Edit detail for Manipulating Expressions revision 3 of 6

The domain InputForm? can be quite useful for manipulating parts of expressions. For example

ex1:=integrate(log(x)+x, x)

(1)

%::InputForm

(2)

ex2:=interpret((%::InputForm).2.2)

(3)

If you would like to do this with a more common type of expression and hide the details, you can define

op(n,x) == interpret((x::InputForm).(n+1))

Then manipulating expressions looks like this:

op(1,ex1)

Compiling function op with type (PositiveInteger,Expression Integer)
       -> Any

(4)

op(1,%)

Compiling function op with type (PositiveInteger,Any) -> Any

(5)

(op(1,op(1,ex1))-op(2,op(1,ex1)))/op(2,ex1)

(6)

Rules and Pattern Matching (from WesterProblemSet?)

Trigonometric manipulations---these are typically difficult for students

r:= cos(3*x)/cos(x)

(7)

=> cos(x)^2 - 3 sin(x)^2 or similar

real(complexNormalize(r))

(8)

=> 2 cos(2 x) - 1

real(normalize(simplify(complexNormalize(r))))

(9)

Use rewrite rules => cos(x)^2 - 3 sin(x)^2

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) )

(10)

sincosAngles r

(11)

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,

isMult(3*x)

(12)

but

isMult(x*y)

(13)

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

isPower(x**y)

(14)

whereas

isPower(x**10)

(15)

In the first case the Integer is assume to be 1.

We have:

isTimes(x*y*z)

(16)

isPlus(x+y+z*y)

(17)

Whereas

isTimes((x+y)*z)

(18)

That is because the expression is internally treated as a MultivariatePolynomial like this:

((x+y)*z)::MPOLY([x,y,z],INT)

(19)

If you say:

isPlus((x+y)*z)

(20)

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.