# 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] n_height =len(height) print (f' HEIGHT FEV') for i in range(0, n_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.plot(height, fev, 'c*') plt.xlabel ('height') plt.ylabel ('fev') plt.title ('height vs fev in boys age 10 to 15')

# 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)

# 5. Plot the interpolating polynomial from a height of 134 cm to 178 cm df = len(fev) -1 coeff = compute_int_coeff (df, height, fev) x1and2 = np.linspace (134, 178); y1and2= [] for i in range (0, len(x1and2)): y1and2.append (eval_int_polynomial(df, height, x1and2[i] , coeff))

plt.plot (height, fev, 'bo') plt.xlabel ('height') plt.ylabel ('fev') plt.plot (x1and2, y1and2, 'm') plt.title (f'{df} degree polynomial interprolating data')

# 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? print (f' predicted fev based on height of 165 cm is', eval_int_polynomial(df, height, 165, coeff)) print (f' predicted fev based on height of 175 cm is', eval_int_polynomial(df, height, 175, coeff)) #175 is a poor prediction because after about the 170 mark is when the graph goes off trend

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

# Plot piecewise linear interpolating polynomial and data; use plt.plot (height, fev, 'r+') plt.xlabel ('height') plt.ylabel ('fev') x_plt = np.linspace(134, 178) y_plt = np.interp(x_plt, height, fev) plt.plot (x_plt, y_plt, 'c') plt.title ('piecewise polynomial predictions')

# 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 print (f' fev prediction based on height of 165 is' , np.interp(165, height, fev)) print (f' fev prediction based on height of 176 is' , np.interp(176, height, fev)) #predicted value for 165 is better because it was closer to the 11th degree polynomial prediction