% Copyright (c) 1991 Marcel Roelofs, University of Twente, Enschede,
% The Netherlands.
%
% $Header: integrator.web,v 0.92 91/12/18 17:39:37 roelofs Exp $
%
\input specification
\def\Version$#1Revision: #2 ${Version #2}
\def\title{INTEGRATOR}
\font\titlefont=cmcsc10 scaled\magstep3
\font\ttitlefont=cmtt10 scaled\magstep4
\def\topofcontents{\null\vfill
\centerline{\titlefont The {\ttitlefont INTEGRATOR} package for REDUCE}
\vskip15pt\centerline{\Version$Revision: 0.92 $}
\vskip15pt\centerline{\sc Marcel Roelofs}\vfill}
\def\enditem{\medskip\noindent\ignorespaces}
\def\pde{p.d.e.}
@*=Introduction. In this \.{WEB} file we shall describe a REDUCE package for
the integration of overdetermined systems of partial differential
equations (p.d.e.'s). This work is mainly based on a similar package by Paul
Kersten for just the determination of symmetry groups and an extension
by myself which also allows the determination of Wahlquist and
Estabrook prolongation algebras.
The main reasons for the implementation of this package, are our
improved insight in the internals of REDUCE, the wish to have one
combined integrator for both cases and the availability of substantially
improved versions of some the procedures used in the former packages.
\medskip
The ``banner line'' defined here is intended for indentification
purposes on loading. It should be changed whenever this file is
modified. System dependent changes, however, should be made in a
separate change file.
@d banner="Integrator package for REDUCE 3.4, $Revision: 0.92 $"
@ We define the following macros for clarity.
@d change_to_symbolic_mode =symbolic
@d change_to_algebraic_mode =algebraic
@d stop_with_error(string_1,expr_1,string_2,expr_2) = @/
msgpri(string_1,expr_1,string_2,expr_2,t) @;
@d message(string_1,expr_1,string_2,expr_2) = @/
msgpri(string_1,expr_1,string_2,expr_2,nil) @;
@d operator_name_of=car
@d arguments_of=cdr
@d first_argument_of=cadr
@d second_argument_of=caddr
@d first_element_of=car
@d rest_of=cdr
@d skip_list=cdr %Skip the |'list| in front of an algebraic list%
@f function = identifier
@ The following macros are intended as common programming idioms.
@d incr(x) = (x:=x+1)@;
@d decr(x) = (x:=x-1)@;
@ A new REDUCE switch can be introduced using the following code.
@d initialize_global(global_name,value)=@/
global '(global_name)$@/
global_name:=value
@d initialize_fluid(fluid_name,value)=@/
fluid '(fluid_name)$@/
fluid_name:=value
@d new_switch(switch_name,value)=@/
initialize_fluid(!* @& switch_name,value)$@/
flag('(switch_name),'switch)
@ We do all initializations in the beginning of the package.
@u
change_to_symbolic_mode$@/
write banner$terpri()$@/
@@/
change_to_algebraic_mode$
@*=Integration of overdetermined systems of \pde's.
For the determination of symmetry groups or prolongation structures of
(systems of) partial differential equations, the defining relations
give rise to an overdetermined system of \pde's. Finding the symmetry
group or prolongation structure boils down to solving such a system.
There are, however, some differences between the determination of a
symmetry group or the determination of a prolongation structure. These
differences are:\medskip
\item{1.} The differential equations for the determination of the
symmetry group are linear, the equations for the determination of a
prolongation structure are nonlinear. This nonlinearity, however, is
of a special kind, namely, the only occuring nonlinear terms are
(possibly nested) liebrackets of the functions to be integrated.
\item{2.} For the determination of symmetry groups, the functions to be
determined integrate to polynomials with constant coefficients. For
the determination of prolongation structures, functions integrate to
polynomials, coefficients of which are generators of some unknown Lie
algebra. The defining relations of this algebra are the remaining
(nonlinear) relations which have no dependency on the independent
variables involved.
\enditem
From the above it is clear that integration has to be treated slightly
different in either of the cases. The differences are however small
enough to allow the implementation of one integrator for both cases.
@ In order to explain all possible p.d.e.'s which can be integrated, we make
the following assumptions:\medskip
\item{1.} Functions are represented by expressions $f(n)$, where $f$ is
some specified operator and $n$ is an integer. Since we intend to use
the package for computations for supersymmetric p.d.e.'s, we shall use
the notion the elements with $n$ positive must integrate to an even
polynomial and elements with negative $n$ must integrate to an odd
polynomial (this is only useful for computations in prolongation
theory, where coefficients can be even or odd Lie algebra generators).
\item{2.} The dependencies of functions are solely listed on the
dependency list, i.e.\ must be stated by the 'depend' statement of REDUCE.
Notice, however, that we do not allow dependencies of odd variables.
The reason for this is a pragmatic one: due to the anticommutivity of
odd variables, $n$ odd variables can only produce $2^n$ different
terms containing these variables, hence can be stated explicitly
provided that $n$ is not too big. On the other hand, if we allow
dependencies of odd variables, a lot of additional operators have to
be implemented to take care of e.g. partial differentation w.r.t. odd
variables.
