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Editor: page Time: 2007/11/19 17:54:25 GMT-8
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From unknown Wed Dec 7 09:11:18 -0600 2005
From: unknown
Date: Wed, 07 Dec 2005 09:11:18 -0600
Subject: szsz
Message-ID: <20051207091118-0600@wiki.axiom-developer.org>

Works with ASCII text output formatting.
\begin{axiom}
)set output tex off
)set output algebra on
\end{axiom}
\begin{axiom}
    solve([x^2 + y^2 - 2*(ax*x + ay*y) = l1, x^2 + y^2 - 2*(cx*x + cy*y) = l2],[x,y])
\end{axiom}

But fails with LaTeX.
\begin{axiom}
)set output tex on
)set output algebra off
\end{axiom}


From kratt6 Tue Jan 3 05:22:52 -0600 2006
From: kratt6
Date: Tue, 03 Jan 2006 05:22:52 -0600
Subject: 0**0
Message-ID: <20060103052252-0600@wiki.axiom-developer.org>

The result of '0**0' depends on the type of '0':

\begin{axiom}
  (0::Float)**(0::Float)
\end{axiom}

The idea was, that defining $0^0$ as 1 is ok whenever there is no notion of limit. However, 

\begin{axiom}
  (0::EXPR INT)**(0::EXPR INT)
\end{axiom}

is not quite in line with this, I think.
There has been some discussion on this subject on axiom-developer.

It is easy to change this behaviour, if we know better...

From unknown Sun Jan 29 13:09:07 -0600 2006
From: unknown
Date: Sun, 29 Jan 2006 13:09:07 -0600
Subject: ruleset
Message-ID: <20060129130907-0600@wiki.axiom-developer.org>

Let's see if the same happens here:
\begin{axiom}
sinCosProducts := rule
  sin (x) * sin (y) == (cos(x-y) - cos(x+y))/2
  cos (x) * cos (y) == (cos(x-y) + cos(x+y))/2
  sin (x) * cos (y) == (sin(x-y) + sin(x+y))/2
\end{axiom}


From BillPage Mon Jan 30 09:00:02 -0600 2006
From: Bill Page
Date: Mon, 30 Jan 2006 09:00:02 -0600
Subject: when typing
Message-ID: <20060130090002-0600@wiki.axiom-developer.org>

When you are typing or when you cut-and-paste commands directly
into the Axiom interpreter you must use an underscore character at
the end of each incomplete line, and you must use the ( ) syntax
instead of identation, like this::

  sinCosProducts := rule (_
  sin (x) * sin (y) == (cos(x-y) - cos(x+y))/2; _
  cos (x) * cos (y) == (cos(x-y) + cos(x+y))/2; _
  sin (x) * cos (y) == (sin(x-y) + sin(x+y))/2)

Alternatively, using a text editor you can enter the commands into a
file called, for example 'sincos.input' exactly as in MathActon above
and the use the command::

  )read sincos.input


From unknown Sat Mar 11 13:19:44 -0600 2006
From: unknown
Date: Sat, 11 Mar 2006 13:19:44 -0600
Subject: 
Message-ID: <20060311131944-0600@wiki.axiom-developer.org>

\begin{axiom}
)lib RINTERPA RINTERP PCDEN GUESS GUESSINT GUESSP
guess(n, [1, 5, 14, 34, 69, 135, 240, 416, 686, 1106], n+->n, [guessRat], [guessSum, guessProduct, guessOne],2)$GuessInteger
\end{axiom}


From BillPage Thu Mar 23 22:21:41 -0600 2006
From: Bill Page
Date: Thu, 23 Mar 2006 22:21:41 -0600
Subject: conversion failed
Message-ID: <20060323222141-0600@wiki.axiom-developer.org>

Unknown wrote:

> z:=sum(myfn(x),x=1..10) -- This fails, why?

The reason this fails is because Axiom tries to evaluate
'myfn(x)' **first**. But 'x' is not yet an 'Integer' so Axiom
cannot compute 'myfn(x)'. I guess you were expecting Axiom
to "wait" and not evaluate 'myfn(x)' until after 'x' has
been assigned the value 1, right? But Axiom does not work
this way.

The solution is to write 'myfn(x)' so that is can be applied
to something symbolic like 'x'. For example something this:

\begin{axiom}
myfn(i : Expression Integer) : Expression Integer == i
myfn(x)
z:=sum(myfn(x),x=1..10)
\end{axiom}

From Bill(Nameomitted) Fri Mar 24 21:45:25 -0600 2006
From: Bill (Name omitted)
Date: Fri, 24 Mar 2006 21:45:25 -0600
Subject: Any hints for multivariate functions?
Message-ID: <20060324214525-0600@wiki.axiom-developer.org>
In-Reply-To: <20060323222141-0600@wiki.axiom-developer.org>

Hi Bill:

Thanks for your quick response.  I tried to respond to this earlier, but didn't see it in the sand box, please forgive me if you get multiple copies.

I tried to simplify the code from my original program, and generated a univariate function, however my actual code has a multivariate function,
and your excellent hint on the use of the Expression qualifier on the parameter and return type which works great for the univariate function case appears to fail for multivarite functions.
Please consider the following example.
\begin{axiom}
a(n : Expression Integer, k : Expression Integer, p : Expression Float) : Expression Float == binomial(n,k) * p**(k) * (1.0-p)**(n-k)
output(a(4,3,0.25)) -- see that the function actually evaluates for sensible values
z := sum(a(4,i,0.25), i=1..3) --- this fails
output(z)
\end{axiom}

I did notice in the Axiom online book, chapter 6.6, around page 241, the recommendation to use untyped functions, which appears to allow Axiom to do
inference on parameter and result type.
\begin{axiom}
b(n, k, p) == binomial(n,k) * p**(k) * (1.0-p)**(n-k)
output(b(4,3,0.25)) -- see that the function actually evaluates for sensible values
z := sum(b(4,i,0.25), i=1..3) --- this fails
output(z)
\end{axiom}
For univariate functions the approach
\begin{axiom}
c(k) == binomial(4,k) * 0.25**k * (1.0 - 0.25)**(4-k) -- This approach is only a test, but is not suitable for my program
output(c(3)) -- test to see if function can be evaluated for sensible arguments
z := sum(c(i), i=1..3) -- still doesn't work
output(z)
\end{axiom}
But interestingly something like
\begin{axiom}
d(k) == 1.5 * k -- coerce uotput to be a Float
z := sum(d(i), i=1..3) -- This works!
output(z)
\end{axiom}
Bill, thanks again for your quick help, unforutnatly I lack a local Axiom expert, any ideas would really be welcome here.

From billpage Sat Mar 25 16:01:07 -0600 2006
From: billpage
Date: Sat, 25 Mar 2006 16:01:07 -0600
Subject: reduce(+,[...]) = sum(...)
Message-ID: <20060325160107-0600@wiki.axiom-developer.org>

Try this
\begin{axiom}
z := reduce(+,[b(4,i,0.25) for  i in 1..3])
\end{axiom}


From Bill(Nameomitted) Mon Mar 27 08:10:26 -0600 2006
From: Bill (Name omitted)
Date: Mon, 27 Mar 2006 08:10:26 -0600
Subject: Handling the result from functions returning a matrix
Message-ID: <20060327081026-0600@wiki.axiom-developer.org>
In-Reply-To: <20060325160107-0600@wiki.axiom-developer.org>

Hi all:

Thanks Bill Page for your help, it is much appreciated (although I used a for loop and not reduce :-)).

I'm having a bit of difficulty getting a Function returning a matrix to work as expected,
perhaps it is just cockpit error, but I don't see the error of my ways.
\begin{axiom}
CFM(Q : Matrix(Float)): Matrix(Float) ==
   x := nrows(Q)
   MyIdentityMatrix : Matrix(Float) := new(x, x, 0)
   for i in 1..nrows(MyIdentityMatrix) repeat
      MyIdnetityMatrix(i,i) := 1.0
   Ninv := MyIdnetityMatrix - Q
   N := inverse(Ninv)
   N

--test ComputeFundamentalMatrix
X := matrix[[0, 0.5, 0],[0.5, 0, 0.5],[0, 0.5, 0]]
output(X)
N := CFM(X)
output(N)
\end{axiom}
Any ideas where I'm blowing it here?  I tried explicitly setting N to be a Matrix type but that failed too.
\begin{axiom}
CFM(Q : Matrix(Float)): Matrix(Float) ==
   x := nrows(Q)
   MyIdentityMatrix : Matrix(Float) := new(x, x, 0)
   for i in 1..nrows(MyIdentityMatrix) repeat
      MyIdnetityMatrix(i,i) := 1.0
   Ninv := MyIdnetityMatrix - Q
   N := inverse(Ninv)
   N

--test ComputeFundamentalMatrix
X := matrix[[0, 0.5, 0],[0.5, 0, 0.5],[0, 0.5, 0]]
output(X)
N : Matrix(Float) := CFM(X)
output(N)
\end{axiom}

Thanks again for all your help.

Regards:

Bill M. (Sorry, my unique last name attracts too much spam).

From billpage Mon Mar 27 09:34:35 -0600 2006
From: billpage
Date: Mon, 27 Mar 2006 09:34:35 -0600
Subject: typo and identity
Message-ID: <20060327093435-0600@wiki.axiom-developer.org>

> although I used a for loop and not reduce :-)

Good thinking. ;)

You have a simple typographical error. You have written both::

  MyIdentityMatrix

and ::

  MyIdnetityMatrix

BTW, instead of the complicated construction of the identify matrix
you should just write::

  Ninv := 1 - Q

For matrices '1' denotes the identity.


