eigs_sym: Find a few eigenvalues and vectors on large, sparse Hermitian...
In PRIMME: Eigenvalues and Singular Values and Vectors from Large Matrices
eigs_sym R Documentation
Find a few eigenvalues and vectors on large, sparse Hermitian matrix
Description
Compute a few eigenpairs from a specified region (the largest, the smallest,
the closest to a point) on a symmetric/Hermitian matrix using PRIMME [1].
Generalized symmetric/Hermitian problem is also supported.
Only the matrix-vector product of the matrix is required. The used method is
usually faster than a direct method (such as eigen) if
seeking a few eigenpairs and the matrix-vector product is cheap. For
accelerating the convergence consider to use preconditioning and/or educated
initial guesses.
Usage
eigs_sym(
A,
NEig = 1,
which = "LA",
targetShifts = NULL,
tol = 1e-06,
x0 = NULL,
ortho = NULL,
prec = NULL,
isreal = NULL,
B = NULL,
...
)
Arguments
A
symmetric/Hermitian matrix or a function with signature f(x) that
returns A %*% x.
NEig
number of eigenvalues and vectors to seek.
which
which eigenvalues to find:
"LA"the largest (rightmost) values;
"SA"the smallest (leftmost) values;
"LM"the farthest from targetShifts;
"SM"the closest to targetShifts;
"CLT"the closest but left to targetShifts;
"CGT"the closest but greater than targetShifts;
- vector of numbers
the closest values to these points.
targetShifts
return the closest eigenvalues to these points as
indicated by target.
tol
the convergence tolerance:
\|A x - x\lambda\| \le tol\|A\|.
x0
matrix whose columns are educated guesses of the eigenvectors to
to find.
ortho
find eigenvectors orthogonal to the space spanned by the
columns of this matrix; useful to avoid finding some eigenvalues or
to find new solutions.
prec
preconditioner used to accelerated the convergence; usually it
is an approximation of the inverse of A - \sigma I if finding
the closest eigenvalues to \sigma. If it is a matrix
it is used as prec %*% x; otherwise it is used as prec(x).
isreal
whether A %*% x always returns real number and not complex.
B
symmetric/Hermitian positive definite matrix or a function with
signature f(x) that returns B %*% x. If given, the function
returns the eigenpairs of (A,B).
...
other PRIMME options (see details).
Details
Optional arguments to pass to PRIMME eigensolver (see further details at [2]):
method used by the solver, one of:
"DYNAMIC" switches dynamically between DEFAULT_MIN_TIME and DEFAULT_MIN_MATVECS
"DEFAULT_MIN_TIME" best method for light matrix-vector product
"DEFAULT_MIN_MATVECS" best method for heavy matrix-vector product or preconditioner
"Arnoldi" an Arnoldi not implemented efficiently
"GD" classical block Generalized Davidson
"GD_plusK" GD+k block GD with recurrence restarting
"GD_Olsen_plusK" GD+k with approximate Olsen preconditioning
"JD_Olsen_plusK" GD+k, exact Olsen (two preconditioner applications per step)
"RQI" Rayleigh Quotient Iteration, also Inverse Iteration
if targetShifts is provided
"JDQR" original block, Jacobi Davidson
"JDQMR" our block JDQMR method (similar to JDCG)
"JDQMR_ETol" slight, but efficient JDQMR modification
"STEEPEST_DESCENT" equivalent to GD(maxBlockSize,2*maxBlockSize)
"LOBPCG_OrthoBasis" equivalent to GD(neig,3*neig)+neig
"LOBPCG_OrthoBasis_Window" equivalent to GD(maxBlockSize,3*maxBlockSize)+maxBlockSize when neig>maxBlockSize
aNormestimation of norm-2 of A, used in convergence test (if not
provided, it is estimated as the largest eigenvalue in magnitude
seen).
maxBlockSizemaximum block size (like in subspace iteration or
LOBPCG).
printLevelmessage level reporting, from 0 (no output) to 5 (show all).
locking1, hard locking; 0, soft locking.
maxBasisSizemaximum size of the search subspace.
minRestartSize minimum Ritz vectors to keep in restarting.
maxMatvecs maximum number of matrix vector multiplications.
maxit maximum number of outer iterations.
scheme the restart scheme (thick restart by default).
maxPrevRetain number of approximate eigenvectors retained from
previous iteration, that are kept after restart.
robustShifts set to true to avoid stagnation.
maxInnerIterations maximum number of inner QMR iterations.
