A First, Minimal Algorithm

We are now in a position to devise a first, very crude program that should, starting from a set of initial guesses, move towards the best fit. Below, a flow diagram is given that represents the basic principle of the Newton-Gauss algorithm  

The crucial part of this algorithm is the computation of J, the derivatives of the residuals with respect to the parameters. It might be best to demonstrate this by an example. 

An exponential curve, including some noise, is generated by the function Data exp, m. The curve is defined by three parameters, the rate, pi, the amplitude p2 and the value at infinity time p3. 

The derivatives of the vector r of residuals with respect to the parameter vector p are given by the following equations. Note that the first column of the Jacobian matrix contains the derivative of the residuals with respect to the first parameter, the second with respect to the second, etc. In this example the derivatives can be computed explicitly., later we will introduce the computation of numerical derivatives. 

The program at this stage is veiy crude and needs several stages of improvements. For the next version we implement two new measures a 

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