# Numerical Reconstruction of Spring-Mass System from Two Nondisjoint Spectra

Authors: Hubert Pickmann; Juan C. Egaña; Ricardo L. Soto
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

A new numerical procedure is presented to reconstruct a fixed-free spring-mass system from two auxiliary spectra, which are nondisjoint. The method is a modification of the fast orthogonal reduction algorithm, which is less computationally expensive than others in the literature. Numerical results are obtained, showing the accuracy of the algorithm.


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## Body

## 1. Introduction

The inverse problems in structures vibration look for determining or estimating the physical properties of a system in vibration (mass density, elastic constants, etc.) from a known dynamic behavior (natural frequencies, electric flux, tension, etc.) (see [1–4]).The model used, which has generated much interest in the literature, as a prototype of structure, is a nonuniform thin rod with one end fixed to a surface (see [2–5]), whose discrete model is a spring-mass system, which consists of n mass mi>0, associated with the masses of each element of the rod and connected by n springs with rigidity constants ki>0 corresponding to the rigidity of each one of these elements (Figure 1).Figure 1
Fixed-free spring-mass system.A spring-mass system in free and longitudinal vibration is governed by a generalized eigenvalue problem of the form (see [1–4])(1)K-λMu=0,λ=ω2,where (2)K=k1+k2-k2-k2k2+k3-k3⋱⋱⋱-kn-1kn-1+kn-kn-knknand M=diag{m1,m2,…,mn} are the stiffness matrix and the mass matrix, respectively, and u is the displacement vector. In this system, the eigenvalues λi,i=1,2,…,n, of (1) are related to the natural frequencies ωi and the eigenvectors ui=(u1i,u2i,…,uni)T represent the vibration modes of the system. The spring-mass system is denoted by (M,K).It is known (see [4]) that the matrices M and K can be uniquely reconstructed if the following information is given: the eigenvalues (λi)1n of the original system (M,K), the eigenvalues (μi)1n-1 of the auxiliary system (M¯,K¯), corresponding to the original system whose last mass is fixed (Figure 2), and an additional factor, for example, the total mass of the system mT=∑i=1nmi.Figure 2
Fixed-fixed spring-mass system.The structural properties of the matricesM and K allow us to reduce the generalized eigenvalue equation (1) to the standard form (see [1–4]) (3)J-λiIvi=0,J=B-1KB-1,vi=Bui,B=diag⁡m11/2,m21/2,…,mr1/2,where the Jacobi matrix J is tridiagonal symmetric positive definite, with the same eigenvalues (λi)1n of the system (M,K), which are real, positive, and distinct. Therefore, a fundamental step to determine the system (M,K) is to reconstruct the matrix J. Without loss of generality, we assume that J is of the following form:(4)J=a1b1b1a2b2⋱⋱⋱bn-2an-1bn-1bn-1an,ai,bi>0.In [6], stable numerical procedures to reconstruct the Jacobi matrix J are discussed. This reconstruction uses as initial spectral information the eigenvalues (λi)1n of J and the eigenvalues (μi)1n-1 of the matrix J¯, which is obtained by deleting the last row and last column of J. A fundamental property in these procedures is the interlacing property (see [1, 4, 6])(5)λ1<μ1<λ2<⋯<μn-1<λn,which is a necessary and sufficient condition for the existence of a physically real system and for constructing J as well.In [7, 8] the authors generalize the reconstruction of the system (M,K) by using the interlaced spectrum corresponding to an auxiliary system that consists in fixing any mass of the system (M,K), other than the extreme masses (Figure 3).Figure 3
Spring-mass system with a fixed interior mass.Clearly, if the auxiliary system(M¯,K¯) is the system with its (r+1)th mass, 1≤r≤n-2, being fixed, then (M¯,K¯) is uncoupled in two auxiliary spring-mass systems, (M¯r,K¯r) and (M¯p,K¯p), with natural frequencies (γi)1r and (ηi)1p, respectively, where p=n-r-1. The structural properties of the matrices M¯r, K¯r, M¯p, and K¯p allow us to partition J as(6)J=Jrbrer0brerTar+1br+1e1T0br+1e1Jp,where the submatrices Jr=Br-1KrBr-1 and Jp=Bp-1KpBp-1, with Br=diag{m11/2,m21/2,…,mr1/2} and Bp=diag⁡{mr+21/2,mr+31/2,…,mn1/2}, are related to the systems (M¯r,K¯r) and (M¯p,K¯p), respectively. As the system (M,K) can be reconstructed from the matrix J, it is enough to reconstruct J from the sets (λi)1n and (μi)1n-1=(γi)1r∪(ηi)1p to obtain (M,K). Thus, the reconstruction of the system (M,K) is reduced to the following problem.Problem 1.
Given the sequence of real numbers(λi)1n, (μi)1n-1=(γi)1r∪(ηi)1p satisfying the interlacing property(7)λi≤μi≤λi+1,1≤i≤n-1,reconstruct the matrix J in (6), such that (8)σJ=λi1n,σJr=γi1r,σJp=ηi1p,p=n-r-1.In this problem two cases arise: in the first one, all natural frequencies(γi)1r and (ηi)1p are distinct; that is, (γi)1r∩(ηi)1p=∅. In terms of the J matrix, the meaning is that no eigenvector vi of J has a node in its coordinate vm+1i; that is, vm+1i≠0,i=1,…,n. In this case, the reconstruction is unique. In the second case, one or more natural frequencies (γi)1r and (ηi)1p, are identical. The meaning of this situation is that some eigenvector of J, let us say vl, has a node in vm+1l; that is, vm+1l=0. In this case, (γi)1r∩(ηi)1p∩(λi)1n≠∅, and a family of isospectral matrices is obtained.In [7], the authors study the first case; that is, they reconstruct the system (M,K), using a modification of the fast orthogonal reduction method, when the auxiliary spectra are separated. In the next section we study the second case, using the same method, and thus the problem is completely solved. This method is less computationally expensive than others in literature [8].


