[Axiom-developer] Fix for bug 294

[Waldek Hebisch]

Below is a fix for bug 294.  Some extra comments:

1) the patch correct problem in goodCoef, there are other bugs in
   integrator, I will work on them separately.
2) AFAICS goodCoef is an optimization, it should be safe to return
   "failed" always (but since we take different path in the other
   part we hit different bugs).  I tried to preserve optimization
   done by goodCoef whenever it is safe.
3) I would like to use higher level constructs, but I keept hitting
   into problems with SPAD compiler, so finally I used a rather low
   level way
4) without the patch the AXIOM recurses infinitely handling the 
   few integrals below (first I previosly gave in the bug report).
   With the patch the first two are correctly handled, the other
   two quickly hit into unimplemented part of Risch algorithm:

integrate(simplify(D((log((x)+1)+a)/(2*((((x*(exp(exp(x)/2)))-(a*(x+exp(-x/2))))*(2/x))-3)),
 x)), x)

integrate(simplify(D((((((sqrt((sqrt((x)/(x)))/(a)))/(sqrt(x)))*((((x)-(1))*(1))-(1)))*(a))*(((log((x)*((a)/(log(sqrt((x)/(((x)*(x))*(x))))))))+(((x)/(sqrt(2)))*(exp(exp(x)))))+((x)-(exp(2)))))+((1)*(2)),
 x)), x)

-- Function not supported by Risch d.e.
integrate(simplify(D(((((((a)/(sqrt((sqrt(x))-(x))))+(x))*(x))+(1))*((((((1)-(log(x)))+(1))*(exp(exp(2))))*(2))*(((1)*((log((a)*(1)))*(a)))*(exp(((exp(x))*(x))-(exp(x)))))))*(a),
 x)), x)
-- intef: failed1 - cannot handle that integrand
integrate(simplify(D((((((((x)+(log(x)))*(((x)/(x))/(sqrt(x))))/((a)*(x)))-(sqrt(sqrt(((exp(x))/((x)-(sqrt(x))))-(log(x))))))-(x))-(((exp((sqrt(log(1)))*((x)*((x)/(log(x))))))/(exp((1)+((2)-((x)-(x))))))*(((((x)/(1))-(log(x)))*(1))+(((2)+(a))/(exp(sqrt(x)))))))/(2),
 x)), x)


------------ The patch ---------------


--- pp/build-improvements/src/algebra/efstruc.spad.pamphlet     2006-09-08 
02:57:48.000000000 +0200
+++ build-improvements.nn5/src/algebra/efstruc.spad.pamphlet    2006-10-07 
19:51:18.000000000 +0200
@@ -347,8 +347,28 @@
       true
 
     goodCoef(v, l, s) ==
-      for i in minIndex v .. maxIndex v for k in l repeat
-        is?(k, s) and
+      -- if k1 is part of k2 we should not express k1 in terms of k2
+      -- (othewise we would get infinite recursion)
+      -- below we impose a stronger condition: we require
+      -- height(k1) to be maximal
+      h:NonNegativeInteger := 0
+      j:Integer := 0
+      ll : List K := [];
+      for k in l repeat
+        if (is?(k, "log"::SY) or is?(k, "exp"::SY)
+            or is?(k, "tan"::SY) or is?(k, "atan"::SY)) then
+              ll := [k, :ll]
+              h := h + 1
+      not (h = (maxIndex(v) - minIndex(v) + 1)) => "failed"
+      h := 0
+      ll := reverse(ll)
+      for i in minIndex v .. maxIndex v for k in ll repeat
+        h1 := height(k)
+        if (h1 > h) then
+          j := i
+          h := h1
+      for i in minIndex v .. maxIndex v for k in ll repeat
+        is?(k, s) and (i >= j) and
            ((r:=recip(qelt(v,i))) case Q) and
             (retractIfCan(r::Q)@Union(Z, "failed") case Z)
               and gdCoef?(qelt(v, i), v) => return([i, k])

-- 
                              Waldek Hebisch
address@hidden