In a recent report Arnold Doray observed that conjugate sometimes
seems to fail when operating on complicated exact complex numbers:
http://lists.gnu.org/archive/html/axiom-mail/2009-06/msg00027.html
To resolve this problem, the following cumulative patch against FriCAS
adds 'conjugate' and 'norm' for symbolic complex expressions to the to
the patch I sent earlier today.
address@hidden:~$ svn diff
~/fricas-src/src/algebra/efstruc.spad.pamphlet > ctrigmnp.patch
address@hidden:~$ cat ctrigmnp.patchIndex:
/home/wspage/fricas-src/src/algebra/efstruc.spad.pamphlet
===================================================================
--- /home/wspage/fricas-src/src/algebra/efstruc.spad.pamphlet (revision 627)
+++ /home/wspage/fricas-src/src/algebra/efstruc.spad.pamphlet (working copy)
@@ -767,6 +767,7 @@
Implementation ==> add
ker2explogs: (KG, List KG, List SY) -> FG
smp2explogs: (PG, List KG, List SY) -> FG
+ nthroot: (GF,GF) -> GF
supexp : (UP, GF, GF, Z) -> GF
GR2GF : GR -> GF
GR2F : GR -> F
@@ -847,13 +848,24 @@
map(x +-> explogs2trigs(x::FG), GR2GF, p)$PolynomialCategoryLifting(
IndexedExponents KG, KG, GR, PG, GF)
- explogs2trigs f ==
- (m := mainKernel f) case "failed" =>
+ -- De Moivre's theorem
+ nthroot(a:GF,n:GF):GF ==
+ r := nthRoot(sqrt(norm(a)),retract(n)@Z)
+ e := exp(complex(0, 1)*argument(a)/n)
+ r*e
+
+ explogs2trigs(f:FG):GF ==
+ m := mainKernel f
+ m case "failed" =>
GR2GF(retract(numer f)@GR) / GR2GF(retract(denom f)@GR)
- op := operator(operator(k := m::KG))$F
+ k := m::KG
+ op := operator(operator(k))$F
arg := [explogs2trigs x for x in argument k]
num := univariate(numer f, k)
den := univariate(denom f, k)
+ is?(op,'nthRoot) =>
+ h := nthroot(first arg, second arg)
+ supexp(num,h,0,0)/supexp(den,h,0,0)
is?(op, 'exp) =>
e := exp real first arg
y := imag first arg
@@ -926,6 +938,10 @@
imag : F -> F
++ imag(f) returns the imaginary part of \spad{f} where \spad{f}
++ is a complex function.
+ conjugate:F-> F
+ ++ conjugate(x+%i*y) returns x-%i*y
+ norm : F -> F
+ ++ norm(f) returns f*conjugate(f)
real? : F -> Boolean
++ real?(f) returns \spad{true} if \spad{f = real f}.
complexForm: F -> Complex F
@@ -952,6 +968,8 @@
real? f == empty?(complexKernels(f).ker)
real f == real complexForm f
imag f == imag complexForm f
+ conjugate f == real(f) - sqrt(-1) * imag(f)
+ norm f == f * conjugate(f)
-- returns [[k1,...,kn], [v1,...,vn]] such that ki should be replaced by vi
complexKernels f ==
@@ -1067,6 +1085,10 @@
imag : F -> FR
++ imag(f) returns the imaginary part of \spad{f} where \spad{f}
++ is a complex function.
+ conjugate:F-> F
+ ++ conjugate(x+%i*y) returns x-%i*y
+ norm : F -> FR
+ ++ norm(f) returns f*conjugate(f)
real? : F -> Boolean
++ real?(f) returns \spad{true} if \spad{f = real f}.
trigs : F -> F
@@ -1087,6 +1109,8 @@
real f == real complexForm f
imag f == imag complexForm f
+ conjugate f == F2FG real(f) - complex(0,1) * F2FG imag(f)
+ norm f == FG2F(f * conjugate f)
rreal? r == zero? imag r
kreal? k == every?(real?, argument k)$List(F)
complexForm f == explogs2trigs f
address@hidden:~$
---
Note: These functions treat symbols appearing in values of type
'Expression Complex R' as variables of type R.
(1) -> conjugate(x+%i*y)
(1) - %i y + x
Type: Expression(Complex(Integer))
(2) -> norm(x+%i*y)
2 2
(2) y + x
Type: Expression(Integer)
(3) -> conjugate(sqrt(x))
+-+
(3) \|x
Type: Expression(Integer)
(4) -> conjugate(sqrt(x*%i))
+--+ +--+
| 2 4| 2
(\|x - %i x)\|x
(4) -------------------
+--+
+-+ | 2
\|2 \|x
Type: Expression(Complex(Integer))
(5) ->
Further testing is encouraged.
Regards,
Bill Page.
------------------------------------------------------------------------
_______________________________________________
Axiom-developer mailing list
address@hidden
http://lists.nongnu.org/mailman/listinfo/axiom-developer