Introduction to Numerical Analysis I

AMSC 466, Fall 2019


Course Information

LectureRoom 4122 CSIC Bldg. #406 TuTh 2-3:15pm
Note special place: Math Bldg. Rm. 3206 on Tue 9/10 and 10/22
Note special place: Math Bldg. Rm. 3206 on Thu 10/24
InstructorProfessor Eitan Tadmor
Contacttel.: x5-0648   Email:
Office HoursBy appointment ( )
CSCAMM 4141 CSIC Bldg. #406
Grader

Michael Rawson Email:

Midterm Thur 10/17 2-3:30pm 4122 CSIC Bldg. #406
NOTE: use of a calculator and your notes is allowed
Final Sat 12/14 10:30-12:30pm 4122 CSIC Bldg. #406 (open material)
Grading25% Homework, 25% Mid-Term, 50% Final


Course Description

  1. DIRECT METHODS for Solving Systems of Linear Equations

    1. Introduction: Cramer's rule; triangular and unitary systems
    2. Gauss elimination: multipliers, operation count
    3. LU decompositions: symmetric, positive definite, banded, Hessenberg,...
    4. Pivoting

      5. Lecture notes: Gaussian elimination and LU decompositions [ pdf file ]
      Assignment #1 [ pdf file ] ... with answers [ pdf file]
    5. Backward error analysis: ill-conditioning, condition number, backward error estimates

      6. Lecture notes: Stability and backward error analysis [ pdf file ]

    6. QR decompositions: Householder reflections, stability, least-squares

      7. Lecture notes: Orthognalization and QR decompositions [ pdf file ]
      Assignment #2 [ pdf file ] with answers [ pdf file]
    7. Beyond LU and QR decompositions: Sherman–Morrison formula, block decomposition, ...
      Circulant matrices and FFT

      8. Lecture notes: Beyond LU and QR - other methods for solving linear systems [ pdf file ]

      • Additional reading: The Fast Fourier Transform and Its Applications [ pdf file]
      Assignment #3 [ pdf file ] ... with answers [ pdf file]

  2. INTERPOLATION

    1. Lagrange and Newton interpolants
      • Error estimates: Runge effect, region of analyticity, Lebesgue constant

      Lecture notes: Lagrange and Newton algebraic interpolants. Error bounds [ pdf file ]

      • Additional reading: M. J. D. Powell, On the Lebesgue constant for Chebyshev nodes [ pdf file]
      Assignment #4 [ pdf file ] ... with answers [ pdf file]
    2. Interpolation with derivatives
      • Hermite interpolation
      • piecewise smooth interpolation: Splines

      Lecture notes: Interpolation with derivatives: Hermite and splines [ pdf file ]

      • Additional reading: 'Splines in Industry' by T. Sauer
      Assignment #5 [ pdf file ] ... with answers [ pdf file]
    3. Interplolation of data in equi-spaced nodes
      • Difference operators; Divided differences
      • Synthetic calculus
      • Forward backward and centered formulae

      - Lecture notes: Interpolation with equi-spaced nodes
        (synthetic calculus of difference opeartors) [ pdf file ]
      Assignment #6 [ pdf file ] with answers: [ pdf file ]
      MID-TERM [ pdf file ] ... and its answers: [ pdf file ]
    4. From trigonometric to near-optimality of Chebyshev nodes
      • Trigonometric interpolation
      • Chebyshev polynomials: min-max property and optimaility

      Lecture notes: Trigonometric interpolation [ pdf file ]

  3. LEAST-SQUARES PROBLEMS

    1. Normal equations
    2. Orthogonal polynomials: Legendre, Chebyshev, ...
    3. Fourier expansions

      4. Lecture notes: The least-squares problem [ pdf file ]
      Assignment #7 [ pdf file ] with answers: [ pdf file ]

  4. NUMERICAL DIFFERETIATION; NUMERICAL INTEGRATION

    1. Numerical differentiation
      • Polynomial and spline interpolants: local stencials; error estimates
      • Equi-spaced points: synthetic calculus; compact stencils; Richardson extrapolation

      Lecture notes: Numerical differenetiation [ pdf file]
      Assignment #8 [ pdf file ] ... with answers [ pdf file]
    2. Gauss quadrature rules
    3. Newton-Cotes rules -- numerical integration with equi-spaced nodes
    4. Composite quadrature rules -- trapezoidal, Simpson, ...
    5. Romberg & adaptive integration

      6. Lecture notes: Gauss, Newton-Cotes, and composite quadrature rules [ pdf file ]
      Lecture notes: Euler-Macluarin formula [ pdf file]
      Assignment #9 [ pdf file ] ... with answers [ pdf file]

  5. ITERATIVE METHODS for Solution of Nonlinear Equations

    1. Fixed point iterations
    2. Scalar equations
      • Newton iterations
      • Approximate Newton method: the secant method, regula falsi, Steffensen's method...,
      • High-order extensions
    3. Accelerations: Aitken's process

      4. Lecture notes: Iterative methods for the solution of nonlinear equations [ pdf file ]
      Assignment #10 [ pdf file ]
    4. Systems of equations
      • Fixed point iterations
      • Newton method revisited
      • Minimizres: gradient descent
      • Continuation methods (homotopy)

    Leonhard Euler [ pdf file ] (by W. Gautschi)


    References

    Recommended reference book (available on UMd bookstore):
    E. Suli and D. Mayers, An INTRODUCTION TO NUMERICAL ANALYSIS, Cambridge Univ. Press, 2013
    and
    W. Gautschi, NUMERICAL ANALYSIS, Birkhäuser Basel (Springer), 2012

    GENERAL TEXTBOOKS

    K. Atkinson, An INTRODUCTION to NUMERICAL ANALYSIS, Wiley, 1987

    S. Conte & C. deBoor, ELEMENTARY NUMERICAL ANALYSIS, McGraw-Hill
    User friendly; Shows how 'it' works; Proofs, exercises and notes

    G. Dahlquist & A. Bjorck, NUMERICAL METHODS, Prentice-Hall,
    User friendly; Shows how 'it' works; Exercises

    E. Isaacson & H. Keller, ANALYSIS of NUMERICAL METHODS, Wiley
    The 'First'; Proofs; out-dated in certain aspects; Encrypted message in Preface

    A. Ralston & P. Rabinowitz, FIRST COURSE in NUMERICAL ANALYSIS, 2nd ed., McGraw-hill,
    Detailed; Scholarly written; Comprehensive; Proofs exercises and notes

    J. Stoer & R. Bulrisch, INTRODUCTION TO NUMERICAL ANALYSIS, 2nd ed., Springer
    detailed account on approximation, linear solvers & eigen-solvers, ODE solvers,..

    B. Wendroff, THEORETICAL NUMERICAL ANALYSIS, Academic Press, 1966
    Only the 'Proofs'; elegant presentation

    APPROXIMATION THEORY

    E. W. Cheney, INTRODUCTION TO APPROXIMATION THEORY
    Classical

    P. Davis, INTERPOLATION & APPROXIMATION, Dover
    Very readable

    T. Rivlin, AN INTRODUCTION to the APPROXIMATION of FUNCTIONS
    Classical

    R. DeVore & G. Lorentz, CONSTRUCTIVE APPROXIMATION, Springer
    A detailed account from classical theory to the modern theory; everything; Proofs exercises and notes

    NUMERICAL INTEGRATION

    F. Davis & P. Rabinowitz, NUMERICAL INTEGRATION,
    Everything...

    Eitan Tadmor