From unknown Sat Apr 8 10:45:29 -0500 2006
From: unknown
Date: Sat, 08 Apr 2006 10:45:29 -0500
Subject: 
Message-ID: <20060408104529-0500@wiki.axiom-developer.org>

\begin{axiom}
)set output tex off
)set output algebra on
FunFun := x**4 - 6* x**3 + 11* x*x + 2* x + 1
radicalSolve(FunFun)
)set output tex on
)set output algebra off

\end{axiom}

Matthias

\begin{axiom}
t:=matrix ([[0,1,1],[1,-2,2],[1,2,-1]])
\end{axiom}

We cat diagonalise t by finding it's eigenvalues.
\begin{axiom}
)set output tex off
)set output algebra on
e:=radicalEigenvectors(t)
d:=diagonalMatrix([e.1.radval,e.2.radval,e.3.radval])
\end{axiom}

Now prove it by constructing the simularity transformation
from the eigenvectors:
\begin{axiom}
p:=horizConcat(horizConcat(e.1.radvect.1,e.2.radvect.1),e.3.radvect.1)
p*d*inverse(p)
)set output tex on
)set output algebra off
\end{axiom}
\end{axiom}


From unknown Fri Apr 28 14:03:28 -0500 2006
From: unknown
Date: Fri, 28 Apr 2006 14:03:28 -0500
Subject: Axiom can't integrame exp(x^4) ;(
Message-ID: <20060428140328-0500@wiki.axiom-developer.org>

Axiom can't integrame exp(x^4) ;(

\begin{axiom}
integrate(exp(x**4),x)
\end{axiom}

But Maple can...

\begin{axiom}
f(x) == (1/4)*x*(-Gamma(1/4,-x**4)*Gamma(3/4)+%pi*sqrt(2))/((-x**4)**(1/4)*Gamma(3/4))
D(f(x),x)
\end{axiom}

From kratt6 Fri Apr 28 16:34:16 -0500 2006
From: kratt6
Date: Fri, 28 Apr 2006 16:34:16 -0500
Subject: Axiom cannot integrate 'e^(4*x)'
Message-ID: <20060428163416-0500@wiki.axiom-developer.org>

This is not a big surprise: note that 'Gamma(x,y)' is not an elementary function.

Martin


From unknown Thu May 18 11:30:21 -0500 2006
From: unknown
Date: Thu, 18 May 2006 11:30:21 -0500
Subject: 
Message-ID: <20060518113021-0500@wiki.axiom-developer.org>

This is both obviously wrong since the integrand is a positive function:
\begin{axiom}
integrate(1/(1+x^4),x=%minusInfinity..%plusInfinity)
numeric(integrate(1/(1+x^4),x=0..1))
\end{axiom}


From unknown Wed May 24 04:31:40 -0500 2006
From: unknown
Date: Wed, 24 May 2006 04:31:40 -0500
Subject: 
Message-ID: <20060524043140-0500@wiki.axiom-developer.org>

\begin{axiom}
)clear co
n := 32

y : FARRAY INT := new(n,1)

n0 := n
n1 := sum(x^1, x=0..n-1)
n2 := sum(x^2, x=0..n-1)
n3 := sum(x^3, x=0..n-1)
n4 := sum(x^4, x=0..n-1)

A := matrix([[n4, n3, n2],_
            [n3, n2, n1],_
            [n2, n1, n0]])

X := vector([x1, x2, x3])
B := vector([sum(x^2* u, x=0..n-1),_
       sum(x*   v, x=0..n-1),_
       sum(     w, x=0..n-1)])

solve([A * X = B], [x1, x2, x3])
\end{axiom}


From unknown Tue May 30 23:51:26 -0500 2006
From: unknown
Date: Tue, 30 May 2006 23:51:26 -0500
Subject: can this be correct?
Message-ID: <20060530235126-0500@wiki.axiom-developer.org>

\begin{axiom}
integrate(1/((x+t)*sqrt(1+(x*t)**2)),t=0..%plusInfinity,"noPole")
subst(%,x=1)
integrate(1/((1+t)*sqrt(1+(1*t)**2)),t=0..%plusInfinity,"noPole")
simplify(%-subst((asinh(x^2)+asinh(1/x^2))/sqrt(1+x^4),x=1))
%::Expression Float
\end{axiom}


From unknown Mon Jul 3 02:07:02 -0500 2006
From: unknown
Date: Mon, 03 Jul 2006 02:07:02 -0500
Subject: 
Message-ID: <20060703020702-0500@wiki.axiom-developer.org>

\begin{axiom}
a := matrix([ [-1,0,0,0,1,0], [0,1,0,0,0,0], [0,0,2,0,0,-2], [0,0,0,4,0,0], [0,0,0,0,3,0], [0,0,-3,0,0,3]]) 
determinant(a)
inverse(a)
\end{axiom}


From unknown Fri Jul 7 11:54:52 -0500 2006
From: unknown
Date: Fri, 07 Jul 2006 11:54:52 -0500
Subject: 
Message-ID: <20060707115452-0500@wiki.axiom-developer.org>

a := matrix([ [-3,1,1,1], [1,1,1,1], [1,1,1,1], [1,1,1,1]])


From unknown Fri Jul 7 13:24:57 -0500 2006
From: unknown
Date: Fri, 07 Jul 2006 13:24:57 -0500
Subject: 
Message-ID: <20060707132457-0500@wiki.axiom-developer.org>
In-Reply-To: <20060707115452-0500@wiki.axiom-developer.org>

\begin{axiom}
As := matrix([ [-3,1,1,1], [1,1,1,1], [1,1,1,1], [1,1,1,1]])
A := subMatrix(As, 2,4,2,4)
ob := orthonormalBasis(A)
P : Matrix(Expression Integer) := new(3,3,0)
setsubMatrix!(P,1,1,ob.3) 
setsubMatrix!(P,1,2,ob.1) 
setsubMatrix!(P,1,3,ob.2)
Pt := transpose(P)
Ps : Matrix(Expression Integer) := new(4,4,0)
Ps(1,1) := 1
setsubMatrix!(Ps,2,2,P)
PsT := transpose(Ps)
PsTAsPs := PsT * As * Ps
b1 := PsTAsPs(2,1)
l1 := PsTAsPs(2,2)
Us : Matrix(Expression Integer) := new(4,4,0)
Us(1,1) := 1
Us(2,2) := 1
Us(3,3) := 1
Us(4,4) := 1
Us(2,1) := -b1 / l1
PsUs := Ps * Us
PsUsT := transpose(PsUs)
PsUsTAsPsUs := PsUsT * As * PsUs
C := inverse(PsUs) 
c := PsUsTAsPsUs(1,1)
gQ := PsUsTAsPsUs / c 
x1 := transpose(matrix([[1,2,3,4]]))
v1 := transpose(x1) * As * x1
x2 := C * x1
v2 := transpose(x2) * PsUsTAsPsUs * x2
\end{axiom}


From unknown Tue Aug 1 02:17:12 -0500 2006
From: unknown
Date: Tue, 01 Aug 2006 02:17:12 -0500
Subject: graphics
Message-ID: <20060801021712-0500@wiki.axiom-developer.org>

\begin{axiom}
draw(y**2/2+(x**2-1)**2/4-1=0, x,y, range ==[-2..2, -1..1])
\end{axiom}


From greg Sat Feb 3 09:35:50 -0600 2007
From: greg
Date: Sat, 03 Feb 2007 09:35:50 -0600
Subject: series test 
Message-ID: <20070203093550-0600@wiki.axiom-developer.org>

\begin{axiom}
f1 := taylor(1 - x**2,x = 0)
asin f1
sin %
\end{axiom}

SandboxMSkuce
\begin{axiom}
1+1
\end{axiom}

From jhnbk Fri Jun 8 03:50:35 -0500 2007
From: jhnbk
Date: Fri, 08 Jun 2007 03:50:35 -0500
Subject: integration
Message-ID: <20070608035035-0500@wiki.axiom-developer.org>

\begin{axiom}
integrate((x-1)/log(x), x)
integrate(x*exp(x)*sin(x),x)
\end{axiom} 

From daneshpajouh Sat Jun 16 07:00:00 -0500 2007
From: daneshpajouh
Date: Sat, 16 Jun 2007 07:00:00 -0500
Subject: Working With Lists
Message-ID: <20070616070000-0500@wiki.axiom-developer.org>

\begin{axiom}
[p for p in primes(2,1000)|(p rem 16)=1]
[p**2+1 for p in primes(2,100)]
\end{axiom}

\begin {axiom}
integrate (2x^2 + 2x, x)
\end {axiom} 


From vv Sat Jul 28 14:00:27 -0500 2007
From: vv
Date: Sat, 28 Jul 2007 14:00:27 -0500
Subject: is it error?
Message-ID: <20070728140027-0500@wiki.axiom-developer.org>
In-Reply-To: <20070616070000-0500@wiki.axiom-developer.org>

\begin{axiom}
radix(36,37)
\end{axiom}

Is it error?


