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How to explain these variations in execution speeds?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg113124] How to explain these variations in execution speeds?
  • From: James Stein <mathgroup at stein.org>
  • Date: Wed, 13 Oct 2010 23:27:27 -0400 (EDT)

Stop your kernel; paste the following 14 lines in a notebook cell; then
evaluate the cell.

(*1*)n=10^6;
(*2*)Timing[Sum[k,{k,n}]]
(*3*)Timing[Sum[k,{k,Evaluate[n]}]]
(*4*)Timing[Sum[k,{k,n+1}]]
(*5*)Timing[Sum[k,{k,n-1}]]
(*6*)Timing[Sum[k,{k,n+2}]]
(*7*)Timing[Sum[k,{k,n-2}]]
(*8*)Timing[Sum[k,{k,n+3}]]
(*9*)Timing[Sum[k,{k,n-3}]]
(*10*)Timing[Sum[k,{k,m}]/.m->n-4]
(*11*)Timing[Sum[k,{k,m}]/.m->Evaluate[n-5]]
(*12*)Timing[Sum[k,{k,m}]/.m->10^6+6]
(*13*)Timing[Sum[k,{k,m}]/.m->10^6-6]

Each sum represents approximately the same amount of "work",
but some sums are computed significantly faster than others.
What results of previous computations are being used? when? where are they
stored? how are they accessed?
Why does the use of 'Rule' (in lines 10-13) speed things up?
Do these examples suggest good ways to speed code in general?
Or are they merely exceptional because 'Sum' is a built-in function?
(If documentation answers these questions, just point me there. I searched
w/o success.)

On my computer, the execution times are approximately:
0.3500 second: Out lines 2,3,5,7,9
0.1000 second: Out lines 4,10
0.0044 second: Out lines 6,8
0.0003 second: Out lines 11,12,13

Some of the puzzle is either explained or compounded (but I'm not sure
which!) by comparing lines 1 and 2 with lines 10 and 11. Why does 'Evaluate'
speed 11 wrt 10, but not 2 wrt 1?

My head began to whirl when I compared these two pairs of lines:

Pair 1: (as above, lines 10 and 11)
Timing[Sum[k, {k, m}] /. m -> n - 4]
Timing[Sum[k, {k, m}] /. m -> Evaluate[n - 4]]

Pair 2: (same as Pair 1, except "dummy" variable 'm' replaced by 'z' in
second line only)
Timing[Sum[k, {k, m}] /. m -> n - 4]
Timing[Sum[k, {k, z}] /. z -> Evaluate[n - 4]]

In each pair (assuming a fresh Kernel for each pair), the second line is
much faster than the first; but in the first pair, the speed increase is
much greater that the speed increase in the second pair.  It seems like the
symbol chosen for the "dummy variable" is somewhere retained, and affects
further evaluations. (If so, how?)


Two ancillary questions:
1. Is there a programmatic way (from a notebook) to stop then restart the
Kernel?
2. Is there a set of commands that effectively return the Kernel to its
initial state?
(i.e., clear all user-defined symbols and history and reclaim memory)



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