In Dyalog:
¯7○¯2
0.5493061443J¯1.570796327
In GNU APL:
¯7○¯2
¯0.5493061443J1.570796327
In Common Lisp:
* (atanh -2)
#C(-0.54930615 1.5707964)
In Wolfram Alpha:
arctanh(-2)
Decimal approximation:
-0.54930614433405484569762261846126285232374527891137472586734716... + 1.5707963267948966192313216916397514420985846996875529104874722... i
Why is Dyalog's result negated and conjugated?
Andrew
On Saturday, October 10, 2020 at 6:24:33 PM UTC-4, Sam Sirlin wrote:
On Friday, October 9, 2020 at 5:37:44 AM UTC-7, Andrew Sengul wrote:
In Dyalog:
¯7○¯2
0.5493061443J¯1.570796327
In GNU APL:
¯7○¯2
¯0.5493061443J1.570796327
In Common Lisp:
* (atanh -2)
#C(-0.54930615 1.5707964)
In Wolfram Alpha:
arctanh(-2)
Decimal approximation:
-0.54930614433405484569762261846126285232374527891137472586734716... + 1.5707963267948966192313216916397514420985846996875529104874722... i
Why is Dyalog's result negated and conjugated?
AndrewA Bug. Is the tanh of the result correct? In J
_7 o. _2
_0.549306j1.5708
a=. _7 o. _2
7 o. a
_2
7 o. -a
2
Also in octave:
octave:2> a = atanh(-2)
a = -5.4931e-01 + 1.5708e+00i
octave:3> tanh(a)
ans = -2.0000e+00 + 1.8370e-16i
octave:4> tanh(-a)
ans = 2.0000e+00 - 1.8370e-16i
Note there is ambiguity on the imaginary part:
octave:12> A = log( 1/sqrt(3))Circular and hyperbolic functions are all periodic, so for each such function f ,
A = -5.4931e-01
octave:16> tanh( A + i*pi/2)
ans = -2.0000e+00 + 1.8370e-16i
octave:17> tanh( A - i*pi/2)
ans = -2.0000e+00 - 1.8370e-16i
there are infinitely many choice for f x for each x. The same is true (in the complex
plane) for the exponential function. It's not possible to define inverses for these
functions that are both single valued and continuous on the complex plane. It is
possible, though too specify where the discontinuities should be, and what values
should be taken at points of discontinuity, in a way that is consistent across all
the inverses.
IIRC the APL Extended standard follows Paul Penfield's APL 81 paper "Principal
values and branch cuts in complex APL" in specifying how these choice are to be
made, where the discontinuities should be and values at the points of discontinuity.
There is a long discussion (which cites and is largely based on the Penfield paper),
with illustrations, in Guy Steele's "Common Lisp, The Language (2nd Ed.)", at section 12.5.3.
IMHO the J implementation is almost certainly correct, because ... Roger.
On Friday, October 9, 2020 at 5:37:44 AM UTC-7, Andrew Sengul wrote:
In Dyalog:
¯7○¯2
0.5493061443J¯1.570796327
In GNU APL:
¯7○¯2
¯0.5493061443J1.570796327
In Common Lisp:
* (atanh -2)
#C(-0.54930615 1.5707964)
In Wolfram Alpha:
arctanh(-2)
Decimal approximation:
-0.54930614433405484569762261846126285232374527891137472586734716... + 1.5707963267948966192313216916397514420985846996875529104874722... i
Why is Dyalog's result negated and conjugated?
AndrewA Bug. Is the tanh of the result correct? In J
_7 o. _2
_0.549306j1.5708
a=. _7 o. _2
7 o. a
_2
7 o. -a
2
Also in octave:
octave:2> a = atanh(-2)
a = -5.4931e-01 + 1.5708e+00i
octave:3> tanh(a)
ans = -2.0000e+00 + 1.8370e-16i
octave:4> tanh(-a)
ans = 2.0000e+00 - 1.8370e-16i
Note there is ambiguity on the imaginary part:
octave:12> A = log( 1/sqrt(3))
A = -5.4931e-01
octave:16> tanh( A + i*pi/2)
ans = -2.0000e+00 + 1.8370e-16i
octave:17> tanh( A - i*pi/2)
ans = -2.0000e+00 - 1.8370e-16i
On Sunday, October 11, 2020 at 12:29:47 PM UTC-4, eki...@gmail.com wrote:
On Saturday, October 10, 2020 at 6:24:33 PM UTC-4, Sam Sirlin wrote:
On Friday, October 9, 2020 at 5:37:44 AM UTC-7, Andrew Sengul wrote:Circular and hyperbolic functions are all periodic, so for each such function f ,
In Dyalog:A Bug. Is the tanh of the result correct? In J
¯7○¯2
0.5493061443J¯1.570796327
In GNU APL:
¯7○¯2
¯0.5493061443J1.570796327
In Common Lisp:
* (atanh -2)
#C(-0.54930615 1.5707964)
In Wolfram Alpha:
arctanh(-2)
Decimal approximation:
-0.54930614433405484569762261846126285232374527891137472586734716... + >>>> 1.5707963267948966192313216916397514420985846996875529104874722... i
Why is Dyalog's result negated and conjugated?
