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Programming Assignment

2013 SIAM Gene Golub SIAM Summer School
10th Shanghai Summer School on Analysis and Numerics in Modern Sciences

Test Matrices
  1. A matrix pencil A-\lambda B arises from solving Korn-Sham equation for C6H6 by SCF. It is a reduced one (of 25×2500) for fast experimenting. The unreduced one is of 14895×14895. Courtesy of Prof. Yunkai Zhou (SMU).
  2. A matrix pencil A-\lambda B arises from solving the Laplacian eigenvalue problem in a barbell shaped domain. Courtesy of Prof. Qiang Ye (U. Kentucky) and Dr. Patrick Quillen (Mathworks).
  3. A linear response eigenvalue problem arises from the linear response analysis for Na2 using plane-waves as a basis set and pseudopotentials. It is generated by the Quantum ESPRESSO. While here both K and M are dense, they are never meant to be formed explicitly, but exist implicitly through matrix-vector product. Nonetheless, they are formed for easy experimenting in MATLAB. Courtesy of Prof. D. Rocca (Universite de Lorraine – CNRS)
MATLAB functions
  1. Steepest Descent Methods (a sample driver):
    • Steepest Descent Method: SDgS.m
    • Block Steepest Descent Method: BSDgS.m
    • Preconditioned Steepest Descent Method: PSDgS.m
    • Block Preconditioned Steepest Descent Method: BPSDgS.m
    • Expert Block Preconditioned Steepest Descent Method: xBPSDgS.m
  2. Conjugate Gradient Methods (a sample driver):
    • Conjugate Gradient Method: CGgS.m
    • Preconditioned Conjugate Gradient Method: PCGgS.m
    • Locally Optimal Conjugate Gradient Method: LOCGgS.m
    • Locally Optimal Block Conjugate Gradient Method: LOBCGgS.m
    • Locally Optimal Preconditioned Conjugate Gradient Method: LOPCGgS.m
    • Locally Optimal Block Preconditioned Conjugate Gradient Method: LOBPCGgS.m
    • Expert Locally Optimal Block Conjugate Gradient Method: xLOBCGgS.m
    • Expert Locally Optimal Block Preconditioned Conjugate Gradient Method: xLOBPCGgS.m
  3. Steepest Descent Methods for the linear response eigenvalue problem: a sample driver and the function.
  4. Conjugate Gradient Methods for the linear response eigenvalue problem: a sample driver and the function. Also needed is KM.m.
  5. Conjugate Gradient Methods for the hyperbolic quadratic eigenvalue problem: a sample driver and the function. Also needed is HQEP.m.
  6. Auxiliary MATLAB functions:
    • Line Search: LNSRCHg.m
    • Modified Gram-Schmidt in B-inner product: MGSg.m
    • Computing the first few smallest eigenvalues of a Hermitian matrix: mineig.m
    • (Linear) Conjugate Gradient method for Ax=b: LinCG.m
Assignment:


Program various extended methods (i.e., enlarge search subspaces to a higher order Krylov subspaces) by modifying the relevant MATLAB functions above.

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