I will show you a representation of the 7x7 form of my matrix, the 10x10
includes a couple additional elements, but has the same overall structure
and layout.
The key point is that the diagonal elements are differences of multiple
values, and each of these values occupies a certain element in the lower
left of the matrix - the upper right is all 0s. The last two columns are
also all 0s.
It is not really possible to simplify it further.
>>> woVIt =
Matrix([[-(k+kcSD),0,0,0,0,0,0],[k,-(kEI+kcED),0,0,0,0,0],[0,kEI,-(kIH+kIR+kcID),0,0,0,0],[0,0,kIH,-(kHHt+kHD+kHR+kcHD),0,0,0],[0,0,0,kHHt,-(kHtD+kHtR),0,0],[kcSD,kcED,kcID,(kHD+kcHD),kHtD,0,0],[0,0,kIR,kHR,kHtR,0,0]])
>>> woVIt.eigenvals()
{0: 2, -kIH - kIR - kcID: 1, -kHD - kHHt - kHR - kcHD: 1, -k - kcSD: 1,
-kEI - kcED: 1, -kHtD - kHtR: 1}
>>> woVIt.eigenvects()
[(0, 2, [Matrix([
[0],
[0],
[0],
[0],
[0],
[1],
[0]]), Matrix([
[0],
[0],
[0],
[0],
[0],
[0],
[1]])]), (-k - kcSD, 1, [Matrix([
[ -(k
+ kcSD)*(k - kEI - kcED + kcSD)*(k - kHtD - kHtR + kcSD)*(k - kIH - kIR -
kcID + kcSD)*(k - kHD - kHHt - kHR - kcHD + kcSD)/(k*kEI*(kHHt*kHtR*kIH -
(kHR*kIH - kIR*(k - kHD - kHHt - kHR - kcHD + kcSD))*(k - kHtD - kHtR +
kcSD)))],
[
(k + kcSD)*(k - kHtD - kHtR + kcSD)*(k - kIH - kIR - kcID + kcSD)*(k - kHD
- kHHt - kHR - kcHD + kcSD)/(kEI*(kHHt*kHtR*kIH - (kHR*kIH - kIR*(k - kHD -
kHHt - kHR - kcHD + kcSD))*(k - kHtD - kHtR + kcSD)))],
[
-(k + kcSD)*(k - kHtD - kHtR + kcSD)*(k - kHD - kHHt - kHR - kcHD +
kcSD)/(kHHt*kHtR*kIH - (kHR*kIH - kIR*(k - kHD - kHHt - kHR - kcHD +
kcSD))*(k - kHtD - kHtR + kcSD))],
[
kIH*(k + kcSD)*(k - kHtD - kHtR + kcSD)/(kHHt*kHtR*kIH - (kHR*kIH - kIR*(k
- kHD - kHHt - kHR - kcHD + kcSD))*(k - kHtD - kHtR + kcSD))],
[
-kHHt*kIH*(k + kcSD)/(kHHt*kHtR*kIH - (kHR*kIH - kIR*(k - kHD - kHHt - kHR
- kcHD + kcSD))*(k - kHtD - kHtR + kcSD))],
[(k*kEI*kHHt*kHtD*kIH - (k*kEI*kIH*(kHD + kcHD) - (k*kEI*kcID - (k*kcED -
kcSD*(k - kEI - kcED + kcSD))*(k - kIH - kIR - kcID + kcSD))*(k - kHD -
kHHt - kHR - kcHD + kcSD))*(k - kHtD - kHtR + kcSD))/(k*kEI*(kHHt*kHtR*kIH
- (kHR*kIH - kIR*(k - kHD - kHHt - kHR - kcHD + kcSD))*(k - kHtD - kHtR +
kcSD)))],
[
1]])]), (-kEI - kcED, 1, [Matrix([
[
0],
[
(kEI + kcED)*(kEI - kHtD - kHtR + kcED)*(kEI - kIH - kIR + kcED -
