Z. Bai and J. Demmel. On swapping diagonal blocks in real Schur form. Lin. Alg. Appl., 186:73--96, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of T (keeping complex conjugate pairs adjacent) there is a real Schur decomposition which achieves that ordering along the diagonal of T . Software for e#ciently updating one real Schur decomposition to another by unitary similarity transformation is widely available. See, for example, [4, 39] or [1, DTREXC] By reordering the eigenvalues along the diagonal of T , a deflation procedure may augment the set of deflatable eigenvalues. Suppose, for example, that the trailing m eigenvalues along the diagonal of T are deflatable, but # = t k m,k m is a simple eigenvalue that cannot be ....

Z. Bai and J. Demmel, On swapping diagonal blocks in real Schur form, Linear Algebra Appl., 186 (1993), pp. 73--95. Also available online as LAPACK Working Note 54 from http://www.netlib.org/lapack/lawns/lawn54.ps and http://www.netlib.org/lapack/ lawnspdf/lawn54.pdf.

....eigenvalues of interest are in the top left hand corner. This is a computation that arises in several applications, including that of computing an invariant subspace corresponding to a given group of eigenvalues. LAPACK uses an improved, guaranteed stable version of an earlier reordering algorithm [8]. For parallel solution of the nonsymmetric eigenvalue problem there is no clear method of choice. The QR algorithm is a fine grained algorithm that has proven difficult to parallelize, though progress has been made in [79] Various interesting alternative approaches have been proposed, which we ....

Zhaojun Bai and James W. Demmel. On swapping diagonal blocks in real Schur form. Linear Algebra and Appl., 186:73--95, 1993.

....the complex case) we refer to Section 7.6. 2 of Golub and Van Loan [8] For a Matlab code based on this algorithm to sort a complex Schur form, see [6] A direct (as opposed to iterative) ordering algorithm for real Schur forms has been described and thoroughly analyzed by Bai and Demmel in [2]. Their procedure will be briefly reviewed in Section 2 of this paper, and implemented in a slightly different form in Matlab in Section 2.3. Numerical experiments are provided that illustrate the success of the code when swapping two two by two blocks having small separation. As a more practical ....

....related packages. We will distinguish between the two essentially different approaches that were described in more detail above. ffl A complete sorted real Schur form is needed. Given an arbitrary real Schur form, 4 we perform the re ordering of the diagonal blocks as given by Bai and Demmel in [2] in a slightly different form. This will be the topic of Section 2. ffl A partial sorted real Schur form of a very large matrix is needed. For this we suppose we have a black box large eigenvalueproblem solver at our disposal. Section 3 contains the details. As mentioned before, Matlab includes ....

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Z. Bai and J.W. Demmel (1993). On swapping diagonal blocks in real Schur form, Lineair Algebra Appl. 186:73--95.

....and q, respectively, how do we transform it to the form # C 11 C 12 0 C 22 # Nm k 2 , 3. 2) where C 11 N k 1 and C 22 Nm 1 have characteristic polynomials q and p, respectively This is a block swapping problem similar to that of swapping two blocks of a matrix in real Schur form [1] or two blocks of a matrix pencil in generalized real Schur form [9] 10] These swaps are accomplished by applying a transformation determined by solving a Sylvester or generalized Sylvester equation, respectively. The method of [9] 10] cannot be applied directly to the pencil (3.1) because ....

....these numbers to zero in order to continue the computation. We can do so without compromising backward stability only if the numbers are tiny, i.e. on the level of the machine precision relative to #A#. It would be nice if we could prove that these numbers are always tiny, but existing results [1] suggest that this may not be possible. The accuracy of the transformation V depends on how accurately the Sylvester like equation (3.10) is solved. An accurate solution can be guaranteed if the Sylvester operator X # B 11 X N 11 XF is well conditioned. We know that this operator is ....

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Z. Bai and J. W. Demmel, On swapping diagonal blocks in real Schur form, Linear Algebra Appl., 186 (1993), pp. 73--95.

....stronger inequality is true) The cost of computing S by (1.12) is not much greater than by (1.7) since we can take C 1 to be the Cholesky factor of (I P P ) and similarly for C 2 . From a numerical point of view it would be better to compute S from a QR factorization of I P . See [1] for a discussion of this point. 2. Matrix Sign Function. In this section we give a method of computing the Fr echet derivative of the matrix sign function, and compare the cost of this method with the cost of computing it by an iterative method (Newton s method) and the cost of estimating it by ....

