Some applications to polynomial matrices

pmkstruct(P; CF1, grade=l, fast = false, atol::Real = 0, rtol::Real = atol>0 ? 0 : n*ϵ) -> KRInfo, iz, ip

Determine the Kronecker-structure and infinite pole-zero structure information of the polynomial matrix P(λ) and return an KRInfo object and the multiplicities of infinite zeros and poles. The computation of the Kronecker-structure employs strong linearizations of P(λ) in either the first companion form, if CF1 = true, or the second companion form, if CF1 = false. The effective grade l to be used for linearization can be specified via the keyword argument grade as grade = l, where l must be chosen equal to or greater than the degree of P(λ). The default value used for l is l = deg(P(λ)).

P(λ) can be specified as a grade k polynomial matrix of the form P(λ) = P_1 + λ P_2 + ... + λ**k P_(k+1), for which the coefficient matrices P_i, i = 1, ..., k+1, are stored in the 3-dimensional matrix P, where P[:,:,i] contains the i-th coefficient matrix P_i (multiplying λ**(i-1)).

P(λ) can also be specified as a matrix, vector or scalar of elements of the Polynomial type provided by the Polynomials package.

The information on the Kronecker-structure consists of the right Kronecker indices rki, left Kronecker indices lki, infinite elementary divisors id, the number of finite eigenvalues nf and normal rank nrank and can be obtained from KRInfo as KRInfo.rki, KRInfo.lki, KRInfo.id, KRInfo.nf and KRInfo.nrank, respectively. For more details, see KRInfo.

The determination of the Kronecker-structure information is performed by building a companion form linearization M-λN of P(λ) and reducing the pencil M-λN to an appropriate Kronecker-like form (KLF) which exhibits the number of finite eigenvalues, the multiplicities of the infinite eigenvalues, the left and right Kronecker indices and the normal rank. The Kronecker-structure information and pole-zero stucture information on P(λ) are recovered from the Kronecker-structure information on M-λN using the results of [1] and [2].

The right Kronecker indices are provided in the integer vector rki. The number of elements of rki is the dimension of the right nullspace of the polynomial matrix P(λ) and their sum is the least degree of a right polynomial nullspace basis.

The left Kronecker indices are provided in the integer vector lki. The number of elements of lki is the dimension of the left nullspace of the polynomial matrix P(λ) and their sum is the least degree of a left polynomial nullspace basis.

The multiplicities of infinite eigenvalues are provided in the integer vector id, where each i-th element id[i] is the order of an infinite elementary divisor (i.e., the multiplicity of an infinite eigenvalue).

The multiplicities of the infinite zeros of P(λ) are returned in iz and represent the positive differences between the multiplicities of the infinite eigenvalues of P(λ) and the effective grade l of P(λ) [2].

The multiplicities of the infinite poles of P(λ) are returned in ip and represent the absolute values of the negative differences between the multiplicities of the infinite eigenvalues of P(λ) and the effective grade l of P(λ) [2].

The reduction to the KLF is performed using orthogonal similarity transformations and involves rank decisions based on rank revealing QR-decompositions with column pivoting, if fast = true, or, the more reliable, SVD-decompositions, if fast = false.

The keyword arguments atol and rtol specify the absolute and relative tolerances for the nonzero coefficients of P(λ), respectively. The default relative tolerance is n*ϵ, where n is the size of the smallest dimension of P(λ), and ϵ is the machine epsilon of the element type of coefficients of P(λ).

[1] F. De Terán, F. M. Dopico, D. S. Mackey, Spectral equivalence of polynomial matrices and the Index Sum Theorem, Linear Algebra and Its Applications, vol. 459, pp. 264-333, 2014.

[2] A. Varga, On computing the Kronecker structure of polynomial matrices using Julia, June 2020, arXiv:2006.06825.

pmeigvals(P; CF1, grade = l, fast = false, atol::Real = 0, rtol::Real = atol>0 ? 0 : n*ϵ) -> (val, KRInfo)

Return the finite and infinite eigenvalues of the polynomial matrix P(λ) in val and information on the Kronecker-structure of P(λ) in the KRInfo object. The computation of eigenvalues and Kronecker-structure employs strong linearizations of P(λ) in either the first companion form, if CF1 = true, or the second companion form, if CF1 = false. The effective grade l to be used for linearization can be specified via the keyword argument grade as grade = l, where l must be chosen equal to or greater than the degree of P(λ).

