Documentation

rationalize

Transform an expression into a rational expression

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

Syntax

rationalize(object, options)

Description

rationalize(object) transforms the expression object into an equivalent rational expression by replacing non-rational subexpressions by newly generated variables.

By default, a rational expression is an expression that contains only sums, products, powers with integer exponents, integers, rational numbers, and identifiers as subexpressions.

The rationalize function returns a sequence (rat, subsSet). The rationalized expression rat contains new variables. The set of substitutions subsSet expresses the new variables by the old ones.

If the original expression contains subexpressions, the rationalize function can rationalize or replace subexpressions or keep them in their original form. Use the options DescendInto, ReplaceType, and StopOn to control the action rationalize takes for particular types of subexpressions.

If FindRelations = ["exp", "_power", "sin"], the rationalize function detects maximal number of algebraic dependencies.

If you call rationalize with any combination of the following three contradicting options, the function chooses the option using the following priorities: ReplaceType, StopOn, DescendInto. For example, if you specify the same type of subexpression with StopOn and DescendInto, the rationalize function uses only the StopOn option for subexpressions of the specified type. If you combine any of these options with the ReplaceType option, rationalize uses only the ReplaceType option.

Examples

Example 1

rationalize operates on single arithmetical expressions, lists, and sets of expressions:

rationalize(2*sqrt(3) + 0.5*x^3)

rationalize([(x - sqrt(2))*(x^2 + sqrt(3)),
             (x - sqrt(2))*(x - sqrt(3))])

Example 2

Use the ApproximateFloats option to replace all floating-point numbers with rational numbers:

rationalize([0.4, 0.333, 0.74], ApproximateFloats)

If you use both ApproximateFloats and ReplaceTypes options, ApproximateFloats does not apply to the types of subexpressions specified in ReplaceTypes:

rationalize(0.4*x^2 + sin(0.33/x),
                ApproximateFloats,
                ReplaceTypes={DOM_FLOAT})

Instead of specifying the value of ReplaceTypes as a sequence of types, you can specify it as a function. The function must return TRUE or FALSE as a result. For example, rationalize the same expression . This time, use the function F to specify the type of subexpressions which you want to replace by variables:

F := X -> testtype(X, DOM_FLOAT):
rationalize(0.4*x^2 + sin(0.33/x),
                ApproximateFloats,
                ReplaceTypes = F)

Example 3

By default, rationalize rationalizes sums, products, bases of integer powers, lists, and sets:

rationalize(ln(sin(x)^2 + cos(x)*exp(x)))

The DescendInto option lets you specify the types of subexpressions that you want to rationalize. Each type can be a domain type, a string as returned by the function type or a Type object. Note that DescendInto overwrites the default types with the types that you specify:

rationalize(ln(sin(x)^2 + cos(x)*exp(x)), DescendInto = {"ln"})

If you want to add new types of subexpressions to the default ones, define the value of DescendInto as a procedure that specifies all required types explicitly. The procedure must return TRUE or FALSE:

F := proc(X)
begin
  hastype(X, {"_plus", "_mult", DOM_SET, DOM_LIST, "ln"}) or
  (hastype(X, "_power") and hastype(op(X, 2), DOM_INT))
end:

rationalize(ln(sin(x)^2 + cos(x)*exp(x)), DescendInto = F)

Example 4

Use the MinimalPolynomials option to find minimal polynomials of irrational expressions:

rationalize(x^(7/6) + x^(3/2), MinimalPolynomials)

Example 5

Use Prefix = s, where s is a string, to specify the prefix for generated variables (the default prefix is X):

rationalize(x^(7/6) + x^(3/2), Prefix = "ABC")

Example 6

Use the ReplaceHardToEval option to replace limits, sums, and integrals with generated variables. Expressions with limits, sums, and integrals tend to be the most computationally expensive:

rationalize(sum(exp(x)/(x^2 + 1), x) +
            limit(sin(cos(1/x))*cos(1/x), x),
                           ReplaceHardToEval)

Example 7

By default, rationalize avoids rationalization of integers, rational numbers, and identifiers:

rationalize(2*sqrt(3) + 0.5*x^3)

The DescendInto option lets you avoid rationalization of particular types of subexpressions. Each type can be specified as a domain type, a string as returned by the function type, or a Type object. For example, rationalize the same expression leaving the subexpression x^3 (of the type "_power") unchanged:

rationalize(2*sqrt(3) + 0.5*x^3, StopOn = {"_power"})

Rationalize the same expression including all subexpressions. Keep floating-point numbers, integers, and identifiers (do not replace them with generated variables):

rationalize(2*sqrt(3) + 0.5*x^3,
            StopOn = {DOM_FLOAT, DOM_INT, DOM_IDENT})

Note that StopOn overwrites the default types with the types that you specify. If you want to add new types of subexpressions to the default ones, specify all the types explicitly:

rationalize(2*sqrt(3) + 0.5*x^3,
            StopOn = {DOM_INT, DOM_IDENT, DOM_RAT, DOM_FLOAT})

rationalize(2*sqrt(3) + 0.5*x^3,
            StopOn = {DOM_INT, DOM_IDENT,
                      DOM_RAT, DOM_FLOAT,
                               "_power"})

The StopOn option also can accept a function as its value. The function must return TRUE or FALSE. For example, use generated variables to replace only subexpressions that contain sin. Keep all other subexpressions intact:

F := X -> not hastype(X, "sin"):
rationalize(sin(x^2) + x^3 + exp(x) + 1/x, StopOn = F)

