from package import redact_ex
from package import \
gauss_seidel_solve, \
check_sys_sols, \
deprox_arr
import numpy as np
EXERCISE_07 = """\
Make a program that is able to solve a system of equations using the
Gauss-Seidel method.\
"""
redact_ex(EXERCISE_07, 7)
a = np.array([[1, 2, -1], [2, 4, 5], [3, -1, -2]], dtype=float)
b = np.array([[2], [25], [-5]], dtype=float)
a = a.reshape(len(a), len(a))
b = b.reshape(len(b), 1)
ab = np.concatenate((a, b), axis=1)
print("Matrix of coefficients A (given):", a, sep='\n\n', end='\n\n')
print("Vector of independent terms B (given):", b, sep='\n\n', end='\n\n')
print("Expanded matrix (AB):", ab, sep='\n\n', end='\n\n')
sols = gauss_seidel_solve(ab)
print("Obtained solutions:", sols, sep='\n\n', end='\n\n')
sols = deprox_arr(sols)
from package import redact_ex
from package import montecarlo_integrate
import numpy as np
EXERCISE_02 = """\
Make a program that is able to compute the integral between 0 and infinity
for f(x) = e**(-x)\
"""
redact_ex(EXERCISE_02, 2)
def f(y):
return (1 / y**2) * np.exp(-1 / y + 1)
interval = [0, 1]
integral, integral_err = montecarlo_integrate(f, interval)
print(f"Integral between {interval!s} for f(x): {integral}")
print(f"Error for the aboveshown integration: {integral_err:.2e}")
from package import redact_ex
from package import dkf
EXERCISE_01 = """\
Make a program that is able to compute the numerical k-order derivative
of a given function using the method of undetermined coefficients.
Apply that to the case f(x) = x**3 - 3*x**2 - x + 3 on x0 = 1.2.\
"""
redact_ex(EXERCISE_01, 1)
def f(x):
return x**3 - 3 * x**2 - x + 3
# df = 3*x**2 - 6*x - 1, on x0 (1.2) = -3.88
# d2f = 6*x - 6, on x0 (1.2) = 1.2
# d3f = 6, on x0 (1.2) = 6
for k in range(1, 6):
for h in [1 / d for d in [10**n for n in range(10)]]:
v = dkf(f, 1.2, k, delta=h)
print('f({k}({x0}) = {v} (h = {h})'.format(k=k, x0=1.2, v=v, h=h),
end='\n\n')
# print(dkf(f, 1.2, 1, delta = 1e-6))
# print(dkf(f, 1.2, 2, delta = 1e-4))
# print(dkf(f, 1.2, 3, delta = 1e-2))
from package import redact_ex
from package import \
gauss_reduce, \
solve_triang_mat, \
check_sys_sols, \
deprox_arr
import numpy as np
EXERCISE_04 = """\
Make a program that is able to solve a system of equations
by the Gauss-Jordan elimination method with full pivoting.\
"""
redact_ex(EXERCISE_04, 4)
a = np.array([[1, 2, -1], [2, 4, 5], [3, -1, -2]], dtype=float)
b = np.array([[2], [25], [-5]], dtype=float)
a = a.reshape(len(a), len(a))
b = b.reshape(len(b), 1)
ab = np.concatenate((a, b), axis=1)
print("Matrix of coefficients A (given):", a, sep='\n\n', end='\n\n')
print("Vector of independent terms B (given):", b, sep='\n\n', end='\n\n')
print("Expanded matrix (AB):", ab, sep='\n\n', end='\n\n')
tmat = gauss_reduce(ab, method='gj-elim', pivoting='total')
print("Triangularized matrix (T):", tmat, sep='\n\n', end='\n\n')
sols = solve_triang_mat(tmat, method='gj-elim')
from package import \
gauss_reduce, \
solve_triang_mat, \
check_sys_sols, \
deprox_arr
import numpy as np
EXERCISE_03 = """\
Make a program that is able to solve a system of equations
by the Gauss elimination method with full pivoting and solution computing
by regressive substitution.\
"""
redact_ex(EXERCISE_03, 3)
a = np.array([[1, 2, -1], [2, 4, 5], [3, -1, -2]], dtype=float)
b = np.array([[2], [25], [-5]], dtype=float)
a = a.reshape(len(a), len(a))
b = b.reshape(len(b), 1)
ab = np.concatenate((a, b), axis=1)
print("Matrix of coefficients A (given):", a, sep='\n\n', end='\n\n')
print("Vector of independent terms B (given):", b, sep='\n\n', end='\n\n')
print("Expanded matrix (AB):", ab, sep='\n\n', end='\n\n')
tmat = gauss_reduce(ab, pivoting='total')
print("Triangularized matrix (T):", tmat, sep='\n\n', end='\n\n')
sols = solve_triang_mat(tmat)
from package import redact_ex
import numpy as np
import random
EXERCISE_06 = """\
Make a program that computes the volume of the part of a torus shown
in the figure (<figure>) using the Monte Carlo method.\
"""
redact_ex(EXERCISE_06, 6)
a, b = 1, 3
npoints = 1e6
n = int(npoints)
inc = 0
for i in range(n):
x, y, z = random.uniform(1, 4), random.uniform(-3, 4), random.uniform(-1, 1)
if (np.sqrt(x**2 + y**2) - b)**2 + z**2 <= a**2:
inc += 1
area = 3*7*2
volume = inc*area/n
print("Obtained volume:", volume, sep = '\n', end = '\n\n')
from package import redact_ex
from package import recursive_integrate
import numpy as np
EXERCISE_05 = """\
Make a program that computes the integral of a function f(x)
in an interval [a, b] using the recursive rule of Simpson 1/3.
Apply that to the case f(x) = (x**2 + x + 1) * cos(x) and a = 0, b = pi/2\
"""
redact_ex(EXERCISE_05, 5)
def f(x):
return (x**2 + x + 1) * np.cos(x)
interval = [0, np.pi / 2]
integral_ab = recursive_integrate(f, interval, method='simpson', prec=1e-12)
print(f"Integrating f(x) in {interval!s} yields (recursive Simpson 1/3):",
integral_ab,
sep='\n')
from package import redact_ex
from package import euler_differentiate
import numpy as np
import warnings
warnings.filterwarnings('ignore')
EXERCISE_00 = """\
Make a program that is able to graphically solve the equation
dx/dt = sin(x) using the Euler method.\
"""
redact_ex(EXERCISE_00, 0)
# dx/dt = sin(x)
def inct(dt, *o):
return dt
def incx(dt, t, x):
return dt*np.sin(x)
print("Computing Euler method...", end='\n\n')
euler_differentiate([inct, incx], bounds = [0, 1], delta = 1e-2, itern = 1e4,
title = r"Euler method for function"
+ r"$\:\:\frac{dx}{dt} = \sin(x)$")
from package import redact_ex
from package import \
euler_differentiate, \
range_kutta_differentiate
import numpy as np
EXERCISE_02i = """\
Make a program that is able to graphically solve the equation
d\u00B2x/dt\u00B2 + b dx/dt + \u03C9_0\u00B2 x = 0 using the Euler method.\
"""
redact_ex(EXERCISE_02i, '2i')
# dx/dt = y
# dy/dt = -b*y - omega
b = 1/2; omega_0 = 2
def inct(dt, *o):
return dt
def incx(dt, t, x, y):
return dt * y
def incy(dt, t, x, y, b = b, omega_0 = omega_0):
return dt * (-b*y - omega_0**2*x)
