Arrays and Plotting

da PowerShell -- jupyter nbconvert --to html notebook.ipynb

NumPy and Array Computing

import numpy as np # module for arrays

def f(x):
    return x**2

n = 5 # number of points
dx = 1.0/(n-1) # x spacing in [0,1]
# memozizzo in due List
xlist = [i*dx for i in range(n)] 
ylist = [f(x) for x in xlist]
# memorizzo in due array
x = np.array(xlist) # turn list xlist into array 
y = np.array(ylist)
# genero array x equispaziato
x = np.linspace(0, 1, n) # n points in [0, 1]
print('x =',x)
y = np.zeros(n) # n zeros (float data type) 
for i in range(n):
    y[i] = f(x[i])

x = [0.   0.25 0.5  0.75 1.  ]

Gli array non accettano "append"

import numpy as np
from math import cos
n = 11
x1 = np.linspace(0,1,n)
y1 = np.zeros(n)    # n zeros (float data type)
for i in range(len(x)):
    y1[i] = cos(x1[i])
# numpy accetta come input un array
z1 = np.cos(x1)     # x1: array, y1: array
#
# math funzioni accettano solo scalari

Un modo compatto, zenza loop for

import numpy as np

n = 100
x = np.linspace(0, 4, n+1)
y = np.exp(-x)*np.sin(2*np.pi*x)

Plotting Curves with Matplotlib

import matplotlib.pyplot as plt
import numpy as np

n = 100
x = np.linspace(0, 4, n+1)
y = np.exp(-x)*np.sin(2*np.pi*x)

plt.figure(figsize=(6, 3), dpi=96)
plt.plot(x, y, label='exp(-x)*sin(2$\pi$ x)')
plt.xlabel('x') # label on the x axis
plt.ylabel('y') # label on the y axis
plt.legend() # mark the curve
plt.axis([0, 4, -0.5, 0.8]) # [tmin, tmax, ymin, ymax] 
plt.title('My First Matplotlib Demo')

#plt.savefig('fig.pdf') # make PDF image for reports 
#plt.savefig('fig.png') # make PNG image for web pages 
plt.show()

Multiple Curves in the Same Plot

import matplotlib.pyplot as plt 
import numpy as np

def f1(x):
    return np.exp(-x)*np.sin(2*np.pi*x)

def f2(x):
    return np.exp(-2*x)*np.sin(4*np.pi*x)

x = np.linspace(0, 8, 401) 
y1 = f1(x)
y2 = f2(x)

plt.figure(figsize=(6, 3), dpi=96)
plt.plot(x, y1, 'r--', label='exp(-x)*sin(2$\pi$ x)') 
plt.plot(x, y2, 'g:', label='exp(-2*x)*sin(4$\pi$ x)')

plt.xlabel('x') 
plt.ylabel('y')
plt.legend()
plt.title('Plotting two curves in the same plot') 
plt.savefig('fig_two_curves.png')
plt.show()

Plotting a user.specified function

import plot_input

Plotting Discontinuous and Piecewise-Defined Functions - ( a tratti )

Così non funziona

import matplotlib.pyplot as plt 
import numpy as np

def H(x):
    if x < 0:
        return 0
    else:
        return 1

x = np.linspace(-10, 10, 5) # few points (simple curve) 
y = H(x)
plot(x, y)

---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
Cell In[12], line 11
      8         return 1
     10 x = np.linspace(-10, 10, 5) # few points (simple curve) 
---> 11 y = H(x)
     12 plot(x, y)

Cell In[12], line 5, in H(x)
      4 def H(x):
----> 5     if x < 0:
      6         return 0
      7     else:

ValueError: The truth value of an array with more than one element is ambiguous. Use a.any() or a.all()

Con un for carico y --> funziona

import numpy as np
import matplotlib.pyplot as plt 
n = 5
x = np.linspace(-5, 5, n+1)
y = np.zeros(n+1)

for i in range(len(x)):
    y[i] = H(x[i])

plt.figure(figsize=(6, 3), dpi=96)
plt.plot(x,y) 
plt.show()

Un altro sistema è quello di usare vectorize

import numpy as np
import matplotlib.pyplot as plt

# funzione scalare con if
def f_scalar(x):
    if x < 0:
        return x**2
    else:
        return np.sqrt(x)

# vettorializzazione
f = np.vectorize(f_scalar)

# dominio
x = np.linspace(-10, 10, 400)

# valutazione
y = f(x)

# grafico
plt.figure(figsize=(6, 3), dpi=96)
plt.plot(x, y)
plt.xlabel("x")
plt.ylabel("f(x)")
plt.title("Funzione con if usando np.vectorize")
plt.grid(True)
plt.show()

Making a Movie of a Plot

$$\large f(x) = \frac{1}{\sqrt{2\pi}} \frac{1}{s} exp\left[ -\frac{1}{2} \left( \frac{x-m}{s} \right)^2 \right] $$

import numpy as np
from vpython import canvas, graph, gcurve, rate, color

def gauss(x, m, s):
    return np.exp(-0.5*((x-m)/s)**2) / (np.sqrt(2*np.pi)*s)

# dominio
m = 0.0
x = np.linspace(-6, 6, 800)

# parametri di sigma
s = 2.0
ds = -0.02
s_min = 0.2

# canvas + graph (fondamentale)
scene = canvas(title="Gaussiana con sigma variabile")

g = graph(
    xtitle="x",
    ytitle="f(x)",
    width=600,
    height=300,
    ymin=0,          # <<< FISSO
    ymax=2           # <<< FISSO
)

# statement fodamentale
curve = gcurve(color=color.blue)

# ciclo temporale (stile VPython puro)
while s > s_min:
    rate(30)

    # costruisco i dati come nel tuo codice
    data = []
    for xi in x:
        data.append([xi, gauss(xi, m, s)])

    curve.data = data    # <<< QUI è la chiave, statement fondamentale

    s += ds

More Useful Array Operations

Copiare un Array - Creare un Array della stessa dimensione di uno dato

import numpy as np
x = np.linspace(0,10,101)
a = np.zeros(x.shape, x.dtype)
# oppure
a = x.copy()
b = np.zeros_like(x) # zeros and same size as x
c = np.asarray(a)

Estrarre elementi da un Array

import numpy as np
a = np.linspace(1,8,8)
print(a)
a[[1,6,7]] = 10      # vario il valore di 3 elementi
print(a)
a[range(2,8,3)] = -2 # vario il valore di 3 elementi - # same as a[2:8:3] = -2
print(a)
print(a<0)
print(a[a<0])        # pick out all negative elements
a[a < 0] = a.max() # if a[i]<0, set a[i]=10
print(a)

[1. 2. 3. 4. 5. 6. 7. 8.]
[ 1. 10.  3.  4.  5.  6. 10. 10.]
[ 1. 10. -2.  4.  5. -2. 10. 10.]
[False False  True False False  True False False]
[-2. -2.]
[ 1. 10. 10.  4.  5. 10. 10. 10.]

import numpy as np
A = np.zeros((3,4)) # 3x4 table of numbers 
A[0,0] = -1
A[1,0] = 1
A[2,0] = 10
A[0,1] = -5
A[2,3] = -100

# can also write (as for nested lists) 
A[2][3] = -100