Cornell University

Department of Physics

Phys 480/680, Astro 690 April 15, 2005

Computational Physics, Spring 2005

Homework Assignment # 8

(Due Thursday, March 21)

Agenda and readings for the weeks of April 18-22:

Goals: Implement numerical minimization using preconditioning, steepest descents, conjugate gradient techniques; solve Kohn-Sham equations of density-functional theory.

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 two weeks of class

• 04/05 Lec 18: Discrete Fourier Transforms: aliasing/Nyquist frequency, Fourier factorization theorem, Fourier inversion theorem
  Reading: NR 12.0, 12.1, 12.4

• 04/07 Lec 19: Eigenvalue problems in $d>1$ dimension: variational principle, analytic continuation for constraints, minimization by steepest descents
  Reading: NR 10.0

• 04/07 Lab 6: FLOP performance of basic operations; FLOP rates in octave; Automated empirical optimization of software (AEOS): automatically tuned linear algebra software (ATLAS), fastest Fourier Transform in the West (FFTW).

• 04/12 Lec 20: 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

• 04/14 Lec 21: Converge of steepest descents; preconditioning; line minimization; motivation for conjugate gradients
  Reading: NR 10.2, 10.3. 10.5

• 04/14 Lab 7: Spectral solution of Poisson's and Schrödinger's equations

Preview of next week of class:

• 04/19 Lec 22: Conjugate gradient algorithm; extraction of eigenstats from variational principle
  Reading: 10.6
• 04/21 Lec 23: FFT
  Reading: 12.2
• 04/21 Lab 8: Converge of steepest descents; preconditioning; line minimization; quantum states of harmonic oscillator, quantum dots

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