## 2. Reconstructing the System from Nondisjoint Spectra

We denote byPn(λ), Qr(λ), and Sp(λ) the characteristic polynomials of the matrices J, Jr, and Jp, respectively; that is,(9)Pnλ=∏i=1nλ-λi;Qrλ=∏i=1rλ-γi;Spλ=∏i=1pλ-ηi.We define the vectors(10)ur=ur,rr,ur,rr-1,…,ur,r1T,wp=wp,11,wp,12,…,wp,1pT,corresponding, respectively, to the last and the first row of the matrices of eigenvectors of Jr and Jp. We also define the diagonal matrices Δr=diagγr,γr-1,…,γ1 and Δp=diagη1,η2,…,ηp.Theorem 2.
Let the real numbers(λi)1n and (μi)1n-1=(γi)1r∪(ηi)1p,p=n-r-1 be given, satisfying the interlacing property (7); that is, (γi)1r∩(ηi)1p∩(λi)1n≠∅. Then there exists an isospectral family of n×n matrices Jα, α∈(0,π/2), of the form (6) such that σ(Jα)=(λi)1n, σ(Jrα)=(γi)1r, and σ(Jpα)=(ηi)1p.Proof.
We suppose that there is a pairγj, ηk of frequencies such that γj=ηk=λm, where m=j+k. From the expansion of det(λIn-Jn), throughout its r+1th row, we find that(11)Pnλ=λ-ar+1QrλSpλ-br2Qr∖rλSpλ-br+12QrλSp∖1λ=-br+12QrλSp∖1λ+Spλλ-ar+1Qrλ-br2Qr∖rλ,where Sp∖1(λ) and Qr∖r(λ) are the characteristic polynomials of Jp and Jr after we delete its first row and column, and rth row and column, respectively.
On the other hand, if we denote byQr+1λ the characteristic polynomial of the principal submatrix obtained from Jr by adding (r+1)th row and column, we have that(12)Qr+1λ=λ-ar+1Qrλ-br2Qr∖rλ.Thus, (11) is(13)Pnλ=SpλQr+1λ-br+12QrλSp∖1λ.Analogously, (11) can be written as(14)Pnλ=-br2Qr∖rλSpλ+Qrλλ-ar+1Spλ-br+12Sp∖1λ.Now, if we denote by Sp+1λ the characteristic polynomial of the principal submatrix obtained from Jp by adding a row and column above Jp, we have(15)Sp+1λ=λ-ar+1Spλ-br+12Sp∖1λ.Then,(16)Pnλ=QrλSp+1λ-br2Qr∖rλSpλ.From (12) we have(17)Qr+1γj=-br2Qr∖rγj,and from (15) we have(18)Sp+1ηk=-br+12Sp∖1ηk.Since the polynomials Pn(λ), Qr(λ), and Sp(λ) in (9) have common factors λ-λm≡λ-ηk≡λ-γj, (13) and (16) are (19)Pn′λm=Sp′ηkQr+1γj-br+12Qr′γjSp∖1ηk,(20)Pn′λm=Qr′γjSp+1ηk-br2Qr∖rγjSp′ηk,respectively, where(21)Qr′γi=∏j=1j≠irγi-γj,Sp′ηi=∏j=1j≠ipηi-ηj,Pn′λm=∏i=1i≠mnλm-λi.Replacing (17) and (18) in these last two equations, we obtain(22)Pn′λm=-br2Qr∖rγjSp′ηk-br+12Qr′γjSp∖1ηkand dividing (22) by Sp′ηkQr′γj, we get(23)Pn′λmSp′ηkQr′γj=-br2Qr∖rγjQr′γj-br+12Sp∖1ηkSp′ηk.It is known that if Ur=[ur(1)ur(2)⋯ur(r)] is the orthogonal matrix of eigenvectors of