## 642:574 Numerical Analysis: Course Syllabus

Material on most of the topics listed below can be found in:

K. Atkinson, An Introduction to Numerical Analysis,,Wiley, 1989 (second edition), referred to as [A].

D. Kincaid and W. Cheney: Mathematics of Scientific Computing, AMS, 2002 (third edition), referred to as [KC].

A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics, Springer, 2004 (second edition), referred to as [QRS]

J. Stoer and R. Bulirsch: Introduction to Numerical Analysis, Springer, 2002, (third edition) referred to as [SB].

### 4: Numerical Solution of Systems of Linear Equations

```4.1 Gaussian elimination [A: 8.1], [KC: 4.1], [QSS: 3.2, 3.3], [SB: 4.1]
4.2 Pivoting and scaling in Gaussian elimination [A: 8.2], [KC: 4.3], [QSS: 3.5]
4.3 Variants of Gaussian elimination [A: 8.3], [KC: 4.2], [QSS: 3.3, 3.4], [SB: 4.2, 4.3]
4.4 Error analysis [A: 8.4], [KC: 4.4], [QSS: 3.1], [SB: 4.4]
4.5 Basic iterative methods: Jacobi, Gauss-Seidel, SOR [A: 8.6], [KC: 4.6], [QSS: 4.2], [SB: 8.1]
4.6 General theory of iterative methods [A: 8.6], [KC: 4.6], [QSS: 4.1], [SB: 8.2]
4.7 Conjugate gradient method [A: 8.9], [KC: 4.7], [QSS: 4.3], [SB: 8.7]
```

### 5: The Matrix Eigenvalue Problem

```5.1 Eigenvalue location, error, and stability results [A: 9.1], [KC: 5.0,5.2], [QSS: 5.1, 5.2], [SB: 6.1, 6.9]
5.2 The power and inverse power methods [A: 9.2], [KC: 5.1], [QSS: 5.3], [SB: 6.6]
5.3 The QR method [A: 9.5], [KC: 5.5], [QSS: 5.6, 5.7], [SB: 6.6]
5.4 The calculation of eigenvectors and inverse iteration [A: 9.6], [QSS: 5.3]
```

### 6: Numerical Solution of Nonlinear Equations

```6.1 Some basic methods: bisection, false position, secant, and Newton's methods
[A: 2.1, 2.2, 2.3], [KC: 3.1, 3.1, 3.2, 3.3], [QSS: 6.2], [SB: 5.1]
6.2 Convergence theory for one-point iteration methods [A: 2.5], [KC: 3.4], [QSS: 6.3], [SB: 5.2]
6.3 Systems of nonlinear equations: Newton's method [A: 2.10, 2.11], [KC: 3.2], [QSS: 7.1], [SB: 5.3]
6.4 Quasi-Newton methods [QSS: 7.1], [SB: 5.4]
6.5 Unconstrained optimization [A: 2.12], [QSS: 7.2], [SB: 5.11]
```

### 7: Two Point Boundary Value Problems: Finite Difference and Finite Element Methods

```7.1 Derivation of finite difference methods [A:6.11], [KC: 8.7, 8.9], [QSS: 12.2], [SB: 7.4]
7.2 Error estimates for finite difference methods [QSS: 12.2], [SB: 7.4]
7.3 Derivation of finite element methods [KC: 8.10], [QSS: 12.4], [SB: 7.5]
7.4 Error estimates for finite element methods [QSS: 12.4], [SB: 7.5]
```

### 8: Finite Difference Methods for Model Problems in Partial Differential Equations

```8.1 Poisson's equation [QSS: 12.6]
8.2 Heat equation [KC: 9.0, 9.1, 9.2, 9.3], [QSS: 13.2, 13.3]
```