A. VAN der SLUIS. Condition numbers and equilibration of matrices. Numer. Math., 14:14--23, 1969. Home/Search Document Not in Database Summary Related Articles Check |
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Relative Perturbation Bound for Invariant Subspaces of.. - Truhar, Slapnicar (1998) (4 citations) (Correct)
....with j = k jAj k k b A k: 11) Since A ii = 1, we have k k ( A) nk k, where (A) j kAk kA denotes the spectral condition number. Also, k jAj k n [14] The diagonal grading matrix D from (10) which is also called the scaling matrix, is almost optimal in the sense that [17] ( b A) n min D ( D H D) n( H ) n(H) where the minimum is taken over all non singular diagonal matrices. Similarly, for more general perturbations of the type jffiH ij j D ii D jj ; 12) 4) holds with j = nk b A k n( b A) 13) Perturbations of the form (9) typically occur ....
A. van der Sluis, Condition numbers and equilibration of matrices, Numer. Math., 14:14--23 (1969).
Combined MEG and EEG Source Imaging by Minimization.. - Baillet, Garnero.. (Correct)
....linearly operating on its rows or its columns. Actually, it can be shown that equalizing the twonorm of the rows of the matrix produces a new linear operator which condition number is no more than a factor away from the smallest condition number that can be achieved with a linear transformation of [30]. A similar theorem stands also for column normalization. It is interesting to note that, though matrix equilibration has strong algebraic roots, the first motivations to do so in the MEEG inverse problem were based on physical considerations. The scaling of MEG and EEG data (like the ....
A. van der Sluis, "Condition numbers and equilibration of matrices," Numer. Math., vol. 14, pp. 14--23, 1969.
Iterative Refinement for Linear Systems and LAPACK - Higham (1996) (4 citations) (Correct)
....LU factorization has an analogue for Cholesky factorization that is at least as strong, yet our analysis does not give a one step is enough result for Cholesky factorization. A scaling argument can be used to replace A in the bounds by H = D AD, where h ii = 1, and a result of Van der Sluis [16] implies that 2 (H) nminf 2 (FAF ) F diagonal g; however, 2 (H) can still be large and the scaling changes the term (jbj jAjjxj) Therefore we pose the open problem: prove that one step is enough for Cholesky factorization, or find a numerical counterexample (with cond(A )f(u; kY k1 ....
A. van der Sluis. Condition numbers and equilibration of matrices. Numer. Math., 14:14--23, 1969.
Numerically Stable Generation of Correlation Matrices and.. - Davies, Higham (1999) (5 citations) (Correct)
....[24, p. 24] They also play an important role in numerical analysis, because of an approximate optimality property. Let A 2 R be symmetric positive definite and write A = DHD; D = diag(A) 1=2 ; 1.1) so that H has unit diagonal and hence is a correlation matrix. A result of van der Sluis [28] states that 2 (H) n min F diagonal 2 (FAF ) 1.2) where 2 (A) kAk 2 kA k 2 . Thus H comes within a factor n of minimizing the 2 norm condition number over all two sided diagonal scalings of A. This property accounts for the appearance of correlation matrices in several contexts in ....
A. van der Sluis. Condition numbers and equilibration of matrices. Numer. Math., 14:14--23, 1969.
Generating Test Matrices for the One- and Two-Sided Jacobi.. - Davies, Higham (1999) (Correct)
.... are normwise backward stable we would expect forward error bounds involving 2 (B) and 2 (A) The presence of the scaled matrices is an improvement in view of the well known results 2 (A) n minf 2 ( e DA e D) e D diagonal g; 2 (B C ) n minf 2 (B e D) e D diagonal g of van der Sluis [21], and similarly for BR . In testing an implementation of a Jacobi method and in investigating the sharpness of the forward error bounds (1.1) and (1.2) it is desirable to be able to generate matrices having the scalings of A, BR and B C and specified condition numbers. One approach, used by ....
