G. S. Ammar and W. B. Gragg, The generalized Schur algorithm for the superfast solution of Toeplitz systems, in Rational Approximation and its Applications in Mathematics and Physics, eds. J. Gilewicz, M. Pindor and W. Siemaszko, Lecture Notes in Mathematics # 1237, Springer, Berlin, 1987, pp. 313--330.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....conditioned. The same is true for another less known pair of inversion formulas that only involve the left biorthogonal Szego polynomials. 1. INTRODUCTION The asymptotically fastest methods for solving Toeplitz systems, which have been proposed by Musicus [23] de Hoog [9] and Ammar and Gragg [3, 2, 4] for Hermitian systems, by Bitmead and Anderson [6] and Morf [22] for strongly regular systems, and by Gutknecht [13] and Gutknecht and Hochbruck [15, 14] for general non Hermitian Toeplitz systems make at the end normally use of inversion formulas for Toeplitz matrices. Two well known such ....

G. S. Ammar and W. B. Gragg, The generalized Schur algorithm for the superfast solution of Toeplitz systems, in Rational Approximation and its Applications in Mathematics and Physics, J. Gilewicz, M. Pindor, and W. Siemaszko, eds., vol. 1237 of Lecture Notes in Mathematics, SpringerVerlag,

....C A : Moreover, it is easy to see that each block C ij is an m Theta m Toeplitz matrix. Thus the problem of computing the product A ne x has been reduced to computing and summing the products C ij x j . Now the product of a Toeplitz matrix of order m and a vector can be computed in O(m log 2 m) [1, 2]. For definiteness let Block Hessenberg Systems 11 us say that it requires Km log 2 m operations. Hence we can compute the product A ne x s with Kp 2 m log 2 m operations. We now have only to insert this bound in the table (3.3) to update the operation count for bhsolve. The result is level ....

G. S. Ammar and W. B. Gragg. The generalized Schur algorithm for the superfast solution of Toeplitz systems. In J. Gilewicz, M. Pindor, and W. Siemaszko, editors, Rational Approximation and Its Applications in Mathematics and Physics. Proceedings, / La'nut, 1985, pages 315--330, New York, 1985. Springer.

....once again, n 2 O(n) flops, and then compute kS Gamma2 sk 2 2 with an additional O(n) flops. Roughly speaking, we can say that, for each increase of k by one, we need to execute an additional n 2 flops. Of course, there are other algorithms such as the fast Toeplitz solvers (see, e.g. [1] and [2] and these could be substituted for the LDA. However, this influences only the complexity of computing our bounds and not the bounds themselves, which are the focus of this work. 4 Approximations As we mentioned before, our bounds will be obtained by the roots of approximations to the ....

Ammar, G.S. and Gragg, W.B. (1987): The generalized Schur algorithm for the superfast solution of Toeplitz systems. In Rational Approximations and its Applications in Mathematics and Physics, J. Gilewicz, M. Pindor and W. Siemaszko, eds., Lecture Notes in Mathematics 1237, Berlin, pp. 315--330.

....C A : Moreover, it is easy to see that each block C ij is an m Theta m Toeplitz matrix. Thus the problem of computing the product A ne x has been reduced to computing and summing the products C ij x j . Now the product of a Toelplitz matrix of order m and a vector can be computed in O(m log 2 m) [1, 2]. For definiteness let us say that it requires Km log 2 m operations. Hence we can compute the product A ne x s with Kp 2 m log 2 m operations. Block Hessenberg Systems 11 We now have only to insert this bound in the table (3.3) to update the operation count for bhsolve. The result is level ....

G. S. Ammar and W. B. Gragg. The generalized Schur algorithm for the superfast solution of Toeplitz systems. In J. Gilewicz, M. Pindor, and W. Siemaszko, editors, Rational Approximation and Its Applications in Mathematics and Physics. Proceedings, / La'nut, 1985, pages 315--330, New York, 1985. Springer.

....matrix which improve their nonsymmetric counterparts considerably. In our numerical tests we used Durbin s algorithm to solve Yule Walker systems and to determine the location of parameters in the spectrum of T n Gamma2 . This information can be gained from superfast Toeplitz solvers (cf. [1], 2] 5] as well. Hence the computational complexity can be reduced to O(n log 2 n) operations. ....