....the condition number for sgn(T ) which is the same as that for sgn(M) we must find an orthogonal V such that W = V TV is upper triangular as well as being of the form required by Theorem 2.1. Using for example the algorithm for ordering the eigenvalues of a triangular matrix given in [1] this can be done in at most 2n 3 flops. Finally, solving AX Gamma XB = 2C takes 11n 3 =16 flops. matrix function condition numbers 9 Let us compare the cost of our method with that of methods based on the scaled Newton s iteration: S k 1 = fl k S k (fl k S k ) Gamma1 ] 2; S 0 = M: ....

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Z. Bai and J. Demmel. On swapping diagonal blocks in the real Schur form. Linear Algebra Appl., 186:73--96, 1993.

....principle, once one has a triangular matrix with the desired singular values and eigenvalues (in any order on the diagonal) one can then use an eigenvalue reordering algorithm to put the eigenvalues in any desired order. However, these algorithms are rather complicated to implement accurately see [3] for the details. Our construction is done by multiplication by unitary diagonal matrices, permutation matrices and rotation matrices. It is numerically stable. This allows one to generate matrices with prescribed singular values and eigenvalues which may be useful in testing numerical linear ....

Z. Bai and J. Demmel, On swapping diagonal blocks in the real Schur form, Linear Algebra Appl. 186 (1993), 73-96.

....the purging problem can be solved by moving the unwanted Ritz values into the southeast corner of the Rayleigh quotient and truncating the decomposition. The process of using unitary similarities to move eigenvalues around in a Schur form has been well studied. The current front running algorithm [2], which has been implemented in the lapack routine xTREXC) is quite reliable far more so than implicit QR. Consequently, our deflation algorithm consists of little more than moving the unwanted Ritz values, which are visible on the diagonals of Sm , to the southeast corner of the Rayleigh ....

Z. Bai and J. W. Demmel. On swapping diagonal blocks in real Schur form. Linear Algebra and Its Applications, 186:73--95, 1993.

....decomposition. Thus the purging problem can be solved by moving the unwanted Ritz values into the southeast corner of the Rayleigh quotient and truncating the decomposition. The process of using unitary similarities to move eigenvalues around in a Schur form has been well studied (see [2] for references and the current front running algorithm, which has been implemented by the lapack routine xTREXC) Consequently, our deflation algorithm consists of little more than moving the unwanted Ritz values, which are visible on the diagonals of Sm , to the southeast corner of the Rayleigh ....

Z. Bai and J. W. Demmel. On swapping diagonal blocks in real Schur form. Linear Algebra and Its Applications, 186:73--95, 1993.

....and unstable eigenvalues. Algorithms that compute the real Schur decomposition of a matrix typically do not partition the diagonal blocks of W according to stability. Instead, given an arbitrary real Schur decomposition M = V W V 0 , one can use the approaches described in either Bai and Demmel (1993) or Stewart (1976) to construct a sequence of orthogonal transformations that reorder the diagonal blocks of W ; while updating V so that M = V W V 0 holds at every step. In summary, the steps for implementing a Schur algorithm are 8 There is also a complex Schur decomposition of a real ....

Bai, Z. and J.W. Demmel (1993). `On Swapping Diagonal Blocks in Real Schur Form'. Linear Algebra and its Applications, Vol. 186, pp. 73--95.

....decomposition. Thus the purging problem can be solved by moving the unwanted Ritz values into the southeast corner of the Rayleigh quotient and truncating the decomposition. The process of using unitary similarities to move eigenvalues around in a Schur form has been well studied (see [2] for references and the current front running algorithm, Draft April 24, 2000 Arnoldi Schur GWS which has been implemented by the lapack routine xTREXC) Consequently, our deflation algorithm consists of little more than moving the unwanted Ritz values, which are visible on the diagonals of Sm ....

Z. Bai and J. W. Demmel. On swapping diagonal blocks in real Schur form. Linear Algebra and Its Applications, 186:73--95, 1993.

....into stable and unstable eigenvalues. Algorithms that compute the real Schur decomposition of a matrix typically do not partition the diagonal blocks of W according to stability. Instead, given an arbitrary real Schur decomposition M = V W V 0 , one can use the approaches described in either Bai and Demmel (1993) or Stewart (1976) to construct a sequence of orthogonal transformations that reorder the diagonal blocks of W ; while updating V so that M = V W V 0 holds at every step. In summary, the steps for implementing a Schur algorithm are 8 There is also a complex Schur decomposition of a real ....

