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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.

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Starting with the basic definition of the problem given in the figure below:

We can use the following simple difference formulas to compute the various derivatives.

One
does have to careful in applying these formulas to take account of roundoff and
truncation error. This is especially
true for the higher derivatives.
Generally speaking the symmetric derivatives are preferred over the
forward or backward derivatives.
Another useful approximation to the derivative is the “5 point formula”,
i.e.,

_{}

which
is equivalent to using a 4^{th} degree polynomial. The following table indicates the errors
associated with various methods of determining the derivative.

In some circumstances, like those depicted below, one may wish to smooth the derivatives to get a more realistic picture of what is happening. That is the case with the analysis of projectile motion.

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**Excel**

** **

The following example shows the use of numerical derivatives in Excel and how they naturally occur in solving problems in analytical chemistry.

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In
Mathematica, the function for numerical derivative is ND. ND[f,x,x_{0}] is the numerical
derivative df/dx at x=x_{0}.
ND[f,{x,n},x_{0}] is the nth derivative

The following example shows how derivatives are taken of List data.

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There are several methods of numerical integration of 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 of these methods.

**Gauss
Quadrature (unequally spaced points).**

** **

The
Cadillac of numerical integration methods is that of Gauss quadrature. Formally the value of the integral I is
approximated as I = c_{1}f(x_{1}) + c_{2}f(x_{2})
+ … c_{n}f(x_{n}) where
the c_{i}are asset of predetermined numerical coefficients. The following table shows these
coefficients.

** **

The
following examples shows the use of the trapezoidal rule and Simpson’s rule in
a spreadsheet The analytic value of the
integral is 2.0000.

** **

The relevant Mathematica function for numerical integration is NIntegrate. The following example shows the use of this Mathematica function.

Thje following program, taken from Nyhoff (Fig 6-7.f) shows a FORTRAN90 implementation of the trapezoidal ruile.

PROGRAM AREA

************************************************************************

* Program to approximate the integral of a function over the interval *

* [A,B] using the trapezoidal method. Identifiers used are: *

* A, B : the endpoints of the interval of integration *

* N : the number of subintervals used *

* I : counter *

* DELX : the length of the subintervals *

* X : a point of subdivision *

* Y : the value of the function at X *

* SUM : the approximating sum *

* F : the integrand *

* *

* Input: A, B, and N *

* Output: Approximation to integral of F on [A, B] *

************************************************************************

REAL F, A, B, DELX, X, Y, SUM

INTEGER N, I

PRINT *, 'ENTER THE INTERVAL ENDPOINTS AND THE # OF SUBINTERVALS'

READ *, A, B, N

* Calculate subinterval length

* and initialize the approximating SUM and X

DELX = (B - A) / REAL(N)

X = A

SUM = 0.0

* Now calculate and display the sum

DO 10 I = 1, N - 1

X = X + DELX

Y = F(X)

SUM = SUM + Y

10 CONTINUE

SUM = DELX * ((F(A) + F(B)) / 2.0 + SUM)

PRINT 20, N, SUM

20 FORMAT (1X, 'APPROXIMATE VALUE USING', I4,

+ ' SUBINTERVALS IS', F10.5)

END

** F(X) *********

* The integrand *

*****************

FUNCTION F(X)

REAL F, X

F = X**2 + 1

END

The following program implements Simpson’s rule in FORTRAN.

The following program implements Gaussian quadrature in FORTRAN.

Another technique of numerical integration, which we discuss in another lesson, is that of Monte Carlo integration. We present the relevant FORTRAN code here for completeness.