Lecture Notes Spring 2024

• Lecture 1 January 9 Intro:errors and Flops

  The answer computer gives you is almost always approximate.

  All the numbers are represented as:

  $$x = \pm\sum\limits_{k=1}^{N}\frac{d_k}{\beta^{k-1}}\cdot \beta^E.$$

  where $\beta$ is base, $N$ is precision, $d_i$ is integer between 0 and $\beta-1$ and $E$ is integer between $L$ and $U$.

  Matlab basics were presented.

• Lecture 2 January 12 Machine Precision

  Machine precision $e_{\rm mach}$. Relative error for presenting normalized number $x$ is given by

  $$\frac{\vert x-{\rm Flop}(x)\vert}{\vert x\vert}<\frac{\epsilon_{\rm mach}}{2}.$$
  Overflow, underflow, gradual underflow, sum, difference, product and division.

• Lecture 3 January 16 End Chapter 1 Start Chapter 2 Smallest representable number, first non-representable integer, Homework discussion. Beginning of Nonlinear Equations. Rate of Convergence and Bisection method .In-Class Matlab examples

• Lecture 4 January 19 Fixed Point iterations, Newton method Review of fixed point iterations, quadratic convergence of the Newton's method

• Lecture 5 January 23 Secant Method, summary and beginning of $A x = b$ Summary of $f(x)=0$ and the start of a new chapter.

• Lecture 6 January 26 Solving $A x = b$ Upper- and lower-triangular matrices, elementary elimination matrices, LU decomposition. Properties of elementary elimination matrices are demonstrated on the attached Matlab demonstration.

• Lecture 7 January 30 Review, LU decomposition

  We had a review of Chapters 1 and 2, and we covered LU decomposition with examples.

  N=12;
  LogTimesInversion=[1:N];LogTimesLU=[1:N];
  for k=[1:N]
   n = 2^k;A=rand(n,n);b=rand(n,1); tic;B=inv(A);
  LogTimesInversion(k)=log(toc);
  tic; x = A\b;LogTimesLU(k)=log(toc);
  end
  plot([1:N],LogTimesInversion,'ro',[1:N],LogTimesLU,'bo',[1:N],LogTimesInversion,[1:N],LogTimesLU)
  

• Lecture 8 February 2 Midterm Exam

• Lecture 9 February 6 Norm of vector and a matrix

  Vector and Matrix 1-norm, 2-norm and infinity-norm.

  $$\vert\vert x\vert\vert _p = \left( \sum\limits_{k=1}^n \vert x_k\vert^p\right)^{\frac{1}{p}}, \ \ p = 1,2,\dots,\infty$$
  and
  $$\vert\vert A\vert\vert = \max\limits_{x} \frac{\vert\vert A x\vert\vert}{\vert\vert x\vert\vert}.$$

• Lecture 10 February 9 Error Estimates for $A x = b$ Vector and Matrix norms, Condition number of a matrix, error estimate in solving $A x = b$. Residual, relative residual, absolute and relative error.
• Lecture 11 February 13 Solving nonlinear Equations Error Estimates, condition number of a matrix, vector form of the Newton's method and the Midterm Review.

• Lecture 12 February 16 Eigenvalue and Eigenvector of a matrix Eigenvalues and eigenvectors of a matrix. Power method to find dominant eigenvalue.
• Lecture 13 February 23 Power Iterations, inverse Power Iterattions, Power Iterations with shift.
• Lecture 14 February 27 Interpolation Interpolation, general basis functions, monomials and linear Lagrange interpolation.

• Lecture 15 March 1 Quadratic Lagrange interpolation, power $n$ Lagrange interpolation, piece-wise linear interpolation
• Lecture 16 March 12 Interpolation: PW linear interpolation example, cubic splines theory and example, s Theorem's and summary

• Lecture 17 March 15 Midterm

• Lecture 18 March 19,22,26 Integration Chapter: Midpoint, Trapezoid and Simson's quadrature rules. Composite Midpoint, composit Trapezoid and composit Simson's quadratures with examples.

• Lecture 21 March 29 Integration Continue Gauss quadrature, error terms for Midpoint, composite midpoint and trapezoid.
• Lecture 22 April 2 Error Terms rror Terms for Simpson's and Trapezoid. Examples of quadratures and summary. http://www.yurilvov.com/NClecture22.pdf
• Lecture 23 April 5 Numerical Differentiation umerical Differentiation: forward, backward, central and one sided finite differences. First and Second derivatives. http://www.yurilvov.com/NClecture23.pdf
• Lecture 24 April 9 Numerical Integration of ODE's orward and Backward Euler and the trapezoid method http://www.yurilvov.com/NClecture24.pdf

• Lecture 25 April 12 Runge-Kutta and Midterm Review unge-Kutta methods of second and fourth order. The review before the Midterm. http://www.yurilvov.com/NClecture25.pdf
• Lecture 26 April 16 Midterm III
• Lecture 27 April 19 Matlab Laboratory
• Lecture 28 April 23 Summary of the NC Semester