Abstract
Circulant matrix families have become an important tool in network engineering. In this paper, two new patterned matrices over which include row skew first-plus-last right circulant matrix and row first-plus-last left circulant matrix are presented. Their basic properties are discussed. Based on Newton-Hensel lifting and Chinese remaindering, two different algorithms are obtained. Moreover, the cost in terms of bit operations for each algorithm is given.
1. Introduction
Circulant matrix families play an important role in network engineering. Basic [1] gave a simple condition for characterizing weighted circulant graphs allowing perfect state transfer in terms of their eigenvalues. Noual et al. [2] showed some preliminary results on the dynamical behaviours of some specific nonmonotone Boolean automata networks that were called xor circulant networks. Using the circulant matrix, the charge transport and the noise of a quantum wire network, made of three semi-infinite external leads attached to a ring crossed by a magnetic flux, were investigated [3]. Based on the circulant adjacency matrices of the networks induced by these interior symmetries, Aguiar and Ruan [4] analyzed the impact of interior symmetries on the multiplicity of the eigenvalues of the Jacobian matrix at a fully synchronous equilibrium for the coupled cell systems associated with homogeneous networks. Involving circulant matrix, the storage of binary cycles in Hopfield-type and other neural networks was investigated [5]. A new structure for the decoupling of circulant symmetric arrays of more than four elements was presented in [6]. Wang and Cheng [7] studied the existence of doubly periodic travelling waves in cellular networks involving the discontinuous Heaviside step function by circulant matrix. Pais et al. [8] proved conditions for the existence of stable limit cycles arising from multiple distinct Hopf bifurcations of the dynamics in the case of circulant fitness matrices. Cho and Chung [9] discussed the routing of a message on circulant networks, that is, a key to the performance of this network. Grassi [10] designed DTCNNs where each trajectory converges to a unique equilibrium point, which depends only on the input and not on the initial state, by exploiting the global asymptotic stability of the equilibrium point of DTCNNs with circulant matrices. Wu [11] obtained the coexistence of multiple large-amplitude wave solutions for the delayed Hopfield-Cohen-Grossberg model of neural networks with a symmetric circulant connection matrix. The system model of the OFDM is based on AF relay networks as well as the strategy of the superimposed training involved circulant matrix [12]. Two-way transmission model was considered in [13] and ensured circular convolution between two frequency selective channels.
In this paper, we give two algorithms for an nonsingular RSFPLR circulant matrix over . The primitive problem is transformed into an equivalent problem over . The first algorithm supposes the factorization of is given and the costs of multiplications and additions over are and , respectively. We obtain the bit complexity bound: where denotes the bit complexity of multiplying -bit integers. The second algorithm does not know the factorization of and its cost is greater, by a factor , than in the first algorithm.
Definition 1. A row skew first-plus-last right (RSFPLR) circulant matrix with the first row over , denoted by , meant a square matrix of the form
Obviously, the RSFPLR circulant matrix over a field is a -circulant matrix [14], and that is neither the extention of circulant matrix over [15] nor its special case, and they are two different families of patterned matrices.
We define as the basic RSFPLR circulant matrix over ; that is,
It is easily verified that has no repeated roots over and is both the minimal polynomial and the characteristic polynomial of the matrix . In addition, is nonderogatory and satisfies and . In view of the structure of the powers of the basic RSFPLR circulant matrix over , it is clear that
Thus, is a RSFPLR circulant matrix over if and only if for some polynomial over . The polynomial will be called the representer of the RSFPLR circulant matrix over . By Definition 1 and (4), it is clear that is a RSFPLR circulant matrix over if and only if commutes with ; that is, .
In addition to the algebraic properties that can be easily derived from the representation (4), we mention that RSFPLR circulant matrices have very nice structure. The product of two RSFPLR circulant matrices is a RSFPLR circulant matrix and is a RSFPLR circulant matrix, too. Furthermore, let . It is a routine to prove that is a commutative ring with the matrix addition and multiplication.
Definition 2. A row first-plus-last left (RSLPFL) circulant matrix with the first row over , denoted by , meant a square matrix of the form
Lemma 3. Let be the matrix of the counteridentity. Then(i) ; (ii) .
Let be a nonsingular matrix over ; we explore the problem of finding a RSFPLR circulant matrix , such that .
Solving is clearly equivalent to finding a polynomial in such that
The congruence modulo follows from the equality . Hence, the problem of solving is equivalent to inversion in the ring .
The following theorem describes a necessary and sufficient condition for the nonsingularity of a RSFPLR circulant matrix over .
Theorem 4. Let be a RSFPLR circulant matrix over ; then the matrix is nonsingular if and only if for , where is the prime powers factorization of and .
Proof. If is nonsingular, by (7), there exists such that, for ,
that is, in .
