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Manipulating Expressions

Edit detail for Manipulating Expressions revision 4 of 6

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Editor: Bill Page Time: 2008/08/21 07:44:13 GMT-7
Note: rewrite rules

changed:
-sincosAngles:= rule ( _
-   cos((n | integer?(n)) * x) == _
-      cos((n - 1)*x) * cos(x) - sin((n - 1)*x) * sin(x); _
-   sin((n | integer?(n)) * x) == _
-      sin((n - 1)*x) * cos(x) + cos((n - 1)*x) * sin(x) )
sincosAngles:= rule
   cos((n | integer?(n)) * x) ==
      cos((n - 1)*x) * cos(x) - sin((n - 1)*x) * sin(x)
   sin((n | integer?(n)) * x) ==
      sin((n - 1)*x) * cos(x) + cos((n - 1)*x) * sin(x)

The domain InputForm can be quite useful for manipulating parts
of expressions. For example
fricas
ex1:=integrate(log(x)+x, x)
\begin{equation}
\label{eq1}{{2 \  x \ {\log \left({x}\right)}}+{{x}^{2}}-{2 \  x}}\over 2\end{equation}
Type: Union(Expression(Integer),...)
fricas
%::InputForm
\begin{equation}
\label{eq2}\left(/ \ {\left(+ \ {\left( \ {\left( \  2 \  x \right)}\ {\left(\log \  x \right)}\right)}\ {\left(+ \ {\left(^\  x \  2 \right)}\ {\left( \  - 2 \  x \right)}\right)}\right)}\  2 \right)\end{equation*}
Type: InputForm?
fricas
ex2:=interpret((%::InputForm).2.2)
\begin{equation}
\label{eq3}2 \  x \ {\log \left({x}\right)}\end{equation}
Type: Expression(Integer)

If you would like to do this with a more common type of
expression and hide the details, you can define
fricas
op(n,x) == interpret((x::InputForm).(n+1))
Type: Void

Then manipulating expressions looks like this:
fricas
op(1,ex1)
fricas
Compiling function op with type (PositiveInteger,Expression(Integer)
      ) -> Any
\begin{equation}
\label{eq4}{2 \  x \ {\log \left({x}\right)}}+{{x}^{2}}-{2 \  x}\end{equation}
Type: Expression(Integer)
fricas
op(1,%)
fricas
Compiling function op with type (PositiveInteger,Any) -> Any
\begin{equation}
\label{eq5}2 \  x \ {\log \left({x}\right)}\end{equation}
Type: Expression(Integer)
fricas
(op(1,op(1,ex1))-op(2,op(1,ex1)))/op(2,ex1)
\begin{equation}
\label{eq6}{{2 \  x \ {\log \left({x}\right)}}-{{x}^{2}}+{2 \  x}}\over 2\end{equation}
Type: Expression(Integer)

Rules and Pattern Matching (from WesterProblemSet?)
  Trigonometric manipulations---these are typically difficult for students
fricas
r:= cos(3*x)/cos(x)
\begin{equation}
\label{eq7}{\cos \left({3 \  x}\right)}\over{\cos \left({x}\right)}\end{equation}
Type: Expression(Integer)

 => cos(x)^2 - 3 sin(x)^2 or similar
fricas
real(complexNormalize(r))
\begin{equation}
\label{eq8}-{2 \ {{\sin \left({x}\right)}^{2}}}+{2 \ {{\cos \left({x}\right)}^{2}}}- 1\end{equation}
Type: Expression(Integer)

=> 2 cos(2 x) - 1
fricas
real(normalize(simplify(complexNormalize(r))))
\begin{equation}
\label{eq9}{2 \ {\cos \left({2 \  x}\right)}}- 1\end{equation}
Type: Expression(Integer)

