Left Prime and Left Coprime Polynomials

Check whether a polynomial is left prime or a pair of two polynomials is left coprime. This check uses a (singular) pencil associated with the polynomial(s). For more details see the vignette Technical Details.

Usage

is.coprime(
  a,
  b = NULL,
  tol = sqrt(.Machine$double.eps),
  only.answer = TRUE,
  debug = FALSE
)

Arguments

a, b

If a is an lmfd or an rmfd object, which represents a left matrix fraction description, say \(p^{-1}(z) q(z)\), or a right MFD, say \(r(z) s^{-1}(z)\), then the procedure tests whether the pair \((p(z),q(z))\) or \((t(r(z)),t(q(z)))\)is left coprime.

Otherwise the arguments a and b (if b is not NULL) must represent two compatible polynomial matrices, i.e. a, b must be polm objects (or objects which may be coerced to polm objects). If b is NULL, the procedures checks whether \(a(z)\) is left prime, otherwise the pair \((a(z),b(z))\) is checked for left coprimeness.

tol

a tolerance parameter, which is used to decide the rank of certain matrices.

only.answer

if TRUE, just return a logical (TRUE or FALSE). Otherwise a list with additional information is returned.

debug

if TRUE, print some diagnostic information.

Value

If only.answer is true then a logical (TRUE or FALSE) is returned. Otherwise, a list with the following slots is returned. A more detailed description of these items is given in the vignette Technical Details.

answer

A boolean as above

A,B

These matrices represent the pencil \((A-Bz)\) (in staircase form) which is used to check the left (co-)prime condition.

m,n

Two integer vectors which code the structure of the staircase form.

zeroes

If available, a vector of zeroes of the matrix \((a(z),b(z))\). If \((a,b)\) have no common zeroes (the left coprime case) then zeroes is an empty numeric vector. The case that \((a(z),b(z))\) is rank deficient for all \(z \in C\) is coded with z=NA.

Note

This procedure returns different objects, depending on the parameter only.answer.

Examples

# Ex 1: Two coprime polynomials ##################################################

# Generate two random (2 x 2) polynomial matrices with degree 2
set.seed(1803)
a = test_polm(dim = c(2,2), degree = 2, random = TRUE, digits = 1)
b = test_polm(dim = c(2,2), degree = 2, random = TRUE, digits = 1)

# Output: "only.answer = TRUE"
is.coprime(a, b, debug = FALSE, only.answer = TRUE)
#> [1] TRUE

# Output: "only.answer = FALSE"
out = is.coprime(a, b, debug = FALSE, only.answer = FALSE)
str(out)
#> List of 6
#>  $ answer: logi TRUE
#>  $ A     : num [1:6, 1:8] -1.67e-17 -1.00 0.00 0.00 0.00 ...
#>  $ B     : num [1:6, 1:8] 0 0 0 0 0 0 0 0 0 0 ...
#>  $ zeroes: num(0) 
#>  $ m     : int [1:3] 2 2 2
#>  $ n     : num [1:3] 2 2 4
out$answer
#> [1] TRUE
out$zeroes
#> numeric(0)

# we could equivalently use the syntax: 
is.coprime(cbind(a,b))
#> [1] TRUE
is.coprime(lmfd(a,b))
#> [1] TRUE

# Ex 2: Two non-coprime polynomials with a finite number of common zeros #############
# Dimensions of a, b, and the common factor r
dim = 3
deg_aa = 1
deg_bb = 1
deg_r = 1

# Generate random polynomial matrices
a0 = a
b0 = b
# generate common factor 
r = test_polm(dim = c(2,2), degree = 1, random = TRUE, digits = 1)

# Generate polynomials with a common factor
a = r %r% a0
b = r %r% b0

out = is.coprime(a, b, debug = FALSE, only.answer = FALSE)
out$answer
#> [1] FALSE
out$zeroes
#> [1] -0.8401447  0.5105993


# Ex 3: Two non-coprime polynomials: Everywhere rank deficient ###################

# generate a common factor of rank 1 
r = test_polm(dim = c(2,1), degree = 1, random = TRUE, digits = 1) %r% 
    test_polm(dim = c(1,2), degree = 1, random = TRUE, digits = 1)

# Rank deficient matrices with common factor
a = r %r% a0
b = r %r% b0

out = is.coprime(a,b, only.answer = FALSE)
out$answer
#> [1] FALSE
out$zeroes
#> [1] NA


# Ex 4: Right-MFD ####

c = test_polm(dim = c(2,2), degree = 2, random = TRUE, digits = 1)
d = test_polm(dim = c(2,2), degree = 2, random = TRUE, digits = 1)

# Output: "only.answer = TRUE"
is.coprime(t(c), t(d), debug = FALSE, only.answer = TRUE)
#> [1] TRUE

# Output: "only.answer = FALSE"
out = is.coprime(t(c), t(d), debug = FALSE, only.answer = FALSE)
str(out)
#> List of 6
#>  $ answer: logi TRUE
#>  $ A     : num [1:6, 1:8] -1.00 3.42e-17 0.00 0.00 0.00 ...
#>  $ B     : num [1:6, 1:8] 0 0 0 0 0 0 0 0 0 0 ...
#>  $ zeroes: num(0) 
#>  $ m     : int [1:3] 2 2 2
#>  $ n     : num [1:3] 2 2 4
out$answer
#> [1] TRUE
out$zeroes
#> numeric(0)

# we could equivalently use the syntax: 
is.coprime(rbind(c,d))
#> [1] FALSE
is.coprime(rmfd(c,d))
#> [1] TRUE

# reset seed
set.seed(NULL)