Name

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        MINF_CONJ_GRAD

Purpose

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      Find the local minimum of a scalar function using conjugate gradient

Explanation

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      Find the local minimum of a scalar function of several variables using
      the Conjugate Gradient method (Fletcher-Reeves-Polak-Ribiere algorithm).
      Function may be anything with computable partial derivatives.
      Each call to minF_conj_grad performs one iteration of algorithm,
      and returns an N-dim point closer to the local minimum of function.
  CALLING EXAMPLE:
      p_min = replicate( 1, N_dim )
      minF_conj_grad, p_min, f_min, conv_factor, FUNC_NAME="name",/INITIALIZE
      while (conv_factor GT 0) do begin
              minF_conj_grad, p_min, f_min, conv_factor, FUNC_NAME="name"
      endwhile

Inputs

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      p_min = vector of independent variables, location of minimum point
              obtained from previous call to minF_conj_grad, (or first guess).

Keywords

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      FUNC_NAME = function name (string)
              Calling mechanism should be: F = func_name( px, gradient )
        where:
              F = scalar value of function at px.
              px = vector of independent variables, input.
              gradient = vector of partial derivatives of the function
                      with respect to independent variables, evaluated at px.
                      This is an optional output parameter:
                      gradient should not be calculated if parameter is not
                      supplied in call (Unless you want to waste some time).
      /INIT must be specified on first call (whenever p_min is a guess),
                      to initialize the iteration scheme of algorithm.
      /USE_DERIV causes the directional derivative of function to be used
                      in the 1-D minimization part of algorithm
                      (default is not to use directional derivative).
      TOLERANCE = desired accuracy of minimum location, default=sqrt(1.e-7).
      /QUADRATIC runs simpler version which works only for quadratic function.

Outputs

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      p_min = vector giving improved solution for location of minimum point.
      f_min = value of function at p_min.
      conv_factor = gives the current rate of convergence (change in value),
                      iteration should be stopped when rate gets near zero.

External Calls

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      pro minF_bracket, to find 3 points which bracket the minimum in 1-D.
      pro minF_parabolic, to find minimum point in 1-D.
      pro minF_parabol_D, to find minimum point in 1-D, using derivatives.

Common Blocks

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      common minf_conj_grad, grad_conj, grad_save, gs_norm
      (to keep conjugate gradient, gradient and norm from previous iteration)

Procedure

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      Algorithm adapted from Numerical Recipes, sec.10.6 (p.305).
      Conjugate gradient is computed from gradient, which then gives
      the best direction (in N-dim space) in which to proceed to find
      the minimum point. The function is then minimized along
      this direction of conjugate gradient (a 1-D minimization).
      The algorithm is repeated starting at the new point by calling again.

Modification History

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