B. Fischer,Polynomial Based Iteration Methods for Symmetric Linear Systems, Wiley, New York, 1996  Home/Search   Document Not in Database   Summary   Related Articles   Check

....parameters i , i and i have to be computed once for a given graph. We de ne an inner product for polynomials p, q by hp; qi : j p( j )q( j ) with j : 1 j . For k = 0; m 1 the polynomials p k are given as the (scaled and shifted) so called kernel polynomials, see [Fis96] They satisfy p 0 (t) 1 p 1 (t) 1 [ 1 t) p 0 (t) p k (t) k [ k t) p k 1 (t) k p k 2 (t) k = 2; m 1 and htp k 1 ; p k 1 i k = k 1 1 = 1 1; k = k 1 k ; k = 2; m 1 : Once these values are computed, OPS can be applied as ....

Fischer, B. 1996 Polynomial Based Iteration Methods for Symmetric Linear Systems. Wiley-Teubner series in advances in numerical mathematics. Wiley-Teubner, Chichester, Stuttgart.

....277 CGNE I N 11 37 164 where O # F N , S and CGNE (Craig s method) cf. 18, p. 239] applied to (MN,f ( f AN (f) MN,f ( f 2 ) MN,f ( f O) 2 x) MN,f ( f 2 b. 6. 2) For both algorithms we have used Matlab implementations of Fischer (see also [12]) In particular, his implementation of preconditioned MINRES avoids the splitting (6.2) In order to make the following computations with MINRES and CGNE comparable, we have stopped both computations if b ANx 2 b 2 10 7 . Example 1. We begin with Hermitian Toeplitz matrices AN ....

B. Fischer, Polynomial Based Iteration Methods for Symmetric Linear Systems, Wiley-- Teubner, New York, 1996.

....we can avoid the condition that A has to be positive definite, as in CG. However, in CR we construct an A orthogonal basis for the Krylov subspace, which leads to a different tridiagonal matrix as in the standard Lanczos method. CR was introduced in [19] for a modern coverage of the method see [6]. With respect to the choice of these methods, as far as numerical stability is concerned, one cannot find much in literature. For SYMMLQ and other approaches like MINRES, it is stated in [13, p. 625] that these approaches are not as accurate as SYMMLQ for the reason that the minimal residual ....

.... like Ap j Gamma1 = Ar j Gamma1 fi j Gamma1 Ap j Gamma2 ; fi j Gamma1 = j Gamma2 Ar j Gamma2 r j = r j Gamma1 Gamma ff j Ap j Gamma1 ; ff j = j Gamma1 A Ap j Gamma1 p j Gamma1 = r j Gamma1 Gamma fi j Gamma1 p j Gamma2 x j = x j Gamma1 ff j p j Gamma1 (28) [19, 6]. Note that the various quantities for CG and CR are different, but we have chosen not to use different notations. The first two recurrences for CR have the same structure as the first two for CG; the main difference is that in CR the r j are updated with an Ap j Gamma1 obtained from a recursion ....

B. Fischer, Polynomial Based Iteration Methods for Symmetric Linear Systems, Wiley-Teubner, Stuttgart 1996.

.... perfected by Freund and Nachtigal [45] and by Brezinski and Redivio Zaglia [12] The theory for this look ahead technique was linked to the theory of Pad e approximations by Gutknecht [59] Other contributions to overcome specific breakdown situations were made by Bank and Chan [9] and Fischer [39]. We will discuss these approaches in our section on QMR. The hybrids were developed primarily in the second half of the last 10 years; the first of these was CGS, published in 1989 by Sonneveld [89] and followed by Bi CGSTAB, by van der Vorst in 1992 [98] and others. The hybrid variants of ....

B. Fischer. Polynomial Based Iteration Methods for Symmetric Linear Systems. Wiley and Teubner, Chichester/Stuttgart, 1996.

