CHAPTER IINUMERICAL APPROXIMATION

BY: MARIA FERNANDA VERGARA M.UNIVERSIDAD INDUSTRIAL DE SANTANDER

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.

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?

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.

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

RELATIVE ERROR

True percent relative error

Relative error is a way to account for the magnitudes of the quantities being evaluated

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.

• 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

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