From pbwagner Mon Sep 10 13:01:48 -0500 2007
From: pbwagner
Date: Mon, 10 Sep 2007 13:01:48 -0500
Subject: (better) example (with axiom markers this time) ;-)
Message-ID: <20070910130148-0500@wiki.axiom-developer.org>

\begin{axiom}
integrate(log(log(x)),x)
\end{axiom}

szsz --unknown, Wed, 07 Dec 2005 09:11:18 -0600 reply

Works with ASCII text output formatting.

axiom

)set output tex off

axiom

)set output algebra on

axiom

solve([x^2 + y^2 - 2*(ax*x + ay*y) = l1, x^2 + y^2 - 2*(cx*x + cy*y) = l2],[x,y])

   (1)
   [
         (- 2cy + 2ay)y - l2 + l1
     [x= ------------------------,
                 2cx - 2ax

             2               2               2      2  2
         (4cy  - 8ay cy + 4cx  - 8ax cx + 4ay  + 4ax )y
       + 
                                                            2            2
             (4cy - 4ay)l2 + (- 4cy + 4ay)l1 + (8ax cx - 8ax )cy - 8ay cx
           + 
             8ax ay cx
        *
           y
       + 
           2                        2        2         2
         l2  + (- 2l1 + 4ax cx - 4ax )l2 + l1  + (- 4cx  + 4ax cx)l1
         =
         0
       ]
     ]

Type: List(List(Equation(Fraction(Polynomial(Integer)))))

But fails with LaTeX?.

axiom

)set output tex on

axiom

)set output algebra off

0**0 --kratt6, Tue, 03 Jan 2006 05:22:52 -0600 reply

The result of 0**0 depends on the type of '0':

axiom

(0::Float)**(0::Float)

   >> Error detected within library code:
   0^0 is undefined

The idea was, that defining as 1 is ok whenever there is no notion of limit. However,

axiom

(0::EXPR INT)**(0::EXPR INT)

$\label{eq1}1$

(1)

Type: Expression(Integer)

is not quite in line with this, I think. There has been some discussion on this subject on axiom-developer.

It is easy to change this behaviour, if we know better...

ruleset --unknown, Sun, 29 Jan 2006 13:09:07 -0600 reply

Let's see if the same happens here:

axiom

sinCosProducts := rule
  sin (x) * sin (y) == (cos(x-y) - cos(x+y))/2
  cos (x) * cos (y) == (cos(x-y) + cos(x+y))/2
  sin (x) * cos (y) == (sin(x-y) + sin(x+y))/2

$\label{eq2}\begin{array}{@{}l} \displaystyle \left\{{{\%B \ {\sin \left({x}\right)}\ {\sin \left({y}\right)}}\mbox{\rm = =}{{-{\%B \ {\cos \left({y + x}\right)}}+{\%B \ {\cos \left({y - x}\right)}}}\over 2}}, \: \right. \ \ \displaystyle \left.{{\%C \ {\cos \left({x}\right)}\ {\cos \left({y}\right)}}\mbox{\rm = =}{{{\%C \ {\cos \left({y + x}\right)}}+{\%C \ {\cos \left({y - x}\right)}}}\over 2}}, \: \right. \ \ \displaystyle \left.{{\%D \ {\cos \left({y}\right)}\ {\sin \left({x}\right)}}\mbox{\rm = =}{{{\%D \ {\sin \left({y + x}\right)}}-{\%D \ {\sin \left({y - x}\right)}}}\over 2}}\right\}$

(2)

Type: Ruleset(Integer,Integer,Expression(Integer))

when typing --Bill Page, Mon, 30 Jan 2006 09:00:02 -0600 reply

When you are typing or when you cut-and-paste commands directly into the Axiom interpreter you must use an underscore character at the end of each incomplete line, and you must use the ( ) syntax instead of identation, like this:

  sinCosProducts := rule (_
  sin (x) * sin (y) == (cos(x-y) - cos(x+y))/2; _
  cos (x) * cos (y) == (cos(x-y) + cos(x+y))/2; _
  sin (x) * cos (y) == (sin(x-y) + sin(x+y))/2)

Alternatively, using a text editor you can enter the commands into a file called, for example sincos.input exactly as in MathActon? above and the use the command:

  )read sincos.input

... --unknown, Sat, 11 Mar 2006 13:19:44 -0600 reply

axiom

)lib RINTERPA RINTERP PCDEN GUESS GUESSINT GUESSP

   RationalInterpolationAlgorithms is now explicitly exposed in frame 
      initial 
   RationalInterpolationAlgorithms will be automatically loaded when 
      needed from /var/zope2/var/LatexWiki/RINTERPA.NRLIB/RINTERPA
   RationalInterpolation is now explicitly exposed in frame initial 
   RationalInterpolation will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/RINTERP.NRLIB/RINTERP
   PolynomialCommonDenominator is now explicitly exposed in frame 
      initial 
   PolynomialCommonDenominator will be automatically loaded when needed
      from /var/zope2/var/LatexWiki/PCDEN.NRLIB/PCDEN
   Guess is now explicitly exposed in frame initial 
   Guess will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/GUESS.NRLIB/GUESS
   GuessInteger is now explicitly exposed in frame initial 
   GuessInteger will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/GUESSINT.NRLIB/GUESSINT
   GuessPolynomial is now explicitly exposed in frame initial 
   GuessPolynomial will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/GUESSP.NRLIB/GUESSP
guess(n, [1, 5, 14, 34, 69, 135, 240, 416, 686, 1106], n+->n, [guessRat], [guessSum, guessProduct, guessOne],2)$GuessInteger

   The function guess is not implemented in GuessInteger .

conversion failed --Bill Page, Thu, 23 Mar 2006 22:21:41 -0600 reply

Unknown wrote:

z:=sum(myfn(x),x=1..10) -- This fails, why?

The reason this fails is because Axiom tries to evaluate myfn(x) first. But x is not yet an Integer so Axiom cannot compute myfn(x). I guess you were expecting Axiom to "wait" and not evaluate myfn(x) until after x has been assigned the value 1, right? But Axiom does not work this way.

The solution is to write myfn(x) so that is can be applied to something symbolic like x. For example something this:

axiom

myfn(i : Expression Integer) : Expression Integer == i

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

Type: Void

axiom

myfn(x)

axiom

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

$\label{eq3}x$

(3)

Type: Expression(Integer)

axiom

z:=sum(myfn(x),x=1..10)

$\label{eq4}55$

(4)

Type: Expression(Integer)

Any hints for multivariate functions? --Bill (Name omitted), Fri, 24 Mar 2006 21:45:25 -0600 reply

Hi Bill:

Thanks for your quick response. I tried to respond to this earlier, but didn't see it in the sand box, please forgive me if you get multiple copies.

I tried to simplify the code from my original program, and generated a univariate function, however my actual code has a multivariate function, and your excellent hint on the use of the Expression qualifier on the parameter and return type which works great for the univariate function case appears to fail for multivarite functions. Please consider the following example.

axiom

a(n : Expression Integer, k : Expression Integer, p : Expression Float) : Expression Float == binomial(n,k) * p**(k) * (1.0-p)**(n-k)

   Function declaration a : (Expression(Integer),Expression(Integer),
      Expression(Float)) -> Expression(Float) has been added to 
      workspace.

Type: Void

axiom

output(a(4,3,0.25)) -- see that the function actually evaluates for sensible values

axiom

Compiling function a with type (Expression(Integer),Expression(
      Integer),Expression(Float)) -> Expression(Float) 
   0.046875

Type: Void

axiom

z := sum(a(4,i,0.25), i=1..3) --- this fails

   There are 6 exposed and 2 unexposed library operations named sum 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                               )display op sum
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named sum 
      with argument type(s) 
                              Expression(Float)
                       SegmentBinding(PositiveInteger)

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

   55

Type: Void

I did notice in the Axiom online book, chapter 6.6, around page 241, the recommendation to use untyped functions, which appears to allow Axiom to do inference on parameter and result type.

axiom

b(n, k, p) == binomial(n,k) * p**(k) * (1.0-p)**(n-k)

Type: Void

axiom

output(b(4,3,0.25)) -- see that the function actually evaluates for sensible values

axiom

Compiling function b with type (PositiveInteger,PositiveInteger,
      Float) -> Float 
   0.046875

Type: Void

axiom

z := sum(b(4,i,0.25), i=1..3) --- this fails

axiom

Compiling function b with type (PositiveInteger,Variable(i),Float)
       -> Expression(Float) 
   There are 6 exposed and 2 unexposed library operations named sum 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                               )display op sum
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named sum 
      with argument type(s) 
                              Expression(Float)
                       SegmentBinding(PositiveInteger)

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

   55

Type: Void

For univariate functions the approach

axiom

c(k) == binomial(4,k) * 0.25**k * (1.0 - 0.25)**(4-k) -- This approach is only a test, but is not suitable for my program

Type: Void

axiom

output(c(3)) -- test to see if function can be evaluated for sensible arguments

axiom

Compiling function c with type PositiveInteger -> Float 
   0.046875

Type: Void

axiom

z := sum(c(i), i=1..3) -- still doesn't work

axiom

Compiling function c with type Variable(i) -> Expression(Float) 
   There are 6 exposed and 2 unexposed library operations named sum 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                               )display op sum
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named sum 
      with argument type(s) 
                              Expression(Float)
                       SegmentBinding(PositiveInteger)

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

   55

Type: Void

But interestingly something like

axiom

d(k) == 1.5 * k -- coerce uotput to be a Float

Type: Void

axiom

z := sum(d(i), i=1..3) -- This works!

axiom

Compiling function d with type Variable(i) -> Polynomial(Float)

$\label{eq5}9.0$

(5)

Type: Fraction(Polynomial(Float))

axiom

output(z)

   9.0

Type: Void

Bill, thanks again for your quick help, unforutnatly I lack a local Axiom expert, any ideas would really be welcome here.

reduce(+,[...]?) = sum(...) --billpage, Sat, 25 Mar 2006 16:01:07 -0600 reply

Try this

axiom

z := reduce(+,[b(4,i,0.25) for  i in 1..3])

$\label{eq6}0.6796875$

(6)

Type: Float

Handling the result from functions returning a matrix --Bill (Name omitted), Mon, 27 Mar 2006 08:10:26 -0600 reply

Hi all:

Thanks Bill Page for your help, it is much appreciated (although I used a for loop and not reduce :-)).

