A Note on the Square Roots of a Class of Circulant Matrices

Abstract

We prove that any 𝑘-circulant matrix and any even order skew 𝑘-circulant matrix are diagonalizable for any 𝑘∈ℂ. Then, we propose two algorithms for computing the square roots of the 𝑘-circulant matrix and the skew 𝑘-circulant matrix, respectively. In particular, we show that the square roots of the 𝑘-circulant matrix are still 𝑘-circulant matrices. Both the theoretical analysis and the numerical experiments show that our algorithms are faster than the standard Schur method.

1. Introduction

Given a matrix 𝐴, a matrix 𝑋 is called a square root of 𝐴 if 𝑋²=𝐴. Matrix square roots appear in a variety of branches of mathematics, such as Markov models of finance, the solution of differential equations, the computation of the polar decomposition, and the matrix sign function [1]. A number of methods have been proposed for computing the square roots of a matrix [2–9]. Among them, the Schur method [7] is the most popular and becomes the standard method for computing the matrix square roots. However, Schur method is not so efficient when the matrix order is relatively high. Thus, it is very desirable to design fast computing methods which can make full use of the particular properties of the matrices when the matrices possess special structures.

Circulant matrices and their generalizations have a wide range of applications in signal processing, coding theory, digital image disposal, self-regress design, Toeplitz systems, and so on [10–14]. A relatively comprehensive survey about circulant matrices can be found in [15]. Recently, Lu and Gu [16] presented two efficient algorithms to compute the square roots of circulant matrices and quasi-skew circulant matrices, respectively. As they are based on LL iteration [17] and the modified Schulz iterative method, the two algorithms are faster than the standard Schur method. Subsequently, Mei generalized those methods and presented algorithms to compute the square roots of 𝑘-circulant matrices and skew 𝑘-circulant matrices [18]. These algorithms are also faster than the standard Schur algorithm, but the work is restricted to the case in which the matrix is of even order and 𝑘∈ℝ and can not be directly extended to compute the 𝑝th root.

In this paper, we first show that 𝑘-circulant matrices of any order for any complex number 𝑘 are diagonalizable and develop an algorithm to compute their principal square roots. Then, we show that skew 𝑘-circulant matrices of even order for any complex number 𝑘 are diagonalizable, and develop an algorithm to compute their principle square roots. Both of our algorithms are theoretically and experimentally proved to be faster than the standard Schur method. Compared with the work in [18], our methods are more general in that they are valid for any 𝑘-circulant matrix and any even order skew 𝑘-circulant matrix, where 𝑘 can be any complex number. The remainder of this paper is organized as follows. In Section 2, we compute the square roots of 𝑘-circulant matrices. In Section 3, we compute the square roots of skew 𝑘-circulant matrices. In Section 4, we present two numerical experiments to exhibit the efficiency of the proposed algorithms in terms of the CPU time.

2. Square Roots of 𝑘-Circulant Matrices

An 𝑛×𝑛 complex matrix Circ_𝑘(𝑎)𝑎∶=⁰𝑎₁𝑎₂⋯𝑎_𝑛−1𝑘𝑎_𝑛−1𝑎₀𝑎₁…𝑎_𝑛−2𝑘𝑎_𝑛−2𝑘𝑎_𝑛−1𝑎₀⋯𝑎_𝑛−3⋮⋮⋮⋱⋮𝑘𝑎₁𝑘𝑎₂𝑘𝑎₃⋯𝑎₀ (1) is called a 𝑘-circulant matrix, where 𝑎=(𝑎₀,𝑎₁,…,𝑎_𝑛−1)∈ℂ^𝑛 and 𝑘∈ℂ. In particular, 1-circulant matrices are circulant matrices, and −1-circulant matrices are skew circulant matrices [19].

Another equivalent definition of a 𝑘-circulant matrix is as follows [18]: let ℂ^𝑛×𝑛 be the set of all 𝑛×𝑛 complex matrices, and then, 𝐴∈ℂ^𝑛×𝑛 is a 𝑘-circulant matrix if and only if 𝐴=𝐺⁻¹𝐴𝐺, where 𝐺=Circ_𝑘([0,1,0,…,0]). In this section, we show that 𝑘-circulant matrices are diagonalizable.

