Approximations and Errors

1
Approximations and Errors

• Application of numerical methods requires use of
  approximations instead of exact values.
• The difference between the approximation and the
  exact value is known as the error

2
Two Major Forms ofNumerical Error

• Roundoff error
• Machine can store only a finite number of
  significant digits
• Truncation error
• Numerical method employs approximation to
  represent exact mathematical operation

3
Accuracy and Precision

• Accuracy
• How closely a computed value agrees with the true
  value
• Precision
• How closely individual values agree with each
  other

4
Numerical Error

• True error
• Et true value approximation
• True percent relative error
• et
• Approx error for iterative methods
• ea
• ea

5
Numerical Error

• Tolerance for error
• ?ea? lt es
• Tolerance for at least n sig-figs
• es (0.5 x 102-n)

6
Round-off Errors

• Not always able to save all the sig-figs in the
  computer
• p, e, 1/3
• Integer
• Limit to magnitude
• Floating point
• Limit to magnitude and sig-figs

7
Floating Point Representation

• Form
• m .b e
• m mantisa
• e exponent
• b base
• Limited range of values
• Underflow
• Overflow

8
Floating Point Representation

• Limit on significant figures
• Single vs. double precision
• Interval between numbers increases as the
  magnitude grows