#
2013 SIAM Gene Golub SIAM Summer School

10th Shanghai Summer School on Analysis and Numerics in Modern Sciences

##
Programming Assignment

###
Test Matrices

- 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)*.
- 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)*.
- 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

- 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

- 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

- Steepest Descent Methods for the
**linear response eigenvalue problem**:
a sample driver and
the function.

- Conjugate Gradient Methods for the
**linear response eigenvalue problem**:
a sample driver and
the function.
Also needed is KM.m.

- Conjugate Gradient Methods for the
**hyperbolic quadratic eigenvalue problem**:
a sample driver and
the function.
Also needed is HQEP.m.

- 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.