Prune (Laurent) Matrix Polynomial

Performs three steps to simplify a matrix polynomial.

1. All leading coefficients where the absolute values of the real and the imaginary parts are less than or equal to tol are set to zero. For Laurent polynomials, the same is happening from the other direction.

2. The zero leading coefficient matrices are dropped. For Laurent polynomials, the same is happening for the coefficient matrices pertaining to low powers.

3. If all the absolute values of the imaginary parts of the coefficients are less than or equal to tol then the coefficients are set to real values.

Empty polynomial matrices (i.e. matrices with zero rows or columns) are set to polynomial of zero degree.

Usage

prune(x, tol = sqrt(.Machine$double.eps), brutal = FALSE)

Arguments

x

polm or lpolm object.

tol

Double. Tolerance parameter. Default set to sqrt(.Machine$double.eps).

brutal

Boolean. Default set to FALSE. If TRUE, all small elements are set to zero (irrespective of whether they are leading coefficients or not).

Value

A Laurent matrix polynomial, i.e. a polm or lpolm object.

Examples

x = polm(array(c(1,0,0,0,
                 0,1,0,0,
                 0,0,1,0,
                 0,0,0,1,
                 0,0,0,0), dim = c(2,2,5)) + 1e-4)
x
#> ( 2 x 2 ) matrix polynomial with degree <= 4 
#>      z^0 [,1]  [,2] z^1 [,1]  [,2] z^2 [,1]   [,2] z^3 [,1]   [,2] z^4 [,1]
#> [1,]   1.0001 1e-04   0.0001 1e-04    1e-04 1.0001    1e-04 0.0001    1e-04
#> [2,]   0.0001 1e-04   1.0001 1e-04    1e-04 0.0001    1e-04 1.0001    1e-04
#>       [,2]
#> [1,] 1e-04
#> [2,] 1e-04
prune(x, tol = 1e-3)
#> ( 2 x 2 ) matrix polynomial with degree <= 3 
#>      z^0 [,1]  [,2] z^1 [,1]  [,2] z^2 [,1]   [,2] z^3 [,1]   [,2]
#> [1,]   1.0001 1e-04   0.0000 1e-04        0 1.0001        0 0.0000
#> [2,]   0.0001 1e-04   1.0001 1e-04        0 0.0001        0 1.0001
prune(x, tol = 1e-3, brutal = TRUE)
#> ( 2 x 2 ) matrix polynomial with degree <= 3 
#>      z^0 [,1]  [,2] z^1 [,1]  [,2] z^2 [,1]   [,2] z^3 [,1]   [,2]
#> [1,]   1.0001     0   0.0000     0        0 1.0001        0 0.0000
#> [2,]   0.0000     0   1.0001     0        0 0.0000        0 1.0001

# Case of complex variables:
x = x + complex(imaginary = 1e-5)
x
#> ( 2 x 2 ) matrix polynomial with degree <= 4 
#>             z^0 [,1]         [,2]  z^1 [,1]     [,2] z^2 [,1]      [,2]
#> [1,] 1.0001+0.00001i 1e-04+1e-05i 0.0001+0i 1e-04+0i 1e-04+0i 1.0001+0i
#> [2,] 0.0001+0.00001i 1e-04+1e-05i 1.0001+0i 1e-04+0i 1e-04+0i 0.0001+0i
#>      z^3 [,1]      [,2] z^4 [,1]     [,2]
#> [1,] 1e-04+0i 0.0001+0i 1e-04+0i 1e-04+0i
#> [2,] 1e-04+0i 1.0001+0i 1e-04+0i 1e-04+0i
prune(x, tol = 1e-3)
#> ( 2 x 2 ) matrix polynomial with degree <= 3 
#>      z^0 [,1]  [,2] z^1 [,1]  [,2] z^2 [,1]   [,2] z^3 [,1]   [,2]
#> [1,]   1.0001 1e-04   0.0000 1e-04        0 1.0001        0 0.0000
#> [2,]   0.0001 1e-04   1.0001 1e-04        0 0.0001        0 1.0001

# also works for constant matrix polynomials (i.e. matrices)
x = polm(array(0:3, dim = c(2,2,1))+1e-4)
x
#> ( 2 x 2 ) matrix polynomial with degree <= 0 
#>      z^0 [,1]   [,2]
#> [1,]   0.0001 2.0001
#> [2,]   1.0001 3.0001
prune(x, tol = 1e-3)
#> ( 2 x 2 ) matrix polynomial with degree <= 0 
#>      z^0 [,1]   [,2]
#> [1,]   0.0000 2.0001
#> [2,]   1.0001 3.0001

# empty polynomials are coerced to polynomials of degree zero 
x = polm(array(0, dim = c(0,2,5)))
x
#> ( 0 x 2 ) matrix polynomial with degree <= 4 
prune(x)
#> ( 0 x 2 ) matrix polynomial with degree <= -1 

# Laurent polynomials:
(lp = lpolm(c(0, 1:3, 0), min_deg = 2))
#> ( 1 x 1 ) Laurent polynomial matrix with degree <= 6, and minimal degree >= 2
#>      z^2 [,1] z^3 [,1] z^4 [,1] z^5 [,1] z^6 [,1]
#> [1,]        0        1        2        3        0
prune(lp)
#> ( 1 x 1 ) Laurent polynomial matrix with degree <= 5, and minimal degree >= 3
#>      z^3 [,1] z^4 [,1] z^5 [,1]
#> [1,]        1        2        3