Naïve Newton Raphson Minimization

Define

The object is to minimize c²with respect to . This requires finding a vector such that for all M components of

First calculating c² and its first derivatives at a value . Use these for a Taylor series expansion about such that near this point the first derivatives at are given by

The Newton-Raphson method for finding the minimum is to truncate the above expansion to the terms shown and solve for the value of such that the first derivatives are all zero.

The first derivative vector at required for this expansion is given by While the second derivative array at is given by

The second sum contains the term , which as the solution is approached oscillates with alternating signs about zero. The first term contains no such oscillating small term. The minimization of c² requires that its partials be set exactly to zero, which means the first derivatives need to be exact. The expansion of this partial is an approximation in which cubic and higher terms are already neglected so that very little is lost by dropping the small term in the second partial to yield

Inverting this matrix allows us to immediately find in the Newton-Raphson approximation