## Copyright (C) 2010, 2011 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/>. ## -*- texinfo -*- ## ## @deftypefn {Function File} {[@var{p}, @var{resid}, @var{cvg}, @var{outp}] =} nonlin_residmin (@var{f}, @var{pin}) ## ## @deftypefnx {Function File} {[@var{p}, @var{resid}, @var{cvg}, @var{outp}] =} nonlin_residmin (@var{f}, @var{pin}, @var{settings}) ## ## Frontend for nonlinear minimization of residuals returned by a model ## function. The functions supplied by the user have a minimal ## interface; any additionally needed constants (e.g. observed values) ## can be supplied by wrapping the user functions into anonymous ## functions. ## ## The following description applies to usage with vector-based ## parameter handling. Differences in usage for structure-based ## parameter handling will be explained in a separate section below. ## ## @var{f}: function returning the array of residuals. It gets a column ## vector of real parameters as argument. In gradient determination, ## this function may be called with an informational second argument, ## whose content depends on the function for gradient determination. ## ## @var{pin}: real column vector of initial parameters. ## ## @var{settings}: structure whose fields stand for optional settings ## referred to below. The fields can be set by @code{optimset()} with ## Octave versions 3.3.55 or greater; with older Octave versions, the ## fields must be set directly as structure-fields in the correct case. ## ## The returned values are the column vector of final parameters ## @var{p}, the final array of residuals @var{resid}, an integer ## @var{cvg} indicating if and how optimization succeeded or failed, and ## a structure @var{outp} with additional information, curently with ## only one field: @var{niter}, the number of iterations. @var{cvg} is ## greater than zero for success and less than or equal to zero for ## failure; its possible values depend on the used backend and currently ## can be @code{0} (maximum number of iterations exceeded), @code{2} ## (parameter change less than specified precision in two consecutive ## iterations), or @code{3} (improvement in objective function --- e.g. ## sum of squares --- less than specified). ## ## @var{settings}: ## ## @code{Algorithm}: String specifying the backend. Default: ## @code{"lm_svd_feasible"}. The latter is currently the only backend ## distributed with this package. It is described in a separate section ## below. ## ## @code{dfdp}: Function computing the jacobian of the residuals with ## respect to the parameters, assuming residuals are reshaped to a ## vector. Default: finite differences. Will be called with the column ## vector of parameters and an informational structure as arguments. The ## structure has the fields @code{f}: value of residuals for current ## parameters, reshaped to a column vector, @code{fixed}: logical vector ## indicating which parameters are not optimized, so these partial ## derivatives need not be computed and can be set to zero, ## @code{diffp}, @code{diff_onesided}, @code{lbound}, @code{ubound}: ## identical to the user settings of this name, @code{plabels}: ## 1-dimensional cell-array of column-cell-arrays, each column with ## labels for all parameters, the first column contains the numerical ## indices of the parameters. The default jacobian function will call ## the model function with the second argument set with fields @code{f}: ## as the @code{f} passed to the jacobian function, @code{plabels}: ## cell-array of 1x1 cell-arrays with the entries of the ## column-cell-arrays of @code{plabels} as passed to the jacobian ## function corresponding to current parameter, @code{side}: @code{0} ## for one-sided interval, @code{1} or @code{2}, respectively, for the ## sides of a two-sided interval, and @code{parallel}: logical scalar ## indicating parallel computation of partial derivatives. ## ## @code{diffp}: column vector of fractional intervals (doubled for ## central intervals) supposed to be used by jacobian functions ## performing finite differencing. Default: @code{.001 * ones (size (parameters))}. The default jacobian function will use these as ## absolute intervals for parameters with value zero. ## ## @code{diff_onesided}: logical column vector indicating that one-sided ## intervals should be used by jacobian functions performing finite ## differencing. Default: @code{false (size (parameters))}. ## ## @code{fixed}: logical column vector indicating which parameters ## should not be optimized, but kept to their inital value. Fixing is ## done independently of the backend, but the backend may choose to fix ## additional parameters under certain conditions. ## ## @code{lbound}, @code{ubound}: column vectors of lower and upper ## bounds for parameters. Default: @code{-Inf} and @code{+Inf}, ## respectively. The bounds are non-strict, i.e. parameters are allowed ## to be exactly equal to a bound. The default jacobian function will ## respect bounds (but no further inequality constraints) in finite ## differencing. ## ## @code{inequc}: Further inequality constraints. Cell-array containing ## up to four entries, two entries for linear inequality constraints ## and/or one or two entries for general inequality constraints. Either ## linear or general constraints may be the first entries, but the two ## entries for linear constraints must be adjacent and, if two entries ## are given for general constraints, they also must be adjacent. The ## two entries for linear constraints are a matrix (say @code{m}) and a ## vector (say @code{v}), specifying linear inequality constraints of ## the form @code{m.' * parameters + v >= 0}. The first entry for ## general constraints must be a differentiable vector valued function ## (say @code{h}), specifying general inequality constraints of the form ## @code{h (p[, idx]) >= 0}; @code{p} is the column vector of optimized ## paraters and the optional argument @code{idx} is a logical index. ## @code{h} has to return the values of all constraints if @code{idx} is ## not given, and has to return only the indexed constraints if ## @code{idx} is given (so computation of the other constraints can be ## spared). In gradient determination, this function may be called with ## an informational third argument, whose content depends on the ## function for gradient determination. If a second entry for general ## inequality constraints is given, it must be a function computing the ## jacobian of the