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Home , Circulant matrix, Sparse matrix, Toeplitz matrix

Circulant decomposition of a and the eigenvalues of Toeplitz type matrices

Hariprasad M., Murugesan Venkatapathi Department of Computational & Data Sciences, Indian Institute of Science, Bangalore - 560012

Abstract We begin by showing that a n×n matrix can be decomposed into a sum of n cir- culant matrices with appropriate relaxations. We use Fast-Fourier-Transform (FFT) operations to perform a sparse similarity transformation representing only the dominant circulant components, to evaluate all eigenvalues of dense Toeplitz, block-Toeplitz and other periodic or quasi-periodic matrices, to a rea- sonable approximation in O(n2) arithmetic operations. This sparse similarity transformation can be exploited for other evaluations as well.

Keywords: Eigenvalues, periodic entries, symbol. AMS subject classifications: 65F15, 15A18, 15B05, 15A04

1. Introduction

Multiple approaches to efficiently and accurately evaluate eigenvalues of sparse and effectively-sparse matrices are known [1, 2, 3, 4, 5]. Efficient methods are equally desirable for dense matrices that occur frequently in the numerical solution of eigenvalue problems; and Toeplitz, block-Toeplitz, and other ma- trices with periodicity in the diagonals are such a class. O(n2) to evaluate the characteristic and its were proposed a few decades ago for Toeplitz [6], block-Toeplitz [7, 8] and Hankel matrices [9]. These algorithms were amalgamated with the Newton methods applied to evaluate only the eigenvalues of interest. On the other hand, the asymptotic behavior of spectra of Toeplitz and block-Toeplitz operators in the limit of large dimen- arXiv:2105.14805v1 [math.NA] 31 May 2021 sions have been well studied [10],[11] and they can also be approximated by an appropriate . An approximation of a finite dimensional using a circulant matrix for speed up of operations was suggested decades ago [12],[13]. Similarly, circulant were constructed for Toeplitz matrices by minimizing

Email addresses: [email protected] (Hariprasad M.), [email protected] (Murugesan Venkatapathi) the Frobenius norm of the residue [14]. Spectral preserving properties of this were shown [15], and block circulant versions were also proposed and analyzed [16]. Note that this preconditioner is the first term in the circulant series representation of a matrix, discussed in this work. Since these precon- ditioners are a projection of the matrix on the low complexity matrix classes, they speed up iterative methods in regularization and optimization. But these preconditioners may not be sufficiently accurate as a similarity transformation of the given matrix. First, using the symbol of a Toeplitz matrix we highlight cases where this single-term circulant approximation is effective for Toeplitz matrices. Later, we broaden the scope of work to matrices with any periodicity along the diagonals. We first recall in remark 1 that any matrix can be decomposed into n cycles that generate its 2n-1 diagonals. These cycles are given by a n term power series of the cyclic with appropriate relaxations. We later show in lemma 1 that when this decomposition is instead applied on an appropriate sim- ilarity transformation of the matrix, it is equivalent to a decomposition of the given matrix into n circulant matrices with appropriate relaxations. By includ- ing only the dominant cycles of the similar matrix, we include the dominant circulant components of the given matrix. The sparse similar matrix can be operated by a non-symmetric Lanczos for (block) tridiagonalization, and one can evaluate all eigenvalues of such dense matrices in O(n2) arithmetic operations. Let In be the of n, and C be the permutation matrix corresponding to the of a full cycle.  0 1 C = . I 0 n−1 n×n Remark 1. Any matrix A can be decomposed into n cycles given by a power Pn−1 j series in C such that A = j=0 ΛjC . In the above, we refer to the non-zero entries of Hadamard product A.Cj, supported on Cj, as the jth cycle of matrix A. Let the permutation matrix given by a flipped identity matrix be   0 ··· 0 0 1   0 ··· 0 1 0   J = 0 ··· 1 0 0 . .  . ..  . . 0 0 0 1 ··· 0 0 0 n×n Remark 2. Eigenvalues of a circulant matrix R are given by the discrete Fourier transform of the first row R(1, j). The corresponding eigenvectors are 1 2π √ i n pq given by the columns of a matrix W with W (p, q) = n e where p, q = 1, 2 ··· n. The matrix W has the property W 2 = JC, so any circulant matrix has an decomposition R = W ΛW †. R is also given by R = W †Λ˜W where Λ˜ = JCΛCT J T .

