octave-optim 1.3.0-1 / usr / share / octave / packages / optim-1.3.0 / doc-cache

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LinearRegression


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 -- Function File: [P,E_VAR,R,P_VAR,Y_VAR] = LinearRegression (F,Y)
 -- Function File: [P,E_VAR,R,P_VAR,Y_VAR] = LinearRegression (F,Y,W)

     general linear regression

     determine the parameters p_j (j=1,2,...,m) such that the function
     f(x) = sum_(i=1,...,m) p_j*f_j(x) is the best fit to the given
     values y_i = f(x_i)

     parameters:
        * F is an n*m matrix with the values of the basis functions at
          the support points.  In column j give the values of f_j at the
          points x_i (i=1,2,...,n)
        * Y is a column vector of length n with the given values
        * W is n column vector of of length n vector with the weights of
          data points

     return values:
        * P is the vector of length m with the estimated values of the
          parameters
        * E_VAR is the estimated variance of the difference between
          fitted and measured values
        * R is the weighted norm of the residual
        * P_VAR is the estimated variance of the parameters p_j
        * Y_VAR is the estimated variance of the dependend variables

     Caution: do NOT request Y_VAR for large data sets, as a n by n
     matrix is generated

     See also: regress,leasqr,nonlin_curvefit,polyfit,wpolyfit,expfit.




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general linear regression



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adsmax


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ADSMAX  Alternating directions method for direct search optimization.
        [x, fmax, nf] = ADSMAX(FUN, x0, STOPIT, SAVIT, P) attempts to
        maximize the function FUN, using the starting vector x0.
        The alternating directions direct search method is used.
        Output arguments:
               x    = vector yielding largest function value found,
               fmax = function value at x,
               nf   = number of function evaluations.
        The iteration is terminated when either
               - the relative increase in function value between successive
                 iterations is <= STOPIT(1) (default 1e-3),
               - STOPIT(2) function evaluations have been performed
                 (default inf, i.e., no limit), or
               - a function value equals or exceeds STOPIT(3)
                 (default inf, i.e., no test on function values).
        Progress of the iteration is not shown if STOPIT(5) = 0 (default 1).
        If a non-empty fourth parameter string SAVIT is present, then
        `SAVE SAVIT x fmax nf' is executed after each inner iteration.
        By default, the search directions are the co-ordinate directions.
        The columns of a fifth parameter matrix P specify alternative search
        directions (P = EYE is the default).
        NB: x0 can be a matrix.  In the output argument, in SAVIT saves,
            and in function calls, x has the same shape as x0.
        ADSMAX(fun, x0, STOPIT, SAVIT, P, P1, P2,...) allows additional
        arguments to be passed to fun, via feval(fun,x,P1,P2,...).
     Reference:
     N. J. Higham, Optimization by direct search in matrix computations,
        SIAM J. Matrix Anal. Appl, 14(2): 317-333, 1993.
     N. J. Higham, Accuracy and Stability of Numerical Algorithms,
        Second edition, Society for Industrial and Applied Mathematics,
        Philadelphia, PA, 2002; sec. 20.5.



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ADSMAX  Alternating directions method for direct search optimization.



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battery


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 battery.m: repeatedly call bfgs using a battery of 
 start values, to attempt to find global min
 of a nonconvex function

 INPUTS:
 func: function to mimimize
 args: args of function
 minarg: argument to minimize w.r.t. (usually = 1)
 startvals: kxp matrix of values to try for sure (don't include all zeros, that's automatic)
 max iters per start value
 number of additional random start values to try

 OUTPUT: theta - the best value found - NOT iterated to convergence



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 battery.m: repeatedly call bfgs using a battery of 
 start values, to attempt t



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bfgsmin


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 bfgsmin: bfgs or limited memory bfgs minimization of function

 Usage: [x, obj_value, convergence, iters] = bfgsmin(f, args, control)

 The function must be of the form
 [value, return_2,..., return_m] = f(arg_1, arg_2,..., arg_n)
 By default, minimization is w.r.t. arg_1, but it can be done
 w.r.t. any argument that is a vector. Numeric derivatives are
 used unless analytic derivatives are supplied. See bfgsmin_example.m
 for methods.

 Arguments:
 * f: name of function to minimize (string)
 * args: a cell array that holds all arguments of the function
 	The argument with respect to which minimization is done
 	MUST be a vector
 * control: an optional cell array of 1-8 elements. If a cell
   array shorter than 8 elements is provided, the trailing elements
   are provided with default values.
 	* elem 1: maximum iterations  (positive integer, or -1 or Inf for unlimited (default))
 	* elem 2: verbosity
 		0 = no screen output (default)
 		1 = only final results
 		2 = summary every iteration
 		3 = detailed information
 	* elem 3: convergence criterion
 		1 = strict (function, gradient and param change) (default)
 		0 = weak - only function convergence required
 	* elem 4: arg in f_args with respect to which minimization is done (default is first)
 	* elem 5: (optional) Memory limit for lbfgs. If it's a positive integer
 		then lbfgs will be use. Otherwise ordinary bfgs is used
 	* elem 6: function change tolerance, default 1e-12
 	* elem 7: parameter change tolerance, default 1e-6
 	* elem 8: gradient tolerance, default 1e-5

 Returns:
 * x: the minimizer
 * obj_value: the value of f() at x
 * convergence: 1 if normal conv, other values if not
 * iters: number of iterations performed

 Example: see bfgsmin_example.m



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 bfgsmin: bfgs or limited memory bfgs minimization of function



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bfgsmin_example


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 initial values



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 initial values




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brent_line_min


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 -- Function File: [S,V,N] brent_line_min ( F,DF,ARGS,CTL )
     Line minimization of f along df

     Finds minimum of f on line x0 + dx*w | a < w < b by bracketing.  a
     and b are passed through argument ctl.

     Arguments
--------------

        * F : string : Name of function.  Must return a real value
        * ARGS : cell : Arguments passed to f or RxC : f's only
          argument.  x0 must be at ARGS{ CTL(2) }
        * CTL : 5 : (optional) Control variables, described below.

     Returned values
--------------------

        * S : 1 : Minimum is at x0 + s*dx
        * V : 1 : Value of f at x0 + s*dx
        * NEV : 1 : Number of function evaluations

     Control Variables
----------------------

        * CTL(1) : Upper bound for error on s Default=sqrt(eps)
        * CTL(2) : Position of minimized argument in args Default= 1
        * CTL(3) : Maximum number of function evaluations Default= inf
        * CTL(4) : a Default=-inf
        * CTL(5) : b Default= inf

     Default values will be used if ctl is not passed or if nan values
     are given.




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Line minimization of f along df



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cauchy


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 -- Function File: cauchy (N, R, X, F )
     Return the Taylor coefficients and numerical differentiation of a
     function F for the first N-1 coefficients or derivatives using the
     fft.  N is the number of points to evaluate, R is the radius of
     convergence, needs to be chosen less then the smallest singularity,
     X is point to evaluate the Taylor expansion or differentiation.
     For example,

     If X is a scalar, the function F is evaluated in a row vector of
     length N.  If X is a column vector, F is evaluated in a matrix of
     length(x)-by-N elements and must return a matrix of the same size.

          d = cauchy(16, 1.5, 0, @(x) exp(x));
          => d(2) = 1.0000 # first (2-1) derivative of function f (index starts from zero)




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Return the Taylor coefficients and numerical differentiation of a
function F for



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cdiff


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 c = cdiff (func,wrt,N,dfunc,stack,dx) - Code for num. differentiation
   = "function df = dfunc (var1,..,dvar,..,varN) .. endfunction
 
 Returns a string of octave code that defines a function 'dfunc' that
 returns the derivative of 'func' with respect to it's 'wrt'th
 argument.

 The derivatives are obtained by symmetric finite difference.

 dfunc()'s return value is in the same format as that of  ndiff()

 func  : string : name of the function to differentiate

 wrt   : int    : position, in argument list, of the differentiation
                  variable.                                Default:1

 N     : int    : total number of arguments taken by 'func'. 
                  If N=inf, dfunc will take variable argument list.
                                                         Default:wrt

 dfunc : string : Name of the octave function that returns the
                   derivatives.                   Default:['d',func]

 stack : string : Indicates whether 'func' accepts vertically
                  (stack="rstack") or horizontally (stack="cstack")
                  arguments. Any other string indicates that 'func'
                  does not allow stacking.                Default:''

 dx    : real   : Step used in the symmetric difference scheme.
                                                  Default:10*sqrt(eps)

 See also : ndiff, eval, todisk



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 c = cdiff (func,wrt,N,dfunc,stack,dx) - Code for num.



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cg_min


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 -- Function File: [X0,V,NEV] cg_min ( F,DF,ARGS,CTL )
     NonLinear Conjugate Gradient method to minimize function F.

     Arguments
--------------

        * F : string : Name of function.  Return a real value
        * DF : string : Name of f's derivative.  Returns a (R*C) x 1
          vector
        * ARGS: cell : Arguments passed to f.
        * CTL : 5-vec : (Optional) Control variables, described below

     Returned values
--------------------

        * X0 : matrix : Local minimum of f
        * V : real : Value of f in x0
        * NEV : 1 x 2 : Number of evaluations of f and of df

     Control Variables
----------------------

        * CTL(1) : 1 or 2 : Select stopping criterion amongst :
        * CTL(1)==0 : Default value
        * CTL(1)==1 : Stopping criterion : Stop search when value
          doesn't improve, as tested by ctl(2) > Deltaf/max(|f(x)|,1)
          where Deltaf is the decrease in f observed in the last
          iteration (each iteration consists R*C line searches).
        * CTL(1)==2 : Stopping criterion : Stop search when updates are
          small, as tested by ctl(2) > max { dx(i)/max(|x(i)|,1) | i in
          1..N } where dx is the change in the x that occured in the
          last iteration.
        * CTL(2) : Threshold used in stopping tests.  Default=10*eps
        * CTL(2)==0 : Default value
        * CTL(3) : Position of the minimized argument in args Default=1
        * CTL(3)==0 : Default value
        * CTL(4) : Maximum number of function evaluations Default=inf
        * CTL(4)==0 : Default value
        * CTL(5) : Type of optimization:
        * CTL(5)==1 : "Fletcher-Reves" method
        * CTL(5)==2 : "Polak-Ribiere" (Default)
        * CTL(5)==3 : "Hestenes-Stiefel" method

     CTL may have length smaller than 4.  Default values will be used if
     ctl is not passed or if nan values are given.

     Example:
-------------

     function r=df( l ) b=[1;0;-1]; r = -( 2*l{1} - 2*b +
     rand(size(l{1}))); endfunction
     function r=ff( l ) b=[1;0;-1]; r = (l{1}-b)' * (l{1}-b);
     endfunction
     ll = { [10; 2; 3] };
     ctl(5) = 3;
     [x0,v,nev]=cg_min( "ff", "df", ll, ctl )

     Comment: In general, BFGS method seems to be better performin in
     many cases but requires more computation per iteration

     See also: bfgsmin,
     http://en.wikipedia.org/wiki/Nonlinear_conjugate_gradient .




