--- /dev/null
+%% Copyright (C) 2007 Paul Kienzle (sort-based lookup in ODE solver)
+%% Copyright (C) 2009 Thomas Treichl <thomas.treichl@gmx.net> (ode23 code)
+%% Copyright (C) 2010 Olaf Till <i7tiol@t-online.de>
+%%
+%% This program is free software; you can redistribute it and/or modify it under
+%% the terms of the GNU General Public License as published by the Free Software
+%% Foundation; either version 3 of the License, or (at your option) any later
+%% version.
+%%
+%% This program is distributed in the hope that it will be useful, but WITHOUT
+%% ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
+%% FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
+%% details.
+%%
+%% You should have received a copy of the GNU General Public License along with
+%% this program; if not, see <http://www.gnu.org/licenses/>.
+
+%% Problems for testing optimizers. Documentation is in the code.
+
+function ret = optim_problems ()
+
+ %% As little external code as possible is called. This leads to some
+ %% duplication of external code. The advantages are that thus these
+ %% problems do not change with evolving external code, and that
+ %% optimization results in Octave can be compared with those in Matlab
+ %% without influence of differences in external code (e.g. ODE
+ %% solvers). Even calling 'interp1 (..., ..., ..., 'linear')' is
+ %% avoided by using an internal subfunction, although this is possibly
+ %% too cautious.
+ %%
+ %% For cross-program comparison of optimizers, the code of these
+ %% problems is intended to be Matlab compatible.
+ %%
+ %% External data may be loaded, which should be supplied in the
+ %% 'private/' subdirectory. Use the variable 'ddir', which contains
+ %% the path to this directory.
+
+ %% Note: The difficulty of problems with dynamic models often
+ %% decisively depends on the the accuracy of the used ODE(DAE)-solver.
+
+ %% Description of the returned structure
+ %%
+ %% According to 3 classes of problems, there are (or should be) three
+ %% fields: 'curve' (curve fitting), 'general' (general optimization),
+ %% and 'zero' (zero finding). The subfields are labels for the
+ %% particular problems.
+ %%
+ %% Under the label fields, there are subfields mostly identical
+ %% between the 3 classes of problems (some may contain empty values):
+ %%
+ %% .f: handle of an internally defined objective function (argument:
+ %% column vector of parameters), meant for minimization, or to a
+ %% 'model function' (arguments: independents, column vector of
+ %% parameters) in the case of curve fitting, where .f should return a
+ %% matrix of equal dimensions as .data.y below.
+ %%
+ %% .dfdp: handle of internally defined function for jacobian of
+ %% objective function or 'model function', respectively.
+ %%
+ %% .init_p: initial parameters, column vector
+ %%
+ %% possibly .init_p_b: two column matrix of ranges to choose initial
+ %% parameters from
+ %%
+ %% possibly .init_p_f: handle of internally defined function which
+ %% returns a column vector of initial parameters unique to the index
+ %% given as function argument; given '0' as function argument,
+ %% .init_p_f returns the maximum index
+ %%
+ %% .result.p: parameters of best known result
+ %%
+ %% possibly .result.obj: value of objective function for .result.p (or
+ %% sum of squared residuals in curve fitting).
+ %%
+ %% .data.x: matrix of independents (curve fitting)
+ %%
+ %% .data.y: matrix of observations, dimensions may be independent of
+ %% .data.x (curve fitting)
+ %%
+ %% .data.wt: matrix of weights, same dimensions as .data.y (curve
+ %% fitting)
+ %%
+ %% .data.cov: covariance matrix of .data.y(:) (not necessarily a
+ %% diagonal matrix, which could be expressed in .data.wt)
+ %%
+ %% .strict_inequc, .non_strict_inequc, .equc: 'strict' inequality
+ %% constraints (<, >), 'non-strict' inequality constraints (<=, >=),
+ %% and equality constraints, respectively. Subfields are: .bounds
+ %% (except in equality constraints): two-column matrix of ranges;
+ %% .linear: cell-array {m, v}, meaning linear constraints m.' *
+ %% parameters + v >|>=|== 0; .general: handle of internally defined
+ %% function h with h (p) >|>=|== 0; possibly .general_dcdp: handle of
+ %% internally defined function (argument: parameters) returning the
+ %% jacobian of the constraints given in .general. For the sake of
+ %% optimizers which can exploit this, the function in subfield
+ %% .general may accept a logical index vector as an optional second
+ %% argument, returning only the indexed constraint values.
+
+
+ %% Please keep the following list of problems current.
+ %%
+ %% .curve.p_1, .curve.p_2, .curve.p_3_d2: from 'Comparison of gradient
+ %% methods for the solution of nonlinear parameter estimation
+ %% problems' (1970), Yonathan Bard, Siam Journal on Numerical Analysis
+ %% 7(1), 157--186. The numbering of problems is the same as in the
+ %% article. Since Bard used strict bounds, testing optimizers which
+ %% used penalization for bounds, the bounds are changed here to allow
+ %% testing with non-strict bounds (<= or >=). .curve.p_3_d2 involves
+ %% dynamic modeling. These are not necessarily difficult problems.
+ %%
+ %% .curve.p_3_d2_noweights: problem .curve.p_3_d2 equivalently
+ %% re-formulated without weights.
+ %%
+ %% .curve.p_r: A seemingly more difficult 'real life' problem with
+ %% dynamic modeling. To assess optimizers, .init_p_f should be used
+ %% with 1:64. There should be two groups of results, indicating the
+ %% presence of two local minima. Olaf Till <olaf.till@uni-jena.de>
+ %%
+ %% .....schittkowski_...: Klaus Schittkowski: 'More test examples for
+ %% nonlinear programming codes.' Lecture Notes in Economics and
+ %% Mathematical Systems 282, Berlin 1987. The published problems are
+ %% numbered from 201 to 395 and may appear here under the fields
+ %% .curve, .general, or .zero.
+ %%
+ %% .general.schittkowski_281: 10 parameters, unconstrained.
+ %%
+ %% .general.schittkowski_289: 30 parameters, unconstrained.
+ %%
+ %% .general.schittkowski_327 and
+ %%
+ %% .curve.schittkowski_327: Two parameters, one general inequality
+ %% constraint, two bounds. The best solution given in the publication
+ %% seems not very good (it probably has been achieved with general
+ %% minimization, not curve fitting) and has been replaced here by a
+ %% better (leasqr).
+ %%
+ %% .curve.schittkowski_372 and
+ %%
+ %% .general.schittkowski_372: 9 parameters, 12 general inequality
+ %% constraints, 6 bounds. Infeasible initial parameters
+ %% (.curve.schittkowski_372.init_p_f(1) provides a set of more or less
+ %% feasible parameters). leasqr sticks at the (feasible) initial
+ %% values. sqp has no problems.
+ %%
+ %% .curve.schittkowski_373: 9 parameters, 6 equality constraints.
+ %% Infeasible initial parameters (.curve.schittkowski_373.init_p_f(1)
+ %% provides a set of more or less feasible parameters). leasqr sticks
+ %% at the (feasible) initial values. sqp has no problems.
+ %%
+ %% .general.schittkowski_391: 30 parameters, unconstrained. The best
+ %% solution given in the publication seems not very good, obviously
+ %% the used routine had not managed to get very far from the starting
+ %% parameters; it has been replaced here by a better (Octaves
+ %% fminunc). The result still varies widely (without much changes in
+ %% objective function) with changes of starting values. Maybe not a
+ %% very good test problem, no well defined minimum ...
+
+ %% needed for some anonymous functions
+ if (exist ('ifelse') ~= 5)
+ ifelse = @ scalar_ifelse;
+ end
+
+ if (~exist ('OCTAVE_VERSION'))
+ NA = NaN;
+ end
+
+ %% determine the directory of this functions file
+ fdir = fileparts (mfilename ('fullpath'));
+ %% data directory
+ ddir = sprintf ('%s%sprivate%s', fdir, filesep, filesep);
+
+ ret.curve.p_1.dfdp = [];
+ ret.curve.p_1.init_p = [1; 1; 1];
+ ret.curve.p_1.data.x = cat (2, ...
