THE PERTURBATION BOUND FOR THE SPECTRAL RADIUS OF A NON-NEGATIVE TENSOR

We study the perturbation bound for the spectral radius of an mth-order n-dimensional nonnegative tensor . The main contribution of this paper is to show that when is perturbed to a nonnegative tensor by , the absolute difference between the spectral radii of and is bounded by the largest magnitude of the ratio of the ith component of and the ith component , where is an eigenvector associated with the largest eigenvalue of in magnitude and its entries are positive. We further derive the bound in terms of the entries of only when is not known in advance. Based on the perturbation analysis, we make use of the NQZ algorithm to estimate the spectral radius of a nonnegative tensor in general. On the other hand, we study the backward error matrix and obtain its smallest error bound for its perturbed largest eigenvalue and associated eigenvector of an irreducible nonnegative tensor. Based on the backward error analysis, we can estimate the stability of computation of the largest eigenvalue of an irreducible nonnegative tensor by the NQZ algorithm. Numerical examples are presented to illustrate the theoretical results of our perturbation analysis.

Theorem 3. Suppose A is an th-order -dimensional nonnegative tensor.
(i) Then there exist  0 ≥ 0 and x ∈ P such that (ii) If A is irreducible, then  0 > 0 and x ∈ int(P).Moreover, if  is an eigenvalue with a nonnegative eigenvector, then  =  0 .If  is an eigenvalue of A, then || ≤  0 .
We call x a Perron vector of a nonnegative tensor corresponding to its largest nonnegative eigenvalue.
Some algorithms for computing the largest eigenvalue of an irreducible tensor were proposed; see, for instance, [8][9][10].However, the perturbation analysis and the backward error analysis for these algorithms have not been studied, which are important to the analysis of the accuracy and stability for computing the largest eigenvalue by these algorithms.
In this paper, we are interested in studying the perturbation bound for the spectral radius of an th-order dimensional nonnegative tensor A. The main contribution of this paper is to show that when A is perturbed to a nonnegative tensor Ã by ΔA and A has a positive Perron vector x, we have where the maximum norm of a tensor is defined as follows: The perturbation bound (6) shows that the absolute difference between the spectral radii of A and Ã is bounded by the largest magnitude of the ratio of the th component of ΔAx −1 and the th component x −1 .We further derive the bound based on the entries of A when x is not necessary to be known.Moreover, there is no convergence result of numerical algorithms [9,10] for computing the spectral radius of a nonnegative tensor in general.We will make use of our perturbation results to estimate the spectral radius of a nonnegative tensor in general via the NQZ algorithm (see [10]).
On the other hand, we will study the backward error matrix ΔA and obtain its smallest error bound of ΔA for such that Ã is an irreducible nonnegative tensor, λ is the largest eigenvalue of Ã, and x is a Perron vector of Ã by the NQZ algorithm.Our theoretical results show that ΔA = (  1 , 2 ,...,  ) can be chosen as follows: where r = λx [−1] − Ax −1 and ‖x‖ 2 denotes the vector 2norm.By using these backward error results, we will evaluate the stability of computation of the largest eigenvalue of an irreducible nonnegative tensor by the NQZ algorithm.The paper is organized as follows.In Section 2, we review the existing results and present the results for the perturbation bound of the spectral radius of a nonnegative tensor.In Section 3, we give the explicit expression of the backward error for the computation of the largest eigenvalue of an irreducible nonnegative tensor by the NQZ algorithm.Finally, concluding remarks are given in Section 4.

The Perturbation Bound of the Spectral Radius
Let us first state some preliminary results of a nonnegative tensor.
Proof.By Theorem 3, we note that Ax −1 =  0 x [−1] and  0 is the largest eigenvalue of A. Therefore, The result follows.
Theorem 7. Suppose A is an th-order -dimensional nonnegative tensor, Ã = A + ΔA is the perturbed nonnegative tensor of A, and Ã has a positive Perron vector z.Then one has min where  z = diag( 1 ,  2 , . . .,   ) and e is a vector of all ones.
Proof.Let us consider the tensor B = Ã − (A)I, where I is the identity tensor.
For an th-order -dimensional identity tensor I = (  1 , 2 ,...,  ), It is clear that the eigenvalue of B is the same as the eigenvalue of B× 1 Σ −(−1) × 2 Σ× 3 ⋅ ⋅ ⋅ ×  Σ for any positive diagonal matrix Σ.Then we obtain By choosing Σ =  z = diag( 1 ,  2 , . . .,   ), where z is a positive Perron vector of Ã, we have, for On the other hand, we can deduce that, for  1 = 1, . . ., , Taking and then by using (20) with  1 =  * , we give the combination of ( 18) and (19) as follows: Since the eigenvalue of A −(−1) z × 2  z × 3 ⋅ ⋅ ⋅ ×   z is the same as the eigenvalue of A, by using Lemma 4, (21) becomes The right hand side of (15) is established.By using the above argument, we can show the left hand side of (15).
By Theorem 7, it is easy to obtain the following corollary.

