Simple Calculations

What follows is a very short interactive introduction covering just the features you need initially. For a more complete introduction to calculations in Mathematica, use the Tour of Mathematica in the Getting Started section of the Help Browser.

Use of RETURN and SHIFT-RETURN

In Mathematica, line breaks are entered by pressing RETURN on the main keyboard; calculations are passed to the kernel by pressing SHIFT-RETURN or ENTER on the numeric keypad.

Terminating Input

There is no need to terminate input. Everything in the current input cell is evaluated as soon as you press SHIFT-RETURN or ENTER. However, terminating the input line with a semicolon suppresses output in Mathematica.

Some programmers switching from other languages may add semicolons out of habit. If you do not get output from Mathematica when you expect it, check to make sure you have not accidentally suppressed it.

Syntax

Mathematica Keywords

Mathematica syntax is cohesive and intuitive. You need to follow only a couple of easy rules:

• Mathematica keywords are usually full English words or concatenated English words. The first letter of each piece of a Mathematica command is capitalized. For example:

  Plot, Plot3D, ListPlot, ListDensityPlot, MeijerG

  Some of the most common operations have the usual mathematical shortcuts (e.g., Tr for trace of a matrix).

• The rules for using brackets are just as simple. Arguments to Mathematica functions are always enclosed in square brackets [ ]. Lists, matrices, and arrays are always enclosed in curly brackets { }. Matrices and arrays are implemented simply as lists of lists.

  Some examples of bracket use:

  

  Most Mathematica functions are automatically threaded over lists.

Multiple Functions

Any Mathematica function can accept any sensible Mathematica output immediately. This means that you can pass function arguments and results directly without having to write external files or conversion code.

For example, this code calculates and plots the eigenvalues of a square random matrix.

Here is the same code written as one step.

Symbolic and Numeric Arguments

Nearly all Mathematica commands accept numeric as well as symbolic input. Mathematica automatically chooses the algorithms based on the kind of input you give.

Symbolic Input Gives Symbolic Results

In general, Mathematica assumes that symbolic arguments are complex numbers and gives the results in complex form.

This assumption is an important safeguard against forgetting or discarding results that depend on branch cuts, for example. However, there are some functions that assume symbols are real numbers, such as PowerExpand, which expands all powers of products and powers.

Other functions that modify expressions of real or complex variables include ComplexExpand and ExpToTrig. These functions can be accessed from the Algebraic Manipulation palette. You may restrict your calculation to real numbers if you are sure you need only real-valued results. The mechanics of the conversion is covered in The Mathematica Book.

Numeric Input Gives Exact Numeric Results

Mathematica keeps numbers and values exact as long as possible to prevent roundoff errors and error propagation. It always returns exact results unless the user instructs it otherwise.

Notice that Mathematica knows that . Programs using approximations can produce results for this example that differ from zero by several hundredths.

All internal Mathematica routines, except dedicated approximate numerical routines like NIntegrate or FindRoot, will return exact results when given integer inputs.

Floating Point Input Gives Floating Point Results to the Required Precision

When it finds a decimal point, Mathematica automatically returns numbers to machine precision.

Forcing Numerical Evaluation

You can force Mathematica to return approximate numerical results to any required precision by using numerical functions or by using N[expression, precision].

An added advantage to using floating point input is speed. When given floating point input with a precision of less than $MachinePrecision, usually 16 significant digits, Mathematica uses fast floating point algorithms instead of exact, but slower bignum algorithms.

Lists

There Is Only One Kind of List

One of the big advantages of Mathematica is that there is only one universal list type. Matrices, tensors, and arrays of any number of dimensions are implemented as lists of lists. Mathematica figures out the appropriate operation for each list on its own. For example, it automatically switches to 2D Fourier transforms if you give the Fourier function a two-dimensional matrix. Therefore, every list can be used as input to any function as long as it is mathematically feasible. The same list can be used as input for normal functions, statistical functions, graphics commands, or any other appropriate function.

A list is always enclosed in curly brackets and the individual elements are separated by commas.

Matrices are simply a list of row vectors, delimited by { }.

To give some examples of how lists work, we can create a noisy dataset using one of the most common ways to create lists, the Table command.

The list can be displayed in different forms.

1	1.1087867880938835`
2	2.3653082990113714`
3	3.5305588298308916`
4	4.15327406955195`
5	5.157271475096308`
6	6.28417990393837`
7	7.448429179488507`
8	8.667842219365623`
9	9.772513060907505`
10	10.883934329215432`
11	11.921947262243041`
12	13.603011457017821`
13	14.704069207924537`
14	15.664297553744325`
15	16.629677451917722`
16	16.945402649876144`
17	19.982942551735594`
18	20.8525720486004`
19	22.338287198467075`
20	21.995644895538305`

Note once again that the list is accepted as input by all functions without change.

Selecting Parts of Lists

Parts of lists, matrices, and multidimensional arrays can be accessed through a variety of commands and shortcuts. The most basic way is by position, using Take and its shortcut [[ ]]. It simply takes the specified ranges and presents them as output.

To take elements from the end of a list, simply use negative ranges.

Other functions for selecting pieces of lists according to their position in the list include functions such as Drop, First, and Part.

Partition and Flatten

To change the form or dimensions of lists, you can use Flatten and Partition.

MatrixForm[Partition[flat, 3]]

Matrix Operations

Mathematica supports all common matrix operations for symbolic and numeric matrices. For a list of the available operations, see the chapters on linear algebra starting at Section 3.7.4 of The Mathematica Book.

Advanced Topic: Selecting Pieces by Value or Criteria

Mathematica also offers a number of ways to select pieces of lists by value or criteria. The easiest way is using Select. To make the most of Select and many other functions, understanding the concept of pure functions is essential. As this goes beyond introductory material, please consult the corresponding sections in The Mathematica Book (2.2, especially Sections 2.2.5 and 2.2.7).