Jr, then UrT(λIr-Jr)Ur=λIr-Λr, and we have(24)λIr-Jr-1=UrλIr-ΛrUrT.The left side in (24) is(25)λIr-Jr-1=1Qrλ·adjλIr-Jr=1Qrλ·∗∗∗∗∗∗∗∗⋮⋮⋮⋮∗∗∗Qr∖rλwhile the right side is(26)UrλIr-ΛrUrT=ur1λ-γ1ur2λ-γ2⋯urrλ-γrUrT.Comparing the entries in position (r,r) in both sides in (24) we find that(27)Qr∖rλQrλ=∑i=1rur,ri2λ-γi.Taking the limit when λ tends to γi, we obtain(28)Qr∖rγjQr′γj=ur,rj2.Analogously, we can obtain(29)Sp∖1ηkSp′ηk=wp,1k2.Then, by replacing (28) and (29) in (23), we get (30)-Pn′λmSp′ηkQr′γj=br2ur,rj2+br+12wp,1k2.For α∈0,π/2 we can define(31)br2αur,rjα2=-Pn′λmSp′ηkQr′γjcos2α,br+12αwp,1kα2=-Pn′λmSp′ηkQr′γjsin2α.Thus, given that(32)br2α=-Pn′γjSp′γjQr′γjcos2⁡α+∑i=1i≠jr-PnγiSpγiQr′γi,(33)br+12α=-Pn′ηkSp′ηkQr′ηksin2⁡α+∑i=1i≠jp-PnηiSp′ηiQrηi,(31) allow us to know ur,rj(α) and wp,1k(α).
Subsequently, once the vectorsur(α) and wp(α) in (10) are known, we can form the (n+1)×(n+1) matrices:(34)Ar+1α=ΛrurαurTαa0r,Ap+1α=a0pwrTαwrαΛp,where the entries a0r and a0p are arbitrary real numbers. Then, we apply the Modified Fast Orthogonal Reduction Algorithm (see [9]) to orthogonally reduce the matrices Ar+1(α) and Ap+1(α) to their tridiagonal form, obtaining in this way the desired matrices Jr(α) and Jp(α). To do this, we first permute the arrowhead matrix Ar+1(α) by applying P=1╱1. We point out that similar relationships are analyzed by Jessup in [10].
Finally, considering that the diagonal entryar+1 of J(α) can be computed as(35)ar+1=∑i=1nλi-∑i=1rγi-∑i=1pηiand the codiagonal entries br(α) and br+1(α) can be computed from (32) and (33), respectively, the matrix J(α) of the form (6) is obtained completely, having a common eigenvalue with Jr(α) and Jp(α).
If we have more common eigenvalues, we can repeat the previous procedure for each pair of common eigenvalues. That is, ifs is the number of identical pairs γjq≡ηkq≡λmq, q=1,…,s, then we have(36)br2αur,riα2=-PnγiSpγiQr′γi,i∗=1,2,…,r,br2αur,rjqα2=-Pn′γjqSp′γjqQr′γjqcos2⁡α,q=1,2,…,s,br+12αwp,1iα2=-PnηiSp′ηiQrηi,i∗=1,2,…,p,br+12αwp,1kqα2=-Pn′ηkqSp′ηkqQr′ηkqsin2⁡α,q=1,2,…,s.Thus,(37)br2α=∑i=1r∗-PnγiSpγiQr′γi+cos2⁡α∑q=1s-Pn′γjqSp′γjqQr′γjq,br+12α=∑i=1p∗-PnηiSp′ηiQrηi+sin2⁡α∑q=1s-Pn′ηkqSp′ηkqQr′ηkq,where ∗ means that s terms i=jq and i=kq,  q=1,…,s, respectively, are omitted. Thus, we obtain an isospectral family of tridiagonal matrices that have identical eigenvalues.