LeftQ use the locked vectors in the left projector.
LeftX use the approx. eigenvector in the left projector.
RightQ use the locked vectors in the right projector.
RightX use the approx. eigenvector in the right projector.
SkewQ use the preconditioned locked vectors in the right projector.
SkewX use the preconditioned approximate eigenvector in the right
projector.
relTolBase a legacy from classical JDQR (recommend not use).
iseed an array of four numbers used as a random seed.
Value
list with the next elements
valuesthe eigenvalues \lambda_i
vectorsthe eigenvectors x_i
rnormsthe residual vector norms
\|A x_i - \lambda_i B x_i\|.
stats$numMatvecsnumber of matrix-vector products performed
stats$numPrecondsnumber of preconditioner applications performed
stats$elapsedTimetime expended by the eigensolver
stats$timeMatvectime expended in the matrix-vector products
stats$timePrecondtime expended in applying the preconditioner
stats$timeOrthotime expended in orthogonalizing
stats$estimateMinEvalestimation of the smallest eigenvalue of A
stats$estimateMaxEvalestimation of the largest eigenvalue of A
stats$estimateANormestimation of the norm of A
References
[1] A. Stathopoulos and J. R. McCombs PRIMME: PReconditioned Iterative
MultiMethod Eigensolver: Methods and software description, ACM
Transaction on Mathematical Software Vol. 37, No. 2, (2010)
21:1-21:30.
[2] https://www.cs.wm.edu/~andreas/software/doc/primmec.html#parameters-guide
See Also
eigen for computing all values;
svds for computing a few singular values
Examples
A <- diag(1:10) # the eigenvalues of this matrix are 1:10 and the
# eigenvectors are the columns of diag(10)
r <- eigs_sym(A, 3);
r$values # the three largest eigenvalues on diag(1:10)
r$vectors # the corresponding approximate eigenvectors
r$rnorms # the corresponding residual norms
r$stats$numMatvecs # total matrix-vector products spend
r <- eigs_sym(A, 3, 'SA') # compute the three smallest values
r <- eigs_sym(A, 3, 2.5) # compute the three closest values to 2.5
r <- eigs_sym(A, 3, 2.5, tol=1e-3); # compute the values with
r$rnorms # residual norm <= 1e-3*||A||
B <- diag(rev(1:10));
r <- eigs_sym(A, 3, B=B); # compute the 3 largest eigenpairs of
# the generalized problem (A,B)
# Build a Jacobi preconditioner (too convenient for a diagonal matrix!)
# and see how reduce the number matrix-vector products
A <- diag(1:1000) # we use a larger matrix to amplify the difference
P <- diag(diag(A) - 2.5)
eigs_sym(A, 3, 2.5, tol=1e-3)$stats$numMatvecs
eigs_sym(A, 3, 2.5, tol=1e-3, prec=P)$stats$numMatvecs
# Passing A and the preconditioner as functions
Af <- function(x) (1:100) * x; # = diag(1:100) %*% x
Pf <- function(x) x / (1:100 - 2.5); # = solve(diag(1:100 - 2.5), x)
r <- eigs_sym(Af, 3, 2.5, tol=1e-3, prec=Pf, n=100)
# Passing initial guesses
A <- diag(1:1000) # we use a larger matrix to amplify the difference
x0 <- diag(1,1000,4) + matrix(rnorm(4000), 1000, 4)/100;
eigs_sym(A, 4, "SA", tol=1e-3)$stats$numMatvecs
eigs_sym(A, 4, "SA", tol=1e-3, x0=x0)$stats$numMatvecs
# Passing orthogonal constrain, in this case, already compute eigenvectors
r <- eigs_sym(A, 4, "SA", tol=1e-3); r$values
eigs_sym(A, 4, "SA", tol=1e-3, ortho=r$vectors)$values
PRIMME documentation built on Oct. 1, 2023, 1:07 a.m.
