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 <[email protected]> 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 <[email protected]> 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]. >> >> > 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. >> > >> > -- >> > 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]. >> > Visit this group at https://groups.google.com/group/sympy. >> > 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. >> > -- > 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]. > Visit this group at https://groups.google.com/group/sympy. > To view this discussion on the web visit > https://groups.google.com/d/msgid/sympy/45532240-28a4-49ac-80d4-4df276ab1f81%40googlegroups.com > <https://groups.google.com/d/msgid/sympy/45532240-28a4-49ac-80d4-4df276ab1f81%40googlegroups.com?utm_medium=email&utm_source=footer> > . > For more options, visit https://groups.google.com/d/optout. > -- 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]. Visit this group at https://groups.google.com/group/sympy. To view this discussion on the web visit https://groups.google.com/d/msgid/sympy/CA%2B01voMMNUm2PXZ76nKto6_wQRmdWGLXaE0wu92YEcjjrT1r5A%40mail.gmail.com. For more options, visit https://groups.google.com/d/optout.
