CLASSIC TWO-STEP DURBIN-TYPE AND LEVINSON-TYPE ALGORITHMS FOR SKEW-SYMMETRIC TOEPLITZ MATRICES

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1 CLASSIC TWO-STEP DURBIN-TYPE AND LEVINSON-TYPE ALGORITHMS FOR SKEW-SYMMETRIC TOEPLITZ MATRICES IYAD T ABU-JEIB Abstract We present ecient classic two-step Durbin-type and Levinsontype algorithms for even order skew-symmetric Toeplitz matrices AMS classications: 65F05, 5A57 Keywords: skew-symmetric Toeplitz matrix; Levinson algorithm; Durbin algorithm; skew-centrosymmetric Introduction We extend the classic Durbin's algorithm and the classic Levinson's algorithm for symmetric Toeplitz matrices to skew-symmetric Toeplitz matrices Levinson's algorithm is for solving the system Ax = b, where A is an n n real symmetric Toeplitz matrix (with some restrictions on A) and x and b are n vectors In our algorithms, we present an O(n 2 ) method to solve HX = B, where H is a nite n n real skew-symmetric Toeplitz matrix (with some restrictions) and B is n 2 (to solve Hx = b, where b is n, simply let b be one of the colummns of B) This method can be used to invert nonsingular skew-symmetric Toeplitz matrices Our algorithms are dierent than the algorithms presented in? They are simple and they use very similar techniques to those used by Durbin and Levinson They are easy to derive and easy to implement They are also two-step In addition, the restrictions we place on the matrix of coecients are less than the restrictions in Durbin's algorithm and in Levinson's algorithm Levinson's algorithm and a twostep version of it can be found in?,?,? We recall here that well-known researchers thought that the classic Durbin's algorithm and the classic Levinson's algorithm can not be generalized to skew-symmetric Toeplitz matrices But, we managed to generalize them Our algorithms were tested on skew-symmetric Toeplitz matrices that appear in Sinc methods They were tested on matrix S n of I n ( ), where I ( ) n where η ij = e i j, e k = 2 + s k, and s k = Thus, I ( ) n sinc(x) = = η ij n i,j= k 0 { sin(πx) πx,, for x = 0 can be expressed in the form I n ( ) = 2 sinc(x)dx, where + S n, for x 0

2 where 2 is the n n matrix whose elements are all equal to 2 algorithms also on matrix I n () of Sinc methods I n () Toeplitz matrix dened as follows I () n = ( ) n n ( ) n n 2 ( ) n 2 We tested the is an n n skew-symmetric ( ) n n ( ) n 2 n 2 ( ) n 3 n 3 n 3 0 Many Sinc methods are dependent on Sinc matrices For more about the matrices of Sinc methods and about Sinc methods, see?,?,?,?,?,? 2 Preliminaries We employ the following notation We denote the transpose of a matrix A by A T As usual, I k denotes the k k identity matrix When counting ops, we treat addition/subtraction the same as multiplication/division By the main counterdiagonal (or simply counterdiagonal) of a square matrix we mean the positions which proceed diagonally from the last entry in the rst row to the rst entry in the last row Denition 2 The counteridentity matrix, denoted J, is the square matrix whose elements are all equal to zero except those on the counterdiagonal, which are all equal to We note that multiplying a matrix A by J from the left results in reversing the rows of A and multiplying A by J from the right results in reversing the columns of A Throughout this paper, we will denote the k k counteridentity matrix by J k Note that multiplying a matrix or a vector by J does not contribute to the running time Denition 22 A matrix A is skew-centrosymmetric if JAJ = A, and Toeplitz if the elements along each diagonal are equal Note that skew-symmetric Toeplitz matrices are skew-symmetric skew-centrosymmetric Note also that if T n is an n n skew-symmetric Toeplitz matrix, then T n has the following form T n = 0 σ σ 2 σ n σ 0 σ σ n 2 σ 2 σ 0 σ n 3 σ n σ n 2 σ n 3 0 The above form is the form we will refer to in the next section Note that the rst row (excluding the rst element) generates (determines) T n, ie the vector h n = σ, σ 2,, σ n T is a generator of T n We will assume that {T n }, n 2Z +, is a family of real skew-symmetric Toeplitz matrices Note that to solve T n X = B, where T n is n n and B is n 2, we need T n+2 To be specic, we need only the elements σ n and σ n+ from the generator h n+2 of T n+2 This is not a shortcome of the algorithm because most of the skew-symmetric Toeplitz matrices

