2.6.7 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 2.7.6, 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.