On Tue, 18 Dec 2007, Art Owen wrote:
> Dear R-ophiles,
>
> I've found something very odd when I apply convolve
> to ever larger vectors. Here is an example below
> with vectors ranging from 2^11 to 2^17. There is
> a funny bump up at 2^12. Then it gets very slow at 2^16.
The time is consumed by fft, which is called with vectors of length
2*2^i-1 when type = 'o'
> system.time( fft(seq_len(2^13-1)) )
user system elapsed
0.31 0.00 0.32
> system.time( fft(seq_len(2^14-1)) )
user system elapsed
0 0 0
>
There are no factors of 2^13-1 or 2^17-1 or 2^18-1
> for (i in 11:20 ) print( c(index=i, nfact=length(which( 0 ==
(2^(i+1)-1)%%(2:trunc(sqrt(2^(i+1)-1)) )))))
index nfact
11 11
index nfact
12 0
index nfact
13 3
index nfact
14 3
index nfact
15 7
index nfact
16 0
index nfact
17 15
index nfact
18 0
index nfact
19 23
index nfact
20 5
>
It looks like the code in fft.c tries to find factors of the series length
and works from there.
So, the size of the problem evidently depends on its factors.
HTH,
Chuck
>
>
> > for( i in 11:20 )print( system.time(convolve(1:2^i,1:2^i,type="o")))
> user system elapsed
> 0.002 0.000 0.002
> user system elapsed
> 0.373 0.002 0.375
> user system elapsed
> 0.014 0.001 0.016
> user system elapsed
> 0.031 0.002 0.034
> user system elapsed
> 0.126 0.004 0.130
> user system elapsed
> 194.095 0.013 194.185
> user system elapsed
> 0.345 0.011 0.356
>
> This example is run on a fedora machine with 64 bits. I hit the same
> wall at 2^16 on a Macbook (Intel processor I think). The fedora machine
> is running R 2.5.0. That's a bit old (April 07) but I saw no mention of
> this speed
> problem in some web searching, and it's not mentioned in the 2.6
> what's new notes.
>
> I've rerun it and found the same bump at 12 and wall at 16.
> The timing at 2^16 can change appreciably. In one
> other case it was about 270 user, 271 elapsed.
> The 2^18 case ran for hours without ever finishing.
>
> At first I thought that this was a memory latency issue. Maybe it
> is. But that makes it hard to explain why 2^17 works better than
> 2^16. I've seen that three times now, so I'm almost ready to call it
> reproducible.
> Also, one of the machines I'm using has lots of memory. Maybe it's
> a cache issue ... but that still does not explain why 2^12 is slower
> than 2^13 or 2^16 is slower than 2^17.
>
> I've checked by running convolve without type="o" and I don't
> see the wall. Similarly fft does not have that problem.
>
> Here's an example without type="open"
> > for( k in 11:20)print(system.time( convolve( 1:2^k,1:2^k)))
> user system elapsed
> 0.001 0.000 0.000
> user system elapsed
> 0.001 0.000 0.001
> user system elapsed
> 0.002 0.000 0.002
> user system elapsed
> 0.004 0.000 0.004
> user system elapsed
> 0.009 0.001 0.010
> user system elapsed
> 0.017 0.001 0.018
> user system elapsed
> 0.138 0.005 0.143
> user system elapsed
> 0.368 0.012 0.389
> user system elapsed
> 1.010 0.032 1.051
> user system elapsed
> 1.945 0.069 2.015
>
> This is more what I expected. Something like N or N log(N) , with
> the difference hard to discern in granularity and noise.
>
> The convolve function is not very big (see below). When type is
> not specified, it defaults to "circular". So my guess is that something
> mysterious might be happening inside the first else clause below,
> at least on some architectures.
>
> -Art Owen
>
>
>
> > convolve
> function (x, y, conj = TRUE, type = c("circular", "open", "filter"))
> {
> type <- match.arg(type)
> n <- length(x)
> ny <- length(y)
> Real <- is.numeric(x) && is.numeric(y)
> if (type == "circular") {
> if (ny != n)
> stop("length mismatch in convolution")
> }
> else {
> n1 <- ny - 1
> x <- c(rep.int(0, n1), x)
> n <- length(y <- c(y, rep.int(0, n - 1)))
> }
> x <- fft(fft(x) * (if (conj)
> Conj(fft(y))
> else fft(y)), inv = TRUE)
> if (type == "filter")
> (if (Real)
> Re(x)
> else x)[-c(1:n1, (n - n1 + 1):n)]/n
> else (if (Real)
> Re(x)
> else x)/n
> }
>
> ______________________________________________
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>
Charles C. Berry (858) 534-2098
Dept of Family/Preventive Medicine
E mailto:[EMAIL PROTECTED] UC San Diego
http://famprevmed.ucsd.edu/faculty/cberry/ La Jolla, San Diego 92093-0901
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