SSMFE_CORE - Sparse Symmetric Matrix-Free Eigensolver (Core Algorithm)¶

Purpose¶

This package computes extreme (leftmost and/or rightmost) eigenpairs \(\{\lambda_i, x_i\}\) of the following eigenvalue problems:

• the standard eigenvalue problem

  \[A x = \lambda x,\]

• the generalized eigenvalue problem

  \[A x = \lambda B x,\]

• the eigenvalue problem

  \[AB x = \lambda x\]

where \(A\) and \(B\) are real symmetric (or Hermitian) matrices and \(B\) is positive definite.

The packages SPRAL_SSMFE and SPRAL_SSMFE_EXPERT provide a simplified interface to this routine, and should be used if the user does not require access to low level features provided in this package.

Usage overview¶

This package implements a block iterative algorithm for simultaneous computation of several eigenpairs for the problems above. The block nature of this algorithm allows the user to benefit from highly optimized linear algebra subroutines and from the ubiquitous multicore architecture of modern computers. It also makes this algorithm more reliable than Krylov-based algorithms employed by e.g. ARPACK in the presence of clustered eigenvalues. However, convergence of the iterations may be slow if the density of the spectrum is high.

Thus, good performance (in terms of speed) is contingent on the following two factors:

1. the number of desired eigenpairs must be substantial (e.g. not fewer than the number of CPU cores), and
2. the employment of a convergence acceleration technique.

The acceleration techniques that can be used are shift-and-invert and preconditioning.

The former rewrites the eigenvalue problem for a matrix \(M\) as \(Ax = \lambda x\) with \(A = (M - \sigma I)^{-1}\), where \(I\) is the identity matrix and \(\sigma\) is a real value near eigenvalues of interest, and the generalized problem \(M x = \mu B x\) as the \(Ax = \lambda B x\) with \(A = (M - \sigma B)^{-1}\).

The latter applies to the case of positive definite \(A\) and requires a matrix or an operator \(T\), called a preconditioner, such that the vector \(v = T f\) is an approximation to the solution \(u\) of the system \(A u = f\) (see the simple example below). Note: This technique is only recommended for experienced users.

For futher detail on the algorithm, see the outline in the method section below, and associated references.

The package SSMFE provides a user-friendly interface to this algorithm, whilst SSMFE_EXPERT provides an interface that allows users to manage their own memory. If this routine is used instead of SSMFE_EXPERT, the user is additionally responsible for deciding when a sufficient number of eigenpairs have been computed to sufficient accuracy. The amount of computation performed by the solver subroutines in this package and the memory they use are negligible. These features facilitate the use of these subroutines for shared-memory, out-of-core and hybrid computation.

Subroutines¶

To use the solver procedures, the user must maintain a workspace of (kw+1) blocks each containing m vectors of size n. For notational convienience we refer to this workspace as a Fortran array W[kw+1][m][n], but the user is free to store it as they wish. Note the block dimension is indexed from zero, not from one. The following table provides minimum values of kw for each setup:

	minAprod=T		minAprod=F
Problem	minBprod=T	minBprod=F	minBprod=T	minBprod=F
standard	7	5	3	3
standard_shift	7	5	N/A	N/A
generalized	7	5	5	3
generalized_shift	7	7	N/A	N/A
buckling	7	7	N/A	N/A

Further, the user must also store the converged eigenvectors \(X\), and (for generalised problems) their \(B\)-images \(BX\) using separate storage, e.g. X[mep][n], BX[mep][n]. In addition to being output, the routine may need to reorthagonalise against these from time to time.

The first (zero-indexed) block holds the eigenvector approximations: the user must fill this block with \(m\) linearly independent vectors before the first call to a solver procedure.

The number of desired eigenpairs may exceed \(m\): whenever converged eigenpairs have been detected, a solver procedure reports the indices of these eigenpairs and they must be moved by the user to separate storage (X(:)).

void spral_ssmfe_core_default_options(struct spral_ssmfe_core_options *options)¶

Intialises members of options structure to default values.

