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LinearRegression


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

     general linear regression

     determine the parameters p_j (j=1,2,...,m) such that the function f(x) =
     sum_(j=1,...,m) p_j*f_j(x) is the best fit to the given values y_i by
     f(x_i) for i=1,...,n, i.e.  minimize sum_(i=1,...,n)(y_i-sum_(j=1,...,m)
     p_j*f_j(x_i))^2 with respect to p_j

     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 a column vector of length n with the weights of the data points.
          1/w_i is expected to be proportional to the estimated uncertainty in
          the y values.  Then the weighted expression
          sum_(i=1,...,n)(w_i^2*(y_i-f(x_i))^2) is minimized.

     return values:
        • P is the vector of length m with the estimated values of the
          parameters
        • E_VAR is the vector of estimated variances of the provided y values.
          If weights are provided, then the product e_var_i * w^2_i is assumed
          to be constant.
        • R is the weighted norm of the residual
        • P_VAR is the vector of estimated variances of the parameters p_j
        • FIT_VAR is the vector of the estimated variances of the fitted
          function values f(x_i)

     To estimate the variance of the difference between future y values and
     fitted y values use the sum of E_VAR and FIT_VAR

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

     See also: ols,gls,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 attemp...



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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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 usage: bfgsmin_example (to run) or edit bfgsmin_example (to examine)

 Shows how to call bfgsmin. There are two objective functions, the first
 supplies the analytic gradient, and the second does not. The true minimizer
 is at "location", a 50x1 vector (0.00, 0.02, 0.04 ..., 1.00).
 Note that limited memory bfgs is faster when the dimension is high.
 Also note that supplying analytic derivatives gives a speedup.

 Six examples are given:
 Example 1: regular bfgs, analytic gradient
 Example 2: same as Example 1, but verbose
 Example 3: limited memory bfgs, analytic gradient
 Example 4: regular bfgs, numeric gradient
 Example 5: limited memory bfgs, numeric gradient
 Example 6: regular bfgs, analytic gradient, minimize wrt second argument



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 usage: bfgsmin_example (to run) or edit bfgsmin_example (to examine)



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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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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. differentiation
   = "...



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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
     http://en.wikipedia.org/wiki/Nonlinear_conjugate_gradient.

     See also: bfgsmin.


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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
m...



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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 derivat...



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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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This is a deprecated wrapper to ‘pronyfit’, please see the documentation of the
latter function.


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This is a deprecated wrapper to ‘pronyfit’, please see the documentation of t...



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fmincon


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 -- Function File: fmincon (OBJF, X0)
 -- Function File: fmincon (OBJF, X0, A, B)
 -- Function File: fmincon (OBJF, X0, A, B, AEQ, BEQ)
 -- Function File: fmincon (OBJF, X0, A, B, AEQ, BEQ, LB, UB)
 -- Function File: fmincon (OBJF, X0, A, B, AEQ, BEQ, LB, UB, NONLCON)
 -- Function File: fmincon (OBJF, X0, A, B, AEQ, BEQ, LB, UB, NONLCON, OPTIONS)
 -- Function File: fmincon (PROBLEM)
 -- Function File: [X, FVAL, CVG, OUTP] = fmincon (...)
     Compatibility frontend for nonlinear minimization of a scalar objective
     function.

     This function is for Matlab compatibility and provides a subset of the
     functionality of ‘nonlin_min’.

     OBJF: objective function.  It gets the real parameters as argument.

     X0: real vector or array of initial parameters.

     A, B: Inequality constraints of the parameters ‘p’ with ‘A * p - b <= 0’.

     AEQ, BEQ: Equality constraints of the parameters ‘p’ with ‘A * p - b = 0’.

     LB, UB: Bounds of the parameters ‘p’ with ‘lb <= p <= ub’.  Vectors or
     arrays.  If the number of elements is smaller than the number of
     parameters, as many bounds as present are applied, starting with the first
     parameter.  This is for compatibility with Matlab.