Andrew
_7 o. _2
_0.549306j1.5708
a=. _7 o. _2
7 o. a
_2
7 o. -a
2
Also in octave:
octave:2> a = atanh(-2)
a = -5.4931e-01 + 1.5708e+00i
octave:3> tanh(a)
ans = -2.0000e+00 + 1.8370e-16i
octave:4> tanh(-a)
ans = 2.0000e+00 - 1.8370e-16i
Note there is ambiguity on the imaginary part:
octave:12> A = log( 1/sqrt(3))
A = -5.4931e-01
octave:16> tanh( A + i*pi/2)
ans = -2.0000e+00 + 1.8370e-16i
octave:17> tanh( A - i*pi/2)
ans = -2.0000e+00 - 1.8370e-16i
there are infinitely many choice for f x for each x. The same is true (in the complex
plane) for the exponential function. It's not possible to define inverses for these
functions that are both single valued and continuous on the complex plane. It is
possible, though too specify where the discontinuities should be, and what values
should be taken at points of discontinuity, in a way that is consistent across all
the inverses.
IIRC the APL Extended standard follows Paul Penfield's APL 81 paper "Principal
values and branch cuts in complex APL" in specifying how these choice are to be
made, where the discontinuities should be and values at the points of discontinuity.
There is a long discussion (which cites and is largely based on the Penfield paper),
with illustrations, in Guy Steele's "Common Lisp, The Language (2nd Ed.)", >> at section 12.5.3.
IMHO the J implementation is almost certainly correct, because ... Roger.
Or maybe not, since J and CL differ.
On 10/11/20 9:45 AM, eki...@gmail.com wrote:
On Sunday, October 11, 2020 at 12:29:47 PM UTC-4, eki...@gmail.com wrote:
On Saturday, October 10, 2020 at 6:24:33 PM UTC-4, Sam Sirlin wrote:
On Friday, October 9, 2020 at 5:37:44 AM UTC-7, Andrew Sengul wrote: >>>> In Dyalog:Circular and hyperbolic functions are all periodic, so for each such function f ,
¯7○¯2A Bug. Is the tanh of the result correct? In J
0.5493061443J¯1.570796327
In GNU APL:
¯7○¯2
¯0.5493061443J1.570796327
In Common Lisp:
* (atanh -2)
#C(-0.54930615 1.5707964)
In Wolfram Alpha:
arctanh(-2)
Decimal approximation:
-0.54930614433405484569762261846126285232374527891137472586734716... + >>>> 1.5707963267948966192313216916397514420985846996875529104874722... i >>>>
Why is Dyalog's result negated and conjugated?
Andrew
_7 o. _2
_0.549306j1.5708
a=. _7 o. _2
7 o. a
_2
7 o. -a
2
Also in octave:
octave:2> a = atanh(-2)
a = -5.4931e-01 + 1.5708e+00i
octave:3> tanh(a)
ans = -2.0000e+00 + 1.8370e-16i
octave:4> tanh(-a)
ans = 2.0000e+00 - 1.8370e-16i
Note there is ambiguity on the imaginary part:
octave:12> A = log( 1/sqrt(3))
A = -5.4931e-01
octave:16> tanh( A + i*pi/2)
ans = -2.0000e+00 + 1.8370e-16i
octave:17> tanh( A - i*pi/2)
ans = -2.0000e+00 - 1.8370e-16i
there are infinitely many choice for f x for each x. The same is true (in the complex
plane) for the exponential function. It's not possible to define inverses for these
functions that are both single valued and continuous on the complex plane. It is
possible, though too specify where the discontinuities should be, and what values
should be taken at points of discontinuity, in a way that is consistent across all
the inverses.
IIRC the APL Extended standard follows Paul Penfield's APL 81 paper "Principal
values and branch cuts in complex APL" in specifying how these choice are to be
made, where the discontinuities should be and values at the points of discontinuity.
