Errors in using null command due to truncation error

Hi,

I'm trying to find eigenvectors of a 9-by-9 square matrix, corresponding to its eigenvalues. The matrix consists of components with complex numbers and one symbolic 'a', so I found nine eigenvalues ('a'), via solving the determinant of the matrix. For some eigenvalues, I used the 'vpa' command since, without 'vpa', they are obtained as a form of 'root(eqn, z, integer)'. Here the issue seems to arise. Due to the truncation error, the 'null' shows empty eigenvectors corresponding to the eigenvalues. FYI, I don't know how to assign a specific variable in 'eig' and it takes forever to run. Is there a breakthrough other than Gauss elimination method with suppressing close-to-zero values?

clc;
close all;
clear;

syms a
R = 0.5234;
r = 0.0054;
s = 0.0084;
for p = 1:3
    M = [a*R*r 1i*r 0 0 0 0 a*R*8.78+1i*(8.78-p*R) 0 0;
        0 0 a*R*s 1i*s 0 0 0 a*R*79.88+1i*(79.88-p*R) 0;
        0 0 0 0 a*R*0.4542 1i*0.4542 -(a*R*291.06+1i*(291.06+p*R)) -(a*R*291.06+1i*(291.06+p*R)) 0;
        a*R*8.78+1i*(8.78-p*R) 0 0 0 0 0 0 0 a*R;
        0 a*R*8.78+1i*(8.78-p*R) 0 0 0 0 0 0 1i;
        0 0 a*R*79.88+1i*(79.88-p*R) 0 0 0 0 0 a*R;
        0 0 0 a*R*79.88+1i*(79.88-p*R) 0 0 0 0 1i;
        0 0 0 0 a*R*291.06+1i*(291.06+p*R) 0 0 0 a*R;
        0 0 0 0 0 a*R*291.06+1i*(291.06+p*R) 0 0 1i];
 
    Det = det(M);
    DetEqn = Det == 0;
    EigenVal1 = solve(DetEqn,a);
    EigVal = vpa(EigenVal1);
    for j=1:rank(M)
        M_temp = subs(M,a,EigVal(j));
        EigVec(:,j) = null(M_temp)
    end
end

Unable to perform assignment because the indices on the left side are not compatible with the size of the right side.

Error in sym/privsubsasgn (line 1168)
L_tilde2 = builtin('subsasgn',L_tilde,struct('type','()','subs',{varargin}),R_tilde);

Error in indexing (line 999)
C = privsubsasgn(L,R,inds{:});

Answer 1

[Walter Roberson]

Your code assumes that the null space is the same size each time, but most of the time the null space is empty. You cannot store an empty vector into a definite vector location.

You need to decide what you want to do when the null space is empty.

[Walter Roberson]

syms a
R = 0.5234;
r = 0.0054;
s = 0.0084;
for p = 1:3
    M = [a*R*r 1i*r 0 0 0 0 a*R*8.78+1i*(8.78-p*R) 0 0;
        0 0 a*R*s 1i*s 0 0 0 a*R*79.88+1i*(79.88-p*R) 0;
        0 0 0 0 a*R*0.4542 1i*0.4542 -(a*R*291.06+1i*(291.06+p*R)) -(a*R*291.06+1i*(291.06+p*R)) 0;
        a*R*8.78+1i*(8.78-p*R) 0 0 0 0 0 0 0 a*R;
        0 a*R*8.78+1i*(8.78-p*R) 0 0 0 0 0 0 1i;
        0 0 a*R*79.88+1i*(79.88-p*R) 0 0 0 0 0 a*R;
        0 0 0 a*R*79.88+1i*(79.88-p*R) 0 0 0 0 1i;
        0 0 0 0 a*R*291.06+1i*(291.06+p*R) 0 0 0 a*R;
        0 0 0 0 0 a*R*291.06+1i*(291.06+p*R) 0 0 1i];
 
