Download - Chapter 2: Numerical Approximation
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CHAPTER IINUMERICAL APPROXIMATION
BY: MARIA FERNANDA VERGARA M.UNIVERSIDAD INDUSTRIAL DE SANTANDER
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NUMERICAL APPROXIMATION
•A numerical approximation is a number X’ that represents another number which its exact value is X. X’ becomes more exact when is closer to the exact value of X
•Is important to take into account this numerical approximation because numerical solutions are not exact, but the main objective is to get a solution really close to the real solution.
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SIGNIFICANT FIGURES• “The concept of a significant figure, or digit, has been
developed to formally designate the reliability of a numerical value. The significant digits of a number are those that can be used with confidence. They correspond to the number of certain digits plus one estimated digit.”-Numerical methods for engineers, CHAPRA-.
Numerical methods yield approximate results, so is important to specify how confident is the result.
Important quantities as π, e, √2, cannot be expressed exactly because computers retain a
only a finite number of significant figures.
Why significant figures are
important in numerical methods?
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ACCURACY AND PRECISION
ACCURACY
•Accuracy refers to how close to the real value is the measure made.
PRECISION
•Precision refers to how close are several measured values from each other.
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ERROR DEFINITIONS• Numerical errors originate when you
approximate to represent exact mathematical quantities or operations. This errors can be: Truncation errors which happen when approximations are used to represent mathemathical procedures; and round-off errors which happen when you use numbers with limited significant figures to express exact numbers.ET=Real Value - Approximation
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RELATIVE ERROR
True percent relative error
Relative error is a way to account for the magnitudes of the quantities being evaluated
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EXAMPLE EXERCISE• The measure of a bridge is 9999cm, and the
measure of a rivet is 9 cm, if the true values are 10.000cm and 10cm, respectively, compute the true error and the true percent relative error for each case.
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• In real world applications, we will not know the true value. So the procedure is to normalize the error using the best avaliable estimate of the true value:
Usin an iterative approach to compute answers, the approximated relative error
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ROUND-OFF ERRORS• This kind of errors originate because computers
can retain a finite number of significant figures, so numbers as e, π, cannot be expressed exactly.
• “Truncation errors are those that result from using an approximation in place of an exact mathematical procedure.”
TRUNCATION ERRORS