Channel number

                The general function is  where x is the channel number.  This needs to be inverted to find x as a function of E.  In the case M=1, there is a mistake.  If M=2, then

 

If M=3, .  This is a quadratic equation  or  

This is the standard quadratic equation, Quadratic.doc .htm, for the case where the quadratic coefficient is much less than the linear coefficient with discrimenent  and solution  

These are implemented in the first two sections of for\Chan.for. 

The last section is at present somewhat theoretical.  In the last version of for\xcaliber.for, the subroutine EPOLY found the energy the cubic spline given at the head of this section.  Earlier versions found it as a quadratic spline.  In any case it has almost never been implemented since RobWin in fitting to isotopes does not go beyond a quadratic polynomial.

This routine solves for x, the channel for which the E(chan) = E, by Newton's method.  That is expand

 

Then solve for x

 

In the interests of speed, this is iterated three times with no check for completion.