Course Information

Course Booklet Information Page

The course meets in the F+ block, from noon - 1:15pm on Tuesdays and Thursdays, in Room BP-1. Due to my travel, we will miss some classes (primarily on Tuesdays), and will try and make these up in the Friday F+ block, from noon - 1:15pm, as our collective schedules permit. My office hours are from 10am - noon on Thursdays and 1:30 - 2:30pm on Fridays, in Room BP-212; I am also available by appointment.

Your grade in this course will be determined entirely by in-class participation. You are expected to lead the discussion, as appropriate, based on the distribution of the papers to be read. You are also expected to participate actively in the discussion in each class (even whem you are not leading the discussion), both asking and answering questions, as appropriate.

Leading the Discussion
For each paper (or group of papers treated together), two students will be assigned to lead the discussion. Together, these students are responsible for outlining the important results included in each paper, as well as for putting these results in context. Typically, one student will be responsible for placing the paper in its historical context, answering how the paper follows from the literature at the time, and for presenting the results discussed within the paper. The second student will be responsible for presenting the results of the paper as covered in the modern literature, and discussing the impact of these results on modern Applied Mathematics.

Participating in the Discussion
All students are expected to have read the paper(s) to be discussed before each class. During the discussion, all students are expected to raise points that they thought were notable about the paper(s) of the day, or ask questions about pieces that were unclear.

Disability Services
If you are requesting an accommodation due to a documented disability, you must register with the Disability Services Office at the beginning of the semester. To do so, call the Student Services Desk at 617-627-2000 to arrange an appointment with Sandra Baer, Program Director of Disability Services.

Learning Objectives
This course addresses the following learning objectives of the Ph.D. Program in Mathematics

  • 1.b. Clear understanding of key hypotheses and conclusions
  • 1.c. Synthesis of formal theory into a comprehensive picture of mathematical phenomena
  • 1.d. Application of general theory to specific examples
  • 1.e. Sharpening of intuition through appropriate counterexamples
  • 3.a. Explanation of key ideas and general strategies
  • 3.b. Motivation of underlying issues
  • 3.c. Clear oral presentation of arguments
  • 3.d. Effective use of visual media
  • 3.f. Thinking on one's feet; fielding questions
  • 4.a. Searching the literature
  • 4.b. Reading and understanding definitions and statements of results
  • 4.c. Organizing information from disparate sources
  • 6.a. Classroom and blackboard techniques
  • 6.b. Organization and presentation of mathematics at a level appropriate to the audience
  • Papers

    This course is based on one originated by Nick Trefethen; you can find information on that course here.

    Over the semester, we will read a series of papers on Numerical Analysis; a superset of the papers that we may read is below.


  • An Algorithm for the Machine Calculation of Complex Fourier Series, by J.W. Cooley and J.W. Tukey, Math. Comput. 19, 297-301 (1965).
  • To be added: Aitken

    Numerical Integration
  • Calculation of Gauss Quadrature Rules, by G.H. Golub and J.H. Welsch, Math. Comput. 23, 221-230 (1969).

  • A Special Stability Problem for Linear Multistep Methods, by G.G. Dahlquist, BIT 3, 27-43 (1963).
  • On the Attainable Order of Runge-Kutta Methods, by J.C. Butcher, Math. Comput. 19, 408-417 (1965).

  • On the Partial Difference Equations of Mathematical Physics, by R. Courant, K. Friedrichs, and H. Lewy, Mathematische Annalen 100, 32-74 (1928).
  • Variational Methods for the Solution of Problems of Equilibrium and Vibrations, by R. Courant, Bulletin of the American Mathematical Society 49, 1-23 (1943).
  • Survey of the Stability of Linear Finite Difference Equations, by P.D. Lax and R.D. Richtmyer, Communications on Pure and Applied Mathematics 9, 267-293 (1956).
  • Systems of Conservation Laws, by P. Lax and B. Wendroff, Communications on Pure and Applied Mathematics 13, 217-237 (1960).