Example 8

Use the FindRelations option to detect algebraic dependencies between exponentials:

rationalize(exp(x/2) + exp(x/3), FindRelations = ["exp"])

Detect algebraic dependencies for different powers of the same base by specifying the type "_power":

rationalize(x^(3/2) + x^(7/4), FindRelations = ["_power"])

Detect algebraic dependencies for trigonometric functions by specifying the type "sin" or "cos":

rationalize(sin(x) + cos(x), FindRelations = ["sin"]);
rationalize(sin(x)^3 + cos(x)^3, FindRelations = ["cos"])

Example 9

For nested exponentials, use the Recursive option to obtain a list of substitutions:

rationalize(exp(exp(x)), FindRelations = ["exp"], Recursive)

The option also works for trigonometric functions:

rationalize(sin(sin(x)), FindRelations = ["sin"], Recursive)

Example 10

The ShowDependencies option shows all original variables upon which each generated variable depends:

rationalize(sin(x)^3, ShowDependencies)

Parameters

object

Any MuPAD® object

Options

ApproximateFloats

When you use the ApproximateFloats option, the rationalize function replaces floating-point numbers with rational numbers. By default, ApproximateFloats= FALSE: the rationalize function replaces all floating-point numbers with the new variables. If you rationalize an expression using both ApproximateFloats and StopOn options, StopOn does not prevent rationalization of floating-point numbers in the specified subexpressions. If you rationalize an expression using both ApproximateFloats and ReplaceTypes options, ApproximateFloats does not apply to the types of subexpressions specified in ReplaceTypes. See Example 2.

DescendInto

When you use the DescendInto option, the rationalize function rationalizes all subexpressions of the specified types. You can specify the value of this option as a set (even if there is only one type) or a procedure that returns TRUE or FALSE. Each type can be

By default, the rationalize function rationalizes the following types of subexpresssions: sums, products, bases of integer powers, lists, and sets. When you specify other types of subexpressions, rationalize uses them instead of the default types. (DescendInto overwrites the default types with the types that you specify.) If you want to extend the set of types of subexpressions retaining the default types, define the value of DescendInto as a procedure that specifies all default and additional types explicitly. See Example 3.

FindRelations

When you use the FindRelations option, the rationalize function detects algebraic dependencies for subexpressions of specified types. This option accepts the types of subexpressions in the form of a list. The following types are available: "sin", "cos", "exp", and "_power". By default, rationalize does not look for dependencies for irrational subexpressions: FindRelations= [].

MinimalPolynomials

When you use the MinimalPolynomials option, the rationalize function returns the minimal polynomials of irrational expressions. The function returns the rationalized expression, the set of substitution equations, and minimal polynomials. By default, MinimalPolynomials= FALSE. See Example 4.

Prefix

Use the Prefix option to specify the prefix for new variables generated by the rationalize function. The value of this option must be a string. By default, Prefix= "X". See Example 5.

Recursive

When you use the Recursive option, the rationalize function recursively rationalizes nested subexpressions, and returns a list of substitution equations. Each generated variable in the returned list can depend on other variables in the list. By default, Recursive= FALSE. See Example 9.

ReplaceHardToEval

When you use the ReplaceHardToEval option, the rationalize function replaces all limits, sums, and integrals by generated variables. Generally, this option allows you to avoid most expensive rationalizations of sums, limits, and integrals. By default, ReplaceHardToEval= FALSE. See Example 6.

ReplaceTypes

When you use the ReplaceTypes option, the rationalize function replaces all subexpressions of the specified types with generated variables. You can specify the value of this option as a set (even if there is only one type) or a procedure that returns TRUE or FALSE. Each type can be

This option allows you to specify and avoid most expensive rationalizations for your particular expression. If you use this option in combination with ReplaceHardToEval, the rationalize function uses generated variables to replace all limits, sums, integrals, and the types that you specify. If ReplaceTypes specifies the same type of subexpression as DescendInto, the ReplaceTypes option prevails. By default, ReplaceTypes= {}.

Alternatively, specify the value of this option as a function that returns TRUE or FALSE. See Example 2.

ShowDependencies

When you use the ShowDependencies option, the rationalize function replaces any irrational subexpression containing the identifiers vars with an expression of the form newvar(vars), showing the dependencies of the generated variables on the original variables. By default, ShowDependencies= FALSE.

StopOn

When you use the StopOn option, the rationalize function does not rationalize the specified types of subexpressions. You can specify the value of this option as a set (even if there is only one type) or a function that returns TRUE or FALSE. Each type can be

By default, the rationalize function does not rationalize or replace integers, rational numbers, and identifiers. When you specify other types of subexpressions, rationalize uses them instead of the default types. (StopOn overwrites the default types with the types that you specify.) If you want to extend the set of types of subexpressions retaining the default types, specify StopOn = {DOM_INT, DOM_IDENT, DOM_RAT, extra types}, where extra types are the additional types of subexpressions that you do not want to rationalize. See Example 7.

If StopOn specifies the same type of subexpression as DescendInto, the StopOn option prevails.

StopOnConstants

When you use the StopOnConstants option, the rationalize function does not rationalize the object of the type Type::Constant: numbers, strings, Boolean constants, NIL, FAIL, PI, EULER, and CATALAN in the set Type::ConstantIdents. By default, StopOnConstants= FALSE.

Return Values

Sequence consisting of the rationalized object and a set of substitution equations. If you use the Recursive option, the rationalize function returns a list of substitution equations instead of a set. If you use the MinimalPolynomials option, the returned value has a third argument: the minimal polynomials.

See Also

MuPAD Functions

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