FEMTEC2009 examples

From SymPy

Integration

<source lang="python"> In [1]: integrate(sin(x)*exp(x), x) In [2]: integrate(sin(x)*exp(2*x), x) In [3]: integrate(1/(x**2+1), x) In [4]: integrate(1/(x**4+1), x) ... >>> integrate(x/(x**2+1), x) >>> integrate(x/(x**2+1), x).diff(x) >>> simplify(integrate(x/(x**2+1), x).diff(x))

In [1]: integrate(LambertW(x), x) In [2]: _.diff(x) In [3]: trim(_) In [4]: var('a b') In [5]: integrate(a*LambertW(b*x), x) In [6]: _.diff(x) In [7]: trim(_) In [8]: integrate(sin(2*sqrt(x)), x) In [9]: f = (x-tan(x))/tan(x)**2 + tan(x) In [10]: integrate(f, x) </source>

Linear Solving

<source lang="python"> In [11]: var('a b c d e f g h i') In [12]: f_1 = a*x + b*y + c*z - 1 In [13]: f_2 = d*x + e*y + f*z - 1 In [14]: f_3 = g*x + h*y + i*z - 1 In [15]: s = solve([f_1, f_2, f_3], x, y, z) In [16]: s In [17]: f_1.subs(s) In [18]: simplify(f_1.subs(s)) In [19]: simplify(f_2.subs(s)) In [20]: simplify(f_3.subs(s)) </source>

Nonlinear Solving

<source lang="python"> In [21]: f_1 = x**2 + y + z - 1 In [22]: f_2 = x + y**2 + z - 1 In [23]: f_3 = x + y + z**2 - 1 In [24]: solve([f_1, f_2, f_3], x, y, z) </source>

Factorization

<source lang="python"> In [1]: factor(x**56-1, x) In [2]: f = x**16+4*x**15+14*x**14+32*x**13+47*x**12+92*x**11+66*x**10+120*x**9 \ -50*x**8-24*x**7-132*x**6-40*x**5-52*x**4-64*x**3-64*x**2-32*x+16

In [3]: f In [4]: factor(f, x) </source>

Term Rewriting

<source lang="python"> In [29]: 36/(x**5-2*x**4-2*x**3+4*x**2+x-2) In [30]: f = apart(36/(x**5-2*x**4-2*x**3+4*x**2+x-2), x) In [31]: f In [32]: f = apart(36/(x**5-2*x**4-2*x**3+4*x**2+x-2), x, evaluate=False) In [33]: f In [34]: Add(*[ g.doit() for g in f.args ]) </source>

Matrices

<source lang="python"> In [1]: M = Matrix(4, 4, lambda i,j: i*j+1) In [2]: M.eigenvals()

In [3]: M = Matrix(4, 4, lambda i,j: i*j+2) In [4]: M.eigenvals()

In [5]: M = Matrix(4, 4, lambda i,j: i*j+3) In [6]: M.eigenvals() In [7]: M

In [35]: M = Matrix(4, 4, lambda i,j: i*j+x) In [36]: M In [37]: M.eigenvals() </source>

General relativity example

<source lang="python"> $ examples/advanced/relativity.py </source>

Integrals table

<source lang="python"> from sympy import *

def derivative_table(functions, x):

   for f in functions:
       s = printing.latex(Eq(Derivative(f, x), diff(f, x)))
       print ":<math>" + s[1:-1] + "</math>", "\n"

def integral_table(functions, x):

   for f in functions:
       s = printing.latex(Eq(Integral(f, x), integrate(f, x)))
       print ":<math>" + s[1:-1] + "</math>", "\n"

var('x')

print "===Derivatives===" derivative_table([cos(x)/(1 + sin(x)**i) for i in range(1, 5)], x)

print "===Integrals===" integral_table([x**i * exp(i*x) for i in range(1, 5)], x) </source>

Million digits of pi

<source lang="python"> In [1]: %time a = pi.evalf(10**6) CPU times: user 7.44 s, sys: 0.06 s, total: 7.50 s Wall time: 7.51 s

In [2]: len(str(a)) Out[2]: 1000001

In [5]: str(a)[:50] Out[5]: 3.141592653589793238462643383279502884197169399375

In [6]: str(a)[-50:] Out[6]: 95678796130331164628399634646042209010610577945815 </source>

Numerical Integration

One-liner:

<source lang="python"> In [1]: Integral(sin(1/x), (x, 0, 1)).transform(x, 1/x).evalf(quad="osc") Out[1]: 0.504067061906928 </source>

Detailed steps:

<source lang="python"> In [1]: e = Integral(sin(1/x), (x, 0, 1)) In [2]: e In [3]: e.transform(x, 1/x) In [4]: e.transform(x, 1/x).evalf(quad="osc") In [5]: e.transform(x, 1/x).evalf(40, quad="osc") </source>

Numerical Summation

quickly convergent series:

<source lang="python"> >>> Sum((2*n**3+1)/factorial(2*n+1), (n, 0, oo)).evalf(1000) </source>

slowly convergent (polynomial rate) series:

<source lang="python"> >>> Sum(n/(n**3+9), (n, 1, oo)).evalf(1000) </source>

Numerical Simplification

<source lang="python"> In [4]: float(1/7) In [5]: nsimplify(_) In [6]: float(1/81) In [7]: nsimplify(_) >>> nsimplify(pi, tolerance=0.01) >>> nsimplify(pi, tolerance=0.001) >>> nsimplify(0.33333, tolerance=1e-4) >>> nsimplify(4.71, [pi], tolerance=0.01) >>> nsimplify(2.0**(1/3.), tolerance=0.001) >>> nsimplify(2.0**(1/3.), tolerance=0.001, full=True) >>> nsimplify(4/(1+sqrt(5)), [GoldenRatio]) >>> nsimplify(2 + exp(2*atan('1/4')*I)) >>> nsimplify((1/(exp(3*pi*I/5)+1))) >>> nsimplify(I**I, [pi]) >>> nsimplify(Sum(1/n**2, (n, 1, oo)), [pi]) >>> nsimplify(gamma('1/4')*gamma('3/4'), [pi]) </source>

Ordinary differential equations

<source lang="python"> In [1]: f = Function("f") In [2]: Eq(f(x).diff(x, x) + 9*f(x) + 1, 1) In [3]: dsolve(_, f(x))

In [11]: Eq(f(x).diff(x, x) + 8*f(x) + 1, 1) In [12]: dsolve(_, f(x)) </source>

Limits

<source lang="python"> $ vim sympy/series/tests/test_demidovich.py </source>

Plotting:

FEMTEC2009 examples

From SymPy

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