This is documentation for Mathematica 4, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.1)
Wolfram Research, Inc.

Evaluation in Patterns, Rules and DefinitionsConditionals

2.5.7 Evaluation in Iteration Functions

The built-in Mathematica iteration functions such as Table and Sum, as well as Plot and Plot3D, evaluate their arguments in a slightly special way.

When evaluating an expression like Table[f, i, imax], the first step, as discussed in Section 2.6.6, is to make the value of i local. Next, the limit imax in the iterator specification is evaluated. The expression f is maintained in an unevaluated form, but is repeatedly evaluated as a succession of values are assigned to i. When this is finished, the global value of i is restored.

The function Random[ ] is evaluated four separate times here, so four different pseudorandom numbers are generated.

In[1]:= Table[Random[ ], {4}]

Out[1]=

This evaluates Random[ ] before feeding it to Table. The result is a list of four identical numbers.

In[2]:= Table[ Evaluate[Random[ ]], {4} ]

Out[2]=

In most cases, it is convenient for the function f in an expression like Table[f, i, imax] to be maintained in an unevaluated form until specific values have been assigned to i. This is true in particular if a complete symbolic form for f valid for any i cannot be found.

This defines fac to give the factorial when it has an integer argument, and to give NaN (standing for "Not a Number") otherwise.

In[3]:= fac[n_Integer] := n! ; fac[x_] := NaN

In this form, fac[i] is not evaluated until an explicit integer value has been assigned to i.

In[4]:= Table[fac[i], {i, 5}]

Out[4]=

Using Evaluate forces fac[i] to be evaluated with i left as a symbolic object.

In[5]:= Table[Evaluate[fac[i]], {i, 5}]

Out[5]=

In cases where a complete symbolic form for f with arbitrary i in expressions such as Table[f, i, imax] can be found, it is often more efficient to compute this form first, and then feed it to Table. You can do this using Table[Evaluate[f], i, imax].

The Sum in this case is evaluated separately for each value of i.

In[6]:= Table[Sum[i^k, {k, 4}], {i, 8}]

Out[6]=

It is however possible to get a symbolic formula for the sum, valid for any value of i.

In[7]:= Sum[i^k, {k, 4}]

Out[7]=

By inserting Evaluate, you tell Mathematica first to evaluate the sum symbolically, then to iterate over i.

In[8]:= Table[Evaluate[Sum[i^k, {k, 4}]], {i, 8}]

Out[8]=

Evaluation in iteration functions.

As discussed in Section 1.9.1, it is convenient to use Evaluate when you plot a graph of a function or a list of functions. This causes the symbolic form of the function or list to be found first, before the iteration begins.

Evaluation in Patterns, Rules and DefinitionsConditionals