3 that arise in applications are of the form we mentioned above (ie they appear as classes/sequences of matrices) as it is the case with the skew-symmetric Toeplitz matrices that appear in Sinc methods For example, in matrix S n of I n ( ) described in the introduction, σ k = k sinc(x)dx, and in matrix I () 0 n, σ k = ( )k k Thus, σ k is dened for all k Z + Also, we recall that Durbin and Levinson used similar teqchniques in their algorithms; ie, to solve H n x = b, where H n is an n n real symmetric Toeplitz matrix, they use H n+ Denition 23 Let A be an n n matrix The leading principal matrix of A of order k is the matrix formed from A by deleting the last n k columns and the last n k rows of A 3 Algorithms for Skew-Symmetric Toeplitz Matrices Throughout the rest of the paper, let k be even, and let T k be a k k real skew-symmetric Toeplitz matrix and assume T i, i {2, 3,, k} 2Z, is nonsingular (ie all leading principal matrices of T k of even order are nonsingular) We recall that Durbin and Levinson have stronger restrictions in their algorithms We note also that it happens sometimes that all even-order matrices of a family of skew-symmetric Toeplitz matrices are non-singular as it is the case with matrix S n of I n ( ) and matrix I n () For the proofs, see?,? Note that odd-order skewsymmetric (and odd-order skew-centrosymmetric matrices) are singular, and hence, it is essential to have a two-step algorithm that skips the odd-order matrices Thus, our algorithm is a two-step algorithm because it moves from order k to order k + 2 instead of moving from order k to order k + Now note that T k+2 can be written as T T k+2 = k J k R k Rk T J, k T 2 σ σ 2 where R k = Once again, in each step we will move from T k to T k+2 σ k σ k+ instead of T k+ We start with T 2 Now, we extend Durbin's algorithm If we know the solution of T k Y = R k, where Y is k 2, then we can know the solution of T k J k R k Z Rk Rk T J =, k T 2 W S k σk+ σ where Z is k 2, W is 2 2, and S k = k+2 Note that T σ k+2 σ k and J k are k+3 k k, R k and Z are k 2, and T 2 and S k are 2 2 Now note that T k Z + J k R k W = R k, and R T k J k Z + T 2 W = S k Thus, Z = Y + J k Y W and W = (T 2 Rk T Y ) (S k + Rk T J ky ) Note that T 2 Rk T Y is nonsingular, because Hk T T T k+2 H k = k 0 Y T J k T k Rk T J k T 2 Rk T Y, where H k = Ik J k Y 0 I 2

