Create objects from class MultiCompanion
mCompanion.RdCreate, generate, or simulate objects from class "MultiCompanion" by
specifying the matrix in several ways.
Arguments
- x
the matrix or, for
mCompaniononly, the top of the matrix or a character string, see section ‘Details’.- misc
information to be stored in the object's pad.
- ...
other arguments to be passed down to generator functions, see section ‘Details’.
- xtop
the top of the matrix.
- mo
the multi-companion order of the matrix.
- n
the dimension.
- mo.col
the top order, meaniing that columns mo.col+1,...,n of the top of the matrix are zeros.
mo.colmay also be set to "detect", in which case it is determined by scanningxtoporx.- ido
the dimension of the identity sub-matrix.
- dimnames
is not used currently.
- detect
controls whether automatic detection of
moandmo.colshould be attempted. The values tested are "mo", "mo.col", "all", and "nothing" with obvious meanings.- .Object
this is set implicitly by package "methods".
Details
Objects from class "MultiCompanion" can be created by calling
mCompanion() or new("MultiCompanion", ...). In the
latter case the “...” arguments are as for the
initialize method, except .Object. Do not call
initialize directly.
mCompanion can generate multi-companion matrices from spectral
information, full or partial, using the methodology developed by
Boshnakov and Iqelan (2009)
. If the
specification is not given in full, the missing information is filled
with suitably simulated values. For example, unspecifies eigenvalues
are generated inside the unit circle, sim_mc.
If argument x is the string "sim" or "gen", then
mCompanion calls sim_mc or
make_mcmatrix, respectively, with the arguments
... and converts the result to class MultiCompanion. See the
documentation of those functions for further details and examples.
The conversion may be the main reason to use mCompanion in this
way rather than call sim_mc and make_mcmatrix directly.
Otherwise, if x is numeric it is taken to specify the top of
the matrix unless detect="mo" in which case it is the whole
matrix. In both cases all arguments are passed down to new, the
only (more or less) change being that x is passed down as
xtop=x and x=x, respectively, see
MultiCompanion.
detect=="gen" signifies that x has the format of the
output from sim_mc or make_mcmatrix, so that mCompanion
may use the additional information in such objects.
The multi-companion order is determined automatically from the content
of the matrix if detect=="mo".
References
Boshnakov GN (2002). “Multi-companion matrices.” Linear Algebra Appl., 354, 53–83. ISSN 0024-3795, doi:10.1016/S0024-3795(01)00475-X .
Boshnakov GN (2007). “Singular value decomposition of multi-companion matrices.” Linear Algebra Appl., 424(2-3), 393–404. ISSN 0024-3795, doi:10.1016/j.laa.2007.02.010 .
Boshnakov GN, Iqelan BM (2009). “Generation of time series models with given spectral properties.” J. Time Series Anal., 30(3), 349–368. ISSN 0143-9782, doi:10.1111/j.1467-9892.2009.00617.x .
