Math

Getting Started with Scilab

Joey Bernard — Thu, 15 Nov 2018 13:15:15 +0000

Introducing one of the larger scientific lab packages for Linux.

Scilab is meant to be an overall package for numerical science, along the lines of Maple, Matlab or Mathematica. Although a lot of built-in functionality exists for all sorts of scientific computations, Scilab also includes its own programming language, which allows you to use that functionality to its utmost. If you prefer, you instead can use this language to extend Scilab's functionality into completely new areas of research. Some of the functionality includes 2D and 3D visualization and optimization tools, as well as statistical functions. Also included in Scilab is Xcos, an editor for designing dynamical systems models.

Several options exist for installing Scilab on your system. Most package management systems should have one or more packages available for Scilab, which also will install several support packages. Or, you simply can download and install a tarball that contains everything you need to be able to run Scilab on your system.

Once it's installed, start the GUI version of Scilab with the scilab command. If you installed Scilab via tarball, this command will be located in the bin subdirectory where you unpacked the tarball.

When it first starts, you should see a full workspace created for your project.

Figure 1. When you first start Scilab, you'll see an empty workspace ready for you to start a new project.

On the left-hand side is a file browser where you can see data files and Scilab scripts. The right-hand side has several panes. The top pane is a variable browser, where you can see what currently exists within the workspace. The middle pane contains a list of commands within that workspace, and the bottom pane has a news feed of Scilab-related news. The center of the workspace is the actual Scilab console where you can interact with the execution engine.

Let's start with some basic mathematics—for example, division:


--> 23/7
 ans  =

   3.2857143

As you can see, the command prompt is -->, where you enter the next command to the execution engine. In the variable browser, you can see a new variable named ans that contains the results of the calculation.

Along with basic arithmetic, there is also a number of built-in functions. One thing to be aware of is that these function names are case-sensitive. For example, the statement sqrt(9) gives the answer of 3, whereas the statement SQRT(9) returns an error.

There also are built-in constants for numbers like e or pi. You can use them in statements, like this command to find the sine of pi/2:

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Introducing Genius, the Advanced Scientific Calculator for Linux

Joey Bernard — Fri, 05 Oct 2018 11:30:00 +0000

by Joey Bernard

Genius is a calculator program that has both a command-line version and a GNOME GUI version. It should available in your distribution's package management system. For Debian-based distributions, the GUI version and the command-line version are two separate packages. Assuming that you want to install both, you can do so with the following command:


sudo apt-get install genius gnome-genius

If you use Ubuntu, be aware that the package gnome-genius doesn't appear to be in Bionic. It's in earlier versions (trusty, xenial and arty), and it appears to be in the next version (cosmic). I ran into this problem, and thought I'd mention it to save you some aggravation.

Starting the command-line version provides an interpreter that should be familiar to Python or R users.

Figure 1. When you start Genius, you get the version and some license information, and then you'll see the interpreter prompt.

If you start gnome-genius, you'll see a graphical interface that is likely to be more comfortable to new users. For the rest of this article, I'm using the GUI version in order to demonstrate some of the things you can do with Genius.

Figure 2. The GUI interface provides easy menu access to most of the functionality within Genius.

You can use Genius just as a general-purpose calculator, so you can do things like:


genius> 4+5
= 9

Along with basic math operators, you also can use trigonometric functions. This command gives the sine of 45 degrees:


genius> sin(45)
= 0.850903524534

These types of calculations can be of essentially arbitrary size. You also can use complex numbers out of the box. Many other standard mathematical functions are available as well, including items like logarithms, statistics, combinatorics and even calculus functions.

Along with functions, Genius also provides control structures like conditionals and looping structures. For example, the following code gives you a basic for loop that prints out the sine of the first 90 degrees:


for i = 1 to 90 do (
   x = sin(i);
   print(x)
)

As you can see, the syntax is almost C-like. At first blush, it looks like the semicolon is being used as a line-ending character, but it's actually a command separator. That's why there is a semicolon on the line with the sine function, but there is no semicolon on the line with the print function. This means you could write the for loop as the following:

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A Look at KDE's KAlgebra

Joey Bernard — Thu, 13 Sep 2018 12:00:00 +0000

by Joey Bernard

Many of the programs I've covered in the past have have been desktop-environment-agnostic—all they required was some sort of graphical display running. This article looks at one of the programs available in the KDE desktop environment, KAlgebra.