I'm having a bit of difficulty getting a Function returning a matrix to work as expected, perhaps it is just cockpit error, but I don't see the error of my ways.

axiom

CFM(Q : Matrix(Float)): Matrix(Float) ==
   x := nrows(Q)
   MyIdentityMatrix : Matrix(Float) := new(x, x, 0)
   for i in 1..nrows(MyIdentityMatrix) repeat
      MyIdnetityMatrix(i,i) := 1.0
   Ninv := MyIdnetityMatrix - Q
   N := inverse(Ninv)
   N

   Function declaration CFM : Matrix(Float) -> Matrix(Float) has been 
      added to workspace.

Type: Void

axiom

--test ComputeFundamentalMatrix
X := matrix[[0, 0.5, 0],[0.5, 0, 0.5],[0, 0.5, 0]]

$\label{eq7}\left[ \begin{array}{ccc} {0.0}&{0.5}&{0.0} \ {0.5}&{0.0}&{0.5} \ {0.0}&{0.5}&{0.0}$

(7)

Type: Matrix(Float)

axiom

output(X)

   +0.0  0.5  0.0+
   |             |
   |0.5  0.0  0.5|
   |             |
   +0.0  0.5  0.0+

Type: Void

axiom

N := CFM(X)

   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.
output(N)

   N

Type: Void

Any ideas where I'm blowing it here? I tried explicitly setting N to be a Matrix type but that failed too.

axiom

CFM(Q : Matrix(Float)): Matrix(Float) ==
   x := nrows(Q)
   MyIdentityMatrix : Matrix(Float) := new(x, x, 0)
   for i in 1..nrows(MyIdentityMatrix) repeat
      MyIdnetityMatrix(i,i) := 1.0
   Ninv := MyIdnetityMatrix - Q
   N := inverse(Ninv)
   N

   Function declaration CFM : Matrix(Float) -> Matrix(Float) has been 
      added to workspace.
   Compiled code for CFM has been cleared.
   1 old definition(s) deleted for function or rule CFM

Type: Void

axiom

--test ComputeFundamentalMatrix
X := matrix[[0, 0.5, 0],[0.5, 0, 0.5],[0, 0.5, 0]]

$\label{eq8}\left[ \begin{array}{ccc} {0.0}&{0.5}&{0.0} \ {0.5}&{0.0}&{0.5} \ {0.0}&{0.5}&{0.0}$

(8)

Type: Matrix(Float)

axiom

output(X)

   +0.0  0.5  0.0+
   |             |
   |0.5  0.0  0.5|
   |             |
   +0.0  0.5  0.0+

Type: Void

axiom

N : Matrix(Float) := CFM(X)

   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.
output(N)

   N is declared as being in Matrix(Float) but has not been given a 
      value.

Thanks again for all your help.

Regards:

Bill M. (Sorry, my unique last name attracts too much spam).

typo and identity --billpage, Mon, 27 Mar 2006 09:34:35 -0600 reply

although I used a for loop and not reduce :-)

Good thinking. ;)

You have a simple typographical error. You have written both:

  MyIdentityMatrix

and :

  MyIdnetityMatrix

BTW, instead of the complicated construction of the identify matrix you should just write:

  Ninv := 1 - Q

For matrices 1 denotes the identity.

... --unknown, Sat, 08 Apr 2006 10:45:29 -0500 reply

axiom

)set output tex off

axiom

)set output algebra on

FunFun := x**4 - 6* x**3 + 11* x*x + 2* x + 1

          4     3      2
   (27)  x  - 6x  + 11x  + 2x + 1

Type: Polynomial(Integer)

axiom

radicalSolve(FunFun)

   (28)
   [
     x =
           -
              ROOT
                              +---------------------+2
                              |     +-+       +----+
                              |2069\|3  + 144\|- 79
                         - 9  |---------------------
                             3|           +-+
                             \|        27\|3
                       + 
                             +---------------------+
                             |     +-+       +----+
                             |2069\|3  + 144\|- 79
                         30  |--------------------- - 169
                            3|           +-+
                            \|        27\|3
                    *
                       ROOT
                                 +---------------------+2
                                 |     +-+       +----+
                                 |2069\|3  + 144\|- 79
                              9  |---------------------
                                3|           +-+
                                \|        27\|3
                            + 
                                  +---------------------+
                                  |     +-+       +----+
                                  |2069\|3  + 144\|- 79
                              15  |--------------------- + 169
                                 3|           +-+
                                 \|        27\|3
                         /
                               +---------------------+
                               |     +-+       +----+
                               |2069\|3  + 144\|- 79
                            9  |---------------------
                              3|           +-+
                              \|        27\|3
                   + 
                            +---------------------+
                            |     +-+       +----+
                            |2069\|3  + 144\|- 79
                     - 144  |---------------------
                           3|           +-+
                           \|        27\|3
                /
                        +---------------------+
                        |     +-+       +----+
                        |2069\|3  + 144\|- 79
                     9  |---------------------
                       3|           +-+
                       \|        27\|3
                  *
                     ROOT
                               +---------------------+2
                               |     +-+       +----+
                               |2069\|3  + 144\|- 79
                            9  |---------------------
                              3|           +-+
                              \|        27\|3
                          + 
                                +---------------------+
                                |     +-+       +----+
                                |2069\|3  + 144\|- 79
                            15  |--------------------- + 169
                               3|           +-+
                               \|        27\|3
                       /
                             +---------------------+
                             |     +-+       +----+
                             |2069\|3  + 144\|- 79
                          9  |---------------------
                            3|           +-+
                            \|        27\|3
         + 
            +-------------------------------------------------------------+
            |   +---------------------+2      +---------------------+
            |   |     +-+       +----+        |     +-+       +----+
            |   |2069\|3  + 144\|- 79         |2069\|3  + 144\|- 79
            |9  |---------------------  + 15  |--------------------- + 169
            |  3|           +-+              3|           +-+
            |  \|        27\|3               \|        27\|3
            |-------------------------------------------------------------  + 3
            |                     +---------------------+
            |                     |     +-+       +----+
            |                     |2069\|3  + 144\|- 79
            |                  9  |---------------------
            |                    3|           +-+
           \|                    \|        27\|3
      /
         2
     ,

     x =
           ROOT
                           +---------------------+2      +---------------------+
                           |     +-+       +----+        |     +-+       +----+
                           |2069\|3  + 144\|- 79         |2069\|3  + 144\|- 79
                      - 9  |---------------------  + 30  |---------------------
                          3|           +-+              3|           +-+
                          \|        27\|3               \|        27\|3
                    + 
                      - 169
                 *
                    ROOT
                              +---------------------+2
                              |     +-+       +----+
                              |2069\|3  + 144\|- 79
                           9  |---------------------
                             3|           +-+
                             \|        27\|3
                         + 
                               +---------------------+
                               |     +-+       +----+
                               |2069\|3  + 144\|- 79
                           15  |--------------------- + 169
                              3|           +-+
                              \|        27\|3
                      /
                            +---------------------+
                            |     +-+       +----+
                            |2069\|3  + 144\|- 79
                         9  |---------------------
                           3|           +-+
                           \|        27\|3
                + 
                         +---------------------+
                         |     +-+       +----+
                         |2069\|3  + 144\|- 79
                  - 144  |---------------------
                        3|           +-+
                        \|        27\|3
             /
                     +---------------------+
                     |     +-+       +----+
                     |2069\|3  + 144\|- 79
                  9  |---------------------
                    3|           +-+
                    \|        27\|3
               *
                 +-------------------------------------------------------------+
                 |   +---------------------+2      +---------------------+
                 |   |     +-+       +----+        |     +-+       +----+
                 |   |2069\|3  + 144\|- 79         |2069\|3  + 144\|- 79
                 |9  |---------------------  + 15  |--------------------- + 169
                 |  3|           +-+              3|           +-+
                 |  \|        27\|3               \|        27\|3
                 |-------------------------------------------------------------
                 |                     +---------------------+
                 |                     |     +-+       +----+
                 |                     |2069\|3  + 144\|- 79
                 |                  9  |---------------------
                 |                    3|           +-+
                \|                    \|        27\|3
         + 
            +-------------------------------------------------------------+
            |   +---------------------+2      +---------------------+
            |   |     +-+       +----+        |     +-+       +----+
            |   |2069\|3  + 144\|- 79         |2069\|3  + 144\|- 79
            |9  |---------------------  + 15  |--------------------- + 169
            |  3|           +-+              3|           +-+
            |  \|        27\|3               \|        27\|3
            |-------------------------------------------------------------  + 3
            |                     +---------------------+
            |                     |     +-+       +----+
            |                     |2069\|3  + 144\|- 79
            |                  9  |---------------------
            |                    3|           +-+
           \|                    \|        27\|3
      /
         2
     ,