Lemma 1 (see [20]). If 𝐴 and 𝐵 are two 𝑘-circulant matrices of the same order, then 𝐴+𝐵 is also a 𝑘-circulant matrix.

Lemma 2 (see [20]). If 𝐴 is a 𝑘-circulant matrix, then for any 𝑙∈ℂ, 𝑙𝐴 is also a 𝑘-circulant matrix.

Lemma 3 (see [20]). Let 𝐺=𝐶𝑖𝑟𝑐_𝑘([0,1,0,…,0]); then, 𝐺^𝑖=0⋯010⋯00⋯001⋯0⋮⋱⋮⋮⋮⋱⋮0⋯000⋯1𝑘⋯000⋯0⋮⋱⋮⋮⋮⋱⋮0⋯𝑘00⋯0 (2) is also a 𝑘-circulant matrix. In particular, 𝐺^𝑛=𝑘𝐸, where 𝐸 is the identity matrix.

Lemma 4 (see [20]). Let 𝐺=𝐶𝑖𝑟𝑐_𝑘([0,1,0,…,0]) and let 𝑘=𝑟𝑒^𝑖𝜃 (𝑟≥0, 0≤𝜃≤2𝜋); then, the eigenvalues of 𝐺 are 𝜃_𝑗=𝑟^1/𝑛𝑒^{𝑖(𝜃+2𝜋(𝑗−1)/𝑛)},𝑗=1,…,𝑛, (3) where 𝑖 is the imaginary unit.

Theorem 5 (see [20]). The matrix 𝐶=𝐶𝑖𝑟𝑐_𝑘(𝑎) is a 𝑘-circulant matrix of the form (1) if and only if 𝐶 can be represented by 𝐶=^𝑛−1_𝑗=0𝑎_𝑗𝐺^𝑗, (4) where 𝑎=(𝑎₀,…,𝑎_𝑛−1), 𝐺=𝐶𝑖𝑟𝑐_𝑘([0,1,0,…,0]), and 𝐺⁰=𝐸.

Theorem 6 (see [20]). Let 𝐶=𝐶𝑖𝑟𝑐_𝑘(𝑎) be a 𝑘-circulant matrix; then, the eigenvalues of 𝐶 are 𝜆_𝑗=𝜙(𝜃_𝑗) (𝑗=1,…,𝑛), where 𝑎=(𝑎₀,…,𝑎_𝑛−1),  𝜙(𝑥)=𝑎₀+𝑎₁𝑥+𝑎₂𝑥²+⋯+𝑎_𝑛−1𝑥^𝑛−1, and 𝜃_𝑗 is defined by (3).

Theorem 7 (see [20]). Any 𝑘-circulant matrix 𝐶𝑖𝑟𝑐_𝑘(𝑎) can be diagonalized as follows: 𝑉⁻¹⋅𝐶𝑖𝑟𝑐_𝑘(𝑎)⋅𝑉=diag𝜙𝜃₁,𝜙𝜃₂,…,𝜙𝜃_𝑛 (5) with the matrix 𝜃𝑉=111⋯1₁𝜃₂𝜃₃⋯𝜃_𝑛𝜃²₁𝜃²₂𝜃²₃⋯𝜃²_𝑛𝜃⋮⋮⋮⋱⋮_1𝑛−1𝜃₂^𝑛−1𝜃₃^𝑛−1⋯𝜃_𝑛^𝑛−1, (6) where 𝜃_𝑗 (𝑗=1,…,𝑛) is defined as (3).

By (5), we can easily obtain the following result.