You can use your distribution's package management system to install it, or you can use Discover, KDE's package manager. After it's installed, you can start it from the command line or the launch menu.

When you first start KAlgebra, you get a blank slate to start doing calculations.

Figure 1. When you start KAlgebra, you get a blank canvas for doing calculations.

The screen layout is a large main pane where all of the calculations and their results are displayed. At the top of this pane are four tabs: Calculator, 2D Graph, 3D Graph and Dictionary. There's also a smaller pane on the right-hand side used for different purposes for each tab.

In the calculator tab, the side pane gives a list of variables, including predefined variables for things like pi or euler, available when you start your new session. You can add new variables with the following syntax:


a := 3

This creates a new variable named a with an initial value of 3. This new variable also will be visible in the list on the right-hand side. Using these variables is as easy as executing them. For example, you can double it with the following:


a * 2

There is a special variable called ans that you can use to get the result from your most recent calculation. All of the standard mathematical operators are available for doing calculations.

Figure 2. KAlgebra lets you create your own variables and functions for even more complex calculations.

There's also a complete set of functions for doing more complex calculations, such as trigonometric functions, mathematical functions like absolute value or floor, and even calculus functions like finding the derivative. For instance, the following lets you find the sine of 45 degrees:


sin(45)

You also can define your own functions using the lambda operator ->. If you want to create a function that calculates cubes, you could do this:


x -> x^3

This is pretty hard to use, so you may want to assign it to a variable name:


cube := x -> x^3

You then can use it just like any other function, and it also shows up in the list of variables on the right-hand side pane.

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Open Axiom

Joey Bernard — Tue, 23 Sep 2014 20:56:14 +0000

by Joey Bernard

Several computer algebra systems are available to Linux users. I even have looked at a few of them in this column, but for this issue, I discuss OpenAxiom. OpenAxiom actually is a fork of Axiom. Axiom originally was developed at IBM under the name ScratchPad. Development started in 1971, so Axiom is as old as I am, and almost as smart. In the 1990s, it was sold off to the Numerical Algorithms Group (NAG). In 2001, NAG removed it from commercial sale and released it as free software. Since then, it has forked into OpenAxiom and FriCAS. Axiom still is available. The system is specified in the book AXIOM: the Scientific Computation System by Richard Jenks and Robert Sutor. This book is available on-line at http://wiki.axiom-developer.org/axiom-website/hyperdoc/axbook/book-contents.xhtml, and it makes up the core documentation for OpenAxiom.

Most Linux distributions should have a package for OpenAxiom. For example, with Debian-based distributions, you can install OpenAxiom with:


sudo apt-get install openaxiom

If you want to build OpenAxiom from source, you need to have a Lisp engine installed. There are several to choose from on Linux, such as CLisp or GNU Common Lisp. Building is a straightforward:


./configure; make; make install

To use OpenAxiom, simply execute open-axiom on the command line. This will give you an interactive OpenAxiom session. If you have a script of commands you want to run as a complete unit, you can do so with:


open-axiom --script myfile.input

where the file "myfile.input" contains the OpenAxiom commands to be executed.

So, what can you actually do with OpenAxiom? OpenAxiom has many different data types. There are algebraic ones (like polynomials, matrices and power series) and data structures (like lists and dictionaries). You can combine them into any reasonable combinations, like polynomials of matrices or matrices of polynomials. These data types are defined by programs in OpenAxiom. These data type programs also include the operations that can be applied to the particular data type. The entire system is polymorphic by design. You also can extend the entire data type system by writing your own data type programs. There are a large number of different numeric types to handle almost any type of operation as well.

The simplest use of OpenAxiom is as a calculator. For example, you can find the cosine of 1.2 with:

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FreeMat—Yet Another MATLAB Replacement

Joey Bernard — Fri, 15 Nov 2013 21:29:56 +0000

by Joey Bernard

Many programs exist that try to serve as a replacement for MATLAB. They all differ in their capabilities—some extending beyond what is available in MATLAB, and others giving subsets of functions that focus on some problem area. In this article, let's look at another available option: FreeMat.

