## Newton's Way of Solving Equations

Newton's method, undoubtedly names after one of the greatest masters of all times, Issac Newton, is a way to solve equations. Well, you might know about the quadratic formula, the cubic formula, or even the quartic formula. And you say all you have to do to solve equations is to plug numbers in.

WRONG.

Try to solve x^5+4x^4+7x^3-10x^2+5=0 for example. There is no "pentagic" formula. No. Nothing to solve the Pentagon. No. Would require something much stronger than that. No. Nothing for the fifth power or above. Then income Mr. Newton and his idea.

Let the function we are to solve be f(x). Then we need to first find a value that makes this function quite small. In other words, we have to first find an approximate solution. Don't ask me how. Guesswork's power is unlimited.

Then, carry out the following steps repeatedly:

WRONG.

Try to solve x^5+4x^4+7x^3-10x^2+5=0 for example. There is no "pentagic" formula. No. Nothing to solve the Pentagon. No. Would require something much stronger than that. No. Nothing for the fifth power or above. Then income Mr. Newton and his idea.

Let the function we are to solve be f(x). Then we need to first find a value that makes this function quite small. In other words, we have to first find an approximate solution. Don't ask me how. Guesswork's power is unlimited.

Then, carry out the following steps repeatedly:

This is, until f(x_n)=0.

Now, why does this work? Try to draw a graph, and draw the tangent lines as derivatives when you carry this out. You will find that the lines are flatter and flatter, until...

Now, why does this work? Try to draw a graph, and draw the tangent lines as derivatives when you carry this out. You will find that the lines are flatter and flatter, until...

*Approximating a Function*

Our friend Euler (pronounced Oil-er) is a great mathematician and one of the "founding fathers" of calculus. He proved to the world how powerful math could be. You are about to witness one of his works.

Suppose I ask you to calculate the square of 40.7892. You may think I'm mad and go for your calculator. However, I knock it out of your hands and tell you the catch: your answer must be within 0.01 of the real answer.

So, what do you do? Take an educated guess. How would you guess? Euler did the following:

He proposed first that f(x+dx)=f(x)+dy, where dx and dy are tiny increments in the x and y direction. Going a step further, he suggested that f(x+dx) is about f(x)+f'(x)dx. This is basically true as f'(x)dx is just the rise of the tangent run on the run of dx. In other words, Euler approximated a little part of the graph with a tangent line.

So, the square of 40.7892 is basically 40^2+0.7892*2. The answer is surprisingly close to the one given by your calculator.

But then you ask: why is Texas Instruments not out of business if every problem can be countered by this method? (the Euler method)

Because sometimes, it gives horrible approximations. Think about the graph of y=x^10. A tiny increment of x=1 occurred. Now let's approximate y(10) by y(9). The answer is off by 1000 at least. Why?

A general rule of thumb says that if a function is increasing very quickly (think about what you have just learned above), the Euler approximation would give a horrible approximation. Vice versa.

There is one other catch. Euler said f(x+dx) is about f(x) +f'(x)dx when dx is tiny, tiny, tiny. This usually means 0.01 or 0.1. Keep this in mind.

Suppose I ask you to calculate the square of 40.7892. You may think I'm mad and go for your calculator. However, I knock it out of your hands and tell you the catch: your answer must be within 0.01 of the real answer.

So, what do you do? Take an educated guess. How would you guess? Euler did the following:

He proposed first that f(x+dx)=f(x)+dy, where dx and dy are tiny increments in the x and y direction. Going a step further, he suggested that f(x+dx) is about f(x)+f'(x)dx. This is basically true as f'(x)dx is just the rise of the tangent run on the run of dx. In other words, Euler approximated a little part of the graph with a tangent line.

So, the square of 40.7892 is basically 40^2+0.7892*2. The answer is surprisingly close to the one given by your calculator.

But then you ask: why is Texas Instruments not out of business if every problem can be countered by this method? (the Euler method)

Because sometimes, it gives horrible approximations. Think about the graph of y=x^10. A tiny increment of x=1 occurred. Now let's approximate y(10) by y(9). The answer is off by 1000 at least. Why?

A general rule of thumb says that if a function is increasing very quickly (think about what you have just learned above), the Euler approximation would give a horrible approximation. Vice versa.

There is one other catch. Euler said f(x+dx) is about f(x) +f'(x)dx when dx is tiny, tiny, tiny. This usually means 0.01 or 0.1. Keep this in mind.