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Editor: 127.0.0.1
Time: 2007年11月16日 05:07:50 GMT-8
Note: make computations live 
changed:
- <ul>
-
- <li>
- <p>
- Some computations. An important thing: everything is
- <em>mathematically typed</em> in Axiom. <br/>
- <pre>
-
-(1) -> 1+1
-
- (1) 2
- Type: PositiveInteger
-(2) -> integrate(1/x^(1/3),x)
-
- 3+-+2
- 3\|x
- (2) ------
- 2
- Type: Union(Expression Integer,...)
- </pre>
- </p>
- Axiom is running under Windows. 
- </li>
-
- <li>
- Some matrix computations under <a href="http://texmacs.org/">TeXmacs</a>.
- <p>
- <img src="/uploads/screenshot3.png">
- </p>
- Please notice the hierarchical editing capabilities of TeXmacs.
- <p>
- <img src="http://axiom-wiki.newsynthesis.org/uploads/screenshot2.jpg" />
- </p>
- </li>
-
- <li>
- <p>Some more complicated computations:
- <pre>
A short demonstration of Axiom
 An important thing: all objects in Axiom have a <em>type</em>. This enables us to give a simple demonstration of the Cayley-Hamilton theorem. 
Let n equal 4. The semicolon at the end of the input tells Axiom not to display the result. Thus, only its type is shown:
\begin{axiom}
n := 4;
\end{axiom}
We define an abbreviation: let 'SM' be the ring of quadratic $n\times n$ matrices with rational functions as entries:
\begin{axiom}
SM ==> SquareMatrix(n, FRAC POLY INT)
\end{axiom}
Let 'M' be a generic 4ドル \times 4$ matrix:
\begin{axiom}
M: SM := matrix [[a[i,j] for j in 1..n] for i in 1..n]
\end{axiom}
Compute the characteristis polynomial of 'M':
\begin{axiom}
P := determinant (M - x * 1);
\end{axiom}
We now interpret 'P' as univariate polynomial in 'x', coefficients being from the 4ドル \times 4$ matrices. The double colon performs this coercion:
\begin{axiom}
Q := P::UP(x, SM);
\end{axiom}
Finally we evaluate this polynomial with the original matrix as argument. In Axiom you do not need to use parenthesis if the function takes only one argument:
\begin{axiom}
Q M
\end{axiom}
Some matrix computations under <a href="http://texmacs.org/">TeXmacs</a>.
<img src="/uploads/screenshot3.png">
Notice the hierarchical editing capabilities of TeXmacs.
<img src="/uploads/screenshot2.jpg" />
Some more complicated computations:
\begin{axiom}
changed:
- 
- All user variables and function definitions have been cleared.
-
-(1) -> Word := OrderedFreeMonoid(Symbol)
- 
-
- (1) OrderedFreeMonoid Symbol
- Type: Domain
-(2) -> poly:= XPR(Integer,Word)
- 
-
- (2) XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
- Type: Domain
-(3) -> p:poly := 2 * x - 3 * y + 1
- 
-
- (3) 1 + 2x - 3y
- Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
-(4) -> q:poly := 2 * x + 1
- 
-
- (4) 1 + 2x
- Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
-(5) -> p + q
- 
-
- (5) 2 + 4x - 3y
- Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
-(6) -> p * q
- 
-
- 2
- (6) 1 + 4x - 3y + 4x - 6y x
- Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
-(7) -> (p +q)^2 -p^2 -q^2 - 2*p*q
- 
-
- (7) - 6x y + 6y x
- Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
-(8) -> M := SquareMatrix(2,Fraction Integer)
- 
-
- (8) SquareMatrix(2,Fraction Integer)
- Type: Domain
-(9) -> poly1:= XPR(M,Word)
- 
-
- (9)
- XPolynomialRing(SquareMatrix(2,Fraction Integer),OrderedFreeMonoid Symbol)
- Type: Domain
-(10) -> m1:M := matrix [[i*j**2 for i in 1..2] for j in 1..2]
- 
-
- +1 2+
- (10) | |
- +4 8+
- Type: SquareMatrix(2,Fraction Integer)
-(11) -> m2:M := m1 - 5/4
- 
-
- + 1 +
- |- - 2 |
- | 4 |
- (11) | |
- | 27|
- | 4 --|
- + 4+
- Type: SquareMatrix(2,Fraction Integer)
-(12) -> m3: M := m2**2
- 
-
- +129 +
- |--- 13 |
- | 16 |
- (12) | |
- | 857|
- |26 ---|
- + 16+
- Type: SquareMatrix(2,Fraction Integer)
-(13) -> pm:poly1 := m1*x + m2*y + m3*z - 2/3
- 
-
- + 2 + + 1 + +129 +
- |- - 0 | |- - 2 | |--- 13 |
- | 3 | +1 2+ | 4 | | 16 |
- (13) | | + | |x + | |y + | |z
- | 2| +4 8+ | 27| | 857|
- | 0 - -| | 4 --| |26 ---|
- + 3+ + 4+ + 16+
-Type: XPolynomialRing(SquareMatrix(2,Fraction Integer),OrderedFreeMonoid Symbol)
-(14) -> qm:poly1 := pm - m1*x
- 
-
- + 2 + + 1 + +129 +
- |- - 0 | |- - 2 | |--- 13 |
- | 3 | | 4 | | 16 |
- (14) | | + | |y + | |z
- | 2| | 27| | 857|
- | 0 - -| | 4 --| |26 ---|
- + 3+ + 4+ + 16+
-Type: XPolynomialRing(SquareMatrix(2,Fraction Integer),OrderedFreeMonoid Symbol)
-(15) -> qm**3
- 
-
- (15)
- + 8 + + 1 8+ +43 52 + + 129 +
- |- -- 0 | |- - -| |-- -- | |- --- - 26 |
- | 27 | | 3 3| | 4 3 | | 8 | 2
- | | + | |y + | |z + | |y
- | 8| |16 | |104 857| | 857|
- | 0 - --| |-- 9| |--- ---| |- 52 - ---|
- + 27+ + 3 + + 3 12+ + 8 +
- + 
- + 3199 831 + + 3199 831 + + 103169 6409 +
- |- ---- - --- | |- ---- - --- | |- ------ - ---- |
- | 32 4 | | 32 4 | | 128 4 | 2
- | |y z + | |z y + | |z
- | 831 26467| | 831 26467| | 6409 820977|
- |- --- - -----| |- --- - -----| | - ---- - ------|
- + 2 32 + + 2 32 + + 2 128 +
- + 
- +3199 831 + +103169 6409 + +103169 6409 +
- |---- --- | |------ ---- | |------ ---- |
- | 64 8 | 3 | 256 8 | 2 | 256 8 |
- | |y + | |y z + | |y z y
- |831 26467| | 6409 820977| | 6409 820977|
- |--- -----| | ---- ------| | ---- ------|
- + 4 64 + + 4 256 + + 4 256 +
- + 
- +3178239 795341 + +103169 6409 + +3178239 795341 +
- |------- ------ | |------ ---- | |------- ------ |
- | 1024 128 | 2 | 256 8 | 2 | 1024 128 |
- | |y z + | |z y + | |z y z
- |795341 25447787| | 6409 820977| |795341 25447787|
- |------ --------| | ---- ------| |------ --------|
- + 64 1024 + + 4 256 + + 64 1024 +
- + 
- +3178239 795341 + +98625409 12326223 +
- |------- ------ | |-------- -------- |
- | 1024 128 | 2 | 4096 256 | 3
- | |z y + | |z
- |795341 25447787| |12326223 788893897|
- |------ --------| |-------- ---------|
- + 64 1024 + + 128 4096 +
-Type: XPolynomialRing(SquareMatrix(2,Fraction Integer),OrderedFreeMonoid Symbol)
- </pre>
- </p>
- </li>
-
- </ul>
-
-</div>
-<div id="credit">$Id: screenshots.html,v 1.8 2003年11月02日 09:43:28 dmentre Exp $
-</div>
-
Word := OrderedFreeMonoid(Symbol)
poly:= XPR(Integer,Word)
p:poly := 2 * x - 3 * y + 1
1 + 2x - 3y
q:poly := 2 * x + 1
p + q
p * q
(p +q)^2 -p^2 -q^2 - 2*p*q
M := SquareMatrix(2,Fraction Integer)
poly1:= XPR(M,Word)
m1:M := matrix [[i*j**2 for i in 1..2] for j in 1..2]
m2:M := m1 - 5/4
m3: M := m2**2
pm:poly1 := m1*x + m2*y + m3*z - 2/3
qm:poly1 := pm - m1*x
qm**3
\end{axiom}

