Hi Jatin,
I would like to give you a quick update. I got help from the GitHub Copilot.
I decided to try using m = symbols('m') in both modules, thus eliminating
HillSeries2.m (replacing it by m).
My code now looks like this:
from sympy.polys.domains import QQ
from sympy.polys.rings import ring
from sympy.polys.ring_series import rs_asin
R, m = ring('m', domain='QQ')
# α = Poly(α)
αP = α.as_poly(domain='QQ')
# α = R(α)
# αP = Poly(α, m)
α0 = αP.coeff_monomial(1) # get constant term
αR = αP - α0 # remove it
αR = R(αR) # define with ring
αR = rs_asin(αR, m, 3) # convert to series, 3 should be 11
α = α0 + αR # restore constant term
print(latex(Eq(symbols('α'), α), order='rev-lex'))
The line αR = R(αR) # define with ring produces an exception:
expected Rational object, got Poly
I’ll search more for that tomorrow.
Tom
(Dr. Thomas S. Ligon)
<mailto:[email protected]> [email protected]
Frohnloher Str. 6a
81475 Muenchen
Germany
Tel. +49(89)74575075
Von: [email protected] <[email protected]>
Gesendet: Mittwoch, 26. Februar 2025 18:53
An: [email protected]
Betreff: AW: [sympy] Re: sympy process out of memory calculating series of asin
Hi Jatin,
thanks again for your valuable help. Here is a long explanation for something
that should be very simple but might be causing confusion.
I have a total of about 10,000 lines of code dealing with power series in
powers of m. m is defined as
m = symbols('m')
Now I have multiple modules and all of them use m, so I want to make sure that
they all use the same thing. I could have repeated the definition m =
symbols('m') in each module but was not sure that would be using the same
thing, so m is defined in HillSeries2.py and when I use it in another module, I
refer to it as HillSeries2.m.
Today, I found documentation of the ring function in
https://docs.sympy.org/latest/modules/polys/domainsref.html#sympy.polys.rings.ring
This documentation, like a lot of SymPy documentation, defines the INPUT
parameters, but provides no complete documentation of the OUTPUT parameters.
For ring, we have:
Construct a polynomial ring returning (ring, x_1, ..., x_n).
But what is the format and meaning of the elements of the output parameter?
It looks like the symbols are defined in the input and repeated in the output,
which seems to be a bit redundant, but there is probably a good reason for that.
You write “However, α is not defined before this line.” Well, of course it is
defined, I just didn’t copy everything into this discussion. In fact, it is
part of the trigonometric formula for solving a cubic equation. If you read
that in Wikipedia, you will find the original cubic equation with coefficients
a, b, and c, and the reduced equation with coefficients p and q, then Cardano’s
formula for the solutions, using p and q. After that, the trigonometric
solution is presented, using p, q, cos, and acos. However, I am not using
Wikipedia, but a different paper, and for that I need asin instead of acos.
α is a function of p and q, which are functions of a, b, and c. In my code, all
are symbolic expressions of power series in m with rational coefficients. I
would be happy to copy the code in here, but that looks to me more like a
distraction than help. Let me know if you want to see that part.
α is a symbolic expression of a partial sum of a power series in m.
I print the current value of α in Latex format, but I call it ‘arg’, because it
is not the final α, but the argument of asin.
