Tecnico
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Publicado: 1 de diciembre de 2012
Numerical Integration and Differentiation
Given the existence of programs like Mathematica, etc, why bother with integration and differentiation? Why “numerical”? The answer is that in science,one frequently encounters functions that are “crazy”, i.e., they do not behave in a regular and predictable manner and thus they are difficult to deal with analytically. Most importantly, however,one needs frequently to integrate (or differentiate) numerical data that is the results of experiments.
Differentiation
Starting with the basic definition of the problem given in the figurebelow:
[pic]
We can use the following simple difference formulas to compute the various derivatives.
[pic]
One does have to careful in applying these formulas to take account of roundoffand truncation error. This is especially true for the higher derivatives. Generally speaking the symmetric derivatives are preferred over the forward or backward derivatives. Another usefulapproximation to the derivative is the “5 point formula”, i.e.,
[pic]
which is equivalent to using a 4th degree polynomial. The following table indicates the errors associated with various methods ofdetermining the derivative.
[pic]
Some improved formulas exist for the higher derivatives
[pic]
In some circumstances, like those depicted below, one may wish to smooth the derivatives toget a more realistic picture of what is happening. That is the case with the analysis of projectile motion.
[pic]
Numerical Integration
There are several methods of numerical integrationof varying accuracy and ease of use. The most commonly used methods are the simplest, the trapezoidal rule and Simpson’s rule. The following handwritten notes summarize some essential features ofthese methods.
[pic]
[pic]
[pic]
[pic]
Gauss Quadrature (unequally spaced points).
The Cadillac of numerical integration methods is that of Gauss quadrature. Formally the...
Given the existence of programs like Mathematica, etc, why bother with integration and differentiation? Why “numerical”? The answer is that in science,one frequently encounters functions that are “crazy”, i.e., they do not behave in a regular and predictable manner and thus they are difficult to deal with analytically. Most importantly, however,one needs frequently to integrate (or differentiate) numerical data that is the results of experiments.
Differentiation
Starting with the basic definition of the problem given in the figurebelow:
[pic]
We can use the following simple difference formulas to compute the various derivatives.
[pic]
One does have to careful in applying these formulas to take account of roundoffand truncation error. This is especially true for the higher derivatives. Generally speaking the symmetric derivatives are preferred over the forward or backward derivatives. Another usefulapproximation to the derivative is the “5 point formula”, i.e.,
[pic]
which is equivalent to using a 4th degree polynomial. The following table indicates the errors associated with various methods ofdetermining the derivative.
[pic]
Some improved formulas exist for the higher derivatives
[pic]
In some circumstances, like those depicted below, one may wish to smooth the derivatives toget a more realistic picture of what is happening. That is the case with the analysis of projectile motion.
[pic]
Numerical Integration
There are several methods of numerical integrationof varying accuracy and ease of use. The most commonly used methods are the simplest, the trapezoidal rule and Simpson’s rule. The following handwritten notes summarize some essential features ofthese methods.
[pic]
[pic]
[pic]
[pic]
Gauss Quadrature (unequally spaced points).
The Cadillac of numerical integration methods is that of Gauss quadrature. Formally the...
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