     NONLCON: Nonlinear constraints.  Function returning the current values of
     nonlinear inequality constraints (constrained to ‘<= 0’) in the first
     output and the current values of nonlinear equality constraints in the
     second output.

     OPTIONS: structure whose fields stand for optional settings referred to
     below.  The fields can be set by ‘optimset()’.

     An argument can be set to ‘[]’ to indicate that its value is not set.

     ‘fmincon’ may also be called with a single structure argument with the
     fields ‘objective’, ‘x0’, ‘Aineq’, ‘bineq’, ‘Aeq’, ‘beq’, ‘lb’, ‘ub’,
     ‘nonlcon’ and ‘options’, resembling the separate input arguments above.
     Additionally, the structure must have the field ‘solver’, set to "fmincon".

     The returned values are the final parameters X, the final value of the
     objective function FVAL, an integer CVG indicating if and how optimization
     succeeded or failed, and a structure OUTP with additional information,
     curently with possible fields: ‘iterations’, the number of iterations,
     ‘funcCount’, the number of objective function calls (indirect calls by
     gradient function not counted), ‘constrviolation’, the maximum of the
     constraint violations.  The backend may define additional fields.  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’ (success without further
     specification of criteria), ‘2’ (parameter change less than specified
     precision in two consecutive iterations), ‘3’ (improvement in objective
     function less than specified), ‘-1’ (algorithm aborted by a user function),
     or ‘-4’ (algorithm got stuck).

     Options:
     ........

     ‘Algorithm’
          ‘interior-point’, ‘sqp’, and ‘sqp-legacy’ are mapped to optims
          ‘lm_feasible’ algorithm (the default) to satisfy constraints
          throughout the optimization.  ‘active-set’ is mapped to ‘octave_sqp’,
          which may perform better if constraints only need to be satisfied for
          the result.  Other algorithms are available with ‘nonlin_min’.

     ‘OutputFcn’
          Similar to the setting ‘user_interaction’ -- see ‘optim_doc()’.
          Differently, ‘OutputFcn’ returns only one output argument, the STOP
          flag.

     ‘GradObj’
          If set to ‘"on"’, OBJF must return the gradient of the objective
          function as a second output.  The default is ‘"off"’.

     ‘GradConstr’
          If set to ‘"on"’, NONLCON must return the Jacobians of the inequality-
          and equality-constraints as third and fourth output, respectively.

     ‘HessianFcn’
          If set to ‘"objective"’, OBJF must not only return the gradient as the
          second, but also the Hessian as the third output.

     ‘Display, FinDiffRelStep, FinDiffType, TypicalX, MaxIter, TolFun, TolX,’
          See documentation of these options in ‘optim_doc()’.

     For description of individual backends, type ‘optim_doc ("scalar
     optimization")’ and choose the backend in the menu.


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Compatibility frontend for nonlinear minimization of a scalar objective
funct...



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fmins


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

     This function is deprecated and will be removed in the future.  It is a
     frontend function, calling adsmax or mdsmax, and previously nmsmax, which
     is now in core Octave with a slightly different interface.

     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 - unused (previously Nelder & Mead simplex)
             if options(6)==1 - Multidirectional search Method (default)
             if options(6)==2 - Alternating Directions search
          options(5)
             unused
          options(10) - Maximum number of function evaluations
     GRAD
          Unused
     P1, P2, ...
          Optional parameters for function F


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This function is deprecated and will be removed in the future.



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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 File: leasqr (X, Y, PIN, F)
 -- Function File: leasqr (X, Y, PIN, F, STOL)
 -- Function File: leasqr (X, Y, PIN, F, STOL, NITER)
 -- Function File: leasqr (X, Y, PIN, F, STOL, NITER, WT)
 -- Function File: leasqr (X, Y, PIN, F, STOL, NITER, WT, DP)
 -- Function File: leasqr (X, Y, PIN, F, STOL, NITER, WT, DP, DFDP)
 -- Function File: leasqr (X, Y, PIN, F, STOL, NITER, WT, DP, DFDP, OPTIONS)
 -- Function File: [F, P, CVG, ITER, CORP, COVP, COVR, STDRESID, Z, R2] = leasqr
          (...)
     Levenberg-Marquardt nonlinear regression.