  • Maximization of a Linear Function of Variables Subject to Linear Inequalities, by G.B. Dantzig, Activity Analysis of Production and Allocation, pp. 339-347. Cowles Commission Monograph No. 13. John Wiley & Sons, Inc., New York, N. Y.; Chapman & Hall, Ltd., London, 1951.
  • Application of the Simplex Method to a Transportation Problem, by G.B. Dantzig, Activity Analysis of Production and Allocation, pp. 359-373. Cowles Commission Monograph No. 13. John Wiley & Sons, Inc., New York, N. Y.; Chapman & Hall, Ltd., London, 1951.
  • A New Polynomial-Time Algorithm for Linear Programming, by N. Karmarkar, Combinatorica 4, 373-395 (1984).
  • The Convergence of a Class of Double-Rank Minimization Algorithms 1. General Considerations, by C.G. Broyden, J. Inst. Maths Applics 6, 76-90 (1970).
  • A New Approach to Variable Metric Algorithms, by R. Fletcher, The Computer Journal 13, 317-322 (1970).
  • A Family of Variable-Metric Methods Derived by Variational Means, by D. Goldfarb, Math. Comp. 24, 23-26 (1970).
  • Conditioning of Quasi-Newton Methods for Function Minimization, by D.F. Shanno, Math. Comp. 24, 647-656 (1970).

    Direct Methods in Linear Algebra
  • Error Analysis of Direct Methods of Matrix Inversion, by J.H. Wilkinson, J. Assoc. Comput. Mach. 8, 281-330 (1961).
  • Computation of Plane Unitary Rotations Transforming a General Matrix to Triangular Form, by W. Givens, J. Soc. Indust. Appl. Math. 6, 26 - 50 (1958).
  • Unitary Triangularization of a Nonsymmetric Matrix, by A.S. Householder, J. Assoc. Comput. Mach. 5, 339-342 (1958).
  • Gaussian Elimination is not Optimal, by V. Strassen, Numer. Math. 13, 354-356 (1969).
  • Nested Dissection of a Regular Finite Element Mesh, by A. George, SIAM J. Numer. Anal. 10, 345-363 (1973).

    Iterative Methods in Linear Algebra
  • Iterative Methods for Solving Partial Difference Equations of Elliptic Type, D. Young, Trans. Amer. Math. Soc. 76, 92-111 (1954).
  • Methods of Conjugate Gradients for Solving Linear Systems, M.R. Hestenes and E. Stiefel, Journal of Research of the National Bureau of Standards 49, 409-436 (1952).
  • A Generalized Conjugate Gradient Method for the Numerical Solution of Elliptic Partial Differential Equations, by P. Concus, G.H. Golub, and D.P. O'Leary, in Sparse Matrix Computations, J.R. Bunch and D.J. Rose, eds., Academic Press, 1976, pp. 309-332.
  • An Iterative Solution Method for Linear Systems of Which the Coefficient Matrix is a Symmetric M-Matrix, by J.A. Meijerink and H.A. van der Vorst, Math. Comp. 31, 148-162 (1977).
  • GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems, by Y. Saad and M.H. Schultz, SIAM J. Sci. Stat. Comput. 7, 856-869 (1986).

    Multigrid Methods
  • A Relaxation Method for Solving Elliptic Difference Equations, by R.P. Fedorenko, Z. Vycisl. Mat. i Mat. Fiz. 1, 922-927, 1961.
  • The Speed of Convergence of One Iterative Process, by R.P. Fedorenko, Z. Vycisl. Mat. i Mat. Fiz. 4, 559-564, 1964.
  • Multi-Level Adaptive Solutions to Boundary-Value Problems, by A. Brandt, Math. Comp. 31, 333-390 (1977).
  • Algebraic Multigrid (AMG) for Automatic Multigrid Solution with Application to Geodetic Computations, by A. Brandt, S. McCormick, and J. Ruge, Tech. Report Inst. Comp. Studies., Colo. State Univ., 1982.
  • Algebraic multigrid (AMG) for sparse matrix equations, by A. Brandt, S. McCormick, and J. Ruge, in Sparsity and its applications (Loughborough, 1983), 257-284, Cambridge Univ. Press, Cambridge, 1985.

