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changed:
-
This is the front page of the SandBox. You can try anything you like
here but keep in mind that other people are also using these pages to
learn and experiment with Axiom and Reduce. Please be curteous to
others if you correct mistakes and try to explain what you are doing.

No Email Notices

Normally, if you 'edit' any page on MathAction and click
'Save' or if you add a comment to a page, a notice of the
change is sent out to all subscribers on the axiom-developer
email list, see the [Axiom Community]. Separate notices are
also sent to those users who 'subscribe' directly to
MathAction.

Use Preview

If you click 'Preview' instead of 'Save', you will get a chance
to see the result of your calculations and LaTeX commands but
**no** email notice is sent out and the result is not saved until
you decide to click 'Save' or not.

Use the !SandBox

On this page or on any other page with a name beginning with
SandBox such as SandBoxJohn2, SandBoxSimple, SandBoxEtc, clicking
'Save' only sends email notices to users who 'subscribe'
directly to that specific SandBox page. Saving and adding
comments does *not* create an email to the email list. You
can safely use these pages for testing without disturbing

New !SandBox Pages

You can also create new SandBox pages as needed just by
below. The link must include at least two uppercase letters
and no spaces or alternatively it can be any phrase written
inside ![ ] brackets as long as it begins with SandBox. When
a blue question mark <font color="blue">?</font> beside it.
Clicking on the blue question mark <font color="blue">?</font>
will ask you if you wish to create a new page.

[SandBox Aldor Foreign] -- Using Aldor to call external C routines

[SandBox Aldor Generator] -- Aldor defines a 'generator' for type Vector

[SandBox Aldor Sieve] -- A prime number sieve in Aldor to count primes <= n.

[SandBox Aldor Testing] -- Using Aldor to write Axiom library routines

[SandBox Arrays] -- How fast is array access in Axiom?

[SandBox Axiom Syntax] -- Syntax of *if* *then* *else*

[SandboxBiblography]

[SandBox Boolean] -- evaluating Boolean expressions and conditions

[SandBox Cast] -- Meaning and use of 'pretend' vs. strong typing

[SandBox Categorical Relativity] -- Special relativity without the Lorentz group

[SandBox Category of Graphs] -- Graph theory in Axiom

[SandBox CL-WEB] -- Tangle operation for literate programming implemented in Common Lisp

[SandBox Combinat] -- A{ld,xi}o{r,m}Combinat

[SandBox Content MathML] -- Content vs. presentation MathML

[SandBox Direct Product] -- A x B

[SandBox DistributedExpression] -- expression in sum-of-products form

[SandBox Domains and Types] -- What is the difference?

[AxiomEmacsMode] -- Beginnings of an Emacs mode for Axiom based off of Jay's work and others

[SandBox Embeded PDF] -- pdf format documents can be displayed inline

[SandBox EndPaper] -- Algebra and Data Structure Hierarchy (lattice) diagrams

[SandBox Folding] -- experiments with DHTML, javascript, etc.

[SandBox Functions] -- How do they work?

[SandBox Functors] -- What are they? In Axiom functors are also called domain constructors.

[SandBox Gamma] -- Numerical evaluation of the incomplete Gamma function

[SandBox GuessingSequence] -- Guessing integer sequences

[SandBox Integration] -- Examples of integration in Axiom and Reduce

[SandBox Kernel] -- What is a "kernel"?

[SandBox kaveh]

[SandBox LaTeX] -- LaTeX commands allowed in MathAction

[SandBox Lisp] -- Using Lisp in Axiom

[SandBox Manip] -- expression manipulations

[SandBox Manipulating Domains] -- testing the domain of an expression

[SandBox Mapping] -- A->B is a type in Axiom

[MathMLFormat]

[SandBox Matrix] -- Examples of working with matrices in Axiom

[SandBox Maxima] -- Testing the Maxima interface

[SandBox Monoid] -- Rings and things

[SandBox Monoid Extend] -- Martin Rubey's beautiful idea about using 'extend'
to add a category to a previously defined domain.

[SandBox Noncommutative Polynomials] -- XPOLY and friends

[SandBox Numerical Integration] -- Simpson method

[SandBox NNI] -- NonNegative Integer without using SubDomain

[SandBox Pamphlet] -- [Literate Programming] support on MathAction

[SandBoxPartialFraction] -- Trigonometric expansion example

[SandBox Polymake] -- an interface between Axiom and PolyMake

[SandBox Polynomials] -- Axiom's polynomial domains are certainly
rich and complex!

