J. Howland. The sign matrix and the separation of matrix eigenvalues. Lin. Alg. Appl., 49:221--232, 1983.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....spectrum has been divided along the imaginary axis. By computing the QR factorization of the sign function of ffA fiI for complex ff and fi, or of (A fiI) ffI for real ff and fi, we can divide the spectrum along other lines in the complex plane, retaining real arithmetic if A is real [7] [88], 122] Using this approach, we can determine the eigenvalues lying within quite general regions of the complex plane. The matrix sign function can be computed using the Newton iteration A i 1 = A i A i ) A 0 = A, which converges globally and quadratically to sign(A) whenever sign(A) is ....

James Lucien Howland. The sign matrix and the separation of matrix eigenvalues. Linear Algebra and Appl., 49:221--232, 1983.

....considerable freedom in implementing SYISDA, in particular with respect to choosing the polynomials p i as well as the method for computing the invariant subspaces. We also mention that any other method that produces invariant subspaces, such as approximation methods for the matrix sign function [12, 13, 20, 2], could be used in the Eigenvalue Smoothing step as well. As in [26] we use predominantly the first incomplete beta function 3x 2 Gamma 2x 3 in our implementation. The experiments in [26] also confirm the numerical robustness of SYISDA. While the SYISDA algorithm can be used to compute a ....

Howland, J. L., The sign matrix and the separation of matrix eigenvalues, Lin. Alg. Appl. 49 (1983), 221-32.

.... Jacobi methods [29, 7, 10, 30] and homotopy methods [25] Parallelizable algorithms for dense nonsymmetric matrices that have been investigated include the QR algorithm [3, 32] Jacobi like methods [31] homotopy methods [24] and the matrix sign function approach to computing invariant subspaces [6, 11, 12, 19, 26, 4]. The purpose of this paper is to present preliminary research results on a new algorithm for finding all the eigenvalues and eigenvectors of a real diagonalizable matrix with real eigenvalues. Although this class of matrices is not completely general, it includes the important class of real ....

....feel that the beta function approach promises more robust, scalable performance than the matrix sign approach for the matrices we are considering. However, for the general nonsymmetric eigenvalue problem where the matrices may have complex eigenvalues, the matrix sign approach is quite promising [6, 11, 12, 19, 26, 4]. 3. Test cases. Testing of the algorithm described was performed on both nonsymmetric and symmetric matrices. Even though the code performs dense computations and does not take advantage of sparsity, we tested our algorithm on both dense and upper Hessenberg matrices, since the reduction to ....

J. L. HOWLAND, The sign matrix and the separation of matrix eigenvalues, Linear Algebra Appl., 49 (1983), pp. 221--232.

....the ways to compute the spectral projector P is to use the matrix sign function. The matrix sign function was introduced by Roberts [31] for solving the algebraic Riccati equation. However, it was soon extended to solving the spectral decomposition problem [5] More recent studies may be found in [28, 3, 23]. The matrix sign function, sign(A) of a matrix A with no eigenvalues on the imaginary axis can be defined via the Jordan canonical form of A (1) where the eigenvalues of J are in the open right halfplane D, and the eigenvalues of J are in the open left halfplane D. Then sign(A) is ....

....solve this matrix equation and derive the following Newton iteration: A j 1 = 1 2 (A j A 1 j ) for j = 0, 1, 2, with A 0 = A. 5) It can be shown that the iteration is globally and ultimately quadratically convergent with lim j## A j = sign(A) provided A has no pure imaginary eigenvalues [31, 23]. The iteration fails otherwise. In finite precision arithmetic, the iteration could converge slowly or not at all if A is close to having pure imaginary eigenvalues. There are many ways to improve the accuracy and convergence rate of this basic iteration [7, 22, 25] For example, if #A 2 ....

J. HOWLAND, The sign matrix and the separation of matrix eigenvalues, Linear Algebra Appl., 49 (1983), pp. 221--232.

....considerable freedom in implementing SYISDA, in particular with respect to choosing the polynomials p i as well as the method for computing the invariant subspaces. We also mention that any other method that produces invariant subspaces, such as approximation methods for the matrix sign function [12, 13, 20, 2], could be used in the Eigenvalue Smoothing step as well. As in [26] we use predominantly the first incomplete beta function 3x 2 Gamma 2x 3 in our implementation. The experiments in [26] also confirm the numerical robustness of SYISDA. While the SYISDA algorithm can be used to compute a ....