The main Web site for FreeMat is hosted on SourceForge. Installation for most Linux distributions should be as easy as using your friendly neighborhood package manager. FreeMat also is available as source code, as well as installation packages for Windows and Mac OS X. Once it's installed, you can go ahead and start it up. This brings up the main window with a working console that should be displaying the license information for FreeMat (Figure 1).

Figure 1. When you first start up FreeMat, you are given a fresh console.

As with most programs like FreeMat, you can do arithmetic right away. All of the arithmetic operations are overloaded to do "the right thing" automatically, based on the data types of the operands. The core data types include integers (8, 16 and 32 bits, both signed and unsigned), floating-point numbers (32 and 64 bits) and complex numbers (64 and 128 bits). Along with those, there is a core data structure, implemented as an N-dimensional array. The default value for N is 6, but is arbitrary. Also, FreeMat supports heterogeneous arrays, where different elements are actually different data types.

Commands in FreeMat are line-based. This means that a command is finished and executed when you press the Enter key. As soon as you do, the command is run, and the output is displayed immediately within your console. You should notice that results that are not saved to a variable are saved automatically to a temporary variable named "ans". You can use this temporary variable to access and reuse the results from previous commands. For example, if you want to find the volume of a cube of side length 3, you could so with:


--> 3 * 3
ans =
 9
--> ans * 3
ans =
 27

Of course, a much faster way would be to do something like this:


--> 3 * 3 * 3
ans =
 27

Or, you could do this:


--> 3^3
ans =
 27

If you don't want to clutter up your console with the output from intermediate calculations, you can tell FreeMat to hide that output by adding a semicolon to the end of the line. So, instead of this:


--> sin(10)
ans =
   -0.5440

you would get this:

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Gnuplot—the Grandfather of Graphing Utilities

Joey Bernard — Mon, 18 Feb 2013 18:48:51 +0000

by Joey Bernard

In these columns, I have covered several different scientific packages for doing calculations in many different areas of research. I also have looked at various packages that handle graphical representation of these calculations. But, one package that I've never looked at before is gnuplot (http://www.gnuplot.info). Gnuplot has been around since the mid-1980s, making it one of the oldest graphical plotting programs around. Because it has been around so long, it's been ported to most of the operating systems that you might conceivably use. This month, I take a look at the basics of gnuplot and show different ways to use it.

Gnuplot is a command-line-driven program. As such, it has been co-opted to provide graphic capabilities in several other applications, such as octave. Thus, you may have used gnuplot without even realizing you were doing so. You can use gnuplot in several ways. It not only can accept input data to plot, but it also can plot functions. Gnuplot can send its output either to the screen (in both a static file format display or an interactive display), or it can send output to any of a large number of file formats. Additionally, lots of functions are available to customize your plots, changing the labels and axes, among other things.

Let's start by installing gnuplot. Binaries are available for many different operating systems. Most Linux distributions also should come with a package for gnuplot, so installation should be a breeze. If you want the latest and greatest features available, you always can download the source code and build gnuplot from scratch.

Once gnuplot is installed, you can start it by executing the command gnuplot. When executed this way, you are launched into an interactive session. Let's start by trying to plot a basic function. You should be able to plot any mathematical function that would be accepted in C, FORTRAN or BASIC. These mathematical expressions can be built up from built-in functions like abs(x), cos(x) or Bessel. You can use integer, real and complex data types as arguments to these functions.

When using gnuplot to generate a plot, you either can have all of the commands in a single file and hand them in to gnuplot as a script, or you can start gnuplot up in interactive mode and issue these commands one at a time in the command environment. To run a gnuplot script, you simply need to add it at the end of the command when you run gnuplot—for example:


gnuplot script_to_run

When you run gnuplot in interactive mode, you can quit your session with the command quit. The two most basic commands are plot and splot. plot generates two-dimensional plots, and splot generates three-dimensional plots. To plot a simple function, you can use:

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Symbolic Math with Python

Joey Bernard — Wed, 19 Dec 2012 19:19:28 +0000

by Joey Bernard

Many programming languages include libraries to do more complicated math. You can do statistics, numerical analysis or handle big numbers. One topic many programming languages have difficulty with is symbolic math. If you use Python though, you have access to sympy, the symbolic math library. Sympy is under constant development, and it's aiming to be a full-featured computer algebra system (CAS). It also is written completely in Python, so you won't need to install any extra requirements. You can download a source tarball or a git repository if you want the latest and greatest. Most distributions also provide a package for sympy for those of you less concerned about being bleeding-edge. Once it is installed, you will be able to access the sympy library in two ways. You can access it like any other library with the import statement. But, sympy also provides a binary called isympy that is modeled after ipython.