     Input arguments:

     X
          Vector or matrix of independent variables.

     Y
          Vector or matrix of observed values.

     PIN
          Vector of initial parameters to be adjusted by leasqr.

     F
          Name of function or function handle.  The function must be of the form
          ‘y = f(x, p)’, with y, x, p of the form Y, X, PIN.

     STOL
          Scalar tolerance on fractional improvement in scalar sum of squares,
          i.e., ‘sum ((WT .* (Y-F))^2)’.  Set to 0.0001 if empty or not given;

     NITER
          Maximum number of iterations.  Set to 20 if empty or not given.

     WT
          Statistical weights (same dimensions as Y).  These should be set to be
          proportional to ‘sqrt (Y) ^-1’, i.e., 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.  Set
          to ‘ones (size (Y))’ if empty or not given.

     DP
          Fractional increment of P for numerical partial derivatives.  Set to
          ‘0.001 * ones (size (PIN))’ if empty or not given.

             • 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 won't change initial
               guess: pin(j)

     DFDP
          Name of partial derivative function in quotes or function handle.  If
          not given or empty, set to ‘dfdp’, a slow but general partial
          derivatives function.  The function must 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 explicitly specified to leasqr (see dfdp.m).

     OPTIONS
          Structure with multiple options.  The following fields are recognized:

          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
               'fract_prec', i.e., ‘all (abs (chg) < abs (options.fract_prec .*
               current_parm_est))’ on two consecutive iterations.  Defaults to
               ‘zeros (size (PIN))’.

          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 'max_fract_change'
               between sucessive iterations, i.e., ‘abs (chg(i)) = abs
               (min([chg(i), options.max_fract_change(i) * current param
               estimate]))’.  Defaults to ‘Inf * ones (size (PIN))’.

          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.

          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.

          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.

          cpiv
               Function for complementary pivot algorithm for inequality
               constraints.  Defaults to cpiv_bard.  No different function is
               supplied.

          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.

     Output:

     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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Levenberg-Marquardt nonlinear regression.



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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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lsqcurvefit


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 -- Function File: lsqcurvefit (FUN, X0, XDATA, YDATA)
 -- Function File: lsqcurvefit (FUN, X0, XDATA, YDATA, LB, UB)
 -- Function File: lsqcurvefit (FUN, X0, XDATA, YDATA, LB, UB, OPTIONS)
 -- Function File: lsqcurvefit (PROBLEM)
 -- Function File: [X, RESNORM, RESIDUAL, EXITFLAG, OUTPUT, LAMBDA, JACOBIAN] =
          lsqcurvefit (...)
     Solve nonlinear least-squares (nonlinear data-fitting) problems
          min [EuclidianNorm (f(x, xdata) - ydata)] .^ 2
           x

     The first four input arguments must be provided with non-empty initial
     guess X0.  For a given input XDATA, YDATA is the observed output.  YDATA
     must be the same size as the vector (or matrix) returned by FUN.  The
     optional bounds LB and UB should be the same size as X0.

     ‘lsqcurvefit’ may also be called with a single structure argument with the
     fields ‘fun’, ‘x0’, ‘xdata’, ‘ydata’, ‘lb’, ‘ub’, and ‘options’, resembling
     the separate input arguments above.  For compatibility reasons, field ‘fun’
     may also be called ‘objective’.  Additionally, the structure must have the
     field ‘solver’, set to "lsqcurvefit".

     OPTIONS can be set with ‘optimset’.  Follwing Matlab compatible options are
     recognized:

     ‘Algorithm’ String specifying backend algorithm.  Currently available
     "lm_svd_feasible" only.

     ‘TolFun’ Minimum fractional improvement in objective function in an
     iteration (termination criterium).  Default: 1e-6.

     ‘TypicalX’ Typical values of x.  Default: 1.

