Course Outline, Homework, Reading, and Exam Dates²

• Part I Fundamentals
  • Lecture 1 (Aug. 23) Matrix-Vector Multiplication, Orthogonal Vectors and Matrices, Norms
  • Lecture 2 (Aug. 25) The Singular Value Decomposition, More on the SVD
• Assignment I (Aug. 30) 1.3, 2.3, 3.5, 4.3, 4.5, 5.3
• Reading I (Sept. 1) Trefethen (1992) - Definition of Numerical Analysis: Appendix of the Textbook.
• Part II QR Factorization and Least Squares
  • Lecture 3 (Aug. 30) Projectors, QR Factorization
  • Lecture 4 (Sept. 1) Gram-Schmidt Orthogonalization, MATLAB
  • Lecture 5 (Sept. 6) Householder Triangularization
  • Lecture 6 (Sept. 8) Least Squares Problems
• Assignment II (Sept. 13) 6.1, 6.5, 7.5, 8.1, 9.3, 10.1, 11.1, 11.3
• Reading II (Sept. 15) Householder (1958) - QR Decomposition: A. S. Householder, "Unitary triangularization of a nonsymmetric matrix," Journal of the Association of Computing Machinery 5 (1958), 339-342.
• Part III Conditioning and Stability
  • Lecture 7 (Sept. 13) Conditioning and Condition Numbers, Floating Point Arithmetic
  • Lecture 8 (Sept. 15) Stability, More on Stability
  • Lecture 9 (Sept. 20) Stability of Householder Triangularization, Stability of Back Substitution
  • Lecture 10 (Sept. 22) Conditioning of Least Squares Problems, Stability of Least Squares Algorithms
• Assignment III (Sept. 27) 12.3, 13.3, 14.1, 15.1, 16.2, 17.1, 18.3, 19.2
• Reading III (Sept. 29) Wilkinson (1961) - Error Analysis for Systems of Eqs.: J.H. Wilkinson, "Error analysis of direct methods of matrix inversion," J. Assoc. Comput. Mach. 8 (1961), 281-330
• EXAM I (Sept. 27)
• Part IV Systems of Equations
  • Lecture 11 (Sept. 29) Gaussian Elimination, Pivoting
  • Lecture 12 (Oct. 4) Stability of Gaussian Elimination
  • Lecture 13 (Oct. 6) Cholesky Factorization
• Assignment IV (Oct. 11) 20.1, 20.3, 21.3, 22.2, 22.3, 23.3
• Reading IV (Oct. 13) Strassen (1969) - Gaussian Elimination is Not Optimal: V. Strassen, "Gaussian elimination is not optimal," Numer. Math. 13 (1969) 354-356.
• Part V Eigenvalues
  • Lecture 14 (Oct. 11) Eigenvalue Problems, Overview of Eigenvalue Algorithms
  • Lecture 15 (Oct. 13) Reduction to Hessenberg or Tridiagonal Form
  • Lecture 16 (Oct. 18) Rayleigh Quotient, Inverse Iteration
  • Lecture 17 (Oct. 20) QR Algorithm without Shifts
  • Lecture 18 (Oct. 25) QR Algorithm with Shifts
• Assignment V (Oct. 27) 24.3, 25.1, 26.2, 27.1, 28.2, 29.1, 30.5, 31.3
• Reading V (Nov. 1) Golub & Kahan (1965)- The Singular Value Decomposition: G. Golub and W. Kahan, "Calculating the singular values and pseudo-inverse of a matrix," SIAM Journal on Numerical Analysis 2 (1965), 205-224.
• EXAM II (Oct. 27)
• Part VI Iterative Methods
  • Lecture 19 (Nov. 1) Overview of Iterative Methods
  • Lecture 20 (Nov. 3) The Arnoldi Iteration
  • Lecture 21 (Nov. 8) How Arnoldi Locates Eigenvalues
  • Lecture 22 (Nov. 10) GMRES
  • Lecture 23 (Nov. 15) The Lanczos Iteration
  • Lecture 24 (Nov. 17) From Lanczos to Gauss Quadrature
  • Lecture 25 (Nov. 22) Conjugate Gradients
  • Lecture 26 (Nov. 29) Biorthogonalization Methods
• Assignment VI (Dec. 1) 33.1, 34.1, 35.3, 36.3, 37.1, 38.3, 38.6, 39.1
• Reading VI (Dec. 1) Hestenes & Stiefel (1952) - The Conjugate Gradient Iteration: Magnus R. Hestenes and Eduard Stiefel, "Methods of conjugate gradients for solving linear systems," Journal of Research of the National Bureau of Standards 49 (1952), 409-436.
• EXAM III (Dec. 1)