    Eigenvalues and SVD:
  • On the Speed of Convergence of Cyclic and Quasicyclic Jacobi Methods for Computing Eigenvalues of Hermitian Matrices, by P. Henrici, J. Soc. Indust. Appl. Math. 6, 144-162 (1958).
  • The QR Transformation: A Unitary Analogue to the LR Transformation - Part 1, by J.G.F. Francis, Comput. J. 4, 265-271 (1961).
  • The QR Transformation - Part 2, by J.G.F. Francis, Comput. J. 4, 332-345 (1962).
  • Calculating the Singular Values and Pseudo-Inverse of a Matrix, by G. Golub and W. Kahan, J. SIAM Numer. Anal. Ser. B 2, 205-224 (1965).

  • Tensor Decompositions and Applications, by T.G. Kolda and B.W. Bader, SIAM Review 51, 455-500 (2009).
  • Schedule

  • 1/19: Introduction
  • 1/24: No Class (Prof. MacLachlan away)
  • 1/26: Dantzig on Linear Programming (2 papers); Prof. MacLachlan leads discussion
  • 1/27: Karmarkar on Linear Programming; Prof. MacLachlan leads discussion
  • 1/31: Golub and Welsch on Gauss Quadrature; Megan and Rozi lead discussion
  • 2/2: Butcher on Runge-Kutta; Tom and Melody lead discussion
  • 2/7: No Class (Prof. MacLachlan away)
  • 2/9: No Class (Prof. MacLachlan away)
  • 2/14: Cooley and Tukey on FFT; Stephen and Dong lead discussion
  • 2/16: Dahlquist on Stability; Melody and Matt lead discussion
  • 2/17: Courant, Friedrichs, and Lewy on CFL condition; Stephanie and Shannon lead discussion
  • 2/21: Courant on Finite Elements; Stephen and Rozi lead discussion
  • 2/23: No Class Monday schedule
  • 2/24: Lax and Richtmyer on Stability and Convergence; Matt and Shannon lead discussion
  • 2/28: Lax and Wendroff on Conservation Laws; Tom and Dong lead discussion
  • 3/1: Householder on Unitary Triangularization; Ning and Stephanie lead discussion
  • 3/6: Givens on Unitary Triangularization; Matt and Tom lead discussion
  • 3/8: Strassen on Matrix Multiplication; Shannon and Melody lead discussion
  • 3/13: Wilkinson on Error Analysis; Ning and Rozi lead discussion
  • 3/15: George on Nested Dissection; Stephen and Dong lead discussion
  • 3/27: No Class (Prof. MacLachlan away)
  • 3/29: No Class (Prof. MacLachlan away)
  • 4/3: Golub and Kahan on SVD; Matt and Stephen lead discussion
  • 4/5: Young on Iterative Methods; Stephanie and Dong lead discussion
  • 4/6: Hestenes and Stiefel on CG; Tom and Ning lead discussion
  • 4/10: Concus, Golub, and O'Leary on PCG; Melody and Rozi lead discussion
  • 4/12: Meijerink and van der Vorst on ILU; Stephanie and Dong lead discussion
  • 4/13: Saad and Schultz on GMRES; Stephen and Shannon lead discussion
  • 4/17: Fedorenko on Multigrid (2 papers); Stephanie and Melody lead discussion
  • 4/19: Brandt on Multigrid; Ning and Matt lead discussion
  • 4/24: Brandt, McCormick, and Ruge on Algebraic Multigrid; Tom and Rozi lead discussion
  • 4/26: Kolda and Bader on Tensor Decompositions; Ning and Shannon lead discussion