SSMFE - Sparse Symmetric Matrix-Free Eigensolver

#include <spral_ssmfe.h> /* or <spral.h> for all packages */

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 buckling problem

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

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

This package provides a user-friendly wrapper around spral_ssmfe_expert, which in turn provides a wrapper around spral_ssmfe_core. If more fine-tuned control of the eigensolver is required, use those modules instead.

Version history

2015-04-20 Version 1.0.0

Initial release

[for detail please see ChangeLog]

Usage overview

The eigensolver subroutines behind this package implement a block iterative algorithm. 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 e.g. by 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 requires the direct solution of linear systems with the matrix \(A\) or its linear combination with \(B\), for which a sparse symmetric indefinite solver (such as HSL_MA97 or SPRAL_SSIDS) can be employed.

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.

Subroutines

void spral_ssmfe_default_options(struct spral_ssmfe_options *options)

Intialises members of options structure to default values.

Parameters:

• options – Structure to be initialised.

void spral_ssmfe_standard_double(struct spral_ssmfe_rcid *rci, int left, int mep, double *lambda, int n, double *x, int ldx, void **keep, const struct spral_ssmfe_options *options, struct spral_ssmfe_inform *inform)

Computes the left-most eigenpairs of the standard eigenvalue problem

\[Ax = \lambda x\]

Optionally uses preconditioning.

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	Restart computation. Non-fatal error, see inform.flag.
-1	None. Computation complete.
1	Calculate \(Y = AX\).
2	Apply preconditioner \(Y = TX\).

The matrices \(X\) and \(Y\) are pointed to by components of rci.

Parameters:

• rci – Reverse communication type. rci.job must be set to 0 before the first call.

• left – Number of left eigenpairs to find.

• mep – Number of working eigenpairs. See method section for guidance on selecting a good value. Must be at least left.

• lambda[mep] – Current eigenvalue estimates in ascending order.

• n – Size of matrix \(A\).

• x[n][ldx] – Current eigenvector estimates corresponding to eigenvalues in lambda. Used to supply initial estimates if options.user_x>0.

• ldx – Leading dimension of x.

• 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_standard_double_complex(struct spral_ssmfe_rciz *rci, int left, int mep, double *lambda, int n, double complex *x, int ldx, void **keep, const struct spral_ssmfe_options *options, struct spral_ssmfe_inform *inform)

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

void spral_ssmfe_standard_shift_double(struct spral_ssmfe_rcid *rci, double sigma, int left, int right, int mep, double *lambda, int n, double *x, int ldx, void **keep, const struct spral_ssmfe_options *options, struct spral_ssmfe_inform *inform)

Computes eigenpairs of the standard eigenvalue problem

\[Ax = \lambda x\]

in the vicinity of a given value \(\sigma\).

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	Restart computation. Non-fatal error, see inform.flag.
-1	None. Computation complete.
1	Calculate \(Y = AX\).
9	Solve \((A-\sigma I)Y = X\) for Y.

The matrices \(X\) and \(Y\) are components of rci.

Parameters:

• rci – Reverse communication type. rci.job must be set to 0 before the first call.

• sigma – Shift value \(sigma\).

• left – Number of left eigenpairs to find.

• right – Number of right eigenpairs to find.

• mep – Number of working eigenpairs. See method section for guidance on selecting a good value. Must be at least left+right.

• lambda[mep] – Current eigenvalue estimates in ascending order.

• n – Size of matrix \(A\).

• x[n][ldx] – Current eigenvector estimates corresponding to eigenvalues in lambda. Used to supply initial estimates if options.user_x>0.

• ldx – Leading dimension of x.

• 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_standard_shift_double_complex(struct spral_ssmfe_rciz *rci, double sigma, int left, int right, int mep, double *lambda, int n, double complex *x, int ldx, void **keep, const struct spral_ssmfe_options *options, struct spral_ssmfe_inform *inform)

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

void spral_ssmfe_generalized_double(struct spral_ssmfe_rcid *rci, int left, int mep, double *lambda, int n, double *x, int ldx, void **keep, const struct spral_ssmfe_options *options, struct spral_ssmfe_inform *inform)

Computes the left-most eigenpairs of the generalized eigenvalue problem

\[Ax = \lambda B x\]

Optionally uses preconditioning.

