Normalize an expression
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normal(f
,options
) normal(object
)
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 nonrational subexpressions
are normalized recursively.
If argument f
contains floatingpoint 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 x^{2} + 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.
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))
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^21)/(x+1)*Y^2 + (x^2(x+1)*(x1))*Y  1, [Y])
normal(%)
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))
If you use the List
option, normal
returns
a list consisting of the numerator and denominator of the input:
normal((x^21)/(x^2+2*x+1), List)
Note that normal(f, List)
is not the
same as [numer(f), denom(f)]
:
[numer, denom]((x^21)/(x^2+2*x+1))
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)
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})
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)
Disable recursive calls to normal
for subexpressions
by using Recursive = FALSE
:
y := sqrt((x^2 + 2*x + 1)/(x + 1)): normal(y, Recursive = FALSE)
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)
 

A polynomial of
type 

Return the numerator and denominator of the normalized expression
in expanded form. See "Details" for more information.
By default, 

Return a list consisting of the numerator and denominator of 

Skip computing common divisors of the numerator and denominator
of 

Option, specified as Cancel out only the specified common divisors 

Option, specified as Perform only basic rationalization of an irrational input expression.
Skip investigating algebraic dependencies. This option works only
in conjunction with the 

Recursively normalize subexpressions of an irrational expression.
By default, 

Option, specified as Specify the number of repeated calls to 
Object of the same type as the input object, or a list of two
arithmetical expressions if the List
option is
used.
object