4 (Note that T k J k Y + J k R k = 0 and Y T J k T k J k Y + Y T R k = 0 also) Hence, det(t k+2 ) = det(t k ) det(t 2 Rk T Y ) Now since we are assuming T k+2 is nonsingular, then T 2 Rk T Y is nonsingular Thus, our O(n2 ) algorithm is (we note that the inverse we have to nd in this algorithm and in all other algorithms is for a 2 2 matrix): Algorithm : Input: n (an even positive integer), σ = σ σ 2 σ n+ T (a generator of a real skew-symmetric Toeplitz matrix T n+2 ) σ2 /σ Y 2 = σ 3 /σ σ 2 /σ 0 σ T 2 = σ 0 for k = 2,, n 2, step 2 Let J k be the counteridentity matrix of order k σ σ 2 R k = σ k σ k+ σk+ σ S k = k+2 σ k+2 σ k+3 P k = (T 2 Rk T Y k) W k = P k (S k + Rk T J ky k ) Z k = Y k + J k Y k W k Zk Y k+2 = W k end for Output: Y n (the solution of T n Y n = R n, where T n is the skew-symmetric Toeplitz matrix whose generator is σ σ 2 σ n T and R n is as before) It is easy to see that the running time of the previous algorithm is n 2 k=2 24k = 6n 2 + O(n) Remark: We remind the reader that we dene a op to be one addition or one multiplication, while some people dene it to be one addition and one multiplication If we dene the op to be one addition and one multiplication, then the running time of the previous algorithm will be almost half of the running time we have above Now we derive a Levinson-type algorithm Let B = b c b 2 c 2 b k c k Assume we have the solution of T k X = B, where X is k 2, and assume also we have the solution (from the previous algorithm) of T k Y = R k, where Y is k 2 Now, we want to solve the next even higher order equation: T k J k R k R T k J k T 2 V M = B C,

5 where V is k 2, M and C are 2 2, and C = T k V + J k R k M = B, R T k J k V + T 2 M = C bk+ c k+ Then, we will have b k+2 c k+2 Thus, V = X + J k Y M and M = (T 2 Rk T Y ) (C + Rk T J kx) Now, all we need to do is to execute the steps above in parallel with the steps for solving T k Y = R k Therefore, our O(n 2 ) algorithm is: Algorithm 2 (Classic Levinson-Type Algorithm): Input: n (an even positive integer), b = b b 2 b n T, c = c c 2 c n T (b and c are the vectors of constant terms), σ = σ σ 2 σ n+ T (a generator of a real skew-symmetric Toeplitz matrix T n+2 ) σ2 /σ Y 2 = σ 3 /σ σ 2 /σ 0 σ T 2 = σ 0 b2 /σ X 2 = c 2 /σ b /σ c /σ for k = 2,, n 2, step 2 Let J k be the counteridentity matrix of order k σ σ 2 R k = σ k σ k+ σk+ σ S k = k+2 σ k+2 σ k+3 P k = (T 2 R T k Y k ) W k = P k (S k + Rk T J ky k ) Z k = Y k + J k Y k W k M k = P k (C k + Rk T J kx k ) V k = X k + J k Y k M k Y k+2 = X k+2 = Zk W k Vk M k end for Output: X n (the solution of T n X n = B n, where B n = b, c and T n is the skew-symmetric Toeplitz matrix whose generator is σ σ 2 σ n T ) It is easy to see that the running time of the previous algorithm is n 2 k=2 40k = 0n 2 + O(n) Note that our algorithm solves for two vectors of constant terms at once Ie if we want to solve T n x = b and T n y = c, where b and c are n and n is even, then we solve the system T n Z = D, where D is an n 2 matrix whose rst column is b and second column is c Then, x will be the rst column of Z and y the second column Solving T n Z = D by our method costs 0n 2 + O(n) Thus, obtaining x costs 5n 2 + O(n) and obtaining y has the same cost 4 Examples In the following example we solve (calculations are done by Octave which is a math-oriented programming language similar to MATLAB) S 6 X = D, where S 6 is

6 the 6 6 matrix of I ( ) mentioned in the introduction, ie S 6 = , D = , and the generator σ of S 8 is T Note that the inputs to the algorithm are 6, b, c, and σ, where b is the rst column of D and c is the second column of D Initializations: Y 2 = T 2 = Iterations: k = 2: X 2 = R 2 = S 2 = C 2 = e e + 00 P 2 = 2872e e W 2 = Z 2 = M 2 = V 2 =

7 k = 4: Y 4 = X 4 = R 4 = S 4 = C 4 = e e + 00 P 4 = 2270e e W 4 = Z 4 = M 4 = V 4 = Y 6 = X 6 =

