Write the codes of newton backward difference in numerical analysis

Answer 1

Newton's backward difference formula is a method used to approximate the value of a function at a given point using backward differences. The formula is as follows:

f(x) = f(x0) + (x-x0)Δf(x0) + (x-x0)(x-x1)Δ^2f(x0) / 2! + ... + (x-x0)...(x-xn)Δ^nf(x0) / n!

Here is the code for Newton's backward difference formula in Python:

```python
def backward_difference(x_list, y_list, xn):
n = len(x_list)
coeffs = []

# Compute the backward differences
for i in range(1, n):
for j in range(n-1, i-1, -1):
y_list[j] = y_list[j] - y_list[j-1]

# Initialize the coefficients list
coeffs.append(y_list[n-1])

# Compute the coefficients for the polynomial
for i in range(1, n):
coeff = 1
for j in range(i):
coeff *= (xn - x_list[n-j-1])
coeff *= y_list[n-i-1] / math.factorial(i)
coeffs.append(coeff)

# Compute the value of the polynomial at xn
fxn = sum(coeffs)

return fxn

# Example usage
x_values = [1, 2, 3, 4, 5]
y_values = [math.log(x) for x in x_values]
xn = 6

result = backward_difference(x_values, y_values, xn)
print("Approximation of f(6) using Newton's backward difference formula:", result)
```

This code defines a function `backward_difference` that takes in a list of x values, a list of y values, and a value `xn` at which to evaluate the polynomial. The function computes the coefficients of the polynomial using backward differences and then evaluates the polynomial at `xn`. Finally, the code calculates an approximation of the function value at `xn` using Newton's backward difference formula.