E. Tyrtyshnikov, Optimal and Super-optimal Circulant Preconditioners, SIAM J. Matrix Anal. Appl., to appear.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of these direct methods for symmetric positive definite matrices are discussed in Bunch [5] Here we will consider solving Toeplitz systems by the preconditioned conjugate gradient squared (PCGS) method. There are many circulant preconditioning strategies for Toeplitz systems, see for instance [23, 11, 16, 15, 26]. The convergence results for these circulant preconditioners are all based on the regularity of the function g( whose Fourier coefficients give the diagonals of T n . The function g( with 2 [ Gamma; is called the generating function of the sequence of Toeplitz matrices T n . A general ....

....i.e. if T n is constant along its diagonals. It is said to be circulant if its diagonals t k further satisfies t n Gammak = t Gammak for 0 k n Gamma 1. The idea of using circulant matrices as preconditioners for Toeplitz matrices has been studied extensively in recent years, see for instance [23, 11, 16, 26, 15]. In this paper, we will concentrate ourselves in the T. Chan circulant preconditioners. The results for the other circulant preconditioners can be obtained similarly, see x5. For a given Toeplitz matrix T n with diagonals ft j g j= Gamma(n Gamma1) the T. Chan circulant preconditioner to T n ....

E. Tyrtyshnikov, Optimal and Super-optimal Circulant Preconditioners, SIAM J. Matrix Anal. Appl., 13 (1992), 459--473.

....quadrature points used in the discretization. We see that (7) is basically a circulant preconditioned Toeplitz system which requires only O(n log n) operations in each iteration by means of Fast Fourier Transforms (FFTs) and the convergence rate of these systems has been analyzed for instances in [3, 11, 15]. One main drawback of using the rectangular rule is that the order of accuracy of the discretized solution y depends only linearly on the number of quadrature points. Thus in order to obtain a reasonable accurate solution for (2) small step size has to be used and hence the dimension of the ....

E. Tyrtyshnikov, Optimal and Super-Optimal Circulant Preconditioners, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 459--473. 20

....functions that are positive functions in the Wiener class, the method has a super linear convergence rate due to the clustering of the eigenvalues of the preconditioned matrices. Several circulant preconditioners have been proposed since then, see for example T. Chan [4] and Tyrtyshinkov [7]. For any n by n matrix A n , the circulant preconditioner proposed in T. Chan [4] called the optimal circulant preconditioner, is defined to be the minimizer of kC n Gamma A n k F over the space of all n by n circulant matrices C n . Here k Delta k F denotes the Frobenius norm. The circulant ....

....preconditioner proposed in T. Chan [4] called the optimal circulant preconditioner, is defined to be the minimizer of kC n Gamma A n k F over the space of all n by n circulant matrices C n . Here k Delta k F denotes the Frobenius norm. The circulant preconditioner given in Tyrtyshinkov [7] is defined to be the minimizer of kI Gamma C n A n k F over the space of all nonsingular circulant matrices C n , and is called the super optimal circulant preconditioner. From the computational point of view, these optimal circulant preconditoners are better than the one proposed in Strang ....

[Article contains additional citation context not shown here]

E. Tyrtyshnikov, Optimal and Super-optimal Circulant Preconditioners. To appear .

....n Gamma S n k 2 = 0 and hence the spectrum of C n A n is also clustered around one for sufficiently large n. These results are generalized to Hermitian positive definite Toeplitz systems in R. Chan [3] and a more precise convergence rate of these methods is given there. Recently, Tyrtyshnikov [8] proposed another circulant preconditioner T n that minimizes kI Gamma T n A n k F over all non singular circulant matrices. In that paper, C n and T n are called optimal and super optimal preconditioners respectively and it is proved that if A n is positive definite, then so are C n and T n ....

....= Q , we see that [C n Delta c(A n ) 0 = c(C n A n ) 0 ; for all 0 n. Thus C n Delta c(A n ) c(C n A n ) By similar arguments, we can prove that c(A n C n ) c(A n ) Delta C n . Theorems 3 and 4 below are just generalization of Theorems 3.1 and 4. 1 in Tyrtyshnikov [8] from the real scalar field to the complex field and their proofs are given in R. Chan, Jin and Yeung [4] Theorem 3. If A n is Hermitian and positive definite, then c(A n ) is Hermitian and positive definite. Moreover we have, 0 min (A n ) min (c(A n ) max (c(A n ) max (A n ) where ....