     x =
           -
              ROOT
                              +---------------------+2
                              |     +-+       +----+
                              |2069\|3  + 144\|- 79
                         - 9  |---------------------
                             3|           +-+
                             \|        27\|3
                       + 
                             +---------------------+
                             |     +-+       +----+
                             |2069\|3  + 144\|- 79
                         30  |--------------------- - 169
                            3|           +-+
                            \|        27\|3
                    *
                       ROOT
                                 +---------------------+2
                                 |     +-+       +----+
                                 |2069\|3  + 144\|- 79
                              9  |---------------------
                                3|           +-+
                                \|        27\|3
                            + 
                                  +---------------------+
                                  |     +-+       +----+
                                  |2069\|3  + 144\|- 79
                              15  |--------------------- + 169
                                 3|           +-+
                                 \|        27\|3
                         /
                               +---------------------+
                               |     +-+       +----+
                               |2069\|3  + 144\|- 79
                            9  |---------------------
                              3|           +-+
                              \|        27\|3
                   + 
                          +---------------------+
                          |     +-+       +----+
                          |2069\|3  + 144\|- 79
                     144  |---------------------
                         3|           +-+
                         \|        27\|3
                /
                        +---------------------+
                        |     +-+       +----+
                        |2069\|3  + 144\|- 79
                     9  |---------------------
                       3|           +-+
                       \|        27\|3
                  *
                     ROOT
                               +---------------------+2
                               |     +-+       +----+
                               |2069\|3  + 144\|- 79
                            9  |---------------------
                              3|           +-+
                              \|        27\|3
                          + 
                                +---------------------+
                                |     +-+       +----+
                                |2069\|3  + 144\|- 79
                            15  |--------------------- + 169
                               3|           +-+
                               \|        27\|3
                       /
                             +---------------------+
                             |     +-+       +----+
                             |2069\|3  + 144\|- 79
                          9  |---------------------
                            3|           +-+
                            \|        27\|3
         + 
            +-------------------------------------------------------------+
            |   +---------------------+2      +---------------------+
            |   |     +-+       +----+        |     +-+       +----+
            |   |2069\|3  + 144\|- 79         |2069\|3  + 144\|- 79
            |9  |---------------------  + 15  |--------------------- + 169
            |  3|           +-+              3|           +-+
            |  \|        27\|3               \|        27\|3
         -  |-------------------------------------------------------------  + 3
            |                     +---------------------+
            |                     |     +-+       +----+
            |                     |2069\|3  + 144\|- 79
            |                  9  |---------------------
            |                    3|           +-+
           \|                    \|        27\|3
      /
         2
     ,

     x =
           ROOT
                           +---------------------+2      +---------------------+
                           |     +-+       +----+        |     +-+       +----+
                           |2069\|3  + 144\|- 79         |2069\|3  + 144\|- 79
                      - 9  |---------------------  + 30  |---------------------
                          3|           +-+              3|           +-+
                          \|        27\|3               \|        27\|3
                    + 
                      - 169
                 *
                    ROOT
                              +---------------------+2
                              |     +-+       +----+
                              |2069\|3  + 144\|- 79
                           9  |---------------------
                             3|           +-+
                             \|        27\|3
                         + 
                               +---------------------+
                               |     +-+       +----+
                               |2069\|3  + 144\|- 79
                           15  |--------------------- + 169
                              3|           +-+
                              \|        27\|3
                      /
                            +---------------------+
                            |     +-+       +----+
                            |2069\|3  + 144\|- 79
                         9  |---------------------
                           3|           +-+
                           \|        27\|3
                + 
                       +---------------------+
                       |     +-+       +----+
                       |2069\|3  + 144\|- 79
                  144  |---------------------
                      3|           +-+
                      \|        27\|3
             /
                     +---------------------+
                     |     +-+       +----+
                     |2069\|3  + 144\|- 79
                  9  |---------------------
                    3|           +-+
                    \|        27\|3
               *
                 +-------------------------------------------------------------+
                 |   +---------------------+2      +---------------------+
                 |   |     +-+       +----+        |     +-+       +----+
                 |   |2069\|3  + 144\|- 79         |2069\|3  + 144\|- 79
                 |9  |---------------------  + 15  |--------------------- + 169
                 |  3|           +-+              3|           +-+
                 |  \|        27\|3               \|        27\|3
                 |-------------------------------------------------------------
                 |                     +---------------------+
                 |                     |     +-+       +----+
                 |                     |2069\|3  + 144\|- 79
                 |                  9  |---------------------
                 |                    3|           +-+
                \|                    \|        27\|3
         + 
            +-------------------------------------------------------------+
            |   +---------------------+2      +---------------------+
            |   |     +-+       +----+        |     +-+       +----+
            |   |2069\|3  + 144\|- 79         |2069\|3  + 144\|- 79
            |9  |---------------------  + 15  |--------------------- + 169
            |  3|           +-+              3|           +-+
            |  \|        27\|3               \|        27\|3
         -  |-------------------------------------------------------------  + 3
            |                     +---------------------+
            |                     |     +-+       +----+
            |                     |2069\|3  + 144\|- 79
            |                  9  |---------------------
            |                    3|           +-+
           \|                    \|        27\|3
      /
         2
     ]

Type: List(Equation(Expression(Integer)))

axiom

)set output tex on

axiom

)set output algebra off

Matthias

axiom

t:=matrix ([[0,1,1],[1,-2,2],[1,2,-1]])

$\label{eq9}\left[ \begin{array}{ccc} 0 & 1 & 1 \ 1 & - 2 & 2 \ 1 & 2 & - 1$

(9)

Type: Matrix(Integer)

We cat diagonalise t by finding it's eigenvalues.

axiom

)set output tex off

axiom

)set output algebra on

e:=radicalEigenvectors(t)

   (30)
   [
                 +-----------------+2     +-----------------+
                 |  +-+    +------+       |  +-+    +------+
                 |3\|3  + \|- 1345        |3\|3  + \|- 1345
              3  |-----------------  - 3  |----------------- + 7
                3|        +-+            3|        +-+
                \|      6\|3             \|      6\|3
     [radval= --------------------------------------------------, radmult= 1,
                                +-----------------+
                                |  +-+    +------+
                                |3\|3  + \|- 1345
                             3  |-----------------
                               3|        +-+
                               \|      6\|3

       radvect =
         [
   [
     [
                     +-----------------+2
                     |  +-+    +------+
                +-+  |3\|3  + \|- 1345
           - 12\|3   |-----------------
                    3|        +-+
                    \|      6\|3
         + 
                                  +-----------------+
                                  |  +-+    +------+
               +-+     +------+   |3\|3  + \|- 1345        +-+     +------+
           (60\|3  + 6\|- 1345 )  |----------------- + 205\|3  + 3\|- 1345
                                 3|        +-+
                                 \|      6\|3
      /
                  +-----------------+2
                  |  +-+    +------+
             +-+  |3\|3  + \|- 1345
         126\|3   |-----------------
                 3|        +-+
                 \|      6\|3
       ]
     ,

     [
                  +-----------------+2
                  |  +-+    +------+
             +-+  |3\|3  + \|- 1345
           6\|3   |-----------------
                 3|        +-+
                 \|      6\|3
         + 
                                   +-----------------+
                                   |  +-+    +------+
                +-+     +------+   |3\|3  + \|- 1345       +-+     +------+
           (117\|3  - 3\|- 1345 )  |----------------- - 71\|3  + 9\|- 1345
                                  3|        +-+
                                  \|      6\|3
      /
                  +-----------------+2
                  |  +-+    +------+
             +-+  |3\|3  + \|- 1345
         126\|3   |-----------------
                 3|        +-+
                 \|      6\|3
       ]
     ,
    [1]]
           ]
       ]
     ,

     [
       radval =
                              +-----------------+2
                              |  +-+    +------+
                  +---+       |3\|3  + \|- 1345
             (- 3\|- 3  - 3)  |-----------------
                             3|        +-+
                             \|      6\|3
           + 
                              +-----------------+
                              |  +-+    +------+
                  +---+       |3\|3  + \|- 1345
             (- 3\|- 3  + 3)  |----------------- + 14
                             3|        +-+
                             \|      6\|3
        /
                          +-----------------+
                          |  +-+    +------+
              +---+       |3\|3  + \|- 1345
           (3\|- 3  - 3)  |-----------------
                         3|        +-+
                         \|      6\|3
       ,
      radmult= 1,

       radvect =
         [
   [
     [
                     +-----------------+2
                     |  +-+    +------+
                +-+  |3\|3  + \|- 1345
           - 24\|3   |-----------------
                    3|        +-+
                    \|      6\|3
         + 
                    +---+       +-+     +------+ +---+     +------+
             ((- 60\|- 3  - 60)\|3  - 6\|- 1345 \|- 3  - 6\|- 1345 )
          *
              +-----------------+
              |  +-+    +------+
              |3\|3  + \|- 1345
              |-----------------
             3|        +-+
             \|      6\|3
         + 
                +---+        +-+     +------+ +---+     +------+
           (205\|- 3  - 205)\|3  + 3\|- 1345 \|- 3  - 3\|- 1345
      /
                  +-----------------+2
                  |  +-+    +------+
             +-+  |3\|3  + \|- 1345
         252\|3   |-----------------
                 3|        +-+
                 \|      6\|3
       ]
     ,

     [
                   +-----------------+2
                   |  +-+    +------+
              +-+  |3\|3  + \|- 1345
           12\|3   |-----------------
                  3|        +-+
                  \|      6\|3
         + 
                     +---+        +-+     +------+ +---+     +------+
             ((- 117\|- 3  - 117)\|3  + 3\|- 1345 \|- 3  + 3\|- 1345 )
          *
              +-----------------+
              |  +-+    +------+
              |3\|3  + \|- 1345
              |-----------------
             3|        +-+
             \|      6\|3
         + 
                 +---+       +-+     +------+ +---+     +------+
           (- 71\|- 3  + 71)\|3  + 9\|- 1345 \|- 3  - 9\|- 1345
      /
                  +-----------------+2
                  |  +-+    +------+
             +-+  |3\|3  + \|- 1345
         252\|3   |-----------------
                 3|        +-+
                 \|      6\|3
       ]
     ,
    [1]]
           ]
       ]
     ,