[SandBox ProblemSolving] -- Test page for educational purposes

[SandBox Qubic] -- Solving cubic polynomials

[SandBox Reduce And MathML] -- Reduce can use MathML for both input and output

[SandBoxRelativeVelocity] -- Slides for IARD 2006: Addition of
Relative Velocites is Associative

[SandBox RenameTitle] -- trying to re-create a crash due to renaming pages

[SandBox Sage] -- This is a test of Sage in MathAction

[SandBox Shortcoming] -- Implementation of solve

[SandBox Solve] -- Solving equations

[SandBox Statistics] -- calculating statistics in Axiom

[SandBox SubDomain] -- What is a SubDomain?

[SandBox Tail Recursion] -- When does Axiom replace recursion with iteration?

[SandBox Text Files] -- How to access text files in Axiom

[SandBox Trace Analysed] -- Tracing can affect output of '1::EXPR INT' or '1::FRAC INT'

[SandBox Tuples Products and Records] -- Basic structured data types in Axiom

[SandBox Units and Dimensions] -- Scientific units and dimensions

[SandBox Speed] -- Compilation speed

[SandBox Zero]

[SandBox Axiom Strengths]

SandBoxJohn2 -- Experiments with matrices and various other stuff

SandBox2 -- Experiments

SandBox3 -- Experiments

SandBox4 -- Experiments

SandBox5 -- Experiments with GraphViz and StructuredTables

SandBox6 -- Differential Equations etc.

[SandBox7] --

[SandBox8] -- Here you can create your own SandBox.

[SandBox9] -- Experiments with JET Bundles

[SandBox10]

[SandBox DoOps] -- used to run Axiom without actually have to have it installed!

[SandBoxKMG]

Click on the <font color="blue">?</font> to create a new page.
You should also edit this page to include a description and a new empty

<hr />

Examples

Here is a simple Axiom command::

!\begin{axiom}
integrate(1/(a+z^3), z=0..1,"noPole")
\end{axiom}

\begin{axiom}
integrate(1/(a+z^3), z=0..1,"noPole")
\end{axiom}

And here is a REDUCE command::

!\begin{reduce}
int(1/(a+z^3), z,0,1);
\end{reduce}

\begin{reduce}
int(1/(a+z^3), z,0,1);
\end{reduce}
<hr />

Common Mistakes

Please review the list of [Common Mistakes] and the list
of [MathAction Problems] if you are have never used
MathAction before. If you are learning to use Axiom and think
that someone must have solved some particular problem before
you, check this list of Common [Axiom Problems].

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

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::

