Numerical Methods: Taylor Series, Root Finding, and More

Taylor Series

f(x+h) = f(x) + f'(x)h + f''(x)h² /2! + f'''(x)h³ /3! + ...

Secant Method (Find Root)

1. Rewrite as f(x)=0
2. Set initial points
3. f'(x₂) = (f(x₂) - f(x₁)) / (x₂-x₁)
4. New point becomes x₂
5. Iterate

LU Decomposition

A = LU

Crout's

Identity top

Doolittle

Identity bottom

1. Set up LU
2. Find components through matrix multiplication and solve for variables

Solving for b matrix

LUX=B

1. Set UX = Y
2. Solve LY = B using algebra
3. Solve UX = Y using algebra

Norm

Absolute value of the largest row sum

Condition

||A||_inf||A^-1||_inf

Gauss-Seidel

1. Set each equation to a variable
2. Use any initial values for the first equation
3. Use the x₁ from the first equation to solve x₂
4. Iterate (Fast version of Jacobi Method)

Discrete Least Squares Approximation

AX = B

1. Make matrix A by [ n ∑(x) ∑(x²) ...]
2. Make matrix B [ ∑(y) ∑(xy) ∑(x²y) ] column
3. Solve for X

Gauss Quadrature

1. Convert integral to limits -1 to 1 by:
   ∫_a^bf(x)dx = ((b-a)/2) ∫_-1¹f(((b-a)/2)x + ((b+a)/2))dx
2. Find the appropriate n-Point function
3. Replace f(x) with the converted f(x). Ex: c₁f(.6x₁+.7)dx + c₂f(.6x₂+.7)dx

Numerical Differentiation

1st Order

• Forward: (f(x+h) - f(x)) / h
• Backward: (f(x) - f(x-h)) / h
• Central: (f(x+h) - f(x-h)) / 2h

2nd Order

• Forward: (f(x+2h) - 2f(x+h) + f(x)) / h²
• Backward: (f(x) - 2f(x-h) + f(x-2h)) / h²
• Central: (f(x+h) - 2f(x) + f(x-h)) / h²

3rd Order

• Forward: (f(x+3h) - 3f(x+2h) + 3f(x+h) - f(x)) / h³
• Backward: (f(x) - 3f(x-h) + 3f(x-2h) - f(x-3h)) / h³
• Central: (f(x+2h) - 2f(x+h) + 2f(x-h) - f(x-2h)) / 2h