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	Restart computation. Non-fatal error, see inform.flag.
-1	None. Computation complete.
1	Calculate \(Y = AX\).
2	Apply preconditioner \(Y = TX\).
3	Calculate \(Y = BX\).

The matrices \(X\) and \(Y\) are components of rci.

Parameters:

• rci – Reverse communication type. rci.job must be set to 0 before the first call.

• left – Number of left eigenpairs to find.

• mep – Number of working eigenpairs. See method section for guidance on selecting a good value. Must be at least left.

• lambda[mep] – Current eigenvalue estimates in ascending order.

• n – Size of matrix \(A\).

• x[n][ldx] – Current eigenvector estimates corresponding to eigenvalues in lambda. Used to supply initial estimates if options.user_x>0.

• ldx – Leading dimension of x.

• 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_generalized_double_complex(struct spral_ssmfe_rciz *rci, int left, int mep, double *lambda, int n, double complex *x, int ldx, void **keep, const struct spral_ssmfe_options *options, struct spral_ssmfe_inform *inform)

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

void spral_ssmfe_generalized_shift_double(struct spral_ssmfe_rcid *rci, double sigma, int left, int right, int mep, double *lambda, int n, double *x, int ldx, void **keep, const struct spral_ssmfe_options *options, struct spral_ssmfe_inform *inform)

Computes eigenpairs of the generalized eigenvalue problem

\[Ax = \lambda B x\]

in the vicinity of a given value \(\sigma\).

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	Restart computation. Non-fatal error, see inform.flag.
-1	None. Computation complete.
1	Calculate \(Y = AX\).
3	Calculate \(Y = BX\).
9	Solve \((A-\sigma B)Y = X\) for Y.

The matrices \(X\) and \(Y\) are components of rci.

Parameters:

• rci – Reverse communication type. rci.job must be set to 0 before the first call.

• sigma – Shift value \(sigma\).

• left – Number of left eigenpairs to find.

• right – Number of right eigenpairs to find.

• mep – Number of working eigenpairs. See method section for guidance on selecting a good value. Must be at least left+right.

• lambda[mep] – Current eigenvalue estimates in ascending order.

• n – Size of matrix \(A\).

• x[n][ldx] – Current eigenvector estimates corresponding to eigenvalues in lambda. Used to supply initial estimates if options.user_x>0.

• ldx – Leading dimension of x.

• 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_generalized_shift_double_complex(struct spral_ssmfe_rciz *rci, double sigma, int left, int right, int mep, double *lambda, int n, double complex *x, int ldx, void **keep, const struct spral_ssmfe_options *options, struct spral_ssmfe_inform *inform)

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

void spral_ssmfe_buckling_double(struct spral_ssmfe_rcid *rci, double sigma, int left, int right, int mep, double *lambda, int n, double *x, int ldx, void **keep, const struct spral_ssmfe_options *options, struct spral_ssmfe_inform *inform)

Computes eigenpairs of the buckling problem

\[Bx = \lambda A x\]

in the vicinity of a given value \(\sigma\).

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	Restart computation. Non-fatal error, see inform.flag.
-1	None. Computation complete.
1	Calculate \(Y = AX\).
3	Calculate \(Y = BX\).
9	Solve \((B-\sigma A)Y = X\) for Y.

The matrices \(X\) and \(Y\) are components of rci.

Parameters:

• rci – Reverse communication type. rci.job must be set to 0 before the first call.

• sigma – Shift value \(sigma\).

• left – Number of left eigenpairs to find.

• right – Number of right eigenpairs to find.

• mep – Number of working eigenpairs. See method section for guidance on selecting a good value. Must be at least left+right.

• lambda[mep] – Current eigenvalue estimates in ascending order.

• n – Size of matrix \(A\).

• x[n][ldx] – Current eigenvector estimates corresponding to eigenvalues in lambda. Used to supply initial estimates if options.user_x>0.