    Det = det(M);
    DetEqn = Det == 0;
    EigenVal1 = solve(DetEqn,a);
    EigVal = (EigenVal1);
    for j=1:rank(M)
        M_temp = subs(M,a,EigVal(j));
        EV = null(M_temp);
        if isempty(EV)
            EigVec(:,j,p) = sym(NaN(size(EV,1),1));
        else
            EigVec(:,j,p) = EV;
        end
    end
end

format long g
EigVec = double(EigVec)

EigVec = 
EigVec(:,:,1) =

  Columns 1 through 3

        -0.0604434080666857 -    0.0568402099138265i        -0.0604434080666857 +    0.0568402099138265i                          0 +                     0i
        -0.0568402099138265 +    0.0604434080666857i        -0.0568402099138265 -    0.0604434080666857i                          0 +                     0i
       -0.00630053702136478 -   0.00625925383312013i       -0.00630053702136478 +   0.00625925383312013i                          0 +                     0i
       -0.00625925383312013 +   0.00630053702136478i       -0.00625925383312013 -   0.00630053702136478i                          0 +                     0i
       -0.00171477249055503 -   0.00171785608816912i       -0.00171477249055503 +   0.00171785608816912i          0.998204973259795 +                     0i
       -0.00171785608816912 +   0.00171477249055503i       -0.00171785608816912 -   0.00171477249055503i                          1 +                     0i
                          0 +                     0i                          0 +                     0i                          0 +                     0i
                          0 +                     0i                          0 +                     0i                          0 +                     0i
                          1 +                     0i                          1 +                     0i                          0 +                     0i

  Columns 4 through 6

                          0 +                     0i           1.06339171087373 +                     0i                        NaN +                     0i
                          0 +                     0i                          1 +                     0i                        NaN +                     0i
           1.00659554466799 +                     0i                          0 +                     0i                        NaN +                     0i
                          1 +                     0i                          0 +                     0i                        NaN +                     0i
                          0 +                     0i                          0 +                     0i                        NaN +                     0i
                          0 +                     0i                          0 +                     0i                        NaN +                     0i
                          0 +                     0i                          0 +                     0i                        NaN +                     0i
                          0 +                     0i                          0 +                     0i                        NaN +                     0i
                          0 +                     0i                          0 +                     0i                        NaN +                     0i

  Columns 7 through 9

                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i
                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i
                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i
                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i
                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i
                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i
                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i
                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i
                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i


EigVec(:,:,2) =

  Columns 1 through 3

        -0.0641387232677706 -    0.0564917511132487i        -0.0641387232677706 +    0.0564917511132487i                          0 +                     0i
        -0.0564917511132487 +    0.0641387232677706i        -0.0564917511132487 -    0.0641387232677706i                          0 +                     0i
       -0.00634195365255762 -   0.00625884452532305i       -0.00634195365255762 +   0.00625884452532305i                          0 +                     0i
       -0.00625884452532305 +   0.00634195365255762i       -0.00625884452532305 -   0.00634195365255762i                          0 +                     0i
       -0.00171169167540565 -   0.00171784779045346i       -0.00171169167540565 +   0.00171784779045346i          0.996416379214725 +                     0i
       -0.00171784779045346 +   0.00171169167540565i       -0.00171784779045346 -   0.00171169167540565i                          1 +                     0i
                          0 +                     0i                          0 +                     0i                          0 +                     0i
                          0 +                     0i                          0 +                     0i                          0 +                     0i
                          1 +                     0i                          1 +                     0i                          0 +                     0i

  Columns 4 through 6

                          0 +                     0i           1.13536440283453 +                     0i                        NaN +                     0i
                          0 +                     0i                          1 +                     0i                        NaN +                     0i
            1.0132786693931 +                     0i                          0 +                     0i                        NaN +                     0i
                          1 +                     0i                          0 +                     0i                        NaN +                     0i
                          0 +                     0i                          0 +                     0i                        NaN +                     0i
                          0 +                     0i                          0 +                     0i                        NaN +                     0i
                          0 +                     0i                          0 +                     0i                        NaN +                     0i
                          0 +                     0i                          0 +                     0i                        NaN +                     0i
                          0 +                     0i                          0 +                     0i                        NaN +                     0i

