# Numerical Error in program

## GOAL

To understand numerical errors in computers.

## Error of input data

Input data error occurs when real number such as infinite decimal numbers are input as finite decimal number by users.

## Approximation error

For example, square root are generated by approximate calculation such as iterative method and bisection method in computers.

## Round-off error

Round-off error is caused because the calculation result is rounded to within the number of significant digits in computers.

1.105982 -> 1.10598
40156.245618 -> 40156.24562

## Truncation error

Truncation error occurs when iterative calculation such as infinite series or iterative method aborted halfway because of time restriction.

## Loss of significance

Loss of significance occurs in calculations of two close numbers using finite-precision arithmetic.

### Example

1.222222- 1.222221 = 0.000001
(significant digits 7) – (significant digits 7) = (significant digits 1)

### Why loss of significance is undesirable?

For example, calculate √1000 – √999 to 7 significant digit.

## Information loss

Information loss occurs when add or subtract big number and small number.

### Example

122.2222 + 0.001234567 = 122.2234 (significant digits 7)