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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

Calling Sequence

convert(p, MatrixPolynomialObject, x)

convert(values, MatrixPolynomialObject, nodes)

type(expr, MatrixPolynomialObject, x)

Parameters

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

Description

• 

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).

Examples

(1)

(2)

(3)

(4)

(5)

Lagrange basis.

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Bernstein Basis: note that the zeros of p are the eigenvalues of the companion matrix pencil of p.

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

A matrix polynomial example.

(22)

(23)

(24)

(25)

See Also

BernsteinBasis

ChebyshevT

ChebyshevU

GegenbauerC

JacobiP

LagrangeBasis

LinearAlgebra[CompanionMatrix]

Matrix

NewtonBasis

OrthogonalSeries

PochhammerBasis

Record