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} SandBoxCS224 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} \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 13:00:06 -0500 2007 From: pbwagner Date: Mon, 10 Sep 2007 13:00:06 -0500 Subject: example from my daughter's college calc Message-ID: <20070910130006-0500@wiki.axiom-developer.org> integrate(log(log(x)),x) 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}  This is the front page of the SandBox. You can try anything you like here but keep in mind that other people are also using these pages to learn and experiment with Axiom and Reduce. Please be curteous to others if you correct mistakes and try to explain what you are doing. No Email Notices Normally, if you edit any page on MathAction and click Save or if you add a comment to a page, a notice of the change is sent out to all subscribers on the axiom-developer email list, see the [Axiom Community]. Separate notices are also sent to those users who subscribe directly to MathAction. Use Preview If you click Preview instead of Save, you will get a chance to see the result of your calculations and LaTeX commands but no email notice is sent out and the result is not saved until you decide to click Save or not. Use the SandBox On this page or on any other page with a name beginning with SandBox such as SandBoxJohn2, SandBoxSimple, SandBoxEtc, clicking Save only sends email notices to users who subscribe directly to that specific SandBox page. Saving and adding comments does not create an email to the email list. You can safely use these pages for testing without disturbing anyone who might not care to know about your experiments. New SandBox Pages You can also create new SandBox pages as needed just by editing this page and adding a link to the list of new page below. The link must include at least two uppercase letters and no spaces or alternatively it can be any phrase written inside [ ] brackets as long as it begins with SandBox. When you Save this page, the link to the new page will appear with a blue question mark ? beside it. Clicking on the blue question mark ? will ask you if you wish to create a new page. [SandBox Aldor Foreign] Using Aldor to call external C routines [SandBox Aldor Generator] Aldor defines a generator for type Vector [SandBox Aldor Sieve] A prime number sieve in Aldor to count primes <= n. [SandBox Aldor Testing] Using Aldor to write Axiom library routines [SandBox Arrays] How fast is array access in Axiom? 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Examples Here is a simple Axiom command: \begin{axiom} integrate(1/(a+z^3), z=0..1,"noPole") \end{axiom} axiomintegrate(1/(a+z^3), z=0..1,"noPole") \begin{equation} \label{eq1}{-{{\sqrt {3}} \ {\log \left( {{{3 \ {a \sp 2} \ {{\root {3} \of {{a \sp 2}}} \sp 2}}+{{\left( -{2 \ {a \sp 3}}+{a \sp 2} \right)} \ {\root {3} \of {{a \sp 2}}}}+{a \sp 4} -{2 \ {a \sp 3}}}} \right)}}+{2 \ {\sqrt {3}} \ {\log \left( {{{{\root {3} \of {{a \sp 2}}} \sp 2}+{2 \ a \ {\root {3} \of {{a \sp 2}}}}+{a \sp 2}}} \right)}}+{{12} \ {\arctan \left( {{{{2 \ {\sqrt {3}} \ {\root {3} \of {{a \sp 2}}}} -{a \ {\sqrt {3}}}} \over {3 \ a}}} \right)}}+{2 \ \pi}} \over {{12} \ {\sqrt {3}} \ {\root {3} \of {{a \sp 2}}}} \end{equation} Type: Union(f1: OrderedCompletion? Expression Integer,...) And here is a REDUCE command: \begin{reduce} load_package sfgamma; load_package defint; int(1/(a+z^3), z,0,1); \end{reduce} \begin{reduce} load_package sfgamma; load_package defint; int(1/(a+z^3), z,0,1); \end{reduce} Common Mistakes Please review the list of [Common Mistakes]? and the list of [MathAction Problems]? if you are have never used MathAction? before. If you are learning to use Axiom and think that someone must have solved some particular problem before you, check this list of Common [Axiom Problems]?. szsz --unknown, Wed, 07 Dec 2005 09:11:18 -0600 replyWorks with ASCII text output formatting. axiom)set output tex off )set output algebra on axiomsolve([x^2 + y^2 - 2*(ax*x + ay*y) = l1, x^2 + y^2 - 2*(cx*x + cy*y) = l2],[x,y]) (2) [ (- 