Howland, J. L., The sign matrix and the separation of matrix eigenvalues, Lin. Alg. Appl. 49 (1983), 221-32.

....as follows: In x2, we discuss the basic building blocks, such as the matrix sign function, which are required for higher level computations. Existing iteration and scaling schemes as well as condition estimates for the matrix sign function are also surveyed. In x3, we discuss the work of Howland [49] and Stickel [70] and show how we can use their results to count the number of eigenvalues in specified domains, or to find the corresponding invariant subspaces. This may be used as the basis either of a divide and conquer or a bisection algorithm. x4 discusses how the choice of algorithm may ....

....positive (or negative) real parts. There are a variety of ways to compute the matrix sign function. The simplest iteration scheme is Newton s method applied to (sign(A) 2 = I : A i 1 = 1 2 (A i A 01 i ) with A 0 = A: 2:1) The iteration is globally and ultimately quadratically convergent [63, 49]. Newton iteration requires the matrix inversion A 01 i which may be expensive or difficult to compute accurately. Other schemes such as Newton Schulz iteration A i 1 = 1 2 A i (3I 0A 2 i ) with A 0 = A require more flops (twice as many per iteration in this case) but use only matrix ....

[Article contains additional citation context not shown here]

J. Howland. The sign matrix and the separation of matrix eigenvalues. Lin. Alg. Appl., 49:221--232, 1983.

.... Jacobi methods [29, 7, 10, 30] and homotopy methods [25] Parallelizable algorithms for dense nonsymmetric matrices that have been investigated include the QR algorithm [3, 32] Jacobilike methods [31] homotopy methods [24] and the matrix sign function approach to computing invariant subspaces [6, 11, 12, 19, 26, 4]. The purpose of this paper is to present preliminary research results on a new algorithm for finding all the eigenvalues and eigenvectors of a real diagonalizable matrix with real eigenvalues. Although this class of matrices is not completely general, it includes the important class of real ....

....feel that the Beta function approach promises more robust, scalable performance than the matrix sign approach for the matrices we are considering. However, for the general nonsymmetric eigenvalue problem where the matrices may have complex eigenvalues, the matrix sign approach is quite promising [6, 11, 12, 19, 26, 4]. 3. Test cases. Testing of the algorithm described was performed on both nonsymmetric and symmetric matrices. Even though the code performs dense computations and does not take advantage of sparsity, we tested our algorithm on both dense and upper Hessenberg matrices, since the reduction to ....

Howland, J. L., The sign matrix and the separation of matrix eigenvalues, Lin. Alg. Appl. 49 (1983), 221-32.

....and circles. 11 The simplest scheme for computing the matrix sign function is the Newton iteration applied to (sign(A) 2 = I : A j 1 = 1 2 (A j A 01 j ) j = 0; 1; 2; with A 0 = A: 3:8) The iteration is globally and ultimately quadratically convergent with lim j 1 A j = sign(A) [47, 36]. The iteration could fail to converge if A has pure imaginary eigenvalues (or, in finite precision, if A is close to having pure imaginary eigenvalues. There are many ways to improve the accuracy and convergence rates of this basic iteration [16, 33, 38] The matrix sign function may also be ....

J. Howland. The sign matrix and the separation of matrix eigenvalues. Lin. Alg. Appl., 49:221--232, 1983.

....method to solve this matrix equation and obtain the following simple iteration: A j 1 = 1 2 (A j A Gamma1 j ) j = 0; 1; 2; with A 0 = A: 2. 5) The iteration is globally and ultimately quadratically convergent with lim j 1 A j = sign(A) provided A has no pure imaginary eigenvalues [46, 35]. The iteration fails otherwise, and in finite precision, the iteration could converge slowly or not at all if A is close to having pure imaginary eigenvalues. There are many ways to improve the accuracy and convergence rate of this basic iteration [12, 33, 37] For example, if kA 2 Gamma Ik ....

