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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.

app_hh APP_HH Apply a Householder transformation. baart BAART Test problem: Fredholm integral equation of the first kind. bidiag BIDIAG Bidiagonalization of an m-times-n matrix with m >= n. bsvd BSVD SVD of a bidiagonal matrix stored in "compact form". cgls CGLS Conjugate gradient algorithm applied implicitly to the normal equations. Contents Regularization Tools. csd CSD CS decomposition. csvd CSVD Compact singular value decomposition. deriv2 DERIV2 Test problem: computation of the second derivative. discrep DISCREP Discrepancy principle criterion for choosing the reg. parameter. dsvd DSVD Damped SVD regularization. fil_fac FIL_FAC Filter factors for some regularization methods. fnder FNDER Dummy function for Regularization Tools foxgood FOXGOOD Severely ill-posed testproblem. gcv GCV Plot the GCV function and find its minimum. gen_form GEN_FORM Transform a standard-form problem back to the general-form setting. gen_hh GEN_HH Generate a Householder transformation. get_l GET_L Compute discrete derivative operators. gsvd GSVD Generalized SVD of a matrix pair. heat HEAT Test problem: inverse heat equation. heb_new HEB_NEW Newton iteration with Hebden model (utility routine for LSQI). ilaplace ILAPLACE Test problem: inverse Laplace transformation. lagrange LAGRANGE Plot the Lagrange function for Tikhonov regularization. lanc_b LANC_B Lanczos bidiagonalization. lsolve LSOLVE Utility routine for "preconditioned" iterative methods. lsqi LSQI Least squares minimizaiton with a quadratic inequality constraint. lsqr LSQR Solution of least squares problems by Lanczos bidiagonalization. ltsolve LTSOLVE Utility routine for "preconditioned" iterative methods. l_corner L_CORNER Locate the "corner" of the L-curve. l_curve L_CURVE Plot the L-curve and find its "corner". maxent MAXENT Maximum entropy regularization. mgs MGS Modified Gram-Schmidt orthonormalization. mtsvd MTSVD Modified truncated SVD regularization. newton NEWTON Newton iteration (utility routine for DISCREP). nu NU Brakhage's nu-method. parallax PARALLAX Stellar parallax problem with 28 fixed, real observations. pcgls PCGLS "Preconditioned" conjugate gradients appl. implicitly to normal equations. phillips PHILLIPS Phillips' "famous" test problem. picard PICARD Visual inspection of the Picard condition. pinit PINIT Utility init.-procedure for "preconditioned" iterative methods. plot_lc PLOT_LC Plot the L-curve. plsqr PLSQR "Preconditioned" version of the LSQR Lanczos bidiagonalization algorithm. pnu PNU "Preconditioned" version of Brakhage's nu-method. ppbrk PPBRK Dummy function for Regularization Tools ppcut PPCUT Dummy function for Regularization Tools ppmak PPMAK Dummy function for Regularization Tools ppual PPUAL Dummy function for Regularization Tools pythag PYTHAG Computes sqrt( y^2 + z^2 ). quasiopt QUASIOPT Quasi-optimality criterion for choosing the regularization parameter. regudemo REGUDEMO Tutorial script for Regularization Tools. shaw SHAW Test problem: one-dimensional iimage restoration model. sorted SORTED Dummy function for Regularization Tools sp2pp SP2PP Dummy function for Regularization Tools spbrk SPBRK Dummy function for Regularization Tools spikes SPIKES Test problem with a "spiky" solution. spleval SPLEVAL Evaluation of a spline or spline curve. spmak SPMAK Dummy function for Regularization Tools sprpp SPRPP Dummy function for Regularization Tools std_form STD_FORM Transform a general-form reg. problem into one in standard form. tgsvd TGSVD Truncated GSVD regularization. tikhonov TIKHONOV Tikhonov regularization. tsvd TSVD Truncated SVD regularization. ttls TTLS Truncated TLS regularization. ursell URSELL Test problem: integral equation wiht no square integrable solution. wing WING Test problem with a discontinuous solution.