.... k and the iterates x k can be recursively computed, thus leading to several algorithms for implementing Lanczos method [36] Among these procedures is the well known biconjugate gradient algorithm [55] which reduces to the conjugate gradient algorithm when A is symmetric and y = r 0 [62] see also [53]. New algorithms can also be deduced [36] Due to the normalization of the formal orthogonal polynomials involved in Lanczos method, some underlying Hankel determinants can be zero. Moreover, some recurrence relations may be impossible to use. In these cases, a division by zero, called a ....

B. Fischer, Polynomial Based Iteration Methods for Symmetric Linear Systems, John Wiley, Chichester and B.G. Teubner, Stuttgart, 1996.

....can be seen more clearly by reviewing related work. Earlier approaches of Eiermann et al. 3] and Shen and Strang [14] involve elementary polynomial transforms: for the special case of two intervals their approaches required a symmetry property not needed in the SCM method. Freund [7] and Fischer [6] turned to a conformal map involving elliptic functions and converted the domain to an annulus. Wathen et al. 15] 16] avoided Green s functions, but employed a perturbation in the two interval case for which a differential equation can be established. The SCM method is more elementary, ....

B. Fischer. Polynomial Based Iteration Methods for Symmetric Linear Systems. Wiley, New York, and Teubner, 1996.

....: N , 0 1;N 2;N : N;N , jjv j;N jj = 1, the eigensystem of the matrix AN and use the spectral decomposition of the rst residual r 0;N = N X j=1 w j;N v j;N ; w j;N = r 0;N ; v j;N ) 2 R: 2. 1) It will be useful to adopt polynomial language in order to describe the CG error [Fi96]. We consider the discrete scalar product with varying weights (p; q) N = N X j=1 w 2 j;N p( j;N )q( j;N ) 2.2) with corresponding norm jjpjj N = p (p; p) N , and denote the corresponding orthonormal polynomials by p 0;N ; p 1;N ; As it is well known [Fi96, Saa96, TrBa97] these ....

....to describe the CG error [Fi96] We consider the discrete scalar product with varying weights (p; q) N = N X j=1 w 2 j;N p( j;N )q( j;N ) 2. 2) with corresponding norm jjpjj N = p (p; p) N , and denote the corresponding orthonormal polynomials by p 0;N ; p 1;N ; As it is well known [Fi96, Saa96, TrBa97], these discrete orthonormal polynomials may be used to describe the error in CG as follows e n;N = 1 p n;N (0) p n;N (A)e 0;N = 1 p n;N (0) A 1 p n;N (A)r 0;N ; 2.3) 6 B. BECKERMANN AND A. B. J. KUIJLAARS and thus ke n;N kAN ke 0;N kAN = kr n;N k A 1 N kr 0;N k A 1 N = 1 jp ....

B. Fischer, Polynomial Based Iteration Methods for Symmetric Linear Systems, Wiley, Teubner, Stuttgart (1996).

....potentials, constrained equilibrium problem. 1 Introduction. The Conjugate Gradient (CG) method is widely used for solving large symmetric positive definite linear systems Ax = b. Error bounds for CG depend on the spectrum (A) of A. The error e n after n steps of CG iteration satisfies (see e.g. [11, 13, 23]) ken kA ke 0 kA En ( A) 1.1) where, for any closed set S ae R, the quantity En (S) is defined by En (S) min p2Pn max 2S jp( j; 1.2) Received September 2000. Revised July 2001. Communicated by y This joint work was partially supported by INTAS project 00 272. The second author is ....

....En (S) Gammag S (0) 1.6) exists, the limit is equal to the negative of the Green function g S (z) associated with S, evaluated at z = 0) the inequality (1.5) leads to the following approximate inequality valid for large n, 1 n log En ( A) Gamma g S (0) 1.7) provided (A) ae S. See [10, 11, 17] for more information on the relation between Green functions from potential theory and error estimates in matrix iteration. Having other additional information about the eigenvalues one may improve on the often crude upper bounds (1.3) 1.4) and (1.7) Indeed, there is an important number of ....