• ldx – Leading dimension of x.

• 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_buckling_double_complex(struct spral_ssmfe_rciz *rci, double sigma, int left, int right, int mep, double *lambda, int n, double complex *x, int ldx, void **keep, const struct spral_ssmfe_options *options, struct spral_ssmfe_inform *inform)

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

void spral_ssmfe_free_double(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.

void spral_ssmfe_free_double_complex(void **keep, struct spral_ssmfe_inform *inform)

As spral_ssmfe_free_double(), but for double complex versions of types.

Derived types

struct spral_ssmfe_rcid

Real-valued reverse communication interface (RCI) type.

int job

Reverse-communication task to perform.

int nx

Number of columns in x and y.

double x[nx][n]

Vector to be transformed by RCI task. Allocated by routine.

double y[nx][n]

Vector to store result of RCI task. Allocated by routine.

struct spral_ssmfe_rciz

Complex-valued reverse communication interface (RCI) type.

int job

Reverse-communication task to perform.

int nx

Number of columns in x and y.

double complex x[nx][n]

Vector to be transformed by RCI task. Allocated by routine.

double complex y[nx][n]

Vector to store result of RCI task. Allocated by routine.

struct spral_ssmfe_options

Options that control the algorithm.

double abs_tol_lambda.

Absolute tolerance for estimated eigenvalue convergence test, see method section. Negative values are treated as the default. Default is 0.0.

double abs_tol_residual

Absolute tolerance for residual convergence test, see method section. Negative values are treated as the default. Default is 0.0.

int max_iterations

Maximum number of iterations. Default is 100.

double rel_tol_lambda

Relative tolerance for estimated eigenvalue error convergence test, see method section. Negative values are treated as the default. Default is 0.0.

double rel_tol_residual

Relative tolerance for residual convergence test, see method section. If both abs_tol_residual and rel_tol_residual are 0.0, then the residual norms are not taken into consideration by the convergence test. Negative values are treated as the default. Default is 0.0.

double tol_x

Tolerance for estimated eigenvector error convergence test, see method section. If tol_x is set to 0.0, the eigenvector error is not estimated. If a negative value is assigned, the tolerance is set to sqrt(DBL_EPSILON). Default is -1.0.

int print_level

Amount of printing. Possible values are:

<0	no printing
0	error and warning messages only
1	the type (standard or generalized) and the size of the problem, the number of eigenpairs requested, the error tolerances and the size of the subspace are printed before the iterations start
2	as above but, for each eigenpair tested for convergence, the iteration number, the index of the eigenpair, the eigenvalue, whether it has converged, the residual norm, and the error estimates are also printed
>2	as 1 but with all eigenvalues, whether converged, residual norms and eigenvalue/eigenvector error estimates printed on each iteration.

Note that for eigenpairs that are far from convergence, ‘rough’ error estimates are printed (the estimates that are actually used by the stopping criteria, see Section [ssmfe:method], only become available on the last few iterations).

Default is 0.

int unit_error

Fortran unit number for error messages. Printing suppressed if negative. Default is 6 (stdout).

int unit_diagnostic

Fortran unit number for diagnostic messages. Printing suppressed if negative. Default is 6 (stdout).

int unit_warning

Fortran unit number for warning messages. Printing suppressed if negative. Default is 6 (stdout).

double left_gap

Minimal acceptable distance between last computed left eigenvalue and rest of spectrum. For spral_ssmfe_standard_double() and spral_ssmfe_generalized_double() the last computed left eigenvalue is the rightmost of those computed. For other routines it is the leftmost. If set to a negative value \(\delta\), the minimal distance is taken as \(|\delta|\) times the average distance between the computed eigenvalues. Note that for this option to have any effect, the value of mep must be larger than left+right. See method section for further explanation. Default is 0.0.

int max_left

Number of eigenvalues to left of \(\sigma\), or a negative value if not known. Default is -1.

int max_right

Number of eigenvalues to right of \(\sigma\), or a negative value if not known. Default is -1.

