Ralphson-Newton Method And Interval Bisection

[Procrastinate]

I know quite well how to do these. However, most of the time, the starting points are just given to me i.e. the a and b values to starting iterating.

I was just wondering if you were to find a starting point for x³-x-1 = 0, what would be the best starting point to use the Ralphson-Newton Method. My notes say 1.5 but I would like to know the reason behind that.

Thanks.

 

[HallsofIvy]

An "educated guess". If x= 1, [itex]x^3- x- 1= 1- 1- 1= -2[/itex] and if x= 2, [itex]x^3- x- 1= 8- 2- 1= 5[/itex]. Since the polynomial changes sign between 1 and 2, there is a root between 1 and 2. 1.5, half way between 1 and 2, is a good starting point. That's really applying the "interval bisection" to the first step. But, in fact, any number in the vicinity of 1 and 2 will work.

You might even do this: y goes from -2 to 5, a difference of 7. To go to 0 from -2 is just a difference of 2 so perhaps 2/7 of the way from 1 to 2, 1.289, would be better. (That's the "secant" method.) But that only changes the number of iterations to get the same accuracy by, maybe, one or two. Frankly, I would have been inclined to start with x= 1.