ChebLanNr, ChebLanTr, ChebSI, RatLanNr and RatLanTr
Polynomial and Rational Filtered Lanczos and subspace iteration algorithms
For Symmetric Eigenvalue problems
Welcome to EVSL. EVSL is a C library for computing the eigenvalues of a symmetric matrix that are located in a given interval. EVSL implements a polynomial filtered Lanczos method (thick restart, no restart) a rational filtered Lanczos method (thick restart, no restart), and a polynomial filtered subspace iteration for solving standard eigenvalue problems A u = λ u and generalized eigenvalue problems A u = λ B u. EVSL also provides tools for spectrum slicing, i.e., the technique of subdividing a given interval into p smaller subintervals and computing the eigenvalues in each subinterval independently. This is achieved by estimating the spectral density of the matrix (or matrix pair for generalized problems) with either the Kernel Polynomial method (KPM) or a Lanczos based method. This release does not yet offer full parallel implementations (trivial openMP test programs are available among the test drivers). Versions 1.1.x have much added functionality relative to version 1.0. Most notably:
- EVSL now handles both standard and generalized problems
- Matrix-free mechanism [whereby user provides own matvec function]
- Interface for FORTRAN users
- Spectrum slicing by KPM as well as Lanczos. Also effective spectrum slicing for Ax=λ Bx
- Interfaces to various direct solvers that are needed by the code
- Recently added: Generalized eigenvalue problems without factorizations [using polynomials in B]
For questions/feedback send e-mail to Yousef Saad [saad@umn.edu]
An article describing EVSL is available from arXiv, which can be cited as follows [bibtex entry]
@article{LiEigenvaluesSlicingLibrary2018a,
archivePrefix = {arXiv},
eprinttype = {arxiv},
eprint = {1802.05215},
primaryClass = {math},
title = {The {{Eigenvalues Slicing Library}} ({{EVSL}}): {{Algorithms}}, {{Implementation}}, and {{Software}}},
url = {http://arxiv.org/abs/1802.05215},
shorttitle = {The {{Eigenvalues Slicing Library}} ({{EVSL}})},
date = {2018-02-14},
keywords = {Mathematics - Numerical Analysis},
author = {Li, Ruipeng and Xi, Yuanzhe and Erlandson, Lucas and Saad, Yousef},
}
A detailed documentation of the code can be found in the `Documentation' folder of the package in EVSL_1.1.x
Library: The user only needs to modify the file makefile.in [see makefile.in.example for samples of files makefile.in that are given for mac-os and for Linux].
cp makefile.in_Linux/MacOS.example makefile.in.
modify makefile.in [provide C compiler and BLAS/LAPACK path]
make clean; make
Test programs: In the directories under TESTS/, you will find a number of directories which contain makefiles to build sample drivers that test a few different situations.
CXSparse CXSparse is the default direct linear solver used in EVSL for the linear systems that arise in rational filtering methods and in generalized eigenvalue problems. CXSparse is included only to allow quick testing. However, be aware that it is significantly slower than other available direct solvers such as those in SuiteSparse for example (see below).
NOTE: The compiler used to compile CXSparse is specified in the makefile at EXTERNAL/CXSparse/Lib/Makefile.
NOTE: CXSparse, which is distributed with EVSL, is Copyrighted by Timothy Davis. As noted above much better performance can be achieved by other existing direct solvers. Refer to CXSparse package for its License. [http://faculty.cse.tamu.edu/davis/suitesparse.html]
SuiteSparse: As a replacement for CXSparse, bindings are provided for the package SuiteSparse. Once SuiteSparse is installed, simply switch the DIRECTSOL variable in the makefile.in, and add the path to EXTERNAL/makefile_suitesparse.in.
EVSL invokes SuiteSparse to solve linear systems with (A-SIGMA I) or (A-SIGMA B), and CHOLMOD for solving linear systems with B.
Other solvers can also be used by providing the same interface as done for SuiteSparse. Follow the examples implemented in EXTERNAL/evsl_suitesparse.c
NOTE: SuiteSparse is NOT distributed with EVSL, and it is Copyrighted by Timothy Davis.
Refer to the SuiteSparse package for its license. [http://faculty.cse.tamu.edu/davis/suitesparse.html]
Pardiso Pardiso bindings are provided. The Pardiso web-site can be accessed at [https://pardiso-project.org/]
UMFPACK and CHOLMOD requires AMD, COLAMD, CCOLAMD and CAMD, and optionally METIS 5.1.0. Compile each of these packages to have the library file in the Lib directory. If SuiteSparse is configured with METIS, give the path to METIS (v 5.1.0) as well to make libmetis.a, in metis-5.1.0/ type
make config; make
Please refer to SuiteSparse and METIS for installation details.