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| eigs_sym | R Documentation |
Find a few eigenvalues and vectors on large, sparse Hermitian matrix
Description
Compute a few eigenpairs from a specified region (the largest, the smallest,
the closest to a point) on a symmetric/Hermitian matrix using PRIMME [1].
Generalized symmetric/Hermitian problem is also supported.
Only the matrix-vector product of the matrix is required. The used method is
usually faster than a direct method (such as eigen) if
seeking a few eigenpairs and the matrix-vector product is cheap. For
accelerating the convergence consider to use preconditioning and/or educated
initial guesses.
Usage
eigs_sym(
A,
NEig = 1,
which = "LA",
targetShifts = NULL,
tol = 1e-06,
x0 = NULL,
ortho = NULL,
prec = NULL,
isreal = NULL,
B = NULL,
...
)
Arguments
A |
symmetric/Hermitian matrix or a function with signature f(x) that
returns |
NEig |
number of eigenvalues and vectors to seek. |
which |
which eigenvalues to find:
|
targetShifts |
return the closest eigenvalues to these points as
indicated by |
tol |
the convergence tolerance:
|
x0 |
matrix whose columns are educated guesses of the eigenvectors to to find. |
ortho |
find eigenvectors orthogonal to the space spanned by the columns of this matrix; useful to avoid finding some eigenvalues or to find new solutions. |
prec |
preconditioner used to accelerated the convergence; usually it
is an approximation of the inverse of |
isreal |
whether A %*% x always returns real number and not complex. |
B |
symmetric/Hermitian positive definite matrix or a function with
signature f(x) that returns |
... |
other PRIMME options (see details). |
Details
Optional arguments to pass to PRIMME eigensolver (see further details at [2]):
methodused by the solver, one of:
"DYNAMIC"switches dynamically between DEFAULT_MIN_TIME and DEFAULT_MIN_MATVECS
"DEFAULT_MIN_TIME"best method for light matrix-vector product
"DEFAULT_MIN_MATVECS"best method for heavy matrix-vector product or preconditioner
"Arnoldi"an Arnoldi not implemented efficiently
"GD"classical block Generalized Davidson
"GD_plusK"GD+k block GD with recurrence restarting
"GD_Olsen_plusK"GD+k with approximate Olsen preconditioning
"JD_Olsen_plusK"GD+k, exact Olsen (two preconditioner applications per step)
"RQI"Rayleigh Quotient Iteration, also Inverse Iteration if
targetShiftsis provided"JDQR"original block, Jacobi Davidson
"JDQMR"our block JDQMR method (similar to JDCG)
"JDQMR_ETol"slight, but efficient JDQMR modification
"STEEPEST_DESCENT"equivalent to GD(
maxBlockSize,2*maxBlockSize)"LOBPCG_OrthoBasis"equivalent to GD(
neig,3*neig)+neig"LOBPCG_OrthoBasis_Window"equivalent to GD(
maxBlockSize,3*maxBlockSize)+maxBlockSizewhen neig>maxBlockSize
aNormestimation of norm-2 of A, used in convergence test (if not provided, it is estimated as the largest eigenvalue in magnitude seen).
maxBlockSizemaximum block size (like in subspace iteration or LOBPCG).
printLevelmessage level reporting, from 0 (no output) to 5 (show all).
locking1, hard locking; 0, soft locking.
maxBasisSizemaximum size of the search subspace.
minRestartSizeminimum Ritz vectors to keep in restarting.
maxMatvecsmaximum number of matrix vector multiplications.
maxitmaximum number of outer iterations.
schemethe restart scheme (thick restart by default).
maxPrevRetainnumber of approximate eigenvectors retained from previous iteration, that are kept after restart.
robustShiftsset to true to avoid stagnation.
maxInnerIterationsmaximum number of inner QMR iterations.