     [
       radval =
                              +-----------------+2
                              |  +-+    +------+
                  +---+       |3\|3  + \|- 1345
             (- 3\|- 3  + 3)  |-----------------
                             3|        +-+
                             \|      6\|3
           + 
                              +-----------------+
                              |  +-+    +------+
                  +---+       |3\|3  + \|- 1345
             (- 3\|- 3  - 3)  |----------------- - 14
                             3|        +-+
                             \|      6\|3
        /
                          +-----------------+
                          |  +-+    +------+
              +---+       |3\|3  + \|- 1345
           (3\|- 3  + 3)  |-----------------
                         3|        +-+
                         \|      6\|3
       ,
      radmult= 1,

       radvect =
         [
   [
     [
                     +-----------------+2
                     |  +-+    +------+
                +-+  |3\|3  + \|- 1345
           - 24\|3   |-----------------
                    3|        +-+
                    \|      6\|3
         + 
                  +---+       +-+     +------+ +---+     +------+
             ((60\|- 3  - 60)\|3  + 6\|- 1345 \|- 3  - 6\|- 1345 )
          *
              +-----------------+
              |  +-+    +------+
              |3\|3  + \|- 1345
              |-----------------
             3|        +-+
             \|      6\|3
         + 
                  +---+        +-+     +------+ +---+     +------+
           (- 205\|- 3  - 205)\|3  - 3\|- 1345 \|- 3  - 3\|- 1345
      /
                  +-----------------+2
                  |  +-+    +------+
             +-+  |3\|3  + \|- 1345
         252\|3   |-----------------
                 3|        +-+
                 \|      6\|3
       ]
     ,

     [
                   +-----------------+2
                   |  +-+    +------+
              +-+  |3\|3  + \|- 1345
           12\|3   |-----------------
                  3|        +-+
                  \|      6\|3
         + 
                   +---+        +-+     +------+ +---+     +------+
             ((117\|- 3  - 117)\|3  - 3\|- 1345 \|- 3  + 3\|- 1345 )
          *
              +-----------------+
              |  +-+    +------+
              |3\|3  + \|- 1345
              |-----------------
             3|        +-+
             \|      6\|3
         + 
               +---+       +-+     +------+ +---+     +------+
           (71\|- 3  + 71)\|3  - 9\|- 1345 \|- 3  - 9\|- 1345
      /
                  +-----------------+2
                  |  +-+    +------+
             +-+  |3\|3  + \|- 1345
         252\|3   |-----------------
                 3|        +-+
                 \|      6\|3
       ]
     ,
    [1]]
           ]
       ]
     ]

Type: List(Record(radval: Expression(Integer),radmult: Integer,radvect: List(Matrix(Expression(Integer)))))

axiom

d:=diagonalMatrix([e.1.radval,e.2.radval,e.3.radval])

   Function definition for d is being overwritten.
   Compiled code for d has been cleared.

   (31)
        +-----------------+2     +-----------------+
        |  +-+    +------+       |  +-+    +------+
        |3\|3  + \|- 1345        |3\|3  + \|- 1345
     3  |-----------------  - 3  |----------------- + 7
       3|        +-+            3|        +-+
       \|      6\|3             \|      6\|3
   [[--------------------------------------------------,0,0],
                       +-----------------+
                       |  +-+    +------+
                       |3\|3  + \|- 1345
                    3  |-----------------
                      3|        +-+
                      \|      6\|3

     [0,

                            +-----------------+2
                            |  +-+    +------+
                +---+       |3\|3  + \|- 1345
           (- 3\|- 3  - 3)  |-----------------
                           3|        +-+
                           \|      6\|3
         + 
                            +-----------------+
                            |  +-+    +------+
                +---+       |3\|3  + \|- 1345
           (- 3\|- 3  + 3)  |----------------- + 14
                           3|        +-+
                           \|      6\|3
      /
                        +-----------------+
                        |  +-+    +------+
            +---+       |3\|3  + \|- 1345
         (3\|- 3  - 3)  |-----------------
                       3|        +-+
                       \|      6\|3
       ,
      0]
     ,

     [0, 0,

                            +-----------------+2
                            |  +-+    +------+
                +---+       |3\|3  + \|- 1345
           (- 3\|- 3  + 3)  |-----------------
                           3|        +-+
                           \|      6\|3
         + 
                            +-----------------+
                            |  +-+    +------+
                +---+       |3\|3  + \|- 1345
           (- 3\|- 3  - 3)  |----------------- - 14
                           3|        +-+
                           \|      6\|3
      /
                        +-----------------+
                        |  +-+    +------+
            +---+       |3\|3  + \|- 1345
         (3\|- 3  + 3)  |-----------------
                       3|        +-+
                       \|      6\|3
       ]
     ]

Type: Matrix(Expression(Integer))

Now prove it by constructing the simularity transformation from the eigenvectors:

axiom

p:=horizConcat(horizConcat(e.1.radvect.1,e.2.radvect.1),e.3.radvect.1)

   (32)
   [
     [
                     +-----------------+2
                     |  +-+    +------+
                +-+  |3\|3  + \|- 1345
           - 12\|3   |-----------------
                    3|        +-+
                    \|      6\|3
         + 
                                  +-----------------+
                                  |  +-+    +------+
               +-+     +------+   |3\|3  + \|- 1345        +-+     +------+
           (60\|3  + 6\|- 1345 )  |----------------- + 205\|3  + 3\|- 1345
                                 3|        +-+
                                 \|      6\|3
      /
                  +-----------------+2
                  |  +-+    +------+
             +-+  |3\|3  + \|- 1345
         126\|3   |-----------------
                 3|        +-+
                 \|      6\|3
       ,

                     +-----------------+2
                     |  +-+    +------+
                +-+  |3\|3  + \|- 1345
           - 24\|3   |-----------------
                    3|        +-+
                    \|      6\|3
         + 
                    +---+       +-+     +------+ +---+     +------+
             ((- 60\|- 3  - 60)\|3  - 6\|- 1345 \|- 3  - 6\|- 1345 )
          *
              +-----------------+
              |  +-+    +------+
              |3\|3  + \|- 1345
              |-----------------
             3|        +-+
             \|      6\|3
         + 
                +---+        +-+     +------+ +---+     +------+
           (205\|- 3  - 205)\|3  + 3\|- 1345 \|- 3  - 3\|- 1345
      /
                  +-----------------+2
                  |  +-+    +------+
             +-+  |3\|3  + \|- 1345
         252\|3   |-----------------
                 3|        +-+
                 \|      6\|3
       ,

                     +-----------------+2
                     |  +-+    +------+
                +-+  |3\|3  + \|- 1345
           - 24\|3   |-----------------
                    3|        +-+
                    \|      6\|3
         + 
                  +---+       +-+     +------+ +---+     +------+
             ((60\|- 3  - 60)\|3  + 6\|- 1345 \|- 3  - 6\|- 1345 )
          *
              +-----------------+
              |  +-+    +------+
              |3\|3  + \|- 1345
              |-----------------
             3|        +-+
             \|      6\|3
         + 
                  +---+        +-+     +------+ +---+     +------+
           (- 205\|- 3  - 205)\|3  - 3\|- 1345 \|- 3  - 3\|- 1345
      /
                  +-----------------+2
                  |  +-+    +------+
             +-+  |3\|3  + \|- 1345
         252\|3   |-----------------
                 3|        +-+
                 \|      6\|3
       ]
     ,

     [
                  +-----------------+2
                  |  +-+    +------+
             +-+  |3\|3  + \|- 1345
           6\|3   |-----------------
                 3|        +-+
                 \|      6\|3
         + 
                                   +-----------------+
                                   |  +-+    +------+
                +-+     +------+   |3\|3  + \|- 1345       +-+     +------+
           (117\|3  - 3\|- 1345 )  |----------------- - 71\|3  + 9\|- 1345
                                  3|        +-+
                                  \|      6\|3
      /
                  +-----------------+2
                  |  +-+    +------+
             +-+  |3\|3  + \|- 1345
         126\|3   |-----------------
                 3|        +-+
                 \|      6\|3
       ,

                   +-----------------+2
                   |  +-+    +------+
              +-+  |3\|3  + \|- 1345
           12\|3   |-----------------
                  3|        +-+
                  \|      6\|3
         + 
                     +---+        +-+     +------+ +---+     +------+
             ((- 117\|- 3  - 117)\|3  + 3\|- 1345 \|- 3  + 3\|- 1345 )
          *
              +-----------------+
              |  +-+    +------+
              |3\|3  + \|- 1345
              |-----------------
             3|        +-+
             \|      6\|3
         + 
                 +---+       +-+     +------+ +---+     +------+
           (- 71\|- 3  + 71)\|3  + 9\|- 1345 \|- 3  - 9\|- 1345
      /
                  +-----------------+2
                  |  +-+    +------+
             +-+  |3\|3  + \|- 1345
         252\|3   |-----------------
                 3|        +-+
                 \|      6\|3
       ,

                   +-----------------+2
                   |  +-+    +------+
              +-+  |3\|3  + \|- 1345
           12\|3   |-----------------
                  3|        +-+
                  \|      6\|3
         + 
                   +---+        +-+     +------+ +---+     +------+
             ((117\|- 3  - 117)\|3  - 3\|- 1345 \|- 3  + 3\|- 1345 )
          *
              +-----------------+
              |  +-+    +------+
              |3\|3  + \|- 1345
              |-----------------
             3|        +-+
             \|      6\|3
         + 
               +---+       +-+     +------+ +---+     +------+
           (71\|- 3  + 71)\|3  - 9\|- 1345 \|- 3  - 9\|- 1345
      /
                  +-----------------+2
                  |  +-+    +------+
             +-+  |3\|3  + \|- 1345
         252\|3   |-----------------
                 3|        +-+
                 \|      6\|3
       ]
     ,
    [1,1,1]]

Type: Matrix(Expression(Integer))

axiom

p*d*inverse(p)

         +0   1    1 +
         |           |
   (33)  |1  - 2   2 |
         |           |
         +1   2   - 1+

Type: Matrix(Expression(Integer))

axiom

)set output tex on

axiom

)set output algebra off

\end{axiom}

Axiom can't integrame exp(x^4) ;( --unknown, Fri, 28 Apr 2006 14:03:28 -0500 reply

Axiom can't integrame exp(x^4) ;(

axiom

integrate(exp(x**4),x)

$\label{eq10}\int^{ \displaystyle x}{{e^{\%A^4}}\ {d \%A}}$

(10)

Type: Union(Expression(Integer),...)