kcID)*(kEI - kHD - kHHt - kHR + kcED - kcHD)/(kEI*(kHHt*kHtR*kIH - (kHR*kIH
- kIR*(kEI - kHD - kHHt - kHR + kcED - kcHD))*(kEI - kHtD - kHtR + kcED)))],
[
-(kEI + kcED)*(kEI - kHtD - kHtR + kcED)*(kEI - kHD - kHHt - kHR + kcED -
kcHD)/(kHHt*kHtR*kIH - (kHR*kIH - kIR*(kEI - kHD - kHHt - kHR + kcED -
kcHD))*(kEI - kHtD - kHtR + kcED))],
[
kIH*(kEI + kcED)*(kEI - kHtD - kHtR + kcED)/(kHHt*kHtR*kIH - (kHR*kIH -
kIR*(kEI - kHD - kHHt - kHR + kcED - kcHD))*(kEI - kHtD - kHtR + kcED))],
[
-kHHt*kIH*(kEI + kcED)/(kHHt*kHtR*kIH - (kHR*kIH - kIR*(kEI - kHD - kHHt -
kHR + kcED - kcHD))*(kEI - kHtD - kHtR + kcED))],
[(kEI*kHHt*kHtD*kIH*(-k + kEI + kcED - kcSD) - (kEI*kIH*(kHD + kcHD)*(-k +
kEI + kcED - kcSD) - (kEI*kcID*(-k + kEI + kcED - kcSD) - kcED*(-k + kEI +
kcED - kcSD)*(kEI - kIH - kIR + kcED - kcID))*(kEI - kHD - kHHt - kHR +
kcED - kcHD))*(kEI - kHtD - kHtR + kcED))/(kEI*(kHHt*kHtR*kIH - (kHR*kIH -
kIR*(kEI - kHD - kHHt - kHR + kcED - kcHD))*(kEI - kHtD - kHtR + kcED))*(-k
+ kEI + kcED - kcSD))],
[
1]])]), (-kHtD - kHtR, 1, [Matrix([
[ 0],
[ 0],
[ 0],
[ 0],
[-(kHtD + kHtR)/kHtR],
[ kHtD/kHtR],
[ 1]])]), (-kIH - kIR - kcID, 1, [Matrix([
[
0],
[
0],
[
-(kIH + kIR + kcID)*(-kHtD - kHtR + kIH + kIR + kcID)*(-kHD - kHHt - kHR +
kIH + kIR - kcHD + kcID)/(kHHt*kHtR*kIH - (kHR*kIH - kIR*(-kHD - kHHt - kHR
+ kIH + kIR - kcHD + kcID))*(-kHtD - kHtR + kIH + kIR + kcID))],
[
kIH*(kIH + kIR + kcID)*(-kHtD - kHtR + kIH + kIR + kcID)/(kHHt*kHtR*kIH -
(kHR*kIH - kIR*(-kHD - kHHt - kHR + kIH + kIR - kcHD + kcID))*(-kHtD - kHtR
+ kIH + kIR + kcID))],
[
-kHHt*kIH*(kIH + kIR + kcID)/(kHHt*kHtR*kIH - (kHR*kIH - kIR*(-kHD - kHHt -
kHR + kIH + kIR - kcHD + kcID))*(-kHtD - kHtR + kIH + kIR + kcID))],
[(kHHt*kHtD*kIH*(-k + kIH + kIR + kcID - kcSD)**2*(-kEI + kIH + kIR - kcED
+ kcID) - (kIH*(kHD + kcHD)*(-k + kIH + kIR + kcID - kcSD)**2*(-kEI + kIH +
kIR - kcED + kcID) - kcID*(-k + kIH + kIR + kcID - kcSD)**2*(-kEI + kIH +
kIR - kcED + kcID)*(-kHD - kHHt - kHR + kIH + kIR - kcHD + kcID))*(-kHtD -
kHtR + kIH + kIR + kcID))/((kHHt*kHtR*kIH - (kHR*kIH - kIR*(-kHD - kHHt -
kHR + kIH + kIR - kcHD + kcID))*(-kHtD - kHtR + kIH + kIR + kcID))*(-k +
kIH + kIR + kcID - kcSD)**2*(-kEI + kIH + kIR - kcED + kcID))],
[
1]])]), (-kHD - kHHt - kHR - kcHD, 1, [Matrix([