LeftQuse the locked vectors in the left projector.
LeftXuse the approx. eigenvector in the left projector.
RightQuse the locked vectors in the right projector.
RightXuse the approx. eigenvector in the right projector.
SkewQuse the preconditioned locked vectors in the right projector.
SkewXuse the preconditioned approximate eigenvector in the right projector.
relTolBasea legacy from classical JDQR (recommend not use).
iseedan array of four numbers used as a random seed.
Value
list with the next elements
valuesthe eigenvalues
\lambda_ivectorsthe eigenvectors
x_irnormsthe residual vector norms
\|A x_i - \lambda_i B x_i\|.stats$numMatvecsnumber of matrix-vector products performed
stats$numPrecondsnumber of preconditioner applications performed
stats$elapsedTimetime expended by the eigensolver
stats$timeMatvectime expended in the matrix-vector products
stats$timePrecondtime expended in applying the preconditioner
stats$timeOrthotime expended in orthogonalizing
stats$estimateMinEvalestimation of the smallest eigenvalue of A
stats$estimateMaxEvalestimation of the largest eigenvalue of A
stats$estimateANormestimation of the norm of A
References
[1] A. Stathopoulos and J. R. McCombs PRIMME: PReconditioned Iterative MultiMethod Eigensolver: Methods and software description, ACM Transaction on Mathematical Software Vol. 37, No. 2, (2010) 21:1-21:30.
[2] https://www.cs.wm.edu/~andreas/software/doc/primmec.html#parameters-guide
See Also
eigen for computing all values;
svds for computing a few singular values
Examples
A <- diag(1:10) # the eigenvalues of this matrix are 1:10 and the
# eigenvectors are the columns of diag(10)
r <- eigs_sym(A, 3);
r$values # the three largest eigenvalues on diag(1:10)
r$vectors # the corresponding approximate eigenvectors
r$rnorms # the corresponding residual norms
r$stats$numMatvecs # total matrix-vector products spend
r <- eigs_sym(A, 3, 'SA') # compute the three smallest values
r <- eigs_sym(A, 3, 2.5) # compute the three closest values to 2.5
r <- eigs_sym(A, 3, 2.5, tol=1e-3); # compute the values with
r$rnorms # residual norm <= 1e-3*||A||
B <- diag(rev(1:10));
r <- eigs_sym(A, 3, B=B); # compute the 3 largest eigenpairs of
# the generalized problem (A,B)
# Build a Jacobi preconditioner (too convenient for a diagonal matrix!)
# and see how reduce the number matrix-vector products
A <- diag(1:1000) # we use a larger matrix to amplify the difference
P <- diag(diag(A) - 2.5)
eigs_sym(A, 3, 2.5, tol=1e-3)$stats$numMatvecs
eigs_sym(A, 3, 2.5, tol=1e-3, prec=P)$stats$numMatvecs
# Passing A and the preconditioner as functions
Af <- function(x) (1:100) * x; # = diag(1:100) %*% x
Pf <- function(x) x / (1:100 - 2.5); # = solve(diag(1:100 - 2.5), x)
r <- eigs_sym(Af, 3, 2.5, tol=1e-3, prec=Pf, n=100)
# Passing initial guesses
A <- diag(1:1000) # we use a larger matrix to amplify the difference
x0 <- diag(1,1000,4) + matrix(rnorm(4000), 1000, 4)/100;
eigs_sym(A, 4, "SA", tol=1e-3)$stats$numMatvecs
eigs_sym(A, 4, "SA", tol=1e-3, x0=x0)$stats$numMatvecs
# Passing orthogonal constrain, in this case, already compute eigenvectors
r <- eigs_sym(A, 4, "SA", tol=1e-3); r$values
eigs_sym(A, 4, "SA", tol=1e-3, ortho=r$vectors)$values
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