How to | Create and Use Rules
Transformation rules in
Mathematica let you set local values for symbols, functions, and all other types of expressions. Using rules provides a powerful and extensible method to replace all or part of another expression with the value you specify.
The short form for a rule uses a right arrow, which you get by typing
(with no space between - and >). The
Mathematica front end automatically converts
into
upon further typing. Either symbol is a short form for
Rule.
Create the following transformation rule, which can be thought of as "
goes to
":
| Out[1]= |  |
By looking at the output for
, you can see that this rule does not do anything: the output is simply the rule itself. This is because rules do not do anything when they are alone. You must use a rule with an expression for it to be of any use.
Rules can be applied to expressions by using
(the short form for
ReplaceAll). The general syntax for this is
.
Use
to use a rule with an expression:
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To use two or more rules with an expression, place them in a list
:
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If you give two rules for the same variable,
Mathematica will use only the first rule:
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You can replace variables with any expression, not just individual values.
Substitute
for
:
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You can also use a rule to replace larger parts of an expression:
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In fact, you can use rules with any expression, including functions.
Substitute
for
:
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Use a rule for
. Note that this rule matches
exactly and does not affect
:
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To replace the function
regardless of its argument, you must use a pattern in the rule.
The rule
can be read as "
goes to
":
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For more information on using patterns, see
"Introduction to Patterns."Rules that are set up using
are immediate rules. That is, the right-hand side is evaluated at the same time as the rule:
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You may need to use delayed rules instead, which are not evaluated until they are used with an expression. Delayed rules are created by using
RuleDelayed.
The short form for a delayed rule is
(with no space between : and >). The
Mathematica front end automatically converts
into
upon typing. Either represents the short form for
RuleDelayed:
| Out[2]= |  |
Consider a problem where you want to use a rule to generate three random real numbers in the range of 0 to 1. Using an immediate rule results in the generation of the same three numbers:
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To generate three different numbers, use a delayed rule:
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Assignments set explicitly using
have a global effect, while rules only affect the expression with which they are used.
Use
to assign
to
, and then evaluate
to see the value:
| Out[2]= |  |
Use a rule to assign a value for
:
Evaluating
, you can see that the value assigned by the rule was not saved:
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You must use a rule with an expression for it to work. However, you can explicitly assign a rule to a symbol and then use that symbol as you would the rule.
Use
to assign the rule
to
, and then use
with an expression:
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Since
is now stored globally as the symbol
, you can continue to use
in place of
.
Similarly, you can explicitly assign an expression to a symbol and then use a rule on the symbol:
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This is especially convenient if you plan to use the expression in more than one computation.