Bai, Z. and J.W. Demmel (1993). `On Swapping Diagonal Blocks in Real Schur Form'. Linear Algebra and its Applications, Vol. 186, pp. 73--95.

....Sohraby, On Computational Aspects of the Invariant Subspace Approach 15 iteration, and reordering the eigenvalues appropriately using orthogonal transformations. For a detailed treatment of the overall procedure and references, we refer the reader to [22, pp. 361 386] For the reordering, see also [8] and the references therein. Compared with the numerical algorithms for finding the eigenvectors, first, the reduction to RSF form is an intermediate step in computing eigenvectors anyway and by definition, it will be faster, and second, this algorithm does not suffer as severely from the ....

Z. Bai and J. W. Demmel. On swapping diagonal blocks in real Schur form. Lin. Alg. Appl., 186:73--95, 1993.

....this equality obviates the need to solve linear systems with R necessary for the similarity transformation. For the error analysis, that follows R Gamma1 is used in a formal sense. Let Z be the computed solution to the Sylvester set of equations. In a similar analysis, Bai and Demmel [2] assume that the QR factorization of S is performed exactly and we do also. The major source of error is that arising from computing Z. DEFLATION TECHNIQUES FOR IMPLICIT RESTARTING 19 Suppose that Q R = Z I j S. Write Z = Z E where E is the error in Z . If QR = S and kR ....

....basis for them is wanted, we must either increase the number of columns of Q used or somehow place them at the top of T . Algorithms for re ordering a Schur form accomplish this task by using orthogonal matrices that move the wanted eigenvalues to the top of T . The recent work of Bai and Demmel [2] attempts to correct the occasional numerical problems encountered by Stewart s algorithm [35] EXCHNG. Their work was motivated by that of Ruhe [29] and that of Dongarra, Hammarling, and Wilkinson [11] Both algorithms swap consecutive 1 Theta 1 and 2 Theta 2 blocks of a quasi triangular matrix ....

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Z. Bai and J. W. Demmel, On swapping diagonal blocks in real Schur form, Linear Algebra and Its Applications, 186 (1993), pp. 73--95.

....is to swap two neighboring 1 Theta 1 or 2 Theta 2 diagonal blocks by an orthogonal transformation. Swapping two 1 Theta 1 blocks or swapping 1 Theta 1 and 2 Theta 2 blocks are well understood [3] Swapping two 2 Theta 2 blocks poses some numerical difficulties. Recently, Bai and Demmel [1] have proposed an algorithm for swapping two 2 Theta 2 blocks which is for all practical purposes backward stable. The algorithm requires the solution of a Sylvester equation associated with the defining 4 Theta 4 matrix. If the algorithm introduces unacceptable rounding error in the (2,1) block ....

Z. Bai and J.W. Demmel, "On swapping diagonal blocks in real Schur form", Lin. Alg. Appl., 186:73-9, 1993.

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Z. Bai and J. Demmel. On swapping diagonal blocks in real Schur form. Lin. Alg. Appl., 186:73--96, 1993.

....such that T 1;1 is closest to . Only if m mmax the sorting of the Schur form has to be be such that all of the mmax leading diagonal elements of T are closest to . For ease of presentation we sorted all diagonal elements here. For a template of an algorithm for the sorting of a Schur form, see [39, 40, 3] and [11, Chap. 6B] S is the m by m matrix with columns s j . 5) The stopping criterion is to accept a Schur vector approximation as soon as the norm of the residual (for the normalized Schur vector approximation) is below ffl. This means that we accept inaccuracies in the order of ffl in the ....

....mmax the sorting of the Schur form has to be be such that all of the mmax leading diagonal elements of T and T represent the mmax harmonic Ritz values closest to 0. For ease of presentation we sorted all diagonal elements here. For an algorithm of the sorting of a generalized Schur form, see [39, 40, 3] and [11, Chap. 6B] The value of needs some attention. We have chosen to compute the Rayleigh quotient # (instead of the harmonic Ritz value) corresponding to the harmonic Ritz vector u (see [33] The Rayleigh quotient follows from the requirement that (A Gamma I)u Gamma #u u instead of ....

Z. J. Bai and J. W. Demmel, On swapping diagonal blocks in real Schur form, Linear Algebra Appl., 186 (1993), pp. 73--95.

No context found.

