Introduction to Numerical Analysis I

AMSC 466, Fall 2026

Course Information

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

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

Lecture notes: Stability and QR decompositions [ pdf file ]


7. Beyond LU and QR decompositions: Sherman–Morrison formula, circulant matrices and FFT

Lecture notes: beyond LU and QR decompositions [ pdf file ]

• Additional reading: Fast Fourier Transform [ Cooley-Tuckey 1969 ][ DeBoor 1990 ]


Assignment #2 [ pdf file ] with answers [ pdf file]

2. LEAST-SQUARES PROBLEMS

1. Normal equations
2. Orthogonal polynomials: Legendre, Chebyshev, ...

Lecture notes: The least-squares problem [ pdf file ]


Assignment #3 [ pdf file ] with answers [ pdf file ]

Lecture notes: The least-squares error: SVD and spectral convergence [ pdf file ]


Assignment #4 [ pdf file ] with answers: [ pdf file ]

MID-TERM [ pdf file ]

3. INTERPOLATION

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

Assignment #5 [ pdf file ]

3. Interpolation with derivatives

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 ]
• Additional reading: The sin(x) subroutine in MATLAB [ pdf file ]
• Additional reading: [Splines in Industry ] by T. Sauer


4. From trigonometric to near-optimality of Chebyshev nodes

Assignment #6

4. NUMERICAL DIFFERETIATION; NUMERICAL INTEGRATION

1. Numerical differentiation

   Assignment #7

2. Gauss quadrature rules
3. Newton-Cotes rules -- numerical integration with equi-spaced nodes
4. Composite quadrature rules -- trapezoidal, Simpson, ...

Assignment #8

5. ITERATIVE METHODS for Solution of Nonlinear Equations

1. Fixed point iterations
2. Scalar equations
3. Systems of equations

Assignment #9

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• Additional reading: Leonhard Euler [ pdf file ] (by W. Gautschi)

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References

Recommended reference book

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...



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Eitan Tadmor