cpviolator / voatol

Variations On A Theme Of Lanczos

VOATOL

(Variations On A Theme Of Lanczos, "vote-ol") 11 Oct 2020

Synopsis

VOATOL is a C++ playground for Lanczos and Arnoldi restarted eigensolvers. Some OMP parallelism is employed, but the purpose of this repository is to give the user an accessible entry point into how restarted eigensolvers work. Ideally, the user would learn from this repository and implement one of the solvers in their own framework.

Build

VOATOL uses CMake. The standard CMake build instructions apply:

1. mkdir VOATOL_build
2. cd VOATOL_build
3. cmake ../VOATOL -DENABLE_OPENMP
4. make -j N

Solvers

VOATOL offers three solvers, located in the 'test' directory. Each will construct a random matrix and solve it using the given parameters. These are the parameters common to all solvers:

• mat_size (int) The rank of the square matrix to solve
• nKr (int) The size of the Krylov space
• nEv (int) The size of the compressed space
• nConv (int) The number of converged eigenvalues to compute
• max_restarts (int) The maximum number of restarts
• diag (double) Add this diagonal constant to the problem matrix
• tol (double) Residual tolerance of the eigenvalues
• spectrum (int) 0=LM, 1=SM, 2=LR, 3=SR, 4=LI, 5=SI with S=Smallest L=Largest M=Modulus R=Real I=Imaginay
• verbosity (int) Give verbose output 1=verbose, 0=quiet
• Eigen Check (int) Cross check VOATOL's results against Eigen 0=false, 1=true

Thick Restarted Lanczos Method (TRLM)

TRLM is used in the following fashion:

./trlm mat_size nKr nEv nConv max-restarts diag tol amin amax polydeg spectrum LU batch verbosity Eigen_Check

TRLM is a symmetric eigensolver, so we allow the use of Chebyshev polynomial acceleration:

• amin (double) The minimum of the polynomial
• amax (double) The maximum of the polynomial
• polydeg (int) The polynomial degree

One can learn more about chebyshev acceleration here: https://github.com/lattice/quda/wiki/QUDA's-eigensolvers#polynomial-acceleration

We optionally use LU batched rotation of the Krylov space in TRLM. One can rotate N vectors using only M extra vectors in memory. This is useful in HPC applications where memory is limited.

• LU (int) 0=false, 1=true
• batch (int) The batch size of the rotation

Block Thick Restarted Lanczos Method (BLKTRLM)

This solver is the Block version of TRLM. It works by ensuring a block of Krylov vectors remain orthonormal during the algorithm. It is useful in HPC applications where one can save time by moving the matrix data to the registers once, while also moving multiple vectors. There is no batched rotation option for BLKTRLM.

BLKTRLM is used in the following fashion:

./block_trlm mat_size nKr nEv nConv max-restarts diag tol amin polydeg spectrum block verbosity Eigen_Check

The extra options are:

• block (int) The size of the Kylov block to employ

Implicilty Restarted Arnoldi Method (IRAM)

Arnoldi can solve hermitian and non-hermitian matrices. As such, we do not allow polynomial acceleration to be employed, but one may solve for an hermitian or non-hermitian matrix. The general usage is:

./iram mat_size nKr nEv nConv max-restarts diag tol spectrum verbosity Eigen Check

The extra options are:

• mat_type (int) Use a symmetric or antisymmetric problem matrix 0=asym, 1=sym

Happy Hacking!