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Editor: Bill Page Time: 2007/09/12 12:23:06 GMT-7
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<H2><A NAME="htoc18">Appendix B</A>&nbsp;&nbsp;Examples</H2>

Polynomials, rational functions:

\begin{reduce}
coeff(X**3 + 3*X**2*Y + 3*X*Y**2 + Y**3,x);
gcd(X**2 + 4*X + 3,X**2 - 2*X - 3);
resultant(X**2 + 4*X + 3,X**2 - 2*X - 3,x);
decompose(x**6+6x**4+x**3+9x**2+3x-5);
factorize(x**6+6x**4+x**3+9x**2+3x);
roots(x**6+6x**4+x**3+9x**2+3x-5);
interpol({0,7,26,63},z,{1,2,3,4});
\end{reduce}

partial fraction decomposition:

\begin{reduce}
pf(2/((x+1)^2*(x+2)),x); 
\end{reduce}

Matrices:

\begin{reduce} 
m:=mat((1,x),(2,y));
1/m;
det m;
\end{reduce}

Ordinary differential equations:

\begin{reduce}
load odesolve;
odesolve(df(y(x),x)=y(x)+x**2+2,y(x),x);
\end{reduce}

Linear system (hidden):

\begin{reduce}
solve({(a*x+y)/(z-1)-3,y+b+z,x-y},
      {x,y,z});
\end{reduce}

Transcendental equations:

\begin{reduce}
solve(a**(2*x)-3*a**x+2,x);
\end{reduce}

Polynomial systems:

\begin{reduce}
solve(
 { a*c1 - b*c1**2 - g*c1*c2 + e*c3,
   -g*c1*c2 + (e+t)*c3 -k*c2,
   g*c1*c2 + k*c2 - (e+t) * c3},
  {c3,c2,c1});
\end{reduce}

Structural analysis:

\begin{reduce}
load_package compact;
compact(s*(1-(sin x**2))
       +c*(1-(cos x)**2)
       +(sin x)**2+(cos x)**2,
   {cos x^2+sin x^2=1});
\end{reduce}

Calculus:

\begin{reduce} 
df(exp(x**2)/x,x,2);
int(x^3*exp(2x),x);
limit(x*sin(1/x),x,infinity);
\end{reduce}

Series:

\begin{reduce}
on rounded;
taylor(sin(x+1),x,0,4);
sum(n,n);
prod(n/(n+2),n);
\end{reduce}

Complex numbers:

\begin{reduce}
w:=(x+3*i)**2;
\end{reduce}

Rounded numbers:

\begin{reduce} 
precision 25;
pi**2;
\end{reduce}

Modular numbers:

\begin{reduce} 
on modular;
setmod 17;
(x-1)**2;
factorize ws;
\end{reduce}

Appendix B Examples

Polynomials, rational functions:

coeff(X**3 + 3*X**2*Y + 3*X*Y**2 + Y**3,x);

reduce

$\displaylines{\qdd \{y^{3}\co 3\cdot y^{2}\co 3\cdot y\co 1 \} \cr}$

gcd(X**2 + 4*X + 3,X**2 - 2*X - 3);

reduce

$\displaylines{\qdd x +1 \cr}$

resultant(X**2 + 4*X + 3,X**2 - 2*X - 3,x);

reduce

$\displaylines{\qdd 0 \cr}$

decompose(x**6+6x**4+x**3+9x**2+3x-5);

reduce

$\displaylines{\qdd \{u^{2} +u -5\co u= \[x^{3} +3\cdot x$

factorize(x**6+6x**4+x**3+9x**2+3x);

reduce

$\displaylines{\qdd \{\{x^{3} +3\cdot x +1\co 1 \} \co \{x^{2} +3\co 1 \} \co \{x\co 1 \} \} \cr}$

roots(x**6+6x**4+x**3+9x**2+3x-5);

reduce

$\displaylines{\qdd \{x= \[-0.775167$

interpol({0,7,26,63},z,{1,2,3,4});

reduce

$\displaylines{\qdd z^{3} -1 \cr}$

partial fraction decomposition:

pf(2/((x+1)^2*(x+2)),x);

reduce

$\displaylines{\qdd \{\frac{2}{ x +2}\co \frac{-2}{ x +1}\co \frac{2}{ x^{2} +2\cdot x +1} \} \cr}$