double right_gap

As left_gap, but for right eigenvalues. Default is 0.0.

int user_x

Number of eigenvectors for which an initial guess is supplied in x(:,:) on the first call. Such eigenvectors must be lineraly independent. Default is 0.

type spral_ssmfe_inform

Information on progress of the algorithm.

int flag

Return status of algorithm. See table below.

int iteration

Number of iterations.

int left

Number of converged left eigenvalues.

double next_left

Upon completion, next left eigenvalue in spectrum (see options.left_gap).

double next_right

Upon completion, next right eigenvalue in spectrum (see options.right_gap).

int non_converged

Number of non-converged eigenpairs.

int right

Number of converged right eigenvalues.

int stat

Fortran allocation status in event of failure

inform.flag
-1	rci.job is out-of-range.
-9	n is out-of-range.
-10	ldx is out-of-range.
-11	left is out-of-range.
-12	right is out-of-range.
-13	mep is less than the number of desired eigenpairs.
-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.
+2	The maximum number of iterations max_iterations has been exceeded. The value of inform.non_converged is set to the number of non-converged eigenpairs.
+3	The solver had run out of storage space for the converged eigenpairs before the gap in the spectrum required by options.left_gap and/or options.right_gap was reached. The value of inform.non_converged is set to the number of non-converged eigenpairs.

If the computation is terminated with the error code 2 or 3, the computation is not complete, but may be restarted with larger values of max_iterations and/or mep. In this case the user should set options.user_x to info.left + info.right and restart the reverse communication loop. An alternative option is to use one of the advanced solver procedures from SSMFE_EXPERT - Sparse Symmetric Matrix-Free Eigensolver (Expert interface) or SSMFE_CORE - Sparse Symmetric Matrix-Free Eigensolver (Core Algorithm) that delegate the storage of computed eigenpairs and the termination of the computation to the user.

Examples

Preconditioning example

The following code computes the 5 leftmost eigenpairs of the matrix \(A\) of order 100 that approximates the two-dimensional Laplacian operator on a 20-by-20 grid. One forward and one backward Gauss-Seidel update are used for preconditioning, which halves the number of iterations compared with solving the same problem without preconditioning. The header laplace2d.h (examples/C/ssmfe/laplace2d.h) supplies a subroutine apply_laplacian() that multiplies a block of vectors by \(A\), and a subroutine apply_gauss_seidel_step() that computes \(y = T x\) for a block of vectors \(x\) by applying one forward and one backward update of the Gauss-Seidel method to the system \(A y = x\).

/* examples/C/ssmfe/precond_ssmfe.c */
/* Laplacian on a square grid (using SPRAL_SSMFE 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 m   = 20;     /* grid points along each side */
   const int n   = m*m;    /* problem size */
   const int nep = 5;      /* eigenpairs wanted */

   double *lambda = malloc(2*nep * sizeof(*lambda)); /* eigenvalues */
   double (*X)[2*nep][n] = malloc(sizeof(*X));       /* eigenvectors */
   struct spral_ssmfe_rcid rci;           /* reverse communication data */
   struct spral_ssmfe_options options;    /* options */
   void *keep;                            /* private data */
   struct spral_ssmfe_inform inform;      /* information */

   /* Initialize options to default values */
   spral_ssmfe_default_options(&options);
   /* gap between the last converged eigenvalue and the rest of the spectrum
    * must be at least 0.1 times average gap between computed eigenvalues */
   options.left_gap = -0.1;

   rci.job = 0; keep = NULL;
   while(true) { /* reverse communication loop */
      spral_ssmfe_standard_double(&rci, nep, 2*nep, lambda, n, &(*X)[0][0], n,
         &keep, &options, &inform);
      switch ( rci.job ) {
      case 1:
         apply_laplacian(m, m, rci.nx, rci.x, rci.y);
         break;
      case 2:
         apply_gauss_seidel_step(m, m, rci.nx, rci.x, rci.y);
         break;
      default:
         goto finished;
      }
   }
finished:
   printf("%d eigenpairs converged in %d iterations\n", inform.left, inform.iteration);
   for(int i=0; i<inform.left; i++)
      printf(" lambda[%1d] = %13.7e\n", i, lambda[i]);
   spral_ssmfe_free_double(&keep, &inform);
   free(lambda);
   free(X);

   /* Success */
   return 0;
}

This code produces the following output:

6 eigenpairs converged in 19 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
 lambda[5] = 2.2040061e-01

Note that the code computed one extra eigenpair because of the insufficient gap between the 5th and 6th eigenvalues.