Corollary 8. The square roots of 𝑘-circulant matrix 𝐶𝑖𝑟𝑐_𝑘(𝑎) are as follows: 𝑉diag𝜙𝜃₁^1/2,𝜙𝜃₂^1/2,…,𝜙𝜃_𝑛^1/2𝑉⁻¹. (7)

Remark 9. We mention that the diagonalization methods [19–22] were used to compute the square roots of 𝑘-circulant matrices, where 𝑘 is restricted to 1 or −1. However, our diagonalization method is valid for any 𝑘∈ℂ.

Next, we show that the square roots of a 𝑘-circulant matrix are still 𝑘-circulant matrices.

Theorem 10. Let 𝑘=𝑟𝑒^𝑖𝜃 (𝑟≥0, 0≤𝜃≤2𝜋), let 𝜃_𝑗=𝑟^1/𝑛𝑒^{𝑖(𝜃+2𝜋(𝑗−1)/𝑛)} (𝑗=1,…,𝑛), and let 𝑉 be defined as (6). For any diagonal matrix Λ=diag(𝜆₁,…,𝜆_𝑛)∈ℂ^𝑛×𝑛, 𝑉Λ𝑉⁻¹ is a 𝑘-circulant matrix 𝐶𝑖𝑟𝑐_𝑘([𝑏₀,…,𝑏_𝑛−1]), where (𝑏₀,…,𝑏_𝑛−1)=(𝜆₁,…,𝜆_𝑛)𝑉⁻¹.

Proof. By Theorem 7, whether 𝑉Λ𝑉⁻¹ is a 𝑘-circulant matrix depends on whether the following system of linear equations with unknown vector (𝑏₀,…,𝑏_𝑛−1)𝑏₀+𝑏₁𝜃₁+𝑏₂𝜃²₁+⋯+𝑏_𝑛−1𝜃₁^𝑛−1=𝜆₁,𝑏₀+𝑏₁𝜃₂+𝑏₂𝜃²₂+⋯+𝑏_𝑛−1𝜃₂^𝑛−1=𝜆₂⋮𝑏₀+𝑏₁𝜃_𝑛+𝑏₂𝜃²_𝑛+⋯+𝑏_𝑛−1𝜃_𝑛^𝑛−1=𝜆_𝑛 (8) is consistent. Obviously the coefficient matrix of (8) is invertible. Thus, we have 𝑏₀,…,𝑏_𝑛−1=𝜆₁,…,𝜆_𝑛𝑉⁻¹. (9) That is to say, 𝑉Λ𝑉⁻¹ is a 𝑘-circulant matrix with the form of Circ_𝑘([𝑏₀,…,𝑏_𝑛−1]).

Remark 11. Theorem 10 provides a method to construct 𝑘-circulant matrix with given eigenvalues. Obviously, the square roots with the form of (7) are 𝑘-circulant matrices Circ_𝑘(𝑏), where 𝑏=(𝜙(𝜃₁)^1/2,…,𝜙(𝜃_𝑛)^1/2)𝑉⁻¹.

Based on Theorem 7 and Corollary 8, we give the following algorithm for computing the principal square root of a 𝑘-circulant matrix.

Algorithm 12. Compute a principal square root of a 𝑘-circulant matrix 𝐴∈ℂ^𝑛×𝑛.

Step 1. Compute the eigenvalues 𝜙(𝜃_𝑗)=∑^𝑛−1_𝑠=0𝑎_𝑠(𝜃_𝑗)^𝑠 (𝑗=1,…,𝑛) of 𝐴.

Step 2. Compute 𝜙(𝜃_𝑗)^1/2 (𝑗=1,…,𝑛) such that arg(𝜙(𝜃_𝑗)^1/2)  (𝑗=1,…,𝑛)∈(−𝜋/2,𝜋/2).

Step 3. Compute the inverse of 𝑉.

Step 4. Compute diag[𝜙(𝜃₁)^1/2,𝜙(𝜃₂)^1/2,…,𝜙(𝜃_𝑛)^1/2]𝑉⁻¹.

Step 5. Compute the square root 𝑉diag[𝜙(𝜃₁)^1/2,𝜙(𝜃₂)^1/2,…,𝜙(𝜃_𝑛)^1/2]𝑉⁻¹ of 𝐴.