## 3. An Optimization Procedure to Find an Objective Jacobi Matrix

In this section we want to find an objective matrix within a family of matrices. First, we observe that the construction procedure depends continuously on the parameterα. Then, by means of an optimization procedure, we find an appropriate α, so that the procedure reconstructs a matrix with a desired structure.Theorem 3.
LetJ~ be a given symmetric tridiagonal matrix partitioned in the following form:(38)J~=J~rb~rer0b~rerTa~r+1b~r+1e1T0b~r+1e1J~p,where σJ~=(λi)1n, σJ~r=(γi)1r, and σJ~p=(ηi)1p, with p=n-r-1 and (γi)1r∩(ηi)1p∩(λi)1n≠∅. For ɛ>0 small enough the function, F:0+ɛ,π/2-ɛ→R, defined by (39)Fα=Jα-J~22=∑jikα-j~ik2,where the matrix Jα is obtained by using the Modified Fast Orthogonal Reduction process, has a minimum in 0+ɛ,π/2-ɛ.Proof.
Givenα∈0,π/2, the Modified Fast Orthogonal Reduction process reconstructs a matrix Jα of the form (38), from its eigenvalues. We show that all the entries of the matrix Jα depend continuously on α. In fact, expressions (37) are clear since cosα and sinα are not zero in 0,π/2. That is, (40)brα≠0,br+1α≠0.The functions (41)ur,riα=-PnγiSpγiQr′γi1br2α,i∗=1,2,…,r,ur,rjqα=-Pn′γjqSp′γjqQr′γjqcos2⁡αbr2α,q=1,2,…,s,wp,1iα=-PnηiSp′ηiQrηi1br+12α,i∗=1,2,…,p,wp,1kqα=-Pn′ηkqSp′ηkqQr′ηkqsin2⁡αbr+12α,q=1,2,…,s,are continuous. Therefore, all the matrices (42)Ar+1α=ΛrαurαurTαa0rα;Ap+1α=a0pαwpTαwpαΛpαhave continuous entries. Since, in the Modified Fast Orthogonal Reduction process, the tridiagonalization matrices have rational entries with nonzero denominators, matrices Jrα,Jpα, and, thus, Jα depend continuously on α∈0,π/2.
Now, ifα=0 and (γi)1r∩(ηi)1p∩(λi)1n=(υi)1p, then br+1α=0. Afterwards, due to the discontinuity of wp,1α, the reconstruction procedure is not completed. Analogously, when α=π/2 and (γ)1r∩(η)1p∩(λ)1n=(γ)1r, we have brα=0, where again a discontinuity of ur,rα is produced. Therefore, by affixing, 0<ɛ≪1, the function (43)Fα=Jα-J~22=∑jikα-j~ik2is defined and is continuous in I=0+ɛ,π/2-ɛ. Then, by Weierstrass’ theorem, F has a minimum in I.