Shift-and-invert example

The following code computes the eigenpairs of the matrix of order 64 that approximates the two-dimensional Laplacian operator on 8-by-8 grid with eigenvalues near the shift sigma=1.0. For the shifted solve, LAPACK subroutines DSYTRS and DSYTRF are used, which perform the LDLT-factorization and the solution of the factorized system respectively. The matrix of the discretized Laplacian is computed by the subroutine set_2d_laplacian_matrix() from the laplace2d.h header (examples/C/ssmfe/laplace2d.h). The header ldltf.h (examples/C/ssmfe/ldltf.h) supplies the function num_neg_D() that counts the number of negative eigenvalues of the D-factor.

/* examples/C/ssmfe/shift_invert.c */
/* Laplacian on a rectangular grid by shift-invert via LDLT factorization */
#include "spral.h"
#include <math.h>
#include <stdlib.h>
#include <stdio.h>
#include <cblas.h>

/* Headers that implements Laplacian and preconditioners and LDLT support */
#include "laplace2d.h"
#include "ldltf.h"

int main(void) {
   const int nx = 8;          /* grid points along x */
   const int ny = 8;          /* grid points along y */
   const int n = nx*ny;       /* problem size */
   const double sigma = 1.0;  /* shift */

   int *ipiv = malloc(n * sizeof(*ipiv));        /* LDLT pivot index */
   double *lambda = malloc(n * sizeof(*lambda)); /* eigenvalues */
   double (*X)[n][n] = malloc(sizeof(*X));       /* eigenvectors */
   double (*A)[n][n] = malloc(sizeof(*A));       /* matrix */
   double (*LDLT)[n][n] = malloc(sizeof(*LDLT)); /* factors */
   double (*work)[n][n] = malloc(sizeof(*work)); /* work array for dsytrf */
   struct spral_ssmfe_options options;    /* eigensolver options */
   struct spral_ssmfe_inform inform;      /* information */
   struct spral_ssmfe_rcid rci;           /* reverse communication data */
   void *keep;                            /* private data */

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

   /* Set up then perform LDLT factorization of the shifted matrix */
   set_laplacian_matrix(nx, ny, n, A);
   for(int j=0; j<n; j++)
   for(int i=0; i<n; i++)
      (*LDLT)[j][i] = (i==j) ? (*A)[j][i] - sigma
                             : (*A)[j][i];
   cwrap_dsytrf('L', n, &(*LDLT)[0][0], n, ipiv, &(*work)[0][0], n*n);

   /* Main loop */
   int left = num_neg_D(n, n, LDLT, ipiv);   /* all evalues to left of sigma */
   int right = 5;                            /* 5 evalues to right of sigma */
   rci.job = 0; keep = NULL;
   while(true) {
      spral_ssmfe_standard_shift_double(&rci, sigma, left, right, n, lambda,
            n, &(*X)[0][0], n, &keep, &options, &inform);
      switch( rci.job ) {
      case 1:
         cblas_dgemm(CblasColMajor, CblasNoTrans, CblasNoTrans, n, rci.nx, n,
            1.0, &(*A)[0][0], n, rci.x, n, 0.0, rci.y, n);
         break;
      case 2:
         // No preconditioning
         break;
      case 9:
         cblas_dcopy(n*rci.nx, rci.x, 1, rci.y, 1);
         cwrap_dsytrs('L', n, rci.nx, &(*LDLT)[0][0], n, ipiv, rci.y, n);
         break;
      default:
         goto finished;
      }
   }
finished:
   printf("Eigenvalues near %e (took %d iterations)\n", sigma, inform.iteration);
   for(int i=0; i<inform.left+inform.right; i++)
      printf(" lambda[%1d] = %13.7e\n", i, lambda[i]);
   spral_ssmfe_free_double(&keep, &inform);
   free(ipiv);
   free(lambda);
   free(X);
   free(A);
   free(LDLT);
   free(work);