     ‘MaxIter’ Maximum number of iterations allowed.  Default: 20.

     ‘Jacobian’ If set to "on", the function FUN must return a second output
     containing a user-specified Jacobian.  The Jacobian is computed using
     finite differences otherwise.  Default: "off"

     ‘FinDiffType’ "centered" or "forward" (Default) type finite differences
     estimation.

     ‘FinDiffRelStep’ Step size factor.  The default is sqrt(eps) for forward
     finite differences, and eps^(1/3) for central finite differences

     ‘OutputFcn’ One or more user-defined functions, either as a function handle
     or as a cell array of function handles that an optimization function calls
     at each iteration.  The function definition has the following form:

     ‘stop = outfun(x, optimValues, state)’

     ‘x’ is the point computed at the current iteration.  ‘optimValues’ is a
     structure containing data from the current iteration in the following
     fields: "iteration"- number of current iteration.  "residual"- residuals.
     ‘state’ is the state of the algorithm: "init" at start, "iter" after each
     iteration and "done" at the end.

     ‘Display’ String indicating the degree of verbosity.  Default: "off".
     Currently only supported values are "off" (no messages) and "iter" (some
     messages after each iteration).

     Returned values:

     X
          Coefficients to best fit the nonlinear function fun(x,xdata) to the
          observed values ydata.

     RESNORM
          Scalar value of objective as squared EuclidianNorm(f(x)).

     RESIDUAL
          Value of solution residuals f(x).

     EXITFLAG
          Status of solution:

          ‘0’
               Maximum number of iterations reached.

          ‘2’
               Change in x was less than the specified tolerance.

          ‘3’
               Change in the residual was less than the specified tolerance.

          ‘-1’
               Output function terminated the algorithm.

     OUTPUT
          Structure with additional information, currently the only field is
          ‘iterations’, the number of used iterations.

     LAMBDA
          Structure containing Lagrange multipliers at the solution X sepatared
          by constraint type (LB and UB).

     JACOBIAN
          m-by-n matrix, where JACOBIAN(i,j) is the partial derivative of FUN(I)
          with respect to X(J) If ‘Jacobian’ is set to "on" in OPTIONS then FUN
          must return a second argument providing a user-sepcified Jacobian.
          Otherwise, lsqnonlin approximates the Jacobian using finite
          differences.

     This function is a compatibility wrapper.  It calls the more general
     ‘nonlin_curvefit’ function internally.

     See also: lsqnonlin, nonlin_residmin, nonlin_curvefit.


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Solve nonlinear least-squares (nonlinear data-fitting) problems
     min [Euc...



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lsqlin


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 -- Function File: lsqlin (C, D, A, B)
 -- Function File: lsqlin (C, D, A, B, AEQ, BEQ, LB, UB)
 -- Function File: lsqlin (C, D, A, B, AEQ, BEQ, LB, UB, X0)
 -- Function File: lsqlin (C, D, A, B, AEQ, BEQ, LB, UB, X0, OPTIONS)
 -- Function File: [X, RESNORM, RESIDUAL, EXITFLAG, OUTPUT, LAMBDA] = lsqlin
          (...)
     Solve the linear least squares program
          min 0.5 sumsq(C*x - d)
          x
     subject to
          A*X <= B,
          AEQ*X = BEQ,
          LB <= X <= UB.

     The initial guess X0 and the constraint arguments (A and B, AEQ and BEQ, LB
     and UB) can be set to the empty matrix (‘[]’) if not given.  If the initial
     guess X0 is feasible the algorithm is faster.

     OPTIONS can be set with ‘optimset’, currently the only option is ‘MaxIter’,
     the maximum number of iterations (default: 200).

     Returned values:

     X
          Position of minimum.

     RESNORM
          Scalar value of objective as sumsq(C*x - d).

     RESIDUAL
          Vector of solution residuals C*x - d.

     EXITFLAG
          Status of solution:

          ‘0’
               Maximum number of iterations reached.

          ‘-2’
               The problem is infeasible.

