def c5p22():
# Given Functions
def force_grav(rho_object, v, g=9.8):
return rho_object * v * g
def force_buoy(rho_medium, v_displaced, g=9.8):
return rho_medium * v_displaced * g
def vol_sphere(rad):
return 4.0 / 3 * np.pi * rad**3
def vol_dry(rad, h):
return (np.pi * h**2 / 3) * (3 * rad - h)
def vol_disp(rad, h):
return vol_sphere(rad) - vol_dry(rad, h)
# Constants
rho_object, rho_medium, rad = 200., 1000., 1.
# args for solutions
f = lambda h: force_grav(rho_object, vol_sphere(rad)) - force_buoy(
rho_medium, vol_disp(rad, h))
a, b = 0., 2.
return {
'bisection': num.bisection(f, a, b),
'false_position': eng_form(num.false_position(f, a, b))
}
def c5p7():
f, a, b, *kwargs = [
lambda x: np.log(x**2) - 0.7, 0.5, 2.0, {
'maxSteps': 3
}
]
return {
'bisection': num.bisection(f, a, b, **kwargs[0]),
'false_position': num.false_position(f, a, b, **kwargs[0])
}
def c6p19():
# Impedance
def z(r, c, l, w):
return np.sqrt( r**(-2) + ( w * c - (w * l)**(-1) )**2 )**(-1)
# Constants
r, c, l, z_desired = 225., 0.6 * 10**(-6), 0.5, 100.
# Args for solution
f = lambda w: z(r, c, l, w) - z_desired
x0, x1, dx = 210, 215, 10**(-4)
return {'newton_raphson': num.newton_raphson( f,x0 ), 'secant_method': num.secant_method( f, x0, x1 ), 'modified_secant': num.modified_secant( f,x1,dx) }
def c6p19():
# Impedance
def z(r, c, l, w):
return np.sqrt(r**(-2) + (w * c - (w * l)**(-1))**2)**(-1)
# Constants
r, c, l, z_desired = 225., 0.6 * 10**(-6), 0.5, 100.
# Args for solution
f = lambda w: z(r, c, l, w) - z_desired
x0, x1, dx = 210, 215, 10**(-4)
return {
'newton_raphson': num.newton_raphson(f, x0),
'secant_method': num.secant_method(f, x0, x1),
'modified_secant': num.modified_secant(f, x1, dx)
}
def c8p3():
A = np.array([[0, -6, 5], [0, 2, 7], [-4, 0, -4]])
b = np.array([[50], [-30], [50]])
return {
'Solution Vector': num.gauss_elimination(A, b),
"Coefficient Transpose": np.transpose(A),
"Coefficient Inverse": inv(A)
}
def c6p21():
# Trajectory
def y(theta, x, v0, y0, g=9.8):
return np.tan(theta) * x - (g * x**2) / (2 * v0**2 *
np.cos(theta)**2) + y0
# Constants
v0, x, y0, y1 = 30., 90, 1.8, 1.
deg = (180. / np.pi)
# Args for solution
f = lambda theta: y(theta, x, v0, y0) - y1
return {
'newton_raphson th1': deg * num.newton_raphson(f, 0.4),
'newton_raphson th2': deg * num.newton_raphson(f, 0.9),
'secant_method th1': deg * num.secant_method(f, 0.4, 0.45),
'secant_method th2': deg * num.secant_method(f, 0.9, 0.95)
}
def c5p16():
# Given functions:
def rho(q, n, mu):
return 1.0 / ( q * n * mu )
def n( N, ni ):
return 0.5 * ( N + np.sqrt( N**2 + 4 * ni**2 ) )
def mu( T, T0, mu0 ):
return mu0 * ( T / T0 )**(-2.42)
# Constants
T0, T, mu0, q, ni, rho_desired = 300., 1000., 1360., 1.7 * 10**(-19), 6.21 * 10**9, 6.5 * 10**6
# args for solutions
f = lambda N: rho( q, n(N,ni), mu(T,T0,mu0) ) - rho_desired
a,b = 0., 2.5 * 10**10
return {'bisection': eng_form( num.bisection( f,a,b ) ),'false_position':eng_form( num.false_position(f,a,b) ) }
def update():
global slider
global ax1
global ax2
global ax3
# create new values that depends on slider
graph = nm.NumMethods(xStart, xFinite, yStart, int(slider.val))
graph.buildPlot()
graph.eulerMethod()
graph.improvedEulerMethod()
graph.rungeKutta()
graph.errors()
# clear subplots
ax1.clear()
ax2.clear()
ax3.clear()
# set the grid
ax1.grid(True)
ax1.set_ylabel("y")
