F. L. Bauer and C. T. Fike. Norms and exclusion theorems. Numerische Mathematik, 2:137--141, 1960.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....kH Gamma1 k 2 k DeltaHk 2 . Remark 3.1. Li [24] also considered extending Theorem 3.1 to diagonalizable matrices under multiplicative perturbations. But the bounds obtained in a recent paper [26] are better. Both Li [24] and Eisenstat and Ipsen [13] extended the classical Bauer Fike theorem [2]. 4. Relative Perturbation Theorems for Singular Value Problems. Throughout the section, B; e B 2 C m Thetan and one is a perturbation of the other. We shall assume without loss of any generality that m n in this section. Denote their singular values by oe(B) foe 1 ; Delta Delta Delta ....

F. L. Bauer and C. T. Fike, Norms and exclusion theorems, Numerische Mathematik, 2 (1960), pp. 137--141.

....an approximate generalized invariant subspace. To estimate all these bounds, p Holder norm and an ad hoc pseudo metric called p Chordal matric which isn t a metric on the Riemann sphere when p 6= 2 are used. x1. Introduction As to perturbations of eigenvalues of matrices, Bauer Fike Theorem[3] is well known, and its generalized version has also been deduced in Kahan et al. 5] and Li[6] so as to it can be applied to non diagonalizable case. Note that the results in Li[6] which also generalize one of the results in Kahan et al. 5] improve slightly the results in [5] On the other hand, ....

Bauer, F. L. and Fike, C. T., Norms and exclusion theorems, Numer., Math., 2(1960), 137-141.

....that pseudospectra are invariant under unitary transformations, and also re ects the extent to which an ill conditioned similarity transformation can alter pseudospectra. When SBS 1 is a diagonalization of A, Theorem 4 is equivalent to the most familiar version of the Bauer Fike Theorem [1]. Theorem 4. A = SBS 1 = A) B) Theorem 4 . A = SBS 1 = A) S) B) Proof. Since (z A) 1 = S(z B) 1 S 1 , k(z A) 1 k (S)k(z B) 1 k. Therefore if k(z A) 1 k 1 , then k(z B) 1 k ( S) 1 . The following theorem makes use of the idea of the ....

F. L. Bauer and C. T. Fike, Norms and exclusion theorems, Numer. Math., 2 (1960), pp. 137{ 141.

....would be interesting to know whether p (X) p ( e X) in inequality (4.6) could be improved to q p (X) p ( e X) as a similar thing happened between (4.4) and (4. 5) Generally, if one of A and e A is diagonalizable and the other is arbitrary, we have the following result due to Bauer and Fike 3 [2, 1960]. 3 One can prove a slightly more stronger inequality than (4.7) j e Gamma j kX Gamma1 ( e A Gamma A)Xk2 : Ren Cang Li: Relative Perturbation Theory 23 Theorem 4.6 (Bauer Fike) Assume A is diagonalizable, i.e. A = X X Gamma1 ; where = diag( 1 ; Delta Delta Delta ; n ) Then ....

....d k q 2 kI Gamma Sigma Gamma1 d k q 2 ; as required. So far we have considered the case when both A and e A are diagonalizable. In what follows, we weaken this assumption by requiring only A to be diagonalizable and derive a relative eigenvalue perturbation bound of Bauer Fike Type [2]. Theorem 6.6 Assume that A 2 C n Thetan is diagonalizable and admits the following decomposition A = X X Gamma1 where = diag( 1 ; Delta Delta Delta ; n ) 6.11) Assume 5 also either e A = DA or e A = AD. Then for any e 2 ( e A) there exists a 2 (A) such that min 2(A) j e ....

F. L. Bauer and C. T. Fike. Norms and exclusion theorems. Numerische Mathematik, 2:137-- 141, 1960.

....i x i y i ffi Ax i ; where x i , and y i are the normalized right and left eigenvectors respectively, corresponding to i , and y i denotes the complex conjugate of y i . The factor 1 y i x i is referred to as the condition number of the ith eigenvalue. The Bauer Fike result (1960) [9], which is actually one of the more famous refinements of Gershgorin s theorem, makes this more precise: the eigenvalues e j of A ffi A lie in discs B i with center i , and radius n kffiAk 2 jy i x i j (for normalized x i and y i ) The Courant Fischer minimax theorem is the basis of ....

F. L. Bauer and C. T. Fike. Norms and exclusion theorems. Numer. Math., 2:137--141, 1960.

....and a closest eigenvalue of A. The matrix A must be diagonalisable, while A E does not have to be. Let A = X X Gamma1 be an eigendecomposition of A, where = 0 B 1 . n 1 C A ; and i are the eigenvalues of A. Let be an eigenvalue of A E. The Bauer Fike Theorem for the two norm (Bauer and Fike 1960, Theorem IIIa) bounds the absolute error, min i j i Gamma j (X) kEk: 2.1) The relative version of the Bauer Fike Theorem below requires in addition that A be non singular. Theorem 2.1 If A is diagonalisable and non singular then min i j i Gamma j j i j (X) kA Gamma1 Ek; where ....