\medskip
@ If $f$ is the operator denoting functions, $x$ the operator denoting
Lie algebra generators (or, in the case of a symmetry group, just
constants), then following the description above, a p.d.e.\
has the following possible terms (any coefficient $c$ is always some polynomial
in the independent variables):\medskip
\item{A.} terms of the form $c_n \hbox{df}(f(n),\dots)$.
\item{B.} terms of the form $c_n f(n)$.
\item{C.} terms of the form $c_{1,2}[z_1,z_2]$ where $z_1,z_2$ are
either functions $f(n)$ or Lie algebra generators $x(n)$.
\item{D.} terms of the form $c_n x(n)$.
\medskip
These possibilities lead, in a natural way, to the following strategy
of solving the p.d.e.'s:\medskip
\item{1.} If there is only one term of type A, we can integrate this
equation homogeneously, i.e. give a polynomial expression for $f(n)$
using the variables involved in the differential term.
\item{2.} If the p.d.e.\ is a polynomial in one or more independent
variables on which none of the occuring functions depend, all
coefficients of this polynomial have to be zero, i.e., the p.d.e.\
splits up into a set of smaller p.d.e.'s.
\item{3.} If there are only terms of type C and D we have a Lie
algebra relation, which can be solved by the LIESUPER package, if
solvable.
\item{4.} If there is a function of type B depending on all variables
occuring in the p.d.e.\ and not occuring in a term of type A, we can
solve for this function.
\item{5.} If there is one term of type A depending on all variables
occuring the p.d.e.\ and the remaining terms are polynomial in the
variables occuring in the derivative, the p.d.e.\ can be integrated
inhomogeneously.
\item{6.} If there is just one function in the p.d.e.\ which depends on a
variable only occuring polynomially in the rest of the p.d.e., such
that the p.d.e.\ can not be integrated inhomogeneously since the
dependencies of the various occuring functions do not match, we can
introduce new equations of type 1 by appropriately differentiating the
p.d.e.
@*1 Initializing an equation set.
The integrator will be implemented in such a way that integration
can be performed on different sets of p.d.e.'s at the same time.
Different sets of p.d.e.'s will be distinguished by the name of the
operator in which they are stored.
For each operator representing a set of p.d.e.'s we must know: the
name of the operator(s) representing the functions and the operator
that must be used to represent constants coefficients during the
integration. If this last operator is of rtype 'algebra\_generator' we
know that we are in the prolongation case and the name of the
associated liebracket can be found on the property list of this
operator.
Moreover, we have to know the total number of equations used, in view of
the additional equations that may be generated and which must be
numbered subsequently.
In connection with the integrations taking place we also have to know the
number of functions, resp. constants (generators) being in use.
This is all taken care of by the procedure |initialize_equations|,
which assigns to an operator |operator_name|, the total number of used
equations |total_used|, the list |variable_list| of all occuring independent
variables, the operator |constant_operator|, elements of
which act as constants, and an arbitrary number of operators
|function_operator| acting as functions.
|constant_operator| and each |function_operator| should be given a an
algebraic list of the form $\{$operator, number of even elements used,
number of odd elements used$\}$.
In order to allow an arbitrary number of parameters we make
|initialize_equations| a |psopfn|. How |psopfn|'s are dealt with
internally is explained in the documentation of either the TOOLS
package or the LIESUPER package.
@=@/
put('initialize_equations,'psopfn,'initialize_equations1)$
@
@u
lisp procedure initialize_equations1 specification_list;
begin scalar operator_name,total_used,variable_list,
specification,even_used,odd_used,
constant_operator,bracketname,function_name,function_list;
if length specification_list<5 then
rederr("INITIALIZE_EQUATIONS: wrong number of parameters");
if not idp(operator_name:=first_element_of specification_list) then
rederr("INITIALIZE_EQUATIONS: equations operator must be identifier");
if not fixp(total_used:=
reval first_element_of(specification_list:=rest_of specification_list))
or total_used<0 then
rederr("INITIALIZE_EQUATIONS: total number of equations must be positive");
put(operator_name,'total_used,total_used);
variable_list:=reval first_element_of(
specification_list:=rest_of specification_list);
if atom variable_list or operator_name_of variable_list neq 'list then
rederr("INITIALIZE_EQUATIONS: variable list must be algebraic list");
put(operator_name,'variable_list,skip_list variable_list);
@;
@;
end$
@ The |constant_operator| can either be of rtype |algebra_generator|
or not. If so, we also have to assign the associated liebracket to
|operator_name| and used the procedure |define_used| to take care of
the assignment of the used dimensions to the liebracket. If
|constant_operator| is not an |algebra_generator|, we store these
dimensions in the same way as happens for liebrackets.