Then, we obtain 𝐵=√𝐴=Circ_𝑘([𝑏₀,…,𝑏_𝑛−1]).

The cost of Step 1 is about 𝑂(𝑛log𝑛) flops by discrete Fourier transform [18]. The cost of Step 2 is 𝑂(𝑛) flops. The cost of Step 3 is about 𝑂(𝑛log²𝑛) flops [23]. The cost of Step 4 is about 𝑂(𝑛²) flops. The cost of Step 5 is about 𝑂(𝑛²log²𝑛) flops [24]. So, it needs about 𝑂(𝑛²log²𝑛) flops in total. The algorithm has the same complexity as the diagonalization methods in [19–21]. But the methods therein are only concerned with the case that 𝑘=1 and 𝑘=−1. If we use the Schur method, it needs about 𝑂(𝑛³) flops in total [7]. We also mention a related work in [18], which only needs about 𝑂(𝑛log𝑛) flops to compute the primary square root of a 𝑘-circulant matrix. However, that work restricts the matrix to be of even order and 𝑘 to be of real number. Those restrictions are not needed in our algorithm.

3. Square Roots of Skew 𝑘-Circulant Matrices

Let 𝑛 be an even number; then, 𝐴∈ℂ^𝑛×𝑛 is a skew 𝑘-circulant matrix if 𝐴=−𝐺⁻¹𝐴𝐺, where 𝐺=Circ_𝑘([0,1,0,…,0]) (see [18]). Let 𝑘=𝑟𝑒^𝑖𝜃 (𝑟≥0, 0≤𝜃≤2𝜋) and let 𝐽=1000⋯000−100⋯000010⋯00000−1⋯00⋮⋮⋮⋮⋱⋮⋮0000⋯100000⋯0−1. (10)

Lemma 13. Let 𝐺=𝐶𝑖𝑟𝑐_𝑘([0,1,0,…,0]). Then,(i)𝐽𝐺𝐽=−𝐺;(ii)𝐽𝐺^𝑗=−𝐺^𝑗𝐽 for odd number 𝑗; 𝐽𝐺^𝑗=𝐺^𝑗𝐽 for even number 𝑗.

Lemma 14. Let 𝑉=𝑣₁,𝑣₂,…,𝑣_2𝑚=𝜃11⋯1₁𝜃₂⋯𝜃_2𝑚𝜃²₁𝜃²₂⋯𝜃²_2𝑚𝜃⋮⋮⋱⋮_12𝑚−1𝜃₂^2𝑚−1⋯𝜃^2𝑚−1_2𝑚, (11) where 𝜃_𝑗=𝑟^1/2𝑚𝑒^{𝑖(𝜃+2𝜋(𝑗−1)/2𝑚)} (𝑗=1,…,2𝑚); then, 𝐽𝑣_𝑡=𝑣_𝑚+𝑡(𝑡=1,…,𝑚).

Proof. The proof is obvious by paying attention to the fact that 𝜃_𝑚+𝑡=𝜃_𝑡𝑒^𝑖𝜋.

Lemma 15. A skew 𝑘-circulant matrix 𝐴 of order 2𝑚 can be written in the form of 𝐽𝐶𝑖𝑟𝑐_𝑘𝑎₀,𝑎₁,…,𝑎_2𝑚−1. (12)

