E. G. Kogbetliantz. Solution of linear equations by diagonalization of coef- cient matrix. Quart. Appl. Math., 13:123-132, 1955. 29  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in Fortran, since arrays are stored columnwise) We can apply two sided Jacobi transformations, in which in each 2 Theta 2 subproblem a pair of rotations is applied from the left and the right to diagonalize the submatrix; this is known as the Kogbetliantz algorithm [8, Section 2.6. 6] [37]. The one sided Jacobi algorithm can also be used to compute the eigensystem of a symmetric positive definite matrix A [19] 39] 50] The idea is to compute a Cholesky factorization A = R R and then apply Algorithm 5.1 to . The SVD R = U Sigma V yields the eigendecomposition A = U ....

E. G. Kogbetliantz. Solution of linear equations by diagonalization of coefficients matrix. Quart. Appl. Math., 13(2):123--132, 1955.

....easy computation of inverse tangents and vector rotations, have proven extremely useful in this context [14, 29, 73] Brent, Luk and VanLoan [6] proposed an expandable array of simple processors for 2 Theta 2 matrices, to compute the SVD of a larger matrix. The array uses the SVDJacobi method [32,49] combined with a parallel ordering [5] scheme to exploit the parallelism inherent in the Jacobi method. Cavallaro and Luk [14] have demonstrated the use of CORDIC in a hardware and performance efficientarchitecture for the 2 Theta 2 processor. Most parallel algorithms and architectures ....

....Luk [5] and later Brent, Luk and VanLoan [6] in the development of systolic arrays where a sweep of the parallel ordering could be executed in O(n) time. Jacobi methods can be readily extended to the SVD of arbitrary matrices. Atwosided Jacobi SVD procedure was first suggested by Kogbetliantz [49]. The theoretical framework for the two sided Jacobi SVD method for square matrices was provided by Forsythe and Henrici [32] The proof and conditions for convergence for the two sided cyclic Jacobi SVD are given in [32] and the asymptotic convergence rate was shown to be quadratic by Wilkinson ....

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E. G. Kogbetliantz. Solution of Linear Equations by Diagonalization of Coefficients Matrix. Quarterly of Applied Mathematics, 14(2):123--132, 1955.

....synchronized multiprocessing capability [76] The primary reasons for the use of systolic arrays in special purpose processing are simple and regular design, concurrency and communication, with balanced computation and I O [74] 1. 3 Jacobi like Algorithms for SVD Jacobi like algorithms for SVD [42, 69], whichwere once found undesirable for computation on the traditional serial architectures, have been revisited for their amenability 5 to parallel implementation. The original iterative method was proposed by Jacobi in 1846 to compute the eigenvalue decomposition of real symmetric matrices [65] ....

....(by norm) off diagonal element. This was followed byatwo sided orthogonal transformation to annihilate the off diagonal elements of the chosen pivot matrix. Jacobi s method can be readily extended to the SVD of arbitrary matrices. A Jacobi like SVD procedure was first suggested by Kogbetliantz [69]. The theoretical framework for the Jacobi SVD method was provided byForsythe and Henrici [42] On serial computers, the search for the 2 Theta 2pivot matrix was replaced by an ordered choice of the pivot matrices (cyclic byrow column) Forsythe and Henrici gavea proof of convergence for the ....

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E. G. Kogbetliantz. Solution of Linear Equations by Diagonalization of Coefficients Matrix. Quarterly of Applied Mathematics, 14(2):123--132, 1955.

....r En and kAk = 1 2 : r where i = ii for i = 1; 2; r. Every decomposition of the form (13) that satis es the above conditions is called a max plus algebraic SVD of A. Proof : This proof is similar to that of Theorem 5. 5, but now we use Kogbetliantz s SVD algorithm [7] to construct a path of SVDs of the matrix valued function A. Afterwards we apply the reverse mapping R to obtain a max plus algebraic SVD of A. See [3] for the details of this proof. 2 Note that for the max plus algebraic SVD we also have an extra condition (kAk = 1 ) that does not ....

E.G. Kogbetliantz, \Solution of linear equations by diagonalization of coecients matrix," Quarterly of Applied Mathematics, vol. 13, no. 2, pp. 123-132, 1955.

....experiments. In the appendix, we include Demmel and Kahan s 2 Theta 2 triangular SVD code, which has not been published in its entirety before, and plays an important role in our algorithm. 2 Paige s GSVD Algorithm To describe Paige s algorithm, we first review the Kogbetliantz algorithm [23] for computing the SVD of an upper triangular matrix A. Then we describe Paige s algorithm for computing the GSVD of A and B with B nonsingular. Finally, we discuss how to generalize the idea to the case where B is ill conditioned or singular. 2.1 Kogbetliantz algorithm for the SVD of a triangular ....