A short demonstration of Axiom

An important thing: all objects in Axiom have a type. This enables us to give a simple demonstration of the Cayley-Hamilton theorem.

Let n equal 4. The semicolon at the end of the input tells Axiom not to display the result. Thus, only its type is shown:

axiom
n := 4;
Type: PositiveInteger?

We define an abbreviation: let SM be the ring of quadratic n\times n matrices with rational functions as entries:

axiom
SM ==> SquareMatrix(n, FRAC POLY INT)
Type: Void

Let M be a generic 4 \times 4 matrix:

axiom
M: SM := matrix [[a[i,j] for j in 1..n] for i in 1..n]
\label{eq1}\left[ \begin{array}{cccc} {a_{1, \: 1}}&{a_{1, \: 2}}&{a_{1, \: 3}}&{a_{1, \: 4}} \ {a_{2, \: 1}}&{a_{2, \: 2}}&{a_{2, \: 3}}&{a_{2, \: 4}} \ {a_{3, \: 1}}&{a_{3, \: 2}}&{a_{3, \: 3}}&{a_{3, \: 4}} \ {a_{4, \: 1}}&{a_{4, \: 2}}&{a_{4, \: 3}}&{a_{4, \: 4}} (1)
Type: SquareMatrix?(4,Fraction(Polynomial(Integer)))

