# Calculating Forward and Backwards error of the sine function

1. Jan 29, 2010

### harrisiqbal

1. The sine function is given by the infinite series
sin(x) = x - x3/3! + x5/5! + x7/7! + .......
a) What are the forward and backward errors if we approximate the sine function
by using only the first term in the series, for x = 0.1, 0.5, 1.0?
b) Using the first two terms.

2. Relevant equations

Forward Error: Fhat - F
Backward Error:xhat - x

Fhat is the function approximation.
Xhat is the input modification used to calculate the backward error.

This is basically from my Numerical Methods Course.

3. The attempt at a solution

I don't have any problem with the Forward Error analysis.

For part a. I simply evaluated the original sin(x) function with x = .1 and got a number. Then evaluated the approximation function Fhat = x

[ this is the sin function expanded into the Taylor series.. but only using the first term of the series]

and then I used the F. Error equation to get the answer...

The problem is the backward error where I have to satisfy this equation:

F(xhat) = Fhat(x) Basically I want a xhat that when put into the original sin function will output my approximation function.

I can't figure this out.. at all. I mean. the only time sin(x) = x is when x = 0? or am i wrong? What am i missing here? It has to be trivial!

I mean if my function was the exponential function then my xhat would be log(Fhat) so e^xhat outputs the Fhat function.. thats easy.. but how is that applied to the sin(x) = x condition?

Thanks!

2. Jan 29, 2010

### harrisiqbal

NVM. I feel so retarded.

I figured it out. Obviously it would be the inverse Sin(x) function that would allow me to calculate the backwards error..

SO obvious wow.

3. Jan 30, 2010

### torquil

You're probably very far from being retarded