....triangular matrix A. Then we describe Paige s algorithm for computing the GSVD of A and B with B nonsingular. Finally, we discuss how to generalize the idea to the case where B is ill conditioned or singular. 2. 1 Kogbetliantz algorithm for the SVD of a triangular matrix The Kogbetliantz algorithm [23] is a kind of Jacobi scheme. Assume that the kth transformation of the algorithm operates on the rows and columns i and j of A, let A ij be the 2 Theta 2 submatrix subtended by rows and columns i and j of A. Let the rotation matrices U k = rot(c u ; s u ) and V k = rot(c v ; s v ) be chosen 3 ....

E. G. Kogbetliantz, Solution of linear equations by diagonalization of coefficients matrix, Quart. Appl. Math. 13:123--132(1955).

....Furthermore, it is obvious that every QR decomposition in R max is also a QR decomposition in S max . 5.3. The max algebraic singular value decomposition. Now we give an alternative proof for the existence theorem of the max algebraic SVD. In this proof we shall use Kogbetliantz s SVD algorithm [20], which can be considered as an extension of Jacobi s method for the computation of the eigenvalue decomposition of THE MAX ALGEBRAIC QRD AND THE MAX ALGEBRAIC SVD 15 a real symmetric matrix. We now state the main properties of this algorithm. The explanation below is mainly based on [4] and ....

E. Kogbetliantz, Solution of linear equations by diagonalization of coecients matrix, Quarterly of Applied Mathematics, 13 (1955), pp. 123-132.

....the results obtained in both implementations. 2 Basic Algorithm Considering A; B 2 n Thetan upper triangular matrices, we want to compute the decomposition U T AQ = Sigma AR; Q T BV = R Gamma1 Sigma B , described in the Introduction. The algorithm is an implicit Kogbetliantz method [5, 8]. If A ij = A i i; j i; j j , two matrix sequences fA k g and fB k g are generated from the expressions: A k 1 = J( i; j) T A k L( i; j) B k 1 = L( i; j) T B k K(OE; i; j) 3) where J and K are plane rotations, chosen such that the matrix C ij = A ij B ij is diagonal. First, the ....

Kogbetliantz, E.: Solution of linear equations by diagonalization of coefficients matrix. Quart. Appl. Math., 13, pp. 123-132 (1955).

....to compute the GSVD directly, without going via computing the CSD. This algorithm has two phases: the first phase is a preprocessing step to reduce the given matrix pair to triangular forms; the second phase is to compute the GSVD of two triangular matrices by a generalized Kogbetliantz algorithm [35]. 5.4.1 Preprocessing By using QR decomposition, two slightly different preprocessing schemes have been proposed by Paige [43] and Bai and Demmel [7] Both of the schemes may result in an irregular triangular pair. Most recently, using the URV decomposition, Bai and Zha [8] propose a new ....

....and B 13 with B 13 nonsingular 2 . This is the focus of the next section. For simplicity of notation, we denote the triangular matrix pair (A 23 ; B 13 ) by (A; B) in the next section. 5.4. 2 Computing the GSVD of two square upper triangular matrices We first review the Kogbetliantz algorithm [35] for computing the SVD of an upper triangular matrix A. Then Paige s algorithm [43] for computing the GSVD can be regarded as a generalization of the Kogbetliantz algorithm. Suppose the kth transformation of Kogbetliantz algorithm operates on rows and columns i and j of A. Let A ij = a ii a ....

E. G. Kogbetliantz, Solution of linear equations by diagonalization of coefficients matrix, Quart. Appl. Math. 13, (1955), pp.123--132.

....data matrix, when new observations (rows) are added continuously. The algorithm may be organized such that it provides at each time a certain approximation for the exact singular value decomposition. It combines a Jacobi type diagonalization scheme, based on two sided orthogonal transformations [10], with QR updates. A systolic implementation for this algorithm is described in [15] Here, we improve upon these results. An alternative algorithm is described, which employs only one sided transformations. Row and column transformations are applied in an alternating fashion. The algorithm is ....

....in a finite number of time steps, e.g. with a sequence of Givens transformations [8] Then an iterative procedure is applied to the triangular factor RA , transforming it into a diagonal matrix. This diagonalization procedure consists in applying a sequence of plane transformations as follows, see [10, 11] for details: R ( RA U ( I V ( I for k = 1; 1 for i = 1; m0 1 2 6 6 4 R ( U T [i;k] 1 R 1 V [i;k] U ( U 1 U [i;k] V ( V 1 V [i;k] end end The parameter i is called the pivot index. The matrices U [i;k] and V [i;k] represent ....

E. Kogbetliantz, `Solution of linear equations by diagonalization of coefficient matrices'. Quart. Appl. Math., 13 (1955), pp 123-132.

