piecewise

Domain of conditionally defined objects

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Syntax

piecewise([cond1, value1], [cond2, value2], …, <[Otherwise, valueN]>, <ExclusiveConditions>)

Description

piecewise([cond1, value1], [cond2, value2], ...) defines a conditional object that equals value1 if cond1 is provably true, value2 if cond2 is provably true, and so on. Typically, such objects define piecewise functions or express solutions based on a case analysis of the free parameters of the mathematical problem. See Example 1.

A pair [condition, value] is called a branch. If condition is provably false, then piecewise discards the entire branch. If condition is provably true, then piecewise returns the corresponding value. If neither condition in a piecewise object is provably true, piecewise returns an object of type piecewise that contains all branches, except for branches with provably false conditions.

If all conditions are provably false, or if you call piecewise without any branches, then piecewise returns undefined. See Example 1.

Conditions do not need to be exhaustive or exclusive. If conditions contain parameters, and you substitute values for the parameters, all conditions can become false. Also, several conditions can become true.

If several conditions are simultaneously true, piecewise returns the value from the first branch that contains the condition recognized as true. Ensure that all values corresponding to the true conditions have the same mathematical meaning. Do not rely on the system to recognize the first mathematically true condition as true. Alternatively, you can use the ExclusiveConditions option to fix the order of the branches.

piecewise([cond1, value1], [cond2, value2], ..., [Otherwise, valueN]) checks the conditions, and, if they are not satisfied, discards them and returns valueN. The Otherwise condition occurs in the last branch. It can occur only once. It remains unchanged as long as there are other branches, but it is treated as true when all other branches are discarded because their conditions are false. See Example 2.

The system checks the truth of the conditions for current values and properties of all involved identifiers each time it evaluates an object of type piecewise. Thus, it simplifies piecewise expressions under various different assumptions.

piecewise objects can be nested: both conditions and values can be piecewise objects themselves. piecewise automatically “flattens” such objects. For example, piecewise([conditionA, piecewise([conditionB, valueC])]) becomes piecewise([conditionA and conditionB, valueC]). See Example 3.

Arithmetical and set-theoretic operations work for piecewise objects, provided these operations are defined for all values contained in the branches. If f is such an operation and p1, p2, ... are piecewise objects, then f(p1, p2, ...) is the piecewise object consisting of all branches of the form [cond1 and cond2 and ..., f(value1, value2, ...)], where [cond1, value1] is a branch of p1, [cond2, value2] is a branch of p2, and so on. In other words, applying f commutes with any assignment to free parameters in the conditions. See Example 4.

piecewise objects can also be mixed with other objects in such operations. In such cases, if p1 is not a piecewise object, the system treats it as a piecewise object with the only branch [TRUE, p1]. See Example 5.

diff, float, limit, int and similar functions handle expressions involving piecewise. When you use a piecewise argument in unary operators and functions with one argument, the system maps the operator or function to the values in each branch. See Example 6, Example 7, and Example 8.

piecewise differs from the if and case branching statements. First, piecewise uses the property mechanism when deciding the truth of the conditions. Therefore, the result depends on the properties of the identifiers that appear in the conditions. Second, piecewise treats conditions mathematically, while if and case evaluate them syntactically. Third, piecewise internally sorts the branches. If conditions in several branches are true, piecewise can return any of these branches. See Example 9.

The ExclusiveConditions option fixes the order of branches in a piecewise expression. If the condition in the first branch returns TRUE, then piecewise returns the value from the first branch. If a true condition appears in any further branch, then piecewise returns the value from that branch and removes all subsequent branches. Thus, piecewise with ExclusiveConditions is very similar to an if-elif-end_if statement. Nevertheless, piecewise with ExclusiveConditions still takes into account assumptions on identifiers and treats conditions mathematically while if-elif-end_if treats them syntactically. See Example 10.

Environment Interactions

piecewise takes into account properties of identifiers.