B. Fischer, Polynomial based iteration methods for symmetric linear systems, WileyTeubner, 1996.

....jjv j;N jj = 1, the eigensystem of the matrix AN and use the spectral decomposition of the first residual r 0;N = N X j=1 w j;N v j;N ; w j;N = r 0;N ; v j;N ) 2 R: 2.1) 6 B. BECKERMANN AND A. B. J. KUIJLAARS It will be useful to adopt polynomial language in order to describe the CG error [Fi96]. We consider the discrete scalar product with varying weights (p; q) N = N X j=1 w 2 j;N p( j;N )q( j;N ) 2.2) with corresponding norm jjpjj N = p (p; p) N , and denote the corresponding orthonormal polynomials by p 0;N ; p 1;N ; As it is well known [Fi96, Saa96, TrBa97] these ....

....to describe the CG error [Fi96] We consider the discrete scalar product with varying weights (p; q) N = N X j=1 w 2 j;N p( j;N )q( j;N ) 2. 2) with corresponding norm jjpjj N = p (p; p) N , and denote the corresponding orthonormal polynomials by p 0;N ; p 1;N ; As it is well known [Fi96, Saa96, TrBa97], these discrete orthonormal polynomials may be used to describe the error in CG as follows e n;N = 1 p n;N (0) p n;N (A)e 0;N = 1 p n;N (0) A Gamma1 p n;N (A)r 0;N ; 2.3) and thus ke n;N kAN ke 0;N kAN = kr n;N k A Gamma1 N kr 0;N k A Gamma1 N = 1 jp n;N (0)j kp n;N ( ....

B. Fischer, Polynomial Based Iteration Methods for Symmetric Linear Systems, Wiley, Teubner, Stuttgart (1996).

....potentials, constrained equilibrium problem. 1 Introduction The Conjugate Gradient (CG) method is widely used for solving large symmetric positive de nite linear systems Ax = b. Error bounds for CG depend on the spectrum (A) of A. The error e n after n steps of CG iteration satis es (see e.g. [11, 13, 23]) ke n k A ke 0 k A E n ( A) 1.1) where, for any closed set S R, the quantity E n (S) is de ned by E n (S) min p2Pn max 2S jp( j; 1.2) and kxkA = p x T Ax is the A norm. In (1.2) the minimum is taken over the class P n of polynomials p of degree at most n with p(0) 1. ....

....n log E n (S) g S (0) 1.6) exists, the limit is equal to the negative of the Green function g S (z) associated with S, evaluated at z = 0) the inequality (1.5) leads to the following approximate inequality valid for large n, 1 n log E n ( A) g S (0) 1.7) provided (A) S. See [10, 11, 17] for more information on the relation between Green functions from potential theory and error estimates in matrix iteration. Having other additional information about the eigenvalues one may improve on the often crude upper bounds (1.3) 1.4) and (1.7) Indeed, there is an important number of ....

B. Fischer, Polynomial based iteration methods for symmetric linear systems, WileyTeubner, 1996.

....algorithm of Paige and Saunders [19] It uses the LQ factorization of the symmetric tridiagonal matrices determined by the Lanczos process. An analogous implementation of the MR method, that is based on the QR factorization instead of the LQ factorization, has recently been described by Fischer [11]. The algorithm requires the matrix A to be symmetric. The matrix may be definite, indefinite or singular. We remark that if the matrix A is known to be positive definite, then the recursions can be simplified, see, e.g. Saad [20, p. 183] for an example of a simpler algorithm for the MR method. ....

B. Fischer, Polynomial Based Iteration Methods for Symmetric Linear Systems, Wiley-Teubner, New York, 1996.

....30E10, 31A99, 41A10, 65F10 1. Introduction. Green s functions in the complex plane are basic tools for the analysis of real and complex polynomial approximations [10,21,24,30,32] which are of central importance in the fields of digital signal processing [16,17,19] and matrix iterations [5,6,11,20,28]. The aim of this article is to show that when the domain of approximation is a collection of real intervals, or more generally symmetric polygons along the real axis, the Green s function can be computed to high accuracy by Schwarz Christoffel conformal mapping. The computation of ....

