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MatrixPolynomialAlgebra[MahlerSystem] - compute the Mahler system of a matrix of polynomials
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Calling Sequence
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MahlerSystem(A, x, vn, vo, returnAll)
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Parameters
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A
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Matrix
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x
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variable name of the polynomial domain
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vn
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list of integers specifying type of Mahler system
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vo
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list of integers specifying order of Mahler system
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returnAll
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(optional) boolean; specify whether to return expression sequence of Mahler system, residual, closest normal point, the order of the Mahler system computed, and a list of indices indicating the nonzero columns of R, or only the Mahler system, residual, and closest normal point
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Description
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The MahlerSystem(A, x, vn, vo) command computes the Mahler system of an m x n rectangular Matrix of univariate polynomials in x over the field of rational numbers Q, or rational expressions over Q (that is, univariate polynomials in x with coefficients in Q(a1,...,an)), its residual R, and its closest normal point v.
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The MahlerSystem(A, x, vn, vo, true) command returns the Mahler system, residual, closest normal point, the order of the Mahler system computed, and a list of indices indicating the nonzero columns of R.
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If M = MahlerSystem(A, x, vn, vo) with the entries of A from ![F[x]](/support/helpjp/helpview.aspx?si=7298/file02409/math116.png) , the columns of M form a module basis for the (mathematical) module
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![{w in F^n[x] | A.w=O(x^vo), degree(w[i])<=vn[i]}](/support/helpjp/helpview.aspx?si=7298/file02409/math124.png)
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in the sense that a module basis consists of ![M[*,i],`...`,(x)^((v)[i]-1) M[*,i]](/support/helpjp/helpview.aspx?si=7298/file02409/math127.png) for  where n is the number of columns of M and v is the closest normal point to vn.
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Examples
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![A := Matrix([[z^5-z^2-1, z^3-2*z^2-1], [z^3-2*z^2+2*z-2, z^3-3*z^2+3*z-4], [z+1, 2-z^3]])](/support/helpjp/helpview.aspx?si=7298/file02409/math165.png)
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![vorder := [3, 5, 4]](/support/helpjp/helpview.aspx?si=7298/file02409/math169.png)
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![M := MahlerSystem(A, z, [1, 3], vorder)](/support/helpjp/helpview.aspx?si=7298/file02409/math173.png)
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![M := Matrix([[-128*z^3, 0], [64*z^3-16*z^4, -128*z^5]])](/support/helpjp/helpview.aspx?si=7298/file02409/math176.png)
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Check the order condition.
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![Matrix([[-128*z^8+64*z^3+96*z^6-16*z^7+16*z^4, -128*z^8+256*z^7+128*z^5], [-16*z^6+16*z^5-16*z^7, -128*z^8+384*z^7-384*z^6+512*z^5], [-160*z^4-64*z^6+16*z^7, -256*z^5+128*z^8]])](/support/helpjp/helpview.aspx?si=7298/file02409/math185.png)
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Return residual and closest normal point.
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![M, R, v, vorder, nonzero := MahlerSystem(A, z, [1, 3], vorder, true)](/support/helpjp/helpview.aspx?si=7298/file02409/math191.png)
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![M, R, v, vorder, nonzero := Matrix([[-128*z^3, 0], [64*z^3-16*z^4, -128*z^5]]), Matrix([[64-128*z^5+96*z^3-16*z^4+16*z, -128*z^5+256*z^4+128*z^2], [16-16*z^2-16*z, 512-128*z^3+384*z^2-384*z], [-160-64*z^2+16*z^3, 128*z^4-256*z]]), Array([3, 5]), Array([3, 5, 4]), [1, 2]](/support/helpjp/helpview.aspx?si=7298/file02409/math194.png)
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Check.
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![W := Matrix(3, 3, proc (i, j) options operator, arrow; if i = j then z^vorder[i] else 0 end if end proc)](/support/helpjp/helpview.aspx?si=7298/file02409/math200.png)
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![Matrix([[0, 0], [0, 0], [0, 0]])](/support/helpjp/helpview.aspx?si=7298/file02409/math207.png)
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References
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Beckermann, B. and Labahn, G. "Fraction-free Computation of Matrix Rational Interpolants and Matrix GCDs." SIAM Journal on Matrix Analysis and Applications. Vol. 22 No. 1, (2000): 114-144.
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