LinearAlgebra[CompanionMatrix] - construct the companion Matrix (pencil) of a monic (or non-monic) polynomial or Matrix polynomial
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Calling Sequence
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CompanionMatrix(P, x, cpt, options)
CompanionMatrix(data1, data2, cpt, options)
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Parameters
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P
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(optional) polynomial or matrix polynomial
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x
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(optional) name; variable to use as the main indeterminate if P is multivariate
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data1
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(optional) list; first coordinate values (nodes) for distinct points in the Lagrange basis which specify the polynomial or matrix polynomial
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data2
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(optional) list; second coordinate values (values) for distinct points in the Lagrange basis which specify the polynomial or matrix polynomial
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cpt
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(optional) BooleanOpt(compact); selects the compact form of the output in the scalar, monic polynomial case
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options
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(optional); constructor options for the result object
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Description
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The CompanionMatrix(P) command returns the companion Matrix associated with the univariate (matrix) polynomial either provided as or specified by distinct data points in the Lagrange basis.
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need not be expressed in the monomial basis; it may be expressed in any of the major orthogonal polynomial bases known to OrthogonalSeries, or in terms of the additional bases BernsteinBasis(k, n, a, b, x) meaning , NewtonBasis(k, [x0, x1, ..., xn], x), or LagrangeBasis(k, [x0, x1, ..., xn], x).
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The CompanionMatrix(P, x) command is used when P is a multivariate polynomial; the main indeterminate of P can be specified by using the optional parameter x.
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If the compact option (cpt) is included in the calling sequence as the symbol compact or in the form compact=true, then the result is built by using a shape function designed to minimize storage. If the option is omitted or entered as compact=false, a full rectangular Matrix is constructed. Generally, if space is not a consideration, the full rectangular form (the default) is more efficient. The compact option does nothing if P is a matrix polynomial.
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Note: If the compact form is selected, any datatype specification in the constructor options is silently ignored. The datatype of any data retrieved from the constructed Matrix is determined by the data used to build the Matrix.
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The constructor options provide additional information (readonly, shape, storage, order, datatype, and attributes) to the Matrix constructor that builds the result. These options may also be provided in the form outputoptions=[...], where [...] represents a Maple list. If a constructor option is provided in both the calling sequence directly and in an outputoptions option, the latter takes precedence (regardless of the order).
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Examples
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Show the Lagrange form of p.
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References
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Corless, Robert M., and Watt, Stephen M. "Bernstein bases are optimal, but, sometimes, Lagrange bases are better." Proceedings of SYNASC. Timisoara. pp. 141-153. MITRON Press, 2004.
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