Question

PLEASE USE PYTHON CODE

The points

x = -2, 1, 4, -1, 3, -4,

y = -1, 2, 59, 4, 24, -53

lie on polynomial. Use the divided difference table of Newton's method to determine the degree of the polynomial.

Answer 1

# Newton divided difference formula  
  
# Function to find the product term  
def proterm(i, value, x):  
    pro = 1;  
    for j in range(i):  
        pro = pro * (value - x[j]);  
    return pro;  
  
# Function for calculating  
# divided difference table  
def dividedDiffTable(x, y, n): 
  
    for i in range(1, n):  
        for j in range(n - i):  
            y[j][i] = ((y[j][i - 1] - y[j + 1][i - 1]) /
                                     (x[j] - x[i + j])); 
    return y; 
  
# Function for applying Newton's  
# divided difference formula  
def applyFormula(value, x, y, n):  
  
    sum = y[0][0];  
  
    for i in range(1, n): 
        sum = sum + (proterm(i, value, x) * y[0][i]);  
      
    return sum;  
# The difference table  
def printDiffTable(y, n):  
  
    for i in range(n):  
        for j in range(n - i):  
            print(round(y[i][j], 4), "\t",  
                               end = " ");  
  
        print("");  
  
# The Driver Code 
  
# The number of inputs given  
n = 6;  
y = [[0 for i in range(10)]  
        for j in range(10)];  
x = [ -2,1,4,-1,3,-4];  
  
# y[][] is used for divided difference  
# table where y[][0] is used for input  
y[0][0] = -1;  
y[1][0] = 2;  
y[2][0] = 59;  
y[3][0] = 4;
y[4][0] = 24;
y[5][0] = -53;
  
# for calculating divided difference table  
y=dividedDiffTable(x, y, n);  
  
# for displaying divided difference table  
printDiffTable(y, n);  
  
# the value to be interpolated  
value = 7;  
# for printing the value  
print("\nValue at", value, "is", 
        round(applyFormula(value, x, y, n), 2)) 

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