Answer to Question #92212 in Quantitative Methods for lee

Question

scientist why the population of Mathsland continues to grow.]
Year: 1930 1940 1950 1960 1970
Populations: In millions 1.0 1.2 1.6 2.8 5.4
Using a Gregory-Newton method, find the population of the country in 1920?
b) In what year does the expected population reach 60.4 million people?

Answer 1

a) Using a Gregory-Newton method, find the population of the country in 1920?

0.4

b) In what year does the expected population reach 60.4 million people?

2020

All the computations were done using python script based on article https://www.geeksforgeeks.org/newton-forward-backward-interpolation/

# Python3 Program to interpolate using
# newton forward interpolation


# calculating u mentioned in the formula



def u_cal(u, n):


    temp = u
    for i in range(1, n):
        temp = temp * (u - i)
    return temp


# calculating factorial of given number n



def fact(n):
    f = 1
    for i in range(2, n + 1):
        f *= i
    return f


# Driver Code



# Number of values given
n = 5
x = [1930, 1940, 1950, 1960, 1970]


# y[][] is used for difference table
# with y[][0] used for input
y = [[0 for i in range(n)]
     for j in range(n)]


y[0][0] = 1.0
y[1][0] = 1.2
y[2][0] = 1.6
y[3][0] = 2.8
y[4][0] = 5.4


# Calculating the forward difference
# table


for i in range(1, n):
    for j in range(n - i):
        y[j][i] = y[j + 1][i - 1] - y[j][i - 1]


# Displaying the forward difference table
for i in range(n):
    print(x[i], end="\t")
    for j in range(n - i):
        print(y[i][j], end="\t")
    print("")


# Value to interpolate at
value = 1920
sum = 0


while sum < 60.4:


    # initializing u and sum
    sum = y[0][0]
    u = (value - x[0]) / (x[1] - x[0])
    for i in range(1, n):
        sum = sum + (u_cal(u, i) * y[0][i]) / fact(i)


    print("\nValue at", value,
        "is", round(sum, 6))
    
    value+=1