Examples

Example 1

Define this rectangular function f. Without additional information about the variable x, the system cannot evaluate the conditions to TRUE or FALSE. Therefore, it returns the piecewise object.

f := x -> piecewise([x < 0 or x > 1, 0], [x >= 0 and x <= 1, 1])

Call the function f with the following arguments. Every time you call this piecewise function, the system checks the conditions in its branches and evaluates the function.

f(0), f(2), f(I)

Example 2

Create this piecewise function using the syntax that includes Otherwise:

pw:= piecewise([x > 0 and x < 1, 1], [Otherwise, 0])

Evaluate pw for these three values:

pw | x = 1/2;
pw | x = 2;
pw | x = I;

For further computations, delete the identifier pw:

delete pw:

Example 3

Create this nested piecewise expression. MuPAD^® flattens nested piecewise objects.

p1 := piecewise([a > 0, a^2], [a <= 0, -a^2]):
piecewise([b > 0, a + b], [b = 0, p1 + b], [b < 0, a + b])

Example 4

Find the sum of these piecewise functions. You can perform most operations on piecewise functions the same way as you would on ordinary arithmetical expressions. The result of an arithmetical operation is only defined at the points where all of the arguments are defined:

piecewise([x > 0, 1], [x < -3, x^2]) + piecewise([x < 2, x])

Example 5

Solve this equation. The solver returns the result as a piecewise set:

S := solve(a*x = 0, x)

You can use set-theoretic operations work for such sets. For example, find the intersection of this set and the interval (3, 5):

S intersect Dom::Interval(3, 5)

Example 6

Many unary functions are overloaded for piecewise by mapping them to the objects in all branches of the input:

f := piecewise([x >= 0, arcsin(x)], [x < 0, arccos(x)]):
sin(f)

Example 7

Find the limit of this piecewise function:

limit(piecewise([a > 0, x],[a < 0 and x > 1, 1/x],
                [a < 0 and x <= 1, -x]), x = infinity)

Example 8

Find the integral of this piecewise function:

int(piecewise([x < 0, x^2], [x > 0, x^3]), x = -1..1)

Example 9

Create this piecewise function. Here, piecewise cannot determine if any branch is true or false. To do that, piecewise needs additional information about the identifier a.

p1 := piecewise([a = 0, 0], [a <> 0, 1/a])

Create a similar structure by using if-then-else. The if-then-else structure evaluates the conditions syntactically. Here, a = 0 is technically false because the identifier a and the integer 0 are different objects.

p2 := (if a = 0 then 0 else 1/a end)

piecewise takes properties of identifiers into account:

p1 := piecewise([a + b = 0, 0], [Otherwise, 1/a]) assuming a + b = 0

if-then-else does not:

p2 := (if a + b = 0 then 0 else 1/a end) assuming a + b = 0

For further computations, delete identifiers a, b, p1, and p2:

delete a, b, p1, p2:

Example 10

Create this piecewise expression:

p := piecewise([x > 0, 1], [y > 0, 2])

Evaluate the expression at y = 1:

p | y = 1

Now, create the piecewise expression with the same branches, but this time use ExclusiveConditions to fix the order of the branches. When you use this option, any branch can be true only if the previous branches are false.

pE := piecewise([x > 0, x], [y > 0, y], ExclusiveConditions)

Evaluate the expression at y = 1:

pE | y = 1

When you use ExclusiveConditions, piecewise acts the same way as an if-then-else statement, but does not ignore properties of identifiers. For example, set the assumption that x = 0:

assume(x = 0)

The piecewise function call returns 0 because it uses the assumption on identifier x:

p := piecewise([x = 0, x], [Otherwise, 1/x^2])

The corresponding if-then-else statement ignores the assumption, and, therefore, returns 1/x^2:

pIf := (if x = 0 then x else 1/x^2 end)

For further computations, delete identifiers p, pE, x, and pIf:

delete p, pE, x, pIf:

Example 11

Find a set of accumulation points of this piecewise function by calling limit with the Intervals option:

limit(piecewise([a > 0, sin(x)], [a < 0 and x > 1, 1/x],
         [a < 0 and x <= 1, -x]), x = infinity, Intervals)

