# Numerical Analysis Aug-Dec 2014

For latest and updated codes, see https://github.com/cpraveen/na

## Students

- Amit Kumar
- Jayesh Vinay Badawaik
- Neelabja Chatterjee
- Prashant Kumar

## References

- K. Atkinson, Introduction to Numerical Analysis, Wiley
- David Kincaid and E. Ward Cheney, Numerical Analysis: Mathematics of scientific computing

## Getting the codes

You can read the codes here

https://github.com/cpraveen/na2014

You can download all the code using git

```
$ git clone https://github.com/cpraveen/na2014
```

Alternately, you can download a zip file containing all the code here.

## Starting ipython

ipython is installed on all the computers in the lab. Open a terminal and type the following to start ipython notebook

```
$ ipython notebook --pylab=inline
```

This should start the notebook in your web browser.

## Introduction

- Sensitivity of polynomial roots
- Unstable iterations
- Unstable iterations: ODE example

## Computer Arithmetic

- Store 1/10 and print it to see the conversion of real number to floating number
- Demonstration of unit round
- Example of overflow error and how to avoid it
- Errors due to subtracting nearly equal quantities [C version]

## Root finding

- Bisection method
- Bisection method: implemented as a function
- Newton method
- Finite difference and round-off error
- Complex variable method for derivative
- Example of fixed point iterations
- Newton method for multiple roots
- Newton method for complex roots
- Newton method for a system of equations

## Interpolation and approximation

- Condition number of vandermonde matrix for uniform points
- Interpolate cosine on uniformly spaced points
- Chebyshev points
- Runge phenomenon
- Function omega in interpolation error
- Interpolate abs(x) on uniformly spaced and chebyshev points
- Barycentric Lagrange interpolation
- Barycentric Lagrange interpolation on Chebyshev points
- Piecewise polynomial interpolation
- Trigonometric interpolation
- Trigonometric interpolation: convergence test

## Solving ODE

- Euler method, convergence test: python
- Euler method, convergence test: python
- Euler method, instability example: python
- Trapezoidal method, convergence test: python
- Absolute stability domains of AB, AM, BDF schemes: python
- Example of stiff ODE: python
- ODE with periodic solution, quadratic invariant: python
- Non-linear BVP using shooting method: python
- Non-linear BVP using finite difference method: python

## Linear Algebra

- LU decomposition
- LU decomposition with row pivoting
- Classical iterative methods: Gauss-Jacobi, Gauss-Seidel, SOR: 1D BVP
- Conjugate gradient method: 1D BVP
- Classical iterative methods: Gauss-Jacobi, Gauss-Seidel, SOR: 2D BVP
- Conjugate gradient method: 2D BVP