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NonLinear Conjugate Gradient method to minimize function F.



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cpiv_bard


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 [lb, idx, ridx, mv] = cpiv_bard (v, m[, incl])

 v: column vector; m: matrix; incl (optional): index. length (v)
 must equal rows (m). Finds column vectors w and l with w == v + m *
 l, w >= 0, l >= 0, l.' * w == 0. Chooses idx, w, and l so that
 l(~idx) == 0, l(idx) == -inv (m(idx, idx)) * v(idx), w(idx) roughly
 == 0, and w(~idx) == v(~idx) + m(idx, ~idx).' * l(idx). idx indexes
 at least everything indexed by incl, but l(incl) may be < 0. lb:
 l(idx) (column vector); idx: logical index, defined above; ridx:
 ~idx & w roughly == 0; mv: [m, v] after performing a Gauss-Jordan
 'sweep' (with gjp.m) on each diagonal element indexed by idx.
 Except the handling of incl (which enables handling of equality
 constraints in the calling code), this is called solving the
 'complementary pivot problem' (Cottle, R. W. and Dantzig, G. B.,
 'Complementary pivot theory of mathematical programming', Linear
 Algebra and Appl. 1, 102--125. References for the current
 algorithm: Bard, Y.: Nonlinear Parameter Estimation, p. 147--149,
 Academic Press, New York and London 1974; Bard, Y., 'An eclectic
 approach to nonlinear programming', Proc. ANU Sem. Optimization,
 Canberra, Austral. Nat. Univ.).



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 [lb, idx, ridx, mv] = cpiv_bard (v, m[, incl])



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curvefit_stat


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 -- Function File: INFO = curvefit_stat (F, P, X, Y, SETTINGS)

     Frontend for computation of statistics for fitting of values,
     computed by a model function, to observed values.

     Please refer to the description of 'residmin_stat'.  The only
     differences to 'residmin_stat' are the additional arguments X
     (independent values) and Y (observations), that the model function
     F, if provided, has a second obligatory argument which will be set
     to X and is supposed to return guesses for the observations (with
     the same dimensions), and that the possibly user-supplied function
     for the jacobian of the model function has also a second obligatory
     argument which will be set to X.

     See also: residmin_stat.




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Frontend for computation of statistics for fitting of values, computed
by a mode



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d2_min


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 [x,v,nev,h,args] = d2_min(f,d2f,args,ctl,code) - Newton-like minimization

 Minimize f(x) using 1st and 2nd derivatives. Any function w/ second
 derivatives can be minimized, as in Newton. f(x) decreases at each
 iteration, as in Levenberg-Marquardt. This function is inspired from the
 Levenberg-Marquardt algorithm found in the book "Numerical Recipes".

 ARGUMENTS :
 f    : string : Cost function's name

 d2f  : string : Name of function returning the cost (1x1), its
                 differential (1xN) and its second differential or it's
                 pseudo-inverse (NxN) (see ctl(5) below) :

                 [v,dv,d2v] = d2f (x).

 args : list   : f and d2f's arguments. By default, minimize the 1st
     or matrix : argument.

 ctl  : vector : Control arguments (see below)
      or struct

 code : string : code will be evaluated after each outer loop that
                 produced some (any) improvement. Variables visible from
                 "code" include "x", the best parameter found, "v" the
                 best value and "args", the list of all arguments. All can
                 be modified. This option can be used to re-parameterize 
                 the argument space during optimization

 CONTROL VARIABLE ctl : (optional). May be a struct or a vector of length
 ---------------------- 5 or less where NaNs are ignored. Default values
                        are written <value>.
 FIELD  VECTOR
 NAME    POS

 ftol, f N/A    : Stop search when value doesn't improve, as tested by

                   f > Deltaf/max(|f(x)|,1)

             where Deltaf is the decrease in f observed in the last
             iteration.                                     <10*sqrt(eps)>

 utol, u N/A    : Stop search when updates are small, as tested by

                   u > max { dx(i)/max(|x(i)|,1) | i in 1..N }

             where  dx is the change in the x that occured in the last
             iteration.                                              <NaN>

 dtol, d N/A    : Stop search when derivative is small, as tested by
 
                   d > norm (dv)                                     <eps>

 crit, c ctl(1) : Set one stopping criterion, 'ftol' (c=1), 'utol' (c=2)
                  or 'dtol' (c=3) to the value of by the 'tol' option. <1>

 tol, t  ctl(2) : Threshold in termination test chosen by 'crit'  <10*eps>

 narg, n ctl(3) : Position of the minimized argument in args           <1>
 maxev,m ctl(4) : Maximum number of function evaluations             <inf>
 maxout,m       : Maximum number of outer loops                      <inf>
 id2f, i ctl(5) : 0 if d2f returns the 2nd derivatives, 1 if           <0>
                  it returns its pseudo-inverse.

 verbose, v N/A : Be more or less verbose (quiet=0)                    <0>



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 [x,v,nev,h,args] = d2_min(f,d2f,args,ctl,code) - Newton-like minimization



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dcdp


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 function prt = dcdp (f, p, dp, func[, bounds])

 This is an interface to __dfdp__.m, similar to dfdp.m, but for
 functions only of parameters 'p', not of independents 'x'. See
 dfdp.m.

 dfpdp is more general and is meant to be used instead of dcdp in
 optimization.



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 function prt = dcdp (f, p, dp, func[, bounds])



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de_min


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 de_min: global optimisation using differential evolution

 Usage: [x, obj_value, nfeval, convergence] = de_min(fcn, control)

 minimization of a user-supplied function with respect to x(1:D),
 using the differential evolution (DE) method based on an algorithm
 by  Rainer Storn (http://www.icsi.berkeley.edu/~storn/code.html)
 See: http://www.softcomputing.net/tevc2009_1.pdf


 Arguments:  
 ---------------
 fcn        string : Name of function. Must return a real value
 control    vector : (Optional) Control variables, described below
         or struct

 Returned values:
 ----------------
 x          vector : parameter vector of best solution
 obj_value  scalar : objective function value of best solution
 nfeval     scalar : number of function evaluations
 convergence       : 1 = best below value to reach (VTR)
                     0 = population has reached defined quality (tol)
                    -1 = some values are close to constraints/boundaries
                    -2 = max number of iterations reached (maxiter)
                    -3 = max number of functions evaluations reached (maxnfe)

 Control variable:   (optional) may be named arguments (i.e. "name",value
 ----------------    pairs), a struct, or a vector, where
                     NaN's are ignored.

 XVmin        : vector of lower bounds of initial population
                *** note: by default these are no constraints ***
 XVmax        : vector of upper bounds of initial population
 constr       : 1 -> enforce the bounds not just for the initial population
 const        : data vector (remains fixed during the minimization)
 NP           : number of population members
 F            : difference factor from interval [0, 2]
 CR           : crossover probability constant from interval [0, 1]
 strategy     : 1 --> DE/best/1/exp           7 --> DE/best/1/bin
                2 --> DE/rand/1/exp           8 --> DE/rand/1/bin
                3 --> DE/target-to-best/1/exp 9 --> DE/target-to-best/1/bin
                4 --> DE/best/2/exp           10--> DE/best/2/bin
                5 --> DE/rand/2/exp           11--> DE/rand/2/bin
                6 --> DEGL/SAW/exp            else  DEGL/SAW/bin
 refresh      : intermediate output will be produced after "refresh"
                iterations. No intermediate output will be produced
                if refresh is < 1
 VTR          : Stopping criterion: "Value To Reach"
                de_min will stop when obj_value <= VTR.
                Use this if you know which value you expect.
 tol          : Stopping criterion: "tolerance"
                stops if (best-worst)/max(1,worst) < tol
                This stops basically if the whole population is "good".
 maxnfe       : maximum number of function evaluations
 maxiter      : maximum number of iterations (generations)

       The algorithm seems to work well only if [XVmin,XVmax] covers the 
       region where the global minimum is expected.
       DE is also somewhat sensitive to the choice of the
       difference factor F. A good initial guess is to choose F from
       interval [0.5, 1], e.g. 0.8.
       CR, the crossover probability constant from interval [0, 1]
       helps to maintain the diversity of the population and is
       rather uncritical but affects strongly the convergence speed.
       If the parameters are correlated, high values of CR work better.
       The reverse is true for no correlation.
       Experiments suggest that /bin likes to have a slightly
       larger CR than /exp.
       The number of population members NP is also not very critical. A
       good initial guess is 10*D. Depending on the difficulty of the
       problem NP can be lower than 10*D or must be higher than 10*D
       to achieve convergence.

 Default Values:
 ---------------
 XVmin = [-2];
 XVmax = [ 2];
 constr= 0;
 const = [];
 NP    = 10 *D
 F     = 0.8;
 CR    = 0.9;
 strategy = 12;
 refresh  = 0;
 VTR   = -Inf;
 tol   = 1.e-3;
 maxnfe  = 1e6;
 maxiter = 1000;


 Example to find the minimum of the Rosenbrock saddle:
 ----------------------------------------------------
 Define f as:
                    function result = f(x);
                      result = 100 * (x(2) - x(1)^2)^2 + (1 - x(1))^2;
                    end
 Then type:

 	ctl.XVmin = [-2 -2];
 	ctl.XVmax = [ 2  2];
 	[x, obj_value, nfeval, convergence] = de_min (@f, ctl);

 Keywords: global-optimisation optimisation minimisation



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 de_min: global optimisation using differential evolution



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deriv


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 -- Function File: DX = deriv (F, X0)
 -- Function File: DX = deriv (F, X0, H)
 -- Function File: DX = deriv (F, X0, H, O)
 -- Function File: DX = deriv (F, X0, H, O, N)
     Calculate derivate of function F.

     F must be a function handle or the name of a function that takes X0
     and returns a variable of equal length and orientation.  X0 must be
     a numeric vector or scalar.

     H defines the step taken for the derivative calculation.  Defaults
     to 1e-7.

     O defines the order of the calculation.  Supported values are 2
     (h^2 order) or 4 (h^4 order).  Defaults to 2.

     N defines the derivative order.  Defaults to the 1st derivative of
     the function.  Can be up to the 4th derivative.

     Reference: Numerical Methods for Mathematics, Science, and
     Engineering by John H. Mathews.




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Calculate derivate of function F.



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dfdp


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 function prt = dfdp (x, f, p, dp, func[, bounds])
 numerical partial derivatives (Jacobian) df/dp for use with leasqr
 --------INPUT VARIABLES---------
 x=vec or matrix of indep var(used as arg to func) x=[x0 x1 ....]
 f=func(x,p) vector initialsed by user before each call to dfdp
 p= vec of current parameter values
 dp= fractional increment of p for numerical derivatives
      dp(j)>0 central differences calculated
      dp(j)<0 one sided differences calculated
      dp(j)=0 sets corresponding partials to zero; i.e. holds p(j) fixed
 func=function (string or handle) to calculate the Jacobian for,
      e.g. to calc Jacobian for function expsum prt=dfdp(x,f,p,dp,'expsum')
 bounds=two-column-matrix of lower and upper bounds for parameters
      If no 'bounds' options is specified to leasqr, it will call
      dfdp without the 'bounds' argument.
----------OUTPUT VARIABLES-------
 prt= Jacobian Matrix prt(i,j)=df(i)/dp(j)
================================

 dfxpdp is more general and is meant to be used instead of dfdp in
 optimization.



