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RE: = versus := and trying to speed up calculations
*To*: mathgroup at smc.vnet.net
*Subject*: [mg35438] RE: [mg35410] = versus := and trying to speed up calculations
*From*: "DrBob" <majort at cox-internet.com>
*Date*: Fri, 12 Jul 2002 04:29:03 -0400 (EDT)
*Reply-to*: <drbob at bigfoot.com>
*Sender*: owner-wri-mathgroup at wolfram.com
Here's a recursive definition for n! that saves answers so they don't
have to be computed again:
ClearAll[f]
f[0] = 1;
f[n_] := f[n] = n f[n - 1]
f[1];
?? f
Global`f
f[0] = 1
f[1] = 1
f[n_] := f[n] = n*f[n - 1]
The rule defined by SetDelayed (":=") is only invoked once for each n;
next time the same n is used, the rule defined by Set ("=") takes
precedence.
The downside is that you're storing a rule for each value of n
encountered. You also have to compute the low-level values ahead of
time. If the above code is followed immediately by
f[2000]
$RecursionLimit::"reclim": "Recursion depth of 256 exceeded."
Instead you have to compute from the bottom up (the first time you want
to reach 2000):
f /@ Range[1, 2000, 256]; f[2000]
In your problem, the memory requirements of this method may be
prohibitive. If so, you might experiment with saving SOME intermediate
results but not others.
Bobby Treat
-----Original Message-----
From: Geoff Tims [mailto:Geoff at swt.edu]
To: mathgroup at smc.vnet.net
Subject: [mg35438] [mg35410] = versus := and trying to speed up calculations
I have the following function and it is called many many times in a
program
I have written. This is a bracket operator for a certain type of Lie
Algebra I am looking at and it must be called over and over to test
whether
or not the Jacobi is actually 0 as it should be. With very low numbers
for
n, my program runs in a second or less. If n is around 20+ or if the
coefficients are large, the program takes nearer a minute or two.
That's
not a long time, but I have a feeling that it's having to calculate the
same
Bracket many times in the program and I'm hoping to get rid of that
somehow.
However, I don't understand the differences between := and = enough.
I've
read the help files, but I don't understand the subtle differences such
as I
can call
Bracket[stuff, something]:= somethingelse
more than once, but it doesn't seem as if I can use = more than once.
Bracket[a_.*e[i_], b_.*e[j_]] :=
Which[
i + j > n, 0,
i == j, 0,
i == 1, a*b*e[j + 1],
i == 2 && j == 3, 14a*b*e[7],
i == 3 && j == 4, 0,
i < j, -a*b*Bracket[e[i + 1], e[j - 1]] +
Bracket[e[1], Bracket[e[i], e[j - 1]]],
i > j, -a*b*Bracket[e[j], e[i]]
];
(* bilinear function *)
Bracket[mul_, expr_Plus] := Map[Bracket[mul, #] &, expr];
Bracket[expr_Plus, mul_] := Map[Bracket[#, mul] &, expr];
Bracket[0, x_] := 0
Bracket[x_, 0] := 0
Any help would be much appreciated.
Geoff Tims
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