Interface for Eigenvalue Interface for Eigenvalue MethodsMethods

John Wu Lawrence Berkeley National Laboratory

john.wu@nersc.govkwu@lbl.gov

OutlineOutline

• Review eigenvalue methodsReview eigenvalue methods

• Examine existing schemesExamine existing schemes

• Explore possible alternatives Explore possible alternatives

Types of Eigenvalue Types of Eigenvalue ProblemsProblems• Standard: Ax = Standard: Ax = xx

• Generalized: Ax = Generalized: Ax = BxBx

• Polynomial: -Ax+Polynomial: -Ax+BB11x+x+22BB22x+…=0x+…=0

• Constrains: Cx=0Constrains: Cx=0

Types of MethodsTypes of Methods

• QR based methods (direct methods)QR based methods (direct methods)– Need to access the elements of the Need to access the elements of the

matrixmatrix

– O(nO(n22) space complexity) space complexity

– O(nO(n33) time complexity) time complexity

• Projection based methods (iterative)Projection based methods (iterative)– Access matrix through MATVECAccess matrix through MATVEC

– Problem dependent space and time needProblem dependent space and time need

Existing SoftwareExisting Software

QR methodsQR methods

• LAPACKLAPACK

• EISPACKEISPACK

• ScaLAPACKScaLAPACK

• PLAPACKPLAPACK

Projection methodsProjection methods

• ARPACKARPACK

• LANSO, LASO, LANSO, LASO, LANZ, LANCZOS …LANZ, LANCZOS …

• Davidson method, Davidson method, JDQZ, …JDQZ, …

• RITZITRITZIT

• CG, CG,

Specifying an Eigenvalue Specifying an Eigenvalue ProblemProblem• Matrices: A, B, AMatrices: A, B, A-1-1, (A-, (A-B)B)-1-1B, BB, B-1-1AA

• Type of eigenvalues: largest, Type of eigenvalues: largest, smallest, around a center, in a smallest, around a center, in a rangerange

• Number of eigenvaluesNumber of eigenvalues

• Are eigenvectors needed as wellAre eigenvectors needed as well

Specifying an Eigenvalue Specifying an Eigenvalue Problem -- Computing Problem -- Computing IssuesIssues• Representations of matrices, Representations of matrices,

eigenvalues and eigenvectorseigenvalues and eigenvectors

• Interface to MATVEC and Interface to MATVEC and preconditionerpreconditioner

• Accuracy (reliability) requirementAccuracy (reliability) requirement

• Resource constrainsResource constrains

• Controlling parametersControlling parameters

netlib.org/lapackCase Study I -- LAPACKCase Study I -- LAPACK

SSYEVX(JOBZ, RANGE, UPLO, N, A, LDA, SSYEVX(JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO)WORK, LWORK, IWORK, IFAIL, INFO)

• QR based methodQR based method

• Representing matrices as arrays (1-D)Representing matrices as arrays (1-D)

• User provide workspaceUser provide workspace

• User must set all optionsUser must set all options

ftp://ftp.caam.rice.edu/pub/software/ARPACK

Case Study II -- ARPACKCase Study II -- ARPACK

DNAUPD(COMM, IDO, BMAT, N, WHICH, DNAUPD(COMM, IDO, BMAT, N, WHICH, NEV, TOL, RESID, NCV, V, LDV, IPARAM, NEV, TOL, RESID, NCV, V, LDV, IPARAM, IPNTR, WORKD, WORKL, LWORKL, INFO)IPNTR, WORKD, WORKL, LWORKL, INFO)

• Implicitly restarted Arnoldi methodImplicitly restarted Arnoldi method

• Reverse communication for MATVECReverse communication for MATVEC

• Arguments are arrays (1-D)Arguments are arrays (1-D)

• User provides workspaceUser provides workspace

lbl.gov/~kwu

Case Study III -- TRLANCase Study III -- TRLAN

TRLAN(OP, INFO, NROW, MEV, EVAL, TRLAN(OP, INFO, NROW, MEV, EVAL, EVEC, LDE)EVEC, LDE)

