JfC chapter 31 has a description of H. that I thought was pretty clear.
Henry Rich
On 1/19/2014 12:08 AM, Ian Clark wrote:
Okay, I know the answer now. But I dare anyone to discover it from the
existing J Help documentation of (H.) !
Including Concrete Math Companion, Vector, and Ewart Shaw's posting !!
Your mission, should you accept it, is to define a verb F that accepts an
argument in (roughly) the syntax of Abramowitz and Stegun (A&S) chapter 15,
viz F(a;b;c;z), and calls (H.) with the correct arguments.
Here's some examples drawn from A&S...
ln=: ^.
arcsin=: _1&o.
arctan=: _3&o.
] z=: 5%~ i.6
0 0.2 0.4 0.6 0.8 1
F(1;1;2;z) NB. [15.1.3]
-(ln 1-z)%z
0 1.11572 1.27706 1.52715 2.0118 _
F(1r2;1;3r2;z^2) NB. [15.1.4]
-:(ln (1+z)%(1-z))%z
0 1.01366 1.05912 1.15525 1.37327 _
F(1r2;1;3r2;-z^2) NB. [15.1.5]
(arctan z) %z
0 0.986978 0.951266 0.900699 0.843426 0.785398
F(1r2;1r2;3r2;z^2) NB. [15.1.6]
(arcsin z) %z
0 1.00679 1.02879 1.0725 1.15912 1.5708
It's 4 instances of the Hypergeometric Series (F) with the functions it is
supposed to approximate when 0<(|z)<1. (So for z=0 and z=1 the results
can't be expected to match. But I've included these values in z anyway.)
Hint: call H. with left argument x=50 (the number of terms of the series to
be summed) as it can take a long time if you let it go to the limit by
calling it monadically.
Just to preempt someone splitting hairs, no my J syntax of A&S's F(a,b;c;z)
isn't quite the same. A&S has a comma as the first separator, whereas I've
a semicolon.
IanClark
On Sat, Jan 18, 2014 at 6:09 PM, Mike Day <[email protected]>wrote:
Ewart Shaw wrote about these, so look for his emails on the subject
failing other channels. He might like to comment for himself, of course,
if his e-address (as I have it, above) is still correct.
Mike
On 18/01/2014 11:01, Ian Clark wrote:
Just one empty stub remains in the Accessible Dictionary (aka NuVoc
--remember it?):
H. (Hypergeometric) Conjunction
http://www.jsoftware.com/jwiki/Vocabulary/hcapdot
Once that's filled-in, then NuVoc is more-or-less ready to go. You can
already see it at http://www.jsoftware.com/jwiki/Vocabulary
But alas, I need help...
The J Dictionary (the old one) references Abramowitz and Stegun (A&S),
Chapter 15: Hypergeometric Functions. Now A&S represent the syntax of the
general case like so:
F(a; b; c; z)
Both NuVoc and the J Dictionary present the syntax of the (H.) primitive
like so:
(m H. n) y
where both m and n are numeric lists.
Now suppose I'm a newbie, and my first sight of:
http://www.jsoftware.com/help/dictionary/dhcapdot.htm
just gives me a dull ache between the eyes. I need clear, unambiguous
instructions for taking any example I choose from A&S and mapping it onto:
(m H. n)y
Let me make a start:
z --> y
That was the easy bit. Now... how should (a; b; c) --> (m; n)?
Or should I be asking: how *best* should (a; b; c) be mapped onto m and n?
Because as I see it, it's ambiguous. Just for starters:
F(a; b; c; z) = F(b; a; c; z) -----[A&S 15.1.1]
Suggestions please.
IanClark
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