- Raphson's Cubic:
Try to read the original paper of Raphson above.
Notice that he is trying to solve the cubic equation
3r^2 x - x^3 = c r^2 for the parameter values c = r = 10.0,
with the initial guess of x = 3.0.
Reproduce his answer for the root in Phaser using Newton-Raphson method.
- sqrt(2) by Sumerians:
On the Sumerian tablet (see the link above), four digits (1)(24)(51)(10) of an approximation to sqrt(2)
are given in base 60. Convert this base60 approximation to base10 (decimal).
Next, compute 12 binary digits of this approximation.
Please show the steps of your computations.
Convert this 12-bit binary number to base 10 (decimal). How many correct decimal digits
of sqrt(2) do you get?
- Can you beat Newton?
Enter the difference equation x1 -> x1 - 0.21*(x1*x1 - 2)
into PHASER. Iterate the initial condition x0 =2.22 and observe that
it approaches to sqrt(2).
Compare the rate of convergence of this method of
computing sqrt(2) with that of Newton's method x1-> 0.5*(x1 + 2/x1).
To make a fair comparison, use the same initial condition
in both iterations. How many iterations are required to get 13 correct digits
of sqrt(2) in each iteration?
- Small angle approximation:
In physics, it is often convenient to use the 'small angle approximation' by
replacing sin(x) with x when x is small. Use Newton-Raphson method
to find a value of x which satisfies the equation sin(x) - x = 0.015.
How many such x can you find? Draw the stair-step diagram of the iteration
process. How many fixed points do you see?