Use rewrite rules => cos(x)^2 - 3 sin(x)^2
fricas
sincosAngles:= rule
   cos((n | integer?(n)) * x) ==
      cos((n - 1)*x) * cos(x) - sin((n - 1)*x) * sin(x)
   sin((n | integer?(n)) * x) ==
      sin((n - 1)*x) * cos(x) + cos((n - 1)*x) * sin(x)
\begin{equation*}
\label{eq10}\begin{array}{@{}l}
\displaystyle
\left\{{{\cos \left({n \  x}\right)}\mbox{\rm = =}{-{{\sin \left({x}\right)}\ {\sin \left({{\left(n - 1 \right)}\  x}\right)}}+{{\cos \left({x}\right)}\ {\cos \left({{\left(n - 1 \right)}\  x}\right)}}}}, \right.
\
\
\displaystyle
\left.\:{{\sin \left({n \  x}\right)}\mbox{\rm = =}{{{\cos \left({x}\right)}\ {\sin \left({{\left(n - 1 \right)}\  x}\right)}}+{{\cos \left({{\left(n - 1 \right)}\  x}\right)}\ {\sin \left({x}\right)}}}}\right\} \end{array}
\end{equation*}
Type: Ruleset(Integer,Integer,Expression(Integer))

fricas
sincosAngles r
\begin{equation}
\label{eq11}-{3 \ {{\sin \left({x}\right)}^{2}}}+{{\cos \left({x}\right)}^{2}}\end{equation}
Type: Expression(Integer)

Other Operations
  The domain FunctionSpace? includes the following operations:
    isExpt(p,f:Symbol) returns [x, n] if p = x**n and n <> 0 and x = f(a)
    isExpt(p,op:BasicOperator) returns [x, n] if p = x**n and n <> 0 and x = op(a)
    isExpt(p) returns [x, n] if p = x**n and n <> 0
    isMult(p) returns [n, x] if p = n * x and n <> 0
    isPlus(p) returns [m1,...,mn] if p = m1 +...+ mn and n > 1
    isPower(p) returns [x, n] if p = x**n and n <> 0
    isTimes(p) returns [a1,...,an] if p = a1*...*an and n > 1


  If these conditions are not met, then the above operations
  return "failed".
For example,
fricas
isMult(3*x)
\begin{equation*}
\label{eq12}\left[{coef = 3}, \:{var = x}\right]?\end{equation*}
Type: Union(Record(coef: Integer,var: Kernel(Expression(Integer))),...)

but
fricas
isMult(x*y)
\begin{equation}
\label{eq13}\mbox{\tt "failed"}\end{equation}
Type: Union("failed",...)

In the context of Expression Integer, or Polynomial Integer
the parameter n must be an Integer. The Symbol y is not an
Integer.
Not exactly analogously
fricas
isPower(x**y)

   There are no library operations named ** 
      Use HyperDoc Browse or issue
                                 )what op **
      to learn if there is any operation containing " ** " in its name.

   Cannot find a definition or applicable library operation named ** 
      with argument type(s) 
                                 Variable(x)
                                 Variable(y)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

whereas
fricas
isPower(x**10)

   There are no library operations named ** 
      Use HyperDoc Browse or issue
                                 )what op **
      to learn if there is any operation containing " ** " in its name.

   Cannot find a definition or applicable library operation named ** 
      with argument type(s) 
                                 Variable(x)
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

In the first case the Integer is assume to be 1.
We have:
fricas
isTimes(x*y*z)
\begin{equation*}
\label{eq14}\left[ z , \: y , \: x \right]?\end{equation*}
Type: Union(List(Polynomial(Integer)),...)
fricas
isPlus(x+y+z*y)
\begin{equation*}
\label{eq15}\left[{y \  z}, \: y , \: x \right]?\end{equation*}
Type: Union(List(Polynomial(Integer)),...)

Whereas 
fricas
isTimes((x+y)*z)
\begin{equation}
\label{eq16}\mbox{\tt "failed"}\end{equation}
Type: Union("failed",...)

That is because the expression is internally treated as a
MultivariatePolynomial like this:
fricas
((x+y)*z)::MPOLY([x,y,z],INT)
\begin{equation}
\label{eq17}{z \  x}+{z \  y}\end{equation}
Type: MultivariatePolynomial?([x,y,z],Integer)

If you say:
fricas
isPlus((x+y)*z)
\begin{equation*}
\label{eq18}\left[{y \  z}, \:{x \  z}\right]?\end{equation*}
Type: Union(List(Polynomial(Integer)),...)

perhaps the result makes sense?
For some of the details of these operations I consulted the
actual algebra code at:
http://axiom-wiki.newsynthesis.org/axiom--test--1/src/algebra/FspaceSpad
Click on pdf or dvi to see the documentation.
You can also enter expressions like isTimes in the search box
on the upper right and see all the places in the algebra where
this operation is defined and used.

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