Quick Demo
system:sage
{{{id=0|
2 + 3
///
5
}}}
{{{id=21|
}}}
{{{id=1|
show(plot(sin(x^2), 0, pi))
}}}
{{{id=22|
}}}
{{{id=2|
a = integrate(sin(x^2),x); a
///
sqrt(pi)*((sqrt(2)*I + sqrt(2))*erf((sqrt(2)*I + sqrt(2))*x/2) + (sqrt(2)*I - sqrt(2))*erf((sqrt(2)*I - sqrt(2))*x/2))/8
}}}
{{{id=3|
show(a)
///
\frac{{\sqrt{ \pi } \cdot \left( {\left( {\sqrt{ 2 } \cdot i} + \sqrt{ 2 } \right) \cdot \left( \text{erf} \left( \frac{{\left( {\sqrt{ 2 } \cdot i} + \sqrt{ 2 } \right) \cdot x}}{2} \right) \right)} + {\left( {\sqrt{ 2 } \cdot i} - \sqrt{ 2 } \right) \cdot \left( \text{erf} \left( \frac{{\left( {\sqrt{ 2 } \cdot i} - \sqrt{ 2 } \right) \cdot x}}{2} \right) \right)} \right)}}{8}
}}}
{{{id=23|
}}}
{{{id=10|
m = random_matrix(QQ,5)^(-1); m
///
[ -3 -2 -3 -2 1]
[ -5 -4 -6 -4 2]
[-3/2 -1 -1 -1 1/2]
[ 0 -1/2 -1/2 0 0]
[ 1 1 1 1 0]
}}}
{{{id=20|
#m.[tab]
}}}
{{{id=11|
latex(m)
///
\left(\begin{array}{rrrrr}
-3&-2&-3&-2&1\\
-5&-4&-6&-4&2\\
-\frac{3}{2}&-1&-1&-1&\frac{1}{2}\\
0&-\frac{1}{2}&-\frac{1}{2}&0&0\\
1&1&1&1&0
\end{array}\right)
}}}
{{{id=12|
show(m)
///
\left(\begin{array}{rrrrr}
-3&-2&-3&-2&1\\
-5&-4&-6&-4&2\\
-\frac{3}{2}&-1&-1&-1&\frac{1}{2}\\
0&-\frac{1}{2}&-\frac{1}{2}&0&0\\
1&1&1&1&0
\end{array}\right)
}}}
{{{id=18|
}}}
{{{id=4|
time n = number_of_partitions(10^8)
///
CPU time: 6.01 s, Wall time: 8.28 s
}}}
{{{id=6|
%mathematica
Timing[N[Log[PartitionsP[10^8]]]]
///
{13.6169, 25630.640277658734642}
}}}
{{{id=17|
}}}
{{{id=13|
e = EllipticCurve([1,2,3,4,5])
print e.aplist(20)
time n = e.aplist(10^6)
///
[1, 0, -3, -1, -1, 1, 5, 4]
Time: CPU 3.68 s, Wall: 5.30 s
}}}
{{{id=5|
%magma
print GetVersion();
E := EllipticCurve([1,2,3,4,5]);
print TracesOfFrobenius(E, 20);
time n := TracesOfFrobenius(E,10^6);
///
2 13 10
[ 1, 0, -3, -1, -1, 1, 5, 4 ]
Time: 8.570
}}}
{{{id=16|
}}}
{{{id=9|
var('a,b,c,X')
s = solve(a*X^2 + b*X + c == 0, X)
show(s[0])
///
X = \frac{-\sqrt{ {b}^{2} - {{4 \cdot a} \cdot c} } - b}{{2 \cdot a}}
}}}
{{{id=8|
%mathematica
Solve[a*X^2 + b*X +c == 0, X]
///
2 2
-b - Sqrt[b - 4 a c] -b + Sqrt[b - 4 a c]
{{X -> ---------------------}, {X -> ---------------------}}
2 a 2 a
}}}
{{{id=7|
# In SAGE
a = random_matrix(ZZ,300, x=-9, y=9)
time f = a.charpoly()
print log(abs(f[0])*1.0)
///
Time: CPU 10.04 s, Wall: 13.03 s
1200.38548128654
}}}
{{{id=14|
%magma
n := 300;
a := MatrixAlgebra(IntegerRing(), n)![Random(-9,9) : i in [1..n^2]];
time f := CharacteristicPolynomial(a);
print Log(AbsoluteValue(Coefficient(f,0))*1.0)
///
Time: 7.640
1214.76576501968242382405874440
}}}
{{{id=15|
# In SAGE
n = 100
a = random_matrix(QQ,n, n+1, num_bound=2^64, den_bound=1)
time a.echelonize()
///
Time: CPU 0.31 s, Wall: 0.41 s
}}}
{{{id=26|
%magma
n := 100;
a := RMatrixSpace(RationalField(), n,n+1)![Random(1,2^64): i in [1..n*(n+1)]];
time e := EchelonForm(a);
///
Time: 6.390
}}}
{{{id=27|
# In SAGE
n = 200
a = random_matrix(QQ,n, n+1, num_bound=2^64, den_bound=1)
time a.echelonize()
///
Time: CPU 1.96 s, Wall: 3.22 s
}}}
{{{id=29|
%magma
n := 200;
a := RMatrixSpace(RationalField(), n,n+1)![Random(1,2^64): i in [1..n*(n+1)]];
time e := EchelonForm(a);
///
Time: 159.920
}}}
{{{id=30|
}}}