   /* Success */
   return 0;
}

This code produces the following output:

Eigenvalues near 1.000000e+00 (took 5 iterations)
 lambda[0] = 2.4122952e-01
 lambda[1] = 5.8852587e-01
 lambda[2] = 5.8852587e-01
 lambda[3] = 9.3582223e-01
 lambda[4] = 1.1206148e+00
 lambda[5] = 1.1206148e+00
 lambda[6] = 1.4679111e+00
 lambda[7] = 1.4679111e+00
 lambda[8] = 1.7733184e+00

Hermitian example

The following code computes the 5 leftmost eigenpairs of the differential operator \(i \frac{d}{dx}\) acting in the space of periodic functions discretized by central differences on a uniform mesh of 80 steps.

/* examples/C/ssmfe/hermitian.c - Example code for SPRAL_SSMFE package */
/* Hermitian operator example */
#include "spral.h"
#include <math.h>
#include <stdlib.h>
#include <stdio.h>
#include <cblas.h>

/* central differences for i d/dx */
void apply_idx(int n, int m, const double complex *x_ptr, double complex *y_ptr) {
   /* Use "variable-modified types" to simplify matrix indexing */
   const double complex (*x)[m][n] = (const double complex (*)[m][n]) x_ptr;
   double complex (*y)[m][n] = (double complex (*)[m][n]) y_ptr;
   for(int j=0; j<m; j++) {
      for(int i=0; i<n; i++) {
         int il = (i==0)   ? n-1 : i-1;
         int ir = (i==n-1) ? 0   : i+1;
         (*y)[j][i] = _Complex_I * ((*x)[j][ir] - (*x)[j][il]);
      }
   }
}

/* main routine */
int main(void) {
   const int n   = 80;                 /* problem size */
   const int nep = 5;                  /* eigenpairs wanted */

   double *lambda = malloc(nep * sizeof(*lambda));   /* eigenvalues */
   double complex (*X)[nep][n] = malloc(sizeof(*X)); /* eigenvectors */
   struct spral_ssmfe_rciz rci;        /* reverse communication data */
   struct spral_ssmfe_options options; /* options */
   void *keep;                         /* private data */
   struct spral_ssmfe_inform inform;   /* information */

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

   rci.job = 0; keep = NULL;
   while(true) { /* reverse communication loop */
      spral_ssmfe_standard_double_complex(&rci, nep, nep, lambda, n,
         &(*X)[0][0], n, &keep, &options, &inform);
      switch ( rci.job ) {
      case 1:
         apply_idx(n, rci.nx, rci.x, rci.y);
         break;
      case 2:
         // No preconditioning
         break;
      default:
         goto finished;
      }
   }
finished:
   printf("%d eigenpairs converged in %d iterations\n", inform.left, inform.iteration);
   for(int i=0; i<inform.left; i++)
      printf(" lambda[%1d] = %13.7e\n", i, lambda[i]);
   spral_ssmfe_free_double_complex(&keep, &inform);
   free(lambda);
   free(X);

   /* Success */
   return 0;
}

This code produces the following output:

5 eigenpairs converged in 25 iterations
 lambda[0] = -2.0000000e+00
 lambda[1] = -1.9938347e+00
 lambda[2] = -1.9938347e+00
 lambda[3] = -1.9753767e+00
 lambda[4] = -1.9753767e+00

Method

spral_ssmfe_core`, upon which spral_ssmfe is built, implements a block iterative algorithm based on the Jacobi-conjugate preconditioned gradients (JCPG) method [2], [3]. This algorithm simultaneously computes \(m < n\) approximate eigenpairs, where the block size \(m\) exceeds the number \(n_e\) of desired eigenpairs for the sake of better convergence, namely, \(m = n_e + \min(10, 0.1 n_e)\).

