1. Introduction.
2. Error analysis.
3. Necessary defnitions and background.
4. Representation of floating-point numbers in computers.
5. Matrix and vector norms.
6. Solving nonlinear equations: the bisection method, the regula falsi method (the false position method), the secant method, and their algorithms.
7. Horner's Method, its algorithm, and its time complexity.
8. Taylor polynomial approximation.
9. Interplolation: Lagrange, Newton, piecewise linear interpolation, cubic splines.
10. Divided differences.
11. Curve fitting: linear least squares, quadratic least squares, power fit.
12. Linearlization.
13. Cubic splines.
14. Numerical integration (quadrature): Newton-Cotes closed and open formulas, the composite trapezoid(al) rule and the composite simpson's rule with their algorithms, changing the interval of integration, Gaussian quadrature, the recursive trapezoid rule, the recursive Simpson rule.
15. Approximation of derivatives: differentiation rules based on Taylor series and polynomial interpolation (two-point forward, backward, and central difference formulas, three-point forward, backward, and central difference formuals.
16. Approximate solutions of initial value problems: Euler's method, Taylor's methods, Midpoint method, Heun's method, Runge_Kutta method of the fourth order, other methods if time permits, local discretization error, global discretization error, final global error, algorithms for the previous methods, examples.
17. Systems of linear equations: review, standard methods, LU factorization (decomposition), Cholesky factorization, DoLittle's algorithm, Crout's algorithm, and Cholesky's algorithm, solving lower triangular systems by forward subsitution, solving upper triangular systems by back subsitution, solving linear systems using LU factorization, existence and uniqueness of LU and Cholesky factorizations, time complexity, space complexity, and stability of the discussed algorithms, solving real symmetric positive linear systems using Cholesky's factorization, examples.

    Next Lecture: Iterative methods for linear systems: the Jacobi method and the Gauss-Seidel method.

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