INDUMAT Procedures (Quiz 2)

Question 1

Steps for LU Decomposition

Answer 1

1. Setup matrices as AX=B
2. Get the upper triangular matrix based on A
3. Put the multipliers in the lower triangular matrix (Make sure to negate or make “+” into “-“ and vice versa)
4. Get y values from LY = B
5. Get x values from UX = Y

Question 2

Steps for Inverse Method

Answer 2

1. Setup the matrices as AX = B
2. Find A^-1
3. X = A^-1 B

Question 3

Steps for Gaussian Elimination

Answer 3

1. Setup augmented matrix
2. Use Elementary Row Operations until you create an upper triangular matrix
3. Make sure the main diagonal values are equal to ‘1’.
4. Convert back into equation form
5. Back substitute

Question 4

Steps for Gauss Jordan Method

Answer 4

1. Setup augmented matrix
2. Use Elementary Row Operations until you create an identity matrix
3. Convert back into equation form

Question 5

Steps for Gauss Jacobi

Answer 5

1. Check for diagonal dominance (rearrange rows if found otherwise)
2. Isolate the variables per row based on the diagonal
3. Initialize/ Get k=0 iteration wherein variables are all equal to zero
4. Substitute values into the equations of the variables using values computed from previous iterations
5. Solve for error values (error = absolute value of old-new)
6. Repeat the process until the calculated answer has an error value below the tolerable error.

Question 6

Steps for Gauss Seidel

Answer 6

1. Check for diagonal dominance (rearrange rows if found otherwise)
2. Isolate the variables per row based on the diagonal
3. Initialize/ Get k=0 iteration wherein variables are all equal to zero
4. Substitute values into the equations of the variables using the most recent values
5. Solve for error values (error = absolute value of old-new)
6. Repeat the process until the calculated answer has an error value below the tolerable error.

Question 7

Steps for SOR

Answer 7

1. Check for diagonal dominance (rearrange rows if found otherwise)
2. Isolate the variables per row based on the diagonal then add the following: (w) + (1-w)x.
3. Conduct the Gauss-Seidel Method in this process
4. Continue until you reach below error value

Question 8

Steps for MOSS

Question 9

Steps for Newton Raphson

Answer 9

1. Check stability criterion which is given by:
   |f(x)f’‘(x)|<[f’(x)]]^2
2. If the stability criterion is valid, new iteration = previous iteration - ratio
3. Substitute the value in f(x) and f’(x)
4. Get ratio: f(x) / f’(x).
5. Get error
6. Repeat process until below tolerable error.

Question 10

Steps for Bisection Method

Answer 10

1. Given a and b, substitute the values into the function
2. Get the slope: m = a+b / 2
3. Substitute m in function
4. if f(m) = positive value, replace b.
5. if f(m) = negative value, replace a.
6. Repeat the process until below error value
7. Make sure to get the m value as the final answer.

Question 11

Steps for Secant Method

Answer 11

1. Given x^0 and x^1 (k=0 and k=1), substitute into function.
2. Conduct iteration method:
   hint: diagonal right - diagonal left / up
3. Repeat until error value

Question 12

Steps for Regula Falsi

Answer 12

1. Given an x+ and x-, substitute value into function
2. Do the iteration (same pattern as secant)
3. Substitute iteration into function
4. If f(x^k+1) = positive value, change x+ and x- keeps the previous iteration value
5. If f(x^k+1) = negative value, change x- and x+ keeps the previous iteration value.