Cornell University

Department of Physics

Phys 480/680, Astro 690 March 25, 2002

Computational Physics, Spring 2002

Homework Assignment # 6

(Due Tuesday, April 2 at 8:40 am in Computer Lab.)

Agenda and readings for the week of March 26:

Goal: Optimize to produce efficient, high-precision software to calculate the electronic structure of atoms within density-functional theory; explore FLOP rates of various operations.

Readings marked NR are from Numerical Recipes: The Art of Scientific Computing, 2 edition (in C). Readings marked LN are from the course lecture notes to be found at http://www.muchomas.lassp.cornell.edu/P480.

Recap of last two weeks of classes:

• Lec12, 03/05 (Tue: mini-lecture, 25mins): Exchange and correlation energy functionals.
  Reading: Section ``The LDA approximation,'' from
  http://math.nist.gov/DFTdata/atomdata/narrative.html
• Lab4, 03/02 (Tue): Full, self-consistent solutions for Hartree theory of atoms
• Lec13, 03/07 (Thu): Solution of many non-linear equations in many unknowns; analysis of convergence of self-consistent iteration schemes; overview of more advanced approaches.
  Reading: NR 9.6-9.7
• Lec14, 03/12 (Tue): Modified Broyden method for large non-linear systems; overview of numerical linear algebra; solution of linear equations by LU decomposition and back-substitution.
  Reading: NR 2.0, 2.3
• Lec15, 03/14 (Thu): Crout's algorithm for LU decomposition (including partial pivoting).
  Reading: NR 2.3

Preview of coming week:

• Lec16, 03/26 (Tue: mini-lecture, 25mins): Optimization -- Amdahl's law, inner loop optimization, rates of common operations
• Lab5, 03/26 (Tue): Full, self-consistent solutions for density functional theory of atoms; profiling and practical optimization.
• Lec17, 03/28 (Thu): Cache optimization; BLAS1-3; blocked matrix multiply; ``Expressive software'' concept.
• Lec18, 04/02 (Tue: mini-lecture, 25 mins): Partial Differential Equations (PDE's); expressive software solution of Poisson's Eq. in d=3 dimensions in a single line of code!
• Lab6, 04/02 (Tue) Fully optimized atomic DFT calculations

• Contents
• Packaging
  • ``getg''
    • Debugging
  • ``ludcmpp480'' and ``lubksbp480''
    • Debugging
• Initial Performance
  • Convergence Rate
  • Run time
• Broyden Solver
  • Bookkeeping
    • Debugging
  • Improved initial guess
  • ``broyden''
    • Matrix G and vector u
      • Declarations:
      • Residual vector :
      • Formation:
    • Solution of G b = u
    • Formation of final Rho[it+1]
  • Convergence rate
• Optimization
  • Calls to dvector() and free_dvector()
  • Calls to rk4p480()
  • Calls to derivs_Schrodinger()
  • Calls to exc() and excp()
• Uranium
• Inner-loop operations
  • timer()
  • Other inner-loop operations
• ``test_getg.c''
• ``test_lu.c''
• ``timer.c''
• timer() for Windows
• About this document ...

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