Lists

ME 443

Mathematica for Engineers

Collecting Objects Together We first encountered lists in previous courses as a way of collecting

numbers together. In this chapter, we shall see many different ways to

use lists. You will find that lists are some of the most flexible and

powerful objects in Mathematica. You will see that lists in

Mathematica represent generalizations of several standard concepts in

mathematics and computer science.

At a basic level, what a Mathematica list essentially does is to provide

a way for you to collect together several expressions of any kind.

The mathematical functions that are built into Mathematica are

mostly set up so that they act separately on each element of a list. This

is, however, not true of all functions in Mathematica. Unless you set it

up specially, a new function f that you introduce will treat lists just as

single objects. In later courses we will describe how you can use Map

to apply a function like this separately to each element in a list.

Making Tables of Values You can use lists as tables of values. You can generate the tables, for

example, by evaluating an expression for a sequence of different

parameter values.

Sometimes you may want to generate a table by evaluating a

particular expression many times, without incrementing any

variables.

All the examples so far have been of tables obtained by varying a

single parameter. You can also make tables that involve several

parameters. These multidimensional tables are specified using the

standard Mathematica iterator notation, discussed in Section 6.6.

The table in this example is a list of lists. The three elements of the

outer list correspond to successive values of i. The pairs of elements

of each inner list correspond to successive values of j, with i fixed.

Getting Pieces of Lists

The functions in this section allow you to pick out pieces that occur at

particular positions in lists. Later, we will see how we can use

functions like Select and Cases to pick out elements of lists based not

on their positions, but instead on their properties.

Testing and Searching List Elements

The previous section discussed how to extract pieces of lists based on

their positions or indices. Mathematica also has functions that search

and test for elements of lists, based on the values of those elements.

The functions Count and Position, as well as MemberQ and FreeQ,

can be used not only to search for particular list elements, but also to

search for classes of elements which match specific "patterns".

Adding, Removing and Modifying List

Elements

Combining Lists

Lists as Sets Mathematica usually keeps the elements of a list in exactly the order

you originally entered them. If you want to treat a Mathematica list

like a mathematical set, however, you may want to ignore the order of

elements in the list.

Rearranging Lists

Grouping and Ungrouping

Elements of Lists

You should realize that because of the way Mathematica stores lists, it

is usually less efficient to add a sequence of elements to a particular

list than to create a nested structure that consists, for example, of lists

of length 2 at each level. When you have built up such a structure, you

can always reduce it to a single list using Flatten.