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	FriCAS Problems
Unevaluated Expressions ←	↑	→ numerical linear algebra

indexed variables

Edit detail for indexed variables revision 2 of 2

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Editor: test1 Time: 2017/02/17 17:19:23 GMT+0
Note:

changed:
-In Axiom 'product' denotes the *formal product* and in this example is
In FriCAS 'product' denotes the *formal product* and in this example is

changed:
-is not what Axiom considers a *variable*. In Axiom $p[k]$ is assumed to be
is not what FriCAS considers a *variable*. In FriCAS $p[k]$ is assumed to be

changed:
-Axiom does what it does. $1+p[k]$ is assumed to be a polynomial over $p[k]$
FriCAS does what it does. $1+p[k]$ is assumed to be a polynomial over $p[k]$

changed:
-In fact, much more complicated indexed symbols can be created in Axiom. For
In fact, much more complicated indexed symbols can be created in FriCAS. For

changed:
-Within this MathAction Axiom wiki it is also possible to create
Within this MathAction FriCAS wiki it is also possible to create

changed:
-protects the \\ which begins a Greek letter, etc. Axiom treats the entire
protects the \\ which begins a Greek letter, etc. FriCAS treats the entire

changed:
-works here and not within stand alone Axiom.
works here and not within stand alone FriCAS.

I am trying to create formulae for products with, say, p sub k as symbols. These are the results from two, at least naively identical formulae:

fricas

(1) -> (1+p[1])*(1+p[2])

$\label{eq1}{{\left({p_{1}}+ 1 \right)}\ {p_{2}}}+{p_{1}}+ 1$

(1)

Type: Polynomial(Integer)

fricas

product(1+p[k],k=1..2)

$\label{eq2}{{p_{k}}^{2}}+{2 \ {p_{k}}}+ 1$

(2)

Type: Expression(Integer)

But perhaps unexpectedly, the 2nd command is interpreted as just .

How could one use "product" (or "sum" for that reason) and an indexed way to denote variables?

--------------

In FriCAS product denotes the formal product and in this example is looking for an expression involving the variable k. For example:

fricas

product(sin(k),k=1..3)

$\label{eq3}{\sin \left({1}\right)}\ {\sin \left({2}\right)}\ {\sin \left({3}\right)}$

(3)

Type: Expression(Integer)

But despite appearances the symbol in in the previous command is not what FriCAS considers a variable. In FriCAS is assumed to be of type symbol, is assumed to be a variable but variables can be coerced to be a symbol. Only variables can be assigned values but any symbol can "stand for" unknown values in expressions.

fricas

p[k]

$\label{eq4}p_{k}$

(4)

Type: Symbol

fricas

1+p[k]

$\label{eq5}{p_{k}}+ 1$

(5)

Type: Polynomial(Integer)

fricas

p[k]:=1

   The form on the left hand side of an assignment must be a single 
      variable, a Tuple of variables or a reference to an entry in an 
      object supporting the setelt operation.

Understanding this distinction is somethimes important to understand why FriCAS does what it does. is assumed to be a polynomial over not .

In fact, much more complicated indexed symbols can be created in FriCAS. For example

fricas

P ==> script(p,[[a,b],[c],[d],[e]])

Type: Void

fricas

reduce(+, [P for a in 1..3])

$\label{eq6}{{}_{e}^{d}p_{3, \: b}^{c}}+{{}_{e}^{d}p_{2, \: b}^{c}}+{{}_{e}^{d}p_{1, \: b}^{c}}$

(6)

Type: Polynomial(Integer)

Within this MathAction FriCAS wiki it is also possible to create symbols that will be displayed as superscripted or subscripted variables which can be assigned values. In fact any LaTeX symbol can be generated by using the underscore charact to "protect" special charactes such as \. For example

fricas

Q:=p_^_\alpha___\beta

$\label{eq7}p^\alpha \<u> \beta$

(7)

Type: Variable(p^\alpha_\beta)

fricas

1+Q

$\label{eq8}p^\alpha \<u> \beta + 1$

(8)

Type: Polynomial(Integer)

In the above the first underscore protects the second underscore which is then handled by LaTeX as the beginning of a subscript and next underscore protects the \ which begins a Greek letter, etc. FriCAS treats the entire thing as a long variable name. Note however that this only works here and not within stand alone FriCAS.

Probably what you should write is

fricas

reduce(*,[1+p[k] for k in 1..2])

$\label{eq9}{{\left({p_{1}}+ 1 \right)}\ {p_{2}}}+{p_{1}}+ 1$

(9)

Type: Polynomial(Integer)

Here the operation of for, creating a sequence, does not treat:

  1+p[k]

as a polynomial over but the final result is a polynomial.

The same comments apply to sum() which represents a formal sum.

Alternatively, and more generally, one can use BasicOperator?

fricas

p:=operator 'p

$\label{eq10}p$

(10)

Type: BasicOperator?

fricas

p(1)

$\label{eq11}p \left({1}\right)$

(11)

Type: Expression(Integer)

fricas

p(k)

$\label{eq12}p \left({k}\right)$

(12)

Type: Expression(Integer)

fricas

product(1+p(k),k=1..2)

$\label{eq13}{{\left({p \left({1}\right)}+ 1 \right)}\ {p \left({2}\right)}}+{p \left({1}\right)}+ 1$

(13)

Type: Expression(Integer)

You can attach a display property so that the argument to p appears as a subscript

fricas

displayOfp(l:List OUTFORM):OUTFORM == subscript('p,l)

   Function declaration displayOfp : List(OutputForm) -> OutputForm has
      been added to workspace.

Type: Void

fricas

display(p,displayOfp)

fricas

Compiling function displayOfp with type List(OutputForm) -> 
      OutputForm

$\label{eq14}p$

(14)

Type: BasicOperator?

fricas

product(1+p(k),k=1..2)

$\label{eq15}{{\left({p_{1}}+ 1 \right)}\ {p_{2}}}+{p_{1}}+ 1$

(15)

Type: Expression(Integer)

You can also attach an evaluation property to handle particular values of the argument

fricas

evalOfp(a:EXPR INT):EXPR INT == {one? a => 1 ; kernel(p,a)}

   Function declaration evalOfp : Expression(Integer) -> Expression(
      Integer) has been added to workspace.

Type: Void

fricas

evaluate(p,evalOfp)

fricas

Compiling function evalOfp with type Expression(Integer) -> 
      Expression(Integer)

$\label{eq16}p$

(16)

Type: BasicOperator?

fricas

[p 1,p 2, p k]

$\label{eq17}\left[ 1, \:{p_{2}}, \:{p_{k}}\right]$

(17)

Type: List(Expression(Integer))

fricas

product(1+p(k),k=1..2)

$\label{eq18}{2 \ {p_{2}}}+ 2$

(18)

Type: Expression(Integer)

likewise for indexing into lists --Christian Sievers, Tue, 23 May 2006 07:02:10 -0500 reply

This is related:

fricas

l:=[1,2,4,8,16]

$\label{eq19}\left[ 1, \: 2, \: 4, \: 8, \:{16}\right]$

(19)

Type: List(PositiveInteger?)

fricas

sum(l.(1+2*k),k=0..2)

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

   Cannot find a definition or applicable library operation named l 
      with argument type(s) 
                             Polynomial(Integer)

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