@d check_valid_function_declaration(op_list,op_name)=@/
if atom op_list or length op_list neq 4 or operator_name_of op_list neq 'list@|
or not idp(op_name:=first_argument_of op_list) or
not fixp(even_used:=reval caddr op_list) @| or
not fixp(odd_used:=reval cadddr op_list)
or even_used<0 or odd_used<0 then @/
stop_with_error("INITIALIZE_EQUATIONS: invalid declaration of",
op_list,nil,nil)
@d put_used_dimensions(op_name,even_used,odd_used)=@/
if get(op_name,'rtype)='algebra_generator then@/
define_used(bracketname,list('list,even_used,odd_used))
else
begin
put(op_name,'even_used,even_used);@/
put(op_name,'odd_used,odd_used);
end
@=@/
specification_list:=rest_of specification_list;
specification:=first_element_of specification_list;
check_valid_function_declaration(specification,constant_operator);
put(operator_name,'constant_operator,constant_operator);
if get(constant_operator,'rtype)='algebra_generator then@/
put(operator_name,'bracketname,
bracketname:=get(constant_operator,'bracketname));
put_used_dimensions(constant_operator,even_used,odd_used)
@
@=@/
for each function_specification in rest_of specification_list do
begin
check_valid_function_declaration(function_specification,function_name);
put_used_dimensions(function_name,even_used,odd_used);
function_list:=function_name . function_list;
end;
put(operator_name,'function_list,function_list)
@ Since we can apparently choose different sets of p.d.e.'s for
solving, we must tell the integrator which set to take. This is done
via a global variable |current_equation_set!*|. We will take the
operator |equ| as the default |current_equation_set!*|.
In this file we will use the abbreviation |ces!*| for
|current_equation_set!*|.
@d ces!*=current_equation_set!*
@=@/
initialize_global(ces!*,'equ)$
@
@u
lisp operator use_equations;@/
lisp procedure use_equations operator_name;
begin
if idp operator_name then
ces!*:=operator_name
else rederr("USE_EQUATIONS: argument must be identifier");
end$
@*1 The integration procedure.
The implementation of the integrator follows the description of all
the possible steps given above.
\noindent For the use of the fluid variable |listpri_depth!*|, see below. Its
local rebinding is necessary for a proper printing of the messages
given by the procedure.
@u
lisp operator integrate_equation;
lisp procedure integrate_equation n;
begin scalar listpri_depth!*,total_used,equation,denominator,
solvable_kernel,solvable_kernels,df_list,df_kernel,
function_list,present_functions_list,variable_list,absent_variables,
polynomial_variables,equations_list,linear_functions_list,constants_list,
bracketname,df_terms,df_functions,@|
linear_functions,functions_and_constants_list,commutator_functions,
present_variables,@|
inhomogeneous_term,nr_of_variables,integration_variables,
forbidden_functions,differentiations_list,polynomial_order;
listpri_depth!*:=200;
terpri!* t;
@;
@;
@;
@;
@;
@;
@;
@;
solved: %Go here when the equation is solved or its type is determined%
end$
@ The part of the equation containing all necessary information is its
numerator. For reasons that will become clear in the sequel we need,
however, also know its denominator. If the equation is zero, no
analysis has to be performed.
@d nullify_equation(n)=@/
setk(list(ces!*,n),0)
@=
if null(total_used:=get(ces!*,'total_used)) or
n>total_used then
stop_with_error("INTEGRATE_EQUATIONS: properly initialize",
ces!*,nil,nil);
if null (equation:=cadr assoc(list(ces!*,n),
get(ces!*,'kvalue))) then
stop_with_error("INTEGRATE_EQUATION:", list(ces!*,n),
"is non-existent",nil);
denominator:=denr(equation:=simp!* equation);
equation:=numr equation;
if null equation then
<>
@*1 Homogeneous integration.
Homogeneous integration must be performed if the equation consists
of just one |df| term. In order to find all possible |df| terms we
apply |split_form| to |equation|. This returns a list the |car| of
which is the part of |equation| independent of the |df| operator, the
|cdr| of which is a list of all linear |df| terms, together with their
coefficients. |split_form| will return with an error if nonlinear |df|
terms occur.
@d independent_part_of=car
@d kc_list_of=cdr
@d kernel_of=car %For use with a kernel-coefficient list%
@d coefficient_of=cdr %For use with a kernel-coefficient list%
@ If there is one |df| term, we only solve it if its coefficient is a
number, by default. This behaviour is governed by the switch
|coefficient_check|, which is |on| by default. In order to check the coefficient
we will use the procedure |find_solvable_kernel| to be explained below.
@=
new_switch(coefficient_check,t)$
@
@d assoc_delete(kernel,assoc_list)=@/
delete(assoc(kernel,assoc_list),assoc_list)
@d successful_message_for(action,kernel)=@/
<>
@d not_a_number_message_for(action,kernel)=@/
<>
@=@/
df_list:=split_form(equation,'(df));
if null independent_part_of df_list and
(kc_list_of df_list) and length(kc_list_of df_list)=1
then
if (solvable_kernel:=find_solvable_kernel(@|
solvable_kernels:=list(kernel_of first_element_of kc_list_of df_list),@|
kc_list_of df_list,denominator)) then
<>
else not_a_number_message_for("Homogeneous integration",
first_element_of solvable_kernels)
@ The procedure |find_solvable_kernel| tries to find the first element
of |kernel_list| which has a number as coefficient.