[Article contains additional citation context not shown here]

E. Tyrtyshnikov, Optimal and Super-optimal Circulant Preconditioners. Tech. Report, Dept. of Numerical Math., USSR Academy of Science, 1990.

....gradient method with circulant matrices as preconditioners for the solution of Toeplitz systems, see Strang [21] Several successful circulant preconditioners have been proposed and analyzed, see for instance Chan [4] T. Chan [8] Huckle [15] Ku and Kuo [18] Tismenetsky [22] and Tyrtyshnikov [24]. In these papers, the Toeplitz matrix A n is assumed to be generated by a generating function f , i.e. the diagonals of A n are given by the Fourier coefficients of f . It has been shown that if f is a positive function in the Wiener class, then these circulant preconditioned systems converge ....

E. Tyrtyshnikov, Optimal and Super-optimal Circulant Preconditioners, SIAM J. Matrix Anal. Appl., V13 (1992), pp. 459--473.

....to be preconditioners for the system Tx = b. Since the convergence rate of the PCG method depends on how good the preconditioner C approximates T , much attention has been focused on searching a circulant matrix C which is close to the matrix T in certain norms, see T. Chan [6] Tyrtyshnikov [22] and Huckle [18] T. Chan in [6] proposed a circulant preconditioner c(T ) which is the minimizer of jjC Gamma T jj F over all circulant matrices C. Here jj Delta jj F denotes the Frobenius norm. He called c(T ) the optimal circulant preconditioner and showed that the first column entries c j of ....

....j = jt Gamma(n Gammaj) n Gamma j)t j ; j = 0; 1; n Gamma 1; where t j are the diagonals of T . It was shown in Chan [7] that if the underlying generating function of T is a positive function in the Wiener class, then the spectrum of c(T ) T is clustered around 1. Tyrtyshnikov in [22] extended the definition of c( Delta) to any general n by n matrix A. Also, he proved that c(A) is symmetric positive definite whenever A is. Note that forming c(A) only requires O(n) operations for Toeplitz matrix A of order n, and O(n ) operations for general n by n matrix A. However, we ....

E. Tyrtyshnikov, Optimal and Super-optimal Circulant Preconditioners, SIAM J. Matrix Anal. Appl., V13 (1992), pp. 459--473.

.... Gamma (I H ) I A )jjj. Following the terminologies used in the study of Toeplitz matrices, we call the first minimizer the optimal preconditioner and denote it by P(A ) see [2] The circulant operator that minimizes the second one will be called the super optimal preconditioner, see [6]. We will prove some of the properties of the operator P. In particular, we will show that for self adjoint operators A , A x; x) where (a; b) j a(t) b(t)dt: Thus if A is a positive operator, then so is P(A ) We also show that the operator norms of P derived from the 2 norm ....

E. Tyrtyshnikov, Optimal and Super-optimal Circulant Preconditioners, SIAM Matrix Anal. Appl., 13 (1992), 459--473.

....of using circulant preconditioners in the PCG for solving square symmetric positive definite Toeplitz systems of equations was first proposed by Strang [25] Since then, several other circulant preconditioning techniques have been proposed, see for instance T. Chan [10] R. Chan [6] Tyrtyshnikov [27], Ku and Kuo [19] and Huckle [18] In particular, when A is an n Theta n Toeplitz matrix, T. Chan s circulant preconditioner (which we denote as c(A) is defined to be the optimal circulant approximation to A in the Frobenius norm. That is, c(A) is the circulant matrix which minimizes jjA Gamma ....

....derive circulant preconditioners for least squares problems. The T. Chan preconditioner c(A) is defined for general square matrices A, not necessarily of Toeplitz form. We note that the operator c preserves the positive definiteness of A. This is stated in the following Lemma due to Tyrtyshnikov [27]. Lemma 3.1. If A is an n Theta n Hermitian matrix, then c(A) is Hermitian. Moreover, we have min (A) min (c(A) max (c(A) max (A) where max ( Delta) and min ( Delta) denote the largest and the smallest eigenvalues, respectively. In particular, if A is positive definite, then c(A) is ....

E. Tyrtyshnikov, Optimal and super-optimal circulant preconditioners, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 459--473.

