#! /bin/sh # This is a shell archive. Remove anything before this line, then unpack # it by saving it into a file and typing "sh file". To overwrite existing # files, type "sh file -c". You can also feed this as standard input via # unshar, or by typing "sh 'AREADME.1ST' <<'END_OF_FILE' X *************************************************************************** X * All the software contained in this library is protected by copyright. * X * Permission to use, copy, modify, and distribute this software for any * X * purpose without fee is hereby granted, provided that this entire notice * X * is included in all copies of any software which is or includes a copy * X * or modification of this software and in all copies of the supporting * X * documentation for such software. * X *************************************************************************** X * THIS SOFTWARE IS BEING PROVIDED "AS IS", WITHOUT ANY EXPRESS OR IMPLIED * X * WARRANTY. IN NO EVENT, NEITHER THE AUTHORS, NOR THE PUBLISHER, NOR ANY * X * MEMBER OF THE EDITORIAL BOARD OF THE JOURNAL "NUMERICAL ALGORITHMS", * X * NOR ITS EDITOR-IN-CHIEF, BE LIABLE FOR ANY ERROR IN THE SOFTWARE, ANY * X * MISUSE OF IT OR ANY DAMAGE ARISING OUT OF ITS USE. THE ENTIRE RISK OF * X * USING THE SOFTWARE LIES WITH THE PARTY DOING SO. * X *************************************************************************** X * ANY USE OF THE SOFTWARE CONSTITUTES ACCEPTANCE OF THE TERMS OF THE * X * ABOVE STATEMENT. * X *************************************************************************** X X AUTHOR: X X P. C. HANSEN X DEPT. OF MATHEMATICAL MODELLING X TECHNICAL UNIVERSITY OF DENMARK X X REFERENCE: X X REGULARIZATION TOOLS: A MATLAB PACKAGE FOR ANALYSIS AND SOLUTION OF X DISCRETE ILL-POSED PROBLEMS, X NUMERICAL ALGORITHMS, 6 (1994), PP. 1-35 X X SOFTWARE REVISION: X X Ver 3.0 APRIL 16, 1998 X X SOFTWARE LANGUAGE: X X MATLAB 5 X X************************************************************************** X XRegularization Tools. XVersion 3.0 16-April-98. XCopyright (c) 1998 by Per Christian Hansen. X XThe installation of Regularization Tools is very simple: X X 1. Unpack the shell archive NA4 by executing the command X /bin/sh na4 X X 2. Remove the file na4 X X 3. The file Manual.ps contains the related manual in PostScript form X X*************************************************************** X* This is Version 3.0 of Regularization Tools for Matlab 5.2 * X*-------------------------------------------------------------* X* Per Christian Hansen, IMM * X*************************************************************** X X02/01/94: XFixed bug in cgls (s -> s2). X X08/03/94: XExpanded stopping criterion in newton. X X08/09/94: XRevised comment lines in maxent. X X10/07/94: XRemoved superfluorus statements in mtsvd. X X11/01/94: XModified get_l slightly such that the sign of L*x is correct. X X02/09/95: XRevised qr in csd, l_curve and mtsvd to compute "economy size" decomposition. XRenamed csd to csdecomp (csd is now a function in the Signal Proc. Toolbox). XRevised gsvd to call csdecomp. X X11/08/95: XFixed bug in csdecomp when p=1. X X03/22/96: XChanged tsvd and tgsvd to allow k=0. X X10/08/96: XChanged tgsvd to allow a square L. X X10/22/96: XChanged tikhonov to allow a square L. X X04/17/97: XReplaced (..