Waldek Hebisch <[EMAIL PROTECTED]> writes:
> > * the patch by Francois Maltey for asin, acos, etc.
> >
>
> Somebody should check (re-check ?) branch cuts and singularities with this
> patch.
(for convenience, I include the patch below)
I guess the right place to check branch cuts and singularities is in
src/input/elemnum.input. I looked at the common lisp hyperspec, the definition
of branchcuts is quite clear there, so I suggest that we take it as definition,
unless you have a better reference.
Now, how to test the branch cuts? I do not really know, but I guess we have to
test both real and complex float and dfloat domains.
Eg., the hyperspec for asin, acos, atan says:
The following definition for arc sine determines the range and branch cuts:
arcsin z = -i log (iz+sqrt(1-z^2))
The branch cut for the arc sine function is in two pieces: one along the
negative real axis to the left of -1 (inclusive), continuous with quadrant II,
and one along the positive real axis to the right of 1 (inclusive), continuous
with quadrant IV. The range is that strip of the complex plane containing
numbers whose real part is between -<PI>/2 and <PI>/2. A number with real part
equal to -<PI>/2 is in the range if and only if its imaginary part is
non-negative; a number with real part equal to <PI>/2 is in the range if and
only if its imaginary part is non-positive.
so we need to test:
real < -1 error
> +1 error
= -1 -%pi/2
= +1 -%pi/2
complex x + 0*%i, x < -1 close to x + 0.1*%i
complex x + 0*%i, x > 1 close to x - 0.1*%i
Is this correct? Are there sensible values for x I could check?
Martin
Index: float.spad.pamphlet
===================================================================
--- float.spad.pamphlet (revision 442)
+++ float.spad.pamphlet (working copy)
@@ -455,7 +455,7 @@
normalize r
atanh x ==
- if x > 1 or x < -1 then error "invalid argument to atanh"
+ if x >= 1 or x <= -1 then error "invalid argument to atanh"
p := min(0,order x)
if zero? x or 2*p < -bits() then return x
inc(5-p); r := log((x+1)/(1-x))/2; dec(5-p)
Index: trigcat.spad.pamphlet
===================================================================
--- trigcat.spad.pamphlet (revision 442)
+++ trigcat.spad.pamphlet (working copy)
@@ -136,7 +136,10 @@
(a := recip x) case "failed" => error "acoth: no reciprocal"
atanh(a::$)
if $ has Field and $ has sqrt: $ -> $ then
- asin x == atan(x/sqrt(1-x**2))
+ asin x ==
+ one?(x) => 2*atan 1
+ one?(-x) => -2*atan 1
+ atan(x/sqrt(1-x**2))
acos x == pi()/2::$ - asin x
acot x == pi()/2::$ - atan x
asinh x == log(x + sqrt(x**2 + 1))
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