Numerical Methods

1. Foundations.


Sample Problems and Numerical Methods. Some Basic Issues. Algorithms and Computer Programs.



2. Solving Equations of One Variable.


Bisection Method. Regular Falsi and Secant Methods. Newton's Method. Muller's Method. Methods of Modern Computing.



3. Solving Systems of Linear Equations: Direct Methods.


Gaussian Elimination. Gaussian Elimination with Row Pivoting. Gaussian Elimination for Tridiagonal Systems. Methods of Modern Computing.



4. LU and QR Factorization.


LU Factorization from Gaussian Elimination. Direct LU Factorization. Applications of LU Factorization. Householder and Givens Transformations. QR Factorization. Methods of Modern Computing.



5. Eigenvalues and Eigenvectors.


Power Method. Inverse Power Method. QR Method. Methods of Modern Computing.



6. Solving Systems of Linear Equations: Iterative Methods.


Jacobi Method. Gauss-Seidel Method. Successive Over Relaxation. Methods of Modern Computing.



7. Nonlinear Functions of Several Variables.


Newton's Method for Systems of Equations. Fixed-Point Iteration for Nonlinear Systems. Minimum of a Nonlinear Function of Several Variables. Methods of Modern Computing.



8. Interpolation.


Polynomial Interpolation. Hermite Interpolation. Rational-Function Interpolation. Spline Interpolation. Methods of Modern Computing.



9. Function Approximation.


Least-Squares Approximation. Continuous Least-Squares Approximation. Function Approximation at a Point. Methods of Modern Computing.



10. Fourier Methods.


Fourier Approximation and Interpolation. Radix-2 Fast Fourier Transforms. General Fast Fourier Transforms. Methods of Modern Computing.



11. Numerical Differentiation and Integration.


Differentiation. Basic Numerical Integration. Better Numerical Integration. Gaussian Quadrature. Methods of Modern Computing.



12. Ordinary Differential Equations: Initial-Value Problems.


Taylor Methods. Runge-Kutta Methods. Multistep Methods. Stability. Methods of Modern Computing.



13. Ordinary Differential Equations: Higher-Order Equations and First-Order Systems.


Higher-Order ODEs. Systems of Two First-Order ODE. Systems of First-Order ODE. Stiff ODE and Ill-conditioned Problems. Methods of Modern Computing.



14. Ordinary Differential Equations: Boundary-Value Problems.


Shooting Method for Linear BVP. Shooting Method Nonlinear BVP. Finite-Difference Method for Linear BVP. Finite-Difference Method for Nonlinear BVP. Methods of Modern Computing.



15. Partial Differential Equations.


Classification of PDE. Heat equation: Parabolic PDE. Wave equation: Hyperbolic PDE. Poisson Equation - Elliptic PDE. Finite Element Method for an Elliptic PDE. Methods of Modern Computing.