P(λ) can be specified as a grade k polynomial matrix of the form P(λ) = P_1 + λ P_2 + ... + λ**k P_(k+1), for which the coefficient matrices P_i, i = 1, ..., k+1, are stored in the 3-dimensional matrix P, where P[:,:,i] contains the i-th coefficient matrix P_i (multiplying λ**(i-1)).

P(λ) can also be specified as a matrix, vector or scalar of elements of the Polynomial type provided by the Polynomials package.

The information on the Kronecker-structure consists of the right Kronecker indices rki, left Kronecker indices lki, infinite elementary divisors id, the number of finite eigenvalues nf and normal rank nrank and can be obtained from KRInfo as KRInfo.rki, KRInfo.lki, KRInfo.id, KRInfo.nf and KRInfo.nrank, respectively. For more details, see KRInfo.

The computation of the eigenvalues is performed by building the companion form based linearization M-λN of the polynomial matrix P(λ) and then reducing the pencil M-λN to an appropriate Kronecker-like form (KLF) which exhibits the number of finite eigenvalues, the multiplicities of the infinite eigenvalues, the left and right Kronecker indices and the normal rank. The left and right Kronecker indices of P(λ) are returned in KRInfo.rki and KRInfo.rki, respectively, and their values are recovered from the left and right Kronecker indices of M-λN using the results of [1]. The multiplicities of the infinite eigenvalues of P(λ) of effective grade l is returned in KRInfo.id. The number of finite eigenvalues in val is equal to the number of finite eigenvalues of M-λN (returned in KRInfo.nf), while the number of infinite eigenvalues in val is the sum of multiplicites returned in KRInfo.id.

The reduction of M-λN to the KLF is performed using orthonal similarity transformations and involves rank decisions based on rank revealing QR-decompositions with column pivoting, if fast = true, or, the more reliable, SVD-decompositions, if fast = false. For efficiency purposes, the reduction is only partially performed, without accumulating the performed orthogonal transformations.

The keyword arguments atol and rtol specify the absolute and relative tolerances for the nonzero coefficients of P(λ), respectively. The default relative tolerance is n*ϵ, where n is the size of the smallest dimension of P(λ), and ϵ is the machine epsilon of the element type of coefficients of P(λ).

[1] F. De Terán, F. M. Dopico, D. S. Mackey, Spectral equivalence of polynomial matrices and the Index Sum Theorem, Linear Algebra and Its Applications, vol. 459, pp. 264-333, 2014.

pmzeros(P; CF1, fast = false, atol::Real = 0, rtol::Real = atol>0 ? 0 : n*ϵ) -> (val, iz, KRInfo)

Return the finite and infinite zeros of the polynomial matrix P(λ) in val, the multiplicities of infinite zeros in iz and information on the Kronecker-structure of P(λ) in the KRInfo object. The computation of zeros and Kronecker-structure employs strong linearizations of P(λ) in either the first companion form, if CF1 = true, or the second companion form, if CF1 = false.

P(λ) can be specified as a grade k polynomial matrix of the form P(λ) = P_1 + λ P_2 + ... + λ**k P_(k+1), for which the coefficient matrices P_i, i = 1, ..., k+1, are stored in the 3-dimensional matrix P, where P[:,:,i] contains the i-th coefficient matrix P_i (multiplying λ**(i-1)).

P(λ) can also be specified as a matrix, vector or scalar of elements of the Polynomial type provided by the Polynomials package.

The information on the Kronecker-structure consists of the right Kronecker indices rki, left Kronecker indices lki, infinite elementary divisors id, the number of finite eigenvalues nf, the normal rank nrank, and can be obtained from KRInfo as KRInfo.rki, KRInfo.lki, KRInfo.id, KRInfo.nf and KRInfo.nrank, respectively. For more details, see KRInfo.

The computation of the zeros is performed by building the companion form based linearization M-λN of the polynomial matrix P(λ) and then reducing the pencil M-λN to an appropriate Kronecker-like form (KLF) which exhibits information on its Kronecker structure. The left and right Kronecker indices of P(λ) are returned in KRInfo.rki and KRInfo.rki, respectively, and their values are recovered from the left and right Kronecker indices of M-λN using the results of [1]. The multiplicities of the infinite zeros of P(λ), returned in iz, are the positive differences between the multiplicities of the infinite eigenvalues of M-λN and the degree of P(λ) [2]. The number of finite zeros in val is equal to the number of finite eigenvalues of M-λN (returned in KRInfo.nf), while the number of infinite zeros in val is the sum of multiplicites in iz.

