Numerical Analysis 2014

References

• 1. K. Atkinson, Introduction to Numerical Analysis, Wiley

  2. David Kincaid and E. Ward Cheney, Numerical Analysis: Mathematics of scientific computing

Getting the codes

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

1. Sensitivity of polynomial roots

2. Unstable iterations

3. Unstable iterations: ODE example

Computer Arithmetic

1. Store 1/10 and print it to see the conversion of real number to floating number

2. Demonstration of unit round

3. Example of overflow error and how to avoid it

   1. Errors due to subtracting nearly equal quantities [C version]

Root finding

1. Bisection method

2. Bisection method: implemented as a function

3. Newton method

4. Finite difference and round-off error

5. Complex variable method for derivative

6. Example of fixed point iterations

7. Newton method for multiple roots

8. Newton method for complex roots

9. Newton method for a system of equations

Interpolation and approximation

1. Condition number of vandermonde matrix for uniform points

2. Interpolate cosine on uniformly spaced points

3. Chebyshev points

4. Runge phenomenon

5. Function omega in interpolation error

6. Interpolate abs(x) on uniformly spaced and chebyshev points

7. Barycentric Lagrange interpolation

8. Barycentric Lagrange interpolation on Chebyshev points

9. Piecewise polynomial interpolation

10. Trigonometric interpolation

11. Trigonometric interpolation: convergence test

Solving ODE

• 1. Euler method, convergence test: python

  2. Euler method, convergence test: python

  3. Euler method, instability example: python

  4. Trapezoidal method, convergence test: python

  5. Absolute stability domains of AB, AM, BDF schemes: python

  6. Example of stiff ODE: python

  7. ODE with periodic solution, quadratic invariant: python

  8. Non-linear BVP using shooting method: python

  9. Non-linear BVP using finite difference method: python

Linear Algebra

1. LU decomposition

   1. LU decomposition with row pivoting 

2. Classical iterative methods: Gauss-Jacobi, Gauss-Seidel, SOR: 1D BVP

3. Conjugate gradient method: 1D BVP

4. Classical iterative methods: Gauss-Jacobi, Gauss-Seidel, SOR: 2D BVP

5. Conjugate gradient method: 2D BVP