Lecture Notes CS350/CSE301/MATH350/ECE391

In addition to the course book, some very useful books are:

§ Advanced Engineering Mathematics, Erwin Kreyszig, Wiley. (multivariate calculus, ODEs and PDEs, some numerics, and linear algebra)

§ Linear Algebra and its Applications, Gilbert Strang, Harcourt, Brace, Jovanovich, Publishers, San Diego. (basic linear algebra)

§ Numerical Linear Algebra, Nick Trefethen and David Bau, SIAM. (numerical linear algebra – what a surprise)

§ Matrix Computations, Golub and Van Loan, Johns Hopkins University Press. (bible of numerical linear algebra)

§ Numerial Mathematics, Alfio Quarteroni, Riccardo Sacco, and Fausto Saleri, Springer, 2000 (same range of topics as Heath, much more material, including more background, but more mathematical/abstract in style)

§ Numerical Methods for Unconstrained Optimization and Nonlinear Equations, J.E. Dennis and R.B. Schnabel, SIAM. (nonlinear equations and optimization)

§ Numerical Solution of Partial Differential Equations, K. W. Morton and D. F. Mayers, Cambridge University Press, 2000. (numerical PDEs, mainly by finite differences, very nice treatment of theoretical concepts such as stability, convergence, consistency, and error analysis)

Lecture 1: Introduction

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Lecture 2: Matlab Introduction

• Read some of the (online) available matlab tutorials
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• matlab1 matlab2 matlab3

Lecture 3: Chapter 1 (part 1): Introduction, Approximations in Scientific Computation

• read Chapter 1, pp. 1-16 (sections 1.1 - 1.2.7)
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Lecture 4: Chapter 1 (part 2): Computer Arithmetic, Mathematical Software

• read Chapter 1, pp. 16-39 (sections 1.4 – 1.5)
• ps_file (from Prof. Heath's lecture notes)
• Matlab example codes used in class (for Computer Problem 1.5):
• Powers of 10 version
• Powers of 2 version (compare powers of 10 version on calculator)
• exp1 version 1 of computing f(x): (exp(x)-1)/x
• exp3 version 2 of computing f(x): (exp(x) – 1)/(log(exp(x))

Lecture 5: Chapter 2 (part 1): Systems of linear equations, Gaussian eliminination, LU

• read Chapter 2, pp. 35-41
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Lecture 6: Chapter 2 (part 2): Pivoting, implementation, complexity, modified problems

• read Chapter 2, pp. 42-52
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Lecture 7: Chapter 2 (part 3): Norms, Condition numbers, and Accuracy

• read Chapter 2, pp. 52-61
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• Matlab examples code

Lecture 8: Chapter 2 (part 4): Special linear systems, efficient implementation

• read Chapter 2, pp. 61-68
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Lecture 9: Example of linear least squares: Medical Imaging

Lecture 10: Chapter 3 (part 1): Linear Least Squares

• read chapter 3, pp. 81-89
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Lecture 11: Chapter 3 (part 2): QR-factorization methods (from Prof. Heath's lecture notes)

· read chapter 3, pp. 89-102

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Lecture 12: Chapter 4 (part 1): Eigenvalues and eigenvectors: Introduction

· read chapter 4, pp. 157-173 (sections 4.1-4.4)

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Lecture 13: Chapter 4 (part 2): Methods for computing a few eigenvalues and -vectors

· chapter 4, pp. 173-

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· Matlab example codes used in class

o mypower.m – power method for two simple problems

o rayleigh.m – rayleigh quotient iteration for same two problems (converges much better)

o mypower2.m - power method with double dominant eigenvalue (nondefective) and complex conjugate pair of ‘dominant’ eigenvalues

Lecture 14: Chapter 4 (part 3): QR Algorithm (for all eigenvalues and -vectors)

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Lecture 15: Chapter 4 (part 4): Generalized eigenvalue problems, QZ Algorithm, SVD (from Prof. Heath's lecture notes)

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Lecture 16: Chapter 4 (part 5): Eigenvalues/vectors from large sparse matrices

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Lecture 17: Chapter 5 (part 1): Nonlinear equations: introduction

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Lecture 18: Chapter 5 (part 2): Nonlinear equations: scalar equations

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Lecture 19: Chapter 5 (part 3): Nonlinear equations: scalar equations and systems of equations

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Lecture 20: Chapter 5 (part 4): Nonlinear equations: systems of equations

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Lecture 21: Chapter 6 (part 1): Optimization: scalar case

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Lecture 22: Chapter 6 (part 2): Optimization: multidimensional case

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Lecture 23: Chapter 7: Interpolation

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Lecture 24: Chapter 8: Numerical Integration (from Prof. Heath's lecture notes)

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Lecture 25: Chapter 8: Numerical Differentiation (from Prof. Heath's lecture notes)

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Lecture 26: Chapter 9 (part 1): Introduction initial value problems, higher ODEs, Examples (pendulums), stability of the (scalar) ODE.

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Lecture 27: Chapter 9 (part 2): Taylor in higher dimensions, review eigenvalues/vectors, stability of systems of ODEs.

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Lecture 28: Chapter 9 (part 3): Numerical solution of ODEs: Euler's method, local and global truncation error, stability, and stepsize control.

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Lecture 29: Chapter 9 (part 4): Implicit methods, stiff differential equations, survey of numerical methods (from M. Heath's lecture notes).

Lecture 30: Chapter 9 (part 5): Survey of numerical methods (from M. Heath's lecture notes = same as book).

Lecture 31: Chapter 10 (part 1): Boundary Value Problems: introduction and finite difference methods

· matlab_example (pendulum)

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Lecture 32: Chapter 10 (part 2): Boundary Value Problems: Galerkin finite element methods

· matlab_example (pendulum)

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Lecture 33: Chapter 10 (part 3): Boundary Value Problems: collocation finite element methods

· also check these examples (from M. Heaths lecture notes).

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Lecture 34: Chapter 11 (part 1): Partial differential equations: introduction, time-dependent semidiscrete methods

Lecture 35: Chapter 11 (part 2) (without the simulations/movies, to be added soon): Partial differential equations - Application: Pattern formation in biology

Some additional slides with simulation results and matlab programs will be added soon

Lecture 36: Chapter 11 (part 3): PDEs: time-dependent fully discrete methods, hyperbolic problems, time-independent problems (from M. Heath's lecture notes)

Lecture 37: Chapter 11 (part 4): PDEs: time-independent problems

Lecture 38: Chapter 11 (part 5): PDEs: Solution of sparse linear systems

Lecture 39: Chapter 11 (part 6): PDEs: Solution of sparse linear systems