The reduction of M-λN to the KLF is performed using orthonal similarity transformations and involves rank decisions based on rank revealing QR-decompositions with column pivoting, if fast = true, or, the more reliable, SVD-decompositions, if fast = false. For efficiency purposes, the reduction is only partially performed, without accumulating the performed orthogonal transformations.

The keyword arguments atol and rtol specify the absolute and relative tolerances for the nonzero coefficients of P(λ), respectively. The default relative tolerance is n*ϵ, where n is the size of the smallest dimension of P(λ), and ϵ is the machine epsilon of the element type of coefficients of P(λ).

[1] F. De Terán, F. M. Dopico, D. S. Mackey, Spectral equivalence of polynomial matrices and the Index Sum Theorem, Linear Algebra and Its Applications, vol. 459, pp. 264-333, 2014.

[2] A. Varga, On computing the Kronecker structure of polynomial matrices using Julia, June 2020, arXiv:2006.06825.

pmzeros1(P; fast = false, atol::Real = 0, rtol::Real = atol>0 ? 0 : n*ϵ) -> (val, iz, KRInfo)

Return the finite and infinite zeros of the polynomial matrix P(λ) in val, the multiplicities of infinite zeros in iz and information on the Kronecker-structure of the underlying strongly irreducible pencil based linearization of P(λ) in the KRInfo object.

The information on the Kronecker-structure of the underlying linearization consists of the right Kronecker indices rki (the same as of P(λ)), left Kronecker indices lki (the same as of P(λ)), infinite elementary divisors id, the number of finite eigenvalues nf (the same as of P(λ)) and the normal rank 'nrank', and can be obtained from KRInfo as KRInfo.rki, KRInfo.lki, KRInfo.id, KRInfo.nf and KRInfo.nrank, respectively. For more details, see KRInfo.

P(λ) can be specified as a grade k polynomial matrix of the form P(λ) = P_1 + λ P_2 + ... + λ**k P_(k+1), for which the coefficient matrices P_i, i = 1, ..., k+1, are stored in the 3-dimensional matrix P, where P[:,:,i] contains the i-th coefficient matrix P_i (multiplying λ**(i-1)).

P(λ) can also be specified as a matrix, vector or scalar of elements of the Polynomial type provided by the Polynomials package.

The computation of the zeros is performed by first building a strongly irreducible pencil based linearization of P(λ) [1] of order ν as a structured system matrix pencil

          | A-λE | B-λF | 
 M - λN = |------|------| ,
          | C-λG | D-λH |

such that P(λ) = (C-λG)*inv(λE-A)*(B-λF)+D-λH, and then reducing the pencil M-λN to an appropriate Kronecker-like form (KLF) which exhibits information on its Kronecker structure. The left and right Kronecker indices of M-λN and P(λ) are the same [2] and are returned in KRInfo.rki and KRInfo.rki, respectively. The multiplicities of the infinite zeros of M-λN and of P(λ) are the same [2] and are returned in iz. The number of finite zeros in val is equal to the number of finite eigenvalues of M-λN (returned in KRInfo.nf), while the number of infinite zeros in val is the sum of multiplicites in iz. The normal rank of P(λ) can be evaluated as KRInfo.nrank - ν.

The reduction of M-λN to the KLF is performed using orthonal similarity transformations and involves rank decisions based on rank revealing QR-decompositions with column pivoting, if fast = true, or, the more reliable, SVD-decompositions, if fast = false. For efficiency purposes, the reduction is only partially performed, without accumulating the performed orthogonal transformations.

The keyword arguments atol and rtol specify the absolute and relative tolerances for the nonzero coefficients of P(λ), respectively. The default relative tolerance is n*ϵ, where n is the size of the smallest dimension of P(λ), and ϵ is the machine epsilon of the element type of coefficients of P(λ).

