REDUCE

16.39 LPDO: Linear Partial Differential Operators

Author: Thomas Sturm

16.39.1 Introduction

Consider the field F = (x1,,xn) of rational functions and a set Δ = {x1,…, xn} of commuting derivations acting on F . That is, for all xi, xj Δ and all f, g F the following properties are satisfied:

 ∂xi(f + g) =   ∂xi(f )+ ∂xi(g),
  ∂x (f ⋅g) =   f ⋅∂x(g) + ∂x(f) ⋅g,          (16.87)
    i                i       i
∂xi(∂xj(f)) =   ∂xj(∂xi(f)).                   (16.88)

Consider now the set F[x1,,∂xn], where the derivations are used as variables. This set forms a non-commutative linear partial differential operator ring with pointwise addition, and multiplication defined as follows: For f F and xi, xj Δ we have for any g F that

  (f∂  )(g)  =  f ⋅∂  (g),
     xi            xi
  (∂xif )(g)  =  ∂xi(f ⋅g),                (16.89)
(∂xi∂xj)(g)  =  ∂xi(∂xj(g)).               (16.90)
Here “ ” denotes the multiplication in F . From (16.90) and (16.88) it follows that xixj = xjxi, and using (16.89) and (16.87) the following commutator can be proved:
∂xif  =  f ∂xi + ∂xi(f ).

A linear partial differential operator (LPDO) of order k is an element

        ∑      j
D   =       aj∂ ∈ F [∂x1,...,∂xn]
        |j|≤k
in canonical form. Here the expression |j|≤ k specifies the set of all tuples of the form j = (j1,,jn) n with i=1nji k, and we define j = x1j1⋅⋅⋅x njn.

A factorization of D is a non-trivial decomposition

D   =   D1⋅⋅⋅Dr ∈ F [∂x1,...,∂xn]
into multiplicative factors, each of which is an LPDO Di of order greater than 0 and less than k. If such a factorization exists, then D is called reducible or factorable, else irreducible.

For the purpose of factorization it is helpful to temporarily consider as regular commutative polynomials certain summands of the LPDO under consideration. Consider a commutative polynomial ring over F in new indeterminates y1, …, yn. Adopting the notational conventions above, for m k the symbol of D of order m is defined as

               ∑      j
Symm  (D)  =       ajy  ∈ F[y1,...,yn].
               |j|=m
For m = k we obtain as a special case the symbol Sym(D) of D.

16.39.2 Operators

16.39.2.1 partial

There is a unary operator partial() denoting .

⟨partial-term ⟩       partial  (  ⟨id ⟩  )  

16.39.2.2 ***

There is a binary operator *** for the non-commutative multiplication involving partials x. All expressions involving *** are implicitly transformed into LPDOs, i.e., into the following normal form:

⟨normalized-lpdo⟩       ⟨normalized- mon ⟩  [+ ⟨normalized-lpdo⟩ ]  
⟨normalized-mon⟩       ⟨F -element⟩  [*** ⟨partial-termprod⟩ ]  
⟨partial- termprod⟩       ⟨partial-term⟩  [***  ⟨partial- termprod⟩ ]  

The summands of the normalized-lpdo are ordered in some canonical way. As an example consider

input: a()***partial(y)***b()***partial(x);  
 
(a()*b()) *** partial(x) *** partial(y) + (a()*diff(b(),y,1)) *** partial(x)

Here the F-elements are polynomials, where the unknowns are of the type constant-operator denoting functions from F :

⟨constant- operator⟩       ⟨id⟩  (  )  

We do not admit division of such constant operators since we cannot exclude that such a constant operator denotes 0.

The operator notation on the one hand emphasizes the fact that the denoted elements are functions. On the other hand it distinguishes a() from the variable a of a rational function, which specifically denotes the corresponding projection. Consider e.g.

input: (x+y)***partial(y)***(x-y)***partial(x);  
 
  2    2  
(x  - y ) *** partial(x) *** partial(y) + ( - x - y) *** partial(x)

Here we use as F-elements specific elements from F = (x,y).

