Modified Gram-Schmidt orthonormalization. newton - Newton-Raphson 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 routine l_corner requires the following routines from the Spline Toolbox: fnder, ppbrk, ppcut, ppmak, sp2pp, spbrk, spmak. If the Spline Toolbox is not available, then dummy functions with these names should reside in the same directory as Regularization Tools.

- bst_splashscreen C:\BrainStorm_2001\Toolbox\bst_splashscreen.m
- bst_static_taskbar C:\BrainStorm_2001\Toolbox\bst_static_taskbar.m
- help_manager C:\BrainStorm_2001\Toolbox\help_manager.m

% Regularization Tools. % Version 2.0 16-March-93. % Copyright (c) 1993 by Per Christian Hansen and UNI-C. % % Demonstration. % regudemo - Tutorial introduction to Regularization Tools. % % Test examples. % baart - Fredholm integral equation of the first kind. % 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 a damped SVD/GSVD solution. % lsqi - Minimizes the residual norm subject to an upper bound % on the (semi-)norm of the solution. % lsqr - 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 - 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||^, % and its derivative. % picard - Plots the (generalized) singular values, % the Fourier coefficient for the right-hand side, and a % (smoothed curve of) the solution 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. % bsvd - Computes the singular values, or the compact SVD, % of a bidiagonal matrix stored in compact form. % csd - Computes the CS decomposition. % 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. % gsvd - Computes the generalized SVD of a matrix pair. % lanc_b - Performs k steps of the Lanczos bidiagonalization % process with/without reorthogonalization. % % 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. % heb_new2 - Ditto, used in discrep. % lsolve - Inversion with A-weighted generalized inverse of L. % ltsolve - Inversion with transposed A-weighted inverse of L. % mgs - Modified Gram-Schmidt orthonormalization. % newton - Newton-Raphson 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 routine l_corner requires the following routines from the Spline % Toolbox: % fnder, ppbrk, ppcut, ppmak, sp2pp, spbrk, spmak. % If the Spline Toolbox is not available, then dummy functions with these % names should reside in the same directory as Regularization Tools.

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