An approximate eigenpair \(\{x,\lambda\}\) is considered to have converged if the following three conditions are all satisfied:

1. if options.abs_tol_lambda and options.rel_tol_lambda are not both equal to zero, then the estimated error in the approximate eigenvalue must be less than or equal to \(\max(\mathrm{options.abs\_tol\_lambda}, \delta*\mathrm{options.rel\_tol\_lambda})\), where \(\delta\) is the estimated average distance between eigenvalues.

2. if options.tol_x is not zero, then the estimated sine of the angle between the approximate eigenvector and the invariant subspace corresponding to the eigenvalue approximated by \(\lambda\) must be less than or equal to options.tol_x.

3. if options.abs_tol_residual and options.rel_tol_residual are not both equal to zero, then the Euclidean norm of the residual, \(\|A x - \lambda B x\|_2\), must be less than or equal to \(\max(\mathrm{options.abs\_tol\_residual}, \mathrm{options.rel\_tol\_residual}*\|\lambda B x\|_2)\).

The extra eigenpairs are not checked for convergence, as their role is purely auxiliary.

If the gap between the last computed eigenvalue and the rest of the spectrum is small, then the accuracy of the corresponding eigenvector may be very low. To prevent this from happening, the user should set the eigenpairs storage size mep to a value that is larger than the number of desired eigenpairs, and set the options options.left_gap and options.right_gap to non-zero values \(\delta_l\) and \(\delta_r\). These values determine the size of the minimal acceptable gaps between the computed eigenvalues and the rest of the spectrum, \(\delta_l\) referring to either leftmost eigenvalues (for ssmfe_standard() and ssmfe_generalized() only) or those to the left of the shift sigma, and \(\delta_r\) to those to the right of the shift sigma. Positive values of \(\delta_l\) and \(\delta_r\) set the gap explicitely, and negative values require the gap to be not less than their absolute value times the average distance between the computed eigenvalues. A recommended value of \(\delta_l\) and \(\delta_r\) is -0.1. The value of mep has little effect on the speed of computation, hence it might be set to any reasonably large value. The larger the value of mep, the larger the size of an eigenvalue cluster for which accurate eigenvectors can be computed, notably: to safeguard against clusters of size up to \(k\), it is sufficient to set mep to the number of desired eigenpairs plus \(k - 1\).

When using the solver procedures that employ the shift-and-invert technique, it is very important to ensure that the numbers of desired eigenvalues each side of the shift do not exceed the actual numbers of these eigenvalues, as the eigenpairs ‘approximating’ non-existing eigenpairs of the problem will not converge. It is therefore strongly recommended that the user employs a linear system solver that performs the \(LDL^T\) factorization of the shifted system, e.g. HSL_MA97 or SPRAL_SSIDS. The \(LDL^T\) factorization of the matrix \(A - \sigma B\) consists in finding a lower triangular matrix \(L\), a block-diagonal matrix \(D\) with \(1\times 1\) and \(2\times 2\) blocks on the diagonal and a permutation matrix \(P\) such that \(P^T(A - \sigma B)P = L D L^T\). By the inertia theorem, the number of eigenvalues to the left and right from the shift \(\sigma\) is equal to the number of negative and positive eigenvalues of \(D\), which allows quick computation of the eigenvalue numbers each side of the shift.

References

[1]

E. E. Ovtchinnikov and J. Reid (2010). A preconditioned block conjugate gradient algorithm for computing extreme eigenpairs of symmetric and Hermitian problems. Technical Report RAL-TR-2010-19.

[2]

E. E. Ovtchinnikov (2008). Jacobi correction equation, line search and conjugate gradients in Hermitian eigenvalue computation I: Computing an extreme eigenvalue. SIAM J. Numer. Anal., 46:2567–2592.

[3]

E. E. Ovtchinnikov (2008). Jacobi correction equation, line search and conjugate gradients in Hermitian eigenvalue computation II: Computing several extreme eigenvalues. SIAM J. Numer. Anal., 46:2593–2619.