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 function prt = dfdp (x, f, p, dp, func[, bounds])
 numerical partial derivative



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dfpdp


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 function jac = dfpdp (p, func[, hook])

 Returns Jacobian of func (p) with respect to p with finite
 differencing. The optional argument hook is a structure which can
 contain the following fields at the moment:

 hook.f: value of func(p) for p as given in the arguments

 hook.diffp: positive vector of fractional steps from given p in
 finite differencing (actual steps may be smaller if bounds are
 given). The default is .001 * ones (size (p)).

 hook.diff_onesided: logical vector, indexing elements of p for
 which only one-sided differences should be computed (faster); even
 if not one-sided, differences might not be exactly central if
 bounds are given. The default is false (size (p)).

 hook.fixed: logical vector, indexing elements of p for which zero
 should be returned instead of the guessed partial derivatives
 (useful in optimization if some parameters are not optimized, but
 are 'fixed').

 hook.lbound, hook.ubound: vectors of lower and upper parameter
 bounds (or -Inf or +Inf, respectively) to be respected in finite
 differencing. The consistency of bounds is not checked.



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 function jac = dfpdp (p, func[, hook])



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dfxpdp


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 function jac = dfxpdp (x, p, func[, hook])

 Returns Jacobian of func (p, x) with respect to p with finite
 differencing. The optional argument hook is a structure which can
 contain the following fields at the moment:

 hook.f: value of func(p, x) for p and x as given in the arguments

 hook.diffp: positive vector of fractional steps from given p in
 finite differencing (actual steps may be smaller if bounds are
 given). The default is .001 * ones (size (p));

 hook.diff_onesided: logical vector, indexing elements of p for
 which only one-sided differences should be computed (faster); even
 if not one-sided, differences might not be exactly central if
 bounds are given. The default is false (size (p)).

 hook.fixed: logical vector, indexing elements of p for which zero
 should be returned instead of the guessed partial derivatives
 (useful in optimization if some parameters are not optimized, but
 are 'fixed').

 hook.lbound, hook.ubound: vectors of lower and upper parameter
 bounds (or -Inf or +Inf, respectively) to be respected in finite
 differencing. The consistency of bounds is not checked.



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 function jac = dfxpdp (x, p, func[, hook])



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expfit


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 USAGE  [alpha,c,rms] = expfit( deg, x1, h, y )

 Prony's method for non-linear exponential fitting

 Fit function:   \sum_1^{deg} c(i)*exp(alpha(i)*x)

 Elements of data vector y must correspond to
 equidistant x-values starting at x1 with stepsize h

 The method is fully compatible with complex linear
 coefficients c, complex nonlinear coefficients alpha
 and complex input arguments y, x1, non-zero h .
 Fit-order deg  must be a real positive integer.

 Returns linear coefficients c, nonlinear coefficients
 alpha and root mean square error rms. This method is
 known to be more stable than 'brute-force' non-linear
 least squares fitting.

 Example
    x0 = 0; step = 0.05; xend = 5; x = x0:step:xend;
    y = 2*exp(1.3*x)-0.5*exp(2*x);
    error = (rand(1,length(y))-0.5)*1e-4;
    [alpha,c,rms] = expfit(2,x0,step,y+error)

  alpha =
    2.0000
    1.3000
  c =
    -0.50000
     2.00000
  rms = 0.00028461

 The fit is very sensitive to the number of data points.
 It doesn't perform very well for small data sets.
 Theoretically, you need at least 2*deg data points, but
 if there are errors on the data, you certainly need more.

 Be aware that this is a very (very,very) ill-posed problem.
 By the way, this algorithm relies heavily on computing the
 roots of a polynomial. I used 'roots.m', if there is
 something better please use that code.

 Demo for a complex fit-function:
 deg= 2; N= 20; x1= -(1+i), x= linspace(x1,1+i/2,N).';
 h = x(2) - x(1)
 y= (2+i)*exp( (-1-2i)*x ) + (-1+3i)*exp( (2+3i)*x );
 A= 5e-2; y+= A*(randn(N,1)+randn(N,1)*i); % add complex noise
 [alpha,c,rms]= expfit( deg, x1, h, y )



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 USAGE  [alpha,c,rms] = expfit( deg, x1, h, y )



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fmin


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 alias for fminbnd



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 alias for fminbnd




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fmins


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 -- Function File: [X] = fmins( F,X0,OPTIONS,GRAD,P1,P2, ...)

     Find the minimum of a funtion of several variables.  By default the
     method used is the Nelder&Mead Simplex algorithm

     Example usage: fmins(inline('(x(1)-5).^2+(x(2)-8).^4'),[0;0])

     *Inputs*
     F
          A string containing the name of the function to minimize
     X0
          A vector of initial parameters fo the function F.
     OPTIONS
          Vector with control parameters (not all parameters are used)
          options(1) - Show progress (if 1, default is 0, no progress)
          options(2) - Relative size of simplex (default 1e-3)
          options(6) - Optimization algorithm
             if options(6)==0 - Nelder & Mead simplex (default)
             if options(6)==1 - Multidirectional search Method
             if options(6)==2 - Alternating Directions search
          options(5)
             if options(6)==0 && options(5)==0 - regular simplex
             if options(6)==0 && options(5)==1 - right-angled simplex
                Comment: the default is set to "right-angled simplex".
                  this works better for me on a broad range of problems,
                  although the default in nmsmax is "regular simplex"
          options(10) - Maximum number of function evaluations
     GRAD
          Unused (For compatibility with Matlab)
     P1,P2, ...
          Optional parameters for function F




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Find the minimum of a funtion of several variables.



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gjp


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 m = gjp (m, k[, l])

 m: matrix; k, l: row- and column-index of pivot, l defaults to k.

 Gauss-Jordon pivot as defined in Bard, Y.: Nonlinear Parameter
 Estimation, p. 296, Academic Press, New York and London 1974. In
 the pivot column, this seems not quite the same as the usual
 Gauss-Jordan(-Clasen) pivot. Bard gives Beaton, A. E., 'The use of
 special matrix operators in statistical calculus' Research Bulletin
 RB-64-51 (1964), Educational Testing Service, Princeton, New Jersey
 as a reference, but this article is not easily accessible. Another
 reference, whose definition of gjp differs from Bards by some
 signs, is Clarke, R. B., 'Algorithm AS 178: The Gauss-Jordan sweep
 operator with detection of collinearity', Journal of the Royal
 Statistical Society, Series C (Applied Statistics) (1982), 31(2),
 166--168.



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 m = gjp (m, k[, l])



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jacobs


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 -- Function File: Df = jacobs (X, F)
 -- Function File: Df = jacobs (X, F, HOOK)
     Calculate the jacobian of a function using the complex step method.

     Let F be a user-supplied function.  Given a point X at which we
     seek for the Jacobian, the function 'jacobs' returns the Jacobian
     matrix 'd(f(1), ..., df(end))/d(x(1), ..., x(n))'.  The function
     uses the complex step method and thus can be applied to real
     analytic functions.

     The optional argument HOOK is a structure with additional options.
     HOOK can have the following fields:
        * 'h' - can be used to define the magnitude of the complex step
          and defaults to 1e-20; steps larger than 1e-3 are not allowed.
        * 'fixed' - is a logical vector internally usable by some
          optimization functions; it indicates for which elements of X
          no gradient should be computed, but zero should be returned.

     For example:

          f = @(x) [x(1)^2 + x(2); x(2)*exp(x(1))];
          Df = jacobs ([1, 2], f)




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Calculate the jacobian of a function using the complex step method.



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leasqr


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function [f,p,cvg,iter,corp,covp,covr,stdresid,Z,r2]=
                   leasqr(x,y,pin,F,{stol,niter,wt,dp,dFdp,options})

 Levenberg-Marquardt nonlinear regression of f(x,p) to y(x).

 Version 3.beta
 Optional parameters are in braces {}.
 x = vector or matrix of independent variables.
 y = vector or matrix of observed values.
 wt = statistical weights (same dimensions as y).  These should be
   set to be proportional to (sqrt of var(y))^-1; (That is, the
   covariance matrix of the data is assumed to be proportional to
   diagonal with diagonal equal to (wt.^2)^-1.  The constant of
   proportionality will be estimated.); default = ones( size (y)).
 pin = vec of initial parameters to be adjusted by leasqr.
 dp = fractional increment of p for numerical partial derivatives;
   default = .001*ones(size(pin))
   dp(j) > 0 means central differences on j-th parameter p(j).
   dp(j) < 0 means one-sided differences on j-th parameter p(j).
   dp(j) = 0 holds p(j) fixed i.e. leasqr wont change initial guess: pin(j)
 F = name of function in quotes or function handle; the function
   shall be of the form y=f(x,p), with y, x, p of the form y, x, pin
   as described above.
 dFdp = name of partial derivative function in quotes or function
 handle; default is 'dfdp', a slow but general partial derivatives
 function; the function shall be of the form
 prt=dfdp(x,f,p,dp,F[,bounds]). For backwards compatibility, the
 function will only be called with an extra 'bounds' argument if the
 'bounds' option is explicitely specified to leasqr (see dfdp.m).
 stol = scalar tolerance on fractional improvement in scalar sum of
   squares = sum((wt.*(y-f))^2); default stol = .0001;
 niter = scalar maximum number of iterations; default = 20;
 options = structure, currently recognized fields are 'fract_prec',
 'max_fract_change', 'inequc', 'bounds', and 'equc'. For backwards
 compatibility, 'options' can also be a matrix whose first and
 second column contains the values of 'fract_prec' and
 'max_fract_change', respectively.
   Field 'options.fract_prec': column vector (same length as 'pin')
   of desired fractional precisions in parameter estimates.
   Iterations are terminated if change in parameter vector (chg)
   relative to current parameter estimate is less than their
   corresponding elements in 'options.fract_prec' [ie. all (abs
   (chg) < abs (options.fract_prec .* current_parm_est))] on two
   consecutive iterations, default = zeros().
   Field 'options.max_fract_change': column vector (same length as
   'pin) of maximum fractional step changes in parameter vector.
   Fractional change in elements of parameter vector is constrained to
   be at most 'options.max_fract_change' between sucessive iterations.
   [ie. abs(chg(i))=abs(min([chg(i)
   options.max_fract_change(i)*current param estimate])).], default =
   Inf*ones().
   Field 'options.inequc': cell-array containing up to four entries,
   two entries for linear inequality constraints and/or one or two
   entries for general inequality constraints. Initial parameters
   must satisfy these 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 m) and a vector
   (say v), specifying linear inequality constraints of the form
   `m.' * parameters + v >= 0'. If the constraints are just bounds,
   it is suggested to specify them in 'options.bounds' instead,
   since then some sanity tests are performed, and since the
   function 'dfdp.m' is guarantied not to violate constraints during
   determination of the numeric gradient only for those constraints
   specified as 'bounds' (possibly with violations due to a certain
   inaccuracy, however, except if no constraints except bounds are
   specified). The first entry for general constraints must be a
   differentiable vector valued function (say h), specifying general
   inequality constraints of the form `h (p[, idx]) >= 0'; p is the
   column vector of optimized paraters and the optional argument idx
   is a logical index. h has to return the values of all constraints
   if idx is not given, and has to return only the indexed
   constraints if idx is given (so computation of the other
   constraints can be spared). If a second entry for general
   constraints is given, it must be a function (say dh) which
   returnes a matrix whos rows contain the gradients of the
   constraint function h with respect to the optimized parameters.
   It has the form jac_h = dh (vh, p, dp, h, idx[, bounds]); p is
   the column vector of optimized parameters, and idx is a logical
   index --- only the rows indexed by idx must be returned (so
   computation of the others can be spared). The other arguments of
   dh are for the case that dh computes numerical gradients: vh is
   the column vector of the current values of the constraint
   function h, with idx already applied. h is a function h (p) to
   compute the values of the constraints for parameters p, it will
   return only the values indexed by idx. dp is a suggestion for
   relative step width, having the same value as the argument 'dp'
   of leasqr above. If bounds were specified to leasqr, they are
   provided in the argument bounds of dh, to enable their
   consideration in determination of numerical gradients. If dh is
   not specified to leasqr, numerical gradients are computed in the
   same way as with 'dfdp.m' (see above). If some constraints are
   linear, they should be specified as linear constraints (or
   bounds, if applicable) for reasons of performance, even if
   general constraints are also specified.
   Field 'options.bounds': two-column-matrix, one row for each
   parameter in 'pin'. Each row contains a minimal and maximal value
   for each parameter. Default: [-Inf, Inf] in each row. If this
   field is used with an existing user-side function for 'dFdp'
   (see above) the functions interface might have to be changed.
   Field 'options.equc': equality constraints, specified the same
   way as inequality constraints (see field 'options.inequc').
   Initial parameters must satisfy these constraints.
   Note that there is possibly a certain inaccuracy in honoring
   constraints, except if only bounds are specified. 
   _Warning_: If constraints (or bounds) are set, returned guesses
   of corp, covp, and Z are generally invalid, even if no constraints
   are active for the final parameters. If equality constraints are
   specified, corp, covp, and Z are not guessed at all.
   Field 'options.cpiv': Function for complementary pivot algorithm
   for inequality constraints, default: cpiv_bard. No different
   function is supplied.