8 End of Iterations The solution X is We note that the solution above is exactly the same as the solution obtained from solving the system using Maple The two solutions even match for a much higher number of decimal places As another example, we solve the system I () 8 X = D, where D is an 8 2 matrix whose rst solumn is the 8 zero vector (call this vector b) and whose second column is c = I () 8 e, where e is the 8 vector of ones; ie (rounded up to 6 digits) c = Note that D was chosen so that the true solution is the matrix whose rst column is the 8 zero vector and whose second column is the 8 vector of ones Solving the system by our algorithm with input 8, b, c, and σ, where σ = /2 /3 /4 /5 /6 /7 /8 /9, gives us the solution

9 5 Improved Algorithms First, we present a 4n 2 +O(n) Durbin-type algorithm The idea here is to reduce the cost of computing Rk T Y k Here we will use the same notation (but we will replace Y by Y k, Z by Z k and W by W k ) we used when we derived the rst two algorithms Now consider Rk T Y k = Rk 2 T Z k 2 ST k 2 W k 2 But, Z k 2 = Y k 2 + J k 2 Y k 2 W k 2 Thus, R T k Y k = R T k 2Y k 2 + (T 2 R T k 2Y k 2 )W 2 k 2 Therefore, we can use the previously computed values of W and R T Y to compute the new value of R T Y which we will call E Algorithm 3 (Classic Durbin-Type Algorithm): Input: n (an even positive integer), σ = σ σ 2 σ n+ T (a generator of a real skew-symmetric Toeplitz matrix T n+2 ) σ2 /σ Y 2 = σ 3 /σ σ 2 /σ 0 σ T 2 = σ 0 σ σ R 2 = 2 E = R 2 T Y W = 0 0 for k = 2,, n 2, step 2 Let J k be the counteridentity matrix of order k σ σ 2 R k = S k = σ k σ k+ σk+ σ k+2 σ k+2 σ k+3 E = E + (T 2 E)W 2 P k = (T 2 E) W = P k (S k + Rk T J ky k ) Z k = Y k + J k Y k W Zk Y k+2 = W end for Output: Y n (the solution of T n Y n = R n, where T n is the skew-symmetric Toeplitz matrix whose generator is σ σ 2 σ n T and R n is as before) It is easy to see that the running time of the previous algorithm is n 2 k=2 6k = 4n 2 + O(n) Algorithm 4 (Improved Classic Levinson-Type Algorithm): Input: n (an even positive integer), b = b b 2 b n T, c = c c 2 c n T (b and c are the vectors of constant terms), σ = σ σ 2 σ n+ T (a generator of a real skew-symmetric Toeplitz matrix T n+2 )

10 σ2 /σ Y 2 = σ 3 /σ σ 2 /σ 0 σ T 2 = σ 0 b2 /σ X 2 = c 2 /σ b /σ c /σ σ σ R 2 = 2 E = R 2 T Y W = 0 0 for k = 2,, n 2, step 2 Let J k be the counteridentity matrix of order k σ σ 2 R k = σ k σ k+ σk+ σ S k = k+2 σ k+2 σ k+3 E = E + (T 2 E)W 2 P k = (T 2 E) W = P k (S k + Rk T J ky k ) Z k = Y k + J k Y k W M k = P k (C k + Rk T J kx k ) V k = X k + J k Y k M k Zk Y k+2 = W Vk X k+2 = M k end for Output: X n (the solution of T n X n = B n, where B n = b, c and T n is the skew-symmetric Toeplitz matrix whose generator is σ σ 2 σ n T ) It is easy to see that the running time of the previous algorithm is n 2 k=2 32k = 8n 2 + O(n) Note that our algorithm solves for two vectors of constant terms at once Ie if we want to solve T n x = b and T n y = c, where b and c are n and n is even, then we solve the system T n Z = D, where D is an n 2 matrix whose rst column is b and second column is c Then, x will be the rst column of Z and y the second column Solving T n Z = D by our method costs 8n 2 + O(n) Thus, obtaining x costs 4n 2 + O(n) and obtaining y has the same cost Solving the rst system we solved in Section?? by the improved Levinson algorithm (using Octave) gives the same solution we got in that section All variables we get here are the same as those in that example except that we do not have W k here, and we have the following additional ones In the initializations part, we have R 2 =

