I am currently trying to see if I can accelerate my code around series
expansions. I really like that series are lazy in FriCAS, but I wonder
what impact multiple operations on the series (or the underlying
stream), like integration etc. have. I would assume that up to memory
requirements that "save" or delay all the performed operations, there
should be no
performance penalty compared to a non-lazy evaluation.
Anyway, while testing I came up with this example, where some expression
is series expanded in two different ways.
)clear completely
)set mes tim on
x := taylor(x');
s1 := x^2 + cos(x) + x^8/ sqrt(x-1);
s2 := taylor(y^2 + cos(y) + y^8/ sqrt(y-1),y=0);
approximate(s1,450);
approximate(s2,450);
approximate(s1,450);
approximate(s2,450);
When running the code (see below) the first method of evaluation is three
times faster than the second. (A second evaluation runs in the same time
for both cases, probably because the results are cached, but that's not
relevant for me).
In my case I start with an Expression(Integer), so I (obviously) can only
use the second version from the ExpressionToUnivariatePowerSeries package.
But my idea is that maybe I can improve the performance if I understand
why the first method is so much faster.
I would have expected that the ExpressionToUnivariatePowerSeries package
does something like "subst(expression, x=series('x))", symbolically
speaking, but then it should be just as fast.
In principle I see two ways to do the series expansions: Compute the
expansion as a whole for the whole expression (for example by taking
derivatives for a Taylor series). Or do it for each function separately,
and use the Cauchy product to multiply the individual series together, or
add them, etc. For the first series expansion example this is clearly
done, but maybe not for ExpressionToUnivariatePowerSeries and this could
be the difference?
Thanks,
Tobias
PS: It's possible that this message appears twice. I tried out the gmane.io
mailinglist <-> NNTP gateway, but posting didn't seem to work, so I send it
directly again.
---- running the code:
approximate(s1,450);
Type: Expression(Integer)
Time: 0 (IN) + 2.20 (EV) + 0 (OT) = 2.20
sec
approximate(s2,450);
Type: Expression(Integer)
Time: 0.00 (IN) + 6.12 (EV) + 0.00 (OT) = 6.13
sec
approximate(s1,450);
Type: Expression(Integer)
Time: 0 (IN) + 1.36 (EV) + 0 (OT) = 1.36
sec
approximate(s2,450);
Type: Expression(Integer)
Time: 0 (IN) + 1.37 (EV) + 0 (OT) = 1.37
sec
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