==NaN) with isnan(..) in bsvd. XAdded initialization of U2t in csdecomp. X X04/21/97: XChanged variable name "case" to "example" in deriv2. XChanged meshdom to meshgrid in spikes, and deleted the flipud command. XChanged variable xi to eta in picard. X X06/30/97: XRemoved function bsvd (obsolete with sparse format of bidiagonal matrices). XChanged to sparse format of bidiagonal matrix in bidiag. XChanged to sparse format of bidiagonal matrix in lanc_b. XAdded function regutm. X X07/02/97: XAdded reorthogonalization of normal eq. residual vectors to cgls and pcgls. XFixed bug in pcgls when computing filter factors. X X07/29/97: XChanged variable name in pinit. XModified lsolve, ltsolve, and std_form according to simpler formulas. X X09/18/97: XAdded blur test problem. XDeleted mgs, and included the MGS process in get_l. X X11/11/97: XModified gen_hh to compensate for Matlab's signum function. X X12/22/97: XReplaced gsvd with cgsvd, and deleted csdecomp. XAdded more output arguments to dsvd, mtsvd, tgsvd, tikhonov, and tsvd. XAdded method = 'ttls' to fil_fac. XImproved the plots in gcv, lagrange, picard, plot_lc, and quasiopt. XAdded input parameter x_0 to tikhonov. X X12/29/97: XAdded call to fmin in gcv, l_curve, and quasiopt. XCorrected bugs in discrep and lsqi. XModified heb_new and newton to work with lambda instead of lambda squared. X X02/05/98: XAdded d==0 to get_l. X X04/16/98: XModified l_corner and spleval to be consistent with Spline Toolbox v. 2.0. X END_OF_FILE if test 4627 -ne `wc -c <'AREADME.1ST'`; then echo shar: \"'AREADME.1ST'\" unpacked with wrong size! fi # end of 'AREADME.1ST' fi if test -f 'Manual.ps' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'Manual.ps'\" else echo shar: Extracting \"'Manual.ps'\" \(2119016 characters\) sed "s/^X//" >'Manual.ps' <<'END_OF_FILE' X%!PS-Adobe-2.0 X%%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software X%%Title: book.dvi X%%Pages: 111 X%%PageOrder: Ascend X%%BoundingBox: 0 0 596 842 X%%EndComments X%DVIPSCommandLine: dvips book X%DVIPSParameters: dpi=300, compressed, comments removed X%DVIPSSource: TeX output 1998.03.26:0819 X%%BeginProcSet: texc.pro X/TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N X/X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 Xmul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} Xifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale 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X%%Page: 4 6 X4 5 bop 64 159 a Fo(4)1486 b(CONTENTS)p 64 178 1767 2 Xv eop X%%Page: 5 7 X5 6 bop 59 546 a Fq(1.)35 b(Intr)n(oduction)59 752 y XFo(Ill-p)q(osed)17 b(problems|and)f(regularization)g(metho)q(ds)f(for)f X(computing)i(stabilized)h(solutions)e(to)g(the)59 809 Xy(ill-p)q(osed)i(problems|o)q(ccur)f(frequen)o(tly)e(enough)h(in)g X(science)h(and)e(engineering)i(to)d(mak)o(e)h(it)h(w)o(orth-)59 X865 y(while)j(to)d(presen)o(t)i(a)f(general)g(framew)o(ork)f(for)h X(their)h(n)o(umerical)g(treatmen)o(t.)22 b(The)16 b(purp)q(ose)h(of)f X(this)59 922 y(pac)o(k)m(age)f(of)g(Matlab)g(routines)h(is)g(to)e(pro)o X(vide)i(the)g(user)f(with)h(easy-to-use)f(routines,)g(based)h(on)f(n)o X(u-)59 978 y(merically)f(robust)f(and)g(e\016cien)o(t)g(algorithms,)g X(for)f(doing)h(exp)q(erimen)o(ts)h(with)f(analysis)g(and)g(solution)59 X1034 y(of)i(discrete)h(ill-p)q(osed)i(problems)e(b)o(y)f(means)g(of)g X(regularization)h(metho)q(ds.)130 1091 y(The)i(theory)h(for)f(ill-p)q X(osed)j(problems)e(is)g(w)o(ell)g(dev)o(elop)q(ed)i(in)e(the)g X(literature.)30 b(W)l(e)19 b(can)g(easily)59 1147 y(illustrate)12 Xb(the)g(main)f(di\016culties)j(asso)q(ciated)d(with)h(suc)o(h)f X(problems)h(b)o(y)g(means)f(of)f(a)h(small)h(n)o(umerical)59 X1204 y(example.)21 b(Consider)16 b(the)f(follo)o(wing)h(least)f X(squares)g(problem)794 1306 y(min)821 1330 y Fi(x)877 X1306 y Fm(k)p Fn(A)8 b Fp(x)h Fm(\000)i Fp(b)p