[
0],
[
0],
[
0],
[
(kHD + kHHt + kHR + kcHD)*(-k + kHD + kHHt + kHR + kcHD - kcSD)*(-kEI + kHD
+ kHHt + kHR - kcED + kcHD)*(kHD + kHHt + kHR - kHtD - kHtR + kcHD)*(kHD +
kHHt + kHR - kIH - kIR + kcHD - kcID)/(kHHt*kHtR*(-k + kHD + kHHt + kHR +
kcHD - kcSD)*(-kEI + kHD + kHHt + kHR - kcED + kcHD)*(kHD + kHHt + kHR -
kIH - kIR + kcHD - kcID) - kHR*(-k + kHD + kHHt + kHR + kcHD - kcSD)*(-kEI
+ kHD + kHHt + kHR - kcED + kcHD)*(kHD + kHHt + kHR - kHtD - kHtR +
kcHD)*(kHD + kHHt + kHR - kIH - kIR + kcHD - kcID))],
[
-kHHt*(kHD + kHHt + kHR + kcHD)*(-k + kHD + kHHt + kHR + kcHD - kcSD)*(-kEI
+ kHD + kHHt + kHR - kcED + kcHD)*(kHD + kHHt + kHR - kIH - kIR + kcHD -
kcID)/(kHHt*kHtR*(-k + kHD + kHHt + kHR + kcHD - kcSD)*(-kEI + kHD + kHHt +
kHR - kcED + kcHD)*(kHD + kHHt + kHR - kIH - kIR + kcHD - kcID) - kHR*(-k +
kHD + kHHt + kHR + kcHD - kcSD)*(-kEI + kHD + kHHt + kHR - kcED +
kcHD)*(kHD + kHHt + kHR - kHtD - kHtR + kcHD)*(kHD + kHHt + kHR - kIH - kIR
+ kcHD - kcID))],
[(kHHt*kHtD*(-k + kHD + kHHt + kHR + kcHD - kcSD)**3*(-kEI + kHD + kHHt +
kHR - kcED + kcHD)**2*(kHD + kHHt + kHR - kIH - kIR + kcHD - kcID) - (kHD +
kcHD)*(-k + kHD + kHHt + kHR + kcHD - kcSD)**3*(-kEI + kHD + kHHt + kHR -
kcED + kcHD)**2*(kHD + kHHt + kHR - kHtD - kHtR + kcHD)*(kHD + kHHt + kHR -
kIH - kIR + kcHD - kcID))/((kHHt*kHtR*(-k + kHD + kHHt + kHR + kcHD -
kcSD)*(-kEI + kHD + kHHt + kHR - kcED + kcHD)*(kHD + kHHt + kHR - kIH - kIR
+ kcHD - kcID) - kHR*(-k + kHD + kHHt + kHR + kcHD - kcSD)*(-kEI + kHD +
kHHt + kHR - kcED + kcHD)*(kHD + kHHt + kHR - kHtD - kHtR + kcHD)*(kHD +
kHHt + kHR - kIH - kIR + kcHD - kcID))*(-k + kHD + kHHt + kHR + kcHD -
kcSD)**2*(-kEI + kHD + kHHt + kHR - kcED + kcHD))],
[
1]])])]
So that is the issue - I cannot really simplify the eigenvalues, and I am
still not sure how to proceed.
Any ideas?
Thank you.
On Thursday, October 4, 2018 at 6:22:54 PM UTC-6, Aaron Meurer wrote:
>
> How sparse is the matrix, and what do the entries look like?
>
> One thing that can help depending on what your matrix looks like is to
> replace large subexpressions with symbols (if there are common
> subexpressions, cse() can help with this). That way the simplification
> algorithms don't get caught up trying to simplify the subexpressions.