### 3.1. Numerical Examples

Here we give some examples which show numerical results obtained in the reconstruction of theJ matrix. In all examples, the reconstructed matrix is the well-known matrix(44)J=21121⋱⋱⋱12112,which has eigenvalues λi=2-2cosπi/n+1. Moreover, it is also known that if we delete the (r+1)th row and column of J, the eigenvalues of the submatrices Jr and Jp are, respectively, γi=2-2cosπi/r+1 and ηi=2-2cosπi/p+1, with p=n-r-1.Example 1.
In Table1, we show the results associated with the reconstructed matrix J^, of the form (44), with n=11, r=3, and p=7 for α=0.5. The given eigenvalues are shown in the second, third, and fourth column. In the fifth and sixth column we list the diagonal and codiagonal entries of J^. In the last column, we show the relative error eλi=λi-λ^i2/λi2 with respect to the exact eigenvalues λi of the matrix J and the eigenvalues λ^i of J^.Table 1
i
λ
i
γ
i
η
i
a
^
i
b
^
i
e
λ
i
×
1
0
-
14
1
0.0681
0.5858
0.1522
2.0000
1.0000
0.4073
2
0.2679
2.0000
0.5858
2.0000
1.0000
0.0622
3
0.5858
3.4142
1.2346
2.0000
1.0748
0.0379
4
1.0000
2.0000
2.0000
0.9191
0.0111
5
1.4824
2.7654
2.0000
1.0000
0.0599
6
2.0000
3.4142
2.0000
1.0000
0.0333
7
2.5176
3.8478
2.0000
1.0880
0.0353
8
3.0000
2.0000
0.9035
0.0148
9
3.4142
2.0000
1.0000
0.0390
10
3.7321
2.0000
1.0000
0.0119
11
3.9319
2.0000
0.0113Example 2.
In Table2, we show the results associated with the reconstructed matrix J^, by considering appropriate orders of J^ and for arbitrary values of α, listed from the first to fourth column. The relative errors ea=loga-a^2/a2 and eb=logb-b^2/b2, with respect to the diagonal and codiagonal entries of J and J^, respectively, are shown in the fifth and sixth column. In the last column, we present the relative errors eλ=logλ-λ^2/λ2, defined as in Example 1.Table 2
n
r
p
α
e
a
e
b
e
λ
9
4
4
0.3
−14.5655
−0.6192
−15.5341
20
5
14
0.6
−13.9291
−2.0497
−15.4525
31
11
19
0.7
−13.6363
−2.1453
−15.4552
41
33
7
1.2
−13.3794
−1.9747
−15.3817
50
17
32
0.8
−13.3984
−1.7963
−15.5074
63
27
35
0.5
−13.2271
−1.6714
−15.3403
71
55
15
1.3
−12.9105
−1.4487
−15.2575
85
49
35
1.1
−13.1134
−1.9090
−15.4762
93
37
55
0.4
−12.9884
−1.8209
−15.4001
101
77
23
0.9
−12.7881
−1.9179
−15.1940Example 3.
In the reconstructed matrixJ^, by considering the same orders of Example 2, we add an optimization process of Golden Section Search [11], of parameter α, obtaining an optimal α, denoted as αopt and listed in the fourth column of Table 3. In the last three columns the relative errors ea,eb, and eλ are shown.Table 3
n
r
p
α
o
p
t
e
a
e
b
e
λ
9
4
4
0.7854
−14.6213
−15.4519
−15.4589
20
5
14
0.5639
−13.9393
−13.8169
−15.5854
31
11
19
0.6591
−13.7277
−13.5305
−15.5647
41
33
7
1.1192
−13.2729
−14.1176
−15.2605
50
17
32
0.6361
−13.2659
−14.2862
−15.2269
63
27
35
0.7227
−13.2019
−14.3889
−15.2199
71
55
15
1.0799
−13.0260
−14.3109
−15.1878
85
49
35
0.8672
−12.9445
−14.1855
−15.1569
93
37
55
0.6891
−12.8941
−14.2323
−15.2115
101
77
23
1.0644
−12.8154
−14.0673
−15.0930Example 4.
In this example, we reconstruct the matrixJ^ for n=101 and 503. In each case we do as many reconstructions of J^ as values r can take. The values of n allow us to have various reconstructions with σJ∩σJr∩σJp≠∅. Figures 4 and 5 show the plots of the relative errors eJ=logJ-J^2/J2 and eλ=logλ-λ^2/λ2. The results of our numerical experiments confirm that our method works quite well.Figure 4
69 of 90 reconstructions ofJ with identical eigenvalues.Figure 5
359 of 501 reconstructions ofJ identical eigenvalues.


