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LinearOperators

  

DEToOrePoly

  

convert a linear ordinary differential equation to an OrePoly structure

  

REToOrePoly

  

convert a linear recurrence equation to an OrePoly structure

  

OrePolyToDE

  

convert an OrePoly structure to a linear ordinary differential equation

  

OrePolyToRE

  

convert an OrePoly structure to a linear recurrence equation

  

FactoredOrePolyToDE

  

convert a FactoredOrePoly structure to a linear ordinary differential equation

  

FactoredOrePolyToRE

  

convert a FactoredOrePoly structure to a linear recurrence equation

  

FactoredOrePolyToOrePoly

  

convert a FactoredOrePoly structure to a OrePoly structure

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

DEToOrePoly(eq,f)

REToOrePoly(eq,f)

OrePolyToDE(L,f)

OrePolyToRE(L,f)

FactoredOrePolyToDE(M,f)

FactoredOrePolyToRE(M,f)

FactoredOrePolyToOrePoly(M,var,case)

Parameters

eq

-

left hand side of a linear equation (either differential or recurrence)

f

-

function from eq, for example, f(x)

L

-

Ore operator

M

-

factored Ore operator

var

-

name of the independent variable

case

-

parameter indicating the case of the equation ('differential' or 'shift')

Description

• 

The LinearOperators[DEToOrePoly] and LinearOperators[REToOrePoly] functions return an Ore operator K such that eq = K(f). The LinearOperators[OrePolyToDE], LinearOperators[OrePolyToRE], LinearOperators[FactoredOrePolyToDE], and LinearOperators[FactoredOrePolyToRE] functions apply the operator (L or M) to the function f. The LinearOperators[FactoredOrePolyToOrePoly] function converts an Ore polynomial in factored form, that is, a FactoredOrePoly structure, to an Ore polynomial in expanded form, that is, an OrePoly structure.

• 

A completely factored Ore operator is represented by a structure that consists of the keyword FactoredOrePoly and a sequence of lists. Each list consists of two elements and describes a first degree factor. The first element provides the zero degree coefficient and the second element provides the first degree coefficient. For example, in the differential case with a differential operator D, FactoredOrePoly([-1, x], [x, 0], [4, x^2], [0, 1]) describes the operator 1+xDxDx2+4D.

• 

An Ore operator is a structure that consists of the keyword OrePoly with a sequence of coefficients starting with the one of degree zero. The coefficients must be rational functions in x. For example, in the differential case with the differential operator D, OrePoly(2/x, x, x+1, 1) represents the operator 2x+xD+x+1D2+D3.

Examples

polyFactoredOrePolyx5,1+x,x,1

poly:=FactoredOrePolyx5,1+x,x,1

(1)

odeLinearOperators[FactoredOrePolyToDE]poly,yx

ode:=yxx6ⅆⅆxyxx5+ⅆⅆxyxx2+yxx+ⅆⅆxyxxⅆ2ⅆx2yxx+yxⅆ2ⅆx2yx

(2)

LinearOperators[DEToOrePoly]ode,yx

OrePolyx6+x+1,x5+x2+x,1x

(3)

lreLinearOperators[FactoredOrePolyToRE]poly,yx

lre:=yxx6y1+xx5+y1+xx2+2xy1+xxy2+x+y1+xy2+x

(4)

LinearOperators[REToOrePoly]lre,yx

OrePolyx6,x5+x2+2x+1,1x

(5)

lreLinearOperators[OrePolyToRE]OrePolyx,x3+x2,x5,yx

lre:=x5y2+x+y1+xx3+xy1+x+yxx2y1+x

(6)

LinearOperators[REToOrePoly]lre,yx

OrePolyx,x3+x2,x5

(7)

LFactoredOrePoly32x,1,12x,1,12x,1

L:=FactoredOrePoly32x,1,12x,1,12x,1

(8)

LinearOperators[FactoredOrePolyToOrePoly]L,x,'differential'

OrePoly18x3,14x2,32x,1

(9)

See Also

DEtools[de2diffop]

DEtools[diffop2de]

LinearOperators

 


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