Applied Linear Algebra and Numerical Analysis

Tentative course syllabus

Chapters, sections or pages refer to the Recktenwald text book.

• MATLAB (Chapter 1, Sections 2.1, 2.2, 3.1--3.3, and parts of 3.4, 3.5)
• Taylor series and truncation errors (Section 5.3)
• Root finding (Chapter 6 through Section 6.4)
  • Bracketing roots
  • Bisection method
  • Fixed point iteration
  • Newton's method
• Linear algebra (Parts of Chapter 7: roughly the sections on pages 293-304, 309-316, 325-333, 335-342, 346-349, 352-355.)
  • vectors, matrices, norms, linear independence, columns space
• Linear systems and Gaussian elimination (Chapter 8 through Section 8.2.3)
  • Consistency of systems, unique solutions
  • forward and back substitution
  • Gaussian elimination
  • LU decomposition
  • ideas of conditioning and stability
  • Finite precision arithmetic (Section 5.2)
• Newton's method for nonlinear systems (Section 8.5.3)
• Curve fitting and least squares (Chapter 9 through page 476, except Sec. 9.1.4)
  • Fit by straight line or linear combination of arbitrary basis functions
  • Setting up and solving the normal equations
  • Exponential fitting by reducing to a linear problem
• Interpolation (Chapter 10 through Section 10.3.4)
  • Polynomial interpolation using monomial basis and Vandermonde matrix
  • Lagrange basis functions
  • Newton basis and divided differences
  • Relation of Newton form to Taylor series (Section 5.3.1)
  • Hermite interpolation of derivatives
  • Piecewise polynomial interpolation
  • Main ideas of cubic splines (overview of 10.3.5)
• Numerical integration or quadrature (Chapter 11 through Section 11.2.2 and 11.4)
  • Piecewise polynomial interpolation and composite rules
  • Trapezoid rule and Simpson's rule
  • Adaptive quadrature
• Ordinary differential equations (Chapter 12 through Section 12.3.3)
  • Euler's method
  • Local and global errors
  • Midpoint method and Runge Kutta methods