Fm(k)1077 X1313 y Fj(2)59 1416 y Fo(with)16 b(co)q(e\016cien)o(t)g(matrix)f XFn(A)g Fo(and)g(righ)o(t-hand)h(side)g Fp(b)f Fo(giv)o(en)h(b)o(y)544 X1565 y Fn(A)d Fo(=)639 1480 y Fh(0)639 1555 y(@)683 1508 Xy Fo(0)p Fn(:)p Fo(16)44 b(0)p Fn(:)p Fo(10)683 1565 Xy(0)p Fn(:)p Fo(17)g(0)p Fn(:)p Fo(11)683 1621 y(2)p XFn(:)p Fo(02)g(1)p Fn(:)p Fo(29)897 1480 y Fh(1)897 1555 Xy(A)941 1565 y Fn(;)98 b Fp(b)13 b Fo(=)1142 1480 y Fh(0)1142 X1555 y(@)1186 1508 y Fo(0)p Fn(:)p Fo(27)1186 1565 y(0)p XFn(:)p Fo(25)1186 1621 y(3)p Fn(:)p Fo(33)1274 1480 y XFh(1)1274 1555 y(A)1333 1565 y Fn(:)59 1713 y Fo(Here,)h(the)g(righ)o X(t-hand)g(side)g Fp(b)g Fo(is)g(generated)g(b)o(y)f(adding)i(a)e(small) Xi(p)q(erturbation)f(to)f(an)g(exact)h(righ)o(t-)59 1770 Xy(hand)i(side)g(corresp)q(onding)g(to)f(the)g(exact)g(solution)984 X1769 y(\026)982 1770 y Fp(x)1010 1753 y Fg(T)1050 1770 Xy Fo(=)d(\(1)j(1\):)523 1921 y Fp(b)e Fo(=)613 1837 y XFh(0)613 1912 y(@)656 1865 y Fo(0)p Fn(:)p Fo(16)45 b(0)p XFn(:)p Fo(10)656 1921 y(0)p Fn(:)p Fo(17)g(0)p Fn(:)p XFo(11)656 1978 y(2)p Fn(:)p Fo(02)g(1)p Fn(:)p Fo(29)871 X1837 y Fh(1)871 1912 y(A)915 1862 y(\022)953 1893 y Fo(1)p XFn(:)p Fo(00)953 1949 y(1)p Fn(:)p Fo(00)1041 1862 y XFh(\023)1082 1921 y Fo(+)1128 1837 y Fh(0)1128 1912 y(@)1207 X1865 y Fo(0)p Fn(:)p Fo(01)1171 1921 y Fm(\000)p Fo(0)p XFn(:)p Fo(03)1207 1978 y(0)p Fn(:)p Fo(02)1295 1837 y XFh(1)1295 1912 y(A)1354 1921 y Fn(:)59 2070 y Fo(The)17 Xb(di\016cult)o(y)i(with)e(this)h(least)f(squares)g(problem)h(is)g(that) Xe(the)h(matrix)g Fn(A)g Fo(is)h(ill-conditione)q(d;)i(its)59 X2127 y(condition)c(n)o(um)o(b)q(er)g(is)f(1)p Fn(:)p XFo(1)9 b Fm(\001)g Fo(10)605 2110 y Fj(3)624 2127 y Fo(.)20 Xb(This)c(implies)h(that)d(the)h(computed)g(solution)h(is)g(p)q(oten)o X(tially)g(v)o(ery)59 2183 y(sensitiv)o(e)j(to)d(p)q(erturbations)i(of)f X(the)g(data.)26 b(Indeed,)19 b(if)f(w)o(e)f(compute)h(the)f(ordinary)h X(least-squares)59 2239 y(solution)e Fp(x)259 2246 y Fj(LSQ)345 X2239 y Fo(b)o(y)f(means)g(of)g(a)g(QR)h(factorization)f(of)g XFn(A)p Fo(,)f(then)i(w)o(e)f(obtain)751 2365 y Fp(x)779 X2372 y Fj(LSQ)863 2365 y Fo(=)911 2305 y Fh(\022)984 X2337 y Fo(7)p Fn(:)p Fo(01)949 2393 y Fm(\000)p Fo(8)p XFn(:)p Fo(40)1073 2305 y Fh(\023)1126 2365 y Fn(:)59 X2485 y Fo(This)i(solution)g(is)g(ob)o(viously)g(w)o(orthless,)e(and)i X(something)f(m)o(ust)g(b)q(e)h(done)f(in)h(order)f(to)g(compute)g(a)59 X2542 y(b)q(etter)f(appro)o(ximation)g(to)g(the)g(exact)g(solution)926 X2541 y(\026)923 2542 y Fp(x)951 2525 y Fg(T)991 2542 Xy Fo(=)e(\(1)h(1\).)130 2598 y(The)i(large)g(condition)i(n)o(um)o(b)q X(er)e(implies)j(that)c(the)h(columns)h(of)f Fn(A)g Fo(are)g(nearly)h X(linearly)h(dep)q(en-)59 2655 y(den)o(t.)k(One)17 b(could)g(therefore)f X(think)h(of)e(replacing)j(the)e(ill-conditioned)k(matrix)15 Xb Fn(A)f Fo(=)h(\()p Fp(a)1628 2662 y Fj(1)1664 2655 Xy Fp(a)1689 2662 y Fj(2)1709 2655 y Fo(\))g(with)59 2711 Xy(either)d(\()p Fp(a)227 2718 y Fj(1)257 2711 y Fp(0)p XFo(\))f(or)g(\()p Fp(0)f(a)443 2718 y Fj(2)463 2711 y XFo(\),)h(b)q(oth)h(of)e(whic)o(h)i(are)e(w)o(ell)i(conditioned.)21 Xb(The)11 b(t)o(w)o(o)e(corresp)q(onding)j(so-called)59 X2768 y(basic)k(solutions)g(are)568 2834 y Fp(x)596 2810 Xy Fj(\(1\))596 2848 y(B)655 2834 y Fo(=)703 2775 y Fh(\022)741 X2806 y Fo(1)p Fn(:)p Fo(65)741 2862 y(0)p Fn(:)p Fo(00)830 X2775 y Fh(\023)883 2834 y Fn(;)98 b Fp(x)1022 2810 y XFj(\(2\))1022 2848 y(B)1082 2834 y Fo(=)1129 2775 y Fh(\022)1168 X2806 y Fo(0)p Fn(:)p Fo(00)1168 2862 y(2)p Fn(:)p Fo(58)1256 X2775 y 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