          OUTPUT VARIABLES
 f = column vector of values computed: f = F(x,p).
 p = column vector trial or final parameters. i.e, the solution.
 cvg = scalar: = 1 if convergence, = 0 otherwise.
 iter = scalar number of iterations used.
 corp = correlation matrix for parameters.
 covp = covariance matrix of the parameters.
 covr = diag(covariance matrix of the residuals).
 stdresid = standardized residuals.
 Z = matrix that defines confidence region (see comments in the source).
 r2 = coefficient of multiple determination, intercept form.

 Not suitable for non-real residuals.

 References:
 Bard, Nonlinear Parameter Estimation, Academic Press, 1974.
 Draper and Smith, Applied Regression Analysis, John Wiley and Sons, 1981.



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function [f,p,cvg,iter,corp,covp,covr,stdresid,Z,r2]=
                   leasqr(



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line_min


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 [a,fx,nev] = line_min (f, dx, args, narg, h, nev_max) - Minimize f() along dx

 INPUT ----------
 f    : string  : Name of minimized function
 dx   : matrix  : Direction along which f() is minimized
 args : cell    : Arguments of f
 narg : integer : Position of minimized variable in args.  Default=1
 h    : scalar  : Step size to use for centered finite difference
 approximation of first and second derivatives. Default=1E-3.
 nev_max : integer : Maximum number of function evaluations.  Default=30

 OUTPUT ---------
 a    : scalar  : Value for which f(x+a*dx) is a minimum (*)
 fx   : scalar  : Value of f(x+a*dx) at minimum (*)
 nev  : integer : Number of function evaluations

 (*) The notation f(x+a*dx) assumes that args == {x}.

 Reference: David G Luenberger's Linear and Nonlinear Programming



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 [a,fx,nev] = line_min (f, dx, args, narg, h, nev_max) - Minimize f() along dx



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linprog


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 -- Function File: X = linprog (F, A, B)
 -- Function File: X = linprog (F, A, B, AEQ, BEQ)
 -- Function File: X = linprog (F, A, B, AEQ, BEQ, LB, UB)
 -- Function File: [X, FVAL] = linprog (...)
     Solve a linear problem.

     Finds

          min (f' * x)

     (both f and x are column vectors) subject to

          A   * x <= b
          Aeq * x  = beq
          lb <= x <= ub

     If not specified, AEQ and BEQ default to empty matrices.

     If not specified, the lower bound LB defaults to minus infinite and
     the upper bound UB defaults to infinite.

     See also: glpk.




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Solve a linear problem.



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mdsmax


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MDSMAX  Multidirectional search method for direct search optimization.
        [x, fmax, nf] = MDSMAX(FUN, x0, STOPIT, SAVIT) attempts to
        maximize the function FUN, using the starting vector x0.
        The method of multidirectional search is used.
        Output arguments:
               x    = vector yielding largest function value found,
               fmax = function value at x,
               nf   = number of function evaluations.
        The iteration is terminated when either
               - the relative size of the simplex is <= STOPIT(1)
                 (default 1e-3),
               - STOPIT(2) function evaluations have been performed
                 (default inf, i.e., no limit), or
               - a function value equals or exceeds STOPIT(3)
                 (default inf, i.e., no test on function values).
        The form of the initial simplex is determined by STOPIT(4):
          STOPIT(4) = 0: regular simplex (sides of equal length, the default),
          STOPIT(4) = 1: right-angled simplex.
        Progress of the iteration is not shown if STOPIT(5) = 0 (default 1).
        If a non-empty fourth parameter string SAVIT is present, then
        `SAVE SAVIT x fmax nf' is executed after each inner iteration.
        NB: x0 can be a matrix.  In the output argument, in SAVIT saves,
            and in function calls, x has the same shape as x0.
        MDSMAX(fun, x0, STOPIT, SAVIT, P1, P2,...) allows additional
        arguments to be passed to fun, via feval(fun,x,P1,P2,...).

 This implementation uses 2n^2 elements of storage (two simplices), where x0
 is an n-vector.  It is based on the algorithm statement in [2, sec.3],
 modified so as to halve the storage (with a slight loss in readability).

 References:
 [1] V. J. Torczon, Multi-directional search: A direct search algorithm for
     parallel machines, Ph.D. Thesis, Rice University, Houston, Texas, 1989.
 [2] V. J. Torczon, On the convergence of the multidirectional search
     algorithm, SIAM J. Optimization, 1 (1991), pp. 123-145.
 [3] N. J. Higham, Optimization by direct search in matrix computations,
     SIAM J. Matrix Anal. Appl, 14(2): 317-333, 1993.
 [4] N. J. Higham, Accuracy and Stability of Numerical Algorithms,
        Second edition, Society for Industrial and Applied Mathematics,
        Philadelphia, PA, 2002; sec. 20.5.



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MDSMAX  Multidirectional search method for direct search optimization.



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minimize


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 [x,v,nev,...] = minimize (f,args,...) - Minimize f

 ARGUMENTS
 f    : string  : Name of function. Must return a real value
 args : list or : List of arguments to f (by default, minimize the first)
        matrix  : f's only argument

 RETURNED VALUES
 x   : matrix  : Local minimum of f. Let's suppose x is M-by-N.
 v   : real    : Value of f in x0
 nev : integer : Number of function evaluations 
     or 1 x 2  : Number of function and derivative evaluations (if
                 derivatives are used)
 

 Extra arguments are either a succession of option-value pairs or a single
 list or struct of option-value pairs (for unary options, the value in the
 struct is ignored).
 
 OPTIONS : DERIVATIVES   Derivatives may be used if one of these options
 ---------------------   uesd. Otherwise, the Nelder-Mean (see
                         nelder_mead_min) method is used.
 
 'd2f', d2f     : Name of a function that returns the value of f, of its
                  1st and 2nd derivatives : [fx,dfx,d2fx] = feval (d2f, x)
                  where fx is a real number, dfx is 1x(M*N) and d2fx is
                  (M*N)x(M*N). A Newton-like method (d2_min) will be used.

 'hess'         : Use [fx,dfx,d2fx] = leval (f, args) to compute 1st and
                  2nd derivatives, and use a Newton-like method (d2_min).

 'd2i', d2i     : Name of a function that returns the value of f, of its
                  1st and pseudo-inverse of second derivatives : 
                  [fx,dfx,id2fx] = feval (d2i, x) where fx is a real
                  number, dfx is 1x(M*N) and d2ix is (M*N)x(M*N).
                  A Newton-like method will be used (see d2_min).

 'ihess'        : Use [fx,dfx,id2fx] = leval (f, args) to compute 1st
                  derivative and the pseudo-inverse of 2nd derivatives,
                  and use a Newton-like method (d2_min).

            NOTE : df, d2f or d2i take the same arguments as f.
 
 'order', n     : Use derivatives of order n. If the n'th order derivative
                  is not specified by 'df', 'd2f' or 'd2i', it will be
                  computed numerically. Currently, only order 1 works.
 
 'ndiff'        : Use a variable metric method (bfgs) using numerical
                  differentiation.

 OPTIONS : STOPPING CRITERIA  Default is to use 'tol'
 ---------------------------
 'ftol', ftol   : Stop search when value doesn't improve, as tested by

              ftol > Deltaf/max(|f(x)|,1)

                 where Deltaf is the decrease in f observed in the last
                 iteration.                                 Default=10*eps

 'utol', utol   : Stop search when updates are small, as tested by

              tol > max { dx(i)/max(|x(i)|,1) | i in 1..N }

                 where  dx is the change in the x that occured in the last
                 iteration.

 'dtol',dtol    : Stop search when derivatives are small, as tested by

              dtol > max { df(i)*max(|x(i)|,1)/max(v,1) | i in 1..N }

                 where x is the current minimum, v is func(x) and df is
                 the derivative of f in x. This option is ignored if
                 derivatives are not used in optimization.

 MISC. OPTIONS
 -------------
 'maxev', m     : Maximum number of function evaluations             <inf>

 'narg' , narg  : Position of the minimized argument in args           <1>
 'isz'  , step  : Initial step size (only for 0 and 1st order method)  <1>
                  Should correspond to expected distance to minimum
 'verbose'      : Display messages during execution

 'backend'      : Instead of performing the minimization itself, return
                  [backend, control], the name and control argument of the
                  backend used by minimize(). Minimimzation can then be
                  obtained without the overhead of minimize by calling, if
                  a 0 or 1st order method is used :

              [x,v,nev] = feval (backend, args, control)
                   
                  or, if a 2nd order method is used :

              [x,v,nev] = feval (backend, control.d2f, args, control)



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 [x,v,nev,...] = minimize (f,args,...) - Minimize f



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nelder_mead_min


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 [x0,v,nev] = nelder_mead_min (f,args,ctl) - Nelder-Mead minimization

 Minimize 'f' using the Nelder-Mead algorithm. This function is inspired
 from the that found in the book "Numerical Recipes".