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 )set output algebra off 0**0 --kratt6, Tue, 03 Jan 2006 05:22:52 -0600 replyThe 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$0^0$as 1 is ok whenever there is no notion of limit. However, axiom(0::EXPR INT)**(0::EXPR INT) \begin{equation} \label{eq2}1 \end{equation} 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 replyLet's see if the same happens here: axiomsinCosProducts := 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 \begin{equation} \label{eq3}\left\{ {{ \%H \ {\sin \left( {x} \right)} \ {\sin \left( {y} \right)}} \mbox{\rm == } {{-{ \%H \ {\cos \left( {{y+x}} \right)}}+{ \%H \ {\cos \left( {{y -x}} \right)}}} \over 2}}, \: {{ \%I \ {\cos \left( {x} \right)} \ {\cos \left( {y} \right)}} \mbox{\rm == } {{{ \%I \ {\cos \left( {{y+x}} \right)}}+{ \%I \ {\cos \left( {{y -x}} \right)}}} \over 2}}, \: {{ \%J \ {\cos \left( {y} \right)} \ {\sin \left( {x} \right)}} \mbox{\rm == } {{{ \%J \ {\sin \left( {{y+x}} \right)}} -{ \%J \ {\sin \left( {{y -x}} \right)}}} \over 2}} \right\} \end{equation} Type: Ruleset(Integer,Integer,Expression Integer) when typing --Bill Page, Mon, 30 Jan 2006 09:00:02 -0600 replyWhen 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 replyaxiom)lib RINTERPA RINTERP PCDEN GUESS GUESSINT GUESSP )library cannot find the file RINTERPA. )library cannot find the file RINTERP. )library cannot find the file PCDEN. )library cannot find the file GUESS. )library cannot find the file GUESSINT. )library cannot find the file GUESSP. guess(n, [1, 5, 14, 34, 69, 135, 240, 416, 686, 1106], n+->n, [guessRat], [guessSum, guessProduct, guessOne],2)$GuessInteger
GuessInteger is not a valid type.
conversion failed --Bill Page,  Thu, 23 Mar 2006 22:21:41 -0600 replyUnknown 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:
axiommyfn(i : Expression Integer) : Expression Integer == i
Function declaration myfn : Expression Integer -> Expression Integer
Type: Void
axiommyfn(x)
axiomCompiling function myfn with type Expression Integer -> Expression
Integer
\begin{equation}
\label{eq4}x
\end{equation}
Type: Expression Integer
axiomz:=sum(myfn(x),x=1..10)
\begin{equation}
\label{eq5}55
\end{equation}
Type: Expression Integer
Any hints for multivariate functions? --Bill (Name omitted),  Fri, 24 Mar 2006 21:45:25 -0600 replyHi 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.
axioma(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
axiomoutput(a(4,3,0.25)) -- see that the function actually evaluates for sensible values
axiomCompiling function a with type (Expression Integer,Expression
Integer,Expression Float) -> Expression Float
0.046875
Type: Void
axiomz := 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
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 "$" to specify which version of the function you need. output(z) 55 Type: Void I did notice in the Axiom online book, chapter 6.6, around page 241, the recommendation to use untyped functions, which appears to allow Axiom to do inference on parameter and result type. axiomb(n, k, p) == binomial(n,k) * p**(k) * (1.0-p)**(n-k) Type: Void axiomoutput(b(4,3,0.25)) -- see that the function actually evaluates for sensible values axiomCompiling function b with type (PositiveInteger,PositiveInteger, Float) -> Float 0.046875 Type: Void axiomz := sum(b(4,i,0.25), i=1..3) --- this fails axiomCompiling function b with type (PositiveInteger,Variable i,Float) -> Expression Float There are 6 exposed and 2 unexposed library operations named sum having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op sum to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named sum with argument type(s) Expression Float SegmentBinding PositiveInteger Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.
output(z)
55
Type: Void
For univariate functions the approach
axiomc(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
axiomoutput(c(3)) -- test to see if function can be evaluated for sensible arguments
axiomCompiling function c with type PositiveInteger -> Float
0.046875
Type: Void