Bindings with Pardiso as a direct solver are also provided -- see the directory EXTERNAL for details.
Rational filtering requires solving linear systems (where the coefficient matrix is the original matrix shifted by a complex shift). A linear solver routine must be provided.
After having computed the rational filter by
find_ratf(intv, &rat),
users can call
SetASigmaBSol(&rat, func, allf, data)
to set the solver functions and associated data for all the poles of
the rational filter.
func is an array of function pointers of length num of poles, i.e., rat->num.
So, func[i] is the function to solve the systems with pole i, the
coefficient matrix of which is A - s_i I(or, B), where s_i = rat->zk[i]
is the complex shift. data is an array of (void*) of the same
length, where data[i] is the data needed by func[i].
Each "func[i]" must be of the following prototype
void SolFuncC(int n, double *br, double *bz, double *xr, double *xz, void *data);
where n is the size of the system, br, bz are the right-hand
side (real and imaginary parts of complex vector), xr, xz will
hold the solution (complex vector) on return, and data contains all the
data needed for the solver.
If all func[i] are the same, one can set func==NULL and set allf to the function
Once SetASigmaBSol is executed, rational filtering Lanczos methods
should be ready to use.
All the eigensolvers in EVSL can be used in `matrix-free' mode. In this mode, users need only to provide their own matrix-vector product function of the following prototype:
void MVFunc(double *x, double *y, void *data);
where y = A * x and data is the pointer to the associated data to
perform the matvec. The (void *) argument is to provide a uniform
interface to all user-specific functions. For a particular Matvec
function, one can pack all data needed by this function into a struct
and pass the pointer of this struct to EVSL (after casting it to (void *)). This function needs to be passed to EVSL as well, so EVSL can
call it to perform all matvecs. The user can also pass needed
matrices in the standard CSR format to EVSL, in which case EVSL will use
its internal MATVEC routine. This can be set by
SetAMatrix(csrMat *A)
SetBMatrix(csrMat *B)
for the matrices A and B
In TESTS/LAP, an example of matvec functions for 2D/3D Laplacian matrices is provided, where the matrix is not explicitly formed but 5pt/7pt stencil is used instead to perform the matvecs. In this example, a struct for matvec is first defined:
typedef struct _lapmv_t {
int nx, ny, nz;
double *stencil;
} lapmv_t;
and the matvec function is implemented as:
void Lap2D3DMatvec(double *x, double *y, void *data) {
lapmv_t *lapmv = (lapmv_t *) data;
int nx = lapmv->nx;
int ny = lapmv->ny;
int nz = lapmv->nz;
double *stencil = lapmv->stencil;
...
}
in which the pointer is first cast and all the data is unpacked. Once these are ready, they can be passed to EVSL by calling
SetAMatvec(n, &Lap2D3DMatvec, (void*) &lapmv) and
SetBMatvec(n, &Lap2D3DMatvec, (void*) &lapmv)
to set the matvec routines for A and B respectively,
where the first input is the size of the "matrix", the second input is
the function pointer and the third one is the data pointer. Once
SetMatvecFunc is called, EVSL will use the registered matvec
function to perform all matvecs with A.
Users should first create a function wrapper of the above type for an external matvec routine. Then, following the steps in the example, it will be straightforward to use it in EVSL.
For solving the generalized eigenproblem A x = λ B x, the users must also provide a solver for the B matrix by calling
SetBSol(SolFuncR func, void *data).
To indicate to EVSL that a generalized eigenvalue problem is being solve, one must call
SetGenEig()
since by default, EVSL assumes that a standard eigenvalue problem is being solved even when a B matrix is provided. Call the function
SetStdEig()
for solving standard eigenvalue problem
To implement spectrum slicing, the current version of EVSL requires system solutions with the matrix L', where B=L*L' is the Cholesky factorization of B. Call the function
SetLTSol(SolFuncR func, void *data)
to set the solver function for L'.
Version 1.1.1 includes drivers for solving certain types of generalized eigenvalue problems by polynomial filtering by methods that avoid matrix factorizations.