16.39.2.3 diff

In our example with constant operators, the transformation into normal form introduces a formal derivative operation diff(,,), which cannot be evaluated. Notice that we do not use the Reduce operator df(,,) here, which for technical reasons cannot smoothly handle our constant operators.

In our second example with rational functions as F-elements, derivative occurring with commutation can be computed such that diff does not occur in the output.

16.39.3 Shapes of F-elements

Besides the generic computations with constant operators, we provide a mechanism to globally fix a certain shape for F-elements and to expand constant operators according to that shape.

16.39.3.1 lpdoset

We give an example for a shape that fixes all constant operators to denote generic bivariate affine linear functions:

input: d := (a()+b())***partial(x1)***partial(x2)**2;  
 
                                                2  
d := (a() + b()) *** partial(x1) *** partial(x2)  
 
input: lpdoset {!#10*x1+!#01*x2+!#00,x1,x2};  
 
{-1}  
 
input: d;  
 
                                                                               2  
(a00 + a01*x2 + a10*x1 + b00 + b01*x2 + b10*x1) *** partial(x1) *** partial(x2)

Notice that the placeholder # must be escaped with !, which is a general convention for Rlisp/Reduce. Notice that lpdoset returns the old shape and that {-1} denotes the default state that there is no shape selected.

16.39.3.2 lpdoweyl

The command lpdoweyl {n,x1,x2,...} creates a shape for generic polynomials of total degree n in variables x1, x2, ….

input: lpdoweyl(2,x1,x2);  
 
                            2                                    2  
{#_00_ + #_01_*x2 + #_02_*x2  + #_10_*x1 + #_11_*x1*x2 + #_20_*x1 ,x1,x2}  
 
input: lpdoset ws;  
 
{#10*x1 + #01*x2 + #00,x1,x2}  
 
input: d;  
 
                            2                                    2  
(a_00_ + a_01_*x2 + a_02_*x2  + a_10_*x1 + a_11_*x1*x2 + a_20_*x1  + b_00_  
 
                       2                                    2  
  + b_01_*x2 + b_02_*x2  + b_10_*x1 + b_11_*x1*x2 + b_20_*x1 ) *** partial(x1)  
 
                2  
 *** partial(x2)

16.39.4 Commands

16.39.4.1 General

The order of an lpdo:

input: lpdoord((a()+b())***partial(x1)***partial(x2)**2+3***partial(x1));  
 
3

Returns the list of derivations (partials) occurring in its argument LPDO d.

input: lpdoptl(a()***partial(x1)***partial(x2)+partial(x4)+diff(a(),x3,1));  
 
{partial(x1),partial(x2),partial(x4)}

That is the smallest set {,∂xi,} such that d is defined in F[,∂xi,]. Notice that formal derivatives are not derivations in that sense.

Given a starting symbol a, a list of variables l, and a degree n, lpdogp(a,l,n) generates a generic (commutative) polynomial of degree n in variables l with coefficients generated from the starting symbol a:

input: lpdogp(a,{x1,x2},2);  
 
                           2                                    2  
a_00_ + a_01_*x2 + a_02_*x2  + a_10_*x1 + a_11_*x1*x2 + a_20_*x1

Given a starting symbol a, a list of variables l, and a degree n, lpdogp(a,l,n) generates a generic differential polynomial of degree n in variables l with coefficients generated from the starting symbol a:

input: lpdogdp(a,{x1,x2},2);  
 
                     2                        2  
a_20_ *** partial(x1)  + a_02_ *** partial(x2)  
 
 + a_11_ *** partial(x1) *** partial(x2) + a_10_ *** partial(x1)  
 
 + a_01_ *** partial(x2) + a_00_

16.39.4.2 Symbols

The symbol of an lpdo. That is the differential monomial of highest order with the partials replaced by corresponding commutative variables:

input: lpdosym((a()+b())***partial(x1)***partial(x2)**2+3***partial(x1));  
 