axiomz := sum(c(i), i=1..3) -- still doesn't work
axiomCompiling 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
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 axiomd(k) == 1.5 * k -- coerce uotput to be a Float Type: Void axiomz := sum(d(i), i=1..3) -- This works! axiomCompiling function d with type Variable i -> Polynomial Float \begin{equation} \label{eq6}9.0 \end{equation} Type: Fraction Polynomial Float axiomoutput(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 replyTry this axiomz := reduce(+,[b(4,i,0.25) for i in 1..3]) \begin{equation} \label{eq7}0.6796875 \end{equation} Type: Float Handling the result from functions returning a matrix --Bill (Name omitted), Mon, 27 Mar 2006 08:10:26 -0600 replyHi 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. axiomCFM(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]] \begin{equation*} \label{eq8}\left[ \begin{array}{ccc} {0.0} & {0.5} & {0.0} \ {0.5} & {0.0} & {0.5} \ {0.0} & {0.5} & {0.0} \end{array} \right] \end{equation*} Type: Matrix Float axiomoutput(X) +0.0 0.5 0.0+ | | |0.5 0.0 0.5| | | +0.0 0.5 0.0+ Type: Void axiomN := 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. axiomCFM(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]] \begin{equation*} \label{eq9}\left[ \begin{array}{ccc} {0.0} & {0.5} & {0.0} \ {0.5} & {0.0} & {0.5} \ {0.0} & {0.5} & {0.0} \end{array} \right] \end{equation*} Type: Matrix Float axiomoutput(X) +0.0 0.5 0.0+ | | |0.5 0.0 0.5| | | +0.0 0.5 0.0+ Type: Void axiomN : 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 replyaxiom)set output tex off )set output algebra on FunFun := x**4 - 6* x**3 + 11* x*x + 2* x + 1 4 3 2 (28) x - 6x + 11x + 2x + 1 Type: Polynomial Integer axiomradicalSolve(FunFun) (29) [ 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 )set output algebra off Matthias axiomt:=matrix ([[0,1,1],[1,-2,2],[1,2,-1]]) \begin{equation*} \label{eq10}\left[ \begin{array}{ccc} 0 & 1 & 1 \ 1 & -2 & 2 \ 1 & 2 & -1 \end{array} \right] \end{equation*} Type: Matrix Integer We cat diagonalise t by finding it's eigenvalues. axiom)set output tex off )set output algebra on e:=radicalEigenvectors(t) (31) [ +-----------------+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) axiomd:=diagonalMatrix([e.1.radval,e.2.radval,e.3.radval]) Function definition for d is being overwritten. Compiled code for d has been cleared. (32) +-----------------+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: axiomp:=horizConcat(horizConcat(e.1.radvect.1,e.2.radvect.1),e.3.radvect.1) (33) [ [ +-----------------+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 axiomp*d*inverse(p) +0 1 1 + | | (34) |1 - 2 2 | | | +1 2 - 1+ Type: Matrix Expression Integer axiom)set output tex on )set output algebra off \end{axiom} Axiom can't integrame exp(x^4) ;( --unknown, Fri, 28 Apr 2006 14:03:28 -0500 replyAxiom can't integrame exp(x^4) ;( axiomintegrate(exp(x**4),x) \begin{equation} \label{eq11}\int \sp{\displaystyle x} {{e \sp { \%Q \sp 4}} \ {d \%Q}} \end{equation} Type: Union(Expression Integer,...) But Maple can... axiomf(x) == (1/4)*x*(-Gamma(1/4,-x**4)*Gamma(3/4)+%pi*sqrt(2))/((-x**4)**(1/4)*Gamma(3/4)) Type: Void axiomD(f(x),x) axiomCompiling function f with type Variable x -> Expression DoubleFloat \begin{equation} \label{eq12}e \sp {x \sp 4} \end{equation} Type: Expression DoubleFloat? Axiom cannot integrate e^(4*x) --kratt6, Fri, 28 Apr 2006 16:34:16 -0500 replyThis 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 replyThis is both obviously wrong since the integrand is a positive function: axiomintegrate(1/(1+x^4),x=%minusInfinity..