B. Fischer, Polynomial Based Iteration Methods for Symmetric Linear Systems, Wiley--Teubner, New York, 1996.

....seen more clearly by reviewing related work. Earlier approaches of Eiermann, Li, and Varga [3] and Shen and Strang [14] involve elementary polynomial transforms: for the special case of 2 intervals their approaches required a symmetry property not needed in the SCM method. Freund [7] and Fischer [6] turned to a conformal map involving elliptic functions and converted the domain to an annulus. Wathen, Fischer, and Silvester [15, 16] avoided Green s functions, but employed a perturbation in the two interval case for which a di erential equation can be established. The SCM method is more ....

B. Fischer. Polynomial Based Iteration Methods for Symmetric Linear Systems. Wiley & Sons and Teubner, 1996.

....found in [3] It should be noted that it is not usually practical to update x LSQR(A Gamma1 ) k at every step, however once y LSQR(A Gamma1 ) k is deemed sufficiently accurate, x LSQR(A Gamma1 ) k can be recovered at a cost of less than one LSQR(A Gamma1 ) step. It can be seen ([2]) that MINRES on the preconditioned system (5.2) requires ffl Matrix vector products 2 (n Theta m) ffl A Gamma1 operations 1 ffl Dot products 2 (n) 2 (m) ffl Additional flops 12n 12m ffl Stored vectors 7 (n) 7 (m) per step, where ( Delta) indicates the dimension of the operation, ....

B. Fischer, Polynomial based iteration methods for symmetric linear systems, Advances in numerical mathematics, Wiley-Teubner, 1996.

....N ; C II N ; S II N g and CGNE (Craig s method) cf. 17, p. 239] applied to (M N;f (jf j; O) Gamma 1 2 AN (f) M N;f (jf j; O) Gamma 1 2 ) M N;f (jf j; O) 1 2 x) M N;f (jf j; O) Gamma 1 2 b : 6.2) For both algorithms we have used MATLAB implementations of B. Fischer. See also [11]. In particular, his implementation of preconditioned MINRES avoids the splitting (6.2) In order to make the following computations with MINRES and CGNE comparable, we have stopped both computations if jjb Gamma ANx (k) jj 2 = jjbjj 2 10 Gamma7 : Example 1. We begin with Hermitian ....

B. Fischer. Polynomial Based Iteration Methods for Symmetric Linear Systems. Wiley-- Teubner, 1996.

....we can avoid the condition that A has to be positive definite, as in CG. However, in CR we construct an A orthogonal basis for the Krylov subspace, which leads to a different tridiagonal matrix as in the standard Lanczos method. CR was introduced in [19] for a modern coverage of the method see [6]. With respect to the choice of these methods, as far as numerical stability is concerned, one cannot find much in literature. For SYMMLQ and other approaches like MINRES, it is stated in [13, p. 625] that these approaches are not as accurate as SYMMLQ for the reason that the minimal residual ....

.... j Gamma2 ; fi j Gamma1 = r T j Gamma1 Ar j Gamma1 r T j Gamma2 Ar j Gamma2 r j = r j Gamma1 Gamma ff j Ap j Gamma1 ; ff j = r T j Gamma1 Ar j Gamma1 p T j Gamma1 A T Ap j Gamma1 p j Gamma1 = r j Gamma1 fi j Gamma1 p j Gamma2 x j = x j Gamma1 ff j p j Gamma1 (28) [19, 6]. Note that the various quantities for CG and CR are different, but we have chosen not to use different notations. The first two recurrences for CR have the same structure as the first two for CG; the main difference is that in CR the r j are updated with an Ap j Gamma1 obtained from a recursion ....

