**Remark 4.**


#### **Algorithm 1** An algorithm for identifying H-tensors

**Step 1.** Set *k*1 := 0, *k*2 := 0, *k*3 := 0 and *s* := 50.

**Step 2.** Given a complex tensor A = (*ai*<sup>1</sup>···*im* ). If *k*3 = *s*, then output *k*1 and *k*2, and stop. Otherwise,

**Step 3.** Compute |*ai*···*<sup>i</sup>*| and *Ri*(A) for all *i* ∈ N.

**Step 4.** If *N*∗ = *N*, then print "A is a H-tensor", and go to Step 5. Otherwise, go to Step 6. **Step 5.** Replace *k*1 by *k*1 + 1, and replace *k*3 by *k*3 + 1, then go to Step 2.

**Step 6.** If *N*∗ = ∅, then print "A is not a H-tensor", and go to Step 7. Otherwise, go to Step 8.

**Step 7.** Replace *k*2 by *k*2 + 1, and replace *k*3 by *k*3 + 1, then go to Step 2.

**Step 8.** Compute |*ai*···*<sup>i</sup>*|, *α*(*i*) N*t* , *α*(*i*) N*t* , |*aj*···*j*|, *α*(*j*) N*s* , *α*(*j*) N*s* for all *i* ∈ N*<sup>t</sup>*, *j* ∈ N*<sup>s</sup>*, 1 ≤ *t* = *s* ≤ *k*. **Step 9.** If Inequality (3) holds, then print "A is a H-tensor", and go to Step 5. Otherwise, **Step 10.** Compute:

$$\sum\_{\substack{i\_2,\dots,i\_{\text{lv}}\in\mathbb{N}^{m-1}\backslash\left(\mathbb{N}\backslash\mathbb{N}\_k\right)^{m-1}\\\delta\_{i i\_2,\dots i\_{\text{lv}}}=0}} \mid a\_{i i\_2,\dots i\_m} \mid \text{ and} \\ \sum\_{\substack{i\_2,\dots,i\_{\text{lv}}\in\left(\mathbb{N}\backslash\mathbb{N}\_k\right)^{m-1}}} \mid a\_{i i i\_2,\dots i\_m} \mid \text{ and} \\ \delta\_{i i\_2,\dots i\_{\text{lv}}} \mid \text{ for}$$

**Step 11.** If Inequality (6) holds, then print "A is not a H-tensor", and go to Step 5. Otherwise,

**Step 12.** Print "Whether A is a strong H-tensor is not checkable", and replace *k*3 by *k*3 + 1. Go to Step 2.