+ (1:15).', ...
+ (15:-1:1).', ...
+ [(1:8).'; (7:-1:1).']);
+ ret.curve.p_1.data.y = [.14; .18; .22; .25; .29; .32; .35; .39; ...
+ .37; .58; .73; .96; 1.34; 2.10; 4.39];
+ ret.curve.p_1.data.wt = [];
+ ret.curve.p_1.data.cov = [];
+ ret.curve.p_1.result.p = [.08241040; 1.133033; 2.343697];
+ ret.curve.p_1.strict_inequc.bounds = [0, 100; 0, 100; 0, 100];
+ ret.curve.p_1.strict_inequc.linear = [];
+ ret.curve.p_1.strict_inequc.general = [];
+ ret.curve.p_1.non_strict_inequc.bounds = ...
+ [eps, 100; eps, 100; eps, 100];
+ ret.curve.p_1.non_strict_inequc.linear = [];
+ ret.curve.p_1.non_strict_inequc.general = [];
+ ret.curve.p_1.equc.linear = [];
+ ret.curve.p_1.equc.general = [];
+ ret.curve.p_1.f = @ f_1;
+
+ ret.curve.p_2.dfdp = [];
+ ret.curve.p_2.init_p = [0; 0; 0; 0; 0];
+ ret.curve.p_2.data.x = [.871, .643, .550; ...
+ .228, .669, .854; ...
+ .528, .229, .170; ...
+ .110, .354, .337; ...
+ .911, .056, .493; ...
+ .476, .154, .918; ...
+ .655, .421, .077; ...
+ .649, .140, .199; ...
+ .995, .045, NA; ...
+ .130, .016, .195; ...
+ .823, .690, .690; ...
+ .768, .992, .389; ...
+ .203, .740, .120; ...
+ .302, .519, .221; ...
+ .991, .450, .249; ...
+ .224, .030, .502; ...
+ .428, .127, .772; ...
+ .552, .494, .110; ...
+ .461, .824, .714; ...
+ .799, .494, .295];
+ ret.curve.p_2.data.y = zeros (20, 3);
+ ret.curve.p_2.data.wt = [];
+ ret.curve.p_2.data.cov = [];
+ ret.curve.p_2.data.misc = [4.36, 5.21, 5.35; ...
+ 4.99, 3.30, 3.10; ...
+ 1.67, NA, 2.75; ...
+ 2.17, 1.48, 1.49; ...
+ 2.98, 4.69, 4.23; ...
+ 4.46, 3.87, 3.15; ...
+ 1.79, 3.18, 3.57; ...
+ 1.71, 3.13, 3.07; ...
+ 3.07, 5.01, 4.58; ...
+ 0.94, 0.93, 0.74; ...
+ 4.97, 5.37, 5.35; ...
+ 4.32, 4.85, 5.46; ...
+ 2.17, 1.78, 2.43; ...
+ 2.22, 2.18, 2.44; ...
+ 2.88, 4.90, 5.11; ...
+ 2.29, 1.94, 1.46; ...
+ 3.76, 3.39, 2.71; ...
+ 1.99, 2.93, 3.31; ...
+ 4.95, 4.08, 4.19; ...
+ 2.96, 4.26, 4.48];
+ ret.curve.p_2.result.p = [.9925145; 2.005293; 3.999732; ...
+ 2.680371; .4977683]; % from maximum
+ % likelyhood optimization
+ ret.curve.p_2.strict_inequc.bounds = [];
+ ret.curve.p_2.strict_inequc.linear = [];
+ ret.curve.p_2.strict_inequc.general = [];
+ ret.curve.p_2.non_strict_inequc.bounds = [];
+ ret.curve.p_2.non_strict_inequc.linear = [];
+ ret.curve.p_2.non_strict_inequc.general = [];
+ ret.curve.p_2.equc.linear = [];
+ ret.curve.p_2.equc.general = [];
+ ret.curve.p_2.f = @ (x, p) f_2 (x, p, ret.curve.p_2.data.misc);
+
+
+
+ ret.curve.p_3_d2.dfdp = [];
+ ret.curve.p_3_d2.init_p = [.01; .01; .001; .001; .02; .001];
+ ret.curve.p_3_d2.data.x = [0; 12.5; 25; 37.5; 50; ...
+ 62.5; 75; 87.5; 100];
+ ret.curve.p_3_d2.data.y=[1 1 0 0 0 ; ...
+ .945757 .961201 .494861 .154976 .111485; ...
+ .926486 .928762 .690492 .314501 .236263; ...
+ .917668 .915966 .751806 .709300 .311747; ...
+ .928987 .917542 .771559 1.19224 .333096; ...
+ .927782 .920075 .780903 1.68815 .340324; ...
+ .925304 .912330 .790539 2.19539 .356787; ...
+ .925083 .917684 .783933 2.74211 .358283; ...
+ .917277 .907529 .779259 3.20025 .361969];
+ ret.curve.p_3_d2.data.y(:, 3) = ...
+ ret.curve.p_3_d2.data.y(:, 3) / 10;
+ ret.curve.p_3_d2.data.y(:, 4:5) = ...
+ ret.curve.p_3_d2.data.y(:, 4:5) / 1000;
+ ret.curve.p_3_d2.data.wt = repmat ([.1, .1, 1, 10, 100], 9, 1);
+ ret.curve.p_3_d2.data.cov = [];
+ ret.curve.p_3_d2.result.p = [.6358247e-2; ...
+ .6774551e-1; ...
+ .5914274e-4; ...
+ .4944010e-3; ...
+ .1018828; ...
+ .4210526e-3];
+ ret.curve.p_3_d2.strict_inequc.bounds = [0, 1; ...
+ 0, 1; ...
+ 0, .1; ...
+ 0, .1; ...
+ 0, 2; ...
+ 0, .1];
+ ret.curve.p_3_d2.strict_inequc.linear = [];
+ ret.curve.p_3_d2.strict_inequc.general = [];
+ ret.curve.p_3_d2.non_strict_inequc.bounds = [eps, 1; ...
+ eps, 1; ...
+ eps, .1; ...
+ eps, .1; ...
+ eps, 2; ...
+ eps, .1];
+ ret.curve.p_3_d2.non_strict_inequc.linear = [];
+ ret.curve.p_3_d2.non_strict_inequc.general = [];
+ ret.curve.p_3_d2.equc.linear = [];
+ ret.curve.p_3_d2.equc.general = [];
+ ret.curve.p_3_d2.f = @ f_3;
+
+ ret.curve.p_3_d2_noweights = ret.curve.p_3_d2;
+ ret.curve.p_3_d2_noweights.data.wt = [];
+ ret.curve.p_3_d2_noweights.data.y(:, 1:2) = ...
+ ret.curve.p_3_d2_noweights.data.y(:, 1:2) * .1;
+ ret.curve.p_3_d2_noweights.data.y(:, 4) = ...
+ ret.curve.p_3_d2_noweights.data.y(:, 4) * 10;
+ ret.curve.p_3_d2_noweights.data.y(:, 5) = ...
+ ret.curve.p_3_d2_noweights.data.y(:, 5) * 100;
+ ret.curve.p_3_d2_noweights.f = @ f_3_noweights;
+
+ ret.curve.p_r.dfdp = [];
+ ret.curve.p_r.init_p = [.3; .03; .003; .7; 1000; .0205];
+ ret.curve.p_r.init_p_b = [.3, .5; ...
+ .03, .05; ...
+ .003, .005; ...
+ .7, .9; ...
+ 1000, 1300; ...