A. van der Sluis. Condition numbers and equilibration of matrices. Numer. Math., 14:14--23, 1969.
Notes on Accuracy and Stability of Algorithms in Numerical Linear .. - Higham (1998) (Correct)
....5.5 shows that Jacobi transformations from the right introduce only small relative errors in the singular values in floating point arithmetic provided that X is well conditioned. It is natural to ask what is the minimum value of 2 (X) over all nonsingular diagonal D. A result of van der Sluis [49] shows that 2 (DRA) n min 2 (DA) Table 5.2: Relative errors in computed singular values. Relative Error True singular value Algorithm 5.1 Matlab s SVD 1.9e 00 2.3e 16 1.2e 16 1.4e 01 1.9e 16 3.8e 16 1.0e 02 6.6e 16 0.0e 00 9.2e 04 2.3e 16 1.1e 15 8.5e 05 8.0e 16 6.4e 16 7.0e 06 ....
A. van der Sluis. Condition numbers and equilibration of matrices. Numer. Math., 14:14--23, 1969.
Perturbation Analyses for the Cholesky Downdating Problem - Chang, Paige (1996) (1 citation) (Correct)
....like to choose D such that c G (R; X;D) and c T (R; X;D) are good approximations to c G (R; X) and c T (R; X) respectively. We see from (3.5) 3.6) and (3.17) that we want to find D such that p 1 i 2 D 2 (D Gamma1 U) approximates its infimum. By a well known result of van der Sluis [21], 2 (D Gamma1 U) will be nearly minimal when the rows of D Gamma1 U are equilibrated. But this could lead to a large i D . So a reasonable compromise is to choose D to equilibrate U as far as possible while keeping i D 1. Specifically, take i 1 = q P n j=1 u 2 1j , i i = q P n j=i u 2 ....
A. van der Sluis, Condition numbers and equilibration of matrices, Numerische Mathematik, 14 (1969), pp. 14--23. 15
On The Sensitivity Of The LU Factorization - Chang, Paige (1997) (Correct)
....L (A) and U (A) in (3.11) directly by the usual approach except when A has some special structure (for example, A is tridiagonal, see Chang and Paige [3] Fortunately we can estimate the condition estimates 0 L (A) and 0 U (A) reasonably efficiently. By a well known result of van der Sluis [10], we have 2 (LD Gamma1 L ) p n inf D2Dn 2 (LD Gamma1 ) where D L j diag(kL( j)k 2 ) This is to say 2 (LD Gamma1 ) will be near its infimum when each column of LD Gamma1 has unit 2 norm. Therefore we have 0 L (A) 0 L (A; DL ) p n 0 L (A) 3.42) So in practice we ....
A. van der Sluis, Condition numbers and equilibration of matrices, Numer. Math., 14 (1969), pp. 14--23.
On the Sensitivity of the SR Decomposition - Chang (1992) (Correct)
....results. The optimization problems (25) and (34) are complicated. In practice we would like to choose D such that R (A; D) is a good approximation to the infimum R (A) and choose another D such that S (A; D) is a good approximation to the infimum S (A) By a well known result of van der Sluis [9], 2 (D Gamma1 R) will be nearly minimal when the rows of D Gamma1 R are equilibrated. But this could lead to a large i D in (22) So a reasonable compromise is to choose D to equilibrate R as far as possible in some sense while keeping i D = 1. There are four obvious possibilities for D: ....
A. van der Sluis, Condition numbers and equilibration of matrices, Numer. Math., 14:14--23 (1969).
Stagnation Of Gmres - Zavorin, O'Leary, Elman (Correct)
....from such extreme matrices and show in particular that two di erent, but equally conditioned, extreme eigenvector distributions have essentially the same range sets S V . Since columns of V are assumed to have Euclidean norm 1, the condition number of V is within a factor of p n of optimal [10], and the singular values satisfy 2 1 2 n = n: 4.1) For extreme matrices V , the stagnation system Y Wy = u has a particularly simple form. Let the singular values of W be 2 j = j = 1; n 1, and 2 n = where is nonnegative and is real. By (4.1) n ....
