Numerical Approximation

When we have no closed form solution, we revert to numerical estimation. The most often used methods are versions of Newton's Method.

Numerical estimation has a simple formulation.

Newton's method uses known properties of a function to fill in the details.

A simple example, finding the square root of 3

A simple SAS program can be used to illustrate the process.

A better version includes convergence criteria.

A different convergence criteria can be used.

A more generalizable program.

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Homework
Write a program that uses Newton's Method to find the cube root of a function. Test the program by calculating the cube root of 3 and of -3.

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Occaisionally, one needs to think about starting values.

Proc Logistic in SAS

Proc Logistic uses Fisher's Scoring which is equivalent to Newton's method. The difference is that the problem is formulated as a weighted least squares problem. You can compare the results using the technique= option in the model statement.

Homework
Note the value for the zeroth iteration in the example. Derive these estimates and write a program that calculates the the initial value used by proc logistic (the zeroth iteration) for the intercept.