Evaluation in Iteration Functions
The built-in
Mathematica iteration functions such as
Table and
Sum evaluate their arguments in a slightly special way.
When evaluating an expression like
Table
, the first step, as discussed in
"Blocks and Local Values", is to make the value of
i local. Next, the limit
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
RandomReal
is evaluated four separate times here, so four different pseudorandom numbers are generated.
| Out[1]= |  |
This evaluates
RandomReal
before feeding it to
Table. The result is a list of four identical numbers.
| Out[2]= |  |
In most cases, it is convenient for the function
f in an expression like
Table
to be maintained in an unevaluated form until specific values have been assigned to
. This is true in particular if a complete symbolic form for
f valid for any
i cannot be found.
This defines
to give the factorial when it has an integer argument, and to give
(standing for "Not a Number") otherwise.
In this form,
is not evaluated until an explicit integer value has been assigned to
.
| Out[4]= |  |
Using
Evaluate forces
to be evaluated with
left as a symbolic object.
| Out[5]= |  |
In cases where a complete symbolic form for
f with arbitrary
i in expressions such as
Table
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
.
| Out[6]= |  |
It is however possible to get a symbolic formula for the sum, valid for any value of
.
| Out[7]= |  |
By inserting
Evaluate, you tell
Mathematica first to evaluate the sum symbolically, then to iterate over
.
| Out[8]= |  |
| Table[f,{i,imax}] | keep f unevaluated until specific values are assigned to i |
| Table[Evaluate[f],{i,imax}] | evaluate f first with i left symbolic |
Evaluation in iteration functions.
