FEAT: skeleton code to test derivatives

Author: aaronjridley

day_6/2023/derivatives_test.py

@@ -0,0 +1,143 @@
+#!/usr/bin/env python
+
+import numpy as np
+import matplotlib.pyplot as plt
+
+# ----------------------------------------------------------------------
+# Take first derivative of a function
+# ----------------------------------------------------------------------
+
+def first_derivative(f, x):
+
+    """ Function that takes the first derivative
+
+    Parameters
+    ----------
+    f - values of a function that is dependent on x -> f(x)
+    x - the location of the point at which f(x) is evaluated
+
+    Returns
+    -------
+    dfdx - the first derivative of f(x)
+
+    Notes
+    -----
+    take the first derivative of f(x) here
+
+    """
+
+    nPts = len(f)
+    
+    dfdx = np.zeros(nPts)
+
+    # do calculation here - need 3 statements:
+    #  1. left boundary ( dfdx(0) = ...)
+    #  2. central region (using spans, like dfdx(1:nPts-2) = ...)
+    #  3. right boundary ( dfdx(nPts-1) = ... )
+
+    return dfdx
+
+# ----------------------------------------------------------------------
+# Take second derivative of a function
+# ----------------------------------------------------------------------
+
+def second_derivative(f, x):
+
+    """ Function that takes the second derivative
+
+    Parameters
+    ----------
+    f - values of a function that is dependent on x -> f(x)
+    x - the location of the point at which f(x) is evaluated
+
+    Returns
+    -------
+    d2fdx2 - the second derivative of f(x)
+
+    Notes
+    -----
+    take the second derivative of f(x) here
+
+    """
+
+    nPts = len(f)
+    
+    d2fdx2 = np.zeros(nPts)
+
+    # do calculation here - need 3 statements:
+    #  1. left boundary ( dfdx(0) = ...)
+    #  2. central region (using spans, like dfdx(1:nPts-2) = ...)
+    #  3. right boundary ( dfdx(nPts-1) = ... )
+
+    return d2fdx2
+
+# ----------------------------------------------------------------------
+# Get the analytic solution to f(x), dfdx(x) and d2fdx2(x)
+# ----------------------------------------------------------------------
+
+def analytic(x):
+
+    """ Function that gets analytic solutions
+
+    Parameters
+    ----------
+    x - the location of the point at which f(x) is evaluated
+
+    Returns
+    -------
+    f - the function evaluated at x
+    dfdx - the first derivative of f(x)
+    d2fdx2 - the second derivative of f(x)
+
+    Notes
+    -----
+    These are analytic solutions!
+
+    """
+
+    f = 4 * x ** 2 - 3 * x -7
+    dfdx = 8 * x - 3
+    d2fdx2 = np.zeros(len(f)) + 8.0
+
+    return f, dfdx, d2fdx2
+
+
+if __name__ == "__main__":
+
+    # get figures:
+    fig = plt.figure(figsize = (10,10))
+    ax1 = fig.add_subplot(311)
+    ax2 = fig.add_subplot(312)
+    ax3 = fig.add_subplot(313)
+    
+    # define dx:
+    dx = np.pi / 8
+    
+    # arange doesn't include last point, so add explicitely:
+    x = np.arange(-2.0 * np.pi, 2.0 * np.pi + dx, dx)
+
+    # get analytic solutions:
+    f, a_dfdx, a_d2fdx2 = analytic(x)
+
+    # get numeric first derivative:
+    n_dfdx = first_derivative(f, x)
+
+    # get numeric first derivative:
+    n_d2fdx2 = second_derivative(f, x)
+
+    # plot:
+
+    ax1.plot(x, f)
+
+    # plot first derivatives:
+    ax2.plot(x, a_dfdx, color = 'black', label = 'Analytic')
+    ax2.plot(x, n_dfdx, color = 'red', label = 'Numeric')
+    ax2.legend()
+
+    # plot second derivatives:
+    
+    plotfile = 'plot.png'
+    print('writing : ',plotfile)    
+    fig.savefig(plotfile)
+    plt.close()
+