On Sat, Dec 02, 2023 at 09:42:03AM +0800, Qian Yun wrote:
>
> On 12/1/23 08:58, Waldek Hebisch wrote:
> >
> > I am not sure if you want to do them. I could do first two and
> > third for rational coefficients. Or maybe you want more general
> > code.
> >
> >
>
> You should do them, and if possible, commit separately for
> different cases -- it's easier for me to testing against Nasser's
> list.
I prefer commit code when reasonably complete and ready.
If you want to test attached is my first shot at x^4 + a case
with negative a. It seems to be rare, but it was easy and
will share most code with case of positive a.
Example:
(1) -> integrate(1/(x^4-2),x)
4+-+
4+-+ 4+-+ x\|8
- log(x\|8 + 2) + log(x\|8 - 2) - 2 atan(-----)
2
(1) -------------------------------------------------
4+-+
4 \|8
Type: Union(Expression(Integer),...)
(before there were two complex logs).
--
Waldek Hebisch
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diff --git a/src/algebra/irexpand.spad b/src/algebra/irexpand.spad
index ead11d7f..7a929584 100644
--- a/src/algebra/irexpand.spad
+++ b/src/algebra/irexpand.spad
@@ -150,10 +150,35 @@ IntegrationResultToFunction(R, F) : Exports == Implementation where
bb := -(r.coef1) k
tantrick(aa * a + bb * b, g) + ilog0(aa, bb, r.coef2, -r.coef1, k)
+ -- Computes (a + b*%i)log(lg(a + b*%i)) + (a - b*%i)log(lg(a - b*%i)
+ -- using Rioboo transformation)
+ root_pair(lg : UP, a : F, b : F, x : Symbol) : F ==
+ lge := quadeval(lg, a, b, -1)
+ f := lge.ans1
+ g := lge.ans2
+ a*log(f*f + g*g) + b*ilog(f, g, x)
+
+ lg2cfunc2 : (UP, UP) -> F
+
+ quartic(p : UP, lg : UP, x : Symbol) : List(F) ==
+ ground?(rp := reductum(p)) =>
+ a := ground(rp)
+ s := sign(a)
+ s case "failed" => [lg2cfunc2(p, lg)]
+ si := s@Integer
+ si = 1 =>
+ [lg2cfunc2(p, lg)]
+ si = -1 =>
+ r1 := rootSimp(zeroOf(p))
+ [cmplex(r1, lg) + cmplex(-r1, lg) + root_pair(lg, 0, r1, x)]
+ error "impossible"
+ [lg2cfunc2(p, lg)]
+
lg2func(lg, x) ==
zero?(d := degree(p := lg.coeff)) => error "poly has degree 0"
(d = 1) => [linear(p, lg.logand)]
d = 2 => quadratic(p, lg.logand, x)
+ d = 4 => quartic(p, lg.logand, x)
odd? d and
((r := retractIfCan(reductum p)@Union(F, "failed")) case F) =>
pairsum([cmplex(alpha := rootSimp zeroOf p, lg.logand)],
@@ -162,8 +187,10 @@ IntegrationResultToFunction(R, F) : Exports == Implementation where
lg.logand], x))
[lg2cfunc lg]
- lg2cfunc lg ==
- +/[cmplex(alpha, lg.logand) for alpha in zerosOf(lg.coeff)]
+ lg2cfunc(lg) == lg2cfunc2(lg.coeff, lg.logand)
+
+ lg2cfunc2(p : UP, lg : UP) ==
+ +/[cmplex(alpha, lg) for alpha in zerosOf(p)]
mkRealFunc(l, x) ==
ans := empty()$List(F)