- Use
EVSLStart()andEVSLFinish()before and after any call to the EVSL functions
-
INC
- evsl.h : user-level function prototypes and constant definitions
- blaslapack.h : C API for BLAS/LAPACK functions used in evsl
- def.h : miscellaneous macros
- struct.h : miscellaneous structs used in evsl
- internal_proto.h : internal function prototypes for SRC/
-
SRC
- cheblanNr.c : Polynomial Filtered no-restart Lanczos
- cheblanTr.c : Polynomial Filtered thick restart Lanczos
- chebpoly.c : Computing and applying polynomial filters
- chebsi.c : Polynomial Filtered Subspace iteration
- dumps.c : Miscellaneous functions for I/O and for debugging
- dos_utils.c : Miscellaneous functions used for DOS based functions.
- evsl.c : Set EVSL solver options and data
- lanbounds.c : Lanczos alg. to compute bounds of spectrum
- landos.c : Lanczos based DOS algorithm for the standard problem
- landosG.c : Lanczos based DOS algorithm for generalized and standard problems
- lanTrbounds.c : A more robust alg. to compute bounds of spectrum based on TR Lanczos
- mactime.c : Timer for mac iOS
- misc_la.c : Miscellaneous linear algebra functions
- ratfilter.c : Computing and applying rational filters
- ratlanNr.c : Rational Filtered no-restart Lanczos
- ratlanTr.c : Rational Filtered thick restart Lanczos
- spmat.c : Sparse matrix routines
- spslice.c : Spectrum slicing functions for Kernel Polynomial Method
- spslice2.c : Spectrum slicing functions for Lanczos
- timing.c : Timer
- vect.c : Vector operations
-
libevsl.a : library
-
TESTS/ : Test drivers
-
Fortran/ : Fortran test drivers
-
PLanR : Polynomial Filtered Thick Restart Lanczos test drivers
- LapPLanR.c : Polynomial filtering T-R Lanczos (Lapliacian)
- MMPLanR.c : Polynomial filtering T-R Lanczos (Matrix Market)
- MMPLanR_omp.c : Polynomial filtering T-R Lanczos (parallelized with OMP for slices) (Matrix Market)
-
RLanR : Rational Filtered Thick Restart Lanczos test drivers
- LapRLanR.c : Rational filtering T-R Lanczos (Laplacian)
- MMRLanR.c : Rational filtering T-R Lanczos (Matrix Market)
-
PLanN : Polynomial Filtered No-Restart Lanczos test drivers
- MMPLanN.c : Polynomial filtering non-restart Lanczos (Matrix Market)
- LapPLanN.c : Polynomial filtering non-restart Lanczos (Laplacian)
- LapPLanN_MatFree.c : "matrix-free" version: not forming matrix but passing mat-vec function (Laplacian)
-
RLanN : Rational Filtered No-Restart Lanczos test drivers
- LapRLanN.c : Rational filtering non-restart Lanczos (Laplacian)
- MMRLanN.c : Rational filtering non-restart Lanczos (Matrix Market)
-
GEN : Test drivers for the generalized eigenvalue problem
- MMsimple.c : A stripped down (no slicing) drivier based on MMPLanN.c
- MMPLanN.c : Polynomial filtering non-restart Lanczos (Matrix Market)
- MMPLanR.c : Polynomial filtering T-R Lanczos (Matrix Market)
- MMRLanN.c : Rational filtering non-restart Lanczos (Matrix Market)
- MRLanR.c : Rational filtering T-R Lanczos (Matrix Market)
-
PSI : Test drivers for polynomial filter subspace iteration
- LapPSI.c : Polynomial filtering subspace iterations (Laplacian)
- MMPSI.c : Polynomial filtering subspace iterations (Matrix Market)
-
COMMON : Routines common to the test drivers
- io.c : parse command-line input parameters
- lapl.c : Build Laplacian matrices and compute the exact eigenvalues of Laplacians
- mmio.c : IO routines for the matrix market format
-
COMMON_GEN : Routines common to the test drivers for the generalized eigenvalue problem
- io.c : parse command-line input parameters
- lapl.c : Build Laplacian matrices and compute the exact eigenvalues of Laplacians
- mmio.c : IO routines for the matrix market format
-
TESTS/Landos : test drivers for the lanDOS related functions.
- LanDos.c : DOS for standard eigenvalue problem using Lanczos
- LanDosG.c : DOS for generalized eigenvalue problem using Lanczos
-
-
EXTERNAL : direct solver interface for generalized eigenvalue problems
- evsl_suitesparse.c : suitesparse UMFPACK and CHOLMOD interface
- evsl_cxsparse.c : cxsparse interface
- evsl_pardiso.c : pardiso interface
- evsl_direct.h : Direct solver interface
-
FORTRAN : Fortran interface
- evsl_f90.c : Fortran interface