Math 136 Numerical Analysis Exam Review

Study Notes

Algorithms are written in a specific programming language
Sources of errors include:
- Truncation errors: replacing continuous models with discrete and finite processes for computability
- Machine epsilon: property of machine representation and operations that affects the accuracy of calculations

Bisection Method:
- Converges to a root if the initial interval contains the root
- Does not depend on the initial iterate
First Order Approximations:
- Chord, Secant, Regula-Falsi, Newton-Raphson, and Steffensen methods
- Newton-Raphson method has a quadratic convergence rate
Fix Point Iterations:
- Used to find roots of a function
- Convergence depends on the initial iterate
Convergence of Root Finding Methods:
- Depends on the initial iterate and the method used
- Newton-Raphson method has a quadratic convergence rate

Linear Systems:
- Triangular Systems: can be solved using back-substitution
- LU Factorization: a method for solving linear systems
- Pivoting: a technique used to improve the accuracy of LU factorization
- Jacobi Method: an iterative method for solving linear systems
- Gauss-Seidel Method: an iterative method for solving linear systems
Nonlinear Systems:
- Newton Method: an iterative method for solving nonlinear systems

Absolute Errors: measure the difference between the exact and approximate values
Relative Errors: measure the difference between the exact and approximate values relative to the exact value
Reduced number of function evaluations: preferred to reduce computational cost and improve efficiency

Horner Method: an efficient method for evaluating polynomials
Python code implementation: requires careful handling of arrays and complex numbers

States that a fixed point iteration converges to a root if the initial iterate is close enough to the root
Proof involves showing that the sequence of iterates converges to the root

Converges to a simple root of a function if the initial iterate is close enough to the root
Quadratic convergence rate: the error decreases quadratically with each iteration