           2  
y_x1_*y_x2_ *(a() + b())

More generally, one can use a second optional arguments to specify a the order of a different differential monomial to form the symbol of:

input: lpdosym((a()+b())***partial(x1)***partial(x2)**2+3***partial(x1),1);  
 
3*y_x1_

Finally, a third optional argument can be used to specify an alternative starting symbol for the commutative variable, which is y by default. Altogether, the optional arguments default like lpdosym()=lpdosym(,lpdoord(),y).

This converts a symbol obtained via lpdosym back into an LPDO resulting in the corresponding differential monomial of the original LPDO.

input: d := a()***partial(x1)***partial(x2)+partial(x3)$  
 
input: s := lpdosym d;  
 
s := a()*y_x1_*y_x2_  
 
input: lpdosym2dp s;  
 
a() *** partial(x1) *** partial(x2)

In analogy to lpdosym there is an optional argument for specifying an alternative starting symbol for the commutative variable, which is y by default.

Given LPDOs p, q and m the function lpdos(p,q,m) computes the commutative polynomial

      ∑   ( ∑n              )
Sm =           pi∂i(qj)+ p0qj  yj.
      |j|=m   i=1
      |j|<k
This is useful for the factorization of LPDOs.
input: p := a()***partial(x1)+b()$  
 
input: q := c()***partial(x1)+d()***partial(x2)$  
 
input: lpdos(p,q,1);  
 
a()*diff(c(),x1,1)*y_x1_ + a()*diff(d(),x1,1)*y_x2_ + b()*c()*y_x1_  
 
 + b()*d()*y_x2_

16.39.4.3 Factorization

Factorize the argument LPDO d. The ground field F must be fixed via lpdoset. The result is a list of lists {,(Ai,Li),}. Ai is is genrally the identifiers true, which indicates reducibility. The respective Li is a list of two differential polynomial factors, the first of which has order 1.

input: bk := (partial(x)+partial(y)+(a10-a01)/2) ***  
       (partial(x)-partial(y)+(a10+a01)/2);  
 
                2             2  
bk := partial(x)  - partial(y)  + a10 *** partial(x) + a01 *** partial(y)  
 
          2      2  
     - a01  + a10  
 + ----------------  
          4  
 
input: lpdoset lpdoweyl(1,x,y);  
 
{#_00_ + #_01_*y + #_10_*x,x,y}  
 
input: lpdofactorize bk;  
 
{{true,  
 
                                 a01 - a10  
  { - partial(x) - partial(y) + -----------,  
                                     2  
 
                                  - a01 - a10  
    - partial(x) + partial(y) + --------------}}}  
                                      2

If the result is the empty list, then this guarantees that there is no approximate factorization possible. In general it is possible to obtain several sample factorizations. Note, however, that the result does not provide a complete list of possible factorizations with a left factor of order 1 but only at least one such sample factorization in case of reducibility.

Furthermore, the procedure might fail due to polynomial degrees exceeding certain bounds for the extended quantifier elimination by virtual substitution used internally. In this case there is the identifier failed returned. This must not be confused with the empty list indicating irreducibility as described above.

Besides

  1. the LPDO d,

lpdofactorizex accepts several optional arguments:

  1. An LPDO of order 1, which serves as a template for the left (linear) factor. The default is a generic linear LPDO with generic coefficient functions according from the ground field specified via lpdoset. The principle idea is to support the factorization by guessing that certain differential monomials are not present.
  2. An LPDO of order ord(d)-1, which serves as a template for the right factor. Similarly to the previous argument the default is fully generic.