+ .0205, .023];
+ ret.curve.p_r.init_p_f = @ (id) pc2 (ret.curve.p_r.init_p_b, id);
+ hook.ns = [84; 84; 85; 86; 84; 84; 84; 84];
+ xb = [0.2, 0.8640; ...
+ 0.2, 0.5320; ...
+ 0.2, 0.4856; ...
+ 0.2, 0.4210; ...
+ 0.2, 0.3328; ...
+ 0.2, 0.2996; ...
+ 0.2, 0.2664; ...
+ 0.2, 0.2498];
+ ns = cat (1, 0, cumsum (hook.ns));
+ x = zeros (ns(end), 1);
+ for id = 1:8
+ x(ns(id) + 1 : ns(id + 1)) = ...
+ linspace (xb(id, 1), xb(id, 2), hook.ns(id)).';
+ end
+ hook.t = x;
+ ret.curve.p_r.data.x = x;
+ ret.curve.p_r.data.y = ...
+ load (sprintf ('%soptim_problems_p_r_y.data', ddir));
+ ret.curve.p_r.data.wt = [];
+ ret.curve.p_r.data.cov = [];
+ ret.curve.p_r.result.p = [4.742909e-01; ...
+ 3.837951e-02; ...
+ 3.652570e-03; ...
+ 7.725986e-01; ...
+ 1.180967e+03; ...
+ 2.107000e-02];
+ ret.curve.p_r.result.obj = 0.2043396;
+ ret.curve.p_r.strict_inequc.bounds = [];
+ ret.curve.p_r.strict_inequc.linear = [];
+ ret.curve.p_r.strict_inequc.general = [];
+ ret.curve.p_r.non_strict_inequc.bounds = [];
+ ret.curve.p_r.non_strict_inequc.linear = [];
+ ret.curve.p_r.non_strict_inequc.general = [];
+ ret.curve.p_r.equc.linear = [];
+ ret.curve.p_r.equc.general = [];
+ hook.mc = [2.0019999999999999e-01, 1.9939999999999999e-01, ...
+ 1.9939999999999999e-01, 1.9780000000000000e-01, ...
+ 2.0080000000000001e-01, 1.9960000000000000e-01, ...
+ 1.9960000000000000e-01, 1.9980000000000001e-01; ...
+ ...
+ 2.0060000000000000e-01, 2.0160000000000000e-01, ...
+ 2.0200000000000001e-01, 2.0200000000000001e-01, ...
+ 2.0180000000000001e-01, 2.0899999999999999e-01, ...
+ 2.0860000000000001e-01, 2.0820000000000000e-01; ...
+ ...
+ 2.1999144799999999e-02, 2.1998803099999999e-02, ...
+ 2.2000449599999999e-02, 2.2000024399999998e-02, ...
+ 2.1998160999999999e-02, 2.1999289000000002e-02, ...
+ 2.1998038800000001e-02, 2.2000270999999998e-02; ...
+ ...
+ -6.8806551999999986e-03, -1.3768898999999999e-02, ...
+ -1.6065479000000001e-02, -2.0657919600000001e-02, ...
+ -3.4479971099999999e-02, -4.5934394099999998e-02, ...
+ -6.9011619100000005e-02, -9.1971348400000000e-02; ...
+ ...
+ 2.3383865100000002e-02, 2.4768462500000001e-02, ...
+ 2.5231915899999999e-02, 2.6155515300000001e-02, ...
+ 2.8933514200000000e-02, 3.1235568599999999e-02, ...
+ 3.5874086299999997e-02, 4.0490560699999997e-02; ...
+ ...
+ -1.8240616806039459e+05, -1.6895474269973661e+03, ...
+ -8.1072652464694931e+02, -7.0113302985566395e+02, ...
+ 1.0929964862867249e+04, 3.5665776039585688e+02, ...
+ 5.7400262910547769e+02, 9.1737316974342252e+02; ...
+ ...
+ 1.0965398741890911e+05, 1.0131334821116490e+03, ...
+ 4.8504892529762208e+02, 4.1801020186158411e+02, ...
+ -6.6178457662355086e+03, -2.2103886018172699e+02, ...
+ -3.5529578864017282e+02, -5.6690686490678263e+02; ...
+ ...
+ -2.1972917026209168e+04, -2.0250659086265861e+02, ...
+ -9.6733175964156985e+01, -8.3069683020988421e+01, ...
+ 1.3356173243752210e+03, 4.5610806266307627e+01, ...
+ 7.3229009073208331e+01, 1.1667126232349770e+02; ...
+ ...
+ 1.4676952576063929e+03, 1.3514357622838521e+01, ...
+ 6.4524906786197480e+00, 5.5245948033669476e+00, ...
+ -8.9827382090060922e+01, -3.1118708128841241e+00, ...
+ -5.0039950796246986e+00, -7.9749636293721071e+00];
+ ret.curve.p_r.f = @ (x, p) f_r (x, p, hook);
+
+ ret.general.schittkowski_281.dfdp = ...
+ @ (p) schittkowski_281_dfdp (p);
+ ret.general.schittkowski_281.init_p = zeros (10, 1);
+ ret.general.schittkowski_281.result.p = ones (10, 1); % 'theoretically'
+ ret.general.schittkowski_281.result.obj = 0; % 'theoretically'
+ ret.general.schittkowski_281.strict_inequc.bounds = [];
+ ret.general.schittkowski_281.strict_inequc.linear = [];
+ ret.general.schittkowski_281.strict_inequc.general = [];
+ ret.general.schittkowski_281.non_strict_inequc.bounds = [];
+ ret.general.schittkowski_281.non_strict_inequc.linear = [];
+ ret.general.schittkowski_281.non_strict_inequc.general = [];
+ ret.general.schittkowski_281.equc.linear = [];
+ ret.general.schittkowski_281.equc.general = [];
+ ret.general.schittkowski_281.f = ...
+ @ (p) (sum (((1:10).') .^ 3 .* (p - 1) .^ 2)) ^ (1 / 3);
+
+ ret.general.schittkowski_289.dfdp = ...
+ @ (p) exp (- sum (p .^ 2) / 60) / 30 * p;
+ ret.general.schittkowski_289.init_p = [-1.03; 1.07; -1.10; 1.13; ...
+ -1.17; 1.20; -1.23; 1.27; ...
+ -1.30; 1.33; -1.37; 1.40; ...
+ -1.43; 1.47; -1.50; 1.53; ...
+ -1.57; 1.60; -1.63; 1.67; ...
+ -1.70; 1.73; -1.77; 1.80; ...
+ -1.83; 1.87; -1.90; 1.93; ...
+ -1.97; 2.00];
+ ret.general.schittkowski_289.result.p = zeros (30, 1); % 'theoretically'
+ ret.general.schittkowski_289.result.obj = 0; % 'theoretically'
+ ret.general.schittkowski_289.strict_inequc.bounds = [];
+ ret.general.schittkowski_289.strict_inequc.linear = [];
+ ret.general.schittkowski_289.strict_inequc.general = [];
+ ret.general.schittkowski_289.non_strict_inequc.bounds = [];
+ ret.general.schittkowski_289.non_strict_inequc.linear = [];
+ ret.general.schittkowski_289.non_strict_inequc.general = [];
+ ret.general.schittkowski_289.equc.linear = [];
+ ret.general.schittkowski_289.equc.general = [];
+ ret.general.schittkowski_289.f = @ (p) 1 - exp (- sum (p .^ 2) / 60);
+
+ ret.curve.schittkowski_327.dfdp = ...
+ @ (x, p) [1 + exp(-p(2) * (x - 8)), ...
+ (p(1) + .49) * (8 - x) .* exp (-p(2) * (x - 8))];
+ ret.curve.schittkowski_327.init_p = [.42; 5];
+ ret.curve.schittkowski_327.data.x = ...
