packages/DiscreteMath/RSolve.m

93 internalSeriesTerm calls InverseZTransform if
237 MethodEGF does not use Apart at all."
265 n is not numerical."
269 n is not numerical."
354 MethodEGF does not use Apart at all."
366 	   method, rsolvec, if, sf, methods,
378                     if inconsistent *)
386         {method, rsolvec, if, sf} =
398 				 rsolvec, if, sf};
419 RSolve::eqn = "`` is not an equation or a system of equations."
443 			          {}, if, sf];
453                 temp still contains that internal symbol (it should not).
487 			         {}, if, sf];
495                 temp still contains that internal symbol (it should not).
550         (* substitute values explicitly given by initial conditions *)
790     Block[{e, e0, sum, if, result},
794 		      e = expr //. {Sum :> sum, If :> if};
799 		      e0 = e0 /. {sum :> Sum, if :> If};
822  Module[{e, sum, if},
824     e = expr //. {Sum :> sum, If :> if};
828     e = e //. {sum :> Sum, if :> If};
1078     Block[{aa, bb, nfree, nfull, s, expand, ips, result, sum, if},
1088 	  nfull = nfull /. {Sum :> sum, If :> if};
1179     Block[{t, tt, if, aa = a, bb = b, result, s, k, sum},
1181 	aa = aa /. {If :> if};
1182 	bb = bb /. {If :> if};
1189 	result /. {if -> If}
1244     Block[{e, e0, sum, if, result},
1248 	      e = expr //. {Sum :> sum, If :> if};
1253 	      e0 = e0 /. {sum :> Sum, if :> If};
1280  Module[{e, sum, if},
1282     e = expr //. {Sum :> sum, If :> if};
1286     e = e //. {sum :> Sum, if :> If};
1351     Block[{aa, bb, e, sum, if},
1353 	e = expr /. {Sum :> sum, If :> if};
1355 	e = e /. {sum :> Sum, if :> If};
1364     Block[{aa, bb, nfree, nfull, s, expand, ieps, result, sum, if},
1369 	nfull = nfull /. {Sum :> sum, If :> if};
1373 	result = result /. {sum :> Sum, if :> If};
1522 GeneratingFunction::eqn = "`` is not an equation or a system of equations."
1585 "`` is not an equation or a system of equations."
1771 SeriesTerm::badopt = "`` -> `` is not a valid option."
1800     Block[{result, oList, assmp, if, sf},
1805 	       (* Issue obsolete messages if obsolete options are in use. *)
1813      	       {assmp, if, sf} =
1816 	       (* Note that if SeriesTerm is called from iMethodGF or
1818 			in oList.  RSolve does not support the Assumptions
1821                                 assmp, if, sf]) =!= $Failed &&
1829 		 do not evaluate using rules for symbolic n. *)
1879 	internalSeriesTerm[#, {z, a, n}, assmp, if, sf]& /@ expr
1890 		and do not belong in the variable list of Block. *)
1891      {integerFunctions, specialFunctions} = {if, sf};
2088 (* Apply iSeriesTerm to sum only if for each term
2128     (* NOTE: returning "ist" is OK if ist is naturally 0 for n < k.
2143 	 abstract n, because Series[z^k f, {z, 0, n}] does not evaluate for
2149     (* NOTE: returning "ist" is OK if ist is naturally 0 for n < k.
