Matrix Exponential using Krylov subspace routines

This package utilises some of the matrix exponential routines included in
EXPOKIT (http://www.maths.uq.edu.au/expokit/), which is software
designed to calculate matrix exponentials for both small dense and large sparse
matrices. The use of Krylov subspace methods implemented within these routines
should result in more efficient computations.

The EXPOKIT software was developed by Roger B. Sidje. Nicholas J. Matzke went on to
adapt this software for the R package Rexpokit, which enables users to make
use of EXPOKIT, a FORTRAN library, within an R environment. Kexpmv uses the
foundations of Rexpokit with the aim of making computing matrix exponentials more
efficient, especially for large sparse matrices which can often be computationally
complex.

Permission to distribute the EXPOKIT source under GPL was obtained from
Roger B. Sidje.

EXPOKIT includes functions for exponentiating both small dense matrices and
large sparse matrices. Fast and efficient matrix exponentiation is needed
for different application areas, i.e. Markov models. This package allows the
user to calculate both the matrix exponential in isolation as well as the
matrix exponential with the product of a vector, which is essential for multi-state
Markov models. Both `expokit_dgexpv`

and `expokit_dmexpv`

functions can compute both
of these calculations with the use of the `vector`

arguement.

When the matrix has large dimensions and is sparse in nature, this means the
matrix has a high volume of elements equal to zero. Similiar to EXPOKIT this package
transforms the matrix into Compressed Row Storage (CRS) format before the
matrix exponentiation is performed. See functions `mat2crs`

and `crs2mat`

for more details.

**Acknowledgements/sources**

**1.** Nicholas
Matzke nickmatzke.ncse@gmail.com helped
greatly with the initial setup of the package. See his
`Rexpokit`

for another R
implementation of EXPOKIT routines.

**2.**
EXPOKIT, original FORTRAN package, by Roger B. Sidje
rbs@maths.uq.edu.au, Department of Mathematics,
University of Queensland, Brisbane, QLD-4072, Australia,
(c) 1996-2013 All Rights Reserved. Sidje has given
permission to include EXPOKIT code in this R package
under the usual GPL license for CRAN R packages. For the
full EXPOKIT copyright and license, see
`expokit_copyright.txt`

under
`inst/notes`

.

**3.** The development of this
package was helped by advice and discussions with my PhD
supervisors Adele Marshall and Karen Cairns.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 | ```
library(kexpmv)
# Construct a square (nxn) matrix
mat=matrix(c(-0.071207, 0.065573, 0.005634, 0, -0.041206, 0.041206, 0, 0, 0),
nrow=3, byrow=TRUE)
# Define value for t
t = 15
# Exponentiate this matrix using DGPADM for t = 15
OutputMat = expokit_dgpadm( mat = mat, t = t, transpose_needed = TRUE)
print(OutputMat)
# Construct column (nx1) vector
v = matrix(0,3,1)
v[1] = 1
# Exponentiate the matrix using DMEXPV for t = 15 with the product of vector v.
OutputMat = expokit_dmexpv( mat = mat, t = t, vector = v, transpose_needed = TRUE,
transform_to_crs = TRUE)
# Print corresponding (nx1) results vector
print(OutputMat$output_probs)
# Print message to determine whether the mxstep value needs increased. If NULL
# then mxstep value is valid.
print(OutputMat$message)
# Exponentiate the matrix using DGEXPV for t = 15.
OutputMat = expokit_dgexpv( mat = mat, t = t, vector = NULL, transpose_needed = TRUE,
transform_to_crs = TRUE)
print(OutputMat$output_mat)
print(OutputMat$message)
# Functions DMEXPV and DGEXPV are very similiar, with the only difference being DMEXPV
# carries out an additional check for Markov chains.
``` |

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