....integral in (26) i.e. the one w.r.t. i) then the resulting matrix is equal to the optimal circulant preconditioner c(A n ) of A n , which is defined to be the minimizer of kB n Gamma A n k F over all circulant matrices B n , see T. Chan [2] Here k Delta k F is the Frobenius norm. Tyrtyshnikov [17] has shown that for a general matrix W n , the entries of its optimal circulant preconditioner c(W n ) are given by [c(W n ) k;l = i Gammaj=k Gammal(mod n) W n ] i;j = W n ] k j) mod n) l j) mod n) 1 k; l n: 27) Theorem 5 Let A n be the Galerkin approximation of A as given by ....

E. Tyrtyshnikov, Optimal and Super-Optimal Circulant Preconditioners, SIAM Matrix Anal. Appl., 13 (1992), 459--473. 19

....form I n U n V n where I n is the identity matrix, U n is a matrix of low rank and V n is a matrix of small 2 norm. Several circulant preconditioners have been proposed and analyzed, see for instance, Chan and Strang [3] Chan [4, 5] Chan, Jin and Yeung [8] Ku and Kuo [17] Tyrtyshinkov [21] and Huckle [16] The convergence rate analysis of these circulant preconditioners depends on an assumption that the diagonals of the Toeplitz matrix A n are Fourier coefficients of a given function called the generating function. One typical convergence result is that if the generating function ....

E. Tyrtyshnikov, Optimal and Super-optimal Circulant Preconditioners, SIAM J. Matrix Anal. Appl., to appear.

....want C n A n to be of the form I n U n W n where I n is the identity matrix, U n is a matrix of low rank and W n is a matrix of small 2 norm. Several circulant preconditioners have been proposed and analysed, see for instance, Chan and Strang [3] T. Chan [10] Chan [4, 5] Tyrtyshinkov [22], Ku and Kuo [17] Chan, Jin and Yeung [6] Huckle [16] and Chan and Jin [8] It has been shown in these papers that if the diagonals a j of the Toeplitz matrix A n are Fourier coefficients of a positive function f in the Wiener class, then the spectrum of the preconditioned system C n A n will ....

E. Tyrtyshnikov, Optimal and Super-optimal Circulant Preconditioners, SIAM J. Matrix Anal. Appl., to appear.

....larger amount of second order information with respect to the OSS OSS methods. In fact, besides and , the updated matrix depends upon the previous Hessian approximation . Notice that the distribution of the eigenvalues of is strictly related to that of . In particular, the following result holds [35], 36] if is Hermitian and is a space of matrices simultaneously diagonalized by a unitary matrix , i.e. then is Hermitian and (20) where , denote the eigenvalues of in nondecreasing order. For this reason, the search direction proposed in QN methods appears to be (at least for the method) ....

....respectively, the search direction and the matrix , or only in the BFGS updating formula defining . Thus, for example, in the QN algorithm the system is replaced by (not preconditioned by . However, an efficient criterion for the choice of the best preconditioner [13] 21] 25] 26] 32] [35] could be useful in the choice of the best QN method, at least in the neighborhood of where can be approximated by a positive quadratic function. In fact, some preliminary experimental results show that if , then QN seems to converge faster. Moreover, a definite, a priori choice of the algebra ....

E. E. Tyrtyshnikov, "Optimal and superoptimal circulant preconditioners, " SIAM J. Matrix Anal. Appl., vol. 13, pp. 459--473, 1992.

....required for convergence is independent of the size of the matrix A n when n is large. In particular, the system A n x = b can be solved in O(n log n) operations. Over the past few years, several other preconditioners have also been proposed, see for instance, T. Chan [9] Chan [5] Tyrtyshinkov [20], Ku and Kuo [16] and Huckle [15] In Chan [4, 5] and Chan, Jin and Yeung [6] we have shown respectively that the preconditioners proposed in [9] 5] and [20] also work for the Wiener class functions, i.e. 1) holds if j ja j j 1. Huckle, on the other hand, has proved in [15] that his ....

....Over the past few years, several other preconditioners have also been proposed, see for instance, T. Chan [9] Chan [5] Tyrtyshinkov [20] Ku and Kuo [16] and Huckle [15] In Chan [4, 5] and Chan, Jin and Yeung [6] we have shown respectively that the preconditioners proposed in [9] 5] and [20] also work for the Wiener class functions, i.e. 1) holds if j ja j j 1. Huckle, on the other hand, has proved in [15] that his preconditioner works for the class of functions with j jja j j 1. We remark that it is the Besov space B 1=2 2 . For T. Chan s preconditioner, Chan and ....

