Thu Aug  1 16:32:10 2024

sympy_test():
  python version: 3.10.12
  numpy version:  1.26.4
  sympy version:  1.9
  sympy() is a symbolic mathematics package.


sympy_derivative():
  Demonstrate the computation of derivatives.


derivative_humps():
  Differentiate humps(x).

  humps(x) =  -6 + 100/((10*x - 3)**2 + 1) + 100/((10*x - 9)**2 + 4)
  d humps /dx =  100*(60 - 200*x)/((10*x - 3)**2 + 1)**2 + 100*(180 - 200*x)/((10*x - 9)**2 + 4)**2
  d2 humps /dx2 =  20000*(4*(10*x - 9)**2/((10*x - 9)**2 + 4)**3 + 4*(10*x - 3)**2/((10*x - 3)**2 + 1)**3 - 1/((10*x - 3)**2 + 1)**2 - 1/((10*x - 9)**2 + 4)**2)

derivative_x_exp_xt():
  Differentiate z(x,t) = x*exp(x*t).

  z(x,t) =  x*exp(t*x)
  dzdx =  t*x*exp(t*x) + exp(t*x)
  dzdt =  x**2*exp(t*x)
  d2zdt2 =  x**3*exp(t*x)
  d2zdtdx =  x*(t*x + 2)*exp(t*x)

sympy_derivative():
  Normal end of execution.


sympy_evaluate():
  Show ways of evaluating the symbolic humps() function.

  humps.subs(x,y) =  -6 + 100/((10*y - 3)**2 + 1) + 100/((10*y - 9)**2 + 4)
  humps.subs(x,cos(y)) =  -6 + 100/((10*cos(y) - 3)**2 + 1) + 100/((10*cos(y) - 9)**2 + 4)
  humps.subs(x,1) =  16
  humps(z) =  [ 5.17647059 19.         16.         -2.81034483 -4.85517241]

sympy_integral():
  Demonstrate the computation of definite and indefinite integrals.

  humps antiderivative =  -6*x + 5*atan(5*x - 9/2) + 10*atan(10*x - 3)
  humps integral 0 to 2 =  -12 + 5*atan(9/2) + 5*atan(11/2) + 10*atan(3) + 10*atan(17)
  humps integral 0 to 2 (numeric) =  29.3262138043911

  f = exp ( - x )
  f antiderivative =  -exp(-x)
  f integral 0 to oo =  1

  double = exp ( - x**2 - y**2 )
  double integral -oo < x < 00, -oo < y < oo =  pi

sympy_limit():
  Demonstrate the sympy limit() operations.

  limit ( sin ( x ) / x ) as x -> 0 =  1
  limit ( log ( x ) / x ) as x -> oo =  0
  limit ( ( 1 + x/n )**n ) as n -> oo =  exp(x)

sympy_linalg():
  Demonstrate sympy linear algebra operations.

  A =  Matrix([[2, -1, 0, 0], [-1, 2, -1, 0], [0, -1, 2, -1], [0, 0, -1, 2]])
  x =  Matrix([[1], [2], [3], [4]])
  b = A*x =  Matrix([[0], [0], [0], [5]])
  Solved A*x=b =  {x1: 1, x2: 2, x3: 3, x4: 4}
  det(A) =  5
  A inverse =  Matrix([[4/5, 3/5, 2/5, 1/5], [3/5, 6/5, 4/5, 2/5], [2/5, 4/5, 6/5, 3/5], [1/5, 2/5, 3/5, 4/5]])
  A * A_inverse =  Matrix([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]])
  A eigenvalues : multiplicity =  {5/2 - sqrt(5)/2: 1, sqrt(5)/2 + 5/2: 1, 3/2 - sqrt(5)/2: 1, sqrt(5)/2 + 3/2: 1}

sympy_ode():
  Demonstrate solving differential equations.

  y-y(1-y)=0 solved by  Eq(y(t), C1/(C1 - exp(t)))
  y"+y=0 solved by  Eq(y(t), C1*sin(t) + C2*cos(t))
  ty'+y-y**2=0 solved by  Eq(y(t), -C1/(-C1 + t))

sympy_plot():
  Plot the humps() function over 0 <= x <= 2.

sympy_polynomial():
  Demonstrate some polynomial operations.

  p(x) =  x**2 - 5*x + 6
  p(7) =  20
  p.factor() =  (x - 3)*(x - 2)
  p**2 =  (x**2 - 5*x + 6)**2
  p**2.expand =  x**4 - 10*x**3 + 37*x**2 - 60*x + 36
  q(x) =  3*x - 2
  p + q =  x**2 - 2*x + 4
  p * q =  (3*x - 2)*(x**2 - 5*x + 6)
  p * q.expand =  3*x**3 - 17*x**2 + 28*x - 12
  p / q =  (x**2 - 5*x + 6)/(3*x - 2)
  apart ( p / q ) =  x/3 - 13/9 + 28/(9*(3*x - 2))
  r(x) =  (x - 3)*(x - 2)*(x + 1)
  r.expand ( ) =  x**3 - 4*x**2 + x + 6

sympy_sample():
  python version: 3.10.12
  numpy version:  1.26.4
  sympy version : 1.9
  sympy() is a symbolic mathematics package.

  Simple calculations:

pi
3.14159265358979
3.14159265358979323846264338328
1.7724538509055160273
10*sqrt(6)
2432902008176640000
29372339821610944823963760

  Manipulate symbolic variables:

sqrt(1 - cos(x)**2)
sqrt(sin(x)**2)
(x + 1)*(x**2 - 4*x + 3)
(x + 1)*(x**2 - 4*x + 3)/(x - 1)

  define a function, plot it, differentiate it:

100*(60 - 200*x)/((10*x - 3)**2 + 1)**2 + 100*(180 - 200*x)/((10*x - 9)**2 + 4)**2

  definite and indefinite integrals:

-6*x + 5*atan(5*x - 9/2) + 10*atan(10*x - 3)
-12 + 5*atan(9/2) + 5*atan(11/2) + 10*atan(3) + 10*atan(17)
29.3262138043911

  polynomials:

  p(x) =  x**2 - 5*x + 6
  p(7) =  20
  p.factor() =  (x - 3)*(x - 2)

  limits:

1

  series:

1 + x + x**2/2 + O(x**4)

  differential equations:

Eq(y(t), C1/(C1 - exp(t)))

  linear algebra:

Matrix([[0], [0], [0], [5]])
{x0: 1, x1: 2, x2: 3, x3: 4}

sympy_sample():
  Normal end of execution.

sympy_series():
  Demonstrate Taylor series expansion.

  Order 4 series around x=0 for exp(sin(x))
1 + x + x**2/2 + O(x**4)

  Order 3 series around x=1 for log(x)
-1 - (x - 1)**2/2 + x + O((x - 1)**3, (x, 1))

  Order 4 series around x=0 for (1+x)e^x
1 + 2*x + 3*x**2/2 + 2*x**3/3 + O(x**4)

  Order 6 series around x=0 for e^x/cos(x)
1 + x + x**2 + 2*x**3/3 + x**4/2 + 3*x**5/10 + O(x**6)

sympy_test():
  Normal end of execution.
Thu Aug  1 16:32:23 2024