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IMSL Name:  UVMGS/DUVMGS (Single/Double precision version)
 
Revised:    May 31, 1991
 
Purpose:    Find the minimum point of a nonsmooth function of a
            single variable.
 
Usage:      CALL UVMGS (F, A, B, TOL, XMIN)
 
Arguments:
   F      - User-supplied FUNCTION to compute the value of the
            function to be minimized.  The form is
            F(X), where
            X      - The point at which the function is
                     evaluated.  (Input)
                     X should not be changed by F.
            F      - The computed function value at the point X.
                     (Output)
            F must be declared EXTERNAL in the calling program.
   A      - On input, A is the lower endpoint of the interval in
            which the minimum of F is to be located.  On output, A
            is the lower endpoint of the interval in which the
            minimum of F is located.  (Input/Output)
   B      - On input, B is the upper endpoint of the interval in
            which the minimum of F is to be located.  On output, B
            is the upper endpoint of the interval in which the
            minimum of F is located.  (Input/Output)
   TOL    - The allowable length of the final subinterval
            containing the minimum point.  (Input)
   XMIN   - The approximate minimum point of the function F on the
            original interval (A,B).  (Output)
 
Remarks:
1. Informational errors
   Type Code
     3   1  TOL is too small to be satisfied.
     4   2  Due to rounding errors F does not appear to be
            unimodal.
 
2. On exit from UVMGS without any error messages, the following
   conditions hold:
   (B-A) .LE. TOL.
   A .LE. XMIN .AND. XMIN .LE. B
   F(XMIN) .LE. F(A) .AND. F(XMIN) .LE. F(B)
 
3. On exit from UVMGS with error code 2, the following conditions
   hold:
   A .LE. XMIN .AND. XMIN .LE. B
   F(XMIN) .GE. F(A) .AND. F(XMIN) .GE. F(B) (only one equality can
   hold).
   Further analysis of the function F is necessary in order to
   determine whether it is not unimodal in the mathematical sense or
   whether it appears to be not unimodal to the routine due to
   rounding errors in which case the A, B, and XMIN returned may be
   acceptable.
 
Keywords:   Golden Section; Unimodal; Optimization; Univariate
            function
 
GAMS:       G1a2
 
Chapter:    MATH/LIBRARY Optimization
 
Page No.:   MATH/LIBRARY User's Manual page 978