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

## Programming Assignment

### 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 25x2500) for fast experimenting. The unreduced one is of 14895x14895. 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.