Proof. Write 𝑛=2𝑚 and 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛. Since 𝐴=−𝐺⁻¹𝐴𝐺 we have 𝐺𝐴=−𝐴𝐺; that is, 𝑎₂₁𝑎₂₂𝑎₂₃⋯𝑎_2𝑛𝑎₃₁𝑎₃₂𝑎₃₃⋯𝑎_3𝑛𝑎⋮⋮⋮⋱⋮_𝑛1𝑎_𝑛2𝑎_𝑛3⋯𝑎_𝑛𝑛𝑘𝑎₁₁𝑘𝑎₁₂𝑘𝑎₁₃⋯𝑘𝑎_1𝑛=−𝑘𝑎_1𝑛−𝑎₁₁−𝑎₁₂⋯−𝑎_1(𝑛−1)−𝑘𝑎_2𝑛−𝑎₂₁−𝑎₂₂⋯−𝑎_2(𝑛−1)⋮⋮⋮⋱⋮−𝑘𝑎_{(𝑛−1)𝑛}−𝑎_(𝑛−1)1−𝑎_(𝑛−1)2⋯−𝑎_{(𝑛−1)(𝑛−1)}−𝑘𝑎_𝑛𝑛−𝑎_𝑛1−𝑎_𝑛2⋯−𝑎_{𝑛(𝑛−1)}. (13) Then, 𝑎𝐴=¹¹𝑎₁₂𝑎₁₃⋯𝑎_1𝑛−𝑘𝑎_1𝑛−𝑎₁₁−𝑎₁₂⋯−𝑎_1(𝑛−1)−𝑘𝑎_2𝑛−𝑎₂₁−𝑎₂₂⋯−𝑎_2(𝑛−1)⋮⋮⋮⋱⋮−𝑘𝑎_{(𝑛−1)𝑛}−𝑎_(𝑛−1)1−𝑎_(𝑛−1)2⋯−𝑎_{(𝑛−1)(𝑛−1)}𝑘𝑎₁₁,𝑘𝑎₁₂,𝑘𝑎₁₃,…,𝑘𝑎_1𝑛=−𝑘𝑎_𝑛𝑛,−𝑎_𝑛1,−𝑎_𝑛2,…,−𝑎_{𝑛(𝑛−1)}. (14) Thus, we have that 𝑎𝐴=¹¹𝑎₁₂𝑎₁₃⋯𝑎_1𝑛−𝑘𝑎_1𝑛−𝑎₁₁−𝑎₁₂⋯−𝑎_1(𝑛−1)𝑘𝑎_1(𝑛−1)𝑘𝑎_1𝑛𝑎₁₁⋯𝑎_1(𝑛−2)⋮⋮⋮⋱⋮−𝑘𝑎₁₂−𝑘𝑎₁₃−𝑘𝑎₁₄⋯−𝑎₁₁=𝐽Circ_𝑘𝑎₁₁,𝑎₁₂,…,𝑎_1𝑛. (15)

Theorem 16. A skew 𝑘-circulant matrix 𝐽𝐶𝑖𝑟𝑐_𝑘([𝑎₀,…,𝑎_2𝑚−1]) of order 𝑛=2𝑚 is diagonalizable.