 ARGUMENTS
 ---------
 f     : string : Name of function. Must return a real value
 args  : list   : Arguments passed to f.
      or matrix : f's only argument
 ctl   : vector : (Optional) Control variables, described below
      or struct

 RETURNED VALUES
 ---------------
 x0  : matrix   : Local minimum of f
 v   : real     : Value of f in x0
 nev : number   : Number of function evaluations
 
 CONTROL VARIABLE : (optional) may be named arguments (i.e. "name",value
 ------------------ pairs), a struct, or a vector of length <= 6, where
                    NaN's are ignored. Default values are written <value>.
  OPT.   VECTOR
  NAME    POS
 ftol,f  N/A    : Stopping criterion : stop search when values at simplex
                  vertices are all alike, as tested by 

                   f > (max_i (f_i) - min_i (f_i)) /max(max(|f_i|),1)

                  where f_i are the values of f at the vertices.  <10*eps>

 rtol,r  N/A    : Stop search when biggest radius of simplex, using
                  infinity-norm, is small, as tested by :

              ctl(2) > Radius                                     <10*eps>

 vtol,v  N/A    : Stop search when volume of simplex is small, tested by
            
              ctl(2) > Vol

 crit,c ctl(1)  : Set one stopping criterion, 'ftol' (c=1), 'rtol' (c=2)
                  or 'vtol' (c=3) to the value of the 'tol' option.    <1>

 tol, t ctl(2)  : Threshold in termination test chosen by 'crit'  <10*eps>

 narg  ctl(3)  : Position of the minimized argument in args            <1>
 maxev ctl(4)  : Maximum number of function evaluations. This number <inf>
                 may be slightly exceeded.
 isz   ctl(5)  : Size of initial simplex, which is :                   <1>

                { x + e_i | i in 0..N } 
 
                Where x == args{narg} is the initial value 
                 e_0    == zeros (size (x)), 
                 e_i(j) == 0 if j != i and e_i(i) == ctl(5)
                 e_i    has same size as x

                Set ctl(5) to the distance you expect between the starting
                point and the minimum.

 rst   ctl(6)   : When a minimum is found the algorithm restarts next to
                  it until the minimum does not improve anymore. ctl(6) is
                  the maximum number of restarts. Set ctl(6) to zero if
                  you know the function is well-behaved or if you don't
                  mind not getting a true minimum.                     <0>

 verbose, v     Be more or less verbose (quiet=0)                      <0>



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 [x0,v,nev] = nelder_mead_min (f,args,ctl) - Nelder-Mead minimization



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nmsmax


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NMSMAX  Nelder-Mead simplex method for direct search optimization.
        [x, fmax, nf] = NMSMAX(FUN, x0, STOPIT, SAVIT) attempts to
        maximize the function FUN, using the starting vector x0.
        The Nelder-Mead direct search method is used.
        Output arguments:
               x    = vector yielding largest function value found,
               fmax = function value at x,
               nf   = number of function evaluations.
        The iteration is terminated when either
               - the relative size of the simplex is <= STOPIT(1)
                 (default 1e-3),
               - STOPIT(2) function evaluations have been performed
                 (default inf, i.e., no limit), or
               - a function value equals or exceeds STOPIT(3)
                 (default inf, i.e., no test on function values).
        The form of the initial simplex is determined by STOPIT(4):
           STOPIT(4) = 0: regular simplex (sides of equal length, the default)
           STOPIT(4) = 1: right-angled simplex.
        Progress of the iteration is not shown if STOPIT(5) = 0 (default 1).
           STOPIT(6) indicates the direction (ie. minimization or 
                   maximization.) Default is 1, maximization.
                   set STOPIT(6)=-1 for minimization
        If a non-empty fourth parameter string SAVIT is present, then
        `SAVE SAVIT x fmax nf' is executed after each inner iteration.
        NB: x0 can be a matrix.  In the output argument, in SAVIT saves,
            and in function calls, x has the same shape as x0.
        NMSMAX(fun, x0, STOPIT, SAVIT, P1, P2,...) allows additional
        arguments to be passed to fun, via feval(fun,x,P1,P2,...).
 References:
 N. J. Higham, Optimization by direct search in matrix computations,
    SIAM J. Matrix Anal. Appl, 14(2): 317-333, 1993.
 C. T. Kelley, Iterative Methods for Optimization, Society for Industrial
    and Applied Mathematics, Philadelphia, PA, 1999.



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NMSMAX  Nelder-Mead simplex method for direct search optimization.



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nonlin_curvefit


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 -- Function File: [P, FY, CVG, OUTP] = nonlin_curvefit (F, PIN, X, Y)
 -- Function File: [P, FY, CVG, OUTP] = nonlin_curvefit (F, PIN, X, Y,
          SETTINGS)
     Frontend for nonlinear fitting of values, computed by a model
     function, to observed values.

     Please refer to the description of 'nonlin_residmin'.  The only
     differences to 'nonlin_residmin' are the additional arguments X
     (independent values, mostly, but not necessarily, an array of the
     same dimensions or the same number of rows as Y) and Y (array of
     observations), the returned value FY (final guess for observed
     values) instead of RESID, that the model function has a second
     obligatory argument which will be set to X and is supposed to
     return guesses for the observations (with the same dimensions), and
     that the possibly user-supplied function for the jacobian of the
     model function has also a second obligatory argument which will be
     set to X.

     See also: nonlin_residmin.




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Frontend for nonlinear fitting of values, computed by a model function,
to obser



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nonlin_min


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 -- Function File: [P, OBJF, CVG, OUTP] = nonlin_min (F, PIN)
 -- Function File: [P, OBJF, CVG, OUTP] = nonlin_min (F, PIN, SETTINGS)

     Frontend for constrained nonlinear minimization of a scalar
     objective function.  The functions supplied by the user have a
     minimal interface; any additionally needed constants 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.

     F: objective function.  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.

     PIN: real column vector of initial parameters.

     SETTINGS: structure whose fields stand for optional settings
     referred to below.  The fields can be set by '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 P,
     the final value of the objective function OBJF, an integer CVG
     indicating if and how optimization succeeded or failed, and a
     structure OUTP with additional information, curently with only one
     field: NITER, the number of iterations.  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 '0'
     (maximum number of iterations exceeded), '1' (fixed number of
     iterations completed, e.g.  in stochastic optimizers), '2'
     (parameter change less than specified precision in two consecutive
     iterations), '3' (improvement in objective function less than
     specified), or '-4' (algorithm got stuck).

     SETTINGS:

     'Algorithm': String specifying the backend.  Currently available
     are '"lm_feasible"' (default) and '"siman"'.  They are described in
     separate sections below.

     'objf_grad': Function computing the gradient of the objective
     function with respect to the parameters.  Will be called with the
     column vector of parameters and an informational structure as
     arguments.  The structure has the fields 'f': value of objective
     function for current parameters, 'fixed': logical vector indicating
     which parameters are not optimized, so these partial derivatives
     need not be computed and can be set to zero, 'diffp',
     'diff_onesided', 'lbound', 'ubound': identical to the user settings
     of this name, '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 gradient function will call the objective function with the
     second argument set with fields 'f': as the 'f' passed to the
     gradient function, 'plabels': cell-array of 1x1 cell-arrays with
     the entries of the column-cell-arrays of 'plabels' as passed to the
     jacobian function corresponding to current parameter, 'side': '0'
     for one-sided interval, '1' or '2', respectively, for the sides of
     a two-sided interval, and 'parallel': logical scalar indicating
     parallel computation of partial derivatives.

     'objf_hessian': Function computing the Hessian of the objective
     function with respect to the parameters.  The default is backend
     specific.  Will be called with the column vector of parameters as
     argument.

     'diffp': column vector of fractional intervals (doubled for central
     intervals) supposed to be used by gradient functions performing
     finite differencing.  Default: '.001 * ones (size (parameters))'.
     The default gradient function will use these as absolute intervals
     for parameters with value zero.

     'diff_onesided': logical column vector indicating that one-sided
     intervals should be used by gradient functions performing finite
     differencing.  Default: 'false (size (parameters))'.

     'complex_step_derivative_objf', 'complex_step_derivative_inequc',
     'complex_step_derivative_equc': logical scalars, default: false.
     Estimate gradient of objective function, general inequality
     constraints, and general equality constraints, respectively, with
     complex step derivative approximation.  Use only if you know that
     your objective function, function of general inequality
     constraints, or function of general equality constraints,
     respectively, is suitable for this.  No user function for the
     respective gradient must be specified.

     'cstep': scalar step size for complex step derivative
     approximation.  Default: 1e-20.

     'parallel_local': logical scalar, default: false.  Estimate
     gradient of objective function and of constraints in parallel
     processes.  Works for default finite difference gradient function
     and for complex step derivatives.  Due to overhead, a speed
     advantage can only be expected if objective function or constraint
     functions are time consuming enough.

     '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.

     'lbound', 'ubound': column vectors of lower and upper bounds for
     parameters.  Default: '-Inf' and '+Inf', respectively.  The bounds
     are non-strict, i.e.  parameters are allowed to be exactly equal to
     a bound.  The default gradient function will respect bounds (but no
     further inequality constraints) in finite differencing.

     '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
     'm') and a vector (say 'v'), specifying linear inequality
     constraints of the form 'm.' * parameters + v >= 0'.  The first
     entry for general constraints must be a differentiable
     column-vector valued function (say 'h'), specifying general
     inequality constraints of the form 'h (p[, idx]) >= 0'; 'p' is the
     column vector of optimized parameters and the optional argument
     'idx' is a logical index.  'h' has to return the values of all
     constraints if 'idx' is not given.  It may choose to return only
     the indexed constraints if 'idx' is given (so computation of the
     other constraints can be spared); in this case, the additional
     setting 'f_inequc_idx' has to be set to 'true'.  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 'dfdp' above applies, with 2
     exceptions: 1) it is called with 3 arguments since it has an
     additional argument 'idx', a logical index, at second position,
     indicating which rows of the jacobian must be returned (if the
     function chooses to return only indexed rows, the additional
     setting 'df_inequc_idx' has to be set to 'true').  2) the default
     jacobian function calls 'h' with 3 arguments, since the argument
     '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.

     'equc': Equality constraints.  Specified the same way as inequality
     constraints (see 'inequc').  The respective additional settings are
     named 'f_equc_idx' and 'df_equc_idx'.

     'cpiv': Function for complementary pivoting, usable in algorithms
     for constraints.  Default:  cpiv_bard.  Only the default function
     is supplied with the package.

     'TolFun': Minimum fractional improvement in objective function in
     an iteration (termination criterium).  Default: .0001.

     'MaxIter': Maximum number of iterations (termination criterium).
     Default: backend-specific.

     'fract_prec': Column Vector, minimum fractional change of
     parameters in an iteration (termination criterium if violated in
     two consecutive iterations).  Default: backend-specific.

     'max_fract_change': Column Vector, enforced maximum fractional
     change in parameters in an iteration.  Default: backend-specific.