## 3.1. Numerical Examples

Here we give some examples which show numerical results obtained in the reconstruction of theJ matrix. In all examples, the reconstructed matrix is the well-known matrix(44)J=21121⋱⋱⋱12112,which has eigenvalues λi=2-2cosπi/n+1. Moreover, it is also known that if we delete the (r+1)th row and column of J, the eigenvalues of the submatrices Jr and Jp are, respectively, γi=2-2cosπi/r+1 and ηi=2-2cosπi/p+1, with p=n-r-1.Example 1.
In Table1, we show the results associated with the reconstructed matrix J^, of the form (44), with n=11, r=3, and p=7 for α=0.5. The given eigenvalues are shown in the second, third, and fourth column. In the fifth and sixth column we list the diagonal and codiagonal entries of J^. In the last column, we show the relative error eλi=λi-λ^i2/λi2 with respect to the exact eigenvalues λi of the matrix J and the eigenvalues λ^i of J^.Table 1
i
λ
i
γ
i
η
i
a
^
i
b
^
i
e
λ
i
×
1
0
-
14
1
0.0681
0.5858
0.1522
2.0000
1.0000
0.4073
2
0.2679
2.0000
0.5858
2.0000
1.0000
0.0622
3
0.5858
3.4142
1.2346
2.0000
1.0748
0.0379
4
1.0000
2.0000
2.0000
0.9191
0.0111
5
1.4824
2.7654
2.0000
1.0000
0.0599
6
2.0000
3.4142
2.0000
1.0000
0.0333
7
2.5176
3.8478
2.0000
1.0880
0.0353
8
3.0000
2.0000
0.9035
0.0148
9
3.4142
2.0000
1.0000
0.0390
10
3.7321
2.0000
1.0000
0.0119
11
3.9319
2.0000
0.0113Example 2.
In Table2, we show the results associated with the reconstructed matrix J^, by considering appropriate orders of J^ and for arbitrary values of α, listed from the first to fourth column. The relative errors ea=loga-a^2/a2 and eb=logb-b^2/b2, with respect to the diagonal and codiagonal entries of J and J^, respectively, are shown in the fifth and sixth column. In the last column, we present the relative errors eλ=logλ-λ^2/λ2, defined as in Example 1.Table 2
n
r
p
α
e
a
e
b
e
λ
9
4
4
0.3
−14.5655
−0.6192
−15.5341
20
5
14
0.6
−13.9291
−2.0497
−15.4525
31
11
19
0.7
−13.6363
−2.1453
−15.4552
41
33
7
1.2
−13.3794
−1.9747
−15.3817
50
17
32
0.8
−13.3984
−1.7963
−15.5074
63
27
35
0.5
−13.2271
−1.6714
−15.3403
71
55
15
1.3
−12.9105
−1.4487
−15.2575
85
49
35
1.1
−13.1134
−1.9090
−15.4762
93
37
55
0.4
−12.9884
−1.8209
−15.4001
101
77
23
0.9
−12.7881
−1.9179
−15.1940Example 3.
In the reconstructed matrixJ^, by considering the same orders of Example 2, we add an optimization process of Golden Section Search [11], of parameter α, obtaining an optimal α, denoted as αopt and listed in the fourth column of Table 3. In the last three columns the relative errors ea,eb, and eλ are shown.Table 3
n
r
p
α
o
p
t
e
a
e
b
e
λ
9
4
4
0.7854
−14.6213
−15.4519
−15.4589
20
5
14
0.5639
−13.9393
−13.8169
−15.5854
31
11
19
0.6591
−13.7277
−13.5305
−15.5647
41
33
7
1.1192
−13.2729
−14.1176
−15.2605
50
17
32
0.6361
−13.2659
−14.2862
−15.2269
63
27
35
0.7227
−13.2019
−14.3889
−15.2199
71
55
15
1.0799
−13.0260
−14.3109
−15.1878
85
49
35
0.8672
−12.9445
−14.1855
−15.1569
93
37
55
0.6891
−12.8941
−14.2323
−15.2115
101
77
23
1.0644
−12.8154
−14.0673
−15.0930Example 4.
In this example, we reconstruct the matrixJ^ for n=101 and 503. In each case we do as many reconstructions of J^ as values r can take. The values of n allow us to have various reconstructions with σJ∩σJr∩σJp≠∅. Figures 4 and 5 show the plots of the relative errors eJ=logJ-J^2/J2 and eλ=logλ-λ^2/λ2. The results of our numerical experiments confirm that our method works quite well.Figure 4
69 of 90 reconstructions ofJ with identical eigenvalues.Figure 5
359 of 501 reconstructions ofJ identical eigenvalues.



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*Source: 350879-2015-07-01.xml*