Example 12

Rewrite the sign function in terms of a piecewise object:

f := rewrite(sign(x), piecewise)

Example 13

Create this piecewise object:

f := piecewise([x > 0, 1], [x < -3, x^2])

Extract a particular condition or object:

piecewise::condition(f, 1), piecewise::expression(f, 2)

The index operator has the same meaning as piecewise::expression and can be typed faster:

f[2]

The piecewise::branch function extracts whole branches:

piecewise::branch(f, 1)

You can form another piecewise object from the branches for which the condition satisfies a given selection criterion, or split the input into two piecewise objects, as the system functions select and split do it for lists:

piecewise::selectConditions(f, has, 0)

piecewise::splitConditions(f, has, 0)

You can also create a copy of f with some branches added or removed:

piecewise::remove(f, 1)

piecewise::insert(f, [x > -3 and x < 0, sin(x)])

Parameters

cond1, cond2, …	Boolean constants, or expressions representing logical formulas
object1, object2, …	Arbitrary objects
Otherwise	Identifier that specifies the last condition. This condition is always treated as a true condition.

Options

ExclusiveConditions

The ExclusiveConditions option fixes the order of branches in a piecewise expression. This option causes piecewise to automatically remove branches with false conditions. Thus, piecewise with ExclusiveConditions is almost equivalent to an if-elif-end_if statement, except that piecewise takes into account assumptions on identifiers. For example, if the condition in the first branch returns TRUE, then piecewise returns the expression from the first branch. If a true condition appears in any further branch, then piecewise returns the expression from that branch and removes all subsequent branches.

Methods

Mathematical Methods

_in — Membership with piecewise on the left side

_in(p, S)

contains — Apply the function contains to the objects in all branches

contains(p, a)

This method overloads the function contains. The values in all branches must be valid first arguments for contains.

diff — (partial) differentiation

diff(p, <x, …>)

If no variables are given, p is returned.

discont — Determine the discontinuities of a piecewise function

discont(p, x, <F>)

discont(p, x = a .. b, <F>)

The values in all branches of p must be arithmetical expressions.

The optional third parameter has the same meaning as for the function discont.

As for the function discont, only discontinuities in the given interval [a,b] are returned when calling piecewise(p, x = a..b).

disregardPoints — Heuristic for simplifying conditions

disregardPoints(p)

expand — Apply the function expand to the objects in all branches

expand(p)

factor — Apply the function factor to the objects in all branches

factor(p)

getElement — Get any element of a piecewise set

getElement(p)

The result is FAIL if no such common element is found.

This method overloads the function solvelib::getElement.

has — Test for the existence of a subobject

has(p, a)

int — Definite and indefinite integration of a piecewise function

int(p, x, <r>)

If a range a..b is given, this method computes the definite integral of p when x runs through that range.

ilaplace — Apply the function ilaplace to the objects in all branches

ilaplace(p, x, t)

isFinite — Test whether a piecewise set is finite

isFinite(p)

This method overloads solvelib::isFinite.

laplace — Apply the function laplace to the objects in all branches

laplace(p, x, t)

limit — Compute the limit of a piecewise function

limit(p, x, <Left | Right | Real>, <Intervals>, <NoWarning>)

limit(p, x = x0, <Left | Right | Real>, <Intervals>, <NoWarning>)

When called with the Intervals option, the method returns the set of accumulation points of a function.

If the method cannot find the function limit and cannot prove the limit does not exist, the function call returns an unevaluated limit function.

If the limit of a function does not exist, the method returns the special value undefined.

This method overloads the function limit.

normal — Apply the function normal to the objects in all branches

normal(p)

partfrac — Apply the function partfrac to the objects in all branches

partfrac(p)

restrict — Impose an additional condition

restrict(p, C)

set2expr — Membership with piecewise on the right side

set2expr(p, x)

The objects in all branches of p must represent sets.