A. van der Sluis, Condition numbers and equilibration of matrices, Numer. Math., 14 (1969), pp. 14-23.
Perturbation Analyses for the Cholesky Factorization with Backward .. - Chang (1996) (Correct)
.... z n ) 2 R n(n 1) 2 Theta n(n 1) 2 : Thus for any symmetric matrix X we have kduvec(X)k 2 = kXkF . Let D n be the set of all n Theta n real positive definite diagonal matrices, and let e = 1; 1; 1) T 2 R n . Now we introduce the following results due to van der Sluis [12], which will be used later. Theorem 2.1. 12] Let S; T 2 R n Thetan and let S be nonsingular, and define D r = diag(kS(i; k 2 ) Then kTD r k 2 kD Gamma1 r Sk 2 p n inf D2Dn kTDk 2 kD Gamma1 Sk 2 : 2.4) Thus if T = S Gamma1 , then 2 (D Gamma1 r S) p n inf D2Dn 2 (D ....
....2 : Thus for any symmetric matrix X we have kduvec(X)k 2 = kXkF . Let D n be the set of all n Theta n real positive definite diagonal matrices, and let e = 1; 1; 1) T 2 R n . Now we introduce the following results due to van der Sluis [12] which will be used later. Theorem 2.1. [12] Let S; T 2 R n Thetan and let S be nonsingular, and define D r = diag(kS(i; k 2 ) Then kTD r k 2 kD Gamma1 r Sk 2 p n inf D2Dn kTDk 2 kD Gamma1 Sk 2 : 2.4) Thus if T = S Gamma1 , then 2 (D Gamma1 r S) p n inf D2Dn 2 (D Gamma1 S) 2.5) In particular, if S is ....
A. van der Sluis. Condition numbers and equilibration of matrices. Numerische Mathematik , 14:14--23, 1969. 18
Perturbation Analyses for the QR Factorization - Chang, Paige, Stewart (1997) (2 citations) (Correct)
....2 ) condition estimator for R in the QR factorization (i.e. an estimator for R (A) is to choose such a D, use a standard condition estimator (see for example [5] to estimate 2 (D Gamma1 R) and take (R; D) in (5.10) as the appropriate estimate. By a well known result of van der Sluis [16], 2 (D Gamma1 R) will be nearly minimal when the rows of D Gamma1 R are equilibrated. But this could lead to a large i D in (5.10) There are three obvious possibilities for D. The first one is choosing D to equilibrate R precisely. Specifically, take ffi i = q P n j=i r 2 ij for i = ....
A. van der Sluis, Condition numbers and equilibration of matrices, Numerische Mathematik, 14 (1969), pp. 14--23.
A Perturbation Analysis for R in the QR Factorization - Chang, Paige (1995) (Correct)
....n(n 1) 4 n 6n Gamma 1) 18. Thus we have the following bound R (A; D) n 1) q n(n 1) 4 n 6n Gamma 1) 6: The practical outcome of this analysis is that we now have an O(n 2 ) condition estimator for the R factor of the QR factorization. By a well known result of van der Sluis [12], 2 ( R) will be nearly optimal when the rows of R are equilibrated. Thus the procedure is to choose D in R = D R so that the rows of R are equilibrated, and use a condition estimator, see for example [3] to estimate 2 ( R) in 1 R (A) R (A) R (A; D) p 2 minfr(T ) c(T )g 2 ....
A. van der Sluis, Condition numbers and equilibration of matrices, Numerische Mathematik, 14 (1969) 14--23.
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A. VAN der SLUIS. Condition numbers and equilibration of matrices. Numer. Math., 14:14--23, 1969.
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A. VAN der SLUIS. Condition numbers and equilibration of matrices. Numer. Math., 14:14-23, 1969.
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A. van der Sluis. Condition numbers and equilibration of matrices. Numer. Math., 14:14--23, 1969.
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A. Van Der Sluis. Condition numbers and equilibration of matrices. Num. Math., 14:14--23, 1969.
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van der Sluis, A., Condition numbers and equilibration of matrices, Num. Math., 14 (1969), pp. 14--23.
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A. van der Sluis. Condition numbers and equilibration of matrices. Numer. Math., 14:14--23, 1969.
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