There is a long discussion (which cites and is largely based on the Penfield paper),
with illustrations, in Guy Steele's "Common Lisp, The Language (2nd Ed.)",
at section 12.5.3.
IMHO the J implementation is almost certainly correct, because ... Roger.
Or maybe not, since J and CL differ.
J and CL are the same: negative real part, positive imaginary.
On 10/11/20 9:45 AM, eki...@gmail.com wrote:
On Sunday, October 11, 2020 at 12:29:47 PM UTC-4, eki...@gmail.com wrote:
On Saturday, October 10, 2020 at 6:24:33 PM UTC-4, Sam Sirlin wrote:
On Friday, October 9, 2020 at 5:37:44 AM UTC-7, Andrew Sengul wrote: >>>> In Dyalog:Circular and hyperbolic functions are all periodic, so for each such function f ,
¯7○¯2A Bug. Is the tanh of the result correct? In J
0.5493061443J¯1.570796327
In GNU APL:
¯7○¯2
¯0.5493061443J1.570796327
In Common Lisp:
* (atanh -2)
#C(-0.54930615 1.5707964)
In Wolfram Alpha:
arctanh(-2)
Decimal approximation:
-0.54930614433405484569762261846126285232374527891137472586734716... + >>>> 1.5707963267948966192313216916397514420985846996875529104874722... i >>>>
Why is Dyalog's result negated and conjugated?
Andrew
_7 o. _2
_0.549306j1.5708
a=. _7 o. _2
7 o. a
_2
7 o. -a
2
Also in octave:
octave:2> a = atanh(-2)
a = -5.4931e-01 + 1.5708e+00i
octave:3> tanh(a)
ans = -2.0000e+00 + 1.8370e-16i
octave:4> tanh(-a)
ans = 2.0000e+00 - 1.8370e-16i
Note there is ambiguity on the imaginary part:
octave:12> A = log( 1/sqrt(3))
A = -5.4931e-01
octave:16> tanh( A + i*pi/2)
ans = -2.0000e+00 + 1.8370e-16i
octave:17> tanh( A - i*pi/2)
ans = -2.0000e+00 - 1.8370e-16i
there are infinitely many choice for f x for each x. The same is true (in the complex
plane) for the exponential function. It's not possible to define inverses for these
functions that are both single valued and continuous on the complex plane. It is
possible, though too specify where the discontinuities should be, and what values
should be taken at points of discontinuity, in a way that is consistent across all
the inverses.
IIRC the APL Extended standard follows Paul Penfield's APL 81 paper "Principal
values and branch cuts in complex APL" in specifying how these choice are to be
made, where the discontinuities should be and values at the points of discontinuity.
There is a long discussion (which cites and is largely based on the Penfield paper),
with illustrations, in Guy Steele's "Common Lisp, The Language (2nd Ed.)",
at section 12.5.3.
IMHO the J implementation is almost certainly correct, because ... Roger.
Or maybe not, since J and CL differ.
J and CL are the same: negative real part, positive imaginary.
On Sunday, October 11, 2020 at 10:55:42 PM UTC-4, Andrew Sengul wrote:
On 10/11/20 9:45 AM, eki...@gmail.com wrote:
On Sunday, October 11, 2020 at 12:29:47 PM UTC-4, eki...@gmail.com wrote:
On Saturday, October 10, 2020 at 6:24:33 PM UTC-4, Sam Sirlin wrote: >>> On Friday, October 9, 2020 at 5:37:44 AM UTC-7, Andrew Sengul wrote: >>>> In Dyalog:
Circular and hyperbolic functions are all periodic, so for each such function f ,¯7○¯2A Bug. Is the tanh of the result correct? In J
0.5493061443J¯1.570796327
In GNU APL:
¯7○¯2
¯0.5493061443J1.570796327
In Common Lisp:
* (atanh -2)
#C(-0.54930615 1.5707964)
In Wolfram Alpha:
arctanh(-2)
Decimal approximation:
-0.54930614433405484569762261846126285232374527891137472586734716... +
1.5707963267948966192313216916397514420985846996875529104874722... i >>>>
Why is Dyalog's result negated and conjugated?