2 i 2πkp Lemma 1. Given diagonal matrices D with Dk(p, p) = e n and any circulant T k matrix R, the matrices RDk and Λ(˜ C ) are similar. Here Λ is a with the non zero entries given by eigenvalues of R, and C is the cyclic permutation matrix. Proof. Using W for a similarity transformation,

† † † WRDkW = WW Λ˜WDkW , † = Λ˜WDkW = Λ(˜ CT )k.

† T k The equality WDkW = (C ) can be deduced by evaluating its (p, l) entry given by

† 1 X i 2πpq i 2πkq −i 2πql WD W (p, l) = e n e n e n , k n q

1 X i 2π(p+k−l)q = e n , n q = 1 when l = p + k and 0 otherwise.

T k Hence, the eigenvalues of RDk are the eigenvalues of the matrix Λ(˜ C ) . The matrix Λ(˜ CT )k is sparse, and its eigenvalues are easy to compute. Mo- tivated by this result, we construct the decomposition of a matrix into these components. Theorem 1. Any n×n A can be represented as sum of n circulant matrices with relaxations taking values from the nth . It is of the form n X A = RkDk k=1

Where Rk is a circulant matrix and Dk is a diagonal matrix with Dk(p, p) = i 2πkp e n . Proof. Consider the matrix B = W AW †. Using remark 1 to decompose it into Pn ˜ T k cycles, and recalling the proof of lemma 1, we have B = k=1 Λk(C ) = Pn ˜ † ˜ k=1 ΛkWDkW , where Λk are diagonal. The corresponding matrix A is then Pn † ˜ Pn A = k=1 W ΛkWDk = k=1 RkDk. Corollary 1. If a matrix A has k particular dominant frequencies on each of its cycles, then the matrix W AW † has only k dominant cycles. Proof. Consider the decomposition of A into cycles.

n−1 X q A = ΛqC q=0

3 where entries of each Λq are given by k dominant frequencies. Using the pro- posed similarity transformation,

n−1 † X q † W AW = W ( ΛqC )W q=0 n−1 † X B = W AW = RqDq q=0

We know each circulant matrix Rq has eigenvalues given by the diagonal matrix Λq. As mentioned in remark 2, entries Rq(1, j) are given by the inverse FFT of Λ, and hence, all rows have k particular dominant entries, with the transformed similar matrix B having only k dominant cycles. Also note that Pn−1 j q=0 Rq(1, j)C Dq is the FFT of the sequence Rq(1, j) for q = 0, 1, 2, ··· n. By Parsevals theorem, a cycle supported on Cj in B has the same energy as the sequence corresponding entries of Rq(1, j) for q = 0, 1, 2, ··· n.

2. Single-term circulant approximation of a Toeplitz matrix and its relation with the symbol A Toeplitz matrix is of the form,   a0 a1 a2 ··· an−1  .   ..   a−1 a0 a1 a2   . .  A =  .. .. .  a−2 a−1 a0   . . . . .   ......   .  a−(n−1) a−(n−2) ······ a0 iθ Pn−1 2πik The function a(e ) = k=−(n−1) ake for θ ∈ [0, 2π) is called the symbol of the matrix A [17]. In this section we consider different types of symbols, the corresponding eigenvalues of the single-term circulant approximation, and the actual spectrum of the Toeplitz matrices. The single-term approximation is given by only the diagonal entries of W AW †. The symbol may be exploited for computing efficiency, when applicable, as it can be evaluated in O(n) arithmetic operations. On the other hand, the single-term circulant approximation is more accurate and robust in general.

2.1. Case 1: Symbol is a product of a polynomial and a trigonometric polynomial Spectrum of A is said to be canonically distributed if the limiting spectrum approaches the range of the symbol. For symbols of the form a(eiθ) = p(θ)q(eiθ) with p and q, we consider the Toeplitz matrices whose entries are from its truncated expansion. Several conditional theorems on the symbol such that the spectrum of T is canonically distributed have been pre- sented [17]. Figure 1 shows the spectrum, single-term circulant approximation, and the symbol for a(θ) = (1 + θ)eiθ.

4 1.5 Actual 1 Range(a) diag(WAW’) 0.5

0

Imag -0.5

-1

-1.5

-2 -2 -1 0 1 2 Real

Figure 1: Range of the symbol, and diagonal entries of W AW † for a Toeplitz matrix of dimension 200 with a symbol a(θ) = (1 + θ)eiθ.

