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indexed variables

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Editor: 127.0.0.1 Time: 2007/11/15 20:12:49 GMT-8
Note: transferred from axiom-developer

changed:
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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:

\begin{axiom}
(1+p[1])*(1+p[2])
product(1+p[k],k=1..2)
\end{axiom}

But perhaps unexpectedly, the 2nd command is interpreted as just $(1+p_k)^2$.

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

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

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

\begin{axiom}
product(sin(k),k=1..3)
\end{axiom}

But despite appearances the symbol $k$ in $p[k]$ in the previous command
is not what Axiom considers a *variable*. In Axiom $p[k]$ is assumed to be
of type symbol, $k$ 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.

\begin{axiom}
p[k]
1+p[k]
p[k]:=1
\end{axiom}

Understanding this distinction is somethimes important to understand why
Axiom does what it does. $1+p[k]$ is assumed to be a polynomial over $p[k]$
not $k$.

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

\begin{axiom}
P ==> script(p,[[a,b],[c],[d],[e]])
reduce(+, [P for a in 1..3])
\end{axiom}

Within this MathAction Axiom 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

\begin{axiom}
Q:=p_^_\alpha___\beta
1+Q
\end{axiom}

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. Axiom treats the entire
thing as a long variable name. Note however that this only
works here and not within stand alone Axiom.

Probably what you should write is

\begin{axiom}
reduce(*,[1+p[k] for k in 1..2])
\end{axiom}

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

  1+p[k]

as a polynomial over $p[k]$ 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
\begin{axiom}
p:=operator 'p
p(1)
p(k)
product(1+p(k),k=1..2)
\end{axiom}

You can attach a display property so that the argument to p appears as a subscript
\begin{axiom}
displayOfp(l:List OUTFORM):OUTFORM == subscript('p,l)
display(p,displayOfp)
product(1+p(k),k=1..2)
\end{axiom}


You can also attach an evaluation property to handle particular values of the argument
\begin{axiom}
evalOfp(a:EXPR INT):EXPR INT == {one? a => 1 ; kernel(p,a)}
evaluate(p,evalOfp)
[p 1,p 2, p k]
product(1+p(k),k=1..2)
\end{axiom}

From ChristianSievers Tue May 23 07:02:10 -0500 2006
From: Christian Sievers
Date: Tue, 23 May 2006 07:02:10 -0500
Subject: likewise for indexing into lists
Message-ID: <20060523070210-0500@wiki.axiom-developer.org>

This is related:

\begin{axiom}
l:=[1,2,4,8,16]
sum(l.(1+2*k),k=0..2)
sum(elt(l,1+2*k),k=0..2)
reduce(+,[l.(1+2*k) for k in 0..2])
\end{axiom}

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+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 Axiom 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 Axiom considers a variable. In Axiom 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 Axiom does what it does. is assumed to be a polynomial over not .

In fact, much more complicated indexed symbols can be created in Axiom. 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? Axiom 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. Axiom treats the entire thing as a long variable name. Note however that this only works here and not within stand alone Axiom.

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.