Cornell University

Department of Physics

Phys 480/680, Astro 690 April 2, 2001

Computational Physics, Spring 2001

Homework Assignment # 6

(Due Friday, March 30 at 8:00 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.ccmr.cornell.edu/~muchomas/P480.

Recap of last week:

• Lec12, 03/05 [1 hr]: Non-spherical states (higher angular momenta); total energy in density functional theory
  Reading: Sections ``Introduction,'' ``Procedure'' (you can skip the relativistic part), ``Results,'' and ``The LDA approximation,'' from
  http://math.nist.gov/DFTdata/atomdata/narrative.html
• Lec13, 03/07 [1 hr]: Forms in the literature for exchange and correlation energy; solution of many non-linear equations in many unknowns (Broyden approaches); our need for a ``Modified'' Broyden's Method
  Reading: 9.6-9.7
• Lab5, 03/09 [1 hr]: Full, self-consistent solutions for density functional theory of atoms; using gprof; avoiding malloc; inlining optimizations.

Preview of coming week:

• Lec14, 03/26 [1 hr]: Relations among self-consistent iteration, charge mixing, and Broyden's method; line optimization
• Lec15, 03/27 [1 hr]: Full, modified Broyden's method; Introduction to Numerical linear algebra; benefits of LAPACK
• Lab 6, 03/29 [1 hr]: Analysis of convergence of charge mixing; condition numbers; exploration of FLOP rates within inner loops.

• 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
• Basic Linear Algebra Subroutines (BLAS)
  • BLAS2
  • BLAS3
• ``test_getg.c''
• ``test_lu.c''
• ``timer.c''
• timer() for Windows
• About this document ...

Postscript Version

PDF Version