Corollary 8. Suppose
A is an th-order -dimensional nonnegative tensor, Ã = A + ΔA is the perturbed nonnegative tensor of A, and Ã has a positive Perron vector z.Then one has where It is noted that A can also be written as the perturbed tensor of Ã; that is, A = Ã − ΔA.Then, by Theorem 7, we have the following bound.Corollary 9. Suppose A is an th-order -dimensional nonnegative tensor with a positive Perron vector x, and Ã = A + ΔA is the perturbed nonnegative tensor of A. Then one has min where According to Corollary 9, we know that the absolute difference between the spectral radii of A and Ã is bounded by the largest magnitude of the ratio of the th component of ΔAx −1 and the th component x −1 .
Remark 10.In the nonnegative matrix case ( = 2), if  is irreducible nonnegative matrix with positive Perron vector , then the perturbation of the eigenvalues of Ã =  + Δ and  is given by min (25) see [12].It is easy to see that our perturbation result in Corollary 9 can be reduced to the bound in (25).
Remark 11.In [6], it has been shown that if A is symmetric and weakly irreducible, then A has a positive Perron vector.In particular, when A is irreducible, A also has a positive Perron vector; see [4][5][6].
In Corollary 9, a Perron vector x must be known in advance so that the perturbation bound can be computed.Here we derive the perturbation bound in terms of the entries of A only.Lemma 12. Suppose A is an th-order ( ≥ 3) dimensional positive tensor.Then one has where x is the positive Perron vector,   = max 1≤≤   , and Proof.Since A is a positive tensor, A must be irreducible and has a positive Perron vector.The left hand side of ( 26) is straightforward.Now we consider We note that, for any positive numbers   ,   , and   ( = 1, 2, . . ., ), we have As A is a positive tensor and x is a positive vector, it follows from the above inequalities and (30) that Because the above upper bound is valid for 2 ≤ ,   ≤ , we take the minimum among them over 2 ≤ ,   ≤  and we obtain the following inequality: The result follows.
Based on Lemma 12, we have the following lemma.
Lemma 13.Suppose A is an th-order -dimensional positive tensor ( ≥ 3), and Ã = A + ΔA is the perturbed nonnegative tensor of A. Then one has where Proof.By using (25) in Corollary 9, we have The result follows. Remark where (⋅) is defined in (35).
Proof.We just switch the roles of A and Ã in Lemma 13.
Let us state the following lemma to handle the case when A is a nonnegative tensor.

Lemma 16. Suppose A is an 𝑚th-order 𝑛-dimensional nonnegative tensor, and
The proof of this lemma is similar to Theorem 2.3 given in [7].
Suppose that A is a nonnegative tensor.Let A  = A + (1/)J.It is clear that A  is a positive tensor.Let Ã = A  + ΔA be a perturbed nonnegative tensor of A  .By applying Corollary 15 to A  and Ã , we have Therefore, when  → ∞, by Lemma 16, we have the following theorem.
Theorem 17. Suppose A is an th-order -dimensional nonnegative tensor, and Ã = A +ΔA is the perturbed nonnegative tensor of A. Then one has provided that (A) > 0.
Example 18.We conduct an experiment to verify the perturbation bound.We randomly construct a positive tensor A, where each entry is generated by uniform distribution [0, 1].
We further check the value of each entry must be greater than zero, and therefore the constructed tensor is positive.In the experiment, A is perturbed to a positive tensor Ã by adding ΔA, where  is a positive number and ΔA is a positive tensor randomly generated by the above-mentioned method.We study the absolute difference | − ρ| between the spectral radii of A and Ã and the perturbation bound in Corollary 9.In Figure 1, we show the results for  = 5, 10 and  = 4.For each point in the figure, we give the average value of | − ρ|, ‖ΔA× 1  −(−1) or (A) ‖ΔA‖ ∞ based on the computed results for 100 randomly constructed positive tensors.The -axis refers to values of : 0.01, 0.005, 0.001, 0.0005, 0.0001, and 0.00005.We see from the figures that the average values (in logarithm scale) depend linearly on  (in logarithm scale).The perturbation bounds ‖ΔA× 1  −(−1) x × 2  x × 3 ⋅ ⋅ ⋅ ×   x ‖ ∞ and (A) ‖ΔA‖ ∞ provide the upper bound of |− ρ|.This result is consistent with our prediction in the theory.It is interesting to note that the bound ‖ΔA −(−1)