     'Display': String indicating the degree of verbosity.  Default:
     '"off"'.  Possible values are currently '"off"' (no messages) and
     '"iter"' (some messages after each iteration).  Support of this
     setting and its exact interpretation are backend-specific.

     'debug': Logical scalar, default: 'false'.  Will be passed to the
     backend, which might print debugging information if true.

     Structure-based parameter handling

     The setting '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.  If
     initial parameters are a structure, it is an error if 'param_order'
     is not given and there are any non-structure-based configuration
     items or functions.

     The initial parameters PIN can be given as a structure containing
     at least all fields named in 'param_order'.  In this case the
     returned parameters 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
     'true': 'objf_pstruct' (objective function), 'grad_objf_pstruct'
     (gradient of objective function), 'hessian_objf_pstruct' (hessian
     of objective function), 'f_inequc_pstruct' (general inequality
     constraints), 'df_inequc_pstruct' (jacobian of general inequality
     constraints), 'f_equc_pstruct' (general equality constraints), and
     'df_equc_pstruct' (jacobian of general equality constraints).  If a
     gradient (jacobian) function is configured in such a way, it must
     return the entries (columns) of the gradient (jacobian) as fields
     of a structure under the respective parameter names.  If the
     hessian function is configured in such a way, it must return a
     structure (say 'h') with fields e.g.  as 'h.a.b = value' for
     'value' being the 2nd partial derivative with respect to 'a' and
     'b'.  There is no need to also specify the field 'h.b.a' in this
     example.

     Similarly, for specifying linear constraints, instead of the matrix
     (called '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 'lbound', 'ubound', 'fixed', 'diffp',
     'diff_onesided', 'fract_prec', and 'max_fract_change' can be
     replaced by the setting 'param_config'.  It is a structure that can
     contain fields named in '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 '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
     '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

     "lm_feasible"

     A Levenberg/Marquardt-like optimizer, attempting to honour
     constraints throughout the course of optimization.  This means that
     the initial parameters must not violate constraints (to find an
     initial feasible set of parameters, e.g.  Octaves 'sqp' can be
     used, by specifying an objective function which is constant or
     which returns the quadratic distance to the initial values).  If
     the constraints need only be honoured in the result of the
     optimization, Octaves 'sqp' may be preferable.  The Hessian is
     either supplied by the user or is approximated by the BFGS
     algorithm.

     Returned value CVG will be '2' or '3' for success and '0' or '-4'
     for failure (see above for meaning).

     Backend-specific defaults are: 'MaxIter': 20, 'fract_prec': 'zeros
     (size (parameters))', 'max_fract_change': 'Inf' for all parameters.

     Interpretation of 'Display': if set to '"iter"', currently only
     information on applying 'max_fract_change' is printed.

     "siman"

     A simulated annealing (stochastic) optimizer, changing all
     parameters at once in a single step, so being suitable for
     non-bound constraints.

     No gradient or hessian of the objective function is used.  The
     settings 'MaxIter', 'fract_prec', 'TolFun', and 'max_fract_change'
     are not honoured.

     Accepts the additional settings 'T_init' (initial temperature,
     default 0.01), 'T_min' (final temperature, default 1.0e-5), 'mu_T'
     (factor of temperature decrease, default 1.005), 'iters_fixed_T'
     (iterations within one temperature step, default 10),
     'max_rand_step' (column vector or structure-based configuration of
     maximum random steps for each parameter, default 0.005 * PIN),
     'stoch_regain_constr' (if 'true', regain constraints after a random
     step, otherwise take new random value until constraints are met,
     default false), 'trace_steps' (set field 'trace' of OUTP with a
     matrix with a row for each step, first column iteration number,
     second column repeat number within iteration, third column value of
     objective function, rest columns parameter values, default false),
     and 'siman_log' (set field 'log' of OUTP with a matrix with a row
     for each iteration, first column temperature, second column value
     of objective function, rest columns numbers of tries with decrease,
     no decrease but accepted, and no decrease and rejected.

     Steps with increase 'diff' of objective function are accepted if
     'rand (1) < exp (- diff / T)', where 'T' is the temperature of the
     current iteration.

     If regaining of constraints failed, optimization will be aborted
     and returned value of CVG will be '0'.  Otherwise, CVG will be '1'.

     Interpretation of 'Display': if set to '"iter"', an informational
     line is printed after each iteration.




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Frontend for constrained nonlinear minimization of a scalar objective
function.



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nonlin_residmin


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 -- Function File: [P, RESID, CVG, OUTP] = nonlin_residmin (F, PIN)
 -- Function File: [P, RESID, CVG, OUTP] = nonlin_residmin (F, PIN,
          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.

     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.

     PIN: real column vector of initial parameters.

     SETTINGS: structure whose fields stand for optional settings
     referred to below.  The fields can be set by '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 P,
     the final array of residuals RESID, an integer CVG indicating if
     and how optimization succeeded or failed, and a structure OUTP with
     additional information, curently with only one field: NITER, the
     number of iterations.  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 '0' (maximum number of
     iterations exceeded), '2' (parameter change less than specified
     precision in two consecutive iterations), or '3' (improvement in
     objective function - e.g.  sum of squares - less than specified).

     SETTINGS:

     'Algorithm': String specifying the backend.  Default:
     '"lm_svd_feasible"'.  The latter is currently the only backend
     distributed with this package.  It is described in a separate
     section below.

     '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 'f': value of residuals
     for current parameters, reshaped to a column vector, 'fixed':
     logical vector indicating which parameters are not optimized, so
     these partial derivatives need not be computed and can be set to
     zero, 'diffp', 'diff_onesided', 'lbound', 'ubound': identical to
     the user settings of this name, '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 'f': as the 'f' passed to the
     jacobian function, 'plabels': cell-array of 1x1 cell-arrays with
     the entries of the column-cell-arrays of 'plabels' as passed to the
     jacobian function corresponding to current parameter, 'side': '0'
     for one-sided interval, '1' or '2', respectively, for the sides of
     a two-sided interval, and 'parallel': logical scalar indicating
     parallel computation of partial derivatives.

     'diffp': column vector of fractional intervals (doubled for central
     intervals) supposed to be used by jacobian functions performing
     finite differencing.  Default: '.001 * ones (size (parameters))'.
     The default jacobian function will use these as absolute intervals
     for parameters with value zero.

     'diff_onesided': logical column vector indicating that one-sided
     intervals should be used by jacobian functions performing finite
     differencing.  Default: 'false (size (parameters))'.

     'complex_step_derivative_f', 'complex_step_derivative_inequc',
     'complex_step_derivative_equc': logical scalars, default: false.
     Estimate Jacobian of model function, general inequality
     constraints, and general equality constraints, respectively, with
     complex step derivative approximation.  Use only if you know that
     your model function, function of general inequality constraints, or
     function of general equality constraints, respectively, is suitable
     for this.  No user function for the respective Jacobian must be
     specified.

     'cstep': scalar step size for complex step derivative
     approximation.  Default: 1e-20.

     'parallel_local': logical scalar, default: false.  Estimate
     Jacobians of model function and of constraints in parallel
     processes.  Works for default finite difference Jacobian function
     and for complex step derivatives.  Due to overhead, a speed
     advantage can only be expected if model function or constraint
     functions are time consuming enough.

     '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.

     'lbound', 'ubound': column vectors of lower and upper bounds for
     parameters.  Default: '-Inf' and '+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.

     '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
     'm') and a vector (say 'v'), specifying linear inequality
     constraints of the form 'm.' * parameters + v >= 0'.  The first
     entry for general constraints must be a differentiable
     column-vector valued function (say 'h'), specifying general
     inequality constraints of the form 'h (p[, idx]) >= 0'; 'p' is the
     column vector of optimized paraters and the optional argument 'idx'
     is a logical index.  'h' has to return the values of all
     constraints if 'idx' is not given.  It may choose to return only
     the indexed constraints if 'idx' is given (so computation of the
     other constraints can be spared); in this case, the additional
     setting 'f_inequc_idx' has to be set to 'true'.  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 'dfdp' above applies, with 2
     exceptions: 1) it is called with 3 arguments since it has an
     additional argument 'idx', a logical index, at second position,
     indicating which rows of the jacobian must be returned (if the
     function chooses to return only indexed rows, the additional
     setting 'df_inequc_idx' has to be set to 'true').  2) the default
     jacobian function calls 'h' with 3 arguments, since the argument
     '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.

     'equc': Equality constraints.  Specified the same way as inequality
     constraints (see 'inequc').

     'cpiv': Function for complementary pivoting, usable in algorithms
     for constraints.  Default:  cpiv_bard.  Only the default function
     is supplied with the package.

     'weights': Array of weights for the residuals.  Dimensions must
     match.

     'TolFun': Minimum fractional improvement in objective function
     (e.g.  sum of squares) in an iteration (termination criterium).
     Default: .0001.

     'MaxIter': Maximum number of iterations (termination criterium).
     Default: backend-specific.

     'fract_prec': Column Vector, minimum fractional change of
     parameters in an iteration (termination criterium if violated in
     two consecutive iterations).  Default: backend-specific.

     'max_fract_change': Column Vector, enforced maximum fractional
     change in parameters in an iteration.  Default: backend-specific.

     'Display': String indicating the degree of verbosity.  Default:
     '"off"'.  Possible values are currently '"off"' (no messages) and
     '"iter"' (some messages after each iteration).  Support of this
     setting and its exact interpretation are backend-specific.

     'plot_cmd': Function enabling backend to plot results or
     intermediate results.  Will be called with current computed
     residuals.  Default: plot nothing.

     'debug': Logical scalar, default: 'false'.  Will be passed to the
     backend, which might print debugging information if true.

     Structure-based parameter handling

     The setting '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.  If
     initial parameters are a structure, it is an error if 'param_order'
     is not given and there are any non-structure-based configuration
     items or functions.

     The initial parameters PIN can be given as a structure containing
     at least all fields named in 'param_order'.  In this case the
     returned parameters 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
     'true': 'f_pstruct' (model function), 'df_pstruct' (jacobian of
     model function), 'f_inequc_pstruct' (general inequality
     constraints), 'df_inequc_pstruct' (jacobian of general inequality
     constraints), 'f_equc_pstruct' (general equality constraints), and
     '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 '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 'lbound', 'ubound', 'fixed', 'diffp',
     'diff_onesided', 'fract_prec', and 'max_fract_change' can be
     replaced by the setting 'param_config'.  It is a structure that can
     contain fields named in '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 '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
     '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 CVG will be '0', '2', or '3'.

     Backend-specific defaults are: 'MaxIter': 20, 'fract_prec': 'zeros
     (size (parameters))', 'max_fract_change': 'Inf' for all parameters.

     Interpretation of 'Display': if set to '"iter"', currently
     'plot_cmd' is evaluated for each iteration, and some further
     diagnostics may be printed.

     Specific option: '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.

     See also: nonlin_curvefit.




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Frontend for nonlinear minimization of residuals returned by a model
function.



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nrm


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 -- Function File: XMIN = nrm( F,X0)
     Using X0 as a starting point find a minimum of the scalar function
     F.  The Newton-Raphson method is used.