[1] F.M. Dopico, M.C. Quintana and P. Van Dooren, Linear system matrices of rational transfer functions, to appear in "Realization and Model Reduction of Dynamical Systems", A Festschrift to honor the 70th birthday of Thanos Antoulas", Springer-Verlag. arXiv:1903.05016

[2] G. Verghese, Comments on ‘Properties of the system matrix of a generalized state-space system’, Int. J. Control, Vol.31(5) (1980) 1007–1009.

  pmzeros2(P; fast = false, atol::Real = 0, rtol::Real = atol>0 ? 0 : n*ϵ) -> (val, iz, KRInfo)

Return the finite and infinite zeros of the polynomial matrix P(λ) in val, the multiplicities of infinite zeros in iz and information on the Kronecker-structure of the underlying irreducible descriptor system based linearization of P(λ) in the KRInfo object.

The information on the Kronecker-structure of the underlying linearization consists of the right Kronecker indices rki (the same as of P(λ)), left Kronecker indices lki (the same as of P(λ)), infinite elementary divisors id, the number of finite eigenvalues nf (the same as of P(λ)) and the normal rank 'nrank', and can be obtained from KRInfo as KRInfo.rki, KRInfo.lki, KRInfo.id, KRInfo.nf and KRInfo.nrank, respectively. For more details, see KRInfo.

P(λ) can be specified as a grade k polynomial matrix of the form P(λ) = P_1 + λ P_2 + ... + λ**k P_(k+1), for which the coefficient matrices P_i, i = 1, ..., k+1, are stored in the 3-dimensional matrix P, where P[:,:,i] contains the i-th coefficient matrix P_i (multiplying λ**(i-1)).

P(λ) can also be specified as a matrix, vector or scalar of elements of the Polynomial type provided by the Polynomials package.

The computation of the zeros is performed by first building an irreducible descriptor system based linearization of P(λ) [1] of order ν as a structured system matrix pencil

          | A-λE | B | 
 M - λN = |------|---| ,
          |  C   | D |

such that P(λ) = C*inv(λE-A)*B+D, and then reducing the pencil M-λN to an appropriate Kronecker-like form (KLF) which exhibits information on its Kronecker structure. The left and right Kronecker indices of M-λN and P(λ) are the same [2] and are returned in KRInfo.rki and KRInfo.rki, respectively. The multiplicities of the infinite zeros of M-λN and of P(λ) are the same [2] and are returned in iz. The number of finite zeros in val is equal to the number of finite eigenvalues of M-λN (returned in KRInfo.nf), while the number of infinite zeros in val is the sum of multiplicites in iz. The normal rank of P(λ) can be evaluated as KRInfo.nrank - ν.

The reduction of M-λN to the KLF is performed using orthonal similarity transformations and involves rank decisions based on rank revealing QR-decompositions with column pivoting, if fast = true, or, the more reliable, SVD-decompositions, if fast = false. For efficiency purposes, the reduction is only partially performed, without accumulating the performed orthogonal transformations.

The keyword arguments atol and rtol specify the absolute and relative tolerances for the nonzero coefficients of P(λ), respectively. The default relative tolerance is n*ϵ, where n is the size of the smallest dimension of P(λ), and ϵ is the machine epsilon of the element type of coefficients of P(λ).

[1] G. Verghese, B. Levy, and T. Kailath, Generalized state-space systems, IEEE Trans. Automat. Control, 26:811-831 (1981).

[2] G. Verghese, Comments on ‘Properties of the system matrix of a generalized state-space system’, Int. J. Control, Vol.31(5) (1980) 1007–1009.

pmroots(P; fast = false, atol::Real = 0, rtol::Real = atol>0 ? 0 : n*ϵ) -> val

Compute the roots of the determinant of the regular polynomial matrix P(λ) in val.

P(λ) can be specified as a grade k polynomial matrix of the form P(λ) = P_1 + λ P_2 + ... + λ**k P_(k+1), for which the coefficient matrices P_i, i = 1, ..., k+1, are stored in the 3-dimensional matrix P, where P[:,:,i] contains the i-th coefficient matrix P_i (multiplying λ**(i-1)).

P(λ) can also be specified as a matrix, vector or scalar of elements of the Polynomial type provided by the Polynomials package.