%plusInfinity) \begin{equation} \label{eq13}0 \end{equation} Type: Union(f1: OrderedCompletion? Expression Integer,...) axiomnumeric(integrate(1/(1+x^4),x=0..1)) \begin{equation} \label{eq14}-{0.2437477471 9968052418} \end{equation} Type: Float ... --unknown, Wed, 24 May 2006 04:31:40 -0500 replyaxiom)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 \begin{equation} \label{eq15}32 \end{equation} Type: PositiveInteger? axiomy : FARRAY INT := new(n,1) \begin{equation*} \label{eq16}\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, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1 \right] \end{equation*} Type: FlexibleArray? Integer axiomn0 := n \begin{equation} \label{eq17}32 \end{equation} Type: PositiveInteger? axiomn1 := sum(x^1, x=0..n-1) \begin{equation} \label{eq18}496 \end{equation} Type: Fraction Polynomial Integer axiomn2 := sum(x^2, x=0..n-1) \begin{equation} \label{eq19}10416 \end{equation} Type: Fraction Polynomial Integer axiomn3 := sum(x^3, x=0..n-1) \begin{equation} \label{eq20}246016 \end{equation} Type: Fraction Polynomial Integer axiomn4 := sum(x^4, x=0..n-1) \begin{equation} \label{eq21}6197520 \end{equation} Type: Fraction Polynomial Integer axiomA := matrix([[n4, n3, n2],_ [n3, n2, n1],_ [n2, n1, n0]]) \begin{equation*} \label{eq22}\left[ \begin{array}{ccc} {6197520} & {246016} & {10416} \ {246016} & {10416} & {496} \ {10416} & {496} & {32} \end{array} \right] \end{equation*} Type: Matrix Fraction Polynomial Integer axiomX := vector([x1, x2, x3]) \begin{equation*} \label{eq23}\left[ x1, \: x2, \: x3 \right] \end{equation*} Type: Vector OrderedVariableList? [x1,x2,x3]? axiomB := vector([sum(x^2* u, x=0..n-1),_ sum(x* v, x=0..n-1),_ sum( w, x=0..n-1)]) \begin{equation*} \label{eq24}\left[ {{10416} \ u}, \: {{496} \ v}, \: {{32} \ w} \right] \end{equation*} Type: Vector Fraction Polynomial Integer axiomsolve([A * X = B], [x1, x2, x3]) There are 20 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 replyaxiomintegrate(1/((x+t)*sqrt(1+(x*t)**2)),t=0..%plusInfinity,"noPole")
\begin{equation}
\label{eq25}{-{\log
\left(
{{{{{\left( {2 \  {x \sp {10}}} -{4 \  {x \sp 8}}+{6 \  {x \sp 6}} -{6 \  {x
\sp 4}}+{4 \  {x \sp 2}} -2
\right)}
\  {\sqrt {{{x \sp 4}+1}}}}+{2 \  {x \sp {12}}} -{4 \  {x \sp {10}}}+{7 \  {x
\sp 8}} -{8 \  {x \sp 6}}+{7 \  {x \sp 4}} -{4 \  {x \sp 2}}+2} \over {x \sp
2}}}
\right)}+{\log
\left(
{{{x \sp 6}+{x \sp 2}}}
\right)}}
\over {2 \  {\sqrt {{{x \sp 4}+1}}}}
\end{equation}
Type: Union(f1: OrderedCompletion? Expression Integer,...)
axiomsubst(%,x=1)
\begin{equation}
\label{eq26}0
\end{equation}
Type: Expression Integer
axiomintegrate(1/((1+t)*sqrt(1+(1*t)**2)),t=0..%plusInfinity,"noPole")
\begin{equation}
\label{eq27}{{\sqrt {2}} \  {\log
\left(
{{{{12} \  {\sqrt {2}}}+{17}}}
\right)}}
\over 4
\end{equation}
Type: Union(f1: OrderedCompletion? Expression Integer,...)
axiomsimplify(%-subst((asinh(x^2)+asinh(1/x^2))/sqrt(1+x^4),x=1))
\begin{equation}
\label{eq28}{{\log
\left(
{{{{12} \  {\sqrt {2}}}+{17}}}
\right)}
-{4 \  {asinh
\left(
{1}
\right)}}}
\over {2 \  {\sqrt {2}}}
\end{equation}
Type: Expression Integer
axiom%::Expression Float
\begin{equation}
\label{eq29}0.0
\end{equation}
Type: Expression Float
... --unknown,  Mon, 03 Jul 2006 02:07:02 -0500 replyaxioma := 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]])
\begin{equation*}
\label{eq30}\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
\end{array}
\right]
\end{equation*}
Type: Matrix Integer
axiomdeterminant(a)
\begin{equation}
\label{eq31}0
\end{equation}
Type: NonNegativeInteger?
axiominverse(a)
\begin{equation}
\label{eq32}\mbox{\tt "failed"}
\end{equation}
Type: Union("failed",...)
... --unknown,  Fri, 07 Jul 2006 11:54:52 -0500 replya := 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 replyaxiomAs := matrix([ [-3,1,1,1], [1,1,1,1], [1,1,1,1], [1,1,1,1]])
\begin{equation*}
\label{eq33}\left[
\begin{array}{cccc}
-3 & 1 & 1 & 1 \
1 & 1 & 1 & 1 \
1 & 1 & 1 & 1 \
1 & 1 & 1 & 1
\end{array}
\right]
\end{equation*}
Type: Matrix Integer
axiomA := subMatrix(As, 2,4,2,4)
\begin{equation*}
\label{eq34}\left[
\begin{array}{ccc}
1 & 1 & 1 \
1 & 1 & 1 \
1 & 1 & 1
\end{array}
\right]
\end{equation*}
Type: Matrix Integer
axiomob := orthonormalBasis(A)
\begin{equation*}
\label{eq35}\left[
{\left[
\begin{array}{c}
-{{\sqrt {2}} \over {2 \  {\sqrt {3}}}} \
{{\sqrt {2}} \over {\sqrt {3}}} \
-{{\sqrt {2}} \over {2 \  {\sqrt {3}}}}
\end{array}
\right]},
\: {\left[
\begin{array}{c}
-{1 \over {\sqrt {2}}} \
0 \
{1 \over {\sqrt {2}}}
\end{array}
\right]},
\: {\left[
\begin{array}{c}