Application to the NQZ Algorithm.
In this subsection, we apply our perturbation results to the NQZ algorithm [10] which is an iterative method for finding the spectral radius of a nonnegative tensor.The NQZ algorithm presented in [10] is given as follows.x Choose u (0) ∈ int(P) and let v = A(u (0) ) −1 .For  = 0, 1, 2, . .., compute It is shown in [10] that the sequences {} and {} converge to some numbers  and , respectively, and we have  ≤ (A) ≤ .If  = , the gap is zero and therefore both the sequences {} and {} converge to (A).However, a positive gap may happen, which can be seen in an example given in [10].Pearson [13] introduced the notion of essentially positive tensors and showed the convergence of the NQZ algorithm for essentially positive tensors.Chang et al. [8] further established the convergence of the NQZ algorithm for primitive tensors.
Definition 19.An th-order -dimensional tensor A is essentially positive if Ay −1 ∈ int(P) for any nonzero y ∈ P.
Definition 20.An th-order -dimensional tensor A is called primitive if there exists a positive integer  such that A  y ∈ int(P) for any nonzero y ∈ P.
An essentially positive tensor is a primitive tensor, and a primitive tensor is an irreducible nonnegative tensor but not vice versa.Now we demonstrate how to use the results in Theorem 7 to estimate the spectral radius of a nonnegative tensor via the NQZ algorithm.For example, when A is a reducible nonnegative tensor, then the NQZ algorithm may not be convergent.Our idea is to take a small perturbation ΔA = J, where  is a very small positive number and J is a tensor with all the entries being equal to one.It is clear that Ã = A + J is essentially positive.Therefore, when we apply the NQZ algorithm to compute the largest eigenvalue of Ã, it is convergent.Without loss of generality, we normalize the output vector u in the NQZ algorithm.In particular, the 1norm of u is employed.The output of the algorithm contains a positive number λ and u > 0 with ‖u‖ 1 = 1 (i.e., ∑  =1   = 1).We note that Then, by Theorem 7, we have Let   = max 1≤≤   and   = min 1≤≤   .By considering  ≥ 3, we know that that is, we have On the other hand, By putting (46) and (47) into (43), we have One may know This shows that we can estimate (A) of a nonnegative tensor A with a specified precision via the NQZ algorithm for the computation of ( Ã).
Remark 21.Liu et al. [9] modified the NQZ algorithm for computing the spectral radius of A + I, where A is an irreducible nonnegative tensor and  is a very small number, and showed that the algorithm converges to (A) + .This fact can be explained by our theoretical analysis in Theorem 7 by setting Δ = I.We just note that the left and right sides of the inequality in (15) are equal to ; that is,  ≤ ( Ã) − (A) ≤ .It implies that ( Ã) = (A) + .However, when the given nonnegative tensor is not irreducible, then their results cannot be valid.However, our approach can still be used to estimate the spectral radius of a nonnegative tensor in general.There is no convergence result of the NQZ algorithm for such A, and the spectral radius of A is equal to √ 2 ≈ 1.414213562373095; see [8].In this example, we take the perturbation ΔA = J.We apply the NQZ algorithm to compute the spectral radius of Ã to approximate the actual one.According to Table 1, the results show that (A) is about ( Ã) + ().
Example 23.We consider the following 3rd order 3dimensional nonnegative tensor: and the other entries are equal to zero.It can be shown that this tensor A is reducible.The spectral radius of A is equal to 3.There is no convergence result of the NQZ algorithm for such reducible nonnegative tensor.It is interesting to note that this tensor satisfies (49).In this example, we take the perturbation ΔA = J.The spectral radii of Ã by the NQZ algorithm are still accurate approximations of (A): 3.090000000000000 ( = 0.01), 3.009000000000000 ( = 0.001), 3.000900000000000 ( = 0.0001), 3.000090000000001 ( = 0.00001), and 3.000009000000000 ( = 0.000001).Indeed, both the values in the left and right sides of (43) are equal to 3.000000000000000 (up to the number of decimals shown in MATLAB).These results show the bounds are very tight and (A) is about ( Ã) + ().
Example 24.Now we consider the following 3rd order 3dimensional nonnegative tensor:  1,1,1 = 1 and the other entries are equal to zero.A is a reducible nonnegative tensor.For such A, we know the  is the spectral radius of A and [1, 0, . . ., 0]  is the corresponding eigenvector (the normalized in 2-norm).In this example, we take the perturbation ΔA = J.Although this tensor does not satisfy (49), the spectral radii of Ã by the NQZ algorithm are still accurate approximate of (A): 1.015565072567277 ( = 0.01), 1.001139501996208 ( = 0.001), 1.000104123060726 ( = 0.0001), 1.000010127700654 ( = 0.00001), and 1.000001004012030 ( = 0.000001).We find that the values in the left and right sides of (43) are close to 1 and 0, respectively, for different values of .It is clear that the bounds are not tight.It is interesting to note that the maximum and minimum values of the entries of the associated eigenvector are 1 and 0; we expect that the error bound in (43) can be very poor.Indeed, the errors between the actual eigenvector and the approximate eigenvector are 0.243064619277186 ( = 0.01), 0.077415571882820 ( = 0.001), 0.024493622285564 ( = 0.0001), 0.007745927468297 ( = 0.00001), and 0.002449488513126 ( = 0.000001).Thus the approximation is more accurate for the largest eigenvalue than for the associated eigenvector.