The roots of det(P(λ)) are computed as the finite eigenvalues of a companion form linearization M-λNof P(λ). The finite eigenvalues are computed by reducing the pencil M-λN to an appropriate Kronecker-like form (KLF) exhibiting the spliting of the infinite and finite eigenvalue structures of the pencil M-λN. The reduction is performed using orthonal similarity transformations and involves rank decisions based on rank revealing QR-decompositions with column pivoting, if fast = true, or, the more reliable, SVD-decompositions, if fast = false. For efficiency purposes, the reduction is only partially performed, without accumulating the performed orthogonal transformations.

The keyword arguments atol and rtol specify the absolute and relative tolerances for the nonzero coefficients of P(λ), respectively. The default relative tolerance is n*ϵ, where n is the size of the smallest dimension of P(λ), and ϵ is the machine epsilon of the element type of coefficients of P(λ).

pmpoles(P; CF1, fast = false, atol::Real = 0, rtol::Real = atol>0 ? 0 : n*ϵ) -> (val, ip, id)

Return the finite and infinite poles of the polynomial matrix P(λ) in val, the multiplicities of infinite poles in ip and the infinite elementary divisors of Q(λ) = [P(λ) I; I 0] in id. The computation of pole-structure employs strong linearizations of Q(λ) in either the first companion form, if CF1 = true, or the second companion form, if CF1 = false.

P(λ) can be specified as a grade k polynomial matrix of the form P(λ) = P_1 + λ P_2 + ... + λ**k P_(k+1), for which the coefficient matrices P_i, i = 1, ..., k+1, are stored in the 3-dimensional matrix P, where P[:,:,i] contains the i-th coefficient matrix P_i (multiplying λ**(i-1)).

P(λ) can also be specified as a matrix, vector or scalar of elements of the Polynomial type provided by the Polynomials package.

The information on the finite and infinite poles of the polynomial matrix P(λ) is obtained from the Kronecker-structure information of the underlying linearization Q(λ) using the results of [1] and [2]. The computation of the poles is performed by building the companion form based linearization M-λN of the extended polynomial matrix Q(λ) = [P(λ) I; I 0] (see [1]) and then reducing the pencil M-λN to an appropriate Kronecker-like form (KLF) which exhibits information on its Kronecker structure. Since Q(λ) is regular, M-λN has no left or right Kronecker indices. The multiplicities of the infinite poles of P(λ), returned in ip, are the positive differences between the multiplicities of the infinite eigenvalues of M-λN and the degree of P(λ) [2]. The number of finite poles in val is equal to the number of finite eigenvalues of M-λN (returned in KRInfo.nf), while the number of infinite poles in val is the sum of multiplicites in ip. The multiplicities of the infinite eigenvalues of Q(λ) are returned in id.

The reduction of M-λN to the KLF is performed using orthonal similarity transformations and involves rank decisions based on rank revealing QR-decompositions with column pivoting, if fast = true, or, the more reliable, SVD-decompositions, if fast = false. For efficiency purposes, the reduction is only partially performed, without accumulating the performed orthogonal transformations.

The keyword arguments atol and rtol specify the absolute and relative tolerances for the nonzero coefficients of P(λ), respectively. The default relative tolerance is n*ϵ, where n is the size of the smallest dimension of P(λ), and ϵ is the machine epsilon of the element type of coefficients of P(λ).

[1] F. De Terán, F. M. Dopico, D. S. Mackey, Spectral equivalence of polynomial matrices and the Index Sum Theorem, Linear Algebra and Its Applications, vol. 459, pp. 264-333, 2014.

[2] A. Varga, On computing the Kronecker structure of polynomial matrices using Julia, June 2020, arXiv:2006.06825.

pmpoles1(P; fast = false, atol::Real = 0, rtol::Real = atol>0 ? 0 : n*ϵ) -> (val, ip, id)

Return the infinite poles of the polynomial matrix P(λ) in val, the multiplicities of infinite poles in ip and the infinite elementary divisors of the pole pencil Sp(λ) of the underlying strongly irreducible pencil based linearization of Q(λ) := [P(λ) I; I 0] in id.

P(λ) can be specified as a grade k polynomial matrix of the form P(λ) = P_1 + λ P_2 + ... + λ**k P_(k+1), for which the coefficient matrices P_i, i = 1, ..., k+1, are stored in the 3-dimensional matrix P, where P[:,:,i] contains the i-th coefficient matrix P_i (multiplying λ**(i-1)).