2.2. Case 2: Symbol of the banded Toeplitz matrix is a trigonometric polynomial

For a banded Toeplitz matrix A, with first row a0, a1, a2, ··· al and first iθ Pl ikθ column entries a0, a−1, ··· a−m, the symbol is given by a(e ) = k=−m ake . Here a(eiθ) is a closed curve in the complex plane. The eigenvalues of such matrices lie inside the curve a(eiθ) for θ ∈ [0, 2π) [18]. We have the diagonal entries of W AW † as FFT of the sequence n − 1 n − 2 n − l n − m n − 1 (a , a , a , ··· a , 0, 0, ··· 0, a , ··· a ). 0 n 1 n 2 n l n −m n −1 Note that these values lie in near proximity to the trace of the symbol in the complex plane when k  n, as shown in Figure 2.

2.3. Case 3: Symbols of the form f(x) + ig(x) Note that, when the symbol is multiplied by a , its range is rotated in the complex plane, however the spectrum remains the same. Figure 3 shows that the diagonal of W AW † well approximates the spectrum in such cases whereas the symbol fails to approximate the spectrum. The spectrum of iθ θ 1 2 the matrix with symbol a(e ) = 2π + i π2 (θ − π) is shown in Figure 4, while iθ θ 1 2 i2θ the case of a(e ) = 2π + i π2 (θ − π) + e is shown in Figure 5.

3. Approximating eigenvalues using the circulant decomposition

Suppose a matrix has only k frequency components (k << n), then B = W AW † has only k dominant cycles. For any matrix A, we can approximate

5 4

2

range(a) 0 diag(WAW’)

imaginary Actual

-2

-4 -6 -4 -2 0 2 4 6 real

Figure 2: Range of the symbol evaluated at unit circle, and diagonal entries of W AW † for a banded Toeplitz matrix of dimension 100, with 7 non-zero entries in each row.

1 Actual 0.8 Range(a) diag(WAW’)

0.6 Imag 0.4

0.2

0 -1.5 -1 -0.5 0 0.5 1 Real

Figure 3: Range of the symbol, and diagonal entries of W AW † for a Toeplitz matrix of θ dimension 200 with a symbol a(θ) = (1 + i) 2π .

6 1 Actual Range(a) 0.8 diag(WAW’)

0.6 Imag 0.4

0.2

0 -0.5 0 0.5 1 1.5 Real

Figure 4: Range of the symbol, and diagonal entries of W AW † for a Toeplitz matrix of dimension 200 with a symbol a(θ) = θ + i 1 (θ − π)2. 2π π2

1.5 Actual Range(a) 1 diag(WAW’)

0.5 Imag 0

-0.5

-1 -0.5 0 0.5 1 1.5 2 Real

Figure 5: Range of the symbol, and diagonal entries of W AW † for a Toeplitz matrix of dimension 200 with a symbol a(θ) = θ + i 1 (θ − π)2 + ei2θ. 2π π2

7 7000 2000 6000

5000 1500

4000 1000 3000 norm of the cycle

2000 norm of the cycle 2 L 2

L 500 1000

0 0 20 40 60 80 100 0 0 20 40 60 80 100 Cycle number Cycle number Figure 6: L norm of cycles of a random Toeplitz ma- 2 Figure 7: L norm of cycles of a random block Toeplitz trix of dimension 100. The dominant cycle number 2 matrix of block-size 5 and dimension 100. 100 is given by the diagonal of W AW †.

k ˜ P ai the matrix B by B = i=1 Λai C by choosing the k dominant frequencies (ai ∈ {1, 2 ··· n}). This is followed by the application of an algorithm suited for the eigenvalues of a .

3.1. Identifying the dominant circulant components The similarity transformation by the matrix W is shown to restrict the larger magnitudes to certain cycles for the Toeplitz and block Toeplitz matrices (Figure 6 and Figure 7). However, the similarity transform fails to show such behavior for random matrices (Figure 8 and Figure 9). A Toeplitz matrix has constant entries along the diagonals, and the similarity transformation produces a higher magnitude for components that represent no variation in the diagonals of the matrix. The main diagonal of the similar matrix B corresponding to a zero frequency, represents the most dominant circulant component. On the other hand, Block Toeplitz matrices of a dimension n having the block size m are almost m-periodic. So its similar matrix B has dominant cycles given by bn/mc and its multiples. Similarly, in the case of a quasi-periodic matrix where we have a random variation of k frequencies along diagonals, the least common multiple of the k periods determines the dominant cycles.

