(Fall 2017 & Fall 2018)

Instructor: Yu-Jiang Wu

Office Room: No. 434, Qiyun Building, 222 Road Tianshui South, Lanzhou City

e-mail: myjaw@lzu.edu.cn

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Target:

This course is taken by first-year graduate students majoring in mathematics, along with graduate students from other subjects in science and a variety of engineering technologies. The numerical approaches involved in the course of modern methods for scientific computing are partially characterized by implementable solutions equipped with modern numerical software MATALB. Contents of the course are presented here by two main parts. Part one deals with steady-state boundary value problems, starting with two-point boundary value problems in one- dimension and then elliptic equations in two or three dimensions. The finite difference approximation gives a large but finite algebraic system of equations. Part two concerns modern treatments of applied numerical linear algebra in which we present fundamental in as elegant a fashion as possible. Techniques for dense, sparse and structured problems will be covered. Some of the standard problems whose numerical solutions will be studied are systems of linear equations, least squares problems, eigenvalue problems, singular value problems, and some of their generalizations and applications..

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Class Schedule:

Four lectures per week. 72 lectures one semester.

A weekly written homework grading in due time.

Final Exam: After the 18th week.

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Textbook:

1. L. N. Trefethen, D. Bau III. Numerical Linear Algebra. SIAM, Philadelphia, PA 1997.

2. R. J. LeVeque. Finite difference methods for ordinary and partial differential equations. Steady-state and time-dependent problems. SIAM, Philadelphia, PA 2007

We will cover Chapters 1-6 of book 1 and Chapters 1-3 of book 2. Some of the exercises are separated from book 2.

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Course Contents:

Part I. Numerical Solution for Boundary Value Problems

Chapter 1. Finite Difference Approximations

1. Truncation errors

2. Deriving finite difference approximations

3. Second order derivatives

4. Higher order derivatives

5. A general approach to deriving the coefficients

Chapter 2. Steady States and Boundary Value Problems

1. The heat equation

2. Boundary conditions

3. The steady-state problem

4. A simple finite difference method

5. Local truncation error

6. Global error

7. Stability

8. Consistency

9. Convergence

10. Stability in 2-norm

11. Neumann boundary conditions

12. Existence and uniqueness

13. Ordering the unknowns and equations

14. A general linear second order equation

Chapter 3. Elliptic Equations

1. Steady-state heat conduction

2. The 5-point stencil for the Laplacian

3. Ordering the unknowns and equations

4. Accuracy and stability

5. The 9-point Laplacian

6. Other elliptic equations

Part II. Numerical Linear Algebra

Chapter 4. Fundamentals

1. Matrix-vector multiplication

2. Orthogonal vectors and matrices

3. Norms

4. The singular value decomposition

5. More on the SVD

Chapter 5. QR Factorization and Least Squares

1. Projections

2. QR factorization

3. Gauss-Schmidt orthogonalization

4. Householder triangularization

5, Least squares problems

Chapter 6. Conditioning and Stability

1. Conditioning and condition numbers

2. Floating point arithmetic

3. Stability

4. More on stability

5. Stability of Householder triangularization

6. Stability of back substitution

7. Conditioning of least squares problems

8. Stability of least squares algorithms

Chapter 7. Systems of Equations

1. Gaussian elimination

2. Pivoting

3. Stability of Gaussian elimination

4, Cholesky factorization

Chapter 8. Eigenvalues

1. Eigenvalue problems

2. Overview of eigenvalue algorithms

3. Reduction to Householder or tridiagonal form

4. Rayleigh quotient, inverse iteration

5. QR algorithm without shift

6. QR algorithm with shift

7. Other eigenvalue algorithms

8. Computing the SVD

Chapter 9. Iterative Methods

1. Overview of iterative methods

2. The Arnoldi iteration

3. How Arnoldi locates eigenvalues

4. GMRES

5. The Lanczos iteration

6. Preconditioning

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Reading materials:

[1]J. W. Demmel. Applied Numerical Linear Algebra. SIAM, Philadelphia, PA, 1997.

[2]A. Quarteroni, R. Sacco, F. Saleri. Numerical Mathematics. Springer-Verlag, Berlin, 2000.

[3]K. W. Morton, D.F. Mayers. Numerical solution of partial differential equations. An introduction. Second edition. Cambridge University Press, Cambridge, 2005

[4]L. Sadun. Applied Linear Algebra. The Decoupling Principle. AMS, Providence, RI, 2008.

[5]L. N. Trefethen. Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations. (Website：http://people.maths.ox.ac.uk/trefethen/pdetext.html)

[6]G. H. Golub, C. F. Van Loan. Matrix computations. Fourth edition. Johns Hopkins University Press, Baltimore, MD,2013.