2165 				integral or not. *)
2187 	simplified if both rules are present *)
2215 	 condition, because it is not otherwise zero for negative n.  *)
2309   Module[{nlist, dlist, plist, k0, k1, expAlpha, beta, omega, sinn, hyperbolic},
2312     If[Length[nlist] == 1, {k0, k1} = {nlist[[1]], 0},
2313         {k0, k1} = nlist];
2343                 nlist = CoefficientList[np, z];  variable
2345                 Length[nlist] <= 2 && (plist = parm[dlist]) =!= $Failed,
2347                 nlist = CoefficientList[z np, z];  variable
2349                 Length[nlist] <= 2 && (plist = parm[dlist]) =!= $Failed,
2351                 nlist = CoefficientList[z^2 np, z];  variable
2353                 Length[nlist] <= 2 && (plist = parm[dlist]) =!= $Failed,
2621 	(* NOTE that the following expression is not zero for n < 0. *)
2685 	  (* ``a'' is a ratio of polynomials in z, ``b'' is not *)
2698    but not for
2858 	   xx1, xx2, yy0, yy1, temp, if, sum},
2862       f = f //. {HoldPattern[If[c_, d_, e_]] :> if[c, d, e],
2871       If[MatchQ[f, x1_. (x2_. + if[K[sd] == y0_, y1_, 0]) /;
2880       temp //. {if[c_, d_, e_] :> If[c, d, e], sum[e___] :> Sum[e]}
2916 	  (* ``a'' is a ratio of polynomials in z, ``b'' is not *)
3117        possibility for uncontrolled loops if it didn't really expand,
3308   Module[{if, sum, result},
3309 	result = expr //. {If :> if, Sum :> sum};
3317                  if[aa_, b_, c_] :> Module[{isolve = ISolve[aa]},
3319 					if[isolve, b, c]
3324                 {if[Odd[m_], if[m_ >= aa_, b_, c_] d_., e_] :>
3325                      if[Odd[m], Expand[b d], e] /;
3327                  if[Even[m_], if[m_ >= aa_, b_, c_] d_., e_] :>
3328                      if[Even[m], Evaluate[Expand[b d]], e] /;
3333 		{a_. if[cond_, b_, c_] + d_ if[cond_, e_, f_] :>
3334 		   a if[cond,
3336 		 a_. if[cond_, b_, c_] + d_ if[cond_, e_, f_] :>
3337 		   if[cond,
3339 		 a_. if[cond_, b_, c_] + a_. if[cond_, e_, f_] :>
3340 		   a if[cond,
3342 		 a_. if[m_ >= 0, if[Even[m_], b_, 0], 0] +
3343 		   e_. if[m_ >= 1, if[Even[m_ - 1], f_, 0], 0] :>
3344 			if[m >= 0, if[Even[m], a b, e f], 0],
3345 		 c1_. if[m_ >= m1_, a1_, 0] + c2_. if[m_ >= m2_, a2_, 0] :>
3346 			if[m >= m1, Simplify[c1 a1 + c2 a2], 0]  /; (m2 > m1 &&
3352 	result = result //. {sum :> Sum, if :> If};
3413   Module[{simplifySumRules, if, sum, result},
3421     sum[if[k_ == m_, a_, b_] c_., {k_, m_, n_}] :>
3426     sum[(if[-k_ + n_ >= 0, a_, b_] c_. + d_.) e_., {k_, m_, n_}] :>
3430     sum[(if[k_ >= m_, a_, b_] c_. + d_.) e_., {k_, m_, n_}] :>
3434     sum[if[Even[p_], a_, 0] if[Even[k_], b_, 0], {k_, m_, n_}] :>
3436 	 if[Even[n], sum[if[Even[k], Simplify[a b], 0],
3439     sum[if[Odd[p_], a_, 0] if[Odd[k_], b_, 0], {k_, m_, n_}] :>
3441 	 if[Even[n], sum[if[Odd[k], Simplify[a b], 0],
3444     sum[if[Even[p_], a_, 0] if[Odd[k_], b_, 0], {k_, m_, n_}] :>
3446 	 if[Odd[n], sum[if[Odd[k], Simplify[a b], 0],
3449     sum[if[Odd[p_], a_, 0] if[Even[k_], b_, 0], {k_, m_, n_}] :>
3451 	 if[Odd[n], sum[if[Even[k], Simplify[a b], 0],
3456 	result = expr //. {If :> if, Sum :> sum};
3458 	result //. {sum :> Sum, if :> If}
3534 		(* NOTE: the above is not needed as long as
3570 matches pattern, or Null if there is no such element.
3577 FreeListQ[expr, list] returns True if none of the elements of list appears
3861 MakeList[expr, head:List] returns the list of arguments of expr if the head
3909 	   nlist, sum, maxindices0, scan, tempRecur,
3937            nlist = Map[Cases[initconds, #[x_] -> x, Infinity]&,  variable
3939 	   nlist = Map[Max, nlist] + 1;  variable
3944 	   diff = nlist-maxindices0;
3946            If[ maxindices0 =!= nlist,
3950 		            If[MaxIndices[tempRecur, n, unknowns] === nlist,
3955                  (* could not find a single shift appropriate for shifting
3962 				 Transpose[{unknowns, nlist}] ]
3969 				 If[MaxIndices[tempRecur, n, unknowns]===nlist,
3974 		    (* could not find a set of shifts appropriate for
3980                                  Transpose[{unknowns, nlist}] ]
3985 	   ]; (* end If maxindices0 =!= nlist *)
4049 		(* Check that equation eq does not contain u[k],
4080 that match the given pattern, provided that expr does not contain Rule[].
4140 SafeFirst[l] returns the first argument of l, or Null if the length of l is
4237 TTQ[cond] returns True if it determines that the information given about the
4320 	(* NOTE: do not want to do Together[Sqrt[4 + x^2] +