          ‘-3’
               The problem is not convex and unbounded.

          ‘1’
               Global solution found.

     OUTPUT
          Structure with additional information, currently the only field is
          ‘iterations’, the number of used iterations.

     LAMBDA
          Structure containing Lagrange multipliers corresponding to the
          constraints.

     This function calls the more general function ‘quadprog’ internally.

     See also: quadprog.


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Solve the linear least squares program
     min 0.5 sumsq(C*x - d)
     x
   ...



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lsqnonlin


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 -- Function File: lsqnonlin (FUN, X0)
 -- Function File: lsqnonlin (FUN, X0, LB, UB)
 -- Function File: lsqnonlin (FUN, X0, LB, UB, OPTIONS)
 -- Function File: [X, RESNORM, RESIDUAL, EXITFLAG, OUTPUT, LAMBDA, JACOBIAN] =
          lsqnonlin (...)
     Solve nonlinear least-squares (nonlinear data-fitting) problems
          min [EuclidianNorm(f(x))] .^ 2
           x

     FUN computes residuals from given parameters.  The initial guess of the
     parameters X0 must be provided while the bounds LB and UB) can be set to
     the empty matrix (‘[]’) if not given.

     ‘lsqnonlin’ may also be called with a single structure argument with the
     fields ‘fun’, ‘x0’, ‘lb’, ‘ub’, and ‘options’, resembling the separate
     input arguments above.  For compatibility reasons, field ‘fun’ may also be
     called ‘objective’.  Additionally, the structure must have the field
     ‘solver’, set to "lsqnonlin".

     OPTIONS can be set with ‘optimset’.  Follwing Matlab compatible options are
     recognized:

     ‘Algorithm’ String specifying backend algorithm.  Currently available
     "lm_svd_feasible" only.

     ‘TolFun’ Minimum fractional improvement in objective function in an
     iteration (termination criterium).  Default: 1e-6.

     ‘TypicalX’ Typical values of x.  Default: 1.

     ‘MaxIter’ Maximum number of iterations allowed.  Default: 400.

     ‘Jacobian’ If set to "on", the function FUN must return a second output
     containing a user-specified Jacobian.  The Jacobian is computed using
     finite differences otherwise.  Default: "off"

     ‘FinDiffType’ "centered" or "forward" (Default) type finite differences
     estimation.

     ‘FinDiffRelStep’ Step size factor.  The default is sqrt(eps) for forward
     finite differences, and eps^(1/3) for central finite differences

     ‘OutputFcn’ One or more user-defined functions, either as a function handle
     or as a cell array of function handles that an optimization function calls
     at each iteration.  The function definition has the following form:

     ‘stop = outfun(x, optimValues, state)’

     ‘x’ is the point computed at the current iteration.  ‘optimValues’ is a
     structure containing data from the current iteration in the following
     fields: "iteration"- number of current iteration.  "residual"- residuals.
     ‘state’ is the state of the algorithm: "init" at start, "iter" after each
     iteration and "done" at the end.

     ‘Display’ String indicating the degree of verbosity.  Default: "off".
     Currently only supported values are "off" (no messages) and "iter" (some
     messages after each iteration).

     Returned values:

     X
          Position of minimum.

     RESNORM
          Scalar value of objective as squared EuclidianNorm(f(x)).

     RESIDUAL
          Value of solution residuals f(x).

     EXITFLAG
          Status of solution:

          ‘0’
               Maximum number of iterations reached.

          ‘2’
               Change in x was less than the specified tolerance.

          ‘3’
               Change in the residual was less than the specified tolerance.

          ‘-1’
               Output function terminated the algorithm.

     OUTPUT
          Structure with additional information, currently the only field is
          ‘iterations’, the number of used iterations.

     LAMBDA
          Structure containing Lagrange multipliers at the solution X sepatared
          by constraint type (LB and UB).

