Introduction to Numerical Analysis I

AMSC 466, Fall 2018


Course Information

LectureRoom 4122 CSIC Bldg. #406 TuTh 2-3:15pm
Note special place: Math Bldg. Rm. 3206 on Tue 10/23
Note special place: Math Bldg. Rm. 3206 on Thu 10/25
InstructorProfessor Eitan Tadmor
Contacttel.: x5-0648   Email:
Office HoursBy appointment ( )
CSCAMM 4141 CSIC Bldg. #406
Grader Michael Rawson Email:
MidtermThu 10/4 2-3:20pm 4122 CSIC Bldg. #406
NOTE: use of a calculator and your notes is allowed
FinalSat 12/15 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 ]

    5. Backward error analysis: ill-conditioning, condition number, backward error estimates

      6. Lecture notes: Backward error analysis [ pdf file ]

      Assignment #1 [ pdf file ] ... with answers [ pdf file]
    6. QR decompositions: householder reflections, stability, least-squares
    7. Additional topics(*): Sherman-Woodbury formula, block decomposition, circulant natrices, conjugate gradient, ...

      8. Lecture notes: Orthognalization and QR decompositions [ pdf file ]
      Lecture notes: Beyond LU and QR -- other methods for solving linear systems [ pdf file ]
      Assignment #2 [ pdf file ] ... with answers [ pdf file]

  2. INTERPOLATION

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

      Lecture notes: Algebraic interpolation. Lagrange and Newton forms [ pdf file ]
      Lecture notes: Interpolation Error [ pdf file ]
      (and additonal reading on Jackson theorem(*) [ pdf file ])
      Assignment #3 [ pdf file ] ... with answers [ pdf file]
    2. Interpolation II. Interpolation with derivatives
      • Hermite interpolation
      • piecewise interpolation: Splines

      Lecture notes: Interpolation with derivatives: Hermite and splines [ pdf file ]
      (and related lecture notes on 'Splines in Industry' by T. Sauer)
      Polynomial vs. spline interpolation
    3. Interplolation III. Equi-spaced points
      • Difference operators; Divided differences
      • Synthetic calculus
      • Forward backward and centered formulae

      Lecture notes: Interpolation with equi-spaced points. Synthetic calculus of difference opeartors [ pdf file ]
      Assignment #4 [ pdf file ] with answers: [ pdf file ]
      MID-TERM [ pdf file ] ... and its answers: [ pdf file ]
    4. Interplolation IV. Chebyshev points
      • Chebyshev polynomials: min-max property and optimaility
      • Trigonometric interpolation

      Lecture notes: Trigonometric interpolation [ pdf file ]

      Additional reading:
      • M. J. D. Powell, On the Lebesgue constant for Chebyshev nodes [ pdf file ]
      • The sin(x) subroutine in MATLAB [ pdf file ]

  3. 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 #5 [ pdf file ] ... with answers [ pdf file]
    2. Gauss Quadrature rule
    3. Numerical integration with equi-spaced points
      • Newton-Cotes formulae
      • Composite Simpson's rule
      • Romberg & adaptive integration
      Assignment #6 [ pdf file ] ... with answers [ pdf file]
      Lecture notes: Numerical integration: Gauss, Newton-Cotes, and related quadrature [ pdf file ]
      Lecture notes: Numerical integration and Euler-Macluarin formula [ pdf file]
      Review of Q&A #7 [ pdf file ]

  4. ITERATIVE METHODS for Solution of Nonlinear Equations

    1. The bisection method
    2. Newton's Method
      • local vs. global convergence, multiple roots
      • Extensions: the secant method, Steffensen's method...,
      • High-order extensions
      Assignment #8 [ pdf file ]
    3. Systems of equations
    4. Fixed point iterations
    5. Accelerations: Aitken's process, Steffenssen's method

    Epilogue - have you been paying attention in your numerical analysis course?

    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

    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