But Maple can...

axiom

f(x) == (1/4)*x*(-Gamma(1/4,-x**4)*Gamma(3/4)+%pi*sqrt(2))/((-x**4)**(1/4)*Gamma(3/4))

Type: Void

axiom

D(f(x),x)

axiom

Compiling function f with type Variable(x) -> Expression(Integer)

$\label{eq11}e^{x^4}$

(11)

Type: Expression(Integer)

Axiom cannot integrate e^(4*x) --kratt6, Fri, 28 Apr 2006 16:34:16 -0500 reply

This is not a big surprise: note that Gamma(x,y) is not an elementary function.

Martin

... --unknown, Thu, 18 May 2006 11:30:21 -0500 reply

This is both obviously wrong since the integrand is a positive function:

axiom

integrate(1/(1+x^4),x=%minusInfinity..%plusInfinity)

$\label{eq12}{\pi \ {\sqrt{2}}}\over 2$

(12)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

axiom

numeric(integrate(1/(1+x^4),x=0..1))

$\label{eq13}0.86697298733991103758$

(13)

Type: Float

... --unknown, Wed, 24 May 2006 04:31:40 -0500 reply

axiom

)clear co

   All user variables and function definitions have been cleared.
   All )browse facility databases have been cleared.
   Internally cached functions and constructors have been cleared.
   )clear completely is finished.
n := 32

$\label{eq14}32$

(14)

Type: PositiveInteger?

axiom

y : FARRAY INT := new(n,1)

$\label{eq15}\begin{array}{@{}l} \displaystyle \left[ 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1, \right. \ \ \displaystyle \left.\: 1, \: 1, \: 1, \: 1, \: 1, \: 1 \right]$

(15)

Type: FlexibleArray?(Integer)

axiom

n0 := n

$\label{eq16}32$

(16)

Type: PositiveInteger?

axiom

n1 := sum(x^1, x=0..n-1)

$\label{eq17}496$

(17)

Type: Fraction(Polynomial(Integer))

axiom

n2 := sum(x^2, x=0..n-1)

$\label{eq18}10416$

(18)

Type: Fraction(Polynomial(Integer))

axiom

n3 := sum(x^3, x=0..n-1)

$\label{eq19}246016$

(19)

Type: Fraction(Polynomial(Integer))

axiom

n4 := sum(x^4, x=0..n-1)

$\label{eq20}6197520$

(20)

Type: Fraction(Polynomial(Integer))

axiom

A := matrix([[n4, n3, n2],_
            [n3, n2, n1],_
            [n2, n1, n0]])

$\label{eq21}\left[ \begin{array}{ccc} {6197520}&{246016}&{10416} \ {246016}&{10416}&{496} \ {10416}&{496}&{32}$

(21)

Type: Matrix(Fraction(Polynomial(Integer)))

axiom

X := vector([x1, x2, x3])

$\label{eq22}\left[ x 1, \: x 2, \: x 3 \right]$

(22)

Type: Vector(OrderedVariableList?([x1,x2,x3]))

axiom

B := vector([sum(x^2* u, x=0..n-1),_
       sum(x*   v, x=0..n-1),_
       sum(     w, x=0..n-1)])

$\label{eq23}\left[{{10416}\ u}, \:{{496}\ v}, \:{{32}\ w}\right]$

(23)

Type: Vector(Fraction(Polynomial(Integer)))

axiom

solve([A * X = B], [x1, x2, x3])

   There are 18 exposed and 3 unexposed library operations named solve 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op solve
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named solve
      with argument type(s) 
            List(Equation(Vector(Fraction(Polynomial(Integer)))))
                    List(OrderedVariableList([x1,x2,x3]))

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

can this be correct? --unknown, Tue, 30 May 2006 23:51:26 -0500 reply

axiom

integrate(1/((x+t)*sqrt(1+(x*t)**2)),t=0..%plusInfinity,"noPole")

$\label{eq24}{\left( \begin{array}{@{}l} \displaystyle - \ \ \displaystyle {\log{{\left({{{{\left({2 \ {x^{10}}}-{4 \ {x^8}}+{6 \ {x^6}}-{6 \ {x^4}}+{4 \ {x^2}}- 2 \right)}\ {\sqrt{{x^4}+ 1}}}+{2 \ {x^{12}}}-{4 \ {x^{10}}}+{7 \ {x^8}}-{8 \ {x^6}}+{7 \ {x^4}}-{4 \ {x^2}}+ 2}\over{x^2}}\right)}}}+ \ \ \displaystyle {\log \left({{x^6}+{x^2}}\right)}$

(24)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

axiom

subst(%,x=1)

$\label{eq25}0$

(25)

Type: Expression(Integer)

axiom

integrate(1/((1+t)*sqrt(1+(1*t)**2)),t=0..%plusInfinity,"noPole")

$\label{eq26}{{\sqrt{2}}\ {\log \left({{{12}\ {\sqrt{2}}}+{17}}\right)}}\over 4$

(26)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

axiom

simplify(%-subst((asinh(x^2)+asinh(1/x^2))/sqrt(1+x^4),x=1))

$\label{eq27}{{\log \left({{{12}\ {\sqrt{2}}}+{17}}\right)}-{4 \ {asinh \left({1}\right)}}}\over{2 \ {\sqrt{2}}}$

(27)

Type: Expression(Integer)

axiom

%::Expression Float

$\label{eq28}0.0$

(28)

Type: Expression(Float)

... --unknown, Mon, 03 Jul 2006 02:07:02 -0500 reply

axiom

a := matrix([ [-1,0,0,0,1,0], [0,1,0,0,0,0], [0,0,2,0,0,-2], [0,0,0,4,0,0], [0,0,0,0,3,0], [0,0,-3,0,0,3]])

$\label{eq29}\left[ \begin{array}{cccccc} - 1 & 0 & 0 & 0 & 1 & 0 \ 0 & 1 & 0 & 0 & 0 & 0 \ 0 & 0 & 2 & 0 & 0 & - 2 \ 0 & 0 & 0 & 4 & 0 & 0 \ 0 & 0 & 0 & 0 & 3 & 0 \ 0 & 0 & - 3 & 0 & 0 & 3$

(29)

Type: Matrix(Integer)

axiom

determinant(a)

$\label{eq30}0$

(30)

Type: NonNegativeInteger?

axiom

inverse(a)

$\label{eq31}\mbox{\tt "failed"}$

(31)

Type: Union("failed",...)

... --unknown, Fri, 07 Jul 2006 11:54:52 -0500 reply

a := matrix([ [-3,1,1,1]?, [1,1,1,1]?, [1,1,1,1]?, [1,1,1,1]]?)

... --unknown, Fri, 07 Jul 2006 13:24:57 -0500 reply

axiom

As := matrix([ [-3,1,1,1], [1,1,1,1], [1,1,1,1], [1,1,1,1]])

$\label{eq32}\left[ \begin{array}{cccc} - 3 & 1 & 1 & 1 \ 1 & 1 & 1 & 1 \ 1 & 1 & 1 & 1 \ 1 & 1 & 1 & 1$

(32)

Type: Matrix(Integer)

axiom

A := subMatrix(As, 2,4,2,4)

$\label{eq33}\left[ \begin{array}{ccc} 1 & 1 & 1 \ 1 & 1 & 1 \ 1 & 1 & 1$

(33)

Type: Matrix(Integer)

axiom

ob := orthonormalBasis(A)

$\label{eq34}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{c} -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}} \ {{\sqrt{2}}\over{\sqrt{3}}} \ -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}$

(34)

Type: List(Matrix(Expression(Integer)))

axiom

P : Matrix(Expression Integer) := new(3,3,0)

$\label{eq35}\left[ \begin{array}{ccc} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0$

(35)

Type: Matrix(Expression(Integer))

axiom

setsubMatrix!(P,1,1,ob.3)

$\label{eq36}\left[ \begin{array}{ccc} {1 \over{\sqrt{3}}}& 0 & 0 \ {1 \over{\sqrt{3}}}& 0 & 0 \ {1 \over{\sqrt{3}}}& 0 & 0$

(36)

Type: Matrix(Expression(Integer))

axiom

setsubMatrix!(P,1,2,ob.1)

$\label{eq37}\left[ \begin{array}{ccc} {1 \over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}& 0 \ {1 \over{\sqrt{3}}}&{{\sqrt{2}}\over{\sqrt{3}}}& 0 \ {1 \over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}& 0$

(37)

Type: Matrix(Expression(Integer))

axiom

setsubMatrix!(P,1,3,ob.2)

$\label{eq38}\left[ \begin{array}{ccc} {1 \over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}& -{1 \over{\sqrt{2}}} \ {1 \over{\sqrt{3}}}&{{\sqrt{2}}\over{\sqrt{3}}}& 0 \ {1 \over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}&{1 \over{\sqrt{2}}}$

(38)

Type: Matrix(Expression(Integer))

axiom

Pt := transpose(P)

$\label{eq39}\left[ \begin{array}{ccc} {1 \over{\sqrt{3}}}&{1 \over{\sqrt{3}}}&{1 \over{\sqrt{3}}} \ -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}&{{\sqrt{2}}\over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}} \ -{1 \over{\sqrt{2}}}& 0 &{1 \over{\sqrt{2}}}$

(39)

Type: Matrix(Expression(Integer))

axiom

Ps : Matrix(Expression Integer) := new(4,4,0)

