Exercise: Newton's Method

Write code to solve the problem.

Question

Write a program to estimate the square root of 612 using Newton’s method, using five iterations. Here are the necessary detail on Newton’s method:

The square root of 612 is the root of the equation $x^2−612 = 0$ .
Suppose $f(x)=x^2−612$ , then its derivative $f^{\prime}(x)=2x$ .
To find the root of $f(x)=0$ , we start with an initial guess $x_1$ as the root, say $x_1=10$ .
With $i=1,2,\cdots$ , we iteratively compute the values $x_{i+1}=x_i-\frac{f(x_i)}{f^{\prime}(x_i)}$

If for any $x_i$ , where $i\in\{1,2,\cdots\}, f(x_i)=0$ that’s our solution. Otherwise, we iterate a few times to get better approximations. Newton’s method does not guarantee termination.

Exercise: Newton's Method

Write code to solve the problem.

Question

Write a program to estimate the square root of 612 using Newton’s method, using five iterations. Here are the necessary detail on Newton’s method:

The square root of 612 is the root of the equation $x^2−612 = 0$ .
Suppose $f(x)=x^2−612$ , then its derivative $f^{\prime}(x)=2x$ .
To find the root of $f(x)=0$ , we start with an initial guess $x_1$ as the root, say $x_1=10$ .
With $i=1,2,\cdots$ , we iteratively compute the values $x_{i+1}=x_i-\frac{f(x_i)}{f^{\prime}(x_i)}$

If for any $x_i$ , where $i\in\{1,2,\cdots\}, f(x_i)=0$ that’s our solution. Otherwise, we iterate a few times to get better approximations. Newton’s method does not guarantee termination.