E. Tyrtyshnikov, Optimal and Super-optimal Circulant Preconditioners, SIAM J. Matrix Anal. Appl., to appear.

....C F are given by c j = ja Gamma(n Gammaj) n Gamma j)a j ; j = 0; 1; n Gamma 1: It was then shown in Chan [3] that the spectrum of C F A n C F is also clustered around one if the underlying generating function of A n is a positive function in the Wiener class. Tyrtyshnikov in [12] extended the definition of C F to any general n by n matrix A. Also, he proved that C F is symmetric positive definite whenever A is. Note that forming C F only needs O(n) operations for Toeplitz matrix A of order n, and O(n ) operations for general n by n matrix A. We note that instead of ....

....C S actually minimizes jjC Gamma A n jj 1 and jjC Gamma A n jj 1 over all Hermitian circulant matrices C, see Chan [2] Since the matrices C A and C are similar, the PCG method converges superlinearly when the eigenvalues of C A are clustered around one. Tyrtyshnikov in [12] therefore proposed using the circulant preconditioner C T that minimizes jjI Gamma C Ajj F 3 over all nonsingular circulant matrices C. Fast algorithms for finding C T take O(n log n) operations for general n by n matrix A and only O(n log n) operations when A is Toeplitz. He proved that ....

E. Tyrtyshnikov, Optimal and Super-optimal Circulant Preconditioners, SIAM J. Matrix Anal. Appl., Vol. 13 (1992), pp. 459-473. 7

....see Chan and Strang [3] It follows that the preconditioned iterations converge very fast. Besides Strang s initial circulant preconditioner, several other successful circulant preconditioners have been proposed and analyzed, see, e.g. T. Chan [9] Huckle [16] Ku and Kuo [18] and Tyrtyshnikov [23]. In these papers, it has been shown that under the same assumptions on the Toeplitz matrices, these circulant preconditioned systems also converge superlinearly. Among these preconditioners, we remark that the T. Chan s circulant preconditioner is defined for general square matrices, not ....

E. Tyrtyshnikov. Optimal and Super-Optimal Circulant Preconditioners. SIAM J. Matrix Anal. Appl., 13:459--473, 1992.

....by its first column which can be obtained easily by taking the arithmetic average of the entries b ff;fi of B. More precisely, the entries fc ffi g ffi=1 in the first column of c(B) are given by c ffi 1 = ff Gammafi=ffi(mod n) b ff;fi ; ffi = 0; n Gamma 1; 1) see Tyrtyshnikov [17]. Using the circulant structure of c(B) the inverse [c(B) of c(B) and the matrix vector multiplication [c(B) x for any vector x can be obtained in O(n log n) operations by using fast Fourier transforms, see for instance Chan and Ng [8] Moreover, c(B) is positive definite whenever B is, ....

.... structure of c(B) the inverse [c(B) of c(B) and the matrix vector multiplication [c(B) x for any vector x can be obtained in O(n log n) operations by using fast Fourier transforms, see for instance Chan and Ng [8] Moreover, c(B) is positive definite whenever B is, see Tyrtyshnikov [17]. This makes c(B) a very attractive choice of preconditioner in the preconditioned conjugate gradient method for solving the system By = b. For then, if B is positive definite, the preconditioned matrix is positive definite. Moreover, the cost of multiplying [c(B) to a vector, which is ....

E. Tyrtyshnikov, Optimal and Super-Optimal Circulant Preconditioners, SIAM Matrix Anal. Appl., 13 (1992), 459--473.

....of using circulant preconditioners in the PCG for solving square symmetric positive definite Toeplitz systems of equations was first proposed by Strang [25] Since then, several other circulant preconditioning techniques have been proposed, see for instance T. Chan [10] R. Chan [6] Tyrtyshnikov [27], Ku and Kuo [19] and Huckle [18] In particular, when A is an nn Toeplitz matrix, T. Chan s circulant preconditioner (which we denote as c(A) is defined to be the optimal circulant approximation to A in the Frobenius norm. That is, c(A) is the circulant matrix which minimizes A C F over ....

