Mercurial > forge
view main/optim/inst/optim_problems.m @ 9930:d30cfca46e8a octave-forge
optim: upgrade license to GPLv3+ and mention on DESCRIPTION the other package licenses
| author | carandraug |
|---|---|
| date | Fri, 30 Mar 2012 15:14:48 +0000 |
| parents | 55c73f24f0ee |
| children | fba8cdd5f9ad |
line wrap: on
line source
%% Copyright (C) 2007 Paul Kienzle (sort-based lookup in ODE solver) %% Copyright (C) 2009 Thomas Treichl <thomas.treichl@gmx.net> %% Copyright (C) 2010 Olaf Till <olaf.till@uni-jena.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)