  Columns 7 through 9

                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i
                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i
                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i
                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i
                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i
                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i
                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i
                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i
                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i


EigVec(:,:,3) =

  Columns 1 through 3

        -0.0680253050205014 -    0.0558597772365388i        -0.0680253050205014 +    0.0558597772365388i                          0 +                     0i
        -0.0558597772365388 +    0.0680253050205014i        -0.0558597772365388 -    0.0680253050205014i                          0 +                     0i
       -0.00638363887049742 -   0.00625815577392186i       -0.00638363887049742 +   0.00625815577392186i                          0 +                     0i
       -0.00625815577392186 +   0.00638363887049742i       -0.00625815577392186 -   0.00638363887049742i                          0 +                     0i
        -0.0017086164151074 -   0.00171783399737567i        -0.0017086164151074 +   0.00171783399737567i           0.99463418334813 +                     0i
       -0.00171783399737567 +    0.0017086164151074i       -0.00171783399737567 -    0.0017086164151074i                          1 +                     0i
                          0 +                     0i                          0 +                     0i                          0 +                     0i
                          0 +                     0i                          0 +                     0i                          0 +                     0i
                          1 +                     0i                          1 +                     0i                          0 +                     0i

  Columns 4 through 6

                          0 +                     0i           1.21778690116231 +                     0i                        NaN +                     0i
                          0 +                     0i                          1 +                     0i                        NaN +                     0i
           1.02005113025445 +                     0i                          0 +                     0i                        NaN +                     0i
                          1 +                     0i                          0 +                     0i                        NaN +                     0i
                          0 +                     0i                          0 +                     0i                        NaN +                     0i
                          0 +                     0i                          0 +                     0i                        NaN +                     0i
                          0 +                     0i                          0 +                     0i                        NaN +                     0i
                          0 +                     0i                          0 +                     0i                        NaN +                     0i
                          0 +                     0i                          0 +                     0i                        NaN +                     0i

  Columns 7 through 9

                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i
                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i
                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i
                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i
                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i
                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i
                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i
                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i
                        NaN +                     0i                        NaN +                     0i                        NaN +                     0i

[Walter Roberson]

Note that in this above code, the null() calculation is working on the symbolic solutions, so there is no truncation error going on.

[Seung Hyeop Hyun]

Walter,

Thank you for leaving your comments. The issue here is that there must be eigenvectors corresponding to the eigenvalues, which can be obtained from the 'null', and also the dimension of the eigenvectors is definitely 9x1. The 'null' cannot find the eigenvectors and provides 0X1 empty vectors. I believe that it's because of the approximation due to using 'vpa'. The 'a' is already replaced by one of eigenvalues, and therefore, the null is not working on the symbolic solutions. Alternatively, I can do 'eig' for M after one of eigenvalues is entered.

[Matt J]

There doesn't appear to be any reason for you to be using symbolic operations for this problem. Eigenvalue problems generally do not have closed form solutions. Also, all of your problem input parameters (e.g. R,r,s) have a priori known values.

[Torsten]

@Seung Hyeop Hyun

I don't know why you talk about "eigenvalues", but I agree that if "a" gives det(M(a)) = 0, null(M(a)) should be at least 1-dimensional and not empty.

[Walter Roberson]

@Seung Hyeop Hyun

You did not take into account that you use floating point constants and that some of the calculations take place in floating point instead of as symbolic numbers.

When you use symbolic numbers consistently then the problem does not show up.