+ [8; 8; 10; 10; 10; 10; 12; 12; 12; 12; 14; 14; 14; 16; 16; 16; ...
+ 18; 18; 20; 20; 20; 22; 22; 22; 24; 24; 24; 26; 26; 26; 28; ...
+ 28; 30; 30; 30; 32; 32; 34; 36; 36; 38; 38; 40; 42];
+ ret.curve.schittkowski_327.data.y= ...
+ [.49; .49; .48; .47; .48; .47; .46; .46; .45; .43; .45; .43; ...
+ .43; .44; .43; .43; .46; .45; .42; .42; .43; .41; .41; .40; ...
+ .42; .40; .40; .41; .40; .41; .41; .40; .40; .40; .38; .41; ...
+ .40; .40; .41; .38; .40; .40; .39; .39];
+ ret.curve.schittkowski_327.data.wt = [];
+ ret.curve.schittkowski_327.data.cov = [];
+ %% This result was given by Schittkowski. No constraint is active
+ %% here. The second parameter is unchanged from initial value.
+ %%
+ %% ret.curve.schittkowski_327.result.p = [.4219; 5];
+ %% ret.curve.schittkowski_327.result.obj = .0307986;
+ %%
+ %% This is the result of leasqr of Octave Forge. The general
+ %% constraint is active here. Both parameters are different from
+ %% initial value. The value of the objective function is better.
+ %%
+ ret.curve.schittkowski_327.result.p = [.4199227; 1.2842958];
+ ret.curve.schittkowski_327.result.obj = .0284597;
+ ret.curve.schittkowski_327.strict_inequc.bounds = [];
+ ret.curve.schittkowski_327.strict_inequc.linear = [];
+ ret.curve.schittkowski_327.strict_inequc.general = [];
+ ret.curve.schittkowski_327.non_strict_inequc.bounds = [.4, Inf; ...
+ .4, Inf];
+ ret.curve.schittkowski_327.non_strict_inequc.linear = [];
+ ret.curve.schittkowski_327.non_strict_inequc.general = ...
+ @ (p, varargin) apply_idx_if_given ...
+ (-.09 - p(1) * p(2) + .49 * p(2), varargin{:});
+ ret.curve.schittkowski_327.equc.linear = [];
+ ret.curve.schittkowski_327.equc.general = [];
+ ret.curve.schittkowski_327.f = ...
+ @ (x, p) p(1) + (.49 - p(1)) * exp (-p(2) * (x - 8));
+
+ ret.general.schittkowski_327.init_p = [.42; 5];
+ ret.general.schittkowski_327.data.x = ...
+ [8; 8; 10; 10; 10; 10; 12; 12; 12; 12; 14; 14; 14; 16; 16; 16; ...
+ 18; 18; 20; 20; 20; 22; 22; 22; 24; 24; 24; 26; 26; 26; 28; ...
+ 28; 30; 30; 30; 32; 32; 34; 36; 36; 38; 38; 40; 42];
+ ret.general.schittkowski_327.data.y= ...
+ [.49; .49; .48; .47; .48; .47; .46; .46; .45; .43; .45; .43; ...
+ .43; .44; .43; .43; .46; .45; .42; .42; .43; .41; .41; .40; ...
+ .42; .40; .40; .41; .40; .41; .41; .40; .40; .40; .38; .41; ...
+ .40; .40; .41; .38; .40; .40; .39; .39];
+ x = ret.general.schittkowski_327.data.x;
+ y = ret.general.schittkowski_327.data.y;
+ ret.general.schittkowski_327.dfdp = ...
+ @ (p) cat (2, ...
+ 2 * sum ((exp (-p(2 * x - 8)) - 1) * ...
+ (y + (p(1) - .49) * ...
+ exp (-p(2) * (x - 8)) - p1)), ...
+ 2 * (p(1) - .49) * ...
+ sum ((8 - x) * exp (-p(2 * x - 8)) * ...
+ (y + (p(1) - .49) * ...
+ exp (-p(2) * (x - 8)) - p1)));
+ %% This result was given by Schittkowski. No constraint is active
+ %% here. The second parameter is unchanged from initial value.
+ %%
+ %% ret.general.schittkowski_327.result.p = [.4219; 5];
+ %% ret.general.schittkowski_327.result.obj = .0307986;
+ %%
+ %% This is the result of leasqr of Octave Forge. The general
+ %% constraint is active here. Both parameters are different from
+ %% initial value. The value of the objective function is better. sqp
+ %% gives a similar result.
+ ret.general.schittkowski_327.result.p = [.4199227; 1.2842958];
+ ret.general.schittkowski_327.result.obj = .0284597;
+ ret.general.schittkowski_327.strict_inequc.bounds = [];
+ ret.general.schittkowski_327.strict_inequc.linear = [];
+ ret.general.schittkowski_327.strict_inequc.general = [];
+ ret.general.schittkowski_327.non_strict_inequc.bounds = [.4, Inf; ...
+ .4, Inf];
+ ret.general.schittkowski_327.non_strict_inequc.linear = [];
+ ret.general.schittkowski_327.non_strict_inequc.general = ...
+ @ (p, varargin) apply_idx_if_given ...
+ (-.09 - p(1) * p(2) + .49 * p(2), varargin{:});
+ ret.general.schittkowski_327.equc.linear = [];
+ ret.general.schittkowski_327.equc.general = [];
+ ret.general.schittkowski_327.f = ...
+ @ (p) sumsq (y - p(1) - (.49 - p(1)) * exp (-p(2) * (x - 8)));
+
+ ret.curve.schittkowski_372.dfdp = ...
+ @ (x, p) cat (2, zeros (6, 3), eye (6));
+ %% given by Schittkowski, not feasible
+ ret.curve.schittkowski_372.init_p = [300; -100; -.1997; -127; ...
+ -151; 379; 421; 460; 426];
+ %% computed with sqp and a constant objective function, (almost)
+ %% feasible
+ ret.curve.schittkowski_372.init_p_f = @ (id) ...
+ ifelse (id == 0, 1, [2.951277e+02; ...
+ -1.058720e+02; ...
+ -9.535824e-02; ...
+ 2.421108e+00; ...
+ 3.191822e+00; ...
+ 3.790000e+02; ...
+ 4.210000e+02; ...
+ 4.600000e+02; ...
+ 4.260000e+02]);
+ ret.curve.schittkowski_372.data.x = (1:6).'; % any different numbers
+ ret.curve.schittkowski_372.data.y= zeros (6, 1);
+ ret.curve.schittkowski_372.data.wt = [];
+ ret.curve.schittkowski_372.data.cov = [];
+ %% recomputed with sqp (i.e. not with curve fitting)
+ ret.curve.schittkowski_372.result.p = [5.2330557804078126e+02; ...
+ -1.5694790476454301e+02; ...
+ -1.9966450018535931e-01; ...
+ 2.9607990282984435e+01; ...
+ 8.6615541706550545e+01; ...
+ 4.7326722338555498e+01; ...
+ 2.6235616534580515e+01; ...
+ 2.2915996663200740e+01; ...
+ 3.9470733973874445e+01];
+ ret.curve.schittkowski_372.result.obj = 13390.1;
+ ret.curve.schittkowski_372.strict_inequc.bounds = [];
+ ret.curve.schittkowski_372.strict_inequc.linear = [];
+ ret.curve.schittkowski_372.strict_inequc.general = [];
+ ret.curve.schittkowski_372.non_strict_inequc.bounds = [-Inf, Inf; ...
+ -Inf, Inf; ...
+ -Inf, Inf; ...
+ 0, Inf; ...
+ 0, Inf; ...
+ 0, Inf; ...
+ 0, Inf; ...
+ 0, Inf; ...
+ 0, Inf];
+ ret.curve.schittkowski_372.non_strict_inequc.linear = [];
+ ret.curve.schittkowski_372.non_strict_inequc.general = ...