Examples
# simulate a 6x6 mc matrix with 2 non-trivial rows
mCompanion("sim", dim = 4, mo = 2)
#> 4 x 4 Matrix of class "MultiCompanion"
#> [,1] [,2] [,3] [,4]
#> [1,] 0.6355033 -0.1410551 -0.08431604 0.0238790
#> [2,] 4.0536568 -0.4811062 -0.73263671 0.1052342
#> [3,] 1.0000000 0.0000000 0.00000000 0.0000000
#> [4,] 0.0000000 1.0000000 0.00000000 0.0000000
# simulate a 6x6 mc matrix with 4 non-trivial rows
mCompanion("sim", dim = 6, mo = 4)
#> 6 x 6 Matrix of class "MultiCompanion"
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] -0.3397734 -4.772355 -2.238653 1.3757630 2.616662 2.144479
#> [2,] 0.7189267 -2.389909 -1.283609 0.5758772 1.343946 1.248197
#> [3,] -0.5484371 4.427777 1.859407 -0.2521208 -1.873112 -1.875732
#> [4,] -1.5138579 -6.183744 -3.174067 2.7274267 3.783945 2.879694
#> [5,] 1.0000000 0.000000 0.000000 0.0000000 0.000000 0.000000
#> [6,] 0.0000000 1.000000 0.000000 0.0000000 0.000000 0.000000
# similar to above but top rows with 2 non-zero columns
mCompanion("sim", dim = 6, mo = 4, mo.col = 2)
#> 6 x 6 Matrix of class "MultiCompanion"
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] -1.127434 0.077625 0 0 0 0
#> [2,] -21.883051 1.049073 0 0 0 0
#> [3,] 34.232417 -2.017858 0 0 0 0
#> [4,] 51.835203 -3.104050 0 0 0 0
#> [5,] 1.000000 0.000000 0 0 0 0
#> [6,] 0.000000 1.000000 0 0 0 0
## specify the non-trivial top rows (as a matrix):
m1 <- matrix(1:24, nrow = 4)
mCompanion(m1) # mc matrix with m1 on top
#> 6 x 6 Matrix of class "MultiCompanion"
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 1 5 9 13 17 21
#> [2,] 2 6 10 14 18 22
#> [3,] 3 7 11 15 19 23
#> [4,] 4 8 12 16 20 24
#> [5,] 1 0 0 0 0 0
#> [6,] 0 1 0 0 0 0
m2 <- rbind(c(1, 2, 0, 0), c(3, 4, 0, 0))
x2a <- mCompanion(m2) # mc matrix with m2 on top
x2a@mo.col # = 4
#> [1] 4
x2 <- mCompanion(m2, mo.col = "detect")
x2@mo.col # = 2, detects the 0 columns in m2
#> [1] 2
mCompanion(m2, mo.col = 2) # same
#> 4 x 4 Matrix of class "MultiCompanion"
#> [,1] [,2] [,3] [,4]
#> [1,] 1 2 0 0
#> [2,] 3 4 0 0
#> [3,] 1 0 0 0
#> [4,] 0 1 0 0
# create manually an mc matrix
(m3 <- rbind(m1, c(1, rep(0, 5)), c(0, 1, rep(0, 4))))
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 1 5 9 13 17 21
#> [2,] 2 6 10 14 18 22
#> [3,] 3 7 11 15 19 23
#> [4,] 4 8 12 16 20 24
#> [5,] 1 0 0 0 0 0
#> [6,] 0 1 0 0 0 0
# turn it into a MultiCompanion object
x3 <- mCompanion(x = m3, detect = "mo")
x3@mo
#> [1] 4
x3 <- mCompanion(m3)
x3@mo
#> [1] 4
m4 <- rbind(c(1, 2, rep(0, 4)), c(3, 4, rep(0, 4)))
x4 <- mCompanion(m4, mo = 2)
x4@mo.col # = 6,
#> [1] 6
## special structure not incorporated in x4,
## eigen and mc_eigen are equiv. in this case
eigen(x4)
#> eigen() decomposition
#> $values
#> [1] 5.3722813 -0.3722813 0.0000000 0.0000000 0.0000000 0.0000000
#>
#> $vectors
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] -0.40871213 -0.10620629 0 0 0.000000e+00 0.000000e+00
#> [2,] -0.89350221 0.07287246 0 0 0.000000e+00 0.000000e+00
#> [3,] -0.07607795 0.28528504 0 0 6.012505e-292 0.000000e+00
#> [4,] -0.16631709 -0.19574567 0 0 0.000000e+00 6.012505e-292
#> [5,] -0.01416120 -0.76631576 1 0 -1.000000e+00 0.000000e+00
#> [6,] -0.03095837 0.52580040 0 1 0.000000e+00 -1.000000e+00
#>