Q = @(v) sym(v);
syms a
R = Q(5234)/Q(10)^4;
r = Q(54)/Q(10)^4;
s = Q(84)/Q(10)^4;

n8_78 = Q(878)/Q(10)^2;
n79_88 = Q(7988)/Q(10)^2;
n_4542 = Q(4542)/Q(10)^4;
n291_06 = Q(29106)/Q(10)^2;
for p = 1:3
    M = [a*R*r 1i*r 0 0 0 0 a*R*n8_78+1i*(n8_78-p*R) 0 0;
        0 0 a*R*s 1i*s 0 0 0 a*R*n79_88+1i*(n79_88-p*R) 0;
        0 0 0 0 a*R*n_4542 1i*n_4542 -(a*R*n291_06+1i*(n291_06+p*R)) -(a*R*n291_06+1i*(n291_06+p*R)) 0;
        a*R*n8_78+1i*(n8_78-p*R) 0 0 0 0 0 0 0 a*R;
        0 a*R*n8_78+1i*(n8_78-p*R) 0 0 0 0 0 0 1i;
        0 0 a*R*n79_88+1i*(n79_88-p*R) 0 0 0 0 0 a*R;
        0 0 0 a*R*n79_88+1i*(n79_88-p*R) 0 0 0 0 1i;
        0 0 0 0 a*R*n291_06+1i*(n291_06+p*R) 0 0 0 a*R;
        0 0 0 0 0 a*R*n291_06+1i*(n291_06+p*R) 0 0 1i];
 
    Det = det(M);
    DetEqn = Det == 0;
    EigenVal1 = solve(DetEqn,a);
    EigVal = EigenVal1;
    for j=1:rank(M)
        M_temp = subs(M,a,EigVal(j));
        EV = null(M_temp);
        if isempty(EV)
            EigVec(:,j,p) = sym(NaN(size(EV,1),1));
        else
            EigVec(:,j,p) = EV;
        end
    end
end

EigVec

EigVec(:,:,1) = 

EigVec(:,:,2) = 

EigVec(:,:,3) = 

format long g
EigVec = double(EigVec)

EigVec = 
EigVec(:,:,1) =

  Columns 1 through 3

        -0.0604434080666857 -    0.0568402099138265i        -0.0604434080666857 +    0.0568402099138265i                          0 +                     0i
        -0.0568402099138265 +    0.0604434080666857i        -0.0568402099138265 -    0.0604434080666857i                          0 +                     0i
       -0.00630053702136478 -   0.00625925383312013i       -0.00630053702136478 +   0.00625925383312013i                          0 +                     0i
       -0.00625925383312013 +   0.00630053702136478i       -0.00625925383312013 -   0.00630053702136478i                          0 +                     0i
       -0.00171477249055503 -   0.00171785608816912i       -0.00171477249055503 +   0.00171785608816912i          0.998204973259795 +                     0i
       -0.00171785608816912 +   0.00171477249055503i       -0.00171785608816912 -   0.00171477249055503i                          1 +                     0i
                          0 +                     0i                          0 +                     0i                          0 +                     0i
                          0 +                     0i                          0 +                     0i                          0 +                     0i
                          1 +                     0i                          1 +                     0i                          0 +                     0i

  Columns 4 through 6

                          0 +                     0i           1.06339171087373 +                     0i          -2.07917517265388 -      0.12335357734933i
                          0 +                     0i                          1 +                     0i           2.08986241502569 +     0.131173171659899i
           1.00659554466799 +                     0i                          0 +                     0i          -1.17431482491886 -      3.23249177899391i
                          1 +                     0i                          0 +                     0i           1.16945872447306 +      3.25381182291119i
                          0 +                     0i                          0 +                     0i          0.389692214298109 -     0.187973926292369i
                          0 +                     0i                          0 +                     0i         -0.392422256869245 +     0.187636508068213i
                          0 +                     0i                          0 +                     0i         0.0467535877540691 +   0.00573056182938062i
                          0 +                     0i                          0 +                     0i          -0.15363302189684 +     0.127699515578846i
                          0 +                     0i                          0 +                     0i                          1 +                     0i

