Introduction to Numerical Methods

Round-Off Errors

The error that is produced when a calculator or computer is used to perform real number calculations is called round-off error. Real numbers are typically represented in computers using floating-point form.

Machine Numbers

Represented in the normalized decimal floating-point form.

Aka. k-digit decimal machine numbers. Any positive real number can be written in the above form.

The floating-point form of , denoted , is obtained by terminating the mantissa of at decimal digits. There are 2 common ways of performing this termination.

• chopping: chop off the digits to produce the floating-point form
• rounding: when , add to to obtain and then chop off all but the first k digits.

Measuring Errors

Suppose is an approximation of .

Absolute Error

Relative Error

Finite-Digit Arithmetic

Machine arithmetic are done on finite-digits and are not exact. Suppose are machine addition, subtraction, multiplication and division.

Due to this, the accuracy is lost to some extent. The accuracy can be increased by rearranging calculations.

Truncating Error

Occurs because of using approximation in place of an exact mathematical procedure.