P(λ) can also be specified as a matrix, vector or scalar of elements of the Polynomial type provided by the Polynomials package.

The information on the infinite poles of the polynomial matrix P(λ) is obtained from the pole-structure information of the underlying pencil based linearization using the results of [1] and [2]. The determination of the pole-structure information is performed by building a strongly minimal pencil based linearization (A-λE,B-λF,C-λG,D-λH) with A-λE a regular pencil, satisfying P(λ) = (C-λG)*inv(λE-A)*(B-λF)+D-λH and reducing the pole system matrix pencil Sp(λ) = [A-λE -λF; -λG -λH] to an appropriate Kronecker-like form (KLF) which exhibits the multiplicities of the infinite eigenvalues (in excess with one to the multiplicities of infinite poles of P(λ)). The multiplicities of the infinite zeros of Sp(λ) and of infinite poles of P(λ) are the same [2] and are returned in ip. The number of infinite poles in val is equal to the sum of multiplicites in ip. The multiplicities of the infinite eigenvalues of Sp(λ) are returned in id.

The reduction of Sp(λ) to the KLF is performed using orthonal similarity transformations and involves rank decisions based on rank revealing QR-decompositions with column pivoting, if fast = true, or, the more reliable, SVD-decompositions, if fast = false. For efficiency purposes, the reduction is only partially performed, without accumulating the performed orthogonal transformations.

The keyword arguments atol and rtol specify the absolute and relative tolerances for the nonzero coefficients of P(λ), respectively. The default relative tolerance is n*ϵ, where n is the size of the smallest dimension of P(λ), and ϵ is the machine epsilon of the element type of coefficients of P(λ).

[1] F.M. Dopico, M.C. Quintana and P. Van Dooren, Linear system matrices of rational transfer functions, to appear in "Realization and Model Reduction of Dynamical Systems", A Festschrift to honor the 70th birthday of Thanos Antoulas", Springer-Verlag. arXiv:1903.05016

[2] G. Verghese, Comments on ‘Properties of the system matrix of a generalized state-space system’, Int. J. Control, Vol.31(5) (1980) 1007–1009.

pmpoles2(P; fast = false, atol::Real = 0, rtol::Real = atol>0 ? 0 : n*ϵ) -> (val, ip, id)

Return the infinite poles of the polynomial matrix P(λ) in val, the multiplicities of infinite poles in ip and the infinite elementary divisors of the pole pencil of the underlying irreducible descriptor system based linearization of P(λ) in id.

P(λ) can be specified as a grade k polynomial matrix of the form P(λ) = P_1 + λ P_2 + ... + λ**k P_(k+1), for which the coefficient matrices P_i, i = 1, ..., k+1, are stored in the 3-dimensional matrix P, where P[:,:,i] contains the i-th coefficient matrix P_i (multiplying λ**(i-1)).

P(λ) can also be specified as a matrix, vector or scalar of elements of the Polynomial type provided by the Polynomials package.

The information on the infinite poles of the polynomial matrix P(λ) is obtained from the pole-structure information of the underlying linearization using the results of [1] and [2]. The determination of the pole-structure information is performed by building an irreducible descriptor system realization (A-λE,B,C,D) with A-λE unimodular, satisfying R(λ) = C*inv(λE-A)*B+D and reducing the pole pencil A-λE to an appropriate Kronecker-like form (KLF) which exhibits the multiplicities of the infinite eigenvalues (in excess with one to the multiplicities of infinite poles of P(λ)). The multiplicities of the infinite zeros of A-λE and of infinite poles of P(λ) are the same [2] and are returned in ip. The number of infinite poles in val is equal to the sum of multiplicites in ip. The multiplicities of the infinite eigenvalues of A-λE are returned in id.

The reduction of A-λE to the KLF is performed using orthonal similarity transformations and involves rank decisions based on rank revealing QR-decompositions with column pivoting, if fast = true, or, the more reliable, SVD-decompositions, if fast = false. For efficiency purposes, the reduction is only partially performed, without accumulating the performed orthogonal transformations.

The keyword arguments atol and rtol specify the absolute and relative tolerances for the nonzero coefficients of P(λ), respectively. The default relative tolerance is n*ϵ, where n is the size of the smallest dimension of P(λ), and ϵ is the machine epsilon of the element type of coefficients of P(λ).