The proof of sufficient condition for nonsingularity will be given in Section 2 (Lemmas 5 and 6).
Review of Bit Complexity Results [15]. The sum of two polynomials in of degree at most can be trivially calculated in bit operations. The product of two such polynomials can be calculated in multiplications and additions or subtractions in . Therefore, the cost of polynomial multiplication is bit operations, where
Let be two polynomials of degree at most over ( prime); we calculate in bit operations, where
2. Finding Inversion in for Factorization of Given
In this section, let be the given prime powers factorization of . In the following, we discuss the inverse of a RSFPLR circulant matrix over by studying the equivalent problem, that is, finding the inversion of a polynomial over . We obtain algorithms of calculating the inverse via Chinese remaindering, the extended Euclidean algorithm, and Newton-Hensel lifting.
Lemma 5. Let be known such that
for and and is a polynomial in .
One can solve such that
and the cost of bit operations is .
Proof. Due to in , we get
Let . Distinctly, for (). Since , we can solve which satisfies (). Let .
By construction, for , we get () and (). Then, for , we obtain . We come to the conclusion that
that is,
The computation of consists in (one for each coefficient) applications of Chinese remaindering. Obviously, the computation of should be done only once. Since integer division has the same asymptotic cost as multiplication, the cost of bit operations for is . Because each is got via an inversion in , the cost of bit operations for is . Finally, the cost of bit operations for calculating is by using . The thesis follows using the inequality
By Lemma 5, we can find the inversion of a polynomial over when is a prime power. The following lemma presents how to solve this special problem.
Lemma 6. Suppose in ; then is invertible in , where is a polynomial in . In this case, the cost of bit operations for the inverse of is , where and are the same as (11) and (10), respectively.
Proof. Suppose in ; by Bezout’s lemma, there exist which satisfy
In the following, we consider Newton-Hensel lifting; that is,
It is easy to verify by induction that . Therefore, the inverse element of in is .
The cost of bit operations for calculating is . Calculating is calculating each modulo . Therefore, the cost of bit operations for the whole sequence is .
By Theorem 4 and Lemmas 5 and 6, we obtain Algorithm 1 for the inversion of a polynomial over . The cost of bit operations for the algorithm is , where and are bounded by log and , respectively. On the side, by using and , we get Particularly, if , the ascendent term is . That is, the cost of calculating the inverse of is gradually bounded by the cost of executing multiplications in .
3. Algorithm of Finding Inversion in for Factorization of Unknown
In this section, we show how to compute the inverse of without knowing the factorization of the modulus. The number of bit operations of the new algorithm is only a factor greater than in the previous case.
Our idea consists in trying to compute in using the gcd algorithm for . Such algorithm requires the inversion of some scalars, which is not a problem in , but it is not always possible if is not prime. Therefore, the computation of may fail. However, if the gcd algorithm terminates, we have solved the problem. In fact, together with the alleged gcd , the algorithm also returns such that in . If , then is the inverse of . If , one can easily prove that is not invertible in . Note that we must force the gcd algorithm to return a monic polynomial.
If the computation of fails, we use recursion. In fact, the gcd algorithm fails if it cannot invert an element . Inversion is done by using the integer gcd algorithm. If is not invertible, the integer gcd algorithm returns , with . Hence, is a nontrivial factor of . We use to compute either a pair , such that and , or a single factor , such that and . In the first case, we invert in and , and we use Chinese remaindering to get the desired result. In the second case, we invert in and we use one step of Newton-Hensel lifting to get the inverse in .
The computation of the factors is done by procedure GetFactors whose correctness is proven by Lemmas 4.1 and 4.2 in [15]. Combining these procedures together, we get Algorithm 2.
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Theorem 7. Suppose is invertible in ; the cost of bit operations for Algorithm 2 which returns the inverse is .
Proof. It is similar to the proof of Theorem 4.3 in [15].
In addition, by Lemma 3 and Algorithms 1 and 2, it is easy to get two algorithms for inverting RSLPFL circulant matrices over , respectively.
4. Conclusion
In this paper, we consider the problem of finding inverse matrix for a RSFPLR circulant matrix with entries over . We present two different algorithms. Our algorithms require different degrees of knowledge of and , and their costs range, roughly, from to operations over . Moreover, for each algorithm, we give the cost in terms of bit operations. Finally, the extended algorithms are used to solve the problem of inverting RSLPFL circulant matrices over . Based on the existing problem in [16–19], we will develop solving these problems by circulant matrices technology.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported by the GRRC program of Gyeonggi province [(GRRC SUWON 2014-B3), Development of Cloud Computing-Based Intelligent Video Security Surveillance System with Active Tracking Technology]. Their support is gratefully acknowledged.