Clenshaw algorithm

Okay kiddo, have you ever tried to solve a big math problem, but it had lots of parts and was really complicated? Sometimes, when we try to do these math problems, we end up with a big mess that's hard to figure out.

One way to solve these big math problems is with something called the Clenshaw algorithm. This is like a special set of instructions that help us organize our work and make the problem easier.

Let's say we're trying to solve a math problem that looks like this: y = 3x^3 + 5x^2 + 2x + 1. This is a big problem with lots of different parts (the 3x^3, 5x^2, 2x, and 1). We can use the Clenshaw algorithm to help us break it down and solve it step by step.

First, we'll write down our special set of instructions, which are called the Clenshaw recursion formula:

b_n = 0
b_{n-1} = a_{n}
b_{i} = 2 x b_{i+1} - b_{i+2} + a_{i} for i=n-2, n-3, ..., 0
y = a_{0} - x b_{1} + b_{0}

Now, we need to follow these instructions in order to solve our math problem. We start by setting b_n to 0 (this is just a starting point for our calculations).

Next, we set b_{n-1} to a_{n}. In our example problem, n is 3, so we would set b_2 to 3.

Then, we use the Clenshaw recursion formula to calculate each of the b_i values. We start with b_{n-2} and work our way down to b_0. Each time we calculate a new b_i value, we use the values we calculated before.

Finally, we use the last line of the Clenshaw recursion formula to calculate our final answer y. This involves plugging in the value we got for b_1, along with the other values we calculated along the way.

That's it! With the Clenshaw algorithm, we can tackle even the biggest, messiest math problems by breaking them down and solving them step by step.