3.2. Computing W AW † The (p, q) entry of the matrix B = W AW † is given by

n n 1 X X i 2πpk −i 2πjq B(p, q) = e n A(k, j)e n . (1) n k=1 j=1 If FFT(A) represents of the columns of matrix A, we have B = FFT(FFT(A)†)†. (2)

8 140 140

120 120

100 100 norm of the cycle norm of the cycle 2 2

L 80 L 80

60 60 0 20 40 60 80 100 0 20 40 60 80 100 Cycle number Cycle number

† Figure 8: L2 norm of cycles of a A Figure 9: L2 norm of cycles of W AW where entries with entries from N (0, 1). of the random A are given by N (0, 1).

Thus W AW † is computed in 2n2 log n steps. Relation with two dimensional FFT : Let us denote the two dimensional FFT of the matrix A by FFT2(A), which is

n X −i 2πpk i 2πjq FFT2(A)(p, q) = e n A(k, j)e n . j=1

Then we have B = J[FFT2(A)], where J is the flipped identity matrix shown earlier. Calculating k frequency components : In the n point FFT butterfly structure there are log2 n stages. When we choose k frequency components at a particular stage, there are 2k points in the previous stage, 4k points earlier, and so on m n  until 2 k = n for some m. We have m = log2 k . So the number of arithmetic operations Ok for including k frequency components is,

m−1 X n O = 2jk + n(log n − log , k 2 2 k j=0 m Ok = k(2 − 1) + n log2 k,

Ok = (n − k) + n log2 k.

2 The total number of required operations is nOk, which is O(n ) for a given k.

3.3. Error in approximations Let the matrix B be approximated by the matrix B˜ by selecting certain ˜ cycles. Let ∆ = B −B˜. Let λi be the eigenvalues of B and λi be the eigenvalues of B˜. Let B = XΛX−1. From Bauer-Fike theorem, ˜ |λi − λi| ≤ κ(X)k∆k2 (3)

9 Algorithm 1 Eigenvalue approximation using a circulant decomposition Output : e a of eigenvalue approximations for the given matrix. initialization: A,W . Sparsification : Compute B˜ = FFT (FFT (A)†)† − ∆. Form the matrix B˜ using the dominant cycles of the similarity transformation. Reduction : Compute the eigenvalues of B˜ using an for sparse matrices (for example, by reducing B˜ into tridiagonal form or block diagonal form).

Here κ(X) =kXk X−1 is the of the eigenvector matrix X. So the eigenvalues are better approximated when the eigenvector matrix X is well conditioned, and k∆k2 is minimized. Similarly the relative error bound when B is non singular and diagonalizable,

|λ − λ˜ | i i ≤ κ(X) B−1∆ . (4) |λi| 2

4. Numerical Results

We begin by showing the advantages of the suggested sparsification over a direct sparsifier, for matrices with periodic properties. Later, we provide quan- titative results using examples of Toeplitz, block Toeplitz and quasi-periodic matrices. Random matrices with N (0, 1) as entries of the matrix or its blocks were used for corresponding the numerical experiments on Toeplitz and block Toeplitz matrices. We note that in applications where matrices are not random, we expect better results. The results show that including a few dominant cycles provides us lower relative errors for eigenvalues of such matrices, compared to the single-term circulant approximation. Also note that while the relative errors may be notable for the smaller eigenvalues, their absolute errors are very small, as expected.

4.1. Comparison with direct sparsifiers The eigenvalues of the approximated Toeplitz matrix using W AW † and a direct sparsifier [19], along with the actual eigenvalues are shown in Figure 10. Similarly, the eigenvalues of a block Toeplitz matrix are shown in Figure 11. In general, the eigenvalues with the direct sparsifier are smaller in magnitude i.e. more centered in the plot, compared to the eigenvalues with the proposed sparsifier in the frequency domain. Intuitively, the information lost by the direct sparsifier due to a threshold of some entries in the matrix to zero, is larger compared to the information lost in the frequency domain by removing the same number of entries in the cycles of the matrix W AW †.

10 30 15 Actual Actual Approximated using WAW’ Approximated using WAW’ 20 10 Direct sparsifier Direct sparsifier

10 5

0 0 Imag Imag

-10 -5

-20 -10

-30 -15 -20 -10 0 10 20 30 40 50 -10 0 10 20 30 40 Real Real

Figure 10: Eigenvalues and approximations for a ran- Figure 11: Eigenvalues and approximations for a ran- dom Toeplitz matrix. dom block Toeplitz matrix of block size 5.