11 555e e 0 E = 8742e 0 860e W = 0 0 When k = 2 in the iterations, we get 555e e 0 E = 8742e 0 860e W = When k = 4 in the iterations, we get 58234e e 0 E = 22553e 0 569e W = An Octave Program for the Improved Levinson-Type Algorithm We note that we do not have to compute the counteridentity matrix, J, in the program below We can write the program without it We included it in the program for the sake of clarity We note also that the program can be shortened if we use Octave's built-in functions and operators But, we decided to write it as above to make it easy to understand for readers who do not know Octave # ; function J = Counter(n) # usage: J = Counter(n) # description: Creates the counteridentity matrix of order n J=zeros(n); for i=:n J(i,n-i+)=; endfor; endfunction function Z = solve(sigma,d) # description : Solves the system T n Z = D, where n is even and # T n is a real skew-symmetric Toeplitz matrix generated by # sigma() sigma(2) sigma(n ) T # The input sigma = sigma() sigma(2) sigma(n + ) T # is the generator of T n+2 It is an (n + ) column vector # The input D is an n 2 matrix (matrix of constant terms) # usage: Z = solve(sigma,d) n = rows(sigma)- ; b = D(:,); # b is the rst column of D c = D(:,2); # c is the second column of D Y = -sigma(2)/sigma(),-sigma(3)/sigma();,sigma(2)/sigma(); T2 = 0,sigma();-sigma(),0;

12 X = -b(2)/sigma(),-c(2)/sigma();b()/sigma(),c()/sigma(); R = sigma(),sigma(2);sigma(2),sigma(3); E = R * Y; # The prime is used in Octave for transpose W = zeros(2,2); # W is the 2 2 zero matrix for k = 2:n-2 if (rem(k, 2)!= 0) continue; endif; R = zeros(k,2); for i=:k R(i,)=sigma(i); R(i,2) = sigma(i+); endfor; S = zeros(2,2); S(,) = sigma(k+); S(,2) = sigma(k+2); S(2,) = sigma(k+2); S(2,2) = sigma(k+3); C = zeros(2,2); C(,) = b(k+); C(2,) = b(k+2); C(,2) = c(k+); C(2,2) = c(k+2); J = Counter(k); E = E + (T2 - E) * W * W; P = inv(t2 - E); W = P * (S + R * J * Y); Z = Y + J * Y * W; M = P * (C + R * J * X); V = X + J * Y * M; Y = Z;W; X = V;M; endfor; Z = X; endfunction; References I T Abu-Jeib and T S Shores, On properties of matrix I ( ) of Sinc methods, New Zealand J Math 32 (2003) -0 2 D Delsarte and Y Genin, The split Levinson algorithm, IEEE Transactions on Acoustics Speech and Signal Processing ASSP-34 (986) P Gierke, PhD thesis, University of Nebraska-Lincoln, G Heinig and K Rost, Fast algorithms for skewsymmetric Toeplitz matrices, Toeplitz matrices and singular integral equations (Pobershau, 200) , Oper Theory Adv Appl, 35, Birkhäuser, Basel, J Lund and K Bowers, Sinc Methods for Quadrature and Dierential Equations, SIAM, Philadelphia, A Melman, The even-odd split Levinson algorithm for Toeplitz systems, SIAM J Matrix Anal Appl 23, (200)

13 7 A Melman, A two step even-odd split Levinson algorithm for Toeplitz systems, Linear Algebra Appl 338 (200) F Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer-Verlag, New York, F Stenger, Collocating convolutions, Math Comp 64 (995) F Stenger, Matrices of sinc methods, J Comput Appl Math 86 (997) Department of Mathematics and Computer Science, Fenton Hall, SUNY College at Fredonia, Fredonia, NY 4063, USA address: abu-jeib@csfredoniaedu