Parameters:	options – Structure to be initialised.

void spral_ssmfe_double(struct spral_ssmfe_rcid *rci, int problem, int left, int right, int m, double *lambda, double *rr, int *ind, void **keep, const struct spral_ssmfe_core_options *options, struct spral_ssmfe_inform *inform)¶

Computes specified number of leftmost and rightmost eigenvalues and corresponding eigenvectors.

Uses reverse-communication. Upon return the user must perform a task specified by the rci parameter and recall the routine. Possible values of rci and associated tasks are:

rci.job	Task to be performed
-3	None. Fatal error, see inform.flag.
-2	Failed to converge, see inform.flag.
-1	None. Computation complete.
1	Calculate \(\bar{V} = AU\).
2	Apply preconditioner \(\bar{V} = TU\). (Copy if T=I).
3	Compute \(\bar{V} = BU\)
4	Test convergence for each of \(m\) eigenvalues: If eigenpair i has converged, set inform.converged[i] to a positive number. Otherwise, leave at current value. Tests may use the estimated eigenvalue errors inform.err_lambda[i] and eigenvector errors inform.err_x[i].
5	Copy converged eigenvectors \(X\) to user storage: If rci.i>0: W[rci.kx][rci.jx:rci.jx+rci.nx-1][:]. Else: W[rci.kx][rci.jx-rci.nx+1:rci.jx][:]. Optionally save their \(B\)-images: If rci.i>0: W[rci.ky][rci.jx:rci.jx+rci.nx-1][:]. Else: W[rci.ky][rci.jx-rci.nx+1:rci.jx][:].
11	If rci.i == 0, copy \(\bar{V} = U\). Otherwise, reorder columns of block rci.kx such that column ind[j] becomes the new column j for j=0, ..., rci.nx-1 Note: if rci.kx == rci.ky, only reorder once.
12	Compute the dot products \[r_{ii} = U_i \cdot \bar{V}_i\]
13	Perform the scalings \[U_i = U_i/\sqrt{(U_i\cdot \bar{V}_i)}\] and \[\bar{V}_i = \bar{V}_i/\sqrt{(U_i\cdot \bar{V}_i)}\] for each column \(U_i\) and \(\bar{V}_i\) of \(U\) and \(\bar{V}\). Note: if rci.kx == rci.ky, only scale once.
14	Perform the updates \[\bar{V}_i = \bar{V}_i + r_{ii} U_i\] for each column \(\bar{V}_i\) of \(\bar{V}\)
15	Perform the update \[R = \alpha U^T V + \beta R\]
16	Perform the update \[V = \alpha U R + \beta V\]
17	Perform the update \[U = \alpha U R\] Note: \(V\) may be used as a workspace
21	\(B\)-orthogonalize columns of \(V\) to all vectors \(X\) by solving \[(X^TBX) Q = X^T \bar{V}\] for \(Q\) and updating \[\begin{split}U & = & U - XQ \\\end{split}\] If \(B\ne I\), the :math:bar{V} must also be updated as \[\begin{split}\bar{V} & = & \bar{V} - BXQ,\end{split}\] or \[\bar{V} = BU\]
22	\(B\)-orthogonalize columns of \(U\) to all vectors \(X\) by solving \[(X^TBX) Q = X^T U\] for \(Q\) and updating \[U = U - BXQ\]
999	Restart: If rci.k>0: Restart suggested with block size m >= rci.nx + rci.i + rci.j, adjusting workspace size to match. Set rci.i=0 and rci.j=0 and recall the routine. If a restart is not desirable, routine may be recalled with no change to parameters. If rci.k=0: Restart required with the same block size. In both cases, the first block W[0][:][:] should retain vectors rci.jx:rci.jx+rci.nx-1, filling remaining vectors randomly such that the entire set of columns is linearly independent from each other and also from the converged eigenvectors.