ax1.set_xlabel("x")
ax2.grid(True)
ax2.set_ylabel("y")
ax2.set_xlabel("x")
ax3.grid(True)
ax3.set_xlabel("x\n\n")
ax3.set_ylabel("y")
# plot new values
ax1.set_title("Numerical Methods")
ax1.plot(graph.x, graph.yEuler, "blue", label="Euler Method")
ax1.legend(loc='upper right')
ax1.plot(graph.x, graph.yImprEuler, "cyan", label="Improved Euler")
ax1.legend(loc='upper right')
ax1.plot(graph.x, graph.yRunge, "red", label="Runge-Kutta")
ax1.legend(loc='lower left')
ax2.set_title("Exact solution")
ax2.plot(graph.x, graph.y, "black", label="Grapth")
ax2.legend(loc='upper center')
ax3.set_title("Errors")
ax3.plot(graph.x, graph.errorEuler, "blue", label="Euler Method")
ax3.plot(graph.x, graph.errorImprEuler, "cyan", label="Improved Euler")
ax3.plot(graph.x, graph.errorRunge, "red", label="Runge-Kutta")
ax1.xaxis.set_major_locator(mplt.ticker.MultipleLocator(base=0.5))
ax1.yaxis.set_major_locator(mplt.ticker.MultipleLocator(base=0.5))
ax2.xaxis.set_major_locator(mplt.ticker.MultipleLocator(base=0.5))
ax2.yaxis.set_major_locator(mplt.ticker.MultipleLocator(base=0.5))
ax3.xaxis.set_major_locator(mplt.ticker.MultipleLocator(base=0.5))
ax3.yaxis.set_major_locator(mplt.ticker.MultipleLocator(base=0.5))
plt.draw()
def c6p21():
# Trajectory
def y(theta, x, v0, y0, g=9.8 ):
return np.tan( theta ) * x - (g * x**2) /( 2 * v0**2 * np.cos( theta )**2 ) + y0
# Constants
v0, x, y0, y1 = 30., 90, 1.8, 1.
deg = ( 180. / np.pi )
# Args for solution
f = lambda theta: y( theta, x, v0, y0 ) - y1
return { 'newton_raphson th1': deg * num.newton_raphson( f,0.4 ), 'newton_raphson th2': deg * num.newton_raphson( f, 0.9 ) ,'secant_method th1': deg * num.secant_method(f,0.4, 0.45 ), 'secant_method th2':deg * num.secant_method( f,0.9,0.95 ) }
def c5p16():
# Given functions:
def rho(q, n, mu):
return 1.0 / (q * n * mu)
def n(N, ni):
return 0.5 * (N + np.sqrt(N**2 + 4 * ni**2))
def mu(T, T0, mu0):
return mu0 * (T / T0)**(-2.42)
# Constants
T0, T, mu0, q, ni, rho_desired = 300., 1000., 1360., 1.7 * 10**(
-19), 6.21 * 10**9, 6.5 * 10**6
# args for solutions
f = lambda N: rho(q, n(N, ni), mu(T, T0, mu0)) - rho_desired
a, b = 0., 2.5 * 10**10
return {
'bisection': eng_form(num.bisection(f, a, b)),
'false_position': eng_form(num.false_position(f, a, b))
}
def c5p22():
# Given Functions
def force_grav( rho_object, v, g=9.8 ):
return rho_object * v * g
def force_buoy( rho_medium, v_displaced, g=9.8 ):
return rho_medium * v_displaced * g
def vol_sphere( rad ):
return 4.0/3 * np.pi * rad**3
def vol_dry( rad, h ):
return (np.pi * h**2 / 3) * (3 * rad - h)
def vol_disp( rad, h ):
return vol_sphere(rad) - vol_dry( rad, h )
# Constants
rho_object, rho_medium, rad = 200., 1000., 1.
# args for solutions
f = lambda h: force_grav( rho_object, vol_sphere( rad )) - force_buoy( rho_medium, vol_disp( rad,h ))
a,b = 0.,2.
return {'bisection': num.bisection( f,a,b ) ,'false_position':eng_form( num.false_position(f,a,b) ) }
def c8p3():
A = np.array([[0,-6,5],[0,2,7],[-4,0,-4]])
b = np.array([[50],[-30],[50]])
return {'Solution Vector':num.gauss_elimination(A,b),"Coefficient Transpose": np.transpose(A),"Coefficient Inverse": inv(A) }
def c5p7():
f,a,b,*kwargs = [lambda x: np.log( x**2 ) - 0.7, 0.5, 2.0, {'maxSteps':3}]
return {'bisection':num.bisection( f,a,b,**kwargs[0] ), 'false_position':num.false_position( f,a,b,**kwargs[0] )}