Cornell University

Department of Physics

Phys 480/680, Astro 690 March 7, 2007

Computational Physics, Spring 2007

Homework Assignment # 5

(Due Wed, March 14 at 5:00pm)

Agenda and readings:

Goal: Building upon the components you have made, you will perform your own full density-functional-theory calculations of atoms. You will reproduce the accepted values to at least as much precision as published by the US National Institute of Standards and Technology, the micro-hartree level!!!

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

Recap:

• Lec 11, 03/01 (Thu) Exchange-correlation energy per particle, electron gas parameter r$_s$; Functional form for exchange-correlation energy; Solution of many non-linear equations in many unknowns; mixing method, convergence rates, condition numbers
  Readings: NR 9.6-9.7, 2.0; Sections ``Introduction,'' ``Procedure'' (you can skip the relativistic part), ``Results,'' and ``The LDA approximation,'' from
  http://physics.nist.gov/PhysRefData/DFTdata/contents.html .

• Lec 12, 03/06 (Tue) Newton-Raphson and Broyden approaches; motivation for a ``modified'' Broyden's method for large systems of equations; ``modified'' Broyden's method

• Lab 4, 03/06 (Tue) ``Hot-wired''solver for atoms within Hartree theory!

Preview:

• Lec 13, 03/08 (Thu) Introduction to numerical linear algebra; Benefits of LAPACK; Solution of linear systems through LU decomposition; Crout's algorithm for LU decomposition
  Reading: NR 2.0, 2.3

• Lec 14, 03/13 (Tue) Crout's Algorithm with pivoting; inner loop tricks
  Reading: NR 2.3

• Lab 5, 03/15 (Thu) *NOTE* NO LECTURE GO DIRECTLY TO LAB: Profiling and real-time optimization of full atomic solver-- using gprof; avoiding malloc; inlining optimizations.

• 03/20, 03/22 SPRING BREAK

• 03/27 (Tue) Cache optimization, avoiding cache misses, BLAS1-3, blocked matrix multiplies; Automated empirical optimization of software (AEOS): automatically tuned linear algebra software (ATLAS), fastest Fourier Transform in the West (FFTW).

• 03/29 (Thu) Solution of Poisson's equation in d>1 dimensions; expressive software; solution to Poisson's equation in a single line of code; choice of plane-wave (complex exponential) basis for periodic boundary conditions.

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