     JACOBIAN
          m-by-n matrix, where JACOBIAN(I,J) is the partial derivative of FUN(I)
          with respect to X(J) Default: lsqnonlin approximates the Jacobian
          using finite differences.  If ‘Jacobian’ is set to "on" in OPTIONS
          then FUN must return a second argument providing a user-sepcified
          Jacobian .

     This function is a compatibility wrapper.  It calls the more general
     ‘nonlin_residmin’ function internally.

     See also: lsqcurvefit, nonlin_residmin, nonlin_curvefit.


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Solve nonlinear least-squares (nonlinear data-fitting) problems
     min [Euc...



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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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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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nlinfit


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 -- Function File: nlinfit (X, Y, MODELFUN, BETA0)
 -- Function File: nlinfit (X, Y, MODELFUN, BETA0, OPTIONS)
 -- Function File: nlinfit (..., NAME, VALUE)
 -- Function File: [BETA, R, J, COVB, MSE] = nlinfit (...)
     Nonlinear Regression.

          min [EuclidianNorm (Y - modelfun (beta, X))] ^ 2
          beta

     X is a matrix of independents, Y is the observed output and MODELFUN is the
     nonlinear regression model function.  MODELFUN should be specified as a
     function handle, which accepts two inputs: an array of coefficients and an
     array of independents - in that order.  The first four input arguments must
     be provided with non-empty initial guess of the coefficients BETA0.  Y and
     X must be the same size as the vector (or matrix) returned by FUN.  OPTIONS
     is a structure containing estimation algorithm options.  It can be set
     using ‘statset’.  Follwing Matlab compatible options are recognized:

     ‘TolFun’ Minimum fractional improvement in objective function in an
     iteration (termination criterium).  Default: 1e-6.

     ‘MaxIter’ Maximum number of iterations allowed.  Default: 400.

     ‘DerivStep’ Step size factor.  The default is eps^(1/3) for finite
     differences gradient calculation.

     ‘Display’ String indicating the degree of verbosity.  Default: "off".
     Currently only supported values are "off" (no messages) and "iter" (some
     messages after each iteration).

     Optional NAME, VALUE pairs can be provided to set additional options.
     Currently the only applicable name-value pair is 'Weights', w, where w is
     the array of real positive weight factors for the squared residuals.

     Returned values:

     BETA
          Coefficients to best fit the nonlinear function modelfun (beta, X) to
          the observed values Y.

     R
          Value of solution residuals: ‘modelfun (beta, X) - Y’.  If observation
          weights are specified then R is the array of weighted residuals: ‘sqrt
          (weights) .* modelfun (beta, X) - Y’.

     J
          A matrix where ‘J(i,j)’ is the partial derivative of ‘modelfun(i)’
          with respect to ‘beta(j)’.  If observation weights are specified, then
          J is the weighted model function Jacobian: ‘diag (sqrt (weights)) *
          J’.

     COVB

          Estimated covariance matrix of the fitted coefficients.

     MSE
          Scalar valued estimate of the variance of error term.  If the model
          Jacobian is full rank, then MSE = (R' * R)/(N-p), where N is the
          number of observations and p is the number of estimated coefficients.

     This function is a compatibility wrapper.  It calls the more general
     ‘nonlin_curvefit’ and ‘curvefit_stat’ functions internally.

     See also: nonlin_residmin, nonlin_curvefit, residmin_stat, curvefit_stat.


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Nonlinear Regression.



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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 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.

     Also, if the setting ‘user_interaction’ is given, additional information is
     passed to these functions.  Type ‘optim_doc ("Common optimization
     options")’ for this setting.

     See also: nonlin_residmin.


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



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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 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 separately.

     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 (if the function accepts it), 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()’.

     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 possible fields: ‘niter’, the number of
     iterations, ‘nobjf’, the number of objective function calls (indirect calls
     by gradient function not counted), ‘lambda’, the lambda of constraints at
     the result, and ‘user_interaction’, information on user stops (see
     settings).  The backend may define additional fields.  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’ (success without further specification of
     criteria), ‘2’ (parameter change less than specified precision in two
     consecutive iterations), ‘3’ (improvement in objective function less than
     specified), ‘-1’ (algorithm aborted by a user function), or ‘-4’ (algorithm
     got stuck).

