NUMERICAL APPROXIMATION METHODS

CYNDY ARGOTE SIERRAINGENIERIA DE PETRÒLEOS

DEFINITIONS• The numerical methods are useful alternative

procedures to solve math problems for which complicates the use of traditional analytical methods and, occasionally, are the only possible solution.

• They are systematic techniques whose results are approximations of the true value that assumes the variable of interest, the consistent repetition of the technique, which is called iteration, is what you closer and closer to the desired value.

WHEN SPEAKING OF APPROACH SHOULD BE TAKEN INTO ACCOUNT :

PRECISION

CONVERGENCE Y ESTABILITY

ACCURACY

SIGNIFICANT FIGURES

PRECISION AND ACCURACY

• Precision refers to the number of significant figures represents a quantity.

• Accuracy refers to the approach of a number or measure the numerical value is supposed to represent.

• Example: π is an irrational number, consisting of an infinite number of digits; 3.141592653589793 ... is an approximation as good of π, which may be considered which is its exact value.

π = 3.15 is vague and inaccurate. π = 3.14 is accurate but imprecise. π = 3.151692 is precise but inaccurate. π = 3.141593 is accurate and precise.

CONVERGENCE AND STABILITY• Convergence is defined as a numerical method

ensuring that, when making a "good number" of iterations, the approximations obtained eventually move closer and closer to the true value sought.

• Stability means of a numerical method the level of assurance of convergence and numerical methods is that some do not always converge and, on the other hand, diverge, ie away from the more desired result.

It is common to find methods that converge quickly, but they are very unstable and, in contrast, very stable models, but slow convergence.

SIGNIFICANT FIGURES• The number of significant figures is the number of digits

t, which can be used with confidence to measure a variable, for example, three significant figures on the speedometer and 7 significant figures on the odometer.

EXAMPLE• The zeros are included in a number are not always

significant figures, for example, the numbers 0.00001845, 0.001845, 184 500 1845 and apparently have four significant figures, but would have to know the context in which they are working on each case, to identify how many and zeros which should be considered as significant figures.

SELECTION OF ALTERNATIVES

TYPE OF PROBLEM TO SOLVE:

•Roots of equations•Systems of simultaneous linear equations• Interpolation, differentiation and

integration•Ordinary Differential Equations•Partial Differential Equations•Other (not covered in this course, seen in

other subjects)

PROBLEM TYPE

•NUMERICAL METHOD

•MATHEMATICS MODEL

SELECTION OF ALTERNATIVES

Numerical method: there is no better, but recommended

• Extent of application• Friendliness• Stability• Fast convergence• Required number of initial values

Be taken into account, besides• Model complexity• Turbulence data• Ingenuity and creativity

BIBLIOGRAFIA

•Aproximaciòn numerica Gustavo Rocha 2005.

•http://www.educaplus.org/formularios/cifrassignificativas.html