The matrices are defined as follows:

• \(U\) = W[rci.kx][rci.jx:rci.jx+rci.nx-1][:]
• \(V\) = W[rci.ky][rci.jy:rci.jy+rci.ny-1][:]
• \(\bar{V}\) = W[rci.ky][rci.jy:rci.jy+rci.nx-1][:]
• \(R\) = rr[rci.k][rci.j:rci.j+rci.ny-1][rci.i:rci.i+rci.nx-1]

and \(\alpha\) and \(\beta\) are given by rci.alpha and rci.beta respectively. We use the notation \(r_{ii}\) to refer to the \(i\)-th diagonal element of \(R\), being rr[rci.k][rci.j+i-1][rci.i+i-1].

Parameters:

rci – Reverse communication type. rci.job must be set to 0 before the first call. problem – Problem to be solved, one of: 0 \(Ax=\lambda x\) >0 \(Ax=\lambda Bx\) <0 \(ABx=\lambda x\) left – Number of left eigenpairs to find. On return with rci.job=5, can be set to zero if sufficient have been found. right – Number of left eigenpairs to find. On return with rci.job=5, can be set to zero if sufficient have been found. lambda[m] – Current eigenvalue estimates in ascending order. m – Block size of workspace W. Must be at least 2. rr[3][2*m][2*m] – reverse communication workspace. ind[m] – reverse communication workspace. keep – Internal workspace used by routine. options – specifies algorithm options to be used. inform – returns information about the exection of the routine.

void spral_ssmfe_double_complex(struct spral_ssmfe_rciz *rci, int problem, int left, int right, int m, double *lambda, double complex *rr, int *ind, void **keep, const struct spral_ssmfe_core_options *options, struct spral_ssmfe_inform *inform)¶

As spral_ssmfe_double(), but types of rci and rr changed to support type double complex.

void spral_ssmfe_largest_double(struct spral_ssmfe_rcid *rci, int problem, int nep, int m, double *lambda, double *rr, int *ind, void **keep, const struct spral_ssmfe_core_options *options, struct spral_ssmfe_inform *inform)¶

Computes specified number of eigenvalues of largest magnitude and corresponding eigenvectors.

Uses reverse-communication. Upon return the user must perform a task specified by the rci parameter and recall the routine. Possible values of rci and associated tasks are as for spral_ssmfe_double() above.

Parameters:

rci – Reverse communication type. rci.job must be set to 0 before the first call. problem – Problem to be solved, one of: 0 \(Ax=\lambda x\) >0 \(Ax=\lambda Bx\) <0 \(ABx=\lambda x\) nep – Number of eigenpairs to find. lambda[m] – Current eigenvalue estimates in ascending order. m – Block size of workspace W. Must be at least 2. rr[3][2*m][2*m] – reverse communication workspace. ind[m] – reverse communication workspace. keep – Internal workspace used by routine. options – specifies algorithm options to be used. inform – returns information about the exection of the routine.

void spral_ssmfe_largest_double_complex(struct spral_ssmfe_rciz *rci, int problem, int nep, int m, double *lambda, spral_double_complex *rr, int *ind, void **keep, const struct spral_ssmfe_core_options *options, struct spral_ssmfe_inform *inform)¶

As spral_ssmfe_largest_double(), but types of rci and rr changed to support type double complex.

void spral_ssmfe_core_free(void **keep, struct spral_ssmfe_inform *inform)¶

Free memory allocated in keep and inform.

Parameters:	keep – Workspace to be freed. inform – Information type to be freed.

Warning

As memory in keep and inform has been allocated using Fortran functions, this routine must be called to avoid a memory leak.

Derived types¶

struct spral_ssmfe_core_options¶

Options that control the algorithm.

int err_est¶

Error estimation scheme, one of:

1	Residual error bounds: modified Davis-Kahan estimate for eigenvector error and Lehmann bounds for eigenvale error (see method section).
2 (default)	Convergence curve-based estimate.

