ECE60 - Numerical Methods Engr. Charmaine C. PaglinawanInstructor

CCPaglinawan

Facebook GroupECE60-Numericals*

Numerical Methods study of algorithms that use numerical approximation to solve problems usually done symbolically like finding the roots, interpolation and extrapolation, solution to system of equations, eigenvalues, regression, evaluating derivatives and integrals, optimization, and differential equations.

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Accuracy and PrecisionAccuracy - comparison of actual (true) value vs measured (computed) valuesErrors: Absolute Error

Relative Error

Round Off ErrorTruncation Error

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Accuracy and Precision (cont.)Precision comparison of several computed (measured) valuesErrors: Absolute Approximate Error

Relative Approximate Error

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Power SeriesPower Series used to represent infinitely integrable/differentiable functions.

General Form:

*where C0, C1, C2, , Cn are constants of the power series

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To determine constants:

..

If a=x*

Thus

..

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Taylor Series ExpansionSubstituting to the General Form

Recall*

Mc Laurin Series ExpansionLet a=0, then

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Mc Laurin Series Expansion (cont.)Substituting to the General Form

Recall

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Example Taylor Series ExpansionDetermine the Taylor Series Expansion of the function f(x)=ln(x+1) near the value a=0.5 and determine its first 4 non-zero terms.SolutionRecall:*

Example- Mc Laurin Series ExpansionDetermine the Mc Laurin Series Expansion of the function f(x)=ex and express the following terms in a power series formula.Solution

Recall:*

SoftwaresExcelMatlab*

*Absolute error is the amount of physical error in a measurement, period. Lets say a meter stick is used to measure a given distance. The error is rather hastily made, but it is good to 1mm. This is the absolute error of the measurement. Relative error gives an indication of how good a measurement is relative to the size of the thing being measured. Lets say that two students measure two objects with a meter stick. One student measures the height of a room and gets a value of 3.215 meters 1mm (0.001m). Another student measures the height of a small cylinder and measures 0.075 meters 1mm (0.001m). Clearly, the overall accuracy of the ceiling height is much better than that ofthe 7.5 cm cylinder. The comparative accuracy of these measurements can be determined by looking at their relative errorsRound-off error, also calledrounding error, is the difference between the calculatedapproximationof a number and its exact mathematical value.Numerical analysisspecifically tries to estimate this error when using approximationequationsand/oralgorithms, especially when using finitely many digits to represent real numbers (which intheoryhave infinitely many digits).Truncation errororlocal truncation erroris error made by numerical algorithms that arises from taking finite number of steps in computation. It is present even with infinite-precision arithmetic, because it is caused by truncation of the infiniteTaylor seriesto form the algorithm.*