• thick-restart Lanczos methodthick-restart Lanczos method

• MATVEC (OP) has a prescribed interfaceMATVEC (OP) has a prescribed interface

• eigenvalues/eigenvectors are arrayseigenvalues/eigenvectors are arrays

• use Fortran 90 TYPE (INFO) to store use Fortran 90 TYPE (INFO) to store controlling parameterscontrolling parameters

Case Study IV -- ARPACK+Case Study IV -- ARPACK+++

#include “areig.h”#include “areig.h”

Nconv = areig(EigValR, EigValI, Nconv = areig(EigValR, EigValI, EigVec, n, nnz, A, inrow, pcol, nev);EigVec, n, nnz, A, inrow, pcol, nev);

• Based on ARPACKBased on ARPACK

• Simple data structures for matrix, Simple data structures for matrix, eigenvalues and eigenvectorseigenvalues and eigenvectors

ftp://ftp.caam.rice.edu/pub/software/ARPACK

Case Study V -- Case Study V -- RogueWaveRogueWave

#include <rw/deig.h>#include <rw/deig.h>

DoubleEigDecomp eig(a);DoubleEigDecomp eig(a);

• QR based methodsQR based methods

• Solution stored in eig objectSolution stored in eig object

• Eig.eigenValues() for eigenvaluesEig.eigenValues() for eigenvalues

www.roguewave.com

Two Basic StrategiesTwo Basic Strategies

Eigenvalue functionEigenvalue function

• Closer to Closer to traditional librariestraditional libraries

• Explicit arguments Explicit arguments for eigenvalue and for eigenvalue and eigenvectoreigenvector

• May be an May be an interface of a interface of a matrix classmatrix class

Decomposition classDecomposition class

• Computation is Computation is hiddenhidden

• Contains functions Contains functions to output to output eigenvalues, etceigenvalues, etc

• Decomposition is Decomposition is an objectan object

Basic Data ElementsBasic Data Elements

Eigenvalue functionEigenvalue function

• MatricesMatrices

• Eigenvalue and Eigenvalue and eigenvector are eigenvector are function arguments function arguments

• Controlling Controlling parameters are parameters are function argumentsfunction arguments

Decomposition classDecomposition class

• MatricesMatrices

• Eigenvalue and Eigenvalue and eigenvector stored eigenvector stored internallyinternally

• Controlling Controlling parameters set by parameters set by member functionmember function

Interaction With Other Interaction With Other ComponentsComponentsEigenvalue functionEigenvalue function

• Reverse Reverse communication or communication or function pointer or function pointer or direct access to direct access to the matricesthe matrices

Decomposition classDecomposition class

• Function pointer Function pointer (functor) or direct (functor) or direct access to matricesaccess to matrices

• External functions External functions must use must use prescribed prescribed interfaceinterface

Feature ComparisonFeature Comparison

Function Classusability Familiar feel,

explicitargument

Pure objects,easy for starters

performance Do not impedeperformance

May need extracopy

externdability Need newinterface

Easy to add newthings

Eigenvalue Class always has a decomposefunction, but it may not be public

Relation With Equation Relation With Equation SolversSolvers• Share the same basic elements: Share the same basic elements:

vector, multi-vector, matricesvector, multi-vector, matrices

• Share the same external/user-Share the same external/user-supplied functions: MATVEC, supplied functions: MATVEC, preconditionerpreconditioner

• Eigenvalue solver may use equation Eigenvalue solver may use equation solversolver

Relation With Existing Relation With Existing SoftwareSoftware• Existing matrix libraries may be Existing matrix libraries may be

used to specify basic data elementsused to specify basic data elements

• Existing Fortran/C libraries may be Existing Fortran/C libraries may be used to perform the actual used to perform the actual computationscomputations

• Should make it easy to interact with Should make it easy to interact with existing frameworks, e.G., POOMAexisting frameworks, e.G., POOMA

Future WorkFuture Work

• Reviewing other OO eigenvalue Reviewing other OO eigenvalue softwaresoftware

• Concrete interface specificationConcrete interface specification

• Data structure of the basic elementsData structure of the basic elements

• Interface of MATVEC and Interface of MATVEC and preconditionerpreconditioner