SCE000406 - c 2000 Society for Industrial and Applied...

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SCHUR-TYPE METHODS FOR SOLVING LEAST SQUARES PROBLEMS WITH TOEPLITZ STRUCTURE HAESUN PARK AND LARS ELD ´ EN SIAM J. S CI. C OMPUT . c ° 2000 Society for Industrial and Applied Mathematics Vol. 22, No. 2, pp. 406–430 Abstract. We give an overview of fast algorithms for solving least squares problems with Toeplitz structure, based on generalization of the classical Schur algorithm, and discuss their stability properties. In order to obtain more accurate triangular factors of a Toeplitz matrix as well as accurate solutions for the least squares problems, methods based on corrected seminormal equations (CSNE) can be used. We show that the applicability of the generalized Schur algorithm is considerably enhanced when the algorithm is used in conjunction with CSNE. Several numerical tests are reported, where diFerent variants of the generalized Schur algorithm and CSNE are compared for their accuracy and speed. Key words. corrected seminormal equations, displacement representation, downdating, Givens transformations, hyperbolic transformations, least squares problems, QR decomposition, Schur algo- rithm, seminormal equations, Toeplitz matrix, updating AMS subject classifcations. 65±05, 65±20, 65±35 PII. S1064827598347423 1. Introduction. We consider the linear least squares (LS) problem min x k Tx b k 2 , (1.1) where T R m × n ,m n , is a Toeplitz matrix with rank( T )= n , T = t 0 t 1 ··· t n +1 t 1 t 0 t 1 . . . . . . t 1 t 0 . . . . . . . . . t 1 . . . t 1 . . . . . . t 0 . . . t 1 . . . . . . t m 1 t m n , and b R m is an arbitrary vector ( b is arbitrary in the sense that neither [ Tb ] nor [ bT ] is Toeplitz). Although the results presented in this paper apply for complex Toeplitz problems as well, we will restrict the discussion to real cases for simplicity. Throughout this paper, the notation T will denote an m × n real Toeplitz matrix. Received by the editors December 4, 1998; accepted for publication (in revised form) November 20, 1999; published electronically July 13, 2000. Department of Computer Science and Engineering, University of Minnesota, Minneapolis, MN 55455 ([email protected]). The research of this author was supported in part by National Science ±oundation grants CCR-9209726 and CCR-9509085. Department of Mathematics, Link¨oping University, S–581 83 Link¨oping, Sweden ([email protected] 406
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TOEPLITZ LEAST SQUARES PROBLEMS 407 It is well known that the LS problem (1.1) can be solved using the QR decompo- sition of the matrix T , T = Q µ R 0 , (1.2) where Q R m × m is orthogonal and R R n × n is upper triangular. To denote the upper triangular factor R for T in (1.2), we will use the notation R = qr( T ) . (1.3) The QR decomposition can be computed in O ( mn 2 ) flops (floating point operations, 1 flop 1 addition and 1 multiplication) in general, using, e.g., Householder trans- formations (see, e.g., [19, Chap. 5]). For a Toeplitz matrix T , several fast algorithms of O ( mn ) flops exist. The ±rst such algorithm was developed by Sweet [42] taking
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SCE000406 - c 2000 Society for Industrial and Applied...

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