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[f, {i, imax}], the first step, as discussed in
"Blocks and Local Values", 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 RandomReal[] is evaluated four separate times here, so four different pseudorandom numbers are generated.
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This evaluates RandomReal[] before feeding it to Table. The result is a list of four identical numbers.
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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 this form, fac[i] is not evaluated until an explicit integer value has been assigned to i.
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Using Evaluate forces fac[i] to be evaluated with i left as a symbolic object.
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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.
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It is however possible to get a symbolic formula for the sum, valid for any value of i.
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By inserting Evaluate, you tell Mathematica first to evaluate the sum symbolically, then to iterate over i.
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| 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.