     For settings, type ‘optim_doc ("nonlin_min")’.

     For desription of structure-based parameter handling, type ‘optim_doc
     ("parameter structures")’.

     For description of individual backends, type ‘optim_doc ("scalar
     optimization")’ and choose the backend in the menu.


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Frontend for 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 separately.

     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 (if the function accepts
     it), 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()’.

     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 the fields: ‘niter’, the number of iterations and
     ‘user_interaction’, information on user stops (see settings).  The backend
     may define additional fields.  If the backend supports it, OUTP has a field
     ‘lambda’ with determined Lagrange multipliers of any constraints, seperated
     into subfields ‘lower’ and ‘upper’ for bounds, ‘eqlin’ and ‘ineqlin’ for
     linear equality and inequality constraints (except bounds), respectively,
     and ‘eqnonlin’ and ‘ineqnonlin’ for general equality and inequality
     constraints, respectively.  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), or ‘-1’ (algorithm aborted by a user
     function).

     For settings, type ‘optim_doc ("nonlin_residmin")’.

     For desription of structure-based parameter handling, type ‘optim_doc
     ("parameter structures")’.

     For description of individual backends (currently only one), type
     ‘optim_doc ("residual optimization")’ and choose the backend in the menu.

     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.


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Using X0 as a starting point find a minimum of the scalar function F.



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optim_doc


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 -- Function File: optim_doc ()
 -- Function File: optim_doc (KEYWORD)
     Show optim package documentation.

     Runs the info viewer Octave is configured with on the documentation in info
     format of the installed optim package.  Without argument, the top node of
     the documentation is displayed.  With an argument, the respective index
     entry is searched for and its node displayed.


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Show optim package documentation.



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optim_problems


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 Problems for testing optimizers. Documentation is in the code.



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 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.

 This function will be removed from future versions of the optim
 package since it is not related to optimization.



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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



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

 ...



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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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pronyfit


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



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



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quadprog


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 -- Function File: quadprog (H, F)
 -- Function File: quadprog (H, F, A, B)
 -- Function File: quadprog (H, F, A, B, AEQ, BEQ)
 -- Function File: quadprog (H, F, A, B, AEQ, BEQ, LB, UB)
 -- Function File: quadprog (H, F, A, B, AEQ, BEQ, LB, UB, X0)
 -- Function File: quadprog (H, F, A, B, AEQ, BEQ, LB, UB, X0, OPTIONS)
 -- Function File: [X, FVAL, EXITFLAG, OUTPUT, LAMBDA] = quadprog (...)
     Solve the quadratic program
          min 0.5 x'*H*x + x'*f
           x
     subject to
          A*X <= B,
          AEQ*X = BEQ,
          LB <= X <= UB.

     The initial guess X0 and the constraint arguments (A and B, AEQ and BEQ, LB
     and UB) can be set to the empty matrix (‘[]’) if not given.  If the initial
     guess X0 is feasible the algorithm is faster.

     OPTIONS can be set with ‘optimset’, currently the only option is ‘MaxIter’,
     the maximum number of iterations (default: 200).

     Returned values:

     X
          Position of minimum.

     FVAL
          Value at the minimum.

     EXITFLAG
          Status of solution:

          ‘0’
               Maximum number of iterations reached.

          ‘-2’
               The problem is infeasible.

          ‘-3’
               The problem is not convex and unbounded

          ‘1’
               Global solution found.

          ‘4’
               Local solution found.

     OUTPUT
          Structure with additional information, currently the only field is
          ‘iterations’, the number of used iterations.

     LAMBDA
          Structure containing Lagrange multipliers corresponding to the
          constraints.  For equality constraints, the sign of the multipliers is
          chosen to satisfy the equation
               0.5 H * x + f + A' * lambda_inequ + Aeq' * lambda_equ = 0 .
          If lower and upper bounds are equal, or so close to each other that
          they are considered equal by the algorithm, only one of these bounds
          is considered active when computing the solution, and a positive
          lambda will be placed only at this bound.