{1 \over {\sqrt {3}}} \
{1 \over {\sqrt {3}}} \
{1 \over {\sqrt {3}}}
\end{array}
\right]}
\right]
\end{equation*}
Type: List Matrix Expression Integer
axiomP : Matrix(Expression Integer) := new(3,3,0)
\begin{equation*}
\label{eq36}\left[
\begin{array}{ccc}
0 & 0 & 0 \
0 & 0 & 0 \
0 & 0 & 0
\end{array}
\right]
\end{equation*}
Type: Matrix Expression Integer
axiomsetsubMatrix!(P,1,1,ob.3)
\begin{equation*}
\label{eq37}\left[
\begin{array}{ccc}
{1 \over {\sqrt {3}}} & 0 & 0 \
{1 \over {\sqrt {3}}} & 0 & 0 \
{1 \over {\sqrt {3}}} & 0 & 0
\end{array}
\right]
\end{equation*}
Type: Matrix Expression Integer
axiomsetsubMatrix!(P,1,2,ob.1)
\begin{equation*}
\label{eq38}\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
\end{array}
\right]
\end{equation*}
Type: Matrix Expression Integer
axiomsetsubMatrix!(P,1,3,ob.2)
\begin{equation*}
\label{eq39}\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}}}
\end{array}
\right]
\end{equation*}
Type: Matrix Expression Integer
axiomPt := transpose(P)
\begin{equation*}
\label{eq40}\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}}}
\end{array}
\right]
\end{equation*}
Type: Matrix Expression Integer
axiomPs : Matrix(Expression Integer) := new(4,4,0)
\begin{equation*}
\label{eq41}\left[
\begin{array}{cccc}
0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 \
0 & 0 & 0 & 0
\end{array}
\right]
\end{equation*}
Type: Matrix Expression Integer
axiomPs(1,1) := 1
\begin{equation}
\label{eq42}1
\end{equation}
Type: Expression Integer
axiomsetsubMatrix!(Ps,2,2,P)
\begin{equation*}
\label{eq43}\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}}}
\end{array}
\right]
\end{equation*}
Type: Matrix Expression Integer
axiomPsT := transpose(Ps)
\begin{equation*}
\label{eq44}\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}}}
\end{array}
\right]
\end{equation*}
Type: Matrix Expression Integer
axiomPsTAsPs := PsT * As * Ps
\begin{equation*}
\label{eq45}\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
\end{array}
\right]
\end{equation*}
Type: Matrix Expression Integer
axiomb1 := PsTAsPs(2,1)
\begin{equation}
\label{eq46}3 \over {\sqrt {3}}
\end{equation}
Type: Expression Integer
axioml1 := PsTAsPs(2,2)
\begin{equation}
\label{eq47}3
\end{equation}
Type: Expression Integer
axiomUs : Matrix(Expression Integer) := new(4,4,0)
\begin{equation*}
\label{eq48}\left[
\begin{array}{cccc}
0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 \
0 & 0 & 0 & 0
\end{array}
\right]
\end{equation*}
Type: Matrix Expression Integer
axiomUs(1,1) := 1
\begin{equation}
\label{eq49}1
\end{equation}
Type: Expression Integer
axiomUs(2,2) := 1
\begin{equation}
\label{eq50}1
\end{equation}
Type: Expression Integer
axiomUs(3,3) := 1
\begin{equation}
\label{eq51}1
\end{equation}
Type: Expression Integer
axiomUs(4,4) := 1
\begin{equation}
\label{eq52}1
\end{equation}
Type: Expression Integer
axiomUs(2,1) := -b1 / l1
\begin{equation}
\label{eq53}-{1 \over {\sqrt {3}}}
\end{equation}
Type: Expression Integer
axiomPsUs := Ps * Us
\begin{equation*}
\label{eq54}\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}}}
\end{array}
\right]
\end{equation*}
Type: Matrix Expression Integer
axiomPsUsT := transpose(PsUs)
\begin{equation*}
\label{eq55}\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}}}
\end{array}
\right]
\end{equation*}
Type: Matrix Expression Integer
axiomPsUsTAsPsUs := PsUsT * As * PsUs
\begin{equation*}
\label{eq56}\left[
\begin{array}{cccc}
-4 & 0 & 0 & 0 \
0 & 3 & 0 & 0 \
0 & 0 & 0 & 0 \
0 & 0 & 0 & 0
\end{array}
\right]
\end{equation*}
Type: Matrix Expression Integer
axiomC := inverse(PsUs)
\begin{equation*}
\label{eq57}\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}
\end{array}
\right]
\end{equation*}
Type: Union(Matrix Expression Integer,...)
axiomc := PsUsTAsPsUs(1,1)
\begin{equation}
\label{eq58}-4
\end{equation}
Type: Expression Integer
axiomgQ := PsUsTAsPsUs / c
\begin{equation*}
\label{eq59}\left[
\begin{array}{cccc}
1 & 0 & 0 & 0 \
0 & -{3 \over 4} & 0 & 0 \
0 & 0 & 0 & 0 \
0 & 0 & 0 & 0
\end{array}
\right]
\end{equation*}
Type: Matrix Expression Integer
