Generic Eigendecomposition of a Matrix

Description

Compute the set of eigenvalues and eigenvectors decomposition of a real rectangular matrix.

Usage

trlan.eigen(X, neig = min(m, n), opts = list(), lambda = NULL, U = NULL)

Arguments

Details

These routines provides an interface to state-of-art implementation of eigensolver. In particular, nu-TRLAN does the thick-restart Lanczos eigendecomposition of a matrix.

'opts' is a list of different options which can be passed to the routines. Note that by default more or less suitable values for these options are set by the routines automatically.

The options for nu-TRLAN are:

kmax

integer, maximum number of iterations.

maxiter

integer. maximum number of matrix-vector products.

tol

numeric, tolerance level.

verbose

integer, verboseness level.

Value

The returned value is a list with components

References

Wu, K. and Simon, H. (2000). Thick-restart Lanczos method for large symmetric eigenvalue problems. SIAM J. Matrix Anal. Appl. 22, 2, 602-616.

Yamazaki, I., Bai, Z., Simon, H., Wang, L.-W., and Wu, K. (2008). Adaptive projection subspace dimension for the thick restart Lanczos method. Tech. rep., Lawrence Berkeley National Laboratory, University of California, One Cyclotron road, Berkeley, California 94720.

Korobeynikov, A. (2010) Computation- and space-efficient implementation of SSA. Statistics and Its Interface, Vol. 3, No. 3, Pp. 257-268

External matrices operations.

Description

A set of routines to operate on "external" matrices.

Usage

is.extmat(X)
extmat.ncol(X)
extmat.nrow(X)
extmat(mul, tmul, nrow, ncol, env = parent.frame())
ematmul(emat, v, transposed = FALSE)

Arguments

Details

These routines checks whether the given external pointer actually points to "external matrix" structure and allow to extract the number of columns and rows respectively.

'extmat' is a convenient wrapper which allows one provide just the routines which will multiply with matrix and the transposed one (e.g. if the matrix is sparse or structured) and allow to use the SVD routines of the package

Value

Object 'extmat' class

See Also

extmat

Examples

## Not run: 
library(Matrix)
f <- function(v) as.numeric(A %*% v) # Convert Matrix object back to vector
tf <- function(v) as.numeric(tA %*% v) # Convert Matrix object back to vector

e <- new.env()
assign("A", USCounties, e)
assign("tA", t(USCounties), e)
environment(f) <- e
environment(tf) <- e

m <-extmat(f, tf, nrow(USCounties), ncol(USCounties))
system.time(v1 <- propack.svd(m, neig = 10))
#   user  system elapsed 
#  0.252   0.007   0.259 
system.time(v2 <- propack.svd(as.matrix(USCounties), neig = 10))
#   user  system elapsed 
#  8.563   0.027   8.590 

## End(Not run)

# The largest eigenvalue and the corresponding eigenvector of a Hilbert matrix
h <- outer(1:5000, 1:5000, function(i, j) 1 / (i + j - 1))
v1 <- trlan.eigen(h, neig = 1)
print(v1$d)

Class "extmat"

Description

'extmat' is a convenient wrapper which allows one provide just the routines which will multiply with matrix and the transposed one (e.g. if the matrix is sparse or structured) and allow to use the SVD routines of the package. This S4 wrapper allows the use of usual matrix operations on such objects.

Objects from the Class

Objects can be created by calls of the form extmat(mul, tmul, nrow, ncol, env = parent.frame()).

See Also

extmat

Examples

## Not run: 
library(Matrix)
f <- function(v) as.numeric(A %*% v) # Convert Matrix object back to vector
tf <- function(v) as.numeric(tA %*% v) # Convert Matrix object back to vector

e <- new.env()
assign("A", USCounties, e)
assign("tA", t(USCounties), e)
environment(f) <- e
environment(tf) <- e

m <-extmat(f, tf, nrow(USCounties), ncol(USCounties))
system.time(v1 <- propack.svd(m, neig = 10))
#   user  system elapsed 
#  0.252   0.007   0.259 
system.time(v2 <- propack.svd(as.matrix(USCounties), neig = 10))
#   user  system elapsed 
#  8.563   0.027   8.590 

## End(Not run)

Generic Singular Value Decomposition of a Matrix

Description

Compute the singular-value decomposition of a real or complex rectangular matrix.

Usage

propack.svd(X, neig = min(m, n), opts = list())
trlan.svd(X, neig = min(m, n), opts = list(), lambda = NULL, U = NULL)
ztrlan.svd(X, neig = min(m, n), opts = list(), lambda = NULL, U = NULL)

Arguments

Details

These routines provides an interface to two state-of-art implementations of truncated SVD.

PROPACK does this via the implicitly restarted Lanczos bidiagonalization with partial reorthogonalization. nu-TRLAN does the thick-restart Lanczos eigendecomposition of cross-product matrix.

'opts' is a list of different options which can be passed to the routines. Note that by default more or less suitable values for these options are set by the routines automatically.

The options for PROPACK are:

kmax

integer, maximum number of iterations.

dim

integer, dimension of Krylov subspace.

p

integer, number of shifts per restart.

maxiter

integer. maximum number of restarts.

tol

numeric, tolerance level.

verbose

logical, if 'TRUE', provide verbose output.

The options for nu-TRLAN are:

kmax

integer, maximum number of iterations.

maxiter

integer. maximum number of matrix-vector products.

tol

numeric, tolerance level.

verbose

integer, verboseness level.

Value

The returned value is a list with components

References

Wu, K. and Simon, H. (2000). Thick-restart Lanczos method for large symmetric eigenvalue problems. SIAM J. Matrix Anal. Appl. 22, 2, 602-616.

Yamazaki, I., Bai, Z., Simon, H., Wang, L.-W., and Wu, K. (2008). Adaptive projection subspace dimension for the thick restart Lanczos method. Tech. rep., Lawrence Berkeley National Laboratory, University of California, One Cyclotron road, Berkeley, California 94720.

Larsen, R. M. (1998). Efficient algorithms for helioseismic inversion. Ph.D. thesis, University of Aarhus, Denmark.

Korobeynikov, A. (2010) Computation- and space-efficient implementation of SSA. Statistics and Its Interface, Vol. 3, No. 3, Pp. 257-268