B. Fischer, Polynomial Based Iteration Methods for Symmetric Linear Systems, Wiley-Teubner, Stuttgart 1996.

....kr k k 2 kr 0 k 2 (V) min p2P k p(0) 1 max 2(A) jp( j: Here, V) j kVk 2 kV Gamma1 k 2 is the 2 norm condition number of the eigenvector matrix V. If A is normal, then (V) 1; if, in addition, the eigenvalues are real, then (EV) reduces to the standard convergence bound for MINRES [13]. If A is non normal, then (V) 1 and determining the optimal value of (V) can be a challenge [20] this task is further complicated if A has repeated eigenvalues. Throughout this work, the columns of V have unit 2 norm; provided each eigenvalue of A is simple, V) with this scaling can be no ....

.... of (PSA) is obtained by underestimating the convergence rate of (A) taking instead the rate associated with the union of two intervals, ffi Gamma ; ffi ] a Gamma ; b ] This rate can be expressed in terms of elliptic integrals, as described and implemented in Matlab by Fischer [13]. The pseudospectral constants are C psa ( 2 (b Gamma a) While a single outlier, as presented in this example, may appear to be an easilyovercome obstacle for the bounds (FOV) and (PSA) the same phenomenon can prove GMRES CONVERGENCE BOUNDS 11 PSfrag replacements (FOV) EV) exact ....

B. Fischer, Polynomial Based Iteration Methods for Symmetric Linear Systems, Wiley-- 22 MARK EMBREE Teubner, Chichester, 1996. For accompanying software, see http://www.math.mu-luebeck.de/workers/fischer/fischer-home.html.

....we can avoid the condition that A has to be positive definite, as in CG. However, in CR we construct an A orthogonal basis for the Krylov subspace, which leads to a different tridiagonal matrix as in the standard Lanczos method. CR was introduced in [19] for a modern coverage of the method see [6]. With respect to the choice of these methods, as far as numerical stability is concerned, one cannot find much in literature. For SYMMLQ and other approaches like MINRES, it is stated in [13, p. 625] that these approaches are not as accurate as SYMMLQ for the reason that the minimal residual ....

....Ar j Gamma1 r T j Gamma2 Ar j Gamma2 r j = r j Gamma1 Gamma ff j Ap j Gamma1 ; ff j = r T j Gamma1 Ar j Gamma1 p T j Gamma1 A T Ap j Gamma1 Gerard L.G. Sleijpen, et al. 11 p j Gamma1 = r j Gamma1 Gamma fi j Gamma1 p j Gamma2 x j = x j Gamma1 ff j p j Gamma1 (28) [19, 6]. Note that the various quantities for CG and CR are different, but we have chosen not to use different notations. The first two recurrences for CR have the same structure as the first two for CG; the main difference is that in CR the r j are updated with an Ap j Gamma1 obtained from a recursion ....

B. Fischer, Polynomial Based Iteration Methods for Symmetric Linear Systems, Wiley-Teubner, Stuttgart 1996.

....m X j=2 j 1 Gamma j p k ( j ) 2 = min p2 Pi k m X j=2 j 1 Gamma j p( j ) 2 ; k = 0; m Gamma 1 : 20) Proof: The relation (19) means that the polynomials p k are orthogonal with respect to h Delta; Deltai. As is well known from numerical analysis [9], such a sequence of orthogonal polynomials exists and it is unique up to scalings with a scalar factor. This scalar factors are uniquely defined for our situation, since we have the additional restriction p k (1) 1 for k = 0; m Gamma 1. Note that it is also known that the orthogonal ....

....for k = 0; m Gamma 1. Note that it is also known that the orthogonal polynomials have all their zeros within the interval [ m ; 2 ] so that none of them vanishes at t = 1. Finally, the recurrence (17) is just the standard three term recurrence for orthogonal polynomials (see again [9]) adapted to the normalization p k (1) 1. To show (20) let us first introduce the notation h Delta; Deltai 0 for the inner product hp; qi 0 = m X j=2 j 1 Gamma j p( j ) q( j ) so that hp; qi = hp; 1 Gamma t)qi 0 . Now, let p 2 Pi k . Since p Gamma p k has a zero at t = 1, ....