G.S. Ammar and W.B. Gragg, The generalized Schur algorithm for the superfast solution of Toeplitz systems. in J. Gilewicz, M. Pindor, W. Siemaszko, eds., Lecture Notes in Mathematics 1237, Berlin, 1987, pp. 315 --- 330,

.... the diagonal matrix from (1:3) The first fast Toeplitz solver of this type was proposed by Bareiss [2] Later on, closely related algorithms were presented in [27, 29, 31] Today, Toeplitz solvers based on (1:4) are usually referred to as Schur type algorithms because of their intimate connection [1, 25] with Schur s seminal work [32] We remark that the matrices R and S in (1:4) are unique only up to a scaling of their columns. It turns out to be convenient to scale R and S so that they both have the same diagonal elements as D. In particular, the triangular matrices in (1:3) and (1:4) are then ....

Ammar, G.S., Gragg, W.B. (1987): The generalized Schur algorithm for the superfast solution of Toeplitz systems. In: J. Gilewicz, M. Pindor, and W. Siemaszko, eds., Rational Approximation and its Applications in Mathematics and Physics. Lecture Notes in Mathematics, Vol. 1237, pp. 315--330. Springer-Verlag, Berlin Heidelberg New York

.... the diagonal matrix from (1:3) The first fast Toeplitz solver of this type was proposed by Bareiss [2] Later on, closely related algorithms were presented in [27, 29, 31] Today, Toeplitz solvers based on (1:4) are usually referred to as Schur type algorithms because of their intimate connection [1, 25] with Schur s seminal work [32] We remark that the matrices R and S in (1:4) are unique only up to a scaling of their columns. It turns out to be convenient to scale R and S so that they both have the same diagonal elements as D. In particular, the triangular matrices in (1:3) and (1:4) are then ....

Ammar, G.S., Gragg, W.B. (1987): The generalized Schur algorithm for the superfast solution of Toeplitz systems. In: J. Gilewicz, M. Pindor, and W. Siemaszko, eds., Rational Approximation and its Applications in Mathematics and Physics. Lecture Notes in Mathematics, vol. 1237, pp. 315--330. Springer, Berlin Heidelberg New York

....considerably. Realistic and rigorous error bounds are obtained at negligible cost. In our numerical tests we used Durbin s algorithm to solve Yule Walker systems and to determine the diagonal matrix in the decomposition (4) These informations can be gained from superfast Toeplitz solvers (cf. [1], 2] 6] as well. Hence, the computational complexity can be reduced to O(n log 2 n) operations. ....

G.S. Ammar and W.B. Gragg, The generalized Schur algorithm for the superfast solution of Toeplitz systems. In J. Gilewicz, M. Pindor, W. Siemaszko (eds.), Rational Approximation and its Applications in Mathematics and Physics. Lecture Notes in Mathematics 1237, pp. 315 --- 330, Berlin 1987

....of positive definite symmetric Toeplitz matrices. Several algorithms for the former problem (see, e.g. 26] as for the latter (see, e.g. 12] 22] use equation (1) combined with the Levinson Durbin algorithm (see, e.g. 10] 18] and [20] or with superfast Toeplitz solvers (see, e.g. [2], 3] and [13] and an appropriate rootfinder. The eigenvectors are then computed as by products at almost no additional computational cost. However, equations (4) and (5) more fully exploit the spectral structure of Toeplitz matrices and should therefore prove more useful. First of all, they ....

Ammar, G.S. and Gragg, W.B. (1987): The generalized Schur algorithm for the superfast solution of Toeplitz systems. In Rational Approximations and its Applications in Mathematics and Physics, J. Gilewicz, M. Pindor and W. Siemaszko, eds., Lecture Notes in Mathematics 1237, Berlin, pp. 315--330.

....to the Lanczos method. Realistic error bounds are obtained at negligible cost. In our numerical tests we used Durbin s algorithm to solve Yule Walker systems and to determine the diagonal matrix in the decomposition (5) These informations can be gained from superfast Toeplitz solvers (cf. [1], 2] 4] as well. Hence, the computational complexity can be reduced to O(n log 2 n) operations. ....