If |coefficient_check| is |off| we can simply take the first element
of |kernel_list|, otherwise we can most conveniently implement a
recursive procedure |first_solvable_kernel|, which finds the first
element of |kernel_list| with a number as coefficient.
@u
lisp procedure find_solvable_kernel(kernel_list,kc_list,denominator);
if !*coefficient_check then first_solvable_kernel(kernel_list,kc_list,denominator)
else first_element_of kernel_list$
@#
lisp procedure first_solvable_kernel(kernel_list,kc_list,denominator);
if kernel_list then @/
(if numberp coefficient_of kc_pair or
numberp !*ff2a(coefficient_of kc_pair,denominator)
then @/ kernel_of kc_pair
else first_solvable_kernel(rest_of kernel_list,kc_list,denominator))
where kc_pair=assoc(first_element_of kernel_list,kc_list)$
@ The equation
\def\dd#1#2{{\partial^{#2}\over\partial{#1}^{#2}}}
$$
\dd{x_1}{k_1}\cdots\dd{x_m}{k_m} f(x_1,\dots,x_n)=0\qquad (m\leq n)
$$
has general solution
$$
f=\sum_{j=1}^{m}\sum_{i_j=0}^{k_j-1}
x_j^{i_j}f_{j,i_j}(x_1,\dots,\hat{x_j},\dots,x_n).
$$
Thus, given a homogenous p.d.e., |homogeneous_integration_of| has to
return the REDUCE equivalent of the last expression.
If $f$ depends on only one variable the $f_{j,i_j}$ are constants,
otherwise they are new functions with dependency on one less variable.
In the Lie algebra case the constants are generators of the Lie
algebra. Since the dimensions of a |liebracket| in REDUCE have to be
given on beforehand, there may not be enough generators left to
generate $f$. In this case, we have to enlarge the |liebracket|.
@d get_dependencies_of(kernel)=@/
((if depl_entry then cdr depl_entry)@| where depl_entry=assoc(kernel,depl!*))
@u
lisp procedure homogeneous_integration_of df_term;
begin scalar df_function,function_number,dependency_list,integration_list,
coefficient_name,bracketname,even_used,odd_used,
integration_variable,@|
number_of_integrations,solution,new_dependency_list;
@;
dependency_list:=get_dependencies_of(df_function);
if length dependency_list=1 then
coefficient_name:=get(ces!*,'constant_operator)
else coefficient_name:=operator_name_of df_function;
@;
integration_list:=rest_of arguments_of df_term;
@;
if bracketname then
@;
@;
return solution
end$
@ We required |df_term| to be of the form |df|($f(k),\dots$) where
$f$ is a function occuring on the |function_list| of |ces!*| and $k$
is an integer not equal to zero.
@=@/
df_function:=first_argument_of df_term;
if not member(operator_name_of df_function,get(ces!*,'function_list)) @|
or not fixp(function_number:=first_argument_of df_function) or function_number=0 then
@/stop_with_error("PERFORM_HOMOGENEOUS_INTEGRATION: integration of",
df_function, "not allowed",nil)
@ In the liebracket case |even_used| and |odd_used| are stored as
properties of |bracketname| instead of |coefficient_name|.
@=
if get(coefficient_name,'rtype)='algebra_generator then
begin bracketname:=get(ces!*,'bracketname);@/
even_used:=get(bracketname,'even_used);
odd_used:=get(bracketname,'odd_used);
end
else
begin
even_used:=get(coefficient_name,'even_used);@/
odd_used:=get(coefficient_name,'odd_used);
end
@ Finding the integration variables is rather straightforward.
@=
if integration_list then integration_variable:=first_element_of
integration_list else integration_variable:=nil;
if integration_variable and (integration_list:=rest_of integration_list) @|
and fixp first_element_of integration_list then
<>
else number_of_integrations:=1
@ If |df_function| depends on only one variable, the number of
constants being introduced is equal to the |number_of_integrations|.
The even and odd dimension of |bracketname| are stored as the
properties |even_dimension| and |odd_dimension|.
@=
if function_number > 0 then @/
(if even_used+number_of_integrations>get(bracketname,'even_dimension) then@/
change_dimensions_of(bracketname,even_used+number_of_integrations,@|
get(bracketname,'odd_dimension)))
else @/
(if odd_used+number_of_integrations>get(bracketname,'odd_dimension) then@/
change_dimensions_of(bracketname,get(bracketname,'even_dimension),
odd_used+number_of_integrations))
@ The actual integration is fairly straightforward by now: for all the
possible integration variables we can simply add new terms to
|solution|.
@d new_coefficient=@/
list(coefficient_name,if function_number>0 then
incr(even_used) else -incr(odd_used))
@d ext_mksq(kernel,power)=@/
if power=0 then 1 ./ 1 else mksq(kernel,power)
@d depend_new_coefficient(dependency_list)=@/
depl!*:= (list(coefficient_name,if function_number>0 then even_used
else -odd_used) . dependency_list) . depl!*;
@=@/
solution:=nil ./ 1;
while integration_variable do
begin new_dependency_list:=delete(integration_variable,dependency_list);
for i:=0:number_of_integrations-1 do
<>;
@
end;
solution:=mk!*sq subs2 solution;@/
put_used_dimensions(coefficient_name,even_used,odd_used)
@*1 Splitting polynomial equations.
For the polynomial behaviour of |equation| we need to know the
dependencies of all the functions occuring in |equation| at any level.
If there occur any other variables in |equation| and |equation| is
polynomial in these variables, the coefficients of this polynomial
give rise to a new set of equations.
@d pc_list_of=kc_list_of %power-coefficient list%
@d powers_of=kernel_of
@=@/
@;
@;
@
@ Finding all the functions in |equation| can be done by applying the
procedure |get_recursive_kernels| of the TOOLS package.
@=@/
function_list:=get(ces!*,'function_list);@/
present_functions_list:=get_recursive_kernels(equation,function_list);@/
variable_list:=get(ces!*,'variable_list);
absent_variables:=variable_list;
for each function in present_functions_list do
for each variable in get_dependencies_of(function) do@/
absent_variables:=delete(variable,absent_variables)
@ In most cases the equations under consideration are polynomial in any
of the variables and therefore we shall by default not test for
polynomial behaviour. This testing is governed by the switch
|polynomial_check| which, be default, is |off|. If it is |on| testing
is done by the procedure |polynomialp| to be defined below.
@=@/
polynomial_variables:=absent_variables;
if !*polynomial_check then@/
polynomial_variables:=for each variable in polynomial_variables join@/
if polynomialp(equation,variable) then list(variable)
@ @=
new_switch(polynomial_check,nil)$
@ Checking a standard form for polynomial behaviour in some kernel can
be done by checking the main variable, the leading coefficient and the
reductum, respectively.
@u
lisp procedure polynomialp(expression,kernel);
if domainp expression then t
else ((main_variable=kernel or not depends(main_variable,kernel)) @|and
polynomialp(lc expression,kernel) and polynomialp(red expression,kernel)) @|
where main_variable=mvar expression$
@ The coefficients of a polynomial can be found by
applying the procedure |multi_split_form| from the TOOLS package.
@=@/
equations_list:=multi_split_form(equation,polynomial_variables);
if length equations_list>1 then
<>
@ In order to get messages in a readable form, we sometimes need to
print lists partially. This is taken care of the following procedures.
@u
lisp procedure partial_list(printed_list,nr_of_items);
'list . broken_list(printed_list,nr_of_items)$
@#
lisp procedure broken_list(list,n);
if list then if n=0 then '(!.!.!.)
else car list . broken_list(cdr list,n-1)$
@*1 Solving Lie algebra relations.
If the first two steps have failed, we need to analyze |equation| in
a more drastic way: we need to find all functions occuring linearly in
|equation|, and if a liebracket is specified, all commutators and
algebra generators occuring in |equation| as well.
Since we have already looked for |df| terms in |equation| in each next
step we only have to examine the independent part of the previous step.
@=@/
linear_functions_list:=split_form(independent_part_of df_list,
function_list);@/
df_list:=kc_list_of df_list;
constants_list:=split_form(independent_part_of linear_functions_list,
list get(ces!*,'constant_operator));@/
linear_functions_list:=kc_list_of linear_functions_list;
if (bracketname:=get(ces!*,'bracketname)) then
@
@ In the Lie algebra case we can try to solve the Lie expression if
there are no |df| terms or linearly occuring functions. Solving Lie
expression can be done using the procedure |relation_analysis| of the
LIESUPER package. |relation_analysis| returns either the kernel for which
the relation is solved or an atom indicating the nature of the
non-solvability.
@=
if length(df_list)=0 and
length(linear_functions_list)=0 then
<<
if atom(solvable_kernel:=
relation_analysis(!*ff2a(equation,denominator),bracketname))
then <>
else <>;
goto solved
>>
@*1 Solving a function.
If |equation| is not a Lie expression, there may be a function or a
constant for which we can solve it. In order to do this we need to
\medskip
\item{$-$} find all variables |present_variables|, on which at
least one of the present functions |recursive_functions_list| depends;
of course it is the complement of |absent_variables| in |variable_list|.
\item{$-$} find all linearly occuring functions |solvable_kernels| which
depend on all of the |present_variables|; these are the possible
candidates for solving. If there are no |present_variables|,
|equation| is apparently a relation between some constants and we can
try to solve one.
\item{$-$} remove all functions from |solvable_kernels|, which also
occur in a |df| term, or in the liebracket case, in a commutator.
\item{$-$} if |coefficient_check| is |on| we must only solve for those
functions which have a number as coefficient. This is checked by the procedure
|find_solvable_kernel|.
\enditem
Before doing anything we shall, however, construct lists containing
all functions occuring in |df| terms, occuring linearly (and the
constants) and, if necessary, occuring in commutators. These lists
will also come in handy in the next steps.
@=
@;
@;
for each kernel in linear_functions do if length
get_dependencies_of(kernel)=nr_of_variables then@/
solvable_kernels:=kernel . solvable_kernels;
for each kernel in append(df_functions,commutator_functions) do @/
solvable_kernels:=delete(kernel,solvable_kernels);
if solvable_kernels then
@
@ Of course we are only interested in |df| terms of functions occuring
on |function_list|.
@=
df_terms:=for each df_term in df_list join
if member(operator_name_of first_argument_of kernel_of df_term,function_list)
then @/list kernel_of df_term;
for each df_term in df_terms do if not member(first_argument_of
df_term,df_functions) then@/ df_functions:=first_argument_of(df_term) . df_functions;
functions_and_constants_list:=append(linear_functions_list,
kc_list_of constants_list);@/
linear_functions:=for each linear_function in
functions_and_constants_list collect kernel_of linear_function;
if bracketname then commutator_functions:=@|
get_recursive_kernels(independent_part_of constants_list,
get(ces!*,'function_list));
@ @=
present_variables:=variable_list;
for each variable in absent_variables do
present_variables:=delete(variable,present_variables);
nr_of_variables:=length present_variables
@ @=
<>
else not_a_number_message_for("Solving a function",
partial_list(solvable_kernels,3))
>>
@*1 Inhomogeneous integration.
For an inhomogeneous integration, we are looking for a maximal |df| term,
i.e. which has dependency on all the |present_variables|, such that
the remaining part of |equation| is polynomial in the
variables, w.r.t.\ which the function in the |df| term is
differentiated, i.e.\ {\it a}) we only have to look at |df| terms
which are differentiated w.r.t.\ variables on which none of the
non-maximally occuring functions in |equation| depend, and {\it b}) if
|polynomial_check| is |on|, we must check explicitly if the rest of
|equation| is polynomial in these variables.
We shall collect the list of ``integrable'' variables in the list
|integration_variables|.
@=
@;
@
@ Finding the |integration_variables| is rather easy using the lists
|df_functions|, |linear_functions| and |commutator_functions|.
Starting with |present_variables| we have to
delete all variables on which on of the |linear_functions| or
|commutator_functions| depend, or one of the |df_functions|, which do
not have maximal dependency, i.e. which do no depend on
|nr_of_variables| variables.
@=@/
integration_variables:=present_variables;
for each kernel in append(linear_functions,commutator_functions) do
for each variable in get_dependencies_of(kernel) do@/
integration_variables:=delete(variable,integration_variables);
for each df_function in df_functions do
if not length get_dependencies_of(df_function)=nr_of_variables then
for each variable in get_dependencies_of(df_function) do@/
integration_variables:=delete(variable,integration_variables)
@ Finding the integrable |df| terms is rather easy know: find all
the |df| terms which have maximal dependency and are only
differentiated w.r.t.\ variables occuring on |integration_variables|.
In order to check this last item we need to know the form of |df|
term: it is a list |'(df @tfunction@> @tdifferentiation\_sequence@>)|, where
differentiation\_sequence is a sequence of variables, each variable optionally
followed by a integer indicating the number of differentiations
w.r.t.\ to that variable. The procedure
|check_differentiation_sequence| checks whether all variables in a
differentiation\_sequence are member of the second argument
|variable_list|.
@u
lisp procedure check_differentiation_sequence(sequence,variable_list);
if null sequence then t
else @+if fixp first_element_of sequence or
member(first_element_of sequence,variable_list) then@/
check_differentiation_sequence(rest_of sequence,variable_list)$
@ @=
@;
@
@ There one situation we have to take care of specifically: if there
are more |df_terms| for the same function, only one of which is
differentiated just w.r.t. |integration_variables|, we are not allowed
to integrate, since the function would be expressed in itself. In this
case, we will make |solvable_kernels| a list of at least length 2
in order to prevent integration.
@=
for each df_term in df_terms do
<>;
@ @=
if solvable_kernels then
if length(solvable_kernels)=1 then
if (solvable_kernel:=find_solvable_kernel(solvable_kernels,df_list,denominator))
then
if (inhomogeneous_term:=linear_solve(mk!*sq(equation ./ 1),solvable_kernel))@|
and (not !*polynomial_check @|or
check_polynomial_integration(solvable_kernel,inhomogeneous_term))
then
<>
else
<>
else not_a_number_message_for("Inhomogeneous integration",
first_element_of solvable_kernels)
else <>
@ Checking that the inhomogeneous term is polynomial in the
integration variables is fairly easy. For all the integration
variables we have to check that the denominator does not depend on it
and the numerator should be polynomial.
@u lisp procedure check_polynomial_integration(df_term,integration_term);
begin scalar numerator,denominator,integration_variables,variable,ok;
numerator:=numr simp integration_term;
denominator:=denr simp integration_term;@/
integration_variables:=
for each argument in rest_of arguments_of df_term join
if not fixp argument then list argument;
ok:=t;
while ok and integration_variables do
<>;
return ok;
end$
@ We can perform the inhomogeneous integration by applying
|multi_split_form| to find all the
polynomial components of the inhomogeneous term and
|homogeneous_integration_of| for solving the homogeneous equation.
@u
lisp procedure inhomogeneous_integration_of(df_term,inhomogeneous_term);
begin scalar df_sequence,integration_variables,int_sequence,
variable,nr_of_integrations,integration_terms,solution,
powers,coefficient,int_factor,solution_term,n,k;
df_sequence:=rest_of arguments_of df_term;
@;
integration_terms:=multi_split_form(numr simp inhomogeneous_term,
integration_variables);
integration_terms:=(nil . independent_part_of integration_terms) .
pc_list_of integration_terms;
%Make |integration_terms| a full blown |pc_list|%
@;
solution:=multsq(solution,1 ./ denr simp inhomogeneous_term);
solution:=mk!*sq subs2 addsq(solution,simp homogeneous_integration_of df_term);
return solution
end$
@ We must analyze |df_sequence| to get all the integration variables,
together with the number of integrations belonging to them.
@=
while df_sequence do
<>
else nr_of_integrations:=1;
integration_variables:=variable . integration_variables;
int_sequence:=(variable . nr_of_integrations) . int_sequence
>>
@ The particular solution of the equation $F^{(k)}(x)=x^n$ is
$$
F(x)={1\over(n+1)\cdots(n+k)}x^{n+k}.
$$
This process has to be performed for all the terms in
|integration_terms| and for all integrations in |int_sequence|.
@=
solution:=nil ./ 1;
for each term in integration_terms do
<>;
solution_term:=multsq(solution_term,coefficient ./ int_factor);
solution:=addsq(solution,solution_term)
>>
@*1 Generation of new equations by differentiation.
As a last method of solving we notice the following: if there is a
variable, such that just one |df| term or just one linearly occuring
function depends on it and all the other terms are polynomial in this
variable, let's say of degree $n$, then we can differentiate
|equation| $n+1$ times to get a new equation of type A.
Experience has proven, however, that applying the above mentioned
method, generally will lead to multiple generation of equivalent
terms in the answer. Therefore we will only generate a new equation if
the switch |allow_differentiation| is |on|, otherwise we will only
generate a message that it is possible to generate a new equation of
type A. Solving of such a new equation is always left to the responsibility
of the user.
@=@/
new_switch(allow_differentiation,nil)$
@ After this introduction it is clear what we have to do for step 6:
@=
@;
@
@ Counting the occurence of variables is rather easy. For all
functions in |df_terms|, |linear_functions| and
|commutator_functions|, we have to count the occurences of all the
variables in their respective entries on the dependency list |depl!*|.
For this purpose we rebuild |present_variables| to an association list
with entries of the form |variable . origin . number_of_occurences|
where |origin| indicates the |df_term|, |linear_function| or
|commutator_function| in which |variable| occured last.
The action of the following macros, which harmlessly make use of the
procedure |rplacd|, is clear.
@d reinitialize_present_variables=@/
present_variables:=for each variable in present_variables collect
(variable . nil . 0)
@d variable_of=car
@d origin_of=cadr
@d counter_of=cddr
@d update_variable(variable,origin)=
rplacd(entry,origin . (counter_of entry + 1))
where entry=assoc(variable,present_variables)
@d update_variables_using(kernel_list,kernel_selector,flag_function)=@/
for each kernel in kernel_list do
for each variable in get_dependencies_of(kernel_selector(kernel)) do@/
update_variable(variable,flag_function(kernel));
@d identity_function(kernel)=kernel
@d empty_function(kernel)=nil
@=@/
reinitialize_present_variables;@/
update_variables_using(df_terms,first_argument_of,identity_function);@/
update_variables_using(linear_functions,identity_function,identity_function);
if bracketname then update_variables_using(commutator_functions,
identity_function,empty_function)
@ After the preceding step we can generate new equations by
differentiating |equation| w.r.t.\ to all those variables which occur
in only one |df_term| or |linear_function| and for which all other
terms of |equation| are polynomial. Using the above code one can check
that these variables are exactly the ones for which the |origin| has a
value and the |counter| is 1.
@=
differentiations_list:=
for each entry in present_variables join
if origin_of entry and counter_of entry=1 @|and
(polynomial_order:=@|get_polynomial_order(
linear_solve(mk!*sq(equation ./ 1),origin_of entry),variable_of entry))@|
then list(variable_of entry . origin_of entry . (polynomial_order+1));
if differentiations_list then
if !*allow_differentiation then
<>
else <<
write "*** ",ces!*,"(",n,
"): Generation of new equations by differentiation possible.";
terpri!* t; write " Solvable with 'on allow_differentiation'";
terpri!* t; goto solved>>
@ An algebraic expression is polynomial in a variable if the
denominator does not depend on it and if the numerator is polynomial
(we only have to check this if |polynomial_check| is |on|).
The polynomial order we can obtain by simply reordering the numerator
w.r.t. the variable involved.
@u
lisp procedure get_polynomial_order(expression,variable);
if not depends(denr(expression:=simp expression),variable) @|and
(not !*polynomial_check or polynomialp(numr expression,variable)) then
begin scalar kord!*;
setkorder list !*a2k variable;
expression:=reorder numr expression;
return @+if mvar expression=variable then ldeg expression @+else 0;
end$
@ If none of the above methods can be applied, we cannot solve
|equation|.
@=@/
write ces!*,"(",n,") not solved"; terpri!* t
@*=Additional tools.
The following procedure are meant for solving more equations at a
time or solving ``exceptional'' equations, which need the least restrictive
setting of the switches |coefficient_check|, |polynomial_check| or
|allow_differentiation|.
@u
algebraic procedure integrate_equations(m,n);
for i:=m:n do integrate_equation(i)$
@#
lisp operator integrate_exceptional_equation;
lisp procedure integrate_exceptional_equation(n);
integrate_equation(n)
where @|
!*coefficient_check=nil,@|
!*polynomial_check=nil,@|
!*allow_differentiation=t$
@ As a last set of tools, we shall give a procedure to print
an equation together with all the functions occuring in it and their
dependencies, and some procedures for showing and changing the properties
of an equation set and a the functions/constants used.
As a side effect the procedure |show_equation| will reassign the shown
equation to its current value.
@u lisp operator show_equation;
lisp procedure show_equation n;
begin scalar equation,total_used,function_list;
if null(total_used:=get(ces!*,'total_used)) or
n>total_used then
stop_with_error("SHOW_EQUATION: properly initialize",
ces!*,nil,nil);
if (equation:=assoc(list(ces!*,n),get(ces!*,'kvalue))) then
begin
equation:=setk(list(ces!*,n),aeval cadr equation);
varpri(equation,list('setk,mkquote list(ces!*,n),mkquote equation),'only);
function_list:=get_recursive_kernels(numr simp equation,
get(ces!*,'function_list));
if function_list then
<>
>>
else terpri!* nil
end
end$
@#
algebraic procedure show_equations(m,n);
for i:=m:n do show_equation i$
@
@u
lisp operator functions_used,put_functions_used,equations_used,put_equations_used;
@#
lisp procedure functions_used function_name;
list('list,get(function_name,'even_used),get(function_name,'odd_used))$
@#
lisp procedure put_functions_used(function_name,even_used,odd_used);
begin
if not fixp even_used or even_used<0 or
not fixp odd_used or odd_used<0 then@/
stop_with_error("PUT_FUNCTIONS_USED: used functions number invalid",nil,nil,nil);
put(function_name,'even_used,even_used);
put(function_name,'odd_used,odd_used);
end$
@#
lisp procedure equations_used;
get(ces!*,'total_used)$
@#
lisp procedure put_equations_used(n);
if not fixp n or n<0 then@/
stop_with_error("PUT_EQUATIONS_USED: used equation number invalid",nil,nil,nil)
else put(ces!*,'total_used,n)$
@ There is one slight detail which we have not dealt with yet: in
prolongation theory differentiation should act as a derivation on the
arguments of a (eventually nested) commutator. In REDUCE 3.4 there is
a hook which can take care of this situation. In the procedure
|diffp|, which takes care of differentiation of standard powers, if
this standard power is an operator kernel, the property |dfform| is
checked for operator concerned. If this property has a value, it
should be a function which takes care of the differentiation of such a
standard power.
@u
lisp operator df_acts_as_derivation_on;
lisp procedure df_acts_as_derivation_on operator_name;
begin
put(operator_name,'dfform,'df_as_derivation);
end$
@ The procedure |df_as_derivation| is quite straightforward: apply
|df| to all the arguments of the operator, one at a time, leaving the
other ones untouched.
@u
lisp procedure df_as_derivation(kernel,variable,power);
begin scalar left_part,right_part,argument,derivative;
if power neq 1 then
stop_with_error("DF_AS_DERIVATION:",kernel,"must occur linearly",nil);
left_part:=list operator_name_of kernel;@/ right_part:=arguments_of kernel;@/
derivative:=nil . 1;
while right_part do
<>;
return derivative;
end$
@ In order to get nice output of some of the messages given by
|integrate_equation| we redefine the print function |listpri| for
algebraic lists. Namely, we want don't want algebraic lists to split
over multiple lines in the messages we give. For this purpose, we
introduce a fluid variable |listpri_depth!*| which governs the depth
for which algebraic lists are split along lines. The default value is
the same as the value in the used in REDUCE.
@=
initialize_fluid(listpri_depth!*,40)$
@ The following procedure can be used at algebraic level to change
|listpri_depth!*|.
@u
lisp operator listlength$
lisp procedure listlength l;
listpri_depth!*:=l$
@ The definition of |listpri| is basically that of |inprint|, except
that it decides when to split at the comma by looking at the size of
the argument, using the global variable |listpri_depth!*|.
@u
symbolic procedure listpri l;
begin scalar orig,split,u;
u := l;
l := cdr l;
prin2!* get('!*lcbkt!*,'prtch);
% Do it this way so table can change%
orig := orig!*;@/
orig!* := if posn!*<18 then posn!* @+else orig!*+3;
if null l then go to b;
split := treesizep(l,listpri_depth!*);
a: maprint(negnumberchk car l,0);
l := cdr l;
if null l then go to b;
oprin '!*comma!*;
if split then terpri!* t;
go to a;
b: prin2!* get('!*rcbkt!*,'prtch);
orig!* := orig;
return u
end$
@ The end of a REDUCE input file must be marked with |end|.
@u end;
@*=Index. This section contains a cross reference index of all
identifiers, together with the numbers of the mdules in which they are
used. Underlined entries correspond to module numbers where the
identifier was declared.