Default is 2.

int extra_left¶

Number of extra approximate eigenvectors corresponding to leftmost eigenvalues used to enhance convergence. Default is 0.

int extra_right¶

Number of extra approximate eigenvectors corresponding to rightmost eigenvalues used to enhance convergence. Default is 0.

bool minAprod¶

If true, minimize number of multiplications with \(A\) by requiring 2 additional blocks of memory for the workspace W[:][:][:]. If false, three returns with rci.job=1 occur per iteration instead of one. Must be true if problem < 0. Default is true.

bool minBprod¶

If true, minimize number of multiplications with \(B\) by requiring 2 additional blocks of memory for the workspace W[:][:][:]. If false, at least three returns with rci.job=3 occur per iteration instead of one. Default is true.

double min_gap¶

Restart sensitivity: if the relative distance between the last eigenvalue of interest on either margin of the spectrum and the rest of the spectrum is smaller than min_gap, the solver procedure suggests restart (rci.job=999). The values rci.i and rci.j are set to the numbers of eigenvalues on the left and right margin of the spectrum that are too close to the eigenvalues of interest, causing slow convergence. The default value of 0.0 means no restart is ever suggested. Must be in the range \([0.0,1.0]\).

double cf_max¶

Stagnation sensitivity: if the value \(q_{ij}\) (see method section) is greater than cf_max for \(i > 5\), the eigenpair is marked as stagnated by setting inform.converged(j) to a negative value. The default value of 1.0 indicates that the estimated asymptotic convergence factor is not used for stagnation detection. Must be in the range \([0.5, 1.0]\).

struct spral_ssmfe_infrom¶

Information on progress of the algorithm.

int converged[mep]

Convergence status.

• If converged[j]>0, the eigenpair (lambda[j], X[j]) converged on iteration converged[j].
• If converged[j]=0, the eigenpair (lambda[j], X[j]) is still converging.
• If converged[j]<0, the eigenpair (lambda[j], X[j]) stagnated at iteration converged[j].

This component is allocated by the routine.

double err_lambda[mep]

Estimated eigenvalue errors for converged and stagnated eigenvalues. This component is allocated by the routine.

double err_x[mep]

Estimated eigenvector errors for converged and stagnated eigenvectors. This component is allocated by the routine.

int flag¶

Return status of algorithm. See table below.

int iteration¶

Number of iterations.

double residual_norms[mep]

Euclidean norms of residuals for (lambda[:], X[:]) on return with rci.job=4, 5. This component is allocated by the routine.

int stat¶

Fortran allocation status in event of failure

inform.flag
-1	m is out-of-range.
-2	rci.job is out-of-range.
-3	options.err_est is out-of-range.
-4	options.minAprod is incompatible with selected routine.
-5	options.extra_left or options.extra_right is out-of-range.
-6	options.min_gap is out-of-range.
-7	options.cf_max is out-of-range.
-11	left is out-of-range (spral_ssmfe_double()) or nep is out-of-range (spral_ssmfe_largest_double()).
-12	right is out-of-range.
-100	Not enough memory; inform.stat contains the value of the Fortran stat parameter.
-200	\(B\) is not positive definite or user_x>0 and linearly dependent initial guesses were supplied.
+1	The iterations have been terminated because no further improvement in accuracy is possible (this may happen if \(B\) or the preconditioner is not positive definite, or if the components of the residual vectors are so small that the round-off errors make them essentially random). The value of inform.non_converged is set to the number of non-converged eigenpairs.

Examples¶

/* examples/C/ssmfe/precond_core.f90 */
/* Laplacian on a square grid (using SPRAL_SSMFE_CORE routines) */
#include "spral.h"
#include <math.h>
#include <stdlib.h>
#include <stdio.h>
#include <cblas.h>

/* Header that implements Laplacian and preconditioners */
#include "laplace2d.h"

int main(void) {
   const int ngrid = 20;         /* grid points along each side */
   const int n = ngrid*ngrid;    /* problem size */
   const int nep = 5;            /* eigenpairs wanted */
   const int m = 3;              /* dimension of the iterated subspace */
   const double tol = 1.e-6;     /* eigenvector tolerance */

   int state = SPRAL_RANDOM_INITIAL_SEED; /* PRNG state */

   int ind[m];                   /* permutation index */
   double lambda[n];             /* eigenvalues */
   double X[n][n];               /* eigenvectors */
   /* Work arrays */
   double lmd[m];
   double rr[3][2*m][2*m];
   double W[7][m][n];
   double U[m][n];

