First k-1 entries of the k th eigenvector for an upper triangular matrix U
is U[:k-1,:k-1]^-1 @ U[:k-1,k], which is a triangular solve since
U[:k-1,:k-1] is a triangular matrix and it can be done in O(k^2) time.
Isuru
On Tue, Oct 9, 2018 at 1:27 PM Jacob Miner <yacobe...@gmail.com> wrote:
> Isuru,
>
> I went into Heath's text to get your reference, and it helps layout the
> method, but can you please clarify what you meant by 'triangular solves'?
>
> Thank you.
>
> On Tue, Oct 9, 2018, 10:45 Aaron Meurer <asmeu...@gmail.com> wrote:
>
>> 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
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>> >>> >> > "sympy" group.
>> >>> >> > To unsubscribe from this group and stop receiving emails from
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>> >>> >> > email to sympy+un...@googlegroups.com.
>> >>> >> > To post to this group, send email to sy...@googlegroups.com.
>> >>> >> > Visit this group at http://groups.google.com/group/sympy.
>> >>> >> > 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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>> >>> > For more options, visit https://groups.google.com/d/optout.
>> >>
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