Round off and Truncation
Errors
CS 431 : Numerical Analysis I
Chapter 4
• Errors are mainly associated with
calculations and measurements
• Accuracy: How closely a computed value
or measured value agrees with true value
• Precision: How individual computed values
or measured values agree with one
another.
• Inaccuracy is also called bias
• Imprecision is also called uncertainty
Figure 4.1
Chapter 4
• Absolute or True Error:
Et = true value – approximation
• True Fractional Relative Error:
Єt = true value – approximate value
true value
• True Percent Relative Error:
Єt = true value – approximate value * 100%
true value
Chapter 4
• When true value is not known
• Approximate relative error:
Єa= approximate error * 100%
approximation
• Numerical methods using iteration
Єa= (present approximate – previous approximate )* 100%
present approximation
• Stopping criterion: Relative error is less than
some tolerance value
Chapter 4
• Accuracy of result up to n significant digits
Єs= (0.5 X 10 2 - n) %
• When |Єa| < Єs then the result is correct
up to n significant digits
ROUND OFF ERRORS
• Size and Precision limitations
– Integer representation
– Floating point representation
– Machine epsilon (machine precision)
• Numerical manipulations are sensitive to
round off errors
– Subtractive cancellation
– Large computations
TRUNCTION ERRORS
• Result of using a approximation instead of
exact mathematical procedure
– Using Taylor series to approximate a
polynomial function
TOTAL NUMERICAL ERROR
• Summation of truncation and roundoff errors