constraints with respect to the parameters. For this ## function, the description of @code{dfdp} above applies, except that ## it is called with 3 arguments since it has an additional argument ## @code{idx} --- a logical index --- at second position, indicating ## which rows of the jacobian must be returned, and except that the ## default function calls @code{h} with 3 arguments, since the argument ## @code{idx} is also supplied. Note that specifying linear constraints ## as general constraints will generally waste performance, even if ## further, non-linear, general constraints are also specified. ## ## @code{equc}: Equality constraints. Specified the same way as ## inequality constraints (see @code{inequc}). ## ## @code{cpiv}: Function for complementary pivoting, usable in ## algorithms for constraints. Default: @ cpiv_bard. Only the default ## function is supplied with the package. ## ## @code{weights}: Array of weights for the residuals. Dimensions must ## match. ## ## @code{TolFun}: Minimum fractional improvement in objective function ## (e.g. sum of squares) in an iteration (abortion criterium). Default: ## .0001. ## ## @code{MaxIter}: Maximum number of iterations (abortion criterium). ## Default: backend-specific. ## ## @code{fract_prec}: Column Vector, minimum fractional change of ## parameters in an iteration (abortion criterium if violated in two ## consecutive iterations). Default: backend-specific. ## ## @code{max_fract_change}: Column Vector, enforced maximum fractional ## change in parameters in an iteration. Default: backend-specific. ## ## @code{Display}: String indicating the degree of verbosity. Default: ## @code{"off"}. Possible values are currently @code{"off"} (no ## messages) and @code{"iter"} (some messages after each iteration). ## Support of this setting and its exact interpretation are ## backend-specific. ## ## @code{plot_cmd}: Function enabling backend to plot results or ## intermediate results. Will be called with current computed ## residualse. Default: plot nothing. ## ## @code{debug}: Logical scalar, default: @code{false}. Will be passed ## to the backend, which might print debugging information if true. ## ## Structure-based parameter handling ## ## The setting @code{param_order} is a cell-array with names of the ## optimized parameters. If not given, and initial parameters are a ## structure, all parameters in the structure are optimized. It is an ## error if @code{param_order} is not given and there are any ## non-structure-based configuration items or functions. ## ## The initial parameters @var{pin} can be given as a structure ## containing at least all fields named in @code{param_order}. In this ## case the returned parameters @var{p} will also be a structure. ## ## Each user-supplied function can be called with the argument ## containing the current parameters being a structure instead of a ## column vector. For this, a corresponding setting must be set to ## @code{true}: @code{f_pstruct} (model function), @code{dfdp_pstruct} ## (jacobian of model function), @code{f_inequc_pstruct} (general ## inequality constraints), @code{df_inequc_pstruct} (jacobian of ## general inequality constraints), @code{f_equc_pstruct} (general ## equality constraints), and @code{df_equc_pstruct} (jacobian of ## general equality constraints). If a jacobian-function is configured ## in such a way, it must return the columns of the jacobian as fields ## of a structure under the respective parameter names. ## ## Similarly, for specifying linear constraints, instead of the matrix ## (called @code{m} above), a structure containing the rows of the ## matrix in fields under the respective parameter names can be given. ## In this case, rows containing only zeros need not be given. ## ## The vector-based settings @code{lbound}, @code{ubound}, ## @code{fixed}, @code{diffp}, @code{diff_onesided}, @code{fract_prec}, ## and @code{max_fract_change} can be replaced by the setting ## @code{param_config}. It is a structure that can contain fields named ## in @code{param_order}. For each such field, there may be subfields ## with the same names as the above vector-based settings, but ## containing a scalar value for the respective parameter. If ## @code{param_config} is specified, none of the above ## vector/matrix-based settings may be used. ## ## Additionally, named parameters are allowed to be non-scalar real ## arrays. In this case, their dimensions are given by the setting ## @code{param_dims}, a cell-array of dimension vectors, each containing ## at least two dimensions; if not given, dimensions are taken from the ## initial parameters, if these are given in a structure. Any ## vector-based settings or not structure-based linear constraints then ## must correspond to an order of parameters with all parameters ## reshaped to vectors and concatenated in the user-given order of ## parameter names. Structure-based settings or structure-based initial ## parameters must contain arrays with dimensions reshapable to those of ## the respective parameters. ## ## Description of backends (currently only one) ## ## "lm_svd_feasible" ## ## A Levenberg/Marquardt algorithm using singular value decomposition ## and featuring constraints which must be met by the initial parameters ## and are attempted to be kept met throughout the optimization. ## ## Parameters with identical lower and upper bounds will be fixed. ## ## Returned value @var{cvg} will be @code{0}, @code{2}, or @code{3}. ## ## Backend-specific defaults are: @code{MaxIter}: 20, @code{fract_prec}: ## @code{zeros (size (parameters))}, @code{max_fract_change}: @code{Inf} ## for all parameters. ## ## Interpretation of @code{Display}: if set to @code{"iter"}, currently ## @code{plot_cmd} is evaluated for each iteration, and some further ## diagnostics may be printed. ## ## Specific option: @code{lm_svd_feasible_alt_s}: if falling back to ## nearly gradient descent, do it more like original Levenberg/Marquardt ## method, with descent in each gradient component; for testing only. ## ## @seealso {nonlin_curvefit} ## ## @end deftypefn function [p, resid, cvg, outp] = nonlin_residmin (varargin) if (nargin == 1) p = __nonlin_residmin__ (varargin{1}); return; endif if (nargin < 2 || nargin > 3) print_usage (); endif if (nargin == 2) varargin{3} = struct (); endif varargin{4} = struct (); [p, resid, cvg, outp] = __nonlin_residmin__ (varargin{:}); endfunction

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