+ @ (p, varargin) apply_idx_if_given ...
+ (cat (1, p(1) + p(2) * exp (-5 * p(3)) + p(4) - 127, ...
+ p(1) + p(2) * exp (-3 * p(3)) + p(5) - 151, ...
+ p(1) + p(2) * exp (-p(3)) + p(6) - 379, ...
+ p(1) + p(2) * exp (p(3)) + p(7) - 421, ...
+ p(1) + p(2) * exp (3 * p(3)) + p(8) - 460, ...
+ p(1) + p(2) * exp (5 * p(3)) + p(9) - 426, ...
+ -p(1) - p(2) * exp (-5 * p(3)) + p(4) + 127, ...
+ -p(1) - p(2) * exp (-3 * p(3)) + p(5) + 151, ...
+ -p(1) - p(2) * exp (-p(3)) + p(6) + 379, ...
+ -p(1) - p(2) * exp (p(3)) + p(7) + 421, ...
+ -p(1) - p(2) * exp (3 * p(3)) + p(8) + 460, ...
+ -p(1) - p(2) * exp (5 * p(3)) + p(9) + 426), ...
+ varargin{:});
+ ret.curve.schittkowski_372.equc.linear = [];
+ ret.curve.schittkowski_372.equc.general = [];
+ ret.curve.schittkowski_372.f = @ (x, p) p(4:9);
+
+ ret.curve.schittkowski_373.dfdp = ...
+ @ (x, p) cat (2, zeros (6, 3), eye (6));
+ %% not feasible
+ ret.curve.schittkowski_373.init_p = [300; -100; -.1997; -127; ...
+ -151; 379; 421; 460; 426];
+ %% feasible
+ ret.curve.schittkowski_373.init_p_f = @ (id) ...
+ ifelse (id == 0, 1, [2.5722721227695763e+02; ...
+ -1.5126681606092043e+02; ...
+ 8.3101871447778766e-02; ...
+ -3.0390506000425454e+01; ...
+ 1.1661334225083069e+01; ...
+ 2.6097719374430665e+02; ...
+ 3.2814725183082305e+02; ...
+ 3.9686840023267564e+02; ...
+ 3.9796353824451995e+02]);
+ ret.curve.schittkowski_373.data.x = (1:6).'; % any different numbers
+ ret.curve.schittkowski_373.data.y= zeros (6, 1);
+ ret.curve.schittkowski_373.data.wt = [];
+ ret.curve.schittkowski_373.data.cov = [];
+ ret.curve.schittkowski_373.result.p = [523.31; ...
+ -156.95; ...
+ -.2; ...
+ 29.61; ...
+ -86.62; ...
+ 47.33; ...
+ 26.24; ...
+ 22.92; ...
+ -39.47];
+ ret.curve.schittkowski_373.result.obj = 13390.1;
+ ret.curve.schittkowski_373.strict_inequc.bounds = [];
+ ret.curve.schittkowski_373.strict_inequc.linear = [];
+ ret.curve.schittkowski_373.strict_inequc.general = [];
+ ret.curve.schittkowski_373.non_strict_inequc.bounds = [];
+ ret.curve.schittkowski_373.non_strict_inequc.linear = [];
+ ret.curve.schittkowski_373.non_strict_inequc.general = [];
+ ret.curve.schittkowski_373.equc.linear = [];
+ ret.curve.schittkowski_373.equc.general = ...
+ @ (p, varargin) apply_idx_if_given ...
+ (cat (1, p(1) + p(2) * exp (-5 * p(3)) + p(4) - 127, ...
+ p(1) + p(2) * exp (-3 * p(3)) + p(5) - 151, ...
+ p(1) + p(2) * exp (-p(3)) + p(6) - 379, ...
+ p(1) + p(2) * exp (p(3)) + p(7) - 421, ...
+ p(1) + p(2) * exp (3 * p(3)) + p(8) - 460, ...
+ p(1) + p(2) * exp (5 * p(3)) + p(9) - 426), ...
+ varargin{:});
+ ret.curve.schittkowski_373.f = @ (x, p) p(4:9);
+
+ ret.general.schittkowski_372.dfdp = ...
+ @ (p) cat (2, zeros (1, 3), 2 * p(4:9));
+ %% not feasible
+ ret.general.schittkowski_372.init_p = [300; -100; -.1997; -127; ...
+ -151; 379; 421; 460; 426];
+ %% recomputed with sqp
+ ret.general.schittkowski_372.result.p = [5.2330557804078126e+02; ...
+ -1.5694790476454301e+02; ...
+ -1.9966450018535931e-01; ...
+ 2.9607990282984435e+01; ...
+ 8.6615541706550545e+01; ...
+ 4.7326722338555498e+01; ...
+ 2.6235616534580515e+01; ...
+ 2.2915996663200740e+01; ...
+ 3.9470733973874445e+01];
+ ret.general.schittkowski_372.result.obj = 13390.1;
+ ret.general.schittkowski_372.strict_inequc.bounds = [];
+ ret.general.schittkowski_372.strict_inequc.linear = [];
+ ret.general.schittkowski_372.strict_inequc.general = [];
+ ret.general.schittkowski_372.non_strict_inequc.bounds = [-Inf, Inf; ...
+ -Inf, Inf; ...
+ -Inf, Inf; ...
+ 0, Inf; ...
+ 0, Inf; ...
+ 0, Inf; ...
+ 0, Inf; ...
+ 0, Inf; ...
+ 0, Inf];
+ ret.general.schittkowski_372.non_strict_inequc.linear = [];
+ ret.general.schittkowski_372.non_strict_inequc.general = ...
+ @ (p, varargin) apply_idx_if_given ...
+ (cat (1, p(1) + p(2) * exp (-5 * p(3)) + p(4) - 127, ...
+ p(1) + p(2) * exp (-3 * p(3)) + p(5) - 151, ...
+ p(1) + p(2) * exp (-p(3)) + p(6) - 379, ...
+ p(1) + p(2) * exp (p(3)) + p(7) - 421, ...
+ p(1) + p(2) * exp (3 * p(3)) + p(8) - 460, ...
+ p(1) + p(2) * exp (5 * p(3)) + p(9) - 426, ...
+ -p(1) - p(2) * exp (-5 * p(3)) + p(4) + 127, ...
+ -p(1) - p(2) * exp (-3 * p(3)) + p(5) + 151, ...
+ -p(1) - p(2) * exp (-p(3)) + p(6) + 379, ...
+ -p(1) - p(2) * exp (p(3)) + p(7) + 421, ...
+ -p(1) - p(2) * exp (3 * p(3)) + p(8) + 460, ...
+ -p(1) - p(2) * exp (5 * p(3)) + p(9) + 426), ...
+ varargin{:});
+ ret.general.schittkowski_372.equc.linear = [];
+ ret.general.schittkowski_372.equc.general = [];
+ ret.general.schittkowski_372.f = @ (p) sumsq (p(4:9));
+
+ ret.general.schittkowski_391.dfdp = [];
+ ret.general.schittkowski_391.init_p = ...
+ -2.8742711 * alpha_391 (zeros (30, 1), 1:30);
+ %% computed with fminunc (Octave)
+ ret.general.schittkowski_391.result.p = [-1.1986682e+18; ...
+ -1.1474574e+07; ...
+ -1.3715802e+07; ...
+ -1.0772255e+07; ...
+ -1.0634232e+07; ...
+ -1.0622915e+07; ...
+ -8.8775399e+06; ...
+ -8.8201496e+06; ...
+ -9.7729975e+06; ...
+ -1.0431808e+07; ...
+ -1.0415089e+07; ...
+ -1.0350400e+07; ...
+ -1.0325094e+07; ...
+ -1.0278561e+07; ...
+ -1.0275751e+07; ...
+ -1.0276546e+07; ...
+ -1.0292584e+07; ...
