MATH 62251/72251: Numerical Analysis I
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Text: Numerical Linear Algebra, by L. N. Trefethen and D. Bau, III, SIAM, Philadelphia, 1997 | |||
| Week | Lecture Material | Relevant Sections in Text | Homework |
|---|---|---|---|
| 1-2 | Matrix-vector multiplication | Lecture 1 | Exercises 1.3 (Hint: Use that I=inv(R)R), 1.4a |
| 1-2 | Orthogonal vectors and matrices | Lecture 2 | Exercises 2.1, 2.3a,b (You may assume that A is real and symmetric.) , 2.4, 2.5a,b, 2.6 |
| 1-2 | Norms | Lecture 3 | Exercise 3.1, 3.3, 3.5 |
| 3-4 | The SVD | Lecture 4 | Exercises 4.1a-e, 4.4 |
| 3-4 | More on the SVD | Lecture 5 | Exercises 5.1, 5.2 |
| 4 | More on the SVD | Lectures 5 & 6 |
Exercise 5.4 Problem 5.5: Let A be a real m by n matrix with m>=n. Use the SVD to show that the union of range(A') and null(A) make up all of R^n, and that the sets range(A') and null(A) are orthogonal. Also show that range(A) and null(A') make up all of R^m and that the sets range(A) and null(A') are orthogonal. Here ' denotes transposition. Exercise 6.3 |
| 5 | Projectors & Gram-Schmidt orthogonalization | Lectures 6-7 |
Exercises 6.1 (algebraic justification suffices), 6.2, 6.4 Implement Algorithm 7.1 in MATLAB (or some other programming language) and apply to some large problems. Check if the columns of the generated matrix Q are orthogonal. Implement Algorithm 8.1 in MATLAB (or some other programming language) and apply to some large problems. Check if the columns of the generated matrix Q are orthogonal. Compare results for Algorithms 7.1 and 8.1. |
| 9 | QR factorization | Lecture 7 | Exercises 7.1, 7.2, 7.5 |
| 9 | Orthogonal matrices and least squares problems | Lectures 10-11 | Exercises 5.8, 5.11, 5.12 from the supplementary material handed out in class. Exercise 5.12 also can be done differently than suggested by the hint. Exercise 11.3 from the textbook. |
| 9 | Householder and Givens matrices, and condition numbers | Lectures 10-12 | 12.1, 12.3 |
| 9 | Floating point arithmetic | Lecture 13 | Exercise 13.3 |
| 9 | Stability | Lecture 14 | Exercises 14.1, 14.2 |
| 10 | More on Stability | Lecture 15 | Exercises 15.1a-f, 15.2a,b |
| 10 | Stability of Householder Triangularization | Lecture 16 | Exercise 16.1a |
| 11 | Conditioning of Least Squares Problems | Lecture 18 | Exercise 18.1a-c |
| 11 | Gaussian elimination | Lecture 20 | Exercises 20.2 |
| 13 | Pivoting | Lecture 21 |
Exercises 21.1c, 21.2
Exercise 21.8: Let the n by n matrix A be nonsingular. Show that Gaussian elimination with partial pivoting can be carried out, i.e., there is a permutation matrix P, a lower triangular matrix L with ones on the diagonal, and an upper triangular matrix U such that PA=LU. |
| 13 | Stability of Gaussian elimination | Lecture 22 | Exercise 22.1 |
| 13 | Cholesky factorization | Lecture 23 | Exercises 23.1 |
| 13 | Eigenvalue problems | Lecture 24 | Exercise 24.1 |
| 14 | Eigenvalue problems | Lectures 24, 25, 28 | Exercises 24.2, 25.1, 28.2 a,b |