normal

Normalize an expression

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Syntax

normal(f, options)
normal(object)

Description

normal(f) returns a normal form of the rational expression f. MuPAD^® regards an expression as normalized when it is a fraction where both numerator and denominator are polynomials whose greatest common divisor is 1.

normal(object) replaces the operands of object with their normalized form.

normal and simplifyFraction are equivalent.

If argument f contains irrational subexpressions such as sin(x), x^(-1/3) etc., then these are replaced by auxiliary variables before normalization. After normalization, these variables are replaced by the normalization of the original subexpressions. Algebraic dependencies of the subexpressions are not taken into account. The operands of the non-rational subexpressions are normalized recursively.

If argument f contains floating-point numbers, then these are replaced by rational approximants (see numeric::rationalize). In the end, float is applied to the result.

With the Expand option, the normal form is unique for rational expressions: it is the quotient of expanded polynomials whose greatest common divisor is 1. If f and g are rational expressions, the following statements are equivalent:

• f and g are mathematically equivalent.

• normal(f, Expand) = normal(g, Expand)

• normal(f - g, Expand) = 0

A normal form generated without the Expand option (which is equivalent to Expand = FALSE) is the quotient of products of powers of expanded polynomials, where all factors of the numerator and the denominator are coprime. MuPAD regards factorized expressions, such as x (x + 1), and equivalent expanded expressions, such as x² + x, as normalized. Therefore, if you do not use Expand, there is no unique normal form of a rational expression.

If f and g are rational expressions, these statements are equivalent:

• f and g are mathematically equivalent.

• normal(f - g) = 0

For special objects, normal is automatically mapped to its operands. In particular, if object is a polynomial of domain type DOM_POLY, then its coefficients are normalized. Further, if object is a set, list, table or array, respectively, then normal is applied to all entries. Further, the left and right sides of equations (type "_equal"), inequalities (type "_unequal"), and relations (type "_less" or "_leequal") are normalized. Further, the operands of ranges (type "_range") are normalized automatically.

Examples

Example 1

Compute the normal form of some rational expressions:

normal(x^2 - (x + 1)*(x - 1))

normal((x^2 - 1)/(x + 1))

normal(1/(x + 1) + 1/(y - 1))

The following expression must be regarded as a rational expression in the “indeterminates” y and sin(x):

normal(1/sin(x)^2 + y/sin(x))

Example 2

Normalize the entries of this list:

[(x^2 - 1)/(x + 1), x^2 - (x + 1)*(x - 1)]

normal(%)

Now, normalize the coefficients of polynomials:

poly((x^2-1)/(x+1)*Y^2 + (x^2-(x+1)*(x-1))*Y - 1, [Y])

normal(%)

Example 3

If you use the Expand option, normal returns a fraction with the expanded numerator and denominator:

normal(x/(x^6 - 1) + x^2/(x^4 - 1), Expand)

Without Expand, a fraction returned by normal can contain factored expressions:

normal(x/(x^6 - 1) + x^2/(x^4 - 1))

Example 4

If you use the List option, normal returns a list consisting of the numerator and denominator of the input:

normal((x^2-1)/(x^2+2*x+1), List)

Note that normal(f, List) is not the same as [numer(f), denom(f)]:

[numer, denom]((x^2-1)/(x^2+2*x+1))

Example 5

To skip calculation of common divisors of the numerator and denominator of an expression, use the NoGcd option:

y := (x^4 - 1)/(x + 1) + 1:
normal(y);
normal(y, NoGcd)

Example 6

To specify common divisors that you want to cancel out, use the ToCancel option:

y := (x^4 - 1)/(x^2 - 1):
normal(y, ToCancel = {x - 1})

Example 7

By default, normal calls the rationalize function in attempt to rationalize the input expression. You might speed up computations by using Rationalize = None in conjunction with the Expand option. This combination of options lets you skip investigating algebraic dependencies and, therefore, saves some time:

n := exp(u):
a := (n^2 + n)/(n + 1) + 1:
normal(a, Expand, Rationalize = None)

Without Rationalize = None, MuPAD analyzes algebraic dependencies and returns this result:

normal(a, Expand)

Example 8

Disable recursive calls to normal for subexpressions by using Recursive = FALSE:

y := sqrt((x^2 + 2*x + 1)/(x + 1)):
normal(y, Recursive = FALSE)

Example 9

Solve this equation, and sum up the fifth powers of the solutions:

solutions := solve(x^3 + x^2 + 1, x, MaxDegree = 3):
f := _plus((solutions[i]^7) $i = 1..3)

Normalizing the result returns:

normal(f)

To limit the number of internally repeated calls to normal due to analysis of algebraic dependencies, use the Iterations option. The default number of iterations is 5. Use the Iterations option to increase or decrease the number of iterations. For example, normalize the result using just one iteration:

normal(f, Iterations = 1)

After two iterations, the result becomes shorter:

normal(f, Iterations = 2)

After three iterations, you get the simplest result:

normal(f, Iterations = 3)

Parameters

f	An arithmetical expression
object	A polynomial of type DOM_POLY, list, set, table, array, equation, inequality, or range

Options

Expand	Return the numerator and denominator of the normalized expression in expanded form. See “Details” for more information. By default, Expand = FALSE.
List	Return a list consisting of the numerator and denominator of f. By default, List = FALSE.
NoGcd	Skip computing common divisors of the numerator and denominator of f. By default, NoGcd = FALSE.
ToCancel	Option, specified as ToCancel = {expr1, expr2, …} Cancel out only the specified common divisors {expr1, expr2,...}.
Rationalize	Option, specified as Rationalize = None Perform only basic rationalization of an irrational input expression. Skip investigating algebraic dependencies. This option works only in conjunction with the Expand option. Otherwise, normal ignores this option. See Example 7.
Recursive	Recursively normalize subexpressions of an irrational expression. By default, Recursive = TRUE.
Iterations	Option, specified as Iterations = n Specify the number of repeated calls to normal. Repeated calls appear when analysis of algebraic dependencies results in new irrational subexpressions. By default, n = 5.

Return Values

Object of the same type as the input object, or a list of two arithmetical expressions if the List option is used.

Overloaded By

object