Name

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      MINF_PARABOLIC

Purpose

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      Minimize a function using Brent's method with parabolic interpolation

Explanation

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      Find a local minimum of a 1-D function up to specified tolerance.
      This routine assumes that the function has a minimum nearby.
      (recommend first calling minF_bracket, xa,xb,xc, to bracket minimum).
      Routine can also be applied to a scalar function of many variables,
      for such case the local minimum in a specified direction is found,
      This routine is called by minF_conj_grad, to locate minimum in the
      direction of the conjugate gradient of function of many variables.
  CALLING EXAMPLES:
      minF_parabolic, xa,xb,xc, xmin, fmin, FUNC_NAME="name" ;for 1-D func.
  or:
      minF_parabolic, xa,xb,xc, xmin, fmin, FUNC="name", $
                                        POINT=[0,1,1], $
                                        DIRECTION=[2,1,1] ;for 3-D func.

Inputs

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      xa,xb,xc = scalars, 3 points which bracket location of minimum,
              that is, f(xb) < f(xa) and f(xb) < f(xc), so minimum exists.
              When working with function of N variables
              (xa,xb,xc) are then relative distances from POINT_NDIM,
              in the direction specified by keyword DIRECTION,
              with scale factor given by magnitude of DIRECTION.

Input Keywords

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      FUNC_NAME = function name (string)
              Calling mechanism should be: F = func_name( px )
              where:
                      px = scalar or vector of independent variables, input.
                      F = scalar value of function at px.
      POINT_NDIM = when working with function of N variables,
              use this keyword to specify the starting point in N-dim space.
              Default = 0, which assumes function is 1-D.
      DIRECTION = when working with function of N variables,
              use this keyword to specify the direction in N-dim space
              along which to bracket the local minimum, (default=1 for 1-D).
              (xa, xb, xc, x_min are then relative distances from POINT_NDIM)
      MAX_ITER = maximum allowed number iterations, default=100.
      TOLERANCE = desired accuracy of minimum location, default=sqrt(1.e-7).

Outputs

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      xmin = estimated location of minimum.
              When working with function of N variables,
              xmin is the relative distance from POINT_NDIM,
              in the direction specified by keyword DIRECTION,
              with scale factor given by magnitude of DIRECTION,
              so that min. Loc. Pmin = Point_Ndim + xmin * Direction.
      fmin = value of function at xmin (or Pmin).

Procedure

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      Brent's method to minimize a function by using parabolic interpolation.
      Based on function BRENT in Numerical Recipes in FORTRAN (Press et al.
      1992), sec.10.2 (p. 397).

Modification History

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      Written, Frank Varosi NASA/GSFC 1992.
      Converted to IDL V5.0 W. Landsman September 1997