## Local truncation error is approximately proportional to square of step size in Euler’s method

Question: Show that the local truncation error in Euler’s method is proportional to the square of the step size.

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## Myth: Error caused by chopping a number is called truncation error

Round off error is the error caused by approximate representation of numbers.

When people talk about round off error, it is the error between the number and its representation. For example 200/3 would be represented as 66.6667 in a six significant digit computer that rounds off the last digit. The last digit has been rounded up from 6 to a 7. The difference between 200/3 and 66.6667, that is, 200/3-66.6667 is the round off error.

If a computer is chopping off as opposed to rounding the last digit, the error caused is still called the round off error (caused by chopping). If a computer is using chopping, then for example, 200/3 would be represented as 66.6666 in a six significant digit computer. The difference between 200/3 and 66.6666, that is, 200/3-66.6666 is the round off error.

Where does the myth come from? Because if one is chopping off the number, students think that we are truncating a number, and hence the resulting error should be truncation error. No! No! That is still round off error. As a side note, there is something called truncating a number – a number if truncated is just the integer part of the number (example: truncating 20.568 gives 20; truncating 20.03 gives 20).

So what then is truncation error?

Truncation error is error caused by truncating a mathematical procedure.

Examples of truncation error abound and include

1. In exact differentiation, you need dx approaching zero; in numerical differentiation we can only choose dx=finite.
2. In exact integration, one would need infinite number of trapezoids to find the integral; in numerical integration, we can only choose a finite number of trapezoids.
3. In the Maclaurin series for transcendental and trigonometric functions, we need infinite number of terms for exact solution; in a numerical solution, we can only choose finite number of terms.

So let us get this straight – round off error is caused by representing numbers approximately; truncation error is caused by approximating mathematical procedures.

For more details, read the textbook chapter on Sources of Error in Numerical Methods at http://nm.mathforcollege.com

This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://nm.mathforcollege.com and the textbook on Numerical Methods with Applications available from the lulu storefront.