Course Outline, Spring 2021

• Lectures One, Two, Three: Chapter ONE Introduction to numerical computing and floating point numbers. Machine precision, overflow, underflow, normalized numbers and subnormal and the $\bf eps$ command. Crazy graph of
  $$f(x)=\frac{1-(1-x)}{x}, \ \ -10^{-15}<x<10^{-15}.$$
  J26: errors, decimals and binaries,
• Lectures Four,Five, Six Chapter TWO - finding roots of nonlinear equations. Bisection, Newton, Secant and Halley methods. Rate of convergence and fzero command. Chapter Three Solving
  $$A x = b$$
  . Forward and Backward substitutions.
• Lecture 8 The Midterm
• Lecture 9-11 Chapter Three Solving the $A x = b$, Gauss elimination and $LU$ decomposition. Vector and Matrix norms, properties of norms. Condition number of a matrix. Error estimates. Residuals. Solving nonlinear system of equations via Newton iterations. q
• Chapter 4 Lectures 12,13 - Eigenvalues, eigenvectors, examples, 'eig' operation in Matlab, Power iterations, Theorem 4.1 and 4.2
• Lecture 14, inverse Power iterations, power iterations with the shift.
• Chapter 5 Lectures 15-17, Interpolation. Monomial, Lagrange, piece-wise linear. Example of interpolating
  $$f(x)=\frac{1}{1+25 x^2}$$
  , for $-1\le x\le 1$. Wiggles. Cubic Splines. Very boring actual calculation of a cubic spline. Theorems 5.1-5.6
• Chapter 6 Lectures 18-22. Numerical Integration: Left sided, Mid Point, trapezoid, Simpson, Gaussian, Adaptive. Special attention to error terms. First lecture: Intro into integration, and numerical integration of $\int_0^1 e^x d x$ by Left-sided, Midpoint, Trapezoid and Simpson quadratures. Second Lecture: error estimates for Left-Sided, Midpoint and composit Midpoint.