# name: <cell-element>
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Using X0 as a starting point find a minimum of the scalar function F.



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# type: sq_string
# elements: 1
# length: 14
optim_problems


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# elements: 1
# length: 64
 Problems for testing optimizers. Documentation is in the code.



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# elements: 1
# length: 33
 Problems for testing optimizers.



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poly_2_ex


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  ex = poly_2_ex (l, f)       - Extremum of a 1-var deg-2 polynomial

 l  : 3 : Values of variable at which polynomial is known.
 f  : 3 : f(i) = Value of the degree-2 polynomial at l(i).
 
 ex : 1 : Value for which f reaches its extremum
 
 Assuming that f(i) = a*l(i)^2 + b*l(i) + c = P(l(i)) for some a, b, c,
 ex is the extremum of the polynome P.




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  ex = poly_2_ex (l, f)       - Extremum of a 1-var deg-2 polynomial



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polyconf


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 [y,dy] = polyconf(p,x,s)

   Produce prediction intervals for the fitted y. The vector p 
   and structure s are returned from polyfit or wpolyfit. The 
   x values are where you want to compute the prediction interval.

 polyconf(...,['ci'|'pi'])

   Produce a confidence interval (range of likely values for the
   mean at x) or a prediction interval (range of likely values 
   seen when measuring at x).  The prediction interval tells
   you the width of the distribution at x.  This should be the same
   regardless of the number of measurements you have for the value
   at x.  The confidence interval tells you how well you know the
   mean at x.  It should get smaller as you increase the number of
   measurements.  Error bars in the physical sciences usually show 
   a 1-alpha confidence value of erfc(1/sqrt(2)), representing
   one standandard deviation of uncertainty in the mean.

 polyconf(...,1-alpha)

   Control the width of the interval. If asking for the prediction
   interval 'pi', the default is .05 for the 95% prediction interval.
   If asking for the confidence interval 'ci', the default is
   erfc(1/sqrt(2)) for a one standard deviation confidence interval.

 Example:
  [p,s] = polyfit(x,y,1);
  xf = linspace(x(1),x(end),150);
  [yf,dyf] = polyconf(p,xf,s,'ci');
  plot(xf,yf,'g-;fit;',xf,yf+dyf,'g.;;',xf,yf-dyf,'g.;;',x,y,'xr;data;');
  plot(x,y-polyval(p,x),';residuals;',xf,dyf,'g-;;',xf,-dyf,'g-;;');



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 [y,dy] = polyconf(p,x,s)



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polyfitinf


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 function [A,REF,HMAX,H,R,EQUAL] = polyfitinf(M,N,K,X,Y,EPSH,MAXIT,REF0)

   Best polynomial approximation in discrete uniform norm

   INPUT VARIABLES:

   M       : degree of the fitting polynomial
   N       : number of data points
   X(N)    : x-coordinates of data points
   Y(N)    : y-coordinates of data points
   K       : character of the polynomial:
                   K = 0 : mixed parity polynomial
                   K = 1 : odd polynomial  ( X(1) must be >  0 )
                   K = 2 : even polynomial ( X(1) must be >= 0 )
   EPSH    : tolerance for leveling. A useful value for 24-bit
             mantissa is EPSH = 2.0E-7
   MAXIT   : upper limit for number of exchange steps
   REF0(M2): initial alternating set ( N-vector ). This is an
             OPTIONAL argument. The length M2 is given by:
                   M2 = M + 2                      , if K = 0
                   M2 = integer part of (M+3)/2    , if K = 1
                   M2 = 2 + M/2 (M must be even)   , if K = 2

   OUTPUT VARIABLES:

   A       : polynomial coefficients of the best approximation
             in order of increasing powers:
                   p*(x) = A(1) + A(2)*x + A(3)*x^2 + ...
   REF     : selected alternating set of points
   HMAX    : maximum deviation ( uniform norm of p* - f )
   H       : pointwise approximation errors
	R		: total number of iterations
   EQUAL   : success of failure of algorithm
                   EQUAL=1 :  succesful
                   EQUAL=0 :  convergence not acheived
                   EQUAL=-1:  input error
                   EQUAL=-2:  algorithm failure

   Relies on function EXCH, provided below.

   Example: 
   M = 5; N = 10000; K = 0; EPSH = 10^-12; MAXIT = 10;
   X = linspace(-1,1,N);   % uniformly spaced nodes on [-1,1]
   k=1; Y = abs(X).^k;     % the function Y to approximate
   [A,REF,HMAX,H,R,EQUAL] = polyfitinf(M,N,K,X,Y,EPSH,MAXIT);
   p = polyval(A,X); plot(X,Y,X,p) % p is the best approximation

   Note: using an even value of M, e.g., M=2, in the example above, makes
   the algorithm to fail with EQUAL=-2, because of collocation, which
   appears because both the appriximating function and the polynomial are
   even functions. The way aroung it is to approximate only the right half
   of the function, setting K = 2 : even polynomial. For example: 

 N = 10000; K = 2; EPSH = 10^-12; MAXIT = 10;  X = linspace(0,1,N);
 for i = 1:2
     k = 2*i-1; Y = abs(X).^k;
     for j = 1:4
         M = 2^j;
         [~,~,HMAX] = polyfitinf(M,N,K,X,Y,EPSH,MAXIT);
         approxerror(i,j) = HMAX;
     end
 end
 disp('Table 3.1 from Approximation theory and methods, M.J.D.POWELL, p. 27');
 disp(' ');
 disp('            n          K=1          K=3'); 
 disp(' '); format short g;
 disp([(2.^(1:4))' approxerror']);

   ALGORITHM:

   Computation of the polynomial that best approximates the data (X,Y)
   in the discrete uniform norm, i.e. the polynomial with the  minimum
   value of max{ | p(x_i) - y_i | , x_i in X } . That polynomial, also
   known as minimax polynomial, is obtained by the exchange algorithm,
   a finite iterative process requiring, at most,
      n
    (   ) iterations ( usually p = M + 2. See also function EXCH ).
      p
   since this number can be very large , the routine  may not converge
   within MAXIT iterations . The  other possibility of  failure occurs
   when there is insufficient floating point precision  for  the input
   data chosen.

   CREDITS: This routine was developed and modified as 
   computer assignments in Approximation Theory courses by 
   Prof. Andrew Knyazev, University of Colorado Denver, USA.

   Team Fall 98 (Revision 1.0):
           Chanchai Aniwathananon
           Crhistopher Mehl
           David A. Duran
           Saulo P. Oliveira

   Team Spring 11 (Revision 1.1): Manuchehr Aminian

   The algorithm and the comments are based on a FORTRAN code written
   by Joseph C. Simpson. The code is available on Netlib repository:
   http://www.netlib.org/toms/501
   See also: Communications of the ACM, V14, pp.355-356(1971)

   NOTES:

   1) A may contain the collocation polynomial
   2) If MAXIT is exceeded, REF contains a new reference set
   3) M, EPSH and REF can be altered during the execution
   4) To keep consistency to the original code , EPSH can be
   negative. However, the use of REF0 is *NOT* determined by
   EPSH< 0, but only by its inclusion as an input parameter.

   Some parts of the code can still take advantage of vectorization.  

   Revision 1.0 from 1998 is a direct human translation of 
   the FORTRAN code http://www.netlib.org/toms/501
   Revision 1.1 is a clean-up and technical update.  
   Tested on MATLAB Version 7.11.0.584 (R2010b) and 
   GNU Octave Version 3.2.4



# name: <cell-element>
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 function [A,REF,HMAX,H,R,EQUAL] = polyfitinf(M,N,K,X,Y,EPSH,MAXIT,REF0)

   B



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powell


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 -- Function File: [ P , OBJ_VALUE, CONVERGENCE, ITERS, NEVS] = powell
          (F, P0, CONTROL)
     Multidimensional minimization (direction-set method).  Implements a
     direction-set (Powell's) method for multidimensional minimization
     of a function without calculation of the gradient [1, 2]

     Arguments
--------------

        * F: name of function to minimize (string or handle), which
          should accept one input variable (see example for how to pass
          on additional input arguments)

        * P0: An initial value of the function argument to minimize

        * OPTIONS: an optional structure, which can be generated by
          optimset, with some or all of the following fields:
             - MaxIter: maximum iterations (positive integer, or -1 or
               Inf for unlimited (default))
             - TolFun: minimum amount by which function value must
               decrease in each iteration to continue (default is 1E-8)
             - MaxFunEvals: maximum function evaluations (positive
               integer, or -1 or Inf for unlimited (default))
             - SearchDirections: an n*n matrix whose columns contain the
               initial set of (presumably orthogonal) directions to
               minimize along, where n is the number of elements in the
               argument to be minimized for; or an n*1 vector of
               magnitudes for the initial directions (defaults to the
               set of unit direction vectors)

     Examples
-------------

          y = @(x, s) x(1) ^ 2 + x(2) ^ 2 + s;
          o = optimset('MaxIter', 100, 'TolFun', 1E-10);
          s = 1;
          [x_optim, y_min, conv, iters, nevs] = powell(@(x) y(x, s), [1 0.5], o); %pass y wrapped in an anonymous function so that all other arguments to y, which are held constant, are set
          %should return something like x_optim = [4E-14 3E-14], y_min = 1, conv = 1, iters = 2, nevs = 24


     Returns:
-------------

        * P: the minimizing value of the function argument
        * OBJ_VALUE: the value of F() at P
        * CONVERGENCE: 1 if normal convergence, 0 if not
        * ITERS: number of iterations performed
        * NEVS: number of function evaluations

     References
---------------

       1. Powell MJD (1964), An efficient method for finding the minimum
          of a function of several variables without calculating
          derivatives, 'Computer Journal', 7 :155-162

       2. Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (1992).
          'Numerical Recipes in Fortran: The Art of Scientific
          Computing' (2nd Ed.).  New York: Cambridge University Press
          (Section 10.5)




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Multidimensional minimization (direction-set method).



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residmin_stat


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 -- Function File: INFO = residmin_stat (F, P, SETTINGS)
     Frontend for computation of statistics for a residual-based
     minimization.

     SETTINGS is a structure whose fields can be set by 'optimset' with
     Octave versions 3.3.55 or greater; with older Octave versions, the
     fields must be set directly and in the correct case.  With SETTINGS
     the computation of certain statistics is requested by setting the
     fields 'ret_<name_of_statistic>' to 'true'.  The respective
     statistics will be returned in a structure as fields with name
     '<name_of_statistic>'.  Depending on the requested statistic and on
     the additional information provided in SETTINGS, F and P may be
     empty.  Otherwise, F is the model function of an optimization (the
     interface of F is described e.g.  in 'nonlin_residmin', please see
     there), and P is a real column vector with parameters resulting
     from the same optimization.

     Currently, the following statistics (or general information) can be
     requested:

     'dfdp': Jacobian of model function with respect to parameters.

     'covd': Covariance matrix of data (typically guessed by applying a
     factor to the covariance matrix of the residuals).

     'covp': Covariance matrix of final parameters.

     'corp': Correlation matrix of final parameters.