J. Howland. The sign matrix and the separation of matrix eigenvalues. Lin. Alg. Appl., 49:221--232, 1983.

....invariant subspaces, parallel algorithm. Typeset by A M S T E X 2 HUSS LEDERMAN et al. dense nonsymmetric matrices that have been investigated include the QR algorithm [3, 32] Jacobilike methods [31] homotopy methods [24] and the matrix sign function approach to computing invariant subspaces [6, 11, 12, 19, 26, 4]. The purpose of this paper is to present preliminary research results on a new algorithm for finding all the eigenvalues and eigenvectors of a real diagonalizable matrix with real eigenvalues. Although this class of matrices is not completely general, it includes the important class of real ....

....feel that the Beta function approach promises more robust, scalable performance than the matrix sign approach for the matrices we are considering. However, for the general nonsymmetric eigenvalue problem where the matrices may have complex eigenvalues, the matrix sign approach is quite promising [6, 11, 12, 19, 26, 4]. 3. Test cases. Testing of the algorithm described was performed on both nonsymmetric and symmetric matrices. Even though the code performs dense computations and does not take advantage of sparsity, we tested our algorithm on both dense and upper Hessenberg matrices, since the reduction to upper ....

Howland, J. L., The sign matrix and the separation of matrix eigenvalues, Lin. Alg. Appl. 49 (1983), 221-32.

.... matrix equation and derive the following Newton iteration: A j 1 = 1 2 (A j A Gamma1 j ) for j = 0; 1; 2; with A 0 = A: 5) It can be shown that the iteration is globally and ultimately quadratically convergent with lim j 1 A j = sign(A) provided A has no pure imaginary eigenvalues [31, 23]. The iteration fails otherwise. In finite precision arithmetic, the iteration could converge slowly or not at all if A is close to having pure imaginary eigenvalues. There are many ways to improve the accuracy and convergence rate of this basic iteration [7, 22, 25] For example, if kA 2 ....

J. Howland. The sign matrix and the separation of matrix eigenvalues. Lin. Alg. Appl., 49:221--232, 1983.

....subspace of B so we get Q T f(B)Q = B 11 B 12 0 0 # and Q T BQ = B 11 B 12 0 B 22 # The problem thus becomes finding functions f(B) that are easy to evaluate and have large null spaces, or which map selected eigenvalues of B to zero. One such function f is the sign function [13, 108, 120, 135, 168, 190] which maps points with positive real part to 1 and those with negative real part to Gamma1; adding 1 to this function then maps eigenvalues in the right half plane to 2 and in the left plane to 0, as desired. The only operations we can easily perform on (dense) matrices are multiplication and ....

....plane to 0, as desired. The only operations we can easily perform on (dense) matrices are multiplication and inversion, so in practice f must be a rational function. A globally, asymptotically quadratically convergent iteration to compute the sign function of B is B i 1 = B i B Gamma1 i ) 2 [108, 168, 190]; this is simply Newton s method applied to B 2 = I , and can also be seen to equivalent to repeated squaring (the power method) of the Cayley transform of B. It converges more slowly as eigenvalues approach the imaginary axis, and is in fact nonconvergent if there are imaginary eigenvalues, as ....

[Article contains additional citation context not shown here]

J. Howland. The sign matrix and the separation of matrix eigenvalues. Lin. Alg. Appl., 49:221--232, 1983.

....In addition, in Section 2, we describe an acceleration technique we have recently developed that substantially reduces the number of iterations required in the Eigenvalue Smoothing step. We note that other functional iterations, such as approximation methods for the matrix sign function [14, 15, 21, 2], can be used in the Eigenvalue Smoothing step as well. The two key primitives of the algorithm are matrixmatrix multiplication, which accounts for the majority of the computation, and computation of the range and null space of a matrix having eigenvalues clustered around 0 and 1. The sequential ....

Howland, J. L., The sign matrix and the separation of matrix eigenvalues, Lin. Alg. Appl. 49 (1983), 221-32.

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J. Howland. The sign matrix and the separation of matrix eigenvalues. Lin. Alg. Appl., 49:221--232, 1983.

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J. L. Howland, The Sign Matrix and the Separation of Matrix Eigenvalues, Lin. Alg. Appl. 49 (1983), 221-232.

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#35# J. Howland. The sign matrix and the separation of matrix eigenvalues. Lin. Alg. Appl.,

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