The code is
print(latex(Eq(symbols('arg'), α), order='rev-lex'))
and the Latex output is
arg = -1 + \frac{867 m^{2}}{128} - \frac{12393 m^{3}}{512} - \frac{649831
m^{4}}{32768} - \frac{124581731 m^{5}}{589824} + \frac{14200641281
m^{6}}{113246208} - \frac{25578912841 m^{7}}{50331648} + \frac{241589534597669
m^{8}}{173946175488} - \frac{2019731412177637 m^{9}}{521838526464} +
\frac{2581283623490644391 m^{10}}{601157982486528} -
\frac{806384032365914837207 m^{11}}{36069478949191680}
Now I have activated the line that makes the symbolic expression α into a
polynomial. With that, I am getting exception
'Poly' object has no attribute 'ring'
instead of
'Add' object has no attribute 'ring'
My IDE shows me that α is now of type Poly and has a value of
Poly(-806384032365914837207/36069478949191680*m**11 +
2581283623490644391/601157982486528*m**10 - 2019731412177637/521838526464*m**9
+ 241589534597669/173946175488*m**8 - 25578912841/50331648*m**7 +
14200641281/113246208*m**6 - 124581731/589824*m**5 - 649831/32768*m**4 -
12393/512*m**3 + 867/128*m**2 - 1, m, domain='QQ')
The code is now
from sympy.polys.domains import QQ
from sympy.polys.rings import ring
from sympy.polys.ring_series import rs_asin
R, HillSeries2.m = ring('HillSeries2.m', domain='QQ')
# α = Poly(α)
α = α.as_poly(domain='QQ')
α = rs_asin(α, HillSeries2.m, 3) # should be 11
Tom
(Dr. Thomas S. Ligon)
<mailto:[email protected]> [email protected]
Frohnloher Str. 6a
81475 Muenchen
Germany
Tel. +49(89)74575075
Von: [email protected] <mailto:[email protected]>
<[email protected] <mailto:[email protected]> > Im Auftrag von Jatin
Bhardwaj
Gesendet: Mittwoch, 26. Februar 2025 14:02
An: sympy <[email protected] <mailto:[email protected]> >
Betreff: Re: [sympy] Re: sympy process out of memory calculating series of asin
Hi Tom,
I noticed a small issue in your implementation. I’m not sure if this is a
standalone piece of code or part of a larger codebase, but I’d be happy to help
clarify.
The ring function returns a tuple containing the ring itself and all variables
defined within it. In your code:
``` R, m = ring('HillSeries2.m', domain=QQ) ```
You’re defining the variable 'HillSeries2.m' and storing it in m. However,
using dots in variable names can sometimes cause confusion. It’s generally a
good practice to use simple, consistent names for better readability.
In the series expansion:
``` α = rs_asin(α, HillSeries2.m, 3) # should be 11 ```
1. The first argument of rs_asin should be a function that is either a
polynomial or a series, and its variable must be part of the ring. However, α
is not defined before this line.
2. The second argument should be the ring variable, but HillSeries2.m is not a
separate variable. Instead, you should use m, since that’s where the variable
was stored when defining the ring.
Take a look at this code. I hope this helps,
```
from symp import QQ, ring
from sympy.polys.ring_series import rs_asin
R, α = ring('α', domain=QQ) # Define the ring with variable α
s = rs_asin(α, α, 8) # Compute the series expansion of asin(α) up to order 8
#output: 5/112*α**7 + 3/40*α**5 + 1/6*α**3 + α
```
Also, check out "rs_asin" document
https://docs.sympy.org/latest/modules/polys/ringseries.html#sympy.polys.ring_series.rs_asin
Jatin
On Wednesday, 26 February 2025 at 14:19:30 UTC+5:30 [email protected]
<mailto:[email protected]> wrote:
Hi Jatin,
thanks, this looks very promising. I haven’t used polynomials much yet, since
simple symbolic expressions do everything I need, and it looks like I have a
deficit there. Here is my current code:
from sympy.polys.domains import QQ
from sympy.polys.rings import ring
from sympy.polys.ring_series import rs_asin
R, m = ring('HillSeries2.m', domain='QQ')
α = rs_asin(α, HillSeries2.m, 3) # should be 11
Exception in the last line:
'Add' object has no attribute 'ring'
It looks like my expression needs to be a polynomial
I added a line
α = α.as_poly(domain='QQ')
but that didn’t help.
In the sample code
R, m = ring('HillSeries2.m', domain='QQ')
is not documented very well. It looks like it is defining a ring R, but not
using it, but my IDE doesn’t give me a warning for that.
I am using documentation
https://docs.sympy.org/latest/modules/polys/ringseries.html#rs-series
Tom
(Dr. Thomas S. Ligon)
[email protected] <mailto:[email protected]>
Frohnloher Str. 6a
81475 Muenchen
Germany
Tel. +49(89)74575075 <tel:+49%2089%2074575075>
Von: [email protected] <mailto:[email protected]>
<[email protected] <mailto:[email protected]> > Im Auftrag von Jatin
Bhardwaj
Gesendet: Montag, 24. Februar 2025 22:08
An: sympy <[email protected] <mailto:[email protected]> >
Betreff: [sympy] Re: sympy process out of memory calculating series of asin
Hi Tom,
Instead of series(asin(α)), you can use SymPy’s ring_series.rs_asin, which is
optimized for power series manipulations and significantly reduces memory usage.