G.S. Ammar and W.B. Gragg, The generalized Schur algorithm for the superfast solution of Toeplitz systems. in Rational Approximation and its Applications in Mathematics and Physics, J. Gilewicz, M. Pindor, W. Siemaszko, eds., Lecture Notes in Mathematics 1237, Berlin, 1987, pp. 315 --- 330,

....fOE j g j=0 with respect to this inner product. These polynomials are uniquely determined by the moments f j g j= Gamman . Several fast algorithms are available for the solution of the Yule Walker equations, such as Levinson s algorithm, Schur s algorithm and superfast Toeplitz solvers; see [3, 4, 12, 18, 27, 28] and references therein. These algorithms do not only provide the solution fff j g j=1 of the Yule Walker equations, but also the recursion coefficients j=1 and auxiliary coefficients foe j g j=1 for the Szego polynomials associated with the moments f j g j= Gamman defined by (2.4) We ....

....] of the filter determined by Algorithm 1 is ae n . The coefficients fff j g j=1 determined by the algorithm not only solve the YuleWalker equations (2.8) but also are coefficients in the power basis of the auxiliary polynomials OE n given by (1. 5) i.e. OE n (z) 1 see, e.g. [3, 27] for details. Assume for the moment that m = n in (2.6) Then the transfer function associated with the operator (2.6) is given by A(z) 1 Gammaj Often one is interested in the transfer function associated with the inverse of the operator (2.6) It is given by H(z) A(z) OE n ....

G. S. Ammar and W. B. Gragg, The generalized Schur algorithm for the superfast solution of Toeplitz systems, in Rational Approximation and its Applications in Mathematics and Physics, eds. J. Gilewicz, M. Pindor and W. Siemaszko, Lecture Notes in Mathematics # 1237, Springer, Berlin, 1987, pp. 313--330.

....theory is not necessary to derive the algorithms, we find the connections illuminating on the overall mathematical structure of the problems being considered. In fact, we have found the classical connections invaluable in understanding and implementing the superfast Toeplitz solver described in [5, 6]. 2 Schur s Algorithm and the Carath eodory Toeplitz Theorem The original mathematical importance of positive definite Toeplitz matrices is in the solution of the classical Carath eodory coefficient problem, proved independently by Toeplitz and Carath eodory in 1911. An analytic mapping w = ....

....to as algorithms of Schur type . See [28, 31] for discussions of this connection as well as wider connections of Schur s algorithm with signal processing and matrix computation. The explicit relationship between Schur s algorithm and Cholesky factorization is given in the following proposition [5]. Proposition 3.1 Let Mn 1 be a Hermitian positive definite Toeplitz matrix. Then the rational function OE 0 ( ff 0 ( fi 0 ( P n Gamma1 Gamma j 1 P n j=0 j (3.2) is a Schur function. Moreover, the entries of the lower triangular matrix LD = j;k ] j;k=0 are given by ....

[Article contains additional citation context not shown here]

G. S. Ammar and W. B. Gragg. The generalized Schur algorithm for the superfast solution of Toeplitz systems. In J. Gilewicz, M. Pindor, and W. Siemaszko, editors, Rational Approximation and its Applications in Mathematics and Physics, number 1237 in Lecture Notes in Mathematics, pages 315--330. Springer-Verlag, Berlin, 1987.

....Gragg and Reichel may already be applied to the solution of Sinc matrix problems, this paper also points to new directions of matrix research. 1 Introduction and Summary Much of W.B. Gragg s mathematically beautiful work investigates the connection of Toeplitz matrices and rational approximation [3]. My area of research Sinc methods is a family of approximation formulas that require matrix methods for their practical implementation [14,15,18,22] Sinc methods are close to optimal approximation methods in Sinc spaces of functions [7,14,21] and in this setting, there is some ....

....work of Ammar and Gragg [4] Gragg [12] and Heinig and Rost [13] applies to these. Indeed, the matrix stemming from the Sinc approximation of the first derivative, and the Sinc matrices stemming from Hilbert transforms are, in fact a Cauchy matries, and the methods of Boros, Kalath and Olshevski [3], and Calvetti and Reichel [8,9] may be applied to the solution of such matrix problems. We also mention here that some of the work of Pereyra (see e.g. 11] has connections with Stenger s work [17] The solution of some elliptic PDE problems via Sinc methods (see [15; 18, x7.4] can also be ....