B. Fischer. Polynomial Based Iteration Methods for Symmetric Linear Systems. Wiley, 1996.

.... perfected by Freund and Nachtigal [45] and by Brezinski and Redivio Zaglia [12] The theory for this look ahead technique was linked to the theory of Pad e approximations by Gutknecht [59] Other contributions to overcome specific breakdown situations were made by Bank and Chan [9] and Fischer [39]. We will discuss these approaches in our section on QMR. The hybrids were developed primarily in the second half of the last 10 years; the first of these was CGS, published in 1989 by Sonneveld [89] and followed by Bi CGSTAB, by van der Vorst in 1992 [98] and others. The hybrid variants of ....

B. Fischer. Polynomial Based Iteration Methods for Symmetric Linear Systems. Wiley and Teubner, Chichester/Stuttgart, 1996.

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B. Fischer, "Polynomial based iteration methods for symmetric linear systems, " Wiley-Teubner, Chichester, 1996.

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B. Fischer, Polynomial based iteration methods for symmetric linear Systems, Chicester, Wiley-Teubner, 1996.

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B. Fischer. Polynomial Based Iteration Methods for Symmetric Linear Systems. WileyTeubner, Chicester, 1996.

....which have in built minimisation properties whilst being cheap to implement and parameter free. Here we consider minimal residual methods, whose iterates minimise the Euclidean norm of the residual r k at each step. For the symmetric indefinite matrix A j , our method of choice is MINRES ([8], 15] while for the nonsymmetric positive semi definite case A Gamma j we use GMRES ( 17] Note that GMRES compares unfavourably with MINRES in terms of operation count and storage requirements. For the systems under consideration, the quasi minimum residual method QMR [10] is precisely as ....

....= x Gamma 2k = x Gamma 2k 1 which is the required result. As well as demonstrating the equivalence of MINRES and GMRES in terms of residual norms, Theorem 3.4 shows that both methods make no progress on every second step. An alternative proof of this result for the case j = 0 is given in [8]. It is readily seen that the proofs presented here rely on the choice of starting vector (2.5) Only for such a starting vector is the odd even symmetry required in the proofs exactly preserved. However, in practical computation when B is ill conditioned and roundoff is significant, this is ....

B. Fischer, Polynomial Based Iteration Methods for Symmetric Linear Systems, WileyTeubner Series in Advances in Numerical Mathematics, Wiley-Teubner, Chichester 1996.

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B. Fischer. Polynomial Based Iteration Methods for Symmetric Linear Systems. Teubner, Stuttgart, 1996.

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B. Fischer,Polynomial Based Iteration Methods for Symmetric Linear Systems, Wiley, New York, 1996

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B. Fischer, Polynomial based iteration methods for symmetric linear systems, Wiley-Teubner Series Advances in Numerical Mathematics, John Wiley & Sons Ltd., Chichester, 1996.

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B. FISCHER, Polynomial Based Iteration Methods for Symmetric Linear Systems, Wiley-Teubner, Sec. Adv. Numer. Math. (1996).

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B. Fischer. Polynomial based iteration methods for symmetric linear systems. Advances in Numerical Mathematics. Wiley and Teubner, Chichester, Stuttgart, 1996.

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B. Fischer, Polynomial Based Iteration Methods for Symmetric Linear Systems, Teubner-Wiley, New York, 1996.

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Bernd Fischer, Polynomial Based Iteration Methods for Symmetric Linear Systems, Wiley, Chichester, 1996

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B. Fischer, Polynomial Based Iteration Methods for Symmetric Linear Systems, Wiley, Teubner, Stuttgart (1996).

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B. Fischer,Polynomial Based Iteration Methods for Symmetric Linear Systems, Wiley, New York, 1996

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B. Fischer, Polynomial Based Iteration Methods for Symmetric Linear Systems, Wiley-Teubner, Stuttgart 1996.

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