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| }}}