Lists—Wolfram Documentation

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Lists

Making Lists of Objects	Adding, Removing, and Modifying List Elements
Collecting Objects Together	Combining Lists
Making Tables of Values	Lists as Sets
Manipulating Elements of Lists	Rearranging Lists
Vectors and Matrices	Grouping and Combining Elements of Lists
Getting Pieces of Lists	Ordering in Lists
Testing and Searching List Elements	Rearranging Nested Lists

Making Lists of Objects

In doing calculations, it is often convenient to collect together several objects, and treat them as a single entity. Lists give you a way to make collections of objects in the Wolfram Language. As you will see later, lists are very important and general structures in the Wolfram Language.

A list such as {3,5,1} is a collection of three objects. But in many ways, you can treat the whole list as a single object. You can, for example, do arithmetic on the whole list at once, or assign the whole list to be the value of a variable.

Here is a list of three numbers:

This squares each number in the list, and adds 1 to it:

This takes differences between corresponding elements in the two lists. The lists must be the same length:

The value of % is the whole list:

You can apply any of the mathematical functions in "Some Mathematical Functions" to whole lists:

Just as you can set variables to be numbers, so also you can set them to be lists.

This assigns v to be a list:

Wherever v appears, it is replaced by the list:

Collecting Objects Together

We first encountered lists in "Making Lists of Objects" as a way of collecting numbers together. Here, we shall see many different ways to use lists. You will find that lists are some of the most flexible and powerful objects in the Wolfram Language. You will see that lists in the Wolfram Language represent generalizations of several standard concepts in mathematics and computer science.

At a basic level, what a Wolfram Language list essentially does is to provide a way for you to collect together several expressions of any kind.

Here is a list of numbers:

This gives a list of symbolic expressions:

You can differentiate these expressions:

And then you can find values when x is replaced with 3:

The mathematical functions that are built into the Wolfram Language are mostly set up to be "listable" so that they act separately on each element of a list. This is, however, not true of all functions in the Wolfram Language. Unless you set it up specially, a new function f that you introduce will treat lists just as single objects. "Applying Functions to Parts of Expressions" and "Structural Operations" will describe how you can use Map and Thread 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.

This gives a table of the values of

, with

running from 1 to 6:

Here is a table of

for

from

This gives the numerical values:

You can also make tables of formulas:

Table uses exactly the same iterator notation as the functions Sum and Product, which are discussed in "Sums and Products":

This makes a table with values of x running from 0 to 1 in steps of 0.25:

You can perform other operations on the lists you get from Table:

TableForm displays lists in a "tabular" format. Notice that both words in the name TableForm begin with capital letters:

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 Wolfram Language iterator notation, discussed in "Sums and Products".

This makes a table of

with

running from 1 to 3 and

running from 1 to 2:

The table in this example is a list of lists. The elements of the outer list correspond to successive values of

. The elements of each inner list correspond to successive values of

, with

fixed.

Sometimes you may want to generate a table by evaluating a particular expression many times, without incrementing any variables.

This creates a list containing four copies of the symbol x:

This gives a list of four pairs of numbers sampled from {1,2,3,4}. Table reevaluates RandomSample[{1,2,3,4},2] for each element in the list, so that you get four different samples:

This evaluates

for each of the values of i in the list {1,4,9,16}:

This creates a 3×2 table:

In this table, the length of the rows depends on the more slowly varying iterator variable, i:

You can use Table to generate arrays with any number of dimensions.

This generates a three‐dimensional 2×2×2 array. It is a list of lists of lists:

Table[f,{i_max}]	give a list of i_max values of f

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