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← Return to Index

Errors and Approximation

Sources of Error in Scientific Computing

  • Modelling Errors
    • Comes from simplifying assumptions (or omitting aspects of the problem)
  • Data Errors
    • Comes from inaccurate and imprecise empirical measurements
    • Can also come from using potentially incorrect results from some previous computational process
  • Computational Errors
    • Can come from truncating a computational process
    • Can come from rounding

Absolute and Relative Error

  • Absolute Error — | Approx. value - True value |
  • Relative Error — Absolute error ÷ true value × 100%

Data Error and Computational Error

  • Let:
    • f be a true function
    • x be a true value to be used in a computation
    • f(x) be the desired result of the true function on the true value
    • g be an approximation of f
    • y be an inexact measurement of x
    • g(y) be the inexact result
  • Then:
    • Computational Error = g(y) - f(y)
    • Propagated Data Error = f(y) - f(x)
    • Total Error = Computational Data Error + Propagated Error = g(y) - f(x)

Forward & Backward Errors

  • Forward Error
    • Discrepancy between the computed and true values
    • inexact output - exact output
    • Quality of results: |inexact output - exact output| is the relative magnitude of the forward error
  • Backward Error
    • Discrepancy between the initial output that produced the discrepancy in the result
    • inexact input - exact input

Sensitivity and Conditioning

  • Sensitivity — qualitative statement of propagated data error
  • Conditioning — quantitative measure of propagated data error
  • A problem is insensitive or well-conditioned if a given relative change in the input data causes a reasonably commensurate relative change in the solution
    • |input a - input b| ~ |output a - output b|
  • Otherwise, it is sensitive or ill-conditioned
    • |input a - input b| >> or << |output a - output b|
  • Condition Number
    • Ratio of relative change
    • | relative forward error | ÷ | relative backward error |
    • Ill-conditioned problems have a condition number larger than 1
    • ~ xf'(x) ÷ f(x)