# 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]} ')

```
HEIGHT FEV
134 1.7
138 1.9
142 2.0
146 2.1
150 2.2
154 2.5
158 2.7
162 3.0
166 3.1
170 3.4
174 3.8
178 3.9
```

# 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

```
predicted fev based on height of 165 cm is 3.1044129974208774
predicted fev based on height of 175 cm is 3.3490242131985744
```

# 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

```
fev prediction based on height of 165 is 3.075
fev prediction based on height of 176 is 3.8499999999999996
```