functions in roots.i -

             f_inverse(f_and_dfdx, y, x0, x1, xerr)  
          or f_inverse(f_and_dfdx, y, x0, x1, xerr)  
 
     Find values of an inverse function by Newton-Raphson iteration,  
     backed up by bisection if the convergence seems poor.  The  
     subroutine F_AND_DFDX must be defined as:  
          func F_AND_DFDX (x, &f, &dfdx)  
     returning both the function value f(x) and derivative dfdx(x).  
     If the input x is an array, the returned f and dfdx must have  
     the same shape as the input x.  If F_AND_DFDX always returns  
     zero dfdx, f_inverse will use bisection.  
     The result x will have the same shape as the input Y values.  
     The values of x are constrained to lie within the interval from  
     X0 to X1; the function value must be on opposite sides of the  
     required Y at these interval endpoints.  The iteration stops  
     when the root is known to within XERR, or to machine precision  
     if XERR is nil or zero.  X0, X1, and XERR may be arrays conformable  
     with Y.  
     f_inverse takes the same number of iterations for every Y value;  
     it does not notice that some may have converged before others.  

             fmin= mnbrent(f, x0, x1, x2)  
          or fmin= mnbrent(f, x0, x1, x2, xmin)  
          or fmin= mnbrent(f, x0, x1, x2, xmin, xerr)  
 
     returns the minimum of the function F (of a single argument x),  
     given three points X0, X1, and X2 such that F(X1) is less than  
     either F(X0) or F(X2), and X1 is between X0 and X2.  If the  
     XMIN argument is provided, it is set to the x value which  
     produced FMIN.  If XERR is supplied, the search stops when  
     a fractional error of XERR in x is reached; note that XERR  
     smaller than the square root of the machine precision (or  
     omitted) will cause convergence to machine precision in FMIN.  
     The algorithm is Brent's method - a combination of inverse  
     parabolic interpolation and golden section search - as adapted  
     from Numerical Recipes Ch. 10 (Press, et. al.).  

             fmax= mxbrent(f, x0, x1, x2)  
          or fmax= mxbrent(f, x0, x1, x2, xmax)  
          or fmax= mxbrent(f, x0, x1, x2, xmax, xerr)  
 
     returns the maximum of the function F (of a single argument x),  
     given three points X0, X1, and X2 such that F(X1) is greater than  
     either F(X0) or F(X2), and X1 is between X0 and X2.  If the  
     XMAX argument is provided, it is set to the x value which  
     produced FMAX.  If XERR is supplied, the search stops when  
     a fractional error of XERR in x is reached; note that XERR  
     smaller than the square root of the machine precision (or  
     omitted) will cause convergence to machine precision in FMAX.  
     The algorithm is Brent's method - a combination of inverse  
     parabolic interpolation and golden section search - as adapted  
     from Numerical Recipes Ch. 10 (Press, et. al.).  

             nraphson(f_and_dfdx, x0, x1)  
          or nraphson(f_and_dfdx, x0, x1, xerr)  
 
     Find a root of a function by Newton-Raphson iteration, backed  
     up by bisection if the convergence seems poor.  The subroutine  
     F_AND_DFDX must be defined as:  
          func F_AND_DFDX (x, &f, &dfdx)  
     returning both the function value f(x) and derivative dfdx(x).  
     If F_AND_DFDX always returns dfdx==0, nraphson uses bisection.  
     The value of x is constrained to lie within the interval from  
     X0 to X1; the function values at these two points must have  
     opposite sign.  The iteration stops when the root is known to  
     within XERR, or to machine precision if XERR is nil or zero.  
     f_inverse is a "vectorized" version of nraphson.  
     Based on rtsafe from Press, et. al. Numerical Recipes, Ch 9.  

             roots.i  
 
       defines:  
     nraphson     - Newton-Raphson/bisection root solver (scalar)  
     f_inverse    - function inverse by Newton-Raphson (vectorized)  
     mnbrent      - Brent's method minimizer (scalar)  
     mxbrent      - Brent's method maximizer (scalar)