Matrices:

m:=mat((1,x),(2,y));

reduce

$\displaylines{\qdd m:= \[\pmatrix{1&x\cr 2&y\cr }$

1/m;

reduce

$\displaylines{\qdd \pmatrix{\frac{-y}{ 2\cdot x -y} & \frac{x}{ 2\cdot x -y}\cr \frac{2}{ 2\cdot x -y}& \frac{-1}{ 2\cdot x -y}\cr } \Nl}$

det m;

reduce

$\displaylines{\qdd -2\cdot x +y \cr}$

Ordinary differential equations:

load odesolve;

odesolve(df(y(x),x)=y(x)+x**2+2,y(x),x);

*** y declared operator

reduce

$\displaylines{\qdd \{y \(x$

Linear system (hidden):

solve({(a*x+y)/(z-1)-3,y+b+z,x-y},
      {x,y,z});

reduce

$\displaylines{\qdd \{\{x= \[\frac{-3\cdot \(b +1$

Transcendental equations:

solve(a**(2*x)-3*a**x+2,x);

reduce

$\displaylines{\qdd \{x= \[\frac{2\cdot arbint \(2$

Polynomial systems:

solve(
 { a*c1 - b*c1**2 - g*c1*c2 + e*c3,
   -g*c1*c2 + (e+t)*c3 -k*c2,
   g*c1*c2 + k*c2 - (e+t) * c3},
  {c3,c2,c1});

reduce

$\displaylines{\qdd \{\{c_{3}= \[\frac{c_{1}\cdot \(-c_{1}^{2}\cdot b\cdot g +c_{1}\cdot a\cdot g -c_{1}\cdot b\cdot k +a\cdot k$

Structural analysis:

load_package compact;

compact(s*(1-(sin x**2))
       +c*(1-(cos x)**2)
       +(sin x)**2+(cos x)**2,
   {cos x^2+sin x^2=1});

reduce

$\displaylines{\qdd \cos \(x$

Calculus:

df(exp(x**2)/x,x,2);

reduce

$\displaylines{\qdd \frac{2\cdot e^{x^{2}}\cdot \(2\cdot x^{4} -x^{2} +1$

int(x^3*exp(2x),x);

reduce

$\displaylines{\qdd \frac{e^{2\cdot x}\cdot \(4\cdot x^{3} -6\cdot x^{2} +6\cdot x -3$

limit(x*sin(1/x),x,infinity);

reduce

$\displaylines{\qdd 1 \cr}$

Series:

on rounded;

taylor(sin(x+1),x,0,4);

reduce

$\displaylines{\qdd 0.841470984808 +0.540302305868\cdot x -0.420735492404\cdot x^{2}\nl -0.0900503843114\cdot x^{3} +0.0350612910337\cdot x^{4} +O \(x^{5}$

sum(n,n);

reduce

$\displaylines{\qdd 0.5\cdot n\cdot \(n +1$

prod(n/(n+2),n);

reduce

$\displaylines{\qdd \frac{2}{ n^{2} +3\cdot n +2} \cr}$

Complex numbers:

w:=(x+3*i)**2;

reduce

$\displaylines{\qdd w:= \[6\cdot i\cdot x +x^{2} -9$

Rounded numbers:

precision 25;

reduce

$\displaylines{\qdd 12 \cr}$

pi**2;

reduce

$\displaylines{\qdd 9.869604401089358618834491 \cr}$

Modular numbers:

on modular;

*** Domain mode rounded changed to modular 

setmod 17;

reduce

$\displaylines{\qdd 1 \cr}$

(x-1)**2;

reduce

$\displaylines{\qdd x^{2} +15\cdot x +1 \cr}$

factorize ws;

reduce

$\displaylines{\qdd \{\{x +16\co 2 \} \} \cr}$