This is a low-level entry point to the factorization lpdofactorize. It accepts the same arguments as lpdofactorize. It generates factorization conditions as a quite large first-order formula over the reals. This can be passed to extended quantifier elimination. For example, consider bk as in the example for lpdofactorize above:

input: faccond := lpdofac bk$  
 
input: rlqea faccond;  
 
{{true,  
 
               a01 - a10  
  {p_00_00_ = -----------,  
                   2  
 
   p_00_01_ = 0, p_00_10_ = 0, p_01_00_ = -1, p_01_01_ = 0, p_01_10_ = 0,  
 
   p_10_00_ = -1, p_10_01_ = 0, p_10_10_ = 0,  
 
                - a01 - a10  
   q_00_00_ = --------------,  
                    2  
 
   q_00_01_ = 0, q_00_10_ = 0, q_01_00_ = 1, q_01_01_ = 0, q_01_10_ = 0,  
 
   q_10_00_ = -1, q_10_01_ = 0, q_10_10_ = 0}}}

The result of the extended quantifier elimination provides coefficient values for generic factor polynomials p and q. These are automatically interpreted and converted into differential polynomials by lpdofactorize.

16.39.4.4 Approximate Factorization

Approximately factorize the argument LPDO d. The ground field F must be fixed via lpdoset. The result is a list of lists {,(Ai,Li),}. Each Ai is quantifier-free formula possibly containing a variable epsilon, which describes the precision of corresponding factorization Li. Li is a list containing two factors, the first of which is linear.

input: off lpdocoeffnorm$  
 
input: lpdoset lpdoweyl(0,x1,x2)$  
 
input: f2 := partial(x1)***partial(x2) + 1$  
 
input: lpdofactorizex f2;  
 
{{epsilon - 1 >= 0,{partial(x1),partial(x2)}},  
 
 {epsilon - 1 >= 0,{partial(x2),partial(x1)}}}

If the result is the empty list, then this guarantees that there is no approximate factorization possible. In our example we happen to obtain two possible factorizations. Note, however, that the result in general does not provide a complete list of factorizations with a left factor of order 1 but only at least one such sample factorization.

Furthermore, the procedure might fail due to polynomial degrees exceeding certain bounds for the extended quantifier elimination by virtual substitution used internally. If this happens, the corresponding Ai will contain existential quantifiers ex, and Li will be meaningless.
Da sollte besser ein failed kommen ...

The first of the two subresults above has the semantics that x1x2 is an approximate factorization of f2 for all ε 1. Formally, ||f2 - x1x2||≤ ε for all ε 1, which is equivalent to ||f2 - x1x2||≤ 1. That is, 1 is an upper bound for the approximation error over 2. Where there are two possible choices for the seminorm ||⋅||:

  1. ...
  2. ...

explain switch lpdocoeffnorm ...

Besides

  1. the LPDO d,

lpdofactorizex accepts several optional arguments:

  1. A Boolean combination ψ of equations, negated equations, and (possibly strict) ordering constraints. This ψ describes a (semialgebraic) region over which to factorize approximately. The default is true specifying the entire n. It is possible to choose ψ parametrically. Then the parameters will in general occur in the conditions Ai in the result.
  2. An LPDO of order 1, which serves as a template for the left (linear) factor, and an LPDO of order ord(d) - 1, which serves as a template for the right factor. See the documentation of lpdofactorize for defaults and details.
  3. A bound ε for describing the desired precision for approximate factorization. The default is the symbol epsilon, i.e., a symbolic choice such that the optimal choice (with respect to parameters in ψ) is obtained during factorization. It is possible to fix ε . This does, however, not considerably simplify the factorization process in most cases.
input: f3 := partial(x1) *** partial(x2) + x1$  
 
input: psi1 := 0<=x1<=1 and 0<=x2<=1$  
 
input: lpdofactorizex(f3,psi1,a()***partial(x1),b()***partial(x2));  
 
{{epsilon - 1 >= 0,{partial(x1),partial(x2)}}}

This is a low-level entry point to the factorization lpdofactorizex. It is analogous to lpdofac for lpdofactorize; see the documentation there for details.