+ -1.0289350e+07; ...
+ -1.0192566e+07; ...
+ -1.0058577e+07; ...
+ -1.0096341e+07; ...
+ -1.0242386e+07; ...
+ -1.0615831e+07; ...
+ -1.1142096e+07; ...
+ -1.1617283e+07; ...
+ -1.2005738e+07; ...
+ -1.2282117e+07; ...
+ -1.2301260e+07; ...
+ -1.2051365e+07; ...
+ -1.1704693e+07];
+ ret.general.schittkowski_391.result.obj = -5.1615468e+20;
+ ret.general.schittkowski_391.strict_inequc.bounds = [];
+ ret.general.schittkowski_391.strict_inequc.linear = [];
+ ret.general.schittkowski_391.strict_inequc.general = [];
+ ret.general.schittkowski_391.non_strict_inequc.bounds = [];
+ ret.general.schittkowski_391.non_strict_inequc.linear = [];
+ ret.general.schittkowski_391.non_strict_inequc.general = [];
+ ret.general.schittkowski_391.equc.linear = [];
+ ret.general.schittkowski_391.equc.general = [];
+ ret.general.schittkowski_391.f = @ (p) sum (alpha_391 (p, 1:30));
+
+ function ret = f_1 (x, p)
+
+ ret = p(1) + x(:, 1) ./ (p(2) * x(:, 2) + p(3) * x(:, 3));
+
+ function ret = f_2 (x, p, y)
+
+ y(3, 2) = p(4);
+ x(9, 3) = p(5);
+ p = p(:);
+ mp = cat (2, p([1, 2, 3]), p([3, 1, 2]), p([3, 2, 1]));
+ ret = x * mp - y;
+
+ function ret = f_3 (x, p)
+
+ ret = fixed_step_rk4 (x.', [1, 1, 0, 0, 0], 1, ...
+ @ (x, t) f_3_xdot (x, t, p));
+ ret = ret.';
+
+ function ret = f_3_noweights (x, p)
+
+ ret = fixed_step_rk4 (x.', [.1, .1, 0, 0, 0], .2, ...
+ @ (x, t) f_3_xdot_noweights (x, t, p));
+ ret = ret.';
+
+ function ret = f_3_xdot (x, t, p)
+
+ ret = zeros (5, 1);
+ tp = p(2) * x(3) - p(1) * x(1) * x(2);
+ ret(1) = tp;
+ ret(2) = tp - p(4) * x(2) * x(3) + p(5) * x(5) - p(6) * x(2) * x(4);
+ ret(3) = - tp - p(3) * x(3) - p(4) * x(2) * x(3);
+ ret(4) = p(3) * x(3) + p(5) * x(5) - p(6) * x(2) * x(4);
+ ret(5) = p(4) * x(2) * x(3) - p(5) * x(5) + p(6) * x(2) * x(4);
+
+ function ret = f_3_xdot_noweights (x, t, p)
+
+ x(1:2) = x(1:2) / .1;
+ x(4) = x(4) / 10;
+ x(5) = x(5) / 100;
+ ret = f_3_xdot (x, t, p);
+ ret(1:2) = ret(1:2) * .1;
+ ret(4) = ret(4) * 10;
+ ret(5) = ret(5) * 100;
+
+ function ret = f_r (x, p, hook)
+
+ n = size (hook.mc, 2);
+ ns = cat (1, 0, cumsum (hook.ns));
+ xdhook.p = p;
+ ret = zeros (1, ns(end));
+ %% temporary variables
+ dls = p(3) ^ 2;
+ dmhp = p(5) * dls / p(4);
+ mhp = dmhp / 2;
+ %%
+ for id = 1:n
+ xdhook.c = hook.mc(:, id);
+ l = xdhook.c(3);
+ x0 = mhp - sqrt (max (0, mhp ^ 2 + dls + (p(6) - l) * dmhp));
+ ids = ns(id) + 1;
+ ide = ns(id + 1);
+
+ tp = odeset ();
+ %% necessary in Matlab (7.1)
+ tp.OutputSave = [];
+ tp.Refine = 0;
+ %%
+ tp.RelTol = 1e-7;
+ tp.AbsTol = 1e-7;
+ [cx, Xcx] = essential_ode23 (@ (t, X) f_r_xdot (X, t, xdhook), ...
+ x([ids, ide]).', x0, tp);
+ X = lin_interp (cx.', Xcx.', x(ids:ide).');
+
+ X = X.';
+ [discarded, lr] = ...
+ f_r_xdot (X, hook.t(ids:ide), xdhook);
+ ret(ids:ide) = max (0, lr - p(6) - X) * p(5);
+ end
+ ret = ret.';
+
+ function [ret, l] = f_r_xdot (x, t, hook)
+
+ %% keep this working with non-scalar x and t
+
+ p = hook.p;
+ c = hook.c;
+ idl = t <= c(1);
+ idg = t >= c(2);
+ idb = ~ (idl | idg);
+ l = zeros (size (t));
+ l(idl) = c(3);
+ l(idg) = c(4) * t(idg) + c(5);
+ l(idb) = polyval (c(6:9), t(idb));
+ dls = max (1e-6, l - p(6) - x);
+ tf = x / p(3);
+ ido = tf >= 1;
+ idx = ~ido;
+ ret(ido) = 0;
+ ret(idx) = - ((p(4) + p(1)) * p(2)) ./ ...
+ ((p(5) * dls(idx)) ./ (1 - tf(idx) .^ 2) + p(1)) + p(2);
+
+ function ret = alpha_391 (p, id)
+
+ %% for .general.schittkowski_391; id is a numeric index(-vector)
+ %% into p
+
+ p = p(:);
+ n = size (p, 1);
+
+ nid = length (id);
+ id = reshape (id, 1, nid);
+
+ v = sqrt (repmat (p .^ 2, 1, nid) + 1 ./ ((1:n).') * id);
+
+ log_v = log (v);
+
+ ret = 420 * p(id) + (id(:) - 15) .^ 3 + ...
+ sum (v .* (sin (log_v) .^ 5 + cos (log_v) .^ 5)).';
+
+ function ret = schittkowski_281_dfdp (p)
+
+ tp = (sum (((1:10).') .^ 3 .* (p - 1) .^ 2)) ^ (- 2 / 3) / 3;
+
+ ret = 2 * ((1:10).') .^ 3 .* (p - 1) * tp;
+
+ function state = fixed_step_rk4 (t, x0, step, f)
+
+ %% minimalistic fourth order ODE-solver, as said to be a popular one
+ %% by Wikipedia (to make these optimization tests self-contained;
+ %% for the same reason 'lookup' and even 'interp1' are not used
+ %% here)
+
+ n = ceil ((t(end) - t(1)) / step) + 1;
+ m = length (x0);
+ tstate = zeros (m, n);
+ tstate(:, 1) = x0;
+ tt = linspace (t(1), t(1) + step * (n - 1), n);
+ for id = 1 : n - 1
+ k1 = f (tstate(:, id), tt(id));
+ k2 = f (tstate(:, id) + .5 * step * k1, tt(id) + .5 * step);
+ k3 = f (tstate(:, id) + .5 * step * k2, tt(id) + .5 * step);
+ k4 = f (tstate(:, id) + step * k3, tt(id + 1));
+ tstate(:, id + 1) = tstate(:, id) + ...
