Easy Newton-Raphson method on a calculator

The Newton-Raphson method for solving algebraic equations f(x)=0:

    \[ x\leftarrow x-\frac{f(x)}{f'(x)} \]

has always presented my students with difficulties.  To solve, for example,

    \[ x^5+x^3-4=0 \]

requires the computation of

    \[ x-\frac{x^5+x^3-4}{5x^4+3x^2} \]

for successive values of x. And in spite of the fact that most of my students won’t be separated from their calculators, they are very naive in their use. They never, for example, use calculator memories to simplify keystrokes. So to calculate the expression for x=1.19052023, they would laboriously enter, key by key:

    \[ 1.19052023-(1.19052023^{\wedge} 5+1.19052023^{\wedge} 3-4)/(5\times 1.19052023^{\wedge}4+3\times 1.19052023^{\wedge}2) \]

and then go through the process all over again. Naturally the possibility of error is huge; and the students have found this process tedious and boring. I can’t help them much with memory usage, as calculators are all different in how they store values in, and access, memories.

But! just recently I found a simple method which can be implemented on almost all calculators, from expensive top-of-the-range CAS models, to cheap supermarket varieties, and even online and smartphone apps. All you need is an ANS key, which holds the result of the previous computation, and an ability to enter algebraic expressions.

For the equation above, and starting with x=1, enter 1 on the calculator, and then enter

    \[ \mbox{ANS}-\frac{\mbox{ANS}^5+\mbox{ANS}^3-4}{5\times\mbox{ANS}^4+3\times\mbox{ANS}^2} \]

or more simply:

    \[ \mbox{ANS}-(\mbox{ANS}^5+\mbox{ANS}^3-4)/(5\times\mbox{ANS}^4+3\times\mbox{ANS}^2). \]

Then simply keep pressing the “=” (or “Enter”) key to produce new iterates. Here, for example, is RealMax calculator on Android and the CASIO FX-991 calculator:

realmax casio_fx991

They both have an ANS (or Ans) key; most calculators do.

This method is simple to implement, available on almost all calculators (I have yet to find one which can’t manage this), and more importantly makes the Newton-Raphson method easily manageable and error-free.



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