Mathematica Tutorial

Applying Functions to Parts of Expressions

If you have a list of elements, it is often important to be able to apply a function separately to each of the elements. You can do this in Mathematica using Map.

This applies f separately to each element in a list.

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This defines a function which takes the first two elements from a list.

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You can use Map to apply take2 to each element of a list.

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Map[f,{a,b,...}]

apply f to each element in a list, giving {f[a], f[b], ...}

Applying a function to each element in a list.

What Map[f, expr] effectively does is to "wrap" the function f around each element of the expression expr. You can use Map on any expression, not just a list.

This applies f to each element in the sum.

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This applies Sqrt to each argument of g.

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Map[f, expr] applies f to the first level of parts in expr. You can use MapAll[f, expr] to apply f to all the parts of expr.

This defines a 2x2 matrix m.

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Map applies f to the first level of m, in this case the rows of the matrix.

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MapAll applies f at all levels in m. If you look carefully at this expression, you will see an f wrapped around every part.

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In general, you can use level specifications as described in "Levels in Expressions" to tell Map to which parts of an expression to apply your function.

This applies f only to the parts of m at level 2.

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Setting the option Heads->True wraps f around the head of each part, as well as its elements.

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Map[f,expr] or f/@expr	apply f to the first-level parts of expr
MapAll[f,expr] or f//@expr	apply f to all parts of expr
Map[f,expr,lev]	apply f to each part of expr at levels specified by lev

Ways to apply a function to different parts of expressions.

Level specifications allow you to tell Map to which levels of parts in an expression you want a function applied. With MapAt, however, you can instead give an explicit list of parts where you want a function applied. You specify each part by giving its indices, as discussed in "Parts of Expressions".

Here is a 2x3 matrix.

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This applies f to parts {1, 2} and {2, 3}.

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This gives a list of the positions at which b occurs in mm.

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You can feed the list of positions you get from Position directly into MapAt.

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To avoid ambiguity, you must put each part specification in a list, even when it involves only one index.

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MapAt[f,expr,{part₁,part₂,...}]

apply f to specified parts of expr

Applying a function to specific parts of an expression.

Here is an expression.

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This is the full form of t.

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You can use MapAt on any expression. Remember that parts are numbered on the basis of the full forms of expressions.

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MapIndexed[f,expr]	apply f to the elements of an expression, giving the part specification of each element as a second argument to f
MapIndexed[f,expr,lev]	apply f to parts at specified levels, giving the list of indices for each part as a second argument to f

Applying a function to parts and their indices.

This applies f to each element in a list, giving the index of the element as a second argument to f.

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This applies f to both levels in a matrix.

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Map allows you to apply a function of one argument to parts of an expression. Sometimes, however, you may instead want to apply a function of several arguments to corresponding parts of several different expressions. You can do this using MapThread.

MapThread[f,{expr₁,expr₂,...}]	apply f to corresponding elements in each of the expr_i
MapThread[f,{expr₁,expr₂,...},lev]	apply f to parts of the expr_i at the specified level

Applying a function to several expressions at once.

This applies f to corresponding pairs of list elements.

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MapThread works with any number of expressions, so long as they have the same structure.

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Functions like Map allow you to create expressions with parts modified. Sometimes you simply want to go through an expression, and apply a particular function to some parts of it, without building a new expression. A typical case is when the function you apply has certain "side effects", such as making assignments, or generating output.

Scan[f,expr]	evaluate f applied to each element of expr in turn
Scan[f,expr,lev]	evaluate f applied to parts of expr on levels specified by lev

Evaluating functions on parts of expressions.

Map constructs a new list in which f has been applied to each element of the list.

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Scan evaluates the result of applying a function to each element, but does not construct a new expression.

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Scan visits the parts of an expression in a depth-first walk, with the leaves visited first.

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