Proof. Write 𝐶=Circ_𝑘([𝑎₀,…,𝑎_2𝑚−1]); then 𝐶=𝑎₀𝐺⁰+𝑎₁𝐺+𝑎₂𝐺²+⋯+𝑎_2𝑚−1𝐺^2𝑚−1. By Lemma 13, 𝐽𝐶=𝑎₀𝐺⁰−𝑎₁𝐺+𝑎₂𝐺²−⋯−𝑎_2𝑚−1𝐺^2𝑚−1𝐽. (16) Thus, we have 𝐶=𝐽𝑎₀𝐺⁰−𝑎₁𝐺+𝑎₂𝐺²−⋯−𝑎_2𝑚−1𝐺^2𝑚−1𝐽. (17) By Theorem 7, there exists an invertible matrix 𝑉 defined in Lemma 14 such that 𝑉⁻¹𝐽𝑎₀𝐺⁰−𝑎₁𝐺+𝑎₂𝐺²−⋯−𝑎_2𝑚−1𝐺^2𝑚−1𝜙𝜃𝐽𝑉=diag₁,𝜙𝜃₂,…,𝜙𝜃_2𝑚, (18) where 𝜙(𝑥)=𝑎₀+𝑎₁𝑥+𝑎₂𝑥²+⋯+𝑎_2𝑚−1𝑥^2𝑚−1. Then, 𝑎₀𝐺⁰−𝑎₁𝐺+𝑎₂𝐺²−⋯−𝑎_2𝑚−1𝐺^2𝑚−1𝜙𝜃𝐽𝑉=𝐽𝑉diag₁,𝜙𝜃₂,…,𝜙𝜃_2𝑚. (19) By (16), we have 𝐽𝐶𝑣₁,𝑣₂,…,𝑣_2𝑚=𝐽𝑣₁,𝐽𝑣₂,…,𝐽𝑣_2𝑚diag𝜙𝜃₁,𝜙𝜃₂,…,𝜙𝜃_2𝑚. (20) In order to obtain the eigenvalues and eigenvectors of 𝐽𝐶, for 𝑡=1,…,𝑚, we assume that there exist 𝑘_𝑡, 𝑘_𝑚+𝑡, and 𝑙∈ℂ, such that 𝐽𝐶𝑘_𝑡𝑣_𝑡+𝑘_𝑚+𝑡𝑣_𝑚+𝑡=𝑙𝑘_𝑡𝑣_𝑡+𝑘_𝑚+𝑡𝑣_𝑚+𝑡. (21) Combining this with (20) gives 𝑘_𝑡𝜙𝜃_𝑡𝐽𝑣_𝑡+𝑘_𝑚+𝑡𝜙𝜃_𝑚+𝑡𝐽𝑣_𝑚+𝑡=𝑙𝑘_𝑡𝑣_𝑡+𝑙𝑘_𝑚+𝑡𝑣_𝑚+𝑡. (22) Now, we use Lemma 14 to get 𝑘_𝑡𝜙𝜃_𝑡𝑣_𝑚+𝑡+𝑘_𝑚+𝑡𝜙𝜃_𝑚+𝑡𝑣_𝑡=𝑙𝑘_𝑚+𝑡𝑣_𝑚+𝑡+𝑙𝑘_𝑡𝑣_𝑡. (23) Since 𝑣_𝑡 and 𝑣_𝑚+𝑡 are linearly independent, we have 𝑘_𝑡𝜙𝜃_𝑡=𝑙𝑘_𝑚+𝑡,𝑘_𝑚+𝑡𝜙𝜃_𝑚+𝑡=𝑙𝑘_𝑡. (24) So, 𝑙=±𝜙(𝜃_𝑡)𝜙(𝜃_𝑚+𝑡), 𝑘_𝑚+𝑡/𝑘_𝑡=±𝜙(𝜃_𝑡)/𝜙(𝜃_𝑚+𝑡). Namely, 𝐽𝐶𝜙𝜃_𝑚+𝑡𝑣_𝑡+𝜙𝜃_𝑡𝑣_𝑚+𝑡=𝜙𝜃_𝑡𝜙𝜃_𝑚+𝑡𝜙𝜃_𝑚+𝑡𝑣_𝑡+𝜙𝜃_𝑡𝑣_𝑚+𝑡,𝐽𝐶𝜙𝜃_𝑚+𝑡𝑣_𝑡−𝜙𝜃_𝑡𝑣_𝑚+𝑡=−𝜙𝜃_𝑡𝜙𝜃_𝑚+𝑡𝜙𝜃_𝑚+𝑡𝑣_𝑡−𝜙𝜃_𝑡𝑣_𝑚+𝑡. (25) Let 𝑊=𝜙𝜃_𝑚+1𝑣₁+𝜙𝜃₁𝑣_𝑚+1,𝜙𝜃_𝑚+1𝑣₁−𝜙𝜃₁𝑣_𝑚+1,…,𝜙𝜃_2𝑚𝑣_𝑚+𝜙𝜃_𝑚𝑣_2𝑚,𝜙𝜃_2𝑚𝑣_𝑚−𝜙𝜃_𝑚𝑣_2𝑚. (26) Then𝐽𝐶𝑊=𝑊𝜙𝜃₁𝜙𝜃_𝑚+1−𝜙𝜃₁𝜙𝜃_𝑚+1⋱𝜙𝜃_𝑚𝜙𝜃_2𝑚−𝜙𝜃_𝑚𝜙𝜃_2𝑚. (27)So,𝑊⁻¹𝐽𝐶𝑊=𝜙𝜃₁𝜙𝜃_𝑚+1−𝜙𝜃₁𝜙𝜃_𝑚+1⋱𝜙𝜃_𝑚𝜙𝜃_2𝑚−𝜙𝜃_𝑚𝜙𝜃_2𝑚. (28)