+ (step / 6) * (k1 + 2 * k2 + 2 * k3 + k4);
+ end
+ state = lin_interp (tt, tstate, t);
+
+ function ret = pc2 (p, id)
+ %% a combination out of 2 possible values for each parameter
+ r = size (p, 1);
+ n = 2 ^ r;
+ if (id < 0 || id > n)
+ error ('no parameter set for this index');
+ end
+ if (id == 0) % return maximum id
+ ret = n;
+ return;
+ end
+ idx = dec2bin (id - 1, r) == '1';
+ nidx = ~idx;
+ ret = zeros (r, 1);
+ ret(nidx) = p(nidx, 1);
+ ret(idx) = p(idx, 2);
+
+ function [varargout] = essential_ode23 (vfun, vslot, vinit, vodeoptions)
+
+ %% This code is taken from the ode23 solver of Thomas Treichl
+ %% <thomas.treichl@gmx.net>, some flexibility of the
+ %% interface has been removed. The idea behind this duplication is
+ %% to have a fixed version of the solver here which runs both in
+ %% Octave and Matlab.
+
+ %% Some of the option treatment has been left out.
+ if (length (vslot) > 2)
+ vstepsizefixed = true;
+ else
+ vstepsizefixed = false;
+ end
+ if (strcmp (vodeoptions.NormControl, 'on'))
+ vnormcontrol = true;
+ else
+ vnormcontrol = false;
+ end
+ if (~isempty (vodeoptions.NonNegative))
+ if (isempty (vodeoptions.Mass))
+ vhavenonnegative = true;
+ else
+ vhavenonnegative = false;
+ end
+ else
+ vhavenonnegative = false;
+ end
+ if (isempty (vodeoptions.OutputFcn) && nargout == 0)
+ vodeoptions.OutputFcn = @odeplot;
+ vhaveoutputfunction = true;
+ elseif (isempty (vodeoptions.OutputFcn))
+ vhaveoutputfunction = false;
+ else
+ vhaveoutputfunction = true;
+ end
+ if (~isempty (vodeoptions.OutputSel))
+ vhaveoutputselection = true;
+ else
+ vhaveoutputselection = false;
+ end
+ if (isempty (vodeoptions.OutputSave))
+ vodeoptions.OutputSave = 1;
+ end
+ if (vodeoptions.Refine > 0)
+ vhaverefine = true;
+ else
+ vhaverefine = false;
+ end
+ if (isempty (vodeoptions.InitialStep) && ~vstepsizefixed)
+ vodeoptions.InitialStep = (vslot(1,2) - vslot(1,1)) / 10;
+ vodeoptions.InitialStep = vodeoptions.InitialStep / ...
+ 10^vodeoptions.Refine;
+ end
+ if (isempty (vodeoptions.MaxStep) && ~vstepsizefixed)
+ vodeoptions.MaxStep = (vslot(1,2) - vslot(1,1)) / 10;
+ end
+ if (~isempty (vodeoptions.Events))
+ vhaveeventfunction = true;
+ else
+ vhaveeventfunction = false;
+ end
+ if (~isempty (vodeoptions.Mass) && ismatrix (vodeoptions.Mass))
+ vhavemasshandle = false;
+ vmass = vodeoptions.Mass;
+ elseif (isa (vodeoptions.Mass, 'function_handle'))
+ vhavemasshandle = true;
+ else
+ vhavemasshandle = false;
+ end
+ if (strcmp (vodeoptions.MStateDependence, 'none'))
+ vmassdependence = false;
+ else
+ vmassdependence = true;
+ end
+
+ %% Starting the initialisation of the core solver ode23
+ vtimestamp = vslot(1,1); %% timestamp = start time
+ vtimelength = length (vslot); %% length needed if fixed steps
+ vtimestop = vslot(1,vtimelength); %% stop time = last value
+ vdirection = sign (vtimestop); %% Flag for direction to solve
+
+ if (~vstepsizefixed)
+ vstepsize = vodeoptions.InitialStep;
+ vminstepsize = (vtimestop - vtimestamp) / (1/eps);
+ else %% If step size is given then use the fixed time steps
+ vstepsize = vslot(1,2) - vslot(1,1);
+ vminstepsize = sign (vstepsize) * eps;
+ end
+
+ vretvaltime = vtimestamp; %% first timestamp output
+ vretvalresult = vinit; %% first solution output
+
+ %% Initialize the OutputFcn
+ if (vhaveoutputfunction)
+ if (vhaveoutputselection) vretout = ...
+ vretvalresult(vodeoptions.OutputSel);
+ else
+ vretout = vretvalresult;
+ end
+ feval (vodeoptions.OutputFcn, vslot.', ...
+ vretout.', 'init');
+ end
+
+ %% Initialize the EventFcn
+ if (vhaveeventfunction)
+ odepkg_event_handle (vodeoptions.Events, vtimestamp, ...
+ vretvalresult.', 'init');
+ end
+
+ vpow = 1/3; %% 20071016, reported by Luis Randez
+ va = [ 0, 0, 0; %% The Runge-Kutta-Fehlberg 2(3) coefficients
+ 1/2, 0, 0; %% Coefficients proved on 20060827
+ -1, 2, 0]; %% See p.91 in Ascher & Petzold
+ vb2 = [0; 1; 0]; %% 2nd and 3rd order
+ vb3 = [1/6; 2/3; 1/6]; %% b-coefficients
+ vc = sum (va, 2);
+
+ %% The solver main loop - stop if the endpoint has been reached
+ vcntloop = 2; vcntcycles = 1; vu = vinit; vk = vu.' * zeros(1,3);
+ vcntiter = 0; vunhandledtermination = true; vcntsave = 2;
+ while ((vdirection * (vtimestamp) < vdirection * (vtimestop)) && ...
+ (vdirection * (vstepsize) >= vdirection * (vminstepsize)))
+
+ %% Hit the endpoint of the time slot exactely
+ if ((vtimestamp + vstepsize) > vdirection * vtimestop)
+ %% if (((vtimestamp + vstepsize) > vtimestop) || ...
+ %% (abs(vtimestamp + vstepsize - vtimestop) < eps))
+ vstepsize = vtimestop - vdirection * vtimestamp;
+ end
+
+ %% Estimate the three results when using this solver
+ for j = 1:3
+ vthetime = vtimestamp + vc(j,1) * vstepsize;
+ vtheinput = vu.' + vstepsize * vk(:,1:j-1) * va(j,1:j-1).';
+ if (vhavemasshandle) %% Handle only the dynamic mass matrix,
+ if (vmassdependence) %% constant mass matrices have already
+ vmass = feval ... %% been set before (if any)
+ (vodeoptions.Mass, vthetime, vtheinput);
+ else %% if (vmassdependence == false)
+ vmass = feval ... %% then we only have the time argument
+ (vodeoptions.Mass, vthetime);
+ end
+ vk(:,j) = vmass \ feval ...
+ (vfun, vthetime, vtheinput);
+ else
+ vk(:,j) = feval ...
+ (vfun, vthetime, vtheinput);
+ end
+ end
+
+ %% Compute the 2nd and the 3rd order estimation
+ y2 = vu.' + vstepsize * (vk * vb2);
+ y3 = vu.' + vstepsize * (vk * vb3);
+ if (vhavenonnegative)
+ vu(vodeoptions.NonNegative) = abs (vu(vodeoptions.NonNegative));
+ y2(vodeoptions.NonNegative) = abs (y2(vodeoptions.NonNegative));
+ y3(vodeoptions.NonNegative) = abs (y3(vodeoptions.NonNegative));
+ end
+ vSaveVUForRefine = vu;
+
+ %% Calculate the absolute local truncation error and the
+ %% acceptable error
+ if (~vstepsizefixed)
+ if (~vnormcontrol)
+ vdelta = abs (y3 - y2);
+ vtau = max (vodeoptions.RelTol * abs (vu.'), ...
+ vodeoptions.AbsTol);
+ else
+ vdelta = norm (y3 - y2, Inf);
+ vtau = max (vodeoptions.RelTol * max (norm (vu.', Inf), ...
+ 1.0), ...
