1. Module 2.2 Errors 03/08/2011

2. Sources of errors Data errors Modeling Implementation errors Absolute and relative errors Round off errors Overflow and underflow Arithmatic errors Error propagation Truncation errors

3. Precision:the number of significant digits Floating point number : a decimal expansion Flops: FLoating point Operations Per Second, a measure of a computer's performance, especially in fields of scientific calculations that make heavy use of floating point calculationscomputerperformancefloating point calculations Exponential notation: a floating point number as a decimal fraction times a power of 10 (a*10^n, 698.043990  10 3 ) a – significand, fractional part, mantissa, n- exponent A normalized number in exp notation has the decimal point immediately preceding the first nonzero digit. The significant digits of a floating point number are all the digits except the leading zeros The significant digits of an integer are all the digits except the leading and trailing zeros

4. Exponential notation Example: 698.043990  10 3 Fractional part or significand? 698043990 Exponent? 3 Normalized? 0.698043990  10 5

5. Significant digits Significant digits of floating point number All digits except leading zeros Number of significant digits in 698.043990  10 3 ? 9 significant digits Precision Number of significant digits

6. Magnitude Power of 10 when number expressed in normalized exponential notation Magnitude of 698.043990  10 3 ? Magnitude of 0.698043990  10 5 ? 10 5

7. Absolute error |correct – result| Example: correct = 0.698043990  10 5 and result = 0.698043  10 5 Absolute error = ? |0.698043990  10 5 - 0.698043  10 5 | = 0.00000990  10 5 = 0.990

8. Relative error |(correct - result) / correct| Example: (correct - result) = 0.990 and correct = 0.698043990  10 5 0.990/(0.698043990  10 5 ) = 1.4182487  10 -5

9. Round-off error Problem of not having enough bits to store entire floating point number(round up, round down) and approximating the result to the nearest number that can be represented Example: 0.698043990  10 5 if only can store 6 significant digits, rounded? 0.698044  10 5 Do not test directly for equality of floating point variables

10. Overflow/underflow Overflow - error condition that occurs when not enough bits to express value in computer Underflow - error condition that occurs when result of computation is too small for computer to represent

11. Arithematic errors;Algebra rules do not necessarily hold x = 355/113, y = 101/112, z = 52/113 All different ways of calculating x - 3y - z should be 0, but is not

12. Addition and subtraction errors If big difference in magnitude of numbers Example: (0.65  10 5 ) + (0.98  10 -5 ) = ? 65000 + 0.0000098 = 65000.0000098 Suppose can store 6 significant digits rounded? 65000.0 = (0.650000  10 5 )

13. Associative property Does not necessarily hold Sum of many small numbers + large number may not equal adding each small number to large number Similarly, distributive property does not necessarily hold

14. Cancellation error Problem of subtracting numbers that are almost equal Answer can have fewer significant digits than original numbers

15. To reduce numerical errors Round-off errors Use maximum number of significant digits If big difference in magnitude of numbers Add from smallest to largest numbers

16. Propagated (accumulation ) round-off errors Example: repeatedly executing t = t + dt Better to repeatedly increment i and calculate t = i * dt

17. Truncation error Error that occurs when truncated, or finite, sum is used as approximation for sum of infinite series