Given the function , where are data points. Find the system of linear equations you may use to calculate Note: Denote the , for j=1,2,…., for the mixed sums use the notations as those shown in class. Problem 2 a) Find the approximation of the function derivatives . Use the forward difference formula for the 1st derivative, backward for the 2nd, and central formula for the 3rd derivative. Apply the formalism: ; and which leads to . b) Write the truncation error of the forward with respect to , backward and the central 1st order derivatives with respect to of the function ? c) What is the truncation error of the second derivative central formula approximation of g(x)?

QUESTION

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Problem 1.

Given the function , where are data points. Find the system of linear equations you may use to calculate

Note: Denote the , for j=1,2,…., for the mixed sums use the notations as those shown in class.

Problem 2

a) Find the approximation of the function derivatives . Use the forward difference formula for the 1^st derivative, backward for the 2^nd, and central formula for the 3^rd derivative. Apply the formalism: ; and which leads to .

b) Write the truncation error of the forward with respect to , backward and the central 1^st order derivatives with respect to of the function ?

c) What is the truncation error of the second derivative central formula approximation of g(x)?

ANSWER

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Problem 1:

Given the function f(x), where x1, x2, …, xn are data points, we need to find the system of linear equations that can be used to calculate the coefficients a0, a1, …, an in the equation:

f(x) = a0 + a1x + a2x^2 + … + anx^n

To determine the system of linear equations, we substitute the given data points into the equation and form a matrix equation:

| f(x1) | | a0 + a1x1 + a2x1^2 + … + anx1^n |
| f(x2) | | a0 + a1x2 + a2x2^2 + … + anx2^n |
| … | = | … |
| f(xn) | | a0 + a1xn + a2xn^2 + … + anxn^n |

We can rewrite this matrix equation as follows:

A * X = B

where A is the matrix of coefficients, X is the vector of unknowns (a0, a1, …, an), and B is the vector of function values at the data points.

The system of linear equations can be solved by finding the inverse of matrix A (if it exists) and multiplying it with vector B:

X = A^(-1) * B

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Problem 2:

a) To approximate the function derivatives, we can use different finite difference formulas:

1. Forward difference formula for the 1st derivative:
f'(x) ≈ (f(x + h) – f(x)) / h

2. Backward difference formula for the 2nd derivative:
f”(x) ≈ (f(x) – f(x – h)) / h

3. Central difference formula for the 3rd derivative:
f”'(x) ≈ (f(x + h) – 2f(x) + f(x – h)) / (h^2)

Here, h represents a small step size.

b) The truncation error of a finite difference formula represents the error introduced by the approximation. Let’s consider the truncation errors for each formula:

1. Truncation error of the forward difference formula for the 1st derivative:
Error = f'(x) – (f(x + h) – f(x)) / h

2. Truncation error of the backward difference formula for the 2nd derivative:
Error = f”(x) – (f(x) – f(x – h)) / h

3. Truncation error of the central difference formula for the 1st order derivative:
Error = f”'(x) – (f(x + h) – 2f(x) + f(x – h)) / (h^2)

c) The truncation error of the second derivative central formula approximation of g(x) depends on the specific function g(x) being approximated. To calculate the truncation error, we need to expand g(x) in a Taylor series around the point x:

g(x + h) = g(x) + h * g'(x) + (h^2 / 2) * g”(x) + …

Then, we substitute this expansion into the central difference formula for the second derivative and compare it with the actual second derivative of g(x). The difference between the two will give us the truncation error.

It is important to note that the truncation error analysis is

a mathematical concept used to analyze the accuracy of numerical approximations. By understanding the truncation error, we can determine the precision and reliability of the finite difference formulas in approximating the derivatives of a given function.

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To summarize:

– Problem 1 requires finding the system of linear equations to calculate the coefficients of a polynomial function based on given data points.
– Problem 2 involves approximating function derivatives using finite difference formulas, analyzing truncation errors for each formula, and determining the truncation error of a specific approximation for the second derivative of a function.

Please let me know if there’s anything else I can help you with!