Andrew
_7 o. _2
_0.549306j1.5708
a=. _7 o. _2
7 o. a
_2
7 o. -a
2
Also in octave:
octave:2> a = atanh(-2)
a = -5.4931e-01 + 1.5708e+00i
octave:3> tanh(a)
ans = -2.0000e+00 + 1.8370e-16i
octave:4> tanh(-a)
ans = 2.0000e+00 - 1.8370e-16i
Note there is ambiguity on the imaginary part:
octave:12> A = log( 1/sqrt(3))
A = -5.4931e-01
octave:16> tanh( A + i*pi/2)
ans = -2.0000e+00 + 1.8370e-16i
octave:17> tanh( A - i*pi/2)
ans = -2.0000e+00 - 1.8370e-16i
there are infinitely many choice for f x for each x. The same is true (in the complex
plane) for the exponential function. It's not possible to define inverses for these
functions that are both single valued and continuous on the complex plane. It is
possible, though too specify where the discontinuities should be, and what values
should be taken at points of discontinuity, in a way that is consistent across all
the inverses.
IIRC the APL Extended standard follows Paul Penfield's APL 81 paper "Principal
values and branch cuts in complex APL" in specifying how these choice are to be
made, where the discontinuities should be and values at the points of discontinuity.
There is a long discussion (which cites and is largely based on the Penfield paper),
with illustrations, in Guy Steele's "Common Lisp, The Language (2nd Ed.)",
at section 12.5.3.
IMHO the J implementation is almost certainly correct, because ... Roger.
Or maybe not, since J and CL differ.
J and CL are the same: negative real part, positive imaginary.My mistake. Back to my original conclusion: J and CL are (and GNU APL, Octave, and W-ALPHA) , with probability VERY close to 1, consistent across the ln / circular / hyperbolic fns, and so "correct" in a way that Dyalog is not.
Has anyone checked APL2?
In Dyalog:
¯7○¯2
0.5493061443J¯1.570796327
In GNU APL:
¯7○¯2
¯0.5493061443J1.570796327
In Common Lisp:
* (atanh -2)
#C(-0.54930615 1.5707964)
In Wolfram Alpha:
arctanh(-2)
Decimal approximation:
-0.54930614433405484569762261846126285232374527891137472586734716... + 1.5707963267948966192313216916397514420985846996875529104874722... i
Why is Dyalog's result negated and conjugated?
Andrew
The Dyalog implementation follows Penfield 1981 (_Principal Values and Branch Cuts in Complex APL_, https://doi.org/10.1145/800142.805368), equation 12:
arctanh←{⍟(1+⍵)×(÷1-⍵*2)*0.5}
arctanh ¯2
¯0.5493061443J¯1.570796327
¯7 ○ ¯2
¯0.5493061443J¯1.570796327
On 10/27/2020 3:58 PM, Roger Hui wrote:
The Dyalog implementation follows Penfield 1981 (_Principal Values and Branch Cuts in Complex APL_, https://doi.org/10.1145/800142.805368), equation 12:
arctanh←{⍟(1+⍵)×(÷1-⍵*2)*0.5}
arctanh ¯2
¯0.5493061443J¯1.570796327
¯7 ○ ¯2There is some disagreement about Paul Penfield's choice in his 1981
¯0.5493061443J¯1.570796327
paper of branch cut direction (CW vs. CCW) on arctan (and by extension, arctanh) with which I agree. For a more detailed analysis, see Kahan[1] (search for "APL") and Steele[2].
The latter reference recommends the definitions
atan ←{((⍟1+⍵×0J1)-⍟1-⍵×0J1)÷0J2}
atanh←{((⍟1+⍵ )-⍟1-⍵ )÷ 2}
More importantly, note that Steele[2] ends with the following sentence:
"Paul Penfield of MIT, after whose article the Common Lisp branch cuts
were originally patterned, endorses this change."
If you accept the argument to use the other branch cut direction on both arctan and arctanh, then the above formulation of arctanh yields
atanh ¯2
¯0.5493061443J1.570796327
which produces the result (on the positive imaginary axis) chosen by
various implementations including NARS2000, APL2, GNU APL, J, Common
Lisp, GSL, MPC, Octave, and Wolfram Alpha. One small reason I prefer
the form in atanh is that it doesn't rely upon the default choice of the positive result of square root that arctanh uses, but this is not a big deal.
I don't know the rationale behind other implementations, but this is why NARS2000 produces that result.
