Your matrix is far simpler than I had imagined (you should have
mentioned that it was triangular). I think as Isuru said we can likely
implement a faster method for triangular matrices. The eigenvalues
themselves (the diagonals) are already computed very quickly.
Aaron Meurer
On Tue, Oct 9, 2018 at 10:36 AM Isuru Fernando <isu...@gmail.com> wrote:
>
> Hi,
>
> For triangular matrices, it's straightforward to calculate eigenvectors. You
> just need triangular solves. See Section 4.4.1 of Heath's Scientific
> Computing 2nd Edition.
>
> Isuru
>
> On Tue, Oct 9, 2018 at 11:27 AM Jacob Miner <yacobe...@gmail.com> wrote:
>>
>> 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 <yaco...@gmail.com> 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 <cacat...@gmail.com> 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
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>>> >> > email to sympy+un...@googlegroups.com.
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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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>>
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