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In this paper we study properties of H-singular values of a positive tensor and present an iterative algorithm for computing the largest H-singular value of the positive tensor. We prove that this method converges for any positive tensors.

Recently, eigenvalue problems for tensors have gained special attention in the realm of numerical multilinear algebra [

In this paper, we focus on the tensor

The paper is organized as follows. In Section 2, we recall some definitions and define H-singular values for a positive tensor, we extend the Perron-Frobenius theorem to H-singular values of positive tensors. In Section 3, we give an algorithm to find the largest singular value of a positive tensor, some numerical experiments are given to show that our algorithm is efficient.

Let

under the constraints that

We obtain the following system at a critical point:

where

If

Let

A vector

Lemma 1. If a tensor

Proof. If

If

Then

Similarly, we can get

Lemma 2. Let a tensor

solution of (2). If

Then

Proof. Define

if and only if

i.e.,

This implies

Remark. If there exists

Then

Theorem 1. Assume that a tensor

system (1), satisfying

genvectors

tive constant,

Proof. Denote

is well defined.

According to the Brouwer Fixed Point Theorem, there exists

where

Let

Then

Let us show:

this contradicts the result of Lemma 1. Therefore,

The uniqueness of the positive singular value with strongly positive left and right eigenvectors now follows from Lemma 2 directly. The uniqueness up to a multiplicative constant of the strongly positive left and right eigenvectors is proved in the same way as in [

Theorem 2. Assume that

where

Proof. Let

Since it is a positively 0-homogeneous function, it can be restricted on

Let

On the other hand, by the definition of

This means

According to Lemma 2, we have

Similarly, we prove the other equality.

Theorem 3. Assume that

Proof. Let

Apply Theorem 2, we can get

Theorem 4. Suppose that

where

Proof. Let

On the other hand, it is easy to check that C is an eigenvalue of A with corresponding eigenvectors

In this section, we propose an iterative algorithm to calculate the largest H-singular value of a positive tensor based on Theorem 2 and Theorem 3. This algorithm is a modified version of the one given in [

For a positive tensor

Algorithm 3.1

Step 0 Choose

Step 1 Compute

Let

Step 2 If

and replace

In the following, we will give a convergence result for Algorithm 3.1.

Theorem 5. Assume that

Proof. By (8),

We now prove for any

For each

Then,

So,

Hence, we get

which means for

Therefore, we get

Similarly, we can prove that

From Theorem 5,

By Theorem 5, we have

The argument used in the following proof is parallel to that in [

Theorem 6. Let

a)

b)

c)

Proof. As

pactness of the unit ball in

By the continuity of

If

By (a) and the continuity of

Then we obtain

By Theorem 6, we can get the largest H-singular value of

In the following, in order to show the viability of Algorithm 3.1, we used Matlab 7.1 to test it with some randomly generated rectangular tensors. For these randomly generated tensors, the value of each entry is be- tween 0 and 10. we set

Our numerical results are shown in

Ite | ||||||
---|---|---|---|---|---|---|

26 | 8.95e−007 | 36.78 | 2.42e−008 | 1.96e−008 | 1.87e−008 | |

27 | 7.68e−007 | 41.08 | 1.18e−008 | 8.30e−009 | 8.86e−009 | |

28 | 6.10e−007 | 46.39 | 2.82e−009 | 2.44e−009 | 1.87e−009 | |

29 | 9.24e−007 | 77.87 | 2.16e−009 | 1.71e−009 | 8.89e−010 | |

30 | 7.27e−007 | 165.51 | 6.59e−009 | 4.04e−009 | 3.57e−009 |

tive tensors.

In this paper, we give some eigenvalues properties about the H-singular value of a positive tensor

I thank the editor and the referee for their comments. The author is funded by the Fundamental Research Funds for Central Universities.