convert/MatrixPolynomialObject
convert a matrix polynomial or scalar polynomial to a standard internal representation
type/MatrixPolynomialObject
test for a MatrixPolynomialObject
Calling Sequence
Parameters
Description
Examples
convert(p, MatrixPolynomialObject, x)
convert(values, MatrixPolynomialObject, nodes)
type(expr, MatrixPolynomialObject, x)
p
-
polynomial expressed in any of a number of polynomial bases
x
name; the variable for the polynomial
values
list of values of the (matrix or scalar) polynomial p at the (distinct) nodes
nodes
list of algebraic expressions representing distinct scalar nodes
expr
arbitrary Maple object
The convert(p, MatrixPolynomialObject, x) function converts the (matrix or scalar) polynomial p into a standard representation, a Record. This allows systematic (conventional) access to the polynomial properties, such as Degree, in a manner independent of the polynomial basis. The bases understood by MatrixPolynomialObject include:
BernsteinBasis
ChebyshevT
ChebyshevU
GegenbauerC
JacobiP
LagrangeBasis
NewtonBasis
PochhammerBasis
and most others understood by the OrthogonalSeries package. This routine is used internally by LinearAlgebra[CompanionMatrix].
If the input polynomial p contains more than one basis, then this (heuristic) conversion will fail.
The type(expr, MatrixPolynomialObject) function checks whether expr is a Record of the type returned by convert(...,MatrixPolynomialObject).
A MatrixPolynomialObject record has the following fields:
Basis - the name of the basis used; either PowerBasis or any of the supported basis names listed above.
BasisParameters - a list of the parameters of the particular basis; e.g. for LagrangeBasis or NewtonBasis these are the nodes; for BernsteinBasis these are the degree n and the left and right ends a and b of the interval.
Coefficient - a procedure to return a specific coefficient matrix. It takes as argument a nonnegative integer less or equal to Degree and returns a Matrix.
Degree - a nonnegative integer; the degree of the polynomial (in the LagrangeBasis or BernsteinBasis case, an upper bound on the degree).
Dimension - a positive integer; the matrix dimension of the matrix polynomial ( if the original polynomial is a scalar polynomial).
IsMonic - a procedure without arguments returning true or false, depending on whether the polynomial is known to be monic (not relevant for Lagrange or Bernstein bases).
OutputOptions - a list of output options for the coefficient Matrices (see MatrixOptions).
Value - a procedure to evaluate the polynomial at any point. It takes as an argument the point (an algebraic expression) and returns a Matrix.
Variable - a name; the original variable used to define the polynomial (which may be unspecified in the LagrangeBasis case).
Lagrange basis.
Bernstein Basis: note that the zeros of p are the eigenvalues of the companion matrix pencil of p.
A matrix polynomial example.
See Also
LinearAlgebra[CompanionMatrix]
Matrix
OrthogonalSeries
Record
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