     This function calls Octave's ‘__qp__’ back-end algorithm internally.


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Solve the quadratic program
     min 0.5 x'*H*x + x'*f
      x
   subject to
...



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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
     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 (the ‘ret_’ is prepended so that the option name is complete):

     ‘ret_dfdp’: Jacobian of model function with respect to parameters.

     ‘ret_covd’: Covariance matrix of data (typically guessed by applying a
     factor to the covariance matrix of the residuals).

     ‘ret_covp’: Covariance matrix of final parameters.

     ‘ret_corp’: Correlation matrix of final parameters.

     For further settings, type ‘optim_doc ("residmin_stat")’.

     For desription of structure-based parameter handling, type ‘optim_doc
     ("parameter structures")’.

     For backend information, type ‘optim_doc ("residual optimization")’ and
     choose the backends type in the menu.

     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. fns.



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statget


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 -- Function File: statget (OPTIONS, PARNAME)
 -- Function File: statget (OPTIONS, PARNAME, DEFAULT)
     Return the specific option PARNAME from the statistics options structure
     OPTIONS created by ‘statset’.

     If PARNAME is not defined then return DEFAULT if supplied, otherwise return
     an empty matrix.

     See also: statset.


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Return the specific option PARNAME from the statistics options structure OPTI...



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statset


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 -- Function File: statset ()
 -- Function File: OPTIONS = statset ()
 -- Function File: OPTIONS = statset (PAR, VAL, ...)
 -- Function File: OPTIONS = statset (OLD, PAR, VAL, ...)
 -- Function File: OPTIONS = statset (OLD, NEW)
     Create options structure for statistics functions.

     When called without any input or output arguments, ‘statset’ prints a list
     of all valid statistics parameters.

     When called with one output and no inputs, return an options structure with
     all valid option parameters initialized to ‘[]’.

     When called with a list of parameter/value pairs, return an options
     structure with only the named parameters initialized.

     When the first input is an existing options structure OLD, the values are
     updated from either the PAR/VAL list or from the options structure NEW.

     Please see individual statistics functions for valid settings.

     See also: statget.


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Create options structure for statistics functions.



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test_min_1


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 Test an optimization function with the same synopsis as bfgs.m 



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 Test an optimization function with the same synopsis as bfgs.m 




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test_min_2


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 test_min_2                   - Test that bfgs works

 Defines some simple functions and verifies that calling
 
 bfgs on them returns the correct minimum.

 Sets 'ok' to 1 if success, 0 otherwise



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 test_min_2                   - Test that bfgs works



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test_min_3


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 ok                     - Test that bfgs works with extra
                          arguments 

 Defines some simple functions and verifies that calling
 bfgs on them returns the correct minimum.

 Sets 'ok' to 1 if success, 0 otherwise



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 ok                     - Test that bfgs works with extra
                   ...



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test_min_4


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 test_bfgs              - Test that bfgs works

 Check that bfgs treats struct options correctly

 Sets 'ok' to 1 if success, 0 otherwise



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 test_bfgs              - Test that bfgs works



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test_nelder_mead_min_1


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 Checks wether the function 'nelder_mead_min' works, by making it minimize a
 quadratic function.



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 Checks wether the function 'nelder_mead_min' works, by making it minimize a
...



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test_nelder_mead_min_2


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 Checks wether the function 'nelder_mead_min' accepts options properly



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 Checks wether the function 'nelder_mead_min' accepts options properly




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test_wpolyfit


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 Tests for wpolyfit.

 Test cases are taken from the NIST Statistical Reference Datasets
    http://www.itl.nist.gov/div898/strd/



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 Tests for wpolyfit.



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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 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.


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A variant of ‘fzero’.  Fi



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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: core Octave ‘polyfit’.

     See also: polyconf.


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Return the coefficients of a polynomial P(X) of degree N that minimizes ‘sums...



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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)