   /* Derived types */
   struct spral_ssmfe_rcid rci;              /* reverse communication data */
   struct spral_ssmfe_core_options options;  /* options */
   void *keep;                               /* private data */
   struct spral_ssmfe_inform inform;         /* information */

   /* Initialize options to default values */
   spral_ssmfe_core_default_options(&options);

   /* Initialize W to lin indep vectors by randomizing */
   for(int i=0; i<n; i++)
   for(int j=0; j<m; j++)
      W[0][j][i] = spral_random_real(&state, true);

   int ncon = 0;        /* number of converged eigenpairs */
   rci.job = 0; keep = NULL;
   while(true) { /* reverse communication loop */
      spral_ssmfe_double(&rci, 0, nep, 0, m, lmd, &rr[0][0][0], ind,
         &keep, &options, &inform);
      switch ( rci.job ) {
      case 1:
         apply_laplacian(
            ngrid, ngrid, rci.nx, &W[rci.kx][rci.jx][0], &W[rci.ky][rci.jy][0]
            );
         break;
      case 2:
         apply_gauss_seidel_step (
            ngrid, ngrid, rci.nx, &W[rci.kx][rci.jx][0], &W[rci.ky][rci.jy][0]
            );
         break;
      case 4:
         for(int j=0; j<m; j++) {
            if ( inform.converged[j] != 0 ) continue;
            if ( inform.err_X[j] > 0 && inform.err_X[j] < tol )
               inform.converged[j] = 1;
         }
         break;
      case 5:
         if ( rci.i < 0 ) continue;
         for(int k=0; k<rci.nx; k++) {
           int j = ncon + k;
           lambda[j] = lmd[rci.jx + k];
           cblas_dcopy( n, &W[0][rci.jx+k][0], 1, &X[j][0], 1 );
         }
         ncon += rci.nx;
         if ( ncon >= nep || inform.iteration > 300 ) goto finished;
         break;
      case 11:
         if ( rci.i == 0 ) {
            if ( rci.kx != rci.ky || rci.jx > rci.jy ) {
               cblas_dcopy(n*rci.nx, &W[rci.kx][rci.jx][0], 1, &W[rci.ky][rci.jy][0], 1);
            } else if ( rci.jx < rci.jy ) {
               for(int j=rci.nx-1; j>=0; j--)
                  cblas_dcopy(n, &W[rci.kx][rci.jx+j][0], 1, &W[rci.ky][rci.jy+j][0], 1);
            }
         } else {
            for(int i=0; i<n; i++) {
               for(int j=0; j<rci.nx; j++)
                  U[j][i] = W[rci.kx][ind[j]][i];
               for(int j=0; j<rci.nx; j++)
                  W[rci.kx][j][i] = U[j][i];
               if(rci.ky != rci.kx) {
                  for(int j=0; j<rci.nx; j++)
                    U[j][i] = W[rci.ky][ind[j]][i];
                  for(int j=0; j<rci.nx; j++)
                    W[rci.ky][j][i] = U[j][i];
               }
            }
         }
         break;
      case 12:
         for(int i=0; i<rci.nx; i++)
           rr[rci.k][rci.j+i][rci.i+i] =
             cblas_ddot(n, &W[rci.kx][rci.jx+i][0], 1, &W[rci.ky][rci.jy+i][0], 1);
         break;
      case 13:
         for(int i=0; i<rci.nx; i++) {
            if( rci.kx == rci.ky ) {
               double s = cblas_dnrm2(n, &W[rci.kx][rci.jx+i][0], 1);
               if( s > 0 )
                  cblas_dscal(n, 1/s, &W[rci.kx][rci.jx+i][0], 1);
            } else {