In its simplest mode, sympy can be used as a calculator. Sympy has built-in support for three numeric types: float, rational and integer. Float and integer are intuitive, but what is a rational? A rational number is made of a numerator and a denominator. So, Rational(5,2) is equivalent to 5/2. There is also support for complex numbers. The imaginary part of a complex number is tagged with the constant I. So, a basic complex number is:


a + b*I

You can get the imaginary part with "im", and the real part with "re". You need to tell functions explicitly when they need to deal with complex numbers. For example, when doing a basic expansion, you get:


exp(I*x).expand() exp(I*x)

To get the actual expansion, you need to tell expand that it is dealing with complex numbers. This would look like:


exp(I*x).expand(complex=True)

All of the standard arithmetic operators, like addition, multiplication and power are available. All of the usual functions also are available, like trigonometric functions, special functions and so on. Special constants, like e and pi, are treated symbolically in sympy. They won't actually evaluate to a number, so something like "1+pi" remains "1+pi". You actually have to use evalf explicitly to get a numeric value. There is also a class, called oo, which represents the concept of infinity—a handy extra when doing more complicated mathematics.

Although this is useful, the real power of a CAS is the ability to do symbolic mathematics, like calculus or solving equations. Most other CASes automatically create symbolic variables when you use them. In sympy, these symbolic entities exist as classes, so you need to create them explicitly. You create them by using:


x = Symbol('x')
y = Symbol('y')

If you have more than one symbol at a time to define, you can use:

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An Introduction to GCC Compiler Intrinsics in Vector Processing

George Koharchik — Fri, 21 Sep 2012 14:00:00 +0000

by George Koharchik

Speed is essential in multimedia, graphics and signal processing. Sometimes programmers resort to assembly language to get every last bit of speed out of their machines. GCC offers an intermediate between assembly and standard C that can get you more speed and processor features without having to go all the way to assembly language: compiler intrinsics. This article discusses GCC's compiler intrinsics, emphasizing vector processing on three platforms: X86 (using MMX, SSE and SSE2); Motorola, now Freescale (using Altivec); and ARM Cortex-A (using Neon). We conclude with some debugging tips and references.

Download the sample code for this article here: http://www.linuxjournal.com/files/linuxjournal.com/code/11108.tar

So, What Are Compiler Intrinsics?

Compiler intrinsics (sometimes called "builtins") are like the library functions you're used to, except they're built in to the compiler. They may be faster than regular library functions (the compiler knows more about them so it can optimize better) or handle a smaller input range than the library functions. Intrinsics also expose processor-specific functionality so you can use them as an intermediate between standard C and assembly language. This gives you the ability to get to assembly-like functionality, but still let the compiler handle details like type checking, register allocation, instruction scheduling and call stack maintenance. Some builtins are portable, others are not--they are processor-specific. You can find the lists of the portable and target specific intrinsics in the GCC info pages and the include files (more about that below). This article focuses on the intrinsics useful for vector processing.

Vectors and Scalars

In this article, a vector is an ordered collection of numbers, like an array. If all the elements of a vector are measures of the same thing, it's said to be a uniform vector. Non-uniform vectors have elements that represent different things, and their elements have to be processed differently. In software, vectors have their own types and operations. A scalar is a single value, a vector of size one. Code that uses vector types and operations is said to be vector code. Code that uses only scalar types and operations is said to be scalar code.

Vector Processing Concepts

Vector processing is in the category of Single Instruction, Multiple Data (SIMD). In SIMD, the same operation happens to all the data (the values in the vector) at the same time. Each value in the vector is computed independently. Vector operations include logic and math. Math within a single vector is called horizontal math. Math between two vectors is called vertical math.

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