Corollary 17. The square roots of an even order skew 𝑘-circulant matrix 𝐽𝐶𝑖𝑟𝑐_𝑘(𝑎) are as follows:𝑊𝜙(𝜃₁)𝜙(𝜃_𝑚+1)−𝜙(𝜃₁)𝜙(𝜃_𝑚+1)⋱𝜙(𝜃_𝑚)𝜙(𝜃_2𝑚)−𝜙(𝜃_𝑚)𝜙(𝜃_2𝑚)^1/2𝑊⁻¹. (29)

Proof. This is a direct result from (28).

Based on Theorem 16 and Corollary 17, we give the following algorithm for computing the principal square root of a skew 𝑘-circulant matrix.

Algorithm 18. Compute a principal square root of a skew 𝑘-circulant matrix 𝐴∈ℂ^2𝑚×2𝑚.

Step 1. Compute 𝜙(𝜃_𝑗)=∑^2𝑚−1_𝑠=0𝑎_𝑠(𝜃_𝑗)^𝑠 (𝑗=1,…,2𝑚).

Step 2. Compute the eigenvalues of 𝐴: 𝜙(𝜃₁)𝜙(𝜃_𝑚+1),−𝜙(𝜃₁)𝜙(𝜃_𝑚+1),…,𝜙(𝜃_𝑚)𝜙(𝜃_2𝑚),−𝜙(𝜃_𝑚)𝜙(𝜃_2𝑚).

Step 3. Compute the square roots of 𝜙(𝜃₁)𝜙(𝜃_𝑚+1),−𝜙(𝜃₁)𝜙(𝜃_𝑚+1),…,𝜙(𝜃_𝑚)𝜙(𝜃_2𝑚),−𝜙(𝜃_𝑚)𝜙(𝜃_2𝑚) whose arguments should be in (−𝜋/2,𝜋/2).

Step 4. Compute (29).

Let 𝑛=2𝑚. The cost of Step 1 is about 𝑂(𝑛log𝑛) flops by discrete Fourier transform [18]. The cost of Step 2 is 𝑂(𝑛). The cost of Step 3 is 𝑂(𝑛). Since the matrix multiplication and inversion are equivalent in computational complexity [25], the cost of Step 4 is about 𝑂(𝑛^2.4) [26]. So, it needs about 𝑂(𝑛^2.4) flops in total. If we use the Schur method [7] or the method in [18], it needs about 𝑂(𝑛³) flops in total. Moreover, our method allows 𝑘 to be a complex number, whereas the method in [18] only permits 𝑘 to be a real number.

4. Numerical Experiments

We present numerical experiments to compare the algorithms presented in this paper and the standard Schur method with respect to the execution time. The code was implemented in MATLAB 7.0 and run on a Windows-based machine with 2 GB of RAM and Intel Pentium Core Duo CPU running at 2.8 GHz.

We gradually increase the order of the matrix by 100 from 100 to 1100 and record the corresponding CPU time. Figure 1 shows the execution time for Algorithm 12 and the standard Schur method. Figure 2 compares the execution time for Algorithm 18 and the standard Schur method. From those results, we can confirm that our algorithms are clearly faster than the standard Schur method for computing the square roots of 𝑘-circulant matrices and skew 𝑘-circulant matrices.

Figure 1 

Comparison of the CPU time for Algorithm 12 and the standard Schur method in logarithmic scale.

Figure 2 

Comparison of the CPU time for Algorithm 18 and the standard Schur method in logarithmic scale.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is partly supported by the National Natural Science Foundation of China (61101228) and by the Fundamental Research Funds for the Central Universities (3132013337).