Check if an object is stable
mcStable.RdCheck if an object is stable.
Details
A stable matrix is a matrix all of whose eigenvalues have moduli less than one. Other objects are stable if the associated matrix is stable.
This is a generic function. The default method works as follows.
x is a square matrix, the method checks if its eigenvalues
satisfy the stability condition and returns the result.
Otherwise, if x is a rectangular matrix with more columns than
rows, it is assumed to be the top of a multi-companion matrix. If
x is a vector, it is assumed to represent the top row of a
companion matrix. In all other cases x is converted to matrix
with as.matrix(x). The result should be a square matrix whose
eigenvalues are checked.
It is an error for the matrix to have more rows than columns.
Note
An argument ... may be a good idea since methods may wish to
provide options. For example, for continuous
time systems, the stability condition is that the real parts of the
eigenvalues are negative.
For example, an option to choose the left half-plane for the stable region, instead of the unit circle, would handle stability for continuous time systems.
Examples
## a simulated matrix (it is stable by default)
mc <- mCompanion("sim", dim=4, mo=2)
mcStable(mc)
#> [1] TRUE
## a square matrix
m <- matrix(1:9, nrow=3)
eigen(m)$values
#> [1] 1.611684e+01 -1.116844e+00 -5.700691e-16
mcStable(m)
#> [1] FALSE
## a 2x4 matrix, taken to be the top of an mc matrix
m <- matrix(1:8, nrow=2)
mcStable(m)
#> [1] FALSE
mCompanion(m)
#> 4 x 4 Matrix of class "MultiCompanion"
#> [,1] [,2] [,3] [,4]
#> [1,] 1 3 5 7
#> [2,] 2 4 6 8
#> [3,] 1 0 0 0
#> [4,] 0 1 0 0
## a vector, taken to be the top row of an mc matrix
v <- 1:4
mcStable(v)
#> [1] FALSE
mCompanion(v)
#> 4 x 4 Matrix of class "MultiCompanion"
#> [,1] [,2] [,3] [,4]
#> [1,] 1 2 3 4
#> [2,] 1 0 0 0
#> [3,] 0 1 0 0
#> [4,] 0 0 1 0
abs(mc_eigen(mCompanion(v))$values)
#> [1] 2.517996 1.179913 1.179913 1.141051
co1 <- cbind(c(1,1,1,1), c(0,1,0,0))
## a matrix with eigenvalues equal to 1
mat2 <- make_mcmatrix(eigval = c(1), co = co1, dim = 4, len.block = c(2))
## mat2 is ordinary matrix, eigenvalues are computed numerically
eigen(mat2)
#> eigen() decomposition
#> $values
#> [1] 1 1 0 0
#>
#> $vectors
#> [,1] [,2] [,3] [,4]
#> [1,] 0.5 -0.5 0 0
#> [2,] 0.5 -0.5 0 0
#> [3,] 0.5 -0.5 1 0
#> [4,] 0.5 -0.5 0 1
#>
mcStable(mat2) # FALSE but in general depends on floating point arithmetic
#> [1] FALSE
mat2a <- mCompanion(x="gen", eigval = c(1), co = co1, dim = 4, len.block = c(2), what.res = "list")
mc_eigen(mat2a)
#> $values
#> [1] 1 0 0
#>
#> $vectors
#> [,1] [,2] [,3] [,4]
#> [1,] 1 0 0 0
#> [2,] 1 1 0 0
#> [3,] 1 0 1 0
#> [4,] 1 0 0 1
#>
#> $len.block
#> [1] 2 1 1
#>
mcStable(mat2a)
#> [1] FALSE
mat2b0 <- make_mcmatrix(eigval = c(1), co = co1, dim = 4, len.block = c(2), what = "list")
mat2b <- mCompanion(mat2b0, "gen")
mc_eigen(mat2b)
#> $values
#> [1] 1 0 0
#>
#> $vectors
#> [,1] [,2] [,3] [,4]
#> [1,] 1 0 0 0
#> [2,] 1 1 0 0
#> [3,] 1 0 1 0
#> [4,] 1 0 0 1
#>
#> $len.block
#> [1] 2 1 1
#>
mcStable(mat2b)
#> [1] FALSE
## mat2c is a MultiCompanion object with the eigenvalues stored in it
mat2c <- mCompanion(x="sim", eigval = c(1,0,0), co = cbind(co1, c(0,0,1,0), c(0,0,0,1)),
dim = 4, len.block = c(2,1,1))
mat2c
#> 4 x 4 Matrix of class "MultiCompanion"
#> [,1] [,2] [,3] [,4]
#> [1,] 0 1 0 0
#> [2,] -1 2 0 0
#> [3,] 0 1 0 0
#> [4,] 0 1 0 0
## since the eigenvalues are directly available here, no need to compute them
mc_eigen(mat2c) # contains a 2x2 Jordan block.
#> $values
#> [1] 1 0 0
#>
#> $vectors
#> [,1] [,2] [,3] [,4]
#> [1,] 1 0 0 0
#> [2,] 1 1 0 0
#> [3,] 1 0 1 0
#> [4,] 1 0 0 1
#>
#> $len.block
#> [1] 2 1 1
#>
mcStable(mat2c)
#> [1] FALSE