....derive circulant preconditioners for least squares problems. The T. Chan preconditioner c(A) is defined for general square matrices A, not necessarily of Toeplitz form. We note that the operator c preserves the positive definiteness of A. This is stated in the following Lemma due to Tyrtyshnikov [27]. Lemma 3.1. If A is an n n Hermitian matrix, then c(A) is Hermitian. Moreover, we have # min (A) # # min (c(A) # #max (c(A) # #max (A) where #max ( and # min ( denote the largest and the smallest eigenvalues, respectively. In particular, if A is positive definite, then c(A) is ....

E. Tyrtyshnikov, Optimal and super-optimal circulant preconditioners, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 459--473.

....the optimal circulant preconditioner introduced by Chan [20] is denoted as c(A) It is easy to show [16] that c is a linear projection operator. Moreover, the following general result holds for any symmetric matrix; in particular, A does not need to be a Toeplitz matrix. Theorem 7 (Tyrtyshnikov [100]) If A is an n Theta n symmetric matrix, then c(A) is symmetric. Moreover, the minimum and maximum eigenvalues of A and c(A) satisfy min (A) min (c(A) max (c(A) max (A) In order to proceed with the analysis, one more lemma is needed. Lemma 8 (Chan, Nagy, Plemmons [18] Let A be an n ....

E. E. Tyrtyshnikov. Optimal and super-optimal circulant preconditioners. SIAM J. Matrix Anal. Appl., 13:459--473, 1992.

....define the optimal circulant preconditioner as c(A) It is easy to show (cf. Chan, Jin and Yeung [6] that c is a linear projection operator. Moreover, the following general result holds for any symmetric matrix; in particular, A does not need to be a Toeplitz matrix. Theorem 3.2. Tyrtyshnikov [31]) If A is an n Theta n symmetric matrix, then c(A) is symmetric. Moreover, the minimum and maximum eigenvalues of A and c(A) satisfy min (A) min (c(A) max (c(A) max (A) In order to proceed with our analysis, one more lemma in needed. Lemma 3.3. Chan, Nagy, Plemmons [7] Let A be an n ....

E. E. Tyrtyshnikov. Optimal and super-optimal circulant preconditioners. SIAM J. Matrix Anal. Appl., 13:459--473, 1992.

No context found.

E. Tyrtyshnikov, Optimal and superoptimal circulant preconditioners, SIAM J. Matrix Anal. Appl. 12(2): 459--473 (1992).

....the Galerkin matrices for any other smooth contour of diameter less than 1, then A n and C n are spectrally equivalent. It implies that C n can correspond to any convenient contour for which C n is easily invertible. The best choice might be a circle, in which case C n are circulant matrices, see [3, 14, 16]. It was shown in [4] that the optimal circulant integral approximate operator (CIAO) also leads to preconditioners spectrally equivalent to A n (some previous papers on CIAOs for the Wiener Hopf equations are [5, 9] Note that the circlebased preconditioner, though not optimal, in the sense of ....

....function c( Gamma t) a( t) 12) and regard C as a circulant integral approximate operator (CIAO) for A. The choice of the CIAO depends on how we understand (12) Some recent constructions [4, 5, 9] were inspired by T. Chan s idea of optimal circulant preconditioners [6] developed further in [14]) A matrix C = c ij ] 0 i; j n Gamma 1, is called a circulant matrix if c ij is constant along any wrapped diagonal i Gamma j = k (modn) The optimal circulant for a matrix A = a ij ] is the minimizer of jjA Gamma Cjj F . It can be shown that the diagonals c j of the optimal circulant ....

[Article contains additional citation context not shown here]

E. Tyrtyshnikov, Optimal and superoptimal circulant preconditioners, SIAM J. Matrix Anal. Appl. 12(2): 459--473 (1992).

No context found.

E. Tyrtyshnikov, Optimal and Super-optimal Circulant Preconditioners, SIAM J. Matrix Anal. Appl., to appear.

No context found.

E. Tyrtyshnikov, Optimal and Super-optimal Circulant Preconditioners, SIAM J. Matrix Anal. Appl., to appear.

No context found.

E. Tyrtyshnikov, Optimal and Super-optimal Circulant Preconditioners, SIAM J. Matrix Anal. Appl., to appear.

No context found.

E. Tyrtyshnikov, Optimal and Super-optimal Circulant Preconditioners, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 459--473.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC