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REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... *** ~ already defined as operator % Tests of PM. % TESTS OF BASIC CONSTRUCTS. operator f, h$ % A "literal" template. m(f(a),f(a)); t % Not literally equal. m(f(a),f(b)); %Nested operators. m(f(a,h(b)),f(a,h(b))); t % A "generic" template. m(f(a,b),f(a,?a)); {?a->b} m(f(a,b),f(?a,?b)); {?a->a,?b->b} % ??a takes "rest" of arguments. m(f(a,b),f(??a)); {??a->[a,b]} % But ?a does not. m(f(a,b),f(?a)); % Conditional matches. m(f(a,b),f(?a,?b _=(?a=?b))); m(f(a,a),f(?a,?b _=(?a=?b))); {?a->a,?b->a} % "plus" is symmetric. m(a+b+c,c+?a+?b); {?a->a,?b->b} %It is also associative. m(a+b+c,c+?a); {?a->a + b} % Note the effect of using multi-generic symbol is different. m(a+b+c,c+??c); {??c->[a,b]} %Flag h as associative. flag('(h),'assoc); m(h(a,b,d,e),h(?a,d,?b)); {?a->h(a,b),?b->e} % Substitution tests. s(f(a,b),f(a,?b)->?b^2); 2 b s(a+b,a+b->a*b); a*b % "associativity" is used to group a+b+c in to (a+b) + c. s(a+b+c,a+b->a*b); a*b + c % Only substitute top at top level. s(a+b+f(a+b),a+b->a*b,inf,0); f(a + b) + a*b % SIMPLE OPERATOR DEFINITIONS. % Numerical factorial. operator nfac$ s(nfac(3),{nfac(0)->1,nfac(?x)->?x*nfac(?x-1)},1); 3*nfac(2) s(nfac(3),{nfac(0)->1,nfac(?x)->?x*nfac(?x-1)},2); 6*nfac(1) si(nfac(3),{nfac(0)->1,nfac(?x)->?x*nfac(?x-1)}); 6 % General factorial. operator gamma,fac; fac(?x _=Natp(?x)) ::- ?x*fac(?x-1); hold(?x*fac(?x - 1)) fac(0) :- 1; 1 fac(?x) :- Gamma(?x+1); gamma(?x + 1) fac(3); 6 fac(3/2); 5 gamma(---) 2 % Legendre polynomials in ?x of order ?n, ?n a natural number. operator legp; legp(?x,0) :- 1; 1 legp(?x,1) :- ?x; ?x legp(?x,?n _=natp(?n)) ::- ((2*?n-1)*?x*legp(?x,?n-1)-(?n-1)*legp(?x,?n-2))/?n; (2*?n - 1)*?x*legp(?x,?n - 1) - (?n - 1)*legp(?x,?n - 2) hold(----------------------------------------------------------) ?n legp(z,5); 4 2 z*(63*z - 70*z + 15) ------------------------ 8 legp(a+b,3); 3 2 2 3 5*a + 15*a *b + 15*a*b - 3*a + 5*b - 3*b --------------------------------------------- 2 legp(x,y); legp(x,y) % TESTS OF EXTENSIONS TO BASIC PATTERN MATCHER. comment *: MSet[?exprn,?val] or ?exprn ::: ?val assigns the value ?val to the projection ?exprn in such a way as to store explicitly each form of ?exprn requested. *; Nosimp('mset,(t t)); Newtok '((!: !: !: !-) Mset); infix :::-; precedence Mset,RSetd; ?exprn :::- ?val ::- (?exprn ::- (?exprn :- ?val )); hold(?exprn::-(?exprn:-?val)) scs := sin(?x)^2 + Cos(?x)^2 -> 1; 2 2 scs := cos(?x) + sin(?x) ->1 % The following pattern substitutes the rule sin^2 + cos^2 into a sum of % such terms. For 2n terms (ie n sin and n cos) the pattern has a worst % case complexity of O(n^3). operator trig,u; trig(?i) :::- Ap(+, Ar(?i,sin(u(?1))^2+Cos(u(?1))^2)); 2 2 hold(trig(?i):-ap(plus,ar(?i,sin(u(?1)) + cos(u(?1)) ))) if si(trig 1,scs) = 1 then write("Pm ok") else Write("PM failed"); Pm ok if si(trig 10,scs) = 10 then write("Pm ok") else Write("PM failed"); Pm ok % The next one takes about 70 seconds on an HP 9000/350, calling UNIFY % 1927 times. % if si(trig 50,scs) = 50 then write("Pm ok") else Write("PM failed"); % Hypergeometric Function simplification. newtok '((!#) !#); *** # redefined flag('(#), 'symmetric); operator #,@,ghg; xx := ghg(4,3,@(a,b,c,d),@(d,1+a-b,1+a-c),1); xx := ghg(4,3,@(a,b,c,d),@(d,a - b + 1,a - c + 1),1) S(xx,sghg(3)); *** sghg declared operator ghg(4,3,@(a,b,c,d),@(d,a - b + 1,a - c + 1),1) s(ws,sghg(2)); ghg(4,3,@(a,b,c,d),@(d,a - b + 1,a - c + 1),1) yy := ghg(3,2,@(a-1,b,c/2),@((a+b)/2,c),1); c a + b yy := ghg(3,2,@(a - 1,b,---),@(-------,c),1) 2 2 S(yy,sghg(1)); c a + b ghg(3,2,@(a - 1,b,---),@(-------,c),1) 2 2 yy := ghg(3,2,@(a-1,b,c/2),@(a/2+b/2,c),1); c a + b yy := ghg(3,2,@(a - 1,b,---),@(-------,c),1) 2 2 S(yy,sghg(1)); c a + b ghg(3,2,@(a - 1,b,---),@(-------,c),1) 2 2 % Some Ghg theorems. flag('(@), 'symmetric); % Watson's Theorem. SGhg(1) := Ghg(3,2,@(?a,?b,?c),@(?d _=?d=(1+?a+?b)/2,?e _=?e=2*?c),1) -> Gamma(1/2)*Gamma(?c+1/2)*Gamma((1+?a+?b)/2)*Gamma((1-?a-?b)/2+?c)/ (Gamma((1+?a)/2)*Gamma((1+?b)/2)*Gamma((1-?a)/2+?c) *Gamma((1-?b)/2+?c)); 1 + ?a + ?b sghg(1) := ghg(3,2,@(?a,?b,?c),@(?d _= ?d=-------------,?e _= ?e=2*?c),1)->( 2 - ?a - ?b + 2*?c + 1 2*?c + 1 gamma(-----------------------)*gamma(----------) 2 2 ?a + ?b + 1 1 - ?a + 2*?c + 1 *gamma(-------------)*gamma(---))/(gamma(------------------) 2 2 2 - ?b + 2*?c + 1 ?a + 1 ?b + 1 *gamma(------------------)*gamma(--------)*gamma(--------)) 2 2 2 % Dixon's theorem. SGhg(2) := Ghg(3,2,@(?a,?b,?c),@(?d _=?d=1+?a-?b,?e _=?e=1+?a-?c),1) -> Gamma(1+?a/2)*Gamma(1+?a-?b)*Gamma(1+?a-?c)*Gamma(1+?a/2-?b-?c)/ (Gamma(1+?a)*Gamma(1+?a/2-?b)*Gamma(1+?a/2-?c)*Gamma(1+?a-?b-?c)); sghg(2) := ghg(3,2,@(?a,?b,?c),@(?d _= ?d=1 + ?a - ?b,?e _= ?e=1 + ?a - ?c),1)-> ?a - 2*?b - 2*?c + 2 (gamma(?a - ?b + 1)*gamma(?a - ?c + 1)*gamma(----------------------) 2 ?a + 2 *gamma(--------))/(gamma(?a - ?b - ?c + 1)*gamma(?a + 1) 2 ?a - 2*?b + 2 ?a - 2*?c + 2 *gamma(---------------)*gamma(---------------)) 2 2 SGhg(3) := Ghg(?p,?q,@(?a,??b),@(?a,??c),?z) -> Ghg(?p-1,?q-1,@(??b),@(??c),?z); sghg(3) := ghg(?p,?q,@(??b,?a),@(??c,?a),?z)->ghg(?p - 1,?q - 1,@(??b),@(??c),?z) SGhg(9) := Ghg(1,0,@(?a),?b,?z ) -> (1-?z)^(-?a); 1 sghg(9) := ghg(1,0,@(?a),?b,?z)->--------------- ?a ( - ?z + 1) SGhg(10) := Ghg(0,0,?a,?b,?z) -> E^?z; ?z sghg(10) := ghg(0,0,?a,?b,?z)->e SGhg(11) := Ghg(?p,?q,@(??t),@(??b),0) -> 1; sghg(11) := ghg(?p,?q,@(??t),@(??b),0)->1 % If one of the bottom parameters is zero or a negative integer the % hypergeometric functions may be singular, so the presence of a % functions of this type causes a warning message to be printed. % Note it seems to have an off by one level spec., so this may need % changing in future. % % Reference: AS 15.1; Slater, Generalized Hypergeometric Functions, % Cambridge University Press,1966. s(Ghg(3,2,@(a,b,c),@(b,c),z),SGhg(3)); ghg(2,1,@(a,b),@(b),z) si(Ghg(3,2,@(a,b,c),@(b,c),z),{SGhg(3),Sghg(9)}); 1 ------------- a ( - z + 1) S(Ghg(3,2,@(a-1,b,c),@(a-b,a-c),1),sghg 2); a - 2*b - 2*c + 1 a + 1 gamma(a - b)*gamma(a - c)*gamma(-------------------)*gamma(-------) 2 2 --------------------------------------------------------------------- a - 2*b + 1 a - 2*c + 1 gamma(a - b - c)*gamma(-------------)*gamma(-------------)*gamma(a) 2 2 end; (TIME: pmrules 650 650)