[Article contains additional citation context not shown here]

G.S. AMMAR and W.B. GRAGG, The Generalized Schur Algorithm for the Superfast Solution of Toeplitz Systems. In J. Gilewicz, M. Pindor, and W. Siemaszko, editors, "Rational Approximation and its Applications in Mathematics and Physics", # 1237 in Lecture Notes in Mathematics, Springer-Verlag, Berlin, (1987) 315--330.

....fOE j g n j=0 with respect to this inner product. These polynomials are uniquely determined by the moments f j g n j= Gamman . Several fast algorithms are available for the solution of the Yule Walker equations, such as Levinson s algorithm, Schur s algorithm and superfast Toeplitz solvers; see [3, 4, 12, 18, 27, 28] and references therein. These algorithms do not only provide the solution fff j g n j=1 of the Yule Walker equations, but also the recursion coefficients ffl j g n j=1 and auxiliary coefficients foe j g n j=1 for the Szego polynomials associated with the moments f j g n j= Gamman defined ....

....by Algorithm 1 is ae n . The coefficients fff j g n j=1 determined by the algorithm not only solve the YuleWalker equations (2.8) but also are coefficients in the power basis of the auxiliary polynomials OE n given by (1. 5) i.e. OE n (z) 1 n X j=1 ff j z j ; see, e.g. [3, 27] for details. Assume for the moment that m = n in (2.6) Then the transfer function associated with the operator (2.6) is given by A(z) 1 n X j=1 ff j z Gammaj : Often one is interested in the transfer function associated with the inverse of the operator (2.6) It is given by H(z) 1 ....

G. S. Ammar and W. B. Gragg, The generalized Schur algorithm for the superfast solution of Toeplitz systems, in Rational Approximation and its Applications in Mathematics and Physics, eds. J. Gilewicz, M. Pindor and W. Siemaszko, Lecture Notes in Mathematics # 1237, Springer, Berlin, 1987, pp. 313--330.

....theory is not necessary to derive the algorithms, we find the connections illuminating on the overall mathematical structure of the problems being considered. In fact, we have found the classical connections invaluable in understanding and implementing the superfast Toeplitz solver described in [5, 6]. 2 Schur s Algorithm and the Carath eodory Toeplitz Theorem The original mathematical importance of positive definite Toeplitz matrices is in the solution of the classical Carath eodory coefficient problem, proved independently by Toeplitz and Carath eodory in 1911. An analytic mapping w = ....

....to as algorithms of Schur type . See [28, 31] for discussions of this connection as well as wider connections of Schur s algorithm with signal processing and matrix computation. The explicit relationship between Schur s algorithm and Cholesky factorization is given in the following proposition [5]. Proposition 3.1 Let Mn 1 be a Hermitian positive definite Toeplitz matrix. Then the rational function OE 0 ( ff 0 ( fi 0 ( P n Gamma1 j=0 Gamma j 1 j P n j=0 j j (3.2) is a Schur function. Moreover, the entries of the lower triangular matrix LD = j;k ] n j;k=0 are ....

[Article contains additional citation context not shown here]

G. S. Ammar and W. B. Gragg. The generalized Schur algorithm for the superfast solution of Toeplitz systems. In J. Gilewicz, M. Pindor, and W. Siemaszko, editors, Rational Approximation and its Applications in Mathematics and Physics, number 1237 in Lecture Notes in Mathematics, pages 315--330. Springer-Verlag, Berlin, 1987.

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Ammar,G.S.,Gragg,W.B.: The Generalized Schur Algorithm for the Superfast Solution of Toeplitz Systems, in Rational Approximation and its Applications in Mathematics and Physics, J.Gilewicz, M.Pindor and W.Siemaszko, eds., Lecture Notes in Mathematics 1237, Springer, berlin, 1987, pp. 315-330.

No context found.

G. S. Ammar and W. B. Gragg. "The Generalized Schur Algorithm for the Superfast Solution of Toeplitz Systems", Rational Approximation and its Applications in Mathematics and Physics, J.Gilewicz, M.Pindor and W.Siemaszko, eds., Lecture Notes in Mathematics 1237, Springer, Berlin, pp. 315-330, 1987.

No context found.

Ammar, G.S. and Gragg, W.B. (1987): The generalized Schur algorithm for the superfast solution of Toeplitz systems. In Rational Approximations and its Applications in Mathematics and Physics, J. Gilewicz, M. Pindor and W. Siemaszko, eds., Lecture Notes in Mathematics 1237, Berlin, pp. 315--330.

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