Compute the characteristis polynomial of 'M':

axiom
P := determinant (M - x * 1);
Type: Fraction(Polynomial(Integer))

We now interpret P as univariate polynomial in x, coefficients being from the 4 \times 4 matrices. The double colon performs this coercion:

axiom
Q := P::UP(x, SM);
Type: UnivariatePolynomial?(x,SquareMatrix?(4,Fraction(Polynomial(Integer))))

Finally we evaluate this polynomial with the original matrix as argument. In Axiom you do not need to use parenthesis if the function takes only one argument:

axiom
Q M
\label{eq2}\left[ \begin{array}{cccc} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 (2)
Type: SquareMatrix?(4,Fraction(Polynomial(Integer)))

Some matrix computations under TeXmacs? .

Notice the hierarchical editing capabilities of TeXmacs?.

Some more complicated computations:

axiom
)cl all

 All user variables and function definitions have been cleared.
Word := OrderedFreeMonoid(Symbol)
\label{eq3}\hbox{\axiomType{OrderedFreeMonoid}\ } (\hbox{\axiomType{Symbol}\ }) (3)
Type: Type
axiom
poly:= XPR(Integer,Word)
\label{eq4}\hbox{\axiomType{XPolynomialRing}\ } (\hbox{\axiomType{Integer}\ } , \hbox{\axiomType{OrderedFreeMonoid}\ } (\hbox{\axiomType{Symbol}\ })) (4)
Type: Type
axiom
p:poly := 2 * x - 3 * y + 1
\label{eq5}1 +{2 \ x}-{3 \ y} (5)
Type: XPolynomialRing?(Integer,OrderedFreeMonoid?(Symbol))
axiom
1 + 2x - 3y

 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.
q:poly := 2 * x + 1
\label{eq6}1 +{2 \ x} (6)
Type: XPolynomialRing?(Integer,OrderedFreeMonoid?(Symbol))
axiom
p + q
\label{eq7}2 +{4 \ x}-{3 \ y} (7)
Type: XPolynomialRing?(Integer,OrderedFreeMonoid?(Symbol))
axiom
p * q
\label{eq8}1 +{4 \ x}-{3 \ y}+{4 \ {x^2}}-{6 \ y \ x} (8)
Type: XPolynomialRing?(Integer,OrderedFreeMonoid?(Symbol))
axiom
(p +q)^2 -p^2 -q^2 - 2*p*q
\label{eq9}-{6 \ x \ y}+{6 \ y \ x} (9)
Type: XPolynomialRing?(Integer,OrderedFreeMonoid?(Symbol))
axiom
M := SquareMatrix(2,Fraction Integer)
\label{eq10}\hbox{\axiomType{SquareMatrix}\ } (2, \hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Integer}\ })) (10)
Type: Type
axiom
poly1:= XPR(M,Word)
\label{eq11}\hbox{\axiomType{XPolynomialRing}\ } (\hbox{\axiomType{SquareMatrix}\ } (2, \hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Integer}\ })) , \hbox{\axiomType{OrderedFreeMonoid}\ } (\hbox{\axiomType{Symbol}\ })) (11)
Type: Type
axiom
m1:M := matrix [[i*j**2 for i in 1..2] for j in 1..2]
\label{eq12}\left[ \begin{array}{cc} 1 & 2 \ 4 & 8 (12)
Type: SquareMatrix?(2,Fraction(Integer))
axiom
m2:M := m1 - 5/4
\label{eq13}\left[ \begin{array}{cc} -{1 \over 4}& 2 \ 4 &{{27}\over 4} (13)
Type: SquareMatrix?(2,Fraction(Integer))
axiom
m3: M := m2**2
\label{eq14}\left[ \begin{array}{cc} {{129}\over{16}}&{13} \ {26}&{{857}\over{16}} (14)
Type: SquareMatrix?(2,Fraction(Integer))
axiom
pm:poly1 := m1*x + m2*y + m3*z - 2/3
\label{eq15}\begin{array}{@{}l} \displaystyle {\left[ \begin{array}{cc} -{2 \over 3}& 0 \ 0 & -{2 \over 3} (15)
Type: XPolynomialRing?(SquareMatrix?(2,Fraction(Integer)),OrderedFreeMonoid?(Symbol))
axiom
qm:poly1 := pm - m1*x
\label{eq16}\begin{array}{@{}l} \displaystyle {\left[ \begin{array}{cc} -{2 \over 3}& 0 \ 0 & -{2 \over 3} (16)
Type: XPolynomialRing?(SquareMatrix?(2,Fraction(Integer)),OrderedFreeMonoid?(Symbol))
axiom
qm**3
\label{eq17}\begin{array}{@{}l} \displaystyle {\left[ \begin{array}{cc} -{8 \over{27}}& 0 \ 0 & -{8 \over{27}} (17)
Type: XPolynomialRing?(SquareMatrix?(2,Fraction(Integer)),OrderedFreeMonoid?(Symbol))

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