Cornell University

Department of Physics

Phys 480/680, Astro 690 April 5, 2006

Computational Physics, Spring 2006

Homework Assignment # 7

(Due Wednesday, April 12)

Agenda and readings for the weeks of April 6-23:

Goal: Learn to use the octave linear algebraic language to express scientific algorithms succinctly; solution of Poisson's equation in a single line of code

  phi=cI(Linv(-4*pi*O(cJ(n))));

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 of last week of class:

• 03/28 Lec 16 (9:15-9:55am): Memory access overhead factor, reuse of data while in cache, blocked matrix multiplication.

• 03/30 Lec 17: BLAS1-3, blocked matrix multiplies; Automated empirical optimization of software (AEOS), automatically tuned linear algebra software (ATLAS), fastest Fourier Transform in the West (FFTW). 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.

• 04/04 Lec 18: Form of operators and transforms in a plane-wave basis; octave/matlab notations.

Preview of coming weeks:

• 04/06 Lec 19: Discrete Fourier Transforms: standard ordering of indices, aliasing/Nyquist frequency, Fourier factorization theorem
  Reading: NR 12.0, 12.1, 12.4
• 04/06 Lab 7: FLOP performance of basic operations; FLOP rates in octave.

• 04/11 Lec 20: Fourier inversion theorem; Eigenvalue problems in $d>1$ dimension: variational principle, analytic continuation for constraints, minimization by steepest descents
  Reading: NR 10.0
• 04/13 Lec 21: DFT++ expressions for energy and gradient for multiple states and for density functional theory
  Reading: Sections 1-4 of http://arxiv.org/abs/cond-mat/9909130
• 04/13 Lab 8: Spectral solution of Poisson's equation

• 04/18 Lec 22: Converge of steepest descents; preconditioning; line minimization; motivation for conjugate gradients
  Reading: NR 10.2, 10.3. 10.5
• 04/20 Lec 23: Conjugate gradient algorithm
  Reading: 10.6
• 04/20 Lab 9: Spectral solution of Schrödinger's equation and Kohn-Sham equations for simple harmonic oscillator.

• 04/25 Lec 24: FFT
  Reading: 12.2
• 04/27 Lec 25: Further optimizations for spectral methods; ab initio pseudopotentials
• 04/27 Lab 10: Performance of numerical minimization techniques; preconditioner for quantum calculations, H$_2$ molecule

• 05/02 Lec 26: Extraction of eigenstates from variational principle
• 05/04 Lab 11: Physics of harmonic oscillator, H2 molecule and germanium solid

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