Course outline

Instructor: Dr. Gantumur Tsogtgerel

Office hours: During regular meeting times, or by appointment.

Prerequisite: MATH 247 or MATH 251; and MATH 387; or permission of the instructor.

Calendar description: Development, analysis and effective use of numerical methods to solve problems arising in applications. Topics include direct and iterative methods for the solution of linear equations (including preconditioning), eigenvalue problems, interpolation, approximation, quadrature, solution of nonlinear systems.

Homework: Both analytical and computational. Assigned and graded roughly every two weeks.

Course project: The course project consists of the student studying an advanced topic, implementing the relevant algorithms, experimenting, writing a report, and giving a presentation.

Software: Python is the official programming language of the class.

Grading: Homework assignments 50% + Course project 50%.

Course delivery method

What would be lectures in normal times will be replaced by pre-recorded lectures and regular Zoom meetings.

Regular Zoom meetings

  • Scheduled during the assigned lecture times.
  • We will discuss concrete examples related to and variations and extensions of the ideas covered in the pre-recorded lectures.
  • Problem solving and Q&A sessions will also be held.
  • Flipped classroom activities are being considered.

    • Noncontact activities

  • Pre-recorded lectures will be available on MyCourses: To be watched before the regular Zoom meetings
  • Past regular Zoom meetings (recorded on MyCourses)
  • Lecture notes and other reading material (on MyCourses)

    • Topics to be covered

  • Direct methods for linear systems: LU factorization and variants, QR factorization, conditioning and stability
  • Iterative methods for linear systems: Jacobi, Gauss-Seidel, SOR, steepest descent, CG, preconditioning
  • Eigenproblems: Hessenberg form, QR algorithm, Jacobi method
  • Least squares problems: normal equations, QR factorization, SVD
  • Nonlinear systems and optimization: gradient descent, fixed point iteration, Newton-Raphson and variants
  • Approximation of functions: interpolation, least squares approximation, uniform approximation, use of polynomial and trigonometric functions
  • Initial value problems: numerical integration, Runge-Kutta methods, consistency, stability, convergence, stiffness, adaptivity
  • If time permits: linear multistep methods, symplectic methods, shooting method
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    Recommended books

  • Alfio Quarteroni, Riccardo Sacco and Fausto Saleri. Numerical mathematics. Springer
  • Endre Süli and David Mayers. An introduction to numerical analysis. Cambridge University
  • Lloyd Nick Trefethen and David Bau III. Numerical linear algebra. SIAM
  • Germund Dahlquist and Ake Bjorck. Numerical methods. Dover
  • Eugene Isaacson and Herbert Bishop Keller. Analysis of numerical methods. Dover
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    Online resources

  • Related courses: Math 578 Fall 2009, Math 387 Winter 2018, Math 387 Winter 2016
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