This method overloads the system function _in.

simplify — Simplify a conditionally defined object

simplify(p)

solve — Solve a conditionally defined equation or inequality

solve(p, x, <option1, option2, …>)

For each branch [condition, value] of p, with value being an equation or inequality, the method determines the set of all values x such that both condition and value become true mathematically, and returns the union of all obtained sets. The return value can be a conditionally defined set.

This method overloads the function solve. See the corresponding help page for a description of the available options and an overview of the types of sets that can be returned.

solveConditions — Isolate a given identifier in all conditions

solveConditions(p, x)

Union — Union of a system of sets

Union(p, x, indexset)

The values in all branches of p must represent sets.

For each branch [condition, value] of p, this method does the following. It substitutes for x in value all values from indexset satisfying condition and takes the union over all obtained sets. Then it returns the union over the resulting sets for all branches.

This method overloads the function solvelib::Union.

Access Methods

_concat — Merge piecewise objects

_concat(p, …)

branch — Nth branch

branch(p, n)

op — Branches

op(p)

op(p, n)

op(p, n) returns the nth branch of p as a list. If n = 0, then piecewise is returned.

setBranch — Replace the ith branch

setBranch(p, i, b)

numberOfBranches — Number of branches

numberOfBranches(p)

condition — Condition in a specific branch

condition(p, i)

setCondition — Replace the condition in a specific branch by another

setCondition(p, i, cond)

expression — Object in a specific branch

expression(p, i)

Instead of piecewise::expression(p, i), the index operator p[i] can be used synonymously.

_index — Object in a specific branch

_index(p, i)

piecewise::expression can be used synonymously.

setExpression — Replace the object in a specific branch by another

setExpression(p, i, a)

insert — Insert a branch

insert(p, b)

b can either be a branch extracted from another conditionally defined object using extop, or a list [condition, object].

See Example 13.

extmap — Apply a function to the objects in all branches

extmap(p, f, <a, …>)

mapConditions — Apply a function to the conditions in all branches

mapConditions(p, f, <a, …>)

map — Apply the function map to the objects in all branches

map(p, f, <a, …>)

map(p, f) is equivalent to piecewise::extmap(p, map, f).

remove — Remove a branch

remove(p, i)

splitBranch — Split a branch into two branches

splitBranch(p, i, newcondition)

selectConditions — Select branches depending on their condition

selectConditions(p, f, <a, …>)

For every condition in p, f(condition a, …) must return a Boolean  constant.

If none of the conditions satisfies the selection criterion, undefined is returned.

selectExpressions — Select branches depending on their value

selectExpressions(p, f, <a, …>)

For every value in p, f(value a, …) must return a Boolean constant.

If none of the objects satisfies the selection criterion, undefined is returned.

splitConditions — Split branches depending on conditions

splitConditions(p, f, <a, …>)

For every condition in p, f(condition a, …) must return a Boolean constant.

See Example 13.

subs — Substitution

subs(p, s, …)

This method overloads the function subs. The calling syntax is identical to that function. See the corresponding help page for a description of the various types that are allowed for s.

zip — Apply a binary operation pointwise

zip(p1, p2, f)

If we regard conditionally defined objects as functions from the set A of parameter values to a set B of objects, this method implements the canonical extension of the binary operation f on B to the binary operation g on the set B^A of all functions from A to B via g(p1, p2)(a) = f(p1(a), p2(a)) for all a in A.

If only one of the first two arguments—p1, say—is of type piecewise, then each branch [condition, value] of p1 is replaced by [condition, f(value, p2)].

If neither p1 nor p2 are of type piecewise, then piecewise::zip(p1, p2, f) returns f(p1, p2).

Algorithms

The operands of a piecewise object (the branches) are pairs consisting of a condition and the value valid under that condition.

Methods overloading system functions always assume that they have been called via overloading, and that there is some conditionally defined object among their arguments. All other methods do not assume that one of their arguments is of type piecewise. This simplifies the use of piecewise: it is always allowed to enter p:=piecewise(...) and to call some method of piecewise with p as an argument. You do not need to care about the special case where p is not of type piecewise because some condition in its definition is true or all conditions are false.