     Further SETTINGS

     The functionality of the interface is similar to 'nonlin_residmin'.
     In particular, structure-based, possibly non-scalar, parameters and
     flagging parameters as fixed are possible.  The following settings
     have the same meaning as in 'nonlin_residmin' (please refer to
     there): 'param_order', 'param_dims', 'f_pstruct', 'df_pstruct',
     'diffp', 'diff_onesided', 'complex_step_derivative_f', 'cstep',
     'fixed', and 'weights'.  Similarly, 'param_config' can be used, but
     only with fields corresponding to the settings 'fixed', 'diffp',
     and 'diff_onesided'.

     'dfdp' can be set in the same way as in 'nonlin_residmin', but
     alternatively may already contain the computed Jacobian of the
     model function at the final parameters in matrix- or
     structure-form.  Users may pass information on the result of the
     optimization in 'residuals' (self-explaining) and 'covd'
     (covariance matrix of data).  In many cases the type of objective
     function of the optimization must be specified in 'objf_type';
     currently, there is only a backend for the type "wls" (weighted
     least squares).

     Backend-specific information

     The backend for 'objf_type == "wls"' (currently the only backend)
     computes 'covd' (due to user request or as a prerequisite for
     'covp' and 'corp') as a diagonal matrix by assuming that the
     variances of data points are proportional to the reciprocal of the
     squared 'weights' and guessing the factor of proportionality from
     the residuals.  If 'covp' is not defined (e.g.  because the
     Jacobian has no full rank), it makes an attempt to still compute
     its uniquely defined elements, if any, and to find the additional
     defined elements (being '1' or '-1'), if any, in 'corp'.

     See also: curvefit_stat.




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Frontend for computation of statistics for a residual-based
minimization.



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rosenbrock


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 Rosenbrock function - used to create example obj. fns.

 Function value and gradient vector of the rosenbrock function
 The minimizer is at the vector (1,1,..,1),
 and the minimized value is 0.



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 Rosenbrock function - used to create example obj.



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samin_example


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# type: sq_string
# elements: 1
# length: 16
 dimensionality



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# type: sq_string
# elements: 1
# length: 16
 dimensionality




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# length: 10
test_min_1


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 [x,v,niter] = feval (optim_func, "testfunc","dtestf", xinit);



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 [x,v,niter] = feval (optim_func, "testfunc","dtestf", xinit);




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test_min_2


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# length: 60
 [xlev,vlev,nlev] = feval(optim_func, "ff", "dff", xinit) ;



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# length: 60
 [xlev,vlev,nlev] = feval(optim_func, "ff", "dff", xinit) ;




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test_min_3


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 [xlev,vlev,nlev] = feval (optim_func, "ff", "dff", xinit, "extra", extra) ;
 [xlev,vlev,nlev] = feval \
     (optim_func, "ff", "dff", list (xinit, obsmat, obses));



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 [xlev,vlev,nlev] = feval (optim_func, "ff", "dff", xinit, "extra", extra) ;
 [x



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test_min_4


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 Plain run, just to make sure ######################################
 Minimum wrt 'x' is y0
 [xlev,vlev,nlev] = feval (optim_func, "ff", "dff", {x0,y0,1});
 ctl.df = "dff";



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 Plain run, just to make sure ######################################
 Minimum wr



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# length: 15
test_minimize_1


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# type: sq_string
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 Plain run, just to make sure ######################################
 Minimum wrt 'x' is y0



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# type: sq_string
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# length: 80
 Plain run, just to make sure ######################################
 Minimum wr



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# length: 22
test_nelder_mead_min_1


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# type: sq_string
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# length: 29
 Use vanilla nelder_mead_min



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# type: sq_string
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# length: 29
 Use vanilla nelder_mead_min




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test_nelder_mead_min_2


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# length: 70

 Test using volume #################################################



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# elements: 1
# length: 70

 Test using volume #################################################




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test_wpolyfit


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          x         y          dy



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# type: sq_string
# elements: 1
# length: 34
          x         y          dy




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vfzero


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 -- Function File: vfzero (FUN, X0)
 -- Function File: vfzero (FUN, X0, OPTIONS)
 -- Function File: [X, FVAL, INFO, OUTPUT] = vfzero (...)
     A variant of 'fzero'.  Finds a zero of a vector-valued multivariate
     function where each output element only depends on the input
     element with the same index (so the Jacobian is diagonal).

     FUN should be a handle or name of a function returning a column
     vector.  X0 should be a two-column matrix, each row specifying two
     points which bracket a zero of the respective output element of
     FUN.

     If X0 is a single-column matrix then several nearby and distant
     values are probed in an attempt to obtain a valid bracketing.  If
     this is not successful, the function fails.  OPTIONS is a structure
     specifying additional options.  Currently, 'vfzero' recognizes
     these options: '"FunValCheck"', '"OutputFcn"', '"TolX"',
     '"MaxIter"', '"MaxFunEvals"'.  For a description of these options,
     see *note optimset: doc-optimset.

     On exit, the function returns X, the approximate zero and FVAL, the
     function value thereof.  INFO is a column vector of exit flags that
     can have these values:

        * 1 The algorithm converged to a solution.

        * 0 Maximum number of iterations or function evaluations has
          been reached.

        * -1 The algorithm has been terminated from user output
          function.

        * -5 The algorithm may have converged to a singular point.

     OUTPUT is a structure containing runtime information about the
     'fzero' algorithm.  Fields in the structure are:

        * iterations Number of iterations through loop.

        * nfev Number of function evaluations.

        * bracketx A two-column matrix with the final bracketing of the
          zero along the x-axis.

        * brackety A two-column matrix with the final bracketing of the
          zero along the y-axis.

     See also: optimset, fsolve.




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A variant of 'fzero'.



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wpolyfit


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 -- Function File: [P, S] = wpolyfit (X, Y, DY, N)
     Return the coefficients of a polynomial P(X) of degree N that
     minimizes 'sumsq (p(x(i)) - y(i))', to best fit the data in the
     least squares sense.  The standard error on the observations Y if
     present are given in DY.

     The returned value P contains the polynomial coefficients suitable
     for use in the function polyval.  The structure S returns
     information necessary to compute uncertainty in the model.

     To compute the predicted values of y with uncertainty use
          [y,dy] = polyconf(p,x,s,'ci');
     You can see the effects of different confidence intervals and
     prediction intervals by calling the wpolyfit internal plot function
     with your fit:
          feval('wpolyfit:plt',x,y,dy,p,s,0.05,'pi')
     Use DY=[] if uncertainty is unknown.

     You can use a chi^2 test to reject the polynomial fit:
          p = 1-chi2cdf(s.normr^2,s.df);
     p is the probability of seeing a chi^2 value higher than that which
     was observed assuming the data are normally distributed around the
     fit.  If p < 0.01, you can reject the fit at the 1% level.

     You can use an F test to determine if a higher order polynomial
     improves the fit:
          [poly1,S1] = wpolyfit(x,y,dy,n);
          [poly2,S2] = wpolyfit(x,y,dy,n+1);
          F = (S1.normr^2 - S2.normr^2)/(S1.df-S2.df)/(S2.normr^2/S2.df);
          p = 1-f_cdf(F,S1.df-S2.df,S2.df);
     p is the probability of observing the improvement in chi^2 obtained
     by adding the extra parameter to the fit.  If p < 0.01, you can
     reject the lower order polynomial at the 1% level.

     You can estimate the uncertainty in the polynomial coefficients
     themselves using
          dp = sqrt(sumsq(inv(s.R'))'/s.df)*s.normr;
     but the high degree of covariance amongst them makes this a
     questionable operation.

 -- Function File: [P, S, MU] = wpolyfit (...)

     If an additional output 'mu = [mean(x),std(x)]' is requested then
     the X values are centered and normalized prior to computing the
     fit.  This will give more stable numerical results.  To compute a
     predicted Y from the returned model use 'y = polyval(p,
     (x-mu(1))/mu(2)'

 -- Function File: wpolyfit (...)

     If no output arguments are requested, then wpolyfit plots the data,
     the fitted line and polynomials defining the standard error range.

     Example
          x = linspace(0,4,20);
          dy = (1+rand(size(x)))/2;
          y = polyval([2,3,1],x) + dy.*randn(size(x));
          wpolyfit(x,y,dy,2);

 -- Function File: wpolyfit (..., 'origin')

     If 'origin' is specified, then the fitted polynomial will go
     through the origin.  This is generally ill-advised.  Use with
     caution.

     Hocking, RR (2003).  Methods and Applications of Linear Models.
     New Jersey: John Wiley and Sons, Inc.

See also: polyfit,polyconf.




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Return the coefficients of a polynomial P(X) of degree N that minimizes
'sumsq (



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wrap_f_dfdp


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 [ret1, ret2] = wrap_f_dfdp (f, dfdp, varargin)

 f and dftp should be the objective function (or "model function" in
 curve fitting) and its jacobian, respectively, of an optimization
 problem. ret1: f (varagin{:}), ret2: dfdp (varargin{:}). ret2 is
 only computed if more than one output argument is given. This
 manner of calling f and dfdp is needed by some optimization
 functions.



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 [ret1, ret2] = wrap_f_dfdp (f, dfdp, varargin)



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wsolve


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 [x,s] = wsolve(A,y,dy)

 Solve a potentially over-determined system with uncertainty in
 the values. 

     A x = y +/- dy

 Use QR decomposition for increased accuracy.  Estimate the 
 uncertainty for the solution from the scatter in the data.

 The returned structure s contains

    normr = sqrt( A x - y ), weighted by dy
    R such that R'R = A'A
    df = n-p, n = rows of A, p = columns of A

 See polyconf for details on how to use s to compute dy.
 The covariance matrix is inv(R'*R).  If you know that the
 parameters are independent, then uncertainty is given by
 the diagonal of the covariance matrix, or 

    dx = sqrt(N*sumsq(inv(s.R'))')

 where N = normr^2/df, or N = 1 if df = 0.

 Example 1: weighted system

    A=[1,2,3;2,1,3;1,1,1]; xin=[1;2;3]; 
    dy=[0.2;0.01;0.1]; y=A*xin+randn(size(dy)).*dy;
    [x,s] = wsolve(A,y,dy);
    dx = sqrt(sumsq(inv(s.R'))');
    res = [xin, x, dx]

 Example 2: weighted overdetermined system  y = x1 + 2*x2 + 3*x3 + e

    A = fullfact([3,3,3]); xin=[1;2;3];
    y = A*xin; dy = rand(size(y))/50; y+=dy.*randn(size(y));
    [x,s] = wsolve(A,y,dy);
    dx = s.normr*sqrt(sumsq(inv(s.R'))'/s.df);
    res = [xin, x, dx]

 Note there is a counter-intuitive result that scaling the
 uncertainty in the data does not affect the uncertainty in
 the fit.  Indeed, if you perform a monte carlo simulation
 with x,y datasets selected from a normal distribution centered
 on y with width 10*dy instead of dy you will see that the
 variance in the parameters indeed increases by a factor of 100.
 However, if the error bars really do increase by a factor of 10
 you should expect a corresponding increase in the scatter of 
 the data, which will increase the variance computed by the fit.



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 [x,s] = wsolve(A,y,dy)