....= 0 B 1 . n 1 C A ; and i are the eigenvalues of A. Let be an eigenvalue of the perturbed matrix D 1 AD 2 and x 6= 0 a corresponding unit eigenvector, D 1 AD 2 ) x = x; kxk = 1; with residual r j Ax Gamma x: This time we use the Bauer Fike Theorem with residual bound (Bauer and Fike 1960, Theorem IIIa) min 1in j i Gamma j (X) krk: 5.1) The relative error bound below for the eigenvalue of the perturbed matrix D 1 AD 2 measures the error relative to the perturbed eigenvalue rather than an exact eigenvalue. Theorem 5.1 If A is diagonalisable then min 1in j i Gamma j ....

F. Bauer and C. Fike (1960), `Norms and exclusion theorems', Numer. Math. 2, 137-- 41.

....bound. Let A be a diagonalisable matrix with eigendecomposition A = X X Gamma1 , where = 0 1 . n 1 A and i are the eigenvalues of A. The Bauer Fike Theorem bounds the absolute error between a perturbed eigenvalue and a closest eigenvalue of A. Theorem 2. 1 (Theorem IIIa in [2]) If A is diagonalisable then min i j i Gamma j (X) kEk; where (X) j kXk kX Gamma1 k. The Bauer Fike Theorem implies the relative bound below, provided A is nonsingular. Corollary 2.2. If A is diagonalisable and non singular, then min i j i Gamma j j i j (X) kA Gamma1 Ek: ....

F. Bauer and C. Fike, Norms and exclusion theorems, Numer. Math., 2 (1960), pp. 137--41.

....turn to concrete examples illustrating the relative merits of the three standard bounds. If the pseudospectra are large, then the field of values and the eigenvalue condition number must also be large, in the sense defined in the following theorems. The first, a version of the Bauer Fike theorem [2], 42] bounds the pseudospectra by (V) The second relates the pseudospectra to the field of values (see Gustafson and Rao [21, x4.6] The third relates the field of values to the eigenvector condition number as a consequence of the basic inequality (A) kAk 2 . Let Delta r j fz 2 C j jzj rg ....

F. L. Bauer and C. T. Fike, Norms and exclusion theorems, Numer. Math., (1960), pp. 137-- 141.

....following theorem is due to Li [21, pp. 225 226] Theorem 4.5 (Li) Under the conditions of Theorem 4.4. Then max 1in j i Gamma e i j p (X) p ( e X)kA Gamma e Ak p ; where 1 p 1. Generally, if one of A and e B is diagonalizable, we have the following result due to Bauer and Fike 1 [2]. Theorem 4.6 (Bauer Fike) Assume A is diagonalizable, i.e. A = X X Gamma1 ; where = diag( 1 ; Delta Delta Delta ; n ) Then for any e 2 ( e A) there exists a 2 (A) such that j e Gamma j (X)k e A Gamma Ak 2 : 4.5) Regarding singular value perturbations, the following theorem was ....

....d k q 2 kI Gamma Sigma Gamma1 d k q 2 ; as required. So far we have considered the case when both A and e A are diagonalizable. In what follows, we weaken this assumption by requiring only A to be diagonalizable and derive relative eigenvalue perturbation bounds of Bauer Fike Type [2]. Theorem 6.6 Assume that A 2 C n Thetan is diagonalizable and admits the following decomposition A = X X Gamma1 where = diag( 1 ; Delta Delta Delta ; n ) 6.11) Assume 2 also either e A = DA or e A = AD. Then for any e 2 ( e A) there exists a 2 (A) such that min 2(A) j e ....

F. L. Bauer and C. T. Fike. Norms and exclusion theorems. Numerische Mathematik, 2:137--141, 1960.

....p of an n vector x = 1 ; 2 ; Delta Delta Delta ; n ) T , and the p operator norm of an n Theta n matrix X are defined by kxk p def = n X i=1 j i j p 1=p ; kXk p def = max kxkp=1 kXxk p ; where 1 p 1. The following theorem known as Bauer Fike theorem was proved in [1]. Theorem 1 (Bauer Fike) Assume A is diagonalizable, i.e. A = X X Gamma1 ; where = diag( 1 ; Delta Delta Delta ; n ) Then for any e 2 ( e A) there exists a 2 (A) such that j e Gamma j (X)k e A Gamma Ak 2 ; 1) where (X) def = kXk 2 kX Gamma1 k 2 is the spectral condition ....

F. L. Bauer and C. T. Fike. Norms and exclusion theorems. Numerische Mathematik, 2:137--141, 1960.

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F. L. Bauer and C. T. Fike. Norms and exclusion theorems. Numerische Mathematik, 2:137--141, 1960.

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F. L. Bauer and C. T. Fike. Norms and exclusion theorems. Numerische Mathematik, 2:137--141, 1960.

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F. L. Bauer and C. T. Fike, Norms and exclusion theorems, Numer. Math., 2 (1960), pp. 137{ 141.

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F. Bauer, C. Fike, Norms and exclusion theorems. Num. Math. 2, 1960.

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F. Bauer, C. Fike, Norms and exclusion theorems. Num. Math. 2, 1960.

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F. Bauer and C. Fike, Norms and exclusion theorems, Numer. Math., 2 (1960), pp. 137--141.

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F. Bauer and C. Fike, Norms and exclusion theorems, Numer. Math., 2 (1960), pp. 137--41.

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F. Bauer and C. Fike, Norms and exclusion theorems, Numer. Math., 2 (1960), pp. 137--41.

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