> However if you expect the subexpressions to cancel each other out in
> the result, this can be detrimental.
>
> I would start with the eigenvalues. Once you can get those, you will
> want to simplify them if possible, before computing the eigenvectors.
>
> Aaron Meurer
> On Thu, Oct 4, 2018 at 6:12 PM Jacob Miner <[email protected]
> <javascript:>> wrote:
> >
> >
> >
> > On Friday, July 10, 2015 at 3:07:17 PM UTC-6, Ondřej Čertík wrote:
> >>
> >> Hi,
> >>
> >> On Fri, Jul 10, 2015 at 7:30 AM, 刘金国 <[email protected]> wrote:
> >> > 4 x 4 is needed ~~
> >> > mathematica runs extremely fast for 4 x 4 matrix as it should be, but
> ...
> >>
> >> Can you post the Mathematica result? So that we know what you are
> >> trying to get and we can then help you get it with SymPy.
> >>
> >> Ondrej
> >>
> >> >
> >> > 在 2014年2月12日星期三 UTC+8上午5:40:19，Vinzent Steinberg写道：
> >> >>
> >> >> On Monday, February 10, 2014 11:27:09 PM UTC-5, monde wilson wrote:
> >> >>>
> >> >>> why eigenvectors very slow
> >> >>>
> >> >>> what is the difference between numpy and sympy when doing matrix
> >> >>> calculation
> >> >>
> >> >>
> >> >> Sympy calculates eigenvectors symbolically (thus exactly), numpy
> >> >> calculates them numerically using floating point arithmetic.
> >> >> In general you don't want to use sympy to calculate the eigenvectors
> for
> >> >> matrices larger than 2x2, because the symbolic results can be very
> >> >> complicated. (IIRC, the eigenvalues are calculated by finding roots
> of the
> >> >> characteristic polynomial, which can lead to nasty expressions for
> dimension
> >> >> 3 and beyond.)
> >> >>
> >> >>>
> >> >>> will numpy faster and more accurately
> >> >>
> >> >>
> >> >> Numpy will be a lot faster, but not more accurate. If you only need
> >> >> numerical results, you probably should use numpy for this.
> >> >>
> >> >> Vinzent
> >> >
> >> > --
> >> > You received this message because you are subscribed to the Google
> Groups
> >> > "sympy" group.
> >> > To unsubscribe from this group and stop receiving emails from it,
> send an
> >> > email to [email protected].
> >> > To post to this group, send email to [email protected].
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> >> > To view this discussion on the web visit
> >> >
> https://groups.google.com/d/msgid/sympy/62a17328-bcd2-4955-9534-ae5358e89041%40googlegroups.com.
>
>
> >> > For more options, visit https://groups.google.com/d/optout.
> >
> >
> >
> > If I wanted to get the eigenvectors (and eigenvalues) of a 10x10
> symbolic matrix that is relatively sparse, is it possible to use sympy to
> solve this issue? Can the eigenvects() operation be parallelized in any
> way?
> >
> > I am trying to use OCTAVE as well (which calls from sympy), but once I
> get above 4x4 the time required to get a solution seems to scale
> geometrically: (2x2 in <1 sec, 3x3 in ~2 sec, 4x4 in ~minutes, 5x5 ~hr, 7x7
> ~12 hr).
> >
> > Is there some code somewhere with a robust eigensolver that can generate
> the eigenfunctions and eigenvalues of a 10x10 symbolic matrix? Based on my
> 7x7 matrix I know the denominators of the solution can be huge, but this is
> an important problem that I need to solve.
> >
> > Thanks.
> >
> > --
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> an email to [email protected] <javascript:>.
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> > To view this discussion on the web visit
> https://groups.google.com/d/msgid/sympy/d95a66fe-9135-4365-9386-6641bf51d9fa%40googlegroups.com.
>
>
> > For more options, visit https://groups.google.com/d/optout.
>
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