Week 8 Practice Notebooks - Duplicate

# 1. Print out data in columns height = [134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178] fev = [1.7, 1.9, 2., 2.1, 2.2, 2.5, 2.7, 3.,3.1, 3.4, 3.8, 3.9] print("Height FEV") print('****** ******') for i in range(len(height)): print(f" {height[i]} {fev[i]}")

# 2. Import libraries - NumPy and matplotlib import numpy as np import matplotlib.pyplot as plt

# 3. Plot the data; add labels to axes and plot title plt.scatter(height,fev) plt.title("FEV vs Height") plt.xlabel("Height (cm)") plt.ylabel("FEV")

# 4. Input functions to find interpolating polynomial def compute_int_coeff (n, x, y) : coeff = [] # initialize list for i in range (0,n+1): # loop to calculate n+1 coefficients numerator = y[i] denominator = 1. # because denominator is a product we must initialize to 1 for j in range (0,n+1): if ( i != j) : # denominator is product of all terms except # when x_i=x_j which would give 0 denominator = denominator * (x[i] - x[j]) coeff.append (numerator / denominator) # add term to list return (coeff) # # def eval_int_polynomial (n, x, x_eval, coeff) : my_sum = 0. # formula is sum of n+1 terms so initialize for i in range (0,n+1) : # loop to form sum of terms product = 1. # initialize product in numerator of L_i for j in range (0, n+1) : # loop to form product of terms in numerator if ( i != j ) : # include all terms except when i = j product = product * (x_eval - x[j]) # when this j loop is complete we have the numerator of L_i for one term yi_times_Li = product * coeff[i] # multiplies numerator times C_i (y_i/ product) my_sum = my_sum + yi_times_Li # keeps running tally of sum of terms return (my_sum) n = len(height)-1 coeff = compute_int_coeff(n,height,fev) x_eval = np.linspace(134,178,1000) y = [] for i in range(0,len(x_eval)): poly = eval_int_polynomial(n,height,x_eval[i],coeff) y.append(poly) print(y)

# 5. Plot the interpolating polynomial from a height of 134 cm to 178 cm plt.plot(x_eval,y) plt.plot(height,fev,"ro") plt.title("Approximation of FEV")

# 6. Use the polynomial to predict the FEV for a height of 165 cm and of 175 cm. # Which do you think is a better prediction? x = [165,175] z = [] for i in range(len(x)): poly = eval_int_polynomial(n,height,x[i],coeff) print(f"Height: {x[i]} FEV: {poly}") z.append(poly) plt.plot(x_eval,y) plt.plot(x,z,'ro') plt.title("Polynomial interpolating") plt.xlabel("Height (cm)") plt.ylabel("FEV") # I would say the 165 cm is a better prediction as it follows the linear trend height vs FEV but, for 175 cm in the # approximation the polynomial starts to go down and then up between 171 cm and 178 cm # Similarly with 134 cm to 138 cm the approximation polynomial seems to be inaccurate near the ends of the polynomial

# Import libraries import matplotlib.pyplot as plt import numpy as np

# Plot piecewise linear interpolating polynomial and data; use x_int = np.linspace(134,178) y_int = np.interp(x_int,height,fev) plt.plot(height,fev,'ro') plt.plot(x_int,y_int) plt.title("Piecewise linear polynomial interpolating") plt.xlabel("Height (cm)") plt.ylabel("FEV")

# Use the piecewise polynomial to predict the FEV for a height of 165 cm and of 176 cm. # How does this compare with 11th degree polynomial which predicted 3.104 for 165 cm and # 2.697 for 176 cm x_int_1 = np.linspace(165,176) y_int_1 = np.interp(x_int_1,height,fev) print(f"Height: {x_int_1[0]} FEV: {y_int_1[0]}") print(f"Height: {x_int_1[-1]} FEV: {y_int_1[-1]}") plt.plot(height,fev,'ro') plt.plot(x_int,y_int) plt.title("Piecewise linear polynomial interpolating") plt.xlabel("Height (cm)") plt.ylabel("FEV") # For 165 cm it is only off by a little but for 176 cm it is off by over 1 # In this case the interpolation from piecewise is linear with height while in polynomial near the ends it drops or peaks