axiomx1 := transpose(matrix([[1,2,3,4]]))
\begin{equation*}
\label{eq60}\left[
\begin{array}{c}
1 \
2 \
3 \
4
\end{array}
\right]
\end{equation*}
Type: Matrix Integer
axiomv1 := transpose(x1) * As * x1
\begin{equation*}
\label{eq61}\left[
\begin{array}{c}
{96}
\end{array}
\right]
\end{equation*}
Type: Matrix Integer
axiomx2 := C * x1
\begin{equation*}
\label{eq62}\left[
\begin{array}{c}
1 \
{{{10} \  {\sqrt {3}}} \over 3} \
0 \
{\sqrt {2}}
\end{array}
\right]
\end{equation*}
Type: Matrix Expression Integer
axiomv2 := transpose(x2) * PsUsTAsPsUs * x2
\begin{equation*}
\label{eq63}\left[
\begin{array}{c}
{96}
\end{array}
\right]
\end{equation*}
Type: Matrix Expression Integer
graphics --unknown,  Tue, 01 Aug 2006 02:17:12 -0500 replyaxiomdraw(y**2/2+(x**2-1)**2/4-1=0, x,y, range ==[-2..2, -1..1])
There are 20 exposed and 18 unexposed library operations named **
having 2 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op **
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 **
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 replyaxiomf1 := taylor(1 - x**2,x = 0) \begin{equation} \label{eq64}1 -{x \sp 2} \end{equation} Type: UnivariateTaylorSeries?(Expression Integer,x,0) axiomasin f1 \begin{equation} \label{eq65}{\pi \over 2} -{{1 \over {\sqrt {2}}} \ {x \sp 2}} -{{1 \over {8 \ {\sqrt {2}}}} \ {x \sp 4}} -{{1 \over {{32} \ {\sqrt {2}}}} \ {x \sp 6}} -{{5 \over {{512} \ {\sqrt {2}}}} \ {x \sp 8}} -{{7 \over {{2048} \ {\sqrt {2}}}} \ {x \sp {10}}}+{O \left( {{x \sp {11}}} \right)} \end{equation} Type: UnivariateTaylorSeries?(Expression Integer,x,0) axiomsin % \begin{equation} \label{eq66}1 -{{1 \over 4} \ {x \sp 4}} -{{1 \over {16}} \ {x \sp 6}} -{{7 \over {768}} \ {x \sp 8}} -{{5 \over {3072}} \ {x \sp {10}}}+{O \left( {{x \sp {11}}} \right)} \end{equation} Type: UnivariateTaylorSeries?(Expression Integer,x,0) SandboxMSkuce? axiom1+1 \begin{equation} \label{eq67}2 \end{equation} Type: PositiveInteger? SandBoxCS224? integration --jhnbk, Fri, 08 Jun 2007 03:50:35 -0500 replyaxiomintegrate((x-1)/log(x), x) \begin{equation} \label{eq68}\int \sp{\displaystyle x} {{{ \%R -1} \over {\log \left( { \%R} \right)}} \ {d \%R}} \end{equation} Type: Union(Expression Integer,...) axiomintegrate(x*exp(x)*sin(x),x) \begin{equation} \label{eq69}{{x \ {e \sp x} \ {\sin \left( {x} \right)}}+{{\left( -x+1 \right)} \ {\cos \left( {x} \right)} \ {e \sp x}}} \over 2 \end{equation} Type: Union(Expression Integer,...) Working With Lists --daneshpajouh, Sat, 16 Jun 2007 07:00:00 -0500 replyaxiom[p for p in primes(2,1000)|(p rem 16)=1] \begin{equation*} \label{eq70}\left[ {977}, \: {929}, \: {881}, \: {769}, \: {673}, \: {641}, \: {593}, \: {577}, \: {449}, \: {433}, \: {401}, \: {353}, \: {337}, \: {257}, \: {241}, \: {193}, \: {113}, \: {97}, \: {17} \right] \end{equation*} Type: List Integer axiom[p**2+1 for p in primes(2,100)] \begin{equation*} \label{eq71}\left[ 5, \: {9410}, \: {7922}, \: {6890}, \: {6242}, \: {5330}, \: {5042}, \: {4490}, \: {3722}, \: {3482}, \: {2810}, \: {2210}, \: {1850}, \: {1682}, \: {1370}, \: {962}, \: {842}, \: {530}, \: {362}, \: {290}, \: {170}, \: {122}, \: {50}, \: {26}, \: {10} \right] \end{equation*} Type: List Integer axiomintegrate (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.
\end {axiom}
>> Error detected within library code:
index out of range
Is it error?
example from my daughter's college calc --pbwagner,  Mon, 10 Sep 2007 13:00:06 -0500 replyintegrate(log(log(x)),x)
(better) example (with axiom markers this time) ;-) --pbwagner,  Mon, 10 Sep 2007 13:01:48 -0500 replyaxiomintegrate(log(log(x)),x)
\begin{equation}
\label{eq72}{x \  {\log
\left(
{{\log
\left(
{x}
\right)}}
\right)}}
-{li
\left(
{x}
\right)}
\end{equation}
Type: Union(Expression Integer,...)

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Some or all expressions may not have rendered properly, because Latex returned the following error:
! LaTeX Error: Environment reduce undefined.
See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
...
l.505 \begin{reduce}
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