[1] G. Verghese, B. Levy, and T. Kailath, Generalized state-space systems, IEEE Trans. Automat. Control, 26:811-831 (1981).

[2] G. Verghese, Comments on ‘Properties of the system matrix of a generalized state-space system’, Int. J. Control, Vol.31(5) (1980) 1007–1009.

   pmrank(P; fastrank = true, atol = 0, rtol = atol > 0 ? 0 : n*ϵ)

Determine the normal rank of a polynomial matrix P(λ).

P(λ) is a grade k polynomial matrix of the form P(λ) = P_1 + λ P_2 + ... + λ**k P_(k+1), for which the coefficient matrices P_i, i = 1, ..., k+1, are stored in the 3-dimensional matrix P, where P[:,:,i] contains the i-th coefficient matrix P_i (multiplying λ**(i-1)). The normal rank of P(λ) is the number of linearly independent rows or columns.

P(λ) can also be specified as a matrix, vector or scalar of elements of the Polynomial type provided by the Polynomials package.

If fastrank = true, the rank is evaluated by counting how many singular values of P(γ) have magnitude greater than max(atol, rtol*σ₁), where σ₁ is the largest singular value of P(γ) and γ is a randomly generated complex value of magnitude equal to one. If fastrank = false, first a structured linearization of P(λ) is built in the form [A-λE B; C D] with A-λE an order n regular subpencil, and then its normal rank nr is determined. The normal rank of P(λ) is nr - n.

The keyword arguments atol and rtol, specify, respectively, the absolute and relative tolerance for the nonzero coefficients of P(λ). The default relative tolerance is n*ϵ, where n is the size of the smallest dimension of P(λ) and ϵ is the machine epsilon of the element type of its coefficients.

ispmregular(P; fastrank = true, atol::Real = 0, rtol::Real = atol>0 ? 0 : n*ϵ) -> Bool

Test whether the polynomial matrix P(λ) is regular (i.e., P(λ) is square and ${\small\det(P(λ)) \not\equiv 0}$). The underlying computational procedure checks that the normal rank of the square P(λ) is equal to its order.

P(λ) is a grade k polynomial matrix of the form P(λ) = P_1 + λ P_2 + ... + λ**k P_(k+1), for which the coefficient matrices P_i, i = 1, ..., k+1, are stored in the 3-dimensional matrix P, where P[:,:,i] contains the i-th coefficient matrix P_i (multiplying λ**(i-1)).

P(λ) can also be specified as a matrix, vector or scalar of elements of the Polynomial type provided by the Polynomials package. f If fastrank = true, the rank is evaluated by counting how many singular values of P(γ) have magnitude greater than max(atol, rtol*σ₁), where σ₁ is the largest singular value of P(γ) and γ is a randomly generated complex value of magnitude equal to one. If fastrank = false, first a linearization of P(λ) is built in a companion form M-λN of order n and then its normal rank nr is determined. P(λ) is regular if nr = n.

The keyword arguments atol and rtol, specify, respectively, the absolute and relative tolerance for the nonzero coefficients of P(λ). The default relative tolerance is n*ϵ, where n is the size of the smallest dimension of P(λ) and ϵ is the machine epsilon of the element type of its coefficients.

ispmunimodular(P; fastrank = true, atol::Real = 0, rtol::Real = atol>0 ? 0 : n*ϵ) -> Bool

Test whether the polynomial matrix P(λ) is unimodular (i.e., P(λ) is square, regular and det(P(λ)) == constant).

The underlying computational procedure checks that P(λ) is square and its first companion form linearization is regular and has no finite eigenvalues.

P(λ) is a grade k polynomial matrix of the form P(λ) = P_1 + λ P_2 + ... + λ**k P_(k+1), for which the coefficient matrices P_i, i = 1, ..., k+1, are stored in the 3-dimensional matrix P, where P[:,:,i] contains the i-th coefficient matrix P_i (multiplying λ**(i-1)).

P(λ) can also be specified as a matrix, vector or scalar of elements of the Polynomial type provided by the Polynomials package.

The keyword arguments atol and rtol, specify, respectively, the absolute and relative tolerance for the nonzero coefficients of P(λ). The default relative tolerance is n*ϵ, where n is the size of the smallest dimension of P(λ) and ϵ is the machine epsilon of the element type of its coefficients.