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# Directory Listing of Routines

# projects/maven/sep/deconvolve

- AMOEBA
  Multidimensional minimization of a function FUNC(X), where

# Routine Descriptions

## AMOEBA

[List of Routines]

     NAME:
    	AMOEBA

     PURPOSE:
    	Multidimensional minimization of a function FUNC(X), where
    	X is an N-dimensional vector, using the downhill simplex
    	method of Nelder and Mead, 1965, Computer Journal, Vol 7, pp 308-313.

    	This routine is based on the AMOEBA routine, Numerical
    	Recipes in C: The Art of Scientific Computing (Second Edition), Page
    	411, and is used by permission.

     CATEGORY:
    	Function minimization/maximization. Simplex method.

     CALLING SEQUENCE:
    	Result = AMOEBA(Ftol, ....)
     INPUTS:
        FTOL:  the fractional tolerance to be achieved in the function
    	value.  e.g. the fractional decrease in the function value in the
    	terminating step.  This should never be less than the
    	machine's single or double precision.
     KEYWORD PARAMETERS:
        FUNCTION_NAME: a string containing the name of the function to
    	be minimized.  If omitted, the function FUNC is minimized.
    	This function must accept an Ndim vector as its only parameter and
    	return a scalar single or double precision floating point value as its
    	result.
        FUNCTION_VALUE: (output) on exit, an Ndim+1 element vector
    	containing the function values at the simplex points.  The first
    	element contains the function minimum.
        NCALLS: (output) the of times the function was evaluated.
        NMAX: the maximum number of function evaluations allowed
    	before terminating.  Default = 5000.
        P0: Initial starting point, an Ndim element vector.  The starting
    	point must be specified using either the keyword SIMPLEX, or P0 and
    	SCALE.  P0 may be either single or double precision floating.
    	For example, in a 3-dimensional problem, if the initial guess
    	is the point [0,0,0], and it is known that the function's
    	minimum value occurs in the interval: -10 <
    	X(0) < 10, -100 < X(1) < 100, -200 < X(2) < 200, specify: P0=[0,0,0],
    	SCALE=[10, 100, 200].
        SCALE: a scalar or Ndim element vector contaiing the problem's
    	characteristic length scale for each dimension.
    	SCALE is used with P0 to form an initial (Ndim+1) point simplex.
    	If all dimensions have the same	scale, specify a scalar.
        SIMPLEX: (output and/or optional input) On input, if P0 and SCALE
    	are not set, SIMPLEX contains the Ndim+1 vertices, each of
    	Ndim elements, of starting simplex, in either single or double
    	precision floating point, in an (Ndim, Ndim+1) array. On output,
    	SIMPLEX contains the simplex, of dimensions (Ndim, Ndim+1), enclosing
    	the function minimum.  The first point, Simplex(*,0), corresponds to
    	the function's minimum.

     OUTPUTS:
       Result: If the minimum is found, an Ndim vector, corresponding to
    	the Function's minimum value is returned.  If a function minimum
    	within the given tolerance, is NOT found in the given number of
    	evaluations, a scalar value of -1 is returned.

     COMMON BLOCKS:
    	None.

     SIDE EFFECTS:
    	None.

     PROCEDURE:
    	This procedure implements the Simplex method, described in
    	Numerical Recipes, Section 10.4.  See also the POWELL procedure.

    	Advantages:  requires only function evaluations, not
    	derivatives, may be more reliable than the POWELL method.
    	Disadvantages: not as efficient as Powell's method, and usually
    	requires more function evaluations.

    	Results are performed in the mode (single or double precision)
    	returned by the user-supplied function.  The mode of the inputs P0,
    	SCALE, or SIMPLEX, should match that returned by the function. The
    	mode of the input vector supplied to the user-written function, is
    	determined by P0, SCALE, or SIMPLEX.

     EXAMPLE:
    	Use Amoeba to find the slope and intercept of a straight line fitting
    	a given set of points minimizing the maximum error:

    	The function to be minimized returns the maximum error,
    	given p(0) = intercept, and p(1) = slope:
     FUNCTION FUNC, p
     COMMON FUNC_XY, x, y
     RETURN, MAX(ABS(y - (p(0) + p(1) * x)))
     END

    	Put the data points into a common block so they are accessible to the
    	function:
     COMMON FUNC_XY, x, y
    	Define the data points:
       x = findgen(17)*5
       y = [ 12.0,  24.3,  39.6,  51.0,  66.5,  78.4,  92.7, 107.8, 120.0, $
            135.5, 147.5, 161.0, 175.4, 187.4, 202.5, 215.4, 229.9]

    	Call the function.  Fractional tolerance = 1 part in 10^5,
    	Initial guess = [0,0], and the minimum should be found within
    	a distance of 100 of that point:
       r = AMOEBA(1.0e-5, SCALE=1.0e2, P0 = [0, 0], FUNCTION_VALUE=fval)

    	Check for convergence:
       if n_elements(r) eq 1 then message,'AMOEBA failed to converge'
    	Print results.
       print, 'Intercept, Slope:', r, 'Function value (max error): ', fval(0)
    Intercept, Slope:      11.4100      2.72800
    Function value:       1.33000

     MODIFICATION HISTORY:
    	DMS, May, 1996.	Written.

(See projects/maven/sep/deconvolve/mvn_sep_amoeba.pro)