  Columns 7 through 9

          -2.07917517265388 +      0.12335357734933i          -1.91125259969198 -     0.520193365405037i          -1.91125259969198 +     0.520193365405037i
           2.08986241502569 -     0.131173171659899i           1.91129494286941 +     0.553169312823223i           1.91129494286941 -     0.553169312823223i
          -1.17431482491886 +      3.23249177899391i         -0.105661537118353 -     0.770245234313409i         -0.105661537118353 +     0.770245234313409i
           1.16945872447306 -      3.25381182291119i         0.0937570861782642 +     0.775325421161631i         0.0937570861782642 -     0.775325421161631i
          0.389692214298109 +     0.187973926292369i         0.0716575680173222 -     0.185960430138347i         0.0716575680173222 +     0.185960430138347i
         -0.392422256869245 -     0.187636508068213i        -0.0749584912828433 +     0.185626626193629i        -0.0749584912828433 -     0.185626626193629i
         0.0467535877540691 -   0.00573056182938062i         0.0363384221349967 +    0.0221560988269227i         0.0363384221349967 -    0.0221560988269227i
          -0.15363302189684 -     0.127699515578846i       -0.00986540081559587 +   0.00258850280209401i       -0.00986540081559587 -   0.00258850280209401i
                          1 +                     0i                          1 +                     0i                          1 +                     0i


EigVec(:,:,2) =

  Columns 1 through 3

        -0.0641387232677706 -    0.0564917511132487i        -0.0641387232677706 +    0.0564917511132487i                          0 +                     0i
        -0.0564917511132487 +    0.0641387232677706i        -0.0564917511132487 -    0.0641387232677706i                          0 +                     0i
       -0.00634195365255762 -   0.00625884452532305i       -0.00634195365255762 +   0.00625884452532305i                          0 +                     0i
       -0.00625884452532305 +   0.00634195365255762i       -0.00625884452532305 -   0.00634195365255762i                          0 +                     0i
       -0.00171169167540565 -   0.00171784779045346i       -0.00171169167540565 +   0.00171784779045346i          0.996416379214726 +                     0i
       -0.00171784779045346 +   0.00171169167540565i       -0.00171784779045346 -   0.00171169167540565i                          1 +                     0i
                          0 +                     0i                          0 +                     0i                          0 +                     0i
                          0 +                     0i                          0 +                     0i                          0 +                     0i
                          1 +                     0i                          1 +                     0i                          0 +                     0i

  Columns 4 through 6

                          0 +                     0i           1.13536440283453 +                     0i          -1.03424396514104 -    0.0577669915193808i
                          0 +                     0i                          1 +                     0i           1.04493120751284 +    0.0655865858299493i
            1.0132786693931 +                     0i                          0 +                     0i         -0.589585462682334 -      1.60558586753832i
                          1 +                     0i                          0 +                     0i           0.58472936223653 +      1.62690591145559i
                          0 +                     0i                          0 +                     0i          0.193481085863487 -    0.0941556722582628i
                          0 +                     0i                          0 +                     0i         -0.196211128434622 +    0.0938182540341066i
                          0 +                     0i                          0 +                     0i         0.0116310771845003 +   0.00138540923306503i
                          0 +                     0i                          0 +                     0i        -0.0380957587743605 +    0.0318852196758995i
                          0 +                     0i                          0 +                     0i                          1 +                     0i

  Columns 7 through 9

          -1.03424396514104 +    0.0577669915193808i         -0.955605128257274 -     0.243608708993425i         -0.955605128257274 +     0.243608708993425i
           1.04493120751284 -    0.0655865858299493i          0.955647471434705 +     0.276584656411611i          0.955647471434705 -     0.276584656411611i
         -0.589585462682334 +      1.60558586753832i         -0.058782994029221 -     0.382582523732593i         -0.058782994029221 +     0.382582523732593i
           0.58472936223653 -      1.62690591145559i         0.0468785430891321 +     0.387662710580816i         0.0468785430891321 -     0.387662710580816i
          0.193481085863487 +    0.0941556722582628i         0.0341783223759006 -    0.0931471170415329i         0.0341783223759006 +    0.0931471170415329i
         -0.196211128434622 -    0.0938182540341066i        -0.0374792456414217 +    0.0928133130968145i        -0.0374792456414217 -    0.0928133130968145i
         0.0116310771845003 -   0.00138540923306503i        0.00912923446797742 +   0.00536880051870816i        0.00912923446797742 -   0.00536880051870816i
        -0.0380957587743605 -    0.0318852196758995i       -0.00244438930219064 +  0.000683128476076533i       -0.00244438930219064 -  0.000683128476076533i
                          1 +                     0i                          1 +                     0i                          1 +                     0i