References
[1] Kahan, W. "Branch Cuts for Complex Elementary
Functions, or Much Ado About Nothing's Sign Bit"
in Iserles and Powell (eds.) "The State of the Art
in Numerical Analysis", pp. 165-211, Clarendon
Press, 1987, https://people.freebsd.org/~das/kahan86branch.pdf
[2] Steele, "Complex ATAN Branch Cut" http://clhs.lisp.se/Issues/iss069_w.htm _________________________________________
Bob Smith
Bob is exactly right. I came up with my definitions for publication within the APL community in 1981. There was a need for a few choices of which side of a branch cut the functions would be continuous. In one or two cases (after all these years I can't remember which) there was already a commonly accepted convention. In others there was none. The arc tanh example under discussion resulted from one of those. When Guy Steele realized that my definitions and the ones made by Kahan had one of those choices different, he sought my advice. Kahan and I talked for several hours one
I had done a VERY thorough literature search and concluded that prior to my work nobody had articulated any preferences. So I made an arbitrary choice and, I hope, explained that arbitraryness in my papers. On the other hand Kahan had a well deservedworld reputation in the related mathematics. When he made his choices and the subsequent recommendations to HP, he was apparently unaware of my work and relied on his extensive experience. But in the final analysis his choices were, I believe, just as arbitrary as mine.
We did not end up with agreement on how to continue. It seemed obvious to me that everyone would be better served by having one definition rather than two when neither was fundamentally superior. If precedence was considered important, then my 1981publications might have been the preferred choice, but I was not and still am not
I believe the LISP community felt similarly. Later, when Guy Steele and I spoke, I had to agree their choice for Common LISP was reasonable, and told him so.changed away from APL to other things including VLSI. Of course that is not a valid excuse. Perhaps this note can now serve that function. It is only 35 years too late.
The one thing that I did not do at the time and should have, was to publicly endorse Kahan's choice and recommend that APL implementations discontinue using mine. I do regret that and can only excuse it by saying that my interests had by that time
On Wednesday, November 11, 2020 at 10:55:03 AM UTC-5, Bob Smith wrote:
On 10/27/2020 3:58 PM, Roger Hui wrote:
The Dyalog implementation follows Penfield 1981 (_Principal Values and Branch Cuts in Complex APL_, https://doi.org/10.1145/800142.805368), equation 12:There is some disagreement about Paul Penfield's choice in his 1981
arctanh?{?(1+?)×(÷1-?*2)*0.5}
arctanh ¯2
¯0.5493061443J¯1.570796327
¯7 ? ¯2
¯0.5493061443J¯1.570796327
paper of branch cut direction (CW vs. CCW) on arctan (and by extension,
arctanh) with which I agree. For a more detailed analysis, see Kahan[1]
(search for "APL") and Steele[2].
The latter reference recommends the definitions
atan ?{((?1+?×0J1)-?1-?×0J1)÷0J2}
atanh?{((?1+? )-?1-? )÷ 2}
More importantly, note that Steele[2] ends with the following sentence:
"Paul Penfield of MIT, after whose article the Common Lisp branch cuts
were originally patterned, endorses this change."
If you accept the argument to use the other branch cut direction on both
arctan and arctanh, then the above formulation of arctanh yields
atanh ¯2
¯0.5493061443J1.570796327
which produces the result (on the positive imaginary axis) chosen by
various implementations including NARS2000, APL2, GNU APL, J, Common
Lisp, GSL, MPC, Octave, and Wolfram Alpha. One small reason I prefer
the form in atanh is that it doesn't rely upon the default choice of the
positive result of square root that arctanh uses, but this is not a big
deal.
I don't know the rationale behind other implementations, but this is why
NARS2000 produces that result.
References
[1] Kahan, W. "Branch Cuts for Complex Elementary
Functions, or Much Ado About Nothing's Sign Bit"
in Iserles and Powell (eds.) "The State of the Art
in Numerical Analysis", pp. 165-211, Clarendon
Press, 1987, https://people.freebsd.org/~das/kahan86branch.pdf
[2] Steele, "Complex ATAN Branch Cut"
http://clhs.lisp.se/Issues/iss069_w.htm
_________________________________________
Bob Smith
On Thu, 12 Nov 2020, J. Clarke wrote:
This is kind of the old story. When there's more than one way to do
something and no sound reason to prefer one over the other, let the user
or the developer decide. We do it with quad-IO, why not with the branch
directions on complex calculations?
FWIW I have always thought ?io was not a great idea. Most people I have >spoken to on the matter feel similarly.
Any publication on APL needs to specify whether it has ?io as 1 or 0. (Or
do the trick where you add/subtract ?io to/from indices, which is if
anything even worse.) It bifurcates the language and makes it harder to >reason about as a unity.
If you need a trig function with branch cut that goes the other way,
define one inline. It's a fairly niche use-case, and not a particularly >burdensome definition; and it alleviates the main language of overloads.
-E
This is kind of the old story. When there's more than one way to do something and no sound reason to prefer one over the other, let the user
or the developer decide. We do it with quad-IO, why not with the branch directions on complex calculations?
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