Z. Bai and J. Demmel, On swapping diagonal blocks in real Schur form, Lin. Alg. Appl. 186 (1993), 73--95.

....that T 1;1 is closest to . Only if m mmax the sorting of the Schur form has to be be such that all of the mmax leading diagonal elements of T are closest to . For ease of presentation we sorted all diagonal elements here. For a template of an algorithm for the sorting of a Schur form, see [39, 40, 3] and [11, Chap. 6B] S is the m by m matrix with columns s j . 5) The stopping criterion is to accept a Schur vector approximation as soon as the norm of the residual (for the normalized Schur vector approximation) is below ffl. This means that we accept inaccuracies in the order of ffl in the ....

....mmax the sorting of the Schur form has to be be such that all of the mmax leading diagonal elements of T A and T represent the mmax harmonic Ritz values closest to 0. For ease of presentation we sorted all diagonal elements here. For an algorithm of the sorting of a generalized Schur form, see [39, 40, 3] and [11, Chap. 6B] The value of needs some attention. We have chosen to compute the Rayleigh quotient # (instead of the harmonic Ritz value) corresponding to the harmonic Ritz vector u (see [33] The Rayleigh quotient follows from the requirement that (A Gamma I)u Gamma #u u instead of ....

Z. J. Bai and J. W. Demmel, On swapping diagonal blocks in real Schur form, Linear Algebra Appl., 186 (1993), pp. 73--95.

....that T 1;1 is closest to . Only if m mmax the sorting of the Schur form has to be be such that all of the mmax leading diagonal elements of T are closest to . For ease of presentation we sorted all diagonal elements here. For a template of an algorithm for the sorting of a Schur form, see [39, 40, 3] and [11, Chap. 6B] S is the m by m matrix with columns s j . 5) The stopping criterion is to accept a Schur vector approximation as soon as the norm of the residual (for the normalized Schur vector approximation) is below ffl. This means that we accept inaccuracies in the order of ffl in the ....

....mmax the sorting of the Schur form has to be be such that all of the mmax leading diagonal elements of T A and T represent the mmax harmonic Ritz values closest to 0. For ease of presentation we sorted all diagonal elements here. For an algorithm of the sorting of a generalized Schur form, see [39, 40, 3] and [11, Chap. 6B] The value of needs some attention. We have chosen to compute the Rayleigh quotient # (instead of the harmonic Ritz value) corresponding to the harmonic Ritz vector u (see [33] The Rayleigh quotient follows from the requirement that (A Gamma I)u Gamma #u u instead of ....

Z. J. Bai and J. W. Demmel, On swapping diagonal blocks in real Schur form, Linear Algebra Appl., 186 (1993), pp. 73--95.

....has to compute them all. If one only wants an invariant subspace corresponding to a specified set of eigenvalues, one has to reduce the matrix completely to Schur form, and then swap the desired eigenvalues along the diagonal to group them together in order to form the desired invariant subspace [1, 4]. In this paper, we propose a collection of tools from which hybrid eigenvalue algorithms may be constructed. The resulting algorithms are easy to parallelize, and need only work on the part of the spectrum of interest to the user. The new tools use the matrix sign function to spectrally divide ....

Z. Bai and J. Demmel. On swapping diagonal block in real Schur form. to appear in Lin. Alg. Appl.

....and conquer problem is to use the QR algorithm (or the QZ algorithm in the generalized case) to reduce the matrix (or pencil) to Schur form, and then to reorder the eigenvalues on the diagonal of the Schur form to put the eigenvalues in D in the upper left corner, as shown in (1.1) and (1. 2) see [7] and the references therein) The approach is numerically stable, although in some extremely ill conditioned cases, the swapping process may fail 1 . However the approach seems be too fine grain to parallelize successfully [22] There are two highly parallel algorithms for the spectral divide ....

Z. Bai and J. Demmel. On swapping diagonal blocks in real Schur form. Lin. Alg. Appl., 186:73--96, 1993.

....by EISPACK subroutine HQR [9] or LAPACK subroutine SHSEQR in that the eigenvalues of the final quasitriangular matrix are ordered. It is essentially the same as the program HQR3 [13] However, instead of using QR iteration to do the diagonal swapping in HQR3, SLAQR3 uses a direct swapping method [2]. 6. Numerical Experiments The program described above has been tested on a number of problems. In this section, we give three examples that illustrate the flexibility of the method and its ability to deal with equimodular or clustered eigenvalues. All the experiments have been run on a SUN Sparc ....

Z. Bai and J. Demmel, On swapping diagonal blocks in real Schur form, submitted to Lin. Alg. Appl. 1992

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