+ vodeoptions.AbsTol);
+ end
+ else %% if (vstepsizefixed == true)
+ vdelta = 1; vtau = 2;
+ end
+
+ %% If the error is acceptable then update the vretval variables
+ if (all (vdelta <= vtau))
+ vtimestamp = vtimestamp + vstepsize;
+ vu = y3.'; %% MC2001: the higher order estimation as 'local
+ %% extrapolation' Save the solution every vodeoptions.OutputSave
+ %% steps
+ if (mod (vcntloop-1,vodeoptions.OutputSave) == 0)
+ vretvaltime(vcntsave,:) = vtimestamp;
+ vretvalresult(vcntsave,:) = vu;
+ vcntsave = vcntsave + 1;
+ end
+ vcntloop = vcntloop + 1; vcntiter = 0;
+
+ %% Call plot only if a valid result has been found, therefore
+ %% this code fragment has moved here. Stop integration if plot
+ %% function returns false
+ if (vhaveoutputfunction)
+ for vcnt = 0:vodeoptions.Refine %% Approximation between told
+ %% and t
+ if (vhaverefine) %% Do interpolation
+ vapproxtime = (vcnt + 1) * vstepsize / ...
+ (vodeoptions.Refine + 2);
+ vapproxvals = vSaveVUForRefine.' + vapproxtime * (vk * ...
+ vb3);
+ vapproxtime = (vtimestamp - vstepsize) + vapproxtime;
+ else
+ vapproxvals = vu.';
+ vapproxtime = vtimestamp;
+ end
+ if (vhaveoutputselection)
+ vapproxvals = vapproxvals(vodeoptions.OutputSel);
+ end
+ vpltret = feval (vodeoptions.OutputFcn, vapproxtime, ...
+ vapproxvals, []);
+ if vpltret %% Leave refinement loop
+ break;
+ end
+ end
+ if (vpltret) %% Leave main loop
+ vunhandledtermination = false;
+ break;
+ end
+ end
+
+ %% Call event only if a valid result has been found, therefore
+ %% this code fragment has moved here. Stop integration if
+ %% veventbreak is true
+ if (vhaveeventfunction)
+ vevent = ...
+ odepkg_event_handle (vodeoptions.Events, vtimestamp, ...
+ vu(:), []);
+ if (~isempty (vevent{1}) && vevent{1} == 1)
+ vretvaltime(vcntloop-1,:) = vevent{3}(end,:);
+ vretvalresult(vcntloop-1,:) = vevent{4}(end,:);
+ vunhandledtermination = false; break;
+ end
+ end
+ end %% If the error is acceptable ...
+
+ %% Update the step size for the next integration step
+ if (~vstepsizefixed)
+ %% 20080425, reported by Marco Caliari vdelta cannot be negative
+ %% (because of the absolute value that has been introduced) but
+ %% it could be 0, then replace the zeros with the maximum value
+ %% of vdelta
+ vdelta(find (vdelta == 0)) = max (vdelta);
+ %% It could happen that max (vdelta) == 0 (ie. that the original
+ %% vdelta was 0), in that case we double the previous vstepsize
+ vdelta(find (vdelta == 0)) = max (vtau) .* (0.4 ^ (1 / vpow));
+
+ if (vdirection == 1)
+ vstepsize = min (vodeoptions.MaxStep, ...
+ min (0.8 * vstepsize * (vtau ./ vdelta) .^ ...
+ vpow));
+ else
+ vstepsize = max (vodeoptions.MaxStep, ...
+ max (0.8 * vstepsize * (vtau ./ vdelta) .^ ...
+ vpow));
+ end
+ else %% if (vstepsizefixed)
+ if (vcntloop <= vtimelength)
+ vstepsize = vslot(vcntloop) - vslot(vcntloop-1);
+ else %% Get out of the main integration loop
+ break;
+ end
+ end
+
+ %% Update counters that count the number of iteration cycles
+ vcntcycles = vcntcycles + 1; %% Needed for cost statistics
+ vcntiter = vcntiter + 1; %% Needed to find iteration problems
+
+ %% Stop solving because the last 1000 steps no successful valid
+ %% value has been found
+ if (vcntiter >= 5000)
+ error (['Solving has not been successful. The iterative', ...
+ ' integration loop exited at time t = %f before endpoint at', ...
+ ' tend = %f was reached. This happened because the iterative', ...
+ ' integration loop does not find a valid solution at this time', ...
+ ' stamp. Try to reduce the value of ''InitialStep'' and/or', ...
+ ' ''MaxStep'' with the command ''odeset''.\n'], vtimestamp, vtimestop);
+ end
+
+ end %% The main loop
+
+ %% Check if integration of the ode has been successful
+ if (vdirection * vtimestamp < vdirection * vtimestop)
+ if (vunhandledtermination == true)
+ error ('OdePkg:InvalidArgument', ...
+ ['Solving has not been successful. The iterative', ...
+ ' integration loop exited at time t = %f', ...
+ ' before endpoint at tend = %f was reached. This may', ...
+ ' happen if the stepsize grows smaller than defined in', ...
+ ' vminstepsize. Try to reduce the value of ''InitialStep'' and/or', ...
+ ' ''MaxStep'' with the command ''odeset''.\n'], vtimestamp, vtimestop);
+ else
+ warning ('OdePkg:InvalidArgument', ...
+ ['Solver has been stopped by a call of ''break'' in', ...
+ ' the main iteration loop at time t = %f before endpoint at', ...
+ ' tend = %f was reached. This may happen because the @odeplot', ...
+ ' function returned ''true'' or the @event function returned ''true''.'], ...
+ vtimestamp, vtimestop);
+ end
+ end
+
+ %% Postprocessing, do whatever when terminating integration
+ %% algorithm
+ if (vhaveoutputfunction) %% Cleanup plotter
+ feval (vodeoptions.OutputFcn, vtimestamp, ...
+ vu.', 'done');
+ end
+ if (vhaveeventfunction) %% Cleanup event function handling
+ odepkg_event_handle (vodeoptions.Events, vtimestamp, ...
+ vu.', 'done');
+ end
+ %% Save the last step, if not already saved
+ if (mod (vcntloop-2,vodeoptions.OutputSave) ~= 0)
+ vretvaltime(vcntsave,:) = vtimestamp;
+ vretvalresult(vcntsave,:) = vu;
+ end
+
+
+ varargout{1} = vretvaltime; %% Time stamps are first output argument
+ varargout{2} = vretvalresult; %% Results are second output argument
+
+ function yi = lin_interp (x, y, xi)
+
+ %% Actually interp1 with 'linear' should behave equally in Octave
+ %% and Matlab, but having this subset of functionality here is being
+ %% on the safe side.
+
+ n = size (x, 2);
+ m = size (y, 1);
+ %% This elegant lookup is from an older version of 'lookup' by Paul
+ %% Kienzle, and had been suggested by Kai Habel <kai.habel@gmx.de>.
+ [v, p] = sort ([x, xi]);
+ idx(p) = cumsum (p <= n);
+ idx = idx(n + 1 : n + size (xi, 2));
+ %%
+ idx(idx == n) = n - 1;
+ yi = y(:, idx) + ...
+ repmat (xi - x(idx), m, 1) .* ...
+ (y(:, idx + 1) - y(:, idx)) ./ ...
+ repmat (x(idx + 1) - x(idx), m, 1);
+
+ function ret = apply_idx_if_given (ret, idx)
+
+ if (nargin > 1)
+ ret = ret(idx);
+ end
+
+ function fval = scalar_ifelse (cond, tval, fval)
+
+ %% needed for some anonymous functions, builtin ifelse only available
+ %% in Octave > 3.2; we need only the scalar case here
+
+ if (cond)
+ fval = tval;
+ end
+
+%!demo
+%! p_t = optim_problems ().curve.p_1;
+%! global verbose;
+%! verbose = false;
+%! [cy, cp, cvg, iter] = leasqr (p_t.data.x, p_t.data.y, p_t.init_p, p_t.f)
+%! disp (p_t.result.p)
+%! sumsq (cy - p_t.data.y)
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