$\label{eq40}\left[ \begin{array}{cccc} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0$

(40)

Type: Matrix(Expression(Integer))

axiom

Ps(1,1) := 1

$\label{eq41}1$

(41)

Type: Expression(Integer)

axiom

setsubMatrix!(Ps,2,2,P)

$\label{eq42}\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 &{1 \over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}& -{1 \over{\sqrt{2}}} \ 0 &{1 \over{\sqrt{3}}}&{{\sqrt{2}}\over{\sqrt{3}}}& 0 \ 0 &{1 \over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}&{1 \over{\sqrt{2}}}$

(42)

Type: Matrix(Expression(Integer))

axiom

PsT := transpose(Ps)

$\label{eq43}\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 &{1 \over{\sqrt{3}}}&{1 \over{\sqrt{3}}}&{1 \over{\sqrt{3}}} \ 0 & -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}&{{\sqrt{2}}\over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}} \ 0 & -{1 \over{\sqrt{2}}}& 0 &{1 \over{\sqrt{2}}}$

(43)

Type: Matrix(Expression(Integer))

axiom

PsTAsPs := PsT * As * Ps

$\label{eq44}\left[ \begin{array}{cccc} - 3 &{3 \over{\sqrt{3}}}& 0 & 0 \ {3 \over{\sqrt{3}}}& 3 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0$

(44)

Type: Matrix(Expression(Integer))

axiom

b1 := PsTAsPs(2,1)

$\label{eq45}3 \over{\sqrt{3}}$

(45)

Type: Expression(Integer)

axiom

l1 := PsTAsPs(2,2)

$\label{eq46}3$

(46)

Type: Expression(Integer)

axiom

Us : Matrix(Expression Integer) := new(4,4,0)

$\label{eq47}\left[ \begin{array}{cccc} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0$

(47)

Type: Matrix(Expression(Integer))

axiom

Us(1,1) := 1

$\label{eq48}1$

(48)

Type: Expression(Integer)

axiom

Us(2,2) := 1

$\label{eq49}1$

(49)

Type: Expression(Integer)

axiom

Us(3,3) := 1

$\label{eq50}1$

(50)

Type: Expression(Integer)

axiom

Us(4,4) := 1

$\label{eq51}1$

(51)

Type: Expression(Integer)

axiom

Us(2,1) := -b1 / l1

$\label{eq52}-{1 \over{\sqrt{3}}}$

(52)

Type: Expression(Integer)

axiom

PsUs := Ps * Us

$\label{eq53}\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ -{1 \over 3}&{1 \over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}& -{1 \over{\sqrt{2}}} \ -{1 \over 3}&{1 \over{\sqrt{3}}}&{{\sqrt{2}}\over{\sqrt{3}}}& 0 \ -{1 \over 3}&{1 \over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}&{1 \over{\sqrt{2}}}$

(53)

Type: Matrix(Expression(Integer))

axiom

PsUsT := transpose(PsUs)

$\label{eq54}\left[ \begin{array}{cccc} 1 & -{1 \over 3}& -{1 \over 3}& -{1 \over 3} \ 0 &{1 \over{\sqrt{3}}}&{1 \over{\sqrt{3}}}&{1 \over{\sqrt{3}}} \ 0 & -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}}&{{\sqrt{2}}\over{\sqrt{3}}}& -{{\sqrt{2}}\over{2 \ {\sqrt{3}}}} \ 0 & -{1 \over{\sqrt{2}}}& 0 &{1 \over{\sqrt{2}}}$

(54)

Type: Matrix(Expression(Integer))

axiom

PsUsTAsPsUs := PsUsT * As * PsUs

$\label{eq55}\left[ \begin{array}{cccc} - 4 & 0 & 0 & 0 \ 0 & 3 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0$

(55)

Type: Matrix(Expression(Integer))

axiom

C := inverse(PsUs)

$\label{eq56}\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ {{\sqrt{3}}\over 3}&{{\sqrt{3}}\over 3}&{{\sqrt{3}}\over 3}&{{\sqrt{3}}\over 3} \ 0 & -{{\sqrt{3}}\over{3 \ {\sqrt{2}}}}&{{2 \ {\sqrt{3}}}\over{3 \ {\sqrt{2}}}}& -{{{\sqrt{2}}\ {\sqrt{3}}}\over 6} \ 0 & -{{\sqrt{2}}\over 2}& 0 &{{\sqrt{2}}\over 2}$

(56)

Type: Union(Matrix(Expression(Integer)),...)

axiom

c := PsUsTAsPsUs(1,1)

$\label{eq57}- 4$

(57)

Type: Expression(Integer)

axiom

gQ := PsUsTAsPsUs / c

$\label{eq58}\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & -{3 \over 4}& 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0$

(58)

Type: Matrix(Expression(Integer))

axiom

x1 := transpose(matrix([[1,2,3,4]]))

$\label{eq59}\left[ \begin{array}{c} 1 \ 2 \ 3 \ 4$

(59)

Type: Matrix(Integer)

axiom

v1 := transpose(x1) * As * x1

$\label{eq60}\left[ \begin{array}{c} {96}$

(60)

Type: Matrix(Integer)

axiom

x2 := C * x1

$\label{eq61}\left[ \begin{array}{c} 1 \ {{{10}\ {\sqrt{3}}}\over 3} \ 0 \ {\sqrt{2}}$

(61)

Type: Matrix(Expression(Integer))

axiom

v2 := transpose(x2) * PsUsTAsPsUs * x2

$\label{eq62}\left[ \begin{array}{c} {96}$

(62)

Type: Matrix(Expression(Integer))

graphics --unknown, Tue, 01 Aug 2006 02:17:12 -0500 reply

axiom

draw(y**2/2+(x**2-1)**2/4-1=0, x,y, range ==[-2..2, -1..1])

   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) 
                           FlexibleArray(Integer)
                               PositiveInteger

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

series test --greg, Sat, 03 Feb 2007 09:35:50 -0600 reply

axiom

f1 := taylor(1 - x**2,x = 0)

$\label{eq63}1 -{x^2}$

(63)

Type: UnivariateTaylorSeries?(Expression(Integer),x,0)

axiom

asin f1

$\label{eq64}\begin{array}{@{}l} \displaystyle {\pi \over 2}-{{2 \over{\sqrt{2}}}\ x}-{{1 \over{6 \ {\sqrt{2}}}}\ {x^3}}-{{3 \over{{80}\ {\sqrt{2}}}}\ {x^5}}-{{5 \over{{448}\ {\sqrt{2}}}}\ {x^7}}- \ \ \displaystyle {{{35}\over{{9216}\ {\sqrt{2}}}}\ {x^9}}+{O \left({x^{11}}\right)}$

(64)

Type: UnivariateTaylorSeries?(Expression(Integer),x,0)

axiom

sin %

$\label{eq65}1 -{x^2}+{O \left({x^{11}}\right)}$

(65)

Type: UnivariateTaylorSeries?(Expression(Integer),x,0)

SandboxMSkuce?

axiom

1+1

$\label{eq66}2$

(66)

Type: PositiveInteger?

integration --jhnbk, Fri, 08 Jun 2007 03:50:35 -0500 reply

axiom

integrate((x-1)/log(x), x)

$\label{eq67}\int^{ \displaystyle x}{{{\%A - 1}\over{\log \left({\%A}\right)}}\ {d \%A}}$

(67)

Type: Union(Expression(Integer),...)

axiom

integrate(x*exp(x)*sin(x),x)

$\label{eq68}{{x \ {e^x}\ {\sin \left({x}\right)}}+{{\left(- x + 1 \right)}\ {\cos \left({x}\right)}\ {e^x}}}\over 2$

(68)

Type: Union(Expression(Integer),...)

Working With Lists --daneshpajouh, Sat, 16 Jun 2007 07:00:00 -0500 reply

axiom

[p for p in primes(2,1000)|(p rem 16)=1]

$\label{eq69}\begin{array}{@{}l} \displaystyle \left[{977}, \:{929}, \:{881}, \:{769}, \:{673}, \:{641}, \:{5 93}, \:{577}, \:{449}, \:{433}, \:{401}, \:{353}, \:{337}, \: \right. \ \ \displaystyle \left.{257}, \:{241}, \:{193}, \:{113}, \:{97}, \:{17}\right]$

(69)

Type: List(Integer)

axiom

[p**2+1 for p in primes(2,100)]

$\label{eq70}\begin{array}{@{}l} \displaystyle \left[ 5, \:{9410}, \:{7922}, \:{6890}, \:{6242}, \:{5330}, \:{5 042}, \:{4490}, \:{3722}, \:{3482}, \:{2810}, \: \right. \ \ \displaystyle \left.{2210}, \:{1850}, \:{1682}, \:{1370}, \:{962}, \:{842}, \:{530}, \:{362}, \:{290}, \:{170}, \:{122}, \:{50}, \: \right. \ \ \displaystyle \left.{26}, \:{10}\right]$

(70)

Type: List(Integer)

axiom

integrate (2x^2 + 2x, x)

   Cannot find a definition or applicable library operation named 2 
      with argument type(s) 
                                 Variable(x)

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

is it error? --vv, Sat, 28 Jul 2007 14:00:27 -0500 reply

axiom

radix(36,37)

$\label{eq71}36$

(71)

Type: RadixExpansion?(37)

Is it error?

(better) example (with axiom markers this time) ;-) --pbwagner, Mon, 10 Sep 2007 13:01:48 -0500 reply

axiom

integrate(log(log(x)),x)

$\label{eq72}{x \ {\log \left({\log \left({x}\right)}\right)}}-{li \left({x}\right)}$

(72)

Type: Union(Expression(Integer),...)