mc_eigen(x4)
#> eigen() decomposition
#> $values
#> [1] 5.3722813 -0.3722813 0.0000000 0.0000000 0.0000000 0.0000000
#>
#> $vectors
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] -0.40871213 -0.10620629 0 0 0.000000e+00 0.000000e+00
#> [2,] -0.89350221 0.07287246 0 0 0.000000e+00 0.000000e+00
#> [3,] -0.07607795 0.28528504 0 0 6.012505e-292 0.000000e+00
#> [4,] -0.16631709 -0.19574567 0 0 0.000000e+00 6.012505e-292
#> [5,] -0.01416120 -0.76631576 1 0 -1.000000e+00 0.000000e+00
#> [6,] -0.03095837 0.52580040 0 1 0.000000e+00 -1.000000e+00
#>
#> $len.block
#> [1] 1 1 1 1 1 1
#>
x4a <- mCompanion(m4, mo = 2, mo.col = 2)
x4a@mo.col # = 2, has Jordan blocks of size > 1
#> [1] 2
## the eigenvectors do not span the space:
eigen(x4a)
#> eigen() decomposition
#> $values
#> [1] 5.3722813 -0.3722813 0.0000000 0.0000000 0.0000000 0.0000000
#>
#> $vectors
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] -0.40871213 -0.10620629 0 0 0.000000e+00 0.000000e+00
#> [2,] -0.89350221 0.07287246 0 0 0.000000e+00 0.000000e+00
#> [3,] -0.07607795 0.28528504 0 0 6.012505e-292 0.000000e+00
#> [4,] -0.16631709 -0.19574567 0 0 0.000000e+00 6.012505e-292
#> [5,] -0.01416120 -0.76631576 1 0 -1.000000e+00 0.000000e+00
#> [6,] -0.03095837 0.52580040 0 1 0.000000e+00 -1.000000e+00
#>
## mc_eigen exploits the Jordan structure, e.g.2x2 Jordan blocks,
## and gives the generalised eigenvectors:
(ev <- mc_eigen(x4a))
#> $values
#> [1] 5.3722813 -0.3722813 0.0000000 0.0000000
#>
#> $vectors
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] -0.41597356 -0.8245648 0 0 0 0
#> [2,] -0.90937671 0.5657675 0 0 0 0
#> [3,] -0.07742959 2.2148971 0 1 0 0
#> [4,] -0.16927198 -1.5197310 0 0 0 1
#> [5,] -0.01441280 -5.9495253 1 0 0 0
#> [6,] -0.03150840 4.0822112 0 0 1 0
#>
#> $len.block
#> [1] 1 1 2 2
#>
x4a %*% ev$vectors
#> 6 x 6 Matrix of class "dgeMatrix"
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] -2.23472698 0.3069701 0 0 0 0
#> [2,] -4.88542751 -0.2106247 0 0 0 0
#> [3,] -0.41597356 -0.8245648 0 0 0 0
#> [4,] -0.90937671 0.5657675 0 0 0 0
#> [5,] -0.07742959 2.2148971 0 1 0 0
#> [6,] -0.16927198 -1.5197310 0 0 0 1
## construct the Jordan matrix of x4a from eigenvalues and eigenvectors
(x4a.j <- Jordan_matrix(ev$values, ev$len.block))
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 5.372281 0.0000000 0 0 0 0
#> [2,] 0.000000 -0.3722813 0 0 0 0
#> [3,] 0.000000 0.0000000 0 1 0 0
#> [4,] 0.000000 0.0000000 0 0 0 0
#> [5,] 0.000000 0.0000000 0 0 0 1
#> [6,] 0.000000 0.0000000 0 0 0 0
## check that AX = XJ and A = XJX^-1, up to numerical precision:
x4a %*% ev$vectors - ev$vectors %*% x4a.j
#> 6 x 6 sparse Matrix of class "dgCMatrix"
#>
#> [1,] -4.440892e-16 . . . . .
#> [2,] . 5.551115e-17 . . . .
#> [3,] . . . . . .
#> [4,] . . . . . .
#> [5,] . . . . . .
#> [6,] . . . . . .
x4a - ev$vectors %*% x4a.j %*% solve(ev$vectors)
#> 6 x 6 sparse Matrix of class "dgCMatrix"
#>
#> [1,] 2.220446e-16 4.440892e-16 . . . 2.960595e-17
#> [2,] . 4.440892e-16 . . . 8.881784e-17
#> [3,] . 9.915992e-17 . . . 2.960595e-17
#> [4,] 3.309340e-17 . . . . -8.960978e-34
#> [5,] 4.440892e-16 -2.220446e-16 . . 3.027881e-18 5.181041e-17
#> [6,] 2.220446e-16 . . . -2.397072e-17 4.440892e-17