May 29, 2012

Newton's method as feedback system

This entry is inspired by the work by K. Kashima and Y. Yamamoto.

To obtain an approximated value of sqrt(a), one can use Newton's method:

with a given real x[0]. This can be easily implemented in computer programs such as MATLAB or SCILAB, but today I will represent this as a feedback system.

First, the iteration can be represented by

where σ denotes the shift operator

In control theory or signal processing, this operator is also denoted by z-1. Next, the iteration is divided into two parts, φ and σ, that is,

Finally, this can be represented by the following block diagram:

Based on this, you can run simulation with XCOS in SCLAB (xcos file is here).

This is a "feedback system" for sqrt(2). You can change the initial value x[0] by double-clicking "1/z" block.  Also, if you want another square root, double-click the "Expression" block and change "2" in the numerator to another positive number. But, if you feel like wilding out, use a negative number instead. In this case, the sequence will not converge, but show some chaotic behavior. The following is the result for "-2."

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