Backward Error Bound
In this section, we study the backward error tensor ΔA and obtain its smallest error bound of ΔA such that where A and Ã = A + ΔA are irreducible nonnegative tensors with ΔA = (  1 , 2 ,...,  ) for 1 ≤  1 ,  2 , . . .,   ≤ , λ is the largest eigenvalue of Ã, and x is a Perron vector of Ã.
The backward error for the eigenpair is defined as follows: where The minimization of ‖Δ‖ 2  is equivalent to finding a matrix  such that ‖rb † ‖ 2  + ‖( − bb † )‖ 2  can be minimized.It is clear that we choose  = 0; that is, min ‖Δ‖ 2  = ‖rb † ‖ 2  .According to the NQZ algorithm stated in Section 2.1, we know that r is nonnegative.As u is positive, both b and b † are also positive.Let us consider two cases.We note that if r is a zero vector, then  is the largest eigenvalue of A, and the problem is solved.Now we consider r is not a zero vector.It follows that at least one row of Δ = rb † must be positive.It can be shown that Ã is irreducible.The argument is that for any nonempty proper index subset J ⊂ {1, 2, . . ., } such that   1 , 2 ,...,  +   1 , 2 ,...,  > 0, ∀ 1 ∈ J, ∀ 2 , . . .,   ∉ J. (62) Hence the result follows.We randomly construct a positive tensor A, where each entry is generated by uniform distribution [0, 1].We further check the value of each entry must be greater than zero, and therefore the constructed tensor is positive.In Table 2, we show the average backward error based on the computed results for 100 randomly constructed positive tensors.We see from the table that the backward error can be very small, but it still depends on the order  of and the dimension  of the tensor.The backward error is large when  or  is large.

Concluding Remarks
In summary, we have studied and derived the perturbation bounds for the spectral radius of a nonnegative tensor.Numerical examples have been given to demonstrate our theoretical results.Also we have investigated the backward error matrix ΔA and obtain its smallest error bound for its perturbed largest eigenvalue and associated eigenvector of an irreducible nonnegative tensor.Numerical examples have shown that the backward error can be very small.These results may show that the NQZ algorithm may be backward stable for the computation of the largest eigenvalue of an irreducible nonnegative tensor.
In this paper, we do not study the perturbation bound of the eigenvector corresponding to the largest eigenvalue of a nonnegative tensor.This would be an interesting research topic for future consideration.

Table 2 :
[14]backward errors for different cases.Remark 27.When  = 2, the results of the backward error are reduced to the results in Theorem 6.3.2 in[14]; that is, (x) =      rx †      .(63)Example28.Let us conduct an experiment to check the backward error when we use the NQZ algorithm for computing the largest eigenvalue of an irreducible nonnegative tensor.