4.2. Toeplitz and block-Toeplitz matrices The average relative error and standard deviation of the errors decrease with an increase in the number of cycles in the approximation of eigenvalues of a Toeplitz matrix, as shown in Figure 12. The cumulative error in the similarity transformation, k∆k /kAk, is plotted in the figures as well. Average of the average of relative error and its deviation over 1000 trials show relatively small errors, as in Figure 14. Block-Toeplitz matrices with random entries from N (0, 1) were used for the numerical experiments. The average relative error and standard deviation of the errors decrease with an increase in the number of cycles in the approximation as shown in Figure 13. The average of the average relative error and its deviation over 1000 trials are shown in Figure 15. Note that inclusion of a few cycles may be sufficient to achieve a reasonable accuracy in approximating all the eigenvalues. The improvements in the accuracy of the eigenvalue estimation for a given k, with the increase in the size of the matrix, is also illustrated in the Figure 17.

4.3. Other periodic and quasi-periodic matrices The above described properties of Toeplitz and block-Toeplitz matrices can be used to describe the properties and the efficacy of sparsification of a quasi- periodic matrix as well. The dominant components for such matrices becomes the corresponding sizes of periodic blocks in the matrix; thus displaying the characteristics similar to that of a block-Toeplitz matrix. It can be used to include only the dominant cycles of the similar matrix B. Average relative error and deviation in the relative errors are plotted with the cycle number for a random periodic matrix in Figure 16. Here, a uniform distribution of periods 4, 5 and 10 were used to generate the entries of A along the diagonals. These result in dominant cycles in the similar matrix B that are multiples of 25, 20 and 10 respectively. Note that error reduces mostly for corresponding cycles

11 0.08 0.65 error mean and deviation 0.16 0.036 0.07 0.6 error mean and deviation 0.14 0.034 0.06 0.55 0.12 0.032 0.05 0.5 0.1 0.04 0.03 ||/||A|| 0.08

0.45 ∆

0.03 || 0.028 ∆ ||/||A|| ||

Relative error 0.06 0.4 Relative error 0.026 0.02 0.04 0.01 0.35 0.02 0.024 0 0.3 0 0.022 5 10 15 2 4 6 8 10 Cycle number Cycle number

Figure 12: Average and deviation of relative error in Figure 13: Average and deviation of relative error in the approximated eigenvalues of a Toeplitz matrix of the approximated eigenvalues of a block Toeplitz ma- dimension 1000 with increase in the number of cy- trix of block size 5 and dimension 1000 with the in- cles considered. Corresponding errors in the similarity crease in number of cycles considered. Corresponding transformation are given by the dashed line and Y-axis errors in the similarity transformation are given by the on the right. dashed line and Y-axis on the right

0.4 error mean and deviation 0.3 error mean and deviation 0.3 0.25 0.2

0.2 0.1 Relative error Relative error 0.15 0

0.1 -0.1 0 2 4 6 8 10 12 0 5 10 15 20 cycles Cycle number

Figure 14: Average of average relative error and its Figure 15: Average of average relative error and its de- deviation in the approximated eigenvalues of 1000 ran- viation in the approximated eigenvalues of 1000 block dom Toeplitz matrices of dimension 100 with the in- Toeplitz matrices of block size 5 and dimension 100 crease in cycles included. with the increase in cycles included.

12 Figure 16: Average and deviation of relative error in Figure 17: Average and deviation of relative error in the approximated eigenvalues of a random periodic the approximated eigenvalues using two dominant cy- matrix of dimension 100, with periods 4, 5, 10 cho- n cles (C0,C 2 ), of block Toeplitz matrices of block-size sen uniformly randomly along the diagonals. In the five and dimensions 100 to 1000. Corresponding errors suggested algorithm, cycles numbering around 20, 40, in the transformation are given by the dashed line and 50, 60 and 80 will be used preferentially in the sparse Y-axis on the right. similarity transformation. numbering around 20, 40, 50, 60 and 80, as in the case of a mixture of block Toeplitz matrices.

5. Conclusion

Exploiting the periodicity of entries in a matrix, we can sparsify the given matrix using appropriate FFT operations for a similarity transformation. The appropriate similarity transformation of the given matrix is identified using its decomposition into n circulant components. To begin with, we compared the efficacy of the symbol of a Toeplitz matrix, and its one-term circulant approx- imation, in approximating the spectra. While the former can be evaluated in O(n) arithmetic operations compared to O(n2) for the latter, the latter was significantly more robust. Using numerical results, we highlighted the efficacy of the sparsification of the periodic matrices using the circulant decomposition, in terms of errors both in approximation of entries of the matrix and also its eigenvalues. Examples of the relative errors in the eigenvalues were produced as a function of the number of circulant matrices included in the approximation. These results show that the suggested sparse similarity transformation is useful in approximating eigenvalues efficiently, and may be exploited for other evaluations as well.

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