```
from sympy.polys.rings import ring
from sympy.polys.ring_series import rs_asin
R, x = ring('x', domain='QQ')
asin_series = rs_asin(x, 11)
print(asin_series)
```
This method is much more efficient and avoids the heavy symbolic
differentiation that series() performs.
You can also check out the documentation for the rs_series
<https://docs.sympy.org/latest/modules/polys/ringseries.html#rs-series> in
`ring series` module.
Jatin Bhardwaj
On Monday, 24 February 2025 at 13:52:57 UTC+5:30 [email protected]
<mailto:[email protected]> wrote:
I am using sympy to analyze and run some code involving power series. In
general, the code is always running on a partial sum of a power series, and
this is a normal sympy symbolic expression. After doing certain calculations,
the data is no longer a series, and I need to run a Taylor expansion in order
to get a series. For example, if I calculate the quotient of two partial sums,
I need Taylor in order to get another partial sum. In sympy, it is very
convenient to call series() to do this.
Now it is clear that this can use a lot of memory, because every additional
derivative makes the expression more complex and longer. It appears as though
series() does not use parallel processing, so the result is 100% usage of one
CPU and increasing use of memory. On my laptop (running Windows 11 and Visual
Studio), I have 16GB RAM and have sometimes run into memory issues.
To solve this problem, I have allocated an Ubuntu Linux virtual machine on the
Internet (Microsoft Azure). The machine has 2 virtual CPUs and 218GB RAM. With
this, I have successfully calculated partial sums up to order 24 or 31. This
sometimes runs at 100% CPU allocation for more than 24 hours and completes
successfully.
Up to now, I have used series() for algebraic expressions of partial sums
(quotient, square root, etc.) but recently started applying this to a
trigonometric expression. This is because I need to calculate the solution of a
cubic equation and the Cardano formula has run into unpleasant situations
involving complex numbers, and I have been given a recommendation to try the
trigonometric formula instead. Up to now, it looks very good, since the primary
solution is real and does not require complex components. By the way, solving a
cubic equation worked will with both the standard formula and via solve(). For
the cubic equation, I have started with the formula because I can see if each
individual step success before proceeding to the next one.
Specifically, my code is
α = asin(α)
α = series(α, m, n=maxK+1) # calculate series up to maxK
On my laptop, the series statement runs for some time and hits the memory limit
of my PC, so I tried running this on the VM, which has more memory and does not
have the overhead of Windows or Visual Studio.
This first attempt ran for somewhere between 24 and 30 hours and the VM was
stopped. This is presumably because the VM is configured to use a “spot price”,
meaning that the administrators can stop it at any time they need additional
capacity for something else.
The second attempt ran for somewhere between 30 and 48 hours and my output file
ended in the statement killed. Running dmesg gave me this information:
[144763.491084]
oom-kill:constraint=CONSTRAINT_NONE,nodemask=(null),cpuset=/,mems_allowed=0,global_oom,task_memcg=/user.slice/user-1000.slice/session-1.scope,task=python3.9,pid=1699,uid=1000
[144763.491101] Out of memory: Killed process 1699 (python3.9)
total-vm:226036448kB, anon-rss:223754920kB, file-rss:2740kB, shmem-rss:0kB,
UID:1000 pgtables:438192kB oom_score_adj:0
[144772.062513] oom_reaper: reaped process 1699 (python3.9), now anon-rss:0kB,
file-rss:0kB, shmem-rss:0kB
In this case, my partial sum is of order 11, and it would be very valuable for
me to achieve this solution, but I don’t need to go past 11.
Is this to be expected? Since the derivative of arcsin is 1/sqrt(1-x**2), I
expect the calculations to be no worse than what I have done in the past.
Is there any reason why series(asin(α)) would be expected to fail?
Should I try anything else?
Tom
(Dr. Thomas S. Ligon)
[email protected] <mailto:[email protected]>
Frohnloher Str. 6a
81475 Muenchen
Germany
Tel. +49(89)74575075 <tel:+49%2089%2074575075>
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