Definitions

• "ODE" means "system of ordinary differential equations".
• "ODE integration" means finding the solution of an ODE.
• "Integrator" means either an algorithm to integrate an ODE or an algorithm to calculate a definite inetegral.
• "ODE integrator" means an algorithm to integrate an ODE.

Examination projects (10 in total)

0. Symmetric multi-processing with OpenMP
   • Pick an obligatory exercise of your choice and make a multi-processing (two-processing, to be precise) version of it using OpenMP pragrmas.
   • There are two processors on `molveno', so make sure your program uses (ideally) 200% (or in any case more than 100%) CPU time when it runs (as shown by the `top' command).
   • Make your code thread safe and beware of the race conditions.
   • Hint: rand() is NOT thread-safe.
1. Matrix diagonalization: classical Jacobi diagonalization with indexing
   • Implement the classical Jacobi diagonalization (zero the largest off-diagonal element) with indexing:
     • Before starting diagonalization find out the largest off-diagonal elements in each row and store their indices in a separate index array. It's O(n²) operations—expensive—but you only do it once in the beginning.
     • Find the largest off-diagonal element from the pre-found largest elements in each row (only O(n) operations) and zero it by a Jacobi rotation.
     • Update your index. Since the Jacobi rotation only affects elements in rows and columns with indices p and q the update is also only O(n) operations.
     • Repeat the last two steps until converged
   • Illustrate.
2. Quasi-random Halton-sequence integration in multi-dimensions
   Implement the Monte-Carlo integrator which employs Halton quasi-random sequence; illustrate; check the asymptotic behavior of the integration error; compare with plain Monte-Carlo and adaptive integrators; etc.
3. ODE integration with complex numbers
   • Implement an integrator (with adaptive step-size control) for systems of ordinary equations, which takes complex-valued functions of complex argument and integrate the system of ordinary equations from a complex start-point a to a complex end-point b.
4. ODE integration: two-and-half-step method
   • Implement an adaptive step-size integrator for ODE using two-and-half-step method (see the corresponding lecture note)
   • Illustrate and compare with your other methods.
5. Minimization: quasi-Newton method with Broyden's update
   • Implement and illustrate.
6. Minimization: quasi-Newton method with SR1 update
   • Implement and illustrate.
7. ODE integration: two-step method with correction
   • Implement an (adaptive step-size) integrator for ODE using two-step method with correction (see the corresponding lecture note).
   • Illustrate and compare with your other methods.
8. Definite integrals with complex numbers
   • Implement an adaptive algorithm to calculate a definite integral of a complex-valued function of complex argument along a path in the compex plane.
   • Hint: the easiest path is the straight line from a to b.
9. Root-finding in multidimensions: Broyden's quasi-Newton method
   • Implement the Broyden's quasi-Newton method in multidimensions (see the corresponding lecture note).
   • Illustrate and compare with your other methods.