EigVec(:,:,3) =

  Columns 1 through 3

        -0.0680253050205014 -    0.0558597772365388i        -0.0680253050205014 +    0.0558597772365388i                          0 +                     0i
        -0.0558597772365388 +    0.0680253050205014i        -0.0558597772365388 -    0.0680253050205014i                          0 +                     0i
       -0.00638363887049742 -   0.00625815577392186i       -0.00638363887049742 +   0.00625815577392186i                          0 +                     0i
       -0.00625815577392186 +   0.00638363887049742i       -0.00625815577392186 -   0.00638363887049742i                          0 +                     0i
        -0.0017086164151074 -   0.00171783399737567i        -0.0017086164151074 +   0.00171783399737567i           0.99463418334813 +                     0i
       -0.00171783399737567 +    0.0017086164151074i       -0.00171783399737567 -    0.0017086164151074i                          1 +                     0i
                          0 +                     0i                          0 +                     0i                          0 +                     0i
                          0 +                     0i                          0 +                     0i                          0 +                     0i
                          1 +                     0i                          1 +                     0i                          0 +                     0i

  Columns 4 through 6

                          0 +                     0i           1.21778690116231 +                     0i         -0.685933562636758 -    0.0359047962427311i
                          0 +                     0i                          1 +                     0i          0.696620805008562 +    0.0437243905532995i
           1.02005113025445 +                     0i                          0 +                     0i         -0.394675675270157 -      1.06328389705312i
                          1 +                     0i                          0 +                     0i          0.389819574824353 +       1.0846039409704i
                          0 +                     0i                          0 +                     0i           0.12807737638528 -    0.0628829209135606i
                          0 +                     0i                          0 +                     0i         -0.130807418956415 +    0.0625455026894044i
                          0 +                     0i                          0 +                     0i        0.00514395587663915 +  0.000594946349441334i
                          0 +                     0i                          0 +                     0i        -0.0167933653854523 +    0.0141531959048798i
                          0 +                     0i                          0 +                     0i                          1 +                     0i

  Columns 7 through 9

         -0.685933562636758 +    0.0359047962427311i         -0.637055971112372 -     0.151413823522888i         -0.637055971112372 +     0.151413823522888i
          0.696620805008562 -    0.0437243905532995i          0.637098314289803 +     0.184389770941074i          0.637098314289803 -     0.184389770941074i
         -0.394675675270157 +      1.06328389705312i        -0.0431568129995103 -     0.253361620205655i        -0.0431568129995103 +     0.253361620205655i
          0.389819574824353 -       1.0846039409704i         0.0312523620594214 +     0.258441807053877i         0.0312523620594214 -     0.258441807053877i
           0.12807737638528 +    0.0628829209135606i         0.0216852404954267 -    0.0622093460092614i         0.0216852404954267 +    0.0622093460092614i
         -0.130807418956415 -    0.0625455026894044i        -0.0249861637609478 +     0.061875542064543i        -0.0249861637609478 -     0.061875542064543i
        0.00514395587663915 -  0.000594946349441334i         0.0040759677296243 +   0.00231048172032896i         0.0040759677296243 -   0.00231048172032896i
        -0.0167933653854523 -    0.0141531959048798i        -0.0010764184837051 +  0.000319388109428762i        -0.0010764184837051 -  0.000319388109428762i
                          1 +                     0i                          1 +                     0i                          1 +                     0i