               double s = sqrt(fabs(cblas_ddot(
                  n, &W[rci.kx][rci.jx+i][0], 1, &W[rci.ky][rci.jy+i][0], 1)
                  ));
               if ( s > 0 ) {
                  cblas_dscal(n, 1/s, &W[rci.kx][rci.jx+i][0], 1);
                  cblas_dscal(n, 1/s, &W[rci.ky][rci.jy+i][0], 1);
               } else {
                  for(int j=0; j<n; j++)
                     W[rci.ky][rci.jy+i][j] = 0.0;
               }
            }
         }
         break;
      case 14:
         for(int i=0; i<rci.nx; i++) {
           double s = -rr[rci.k][rci.j+i][rci.i+i];
           cblas_daxpy(n, s, &W[rci.kx][rci.jx+i][0], 1, &W[rci.ky][rci.jy+i][0], 1);
         }
         break;
      case 15:
         if ( rci.nx > 0 && rci.ny > 0 )
            cblas_dgemm(
               CblasColMajor, CblasTrans, CblasNoTrans, rci.nx, rci.ny, n,
               rci.alpha, &W[rci.kx][rci.jx][0], n, &W[rci.ky][rci.jy][0], n,
               rci.beta, &rr[rci.k][rci.j][rci.i], 2*m
               );
         break;
      case 16: // Fall through to 17
      case 17:
         if( rci.ny < 1 ) continue;
         if( rci.nx < 1 ) {
            if( rci.job == 17 ) continue;
            if( rci.beta == 1.0 ) continue;
            for(int j=rci.jy; j<rci.jy+rci.ny; j++)
               cblas_dscal(n, rci.beta, &W[rci.ky][j][0], 1);
            continue;
         }
         if( rci.job == 17 ) {
            cblas_dgemm(
               CblasColMajor, CblasNoTrans, CblasNoTrans, n, rci.ny, rci.nx,
               1.0, &W[rci.kx][rci.jx][0], n, &rr[rci.k][rci.j][rci.i], 2*m,
               0.0, &W[rci.ky][rci.jy][0], n
               );
            cblas_dcopy(n*rci.ny, &W[rci.ky][rci.jy][0], 1, &W[rci.kx][rci.jx][0], 1);
         } else {
            cblas_dgemm(
               CblasColMajor, CblasNoTrans, CblasNoTrans, n, rci.ny, rci.nx,
               rci.alpha, &W[rci.kx][rci.jx][0], n, &rr[rci.k][rci.j][rci.i],
               2*m, rci.beta, &W[rci.ky][rci.jy][0], n
               );
         }
         break;
      case 21: // Fall through to 22
      case 22:
         if( ncon > 0 ) {
            cblas_dgemm(
               CblasColMajor, CblasTrans, CblasNoTrans, ncon, rci.nx, n,
               1.0, &X[0][0], n, &W[rci.ky][rci.jy][0], n, 0.0, &U[0][0], n
               );
            cblas_dgemm(
               CblasColMajor, CblasNoTrans, CblasNoTrans, n, rci.nx, ncon,
               -1.0, &X[0][0], n, &U[0][0], n, 1.0, &W[rci.kx][rci.jx][0], n
               );
         }
         break;
      default:
         goto finished;
      }
   }
finished:
   if(inform.flag != 0) printf("inform.flag = %d\n", inform.flag);
   printf("%3d eigenpairs converged in %d iterations\n", ncon, inform.iteration);
   for(int i=0; i<ncon; i++)
      printf(" lambda[%1d] = %13.7e\n", i, lambda[i]);
   spral_ssmfe_core_free(&keep, &inform);

   /* Success */
   return 0;
}

 5 eigenpairs converged in 72 iterations
lambda[0] = 4.4676695e-02
lambda[1] = 1.1119274e-01
lambda[2] = 1.1119274e-01
lambda[3] = 1.7770878e-01
lambda[4] = 2.2040061e-01

Method¶