Newton Method

[Dieter]

(05-14-2016 07:49 PM)bshoring Wrote:  Namir, I plugged your formula into the above program. Using the example equation given above, with guesses of 0 & -1, it reached the same result with the same number of iterations (5). However with a guess of 1, yours converged faster, cutting the number of iterations from 25 to 18. Interesting!

At x=1 the function in the example has a local mininum, i.e. at that point the derivative f'(x) is zero. If f'(x) was calculated exactly the Newton method would cause a division by zero and the iteration would stop with an error.

In our case f'(x) is approximated by a small value: h=0,002 yields f'=0,004... and h=0,0001 leads to f' = 0,0002. Dividing f(1) by such small values results in a first approximation that is waaaayyyy off: ~125 with h=0,002 resp. ~2500 with h=0,0001 ...and +infinity with an exact derivative. ;-)

In other words: starting with x=1 should throw an error due to division by zero. Since the derivative is not exact, the algorithm continues with a very large value. Which is why so many iterations are required.

Dieter