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Fortran 90 is the focus of this chapter, including syntax, program layout, control structures, memory allocation, modular programming, and compilation.

This chapter also includes various sample codes which illustrate proper programming and algorithm structure.

We have already discussed some of these functions in the Arrays, Characters and String chapters.

In the following section we provide brief descriptions of all these functions from other categories.

In other words, it returns the smallest value such that REAL( 1.0, KIND(X)) EPSILON(X) is not equal to REAL( 1.0, KIND(X)).

Here is how we do it, using scipy.special.jn(v,x) where v is the (real) order of the Bessel function.

Many of the SciPy routines are Python wrappers, that is, Python routines that provide a Python interface for numerical libraries and routines originally written in Fortran, C, or C.

Thus, SciPy lets you take advantage of the decades of work that has gone into creating and optimizing numerical routines for science and engineering.

Because the Fortran, C, or C code that Python accesses is compiled, these routines typically run very fast.

Therefore, there is no real downsideno speed penaltyfor using Python in these cases.

Here we will provide a further introduction to a number of other SciPy packages, in particular those on special functions, numerical integration, including routines for numerically solving ordinary differential equations (ODEs), discrete Fourier transforms, linear algebra, and solving non-linear equations.

Our introduction to these capabilities does not include extensive background on the numerical methods employed; that is a topic for another text.

Here we simply introduce the SciPy routines for performing some of the more frequently required numerical tasks.

Most are contained in the scipy.special library, and each has its own special arguments and syntax, depending on the vagaries of the particular function.

We demonstrate a number of them in the code below that produces a plot of the different functions called.

For more information, you should consult the SciPy web site on the scipy.special library.

For example, the first argument of the two types of Bessel functions called in lines 10-13 is the so-called order of the Bessel function, and the second argument is the independent variable.

The Gamma and Error functions take one argument each and produce one output.

The Airy function takes only one input argument, but returns four outputs, which correspond the two Airy functions, normally designated and, and their derivatives and.

If p is a list or array of N numbers and x is an array, then.

SciPy has a number of routines for performing numerical integration.

Most of them are found in the same scipy.integrate library.

We list them here for reference.

Numerical integration is sometimes called quadrature, hence the name.

It is normally the default choice for performing single integrals of a function over a given fixed range from to.

The routine uses adaptive quadrature methods to numerically evaluate integrals, meaning it successively refines the subintervals (makes them smaller) until a desired level of numerical precision is achieved.

For the quad routine, this is about, although it usually does even better.

The first is 0.7468., which is the value of the integral, and the second is 8.29.e-15, which is an estimate of the absolute error in the value of the integral, which we see is quite small compared to 0.7468.

In fact, we can just put the lambda expression directly into the first argument, as illustrated here.

The absolute errors are somewhat larger but still well within acceptable bounds for practical work.

Complex Error Function In Fortran