....a constant number of steps, e.g. with a sequence of Givens transformations [3, 5] Then an iterative procedure is applied to the triangular factor RZ , transforming it into a diagonal matrix. This diagonalization procedure consists in applying a sequence of plane transformations as follows, see [6, 7] for details: R ( RZ U ( I V ( I for j = 1; 1 for i = 1; m01 [ i is called the pivot index] R ( 5 [i;i 1] U H [i;j] 1 R 1 V [i;j] 5 [i;i 1] U ( U 1 U [i;j] 5 [i;i 1] V ( V 1 V [i;j] 5 [i;i 1] end end The matrices U [i;j] and V [i;j] are unitary transformations in the (i; ....

....i 1) entry in R, while R still remains in upper triangular form. Each iteration thus essentially reduces to performing an SVD of a 222 block somewhere on the main diagonal 6 . The permutations are necessary to guarantee convergence to diagonal form . For more details, the reader is referred to [6, 7]. After each iteration we have RZ = U 1 R 1 V H : Eventually, R converges to a diagonal matrix, equal to SZ up to a reordering on the diagonal, i.e. R = 5 T SZ 5 for j = 1, where 5 is a permutation matrix 7 . This results in the required SVD, up to the reordering : Z = QZ 1 RZ = QZ 1 U ....

E. Kogbetliantz, `Solution of linear equations by diagonalization of coefficient matrices'. Quart. Appl. Math., 13 (1955), pp 123-132.

....the Jacobi algorithm may be applied to compute the OSVD by letting U (and V) equal Q and Sigma T Sigma = For a non symmetric matrix the Jacobi algorithm may also apply by first pre multiplying A with a unitary matrix, making it symmetric. This approach was suggested by Kogbetliantz [21], and hence, is known as the Kogbetliantz s method. Formally, A (k) J T (i; j; S T (i; j; OE)A (k Gamma1) J(i; j; 3.10) OE must be chosen such that S(i; j; OE) makes the 2 Theta 2 matrix symmetric. This is fulfilled by: tan(OE) a i;j Gamma a j;i a i;i a j;j (3.11) Next, ....

E. G. Kogbetliantz. Solution of Linear Equations by Diagonalization of Coefficients Matrix. Quart. Applied Mathematics, 13, 1955.

....Furthermore, it is obvious that every QR decomposition in R max is also a QR decomposition in S max . 5.3. The max algebraic singular value decomposition. Now we give an alternative proof for the existence theorem of the max algebraic SVD. In this proof we shall use Kogbetliantz s SVD algorithm [20], which can be considered as an extension of Jacobi s method for the computation of the eigenvalue decomposition of THE MAX ALGEBRAIC QRD AND THE MAX ALGEBRAIC SVD 15 a real symmetric matrix. We now state the main properties of this algorithm. The explanation below is mainly based on [4] and ....

E. Kogbetliantz, Solution of linear equations by diagonalization of coefficients matrix, Quarterly of Applied Mathematics, 13 (1955), pp. 123--132.

.... Phi = oe 1 oe 2 : oe r where oe i = Sigma) ii for i = 1; 2; r. Every decomposition of the form (13) that satisfies the above conditions is called a max plus algebraic SVD of A. Proof : This proof is similar to that of Theorem 5. 5, but now we use Kogbetliantz s SVD algorithm [7] to construct a path of SVDs of the matrix valued function A. Afterwards we apply the reverse mapping R to obtain a max plus algebraic SVD of A. See [3] for the details of this proof. 2 Note that for the max plus algebraic SVD we also have an extra condition (kAk Phi = oe 1 ) that does not ....

E.G. Kogbetliantz, "Solution of linear equations by diagonalization of coefficients matrix," Quarterly of Applied Mathematics, vol. 13, no. 2, pp. 123--132, 1955.

.... this link and used it to define the singular value decomposition (SVD) in the extended max algebra, which is a kind of symmetrization of the max algebra [8, 12] In this paper we present an alternative proof for the existence theorem of the max algebraic SVD based on Kogbetliantz s SVD algorithm [3, 10]. Furthermore, we prove the existence of a kind of QR decomposition (QRD) in the extended max algebra. We also propose possible extensions of the max algebraic SVD. This paper is organized as follows: In Section 1 we recapitulate the most important concepts, definitions and properties of [6] and ....

....of B( Delta) are in S e . If I ae R then U(s) Psi(s) V T (s) is a (constant) SVD of B(s) for each s 2 I if and only if U(s) Sigma(s) V T (s) with Sigma(s) e (c 1)s Psi(s) is a (constant) SVD of A(s) for each s 2 I . We shall apply Kogbetliantz s SVD algorithm [3, 10] on B( Delta) This algorithm can be considered as an extension of Jacobi s method for the calculation of the eigenvalue decomposition of a symmetric matrix. For a matrix B 2 R m Thetan (with m 6 n) a sequence of matrices is generated as follows: X 0 = I m ; Y 0 = I n ; S 0 = B X k = G k X ....

E.G. Kogbetliantz, "Solution of linear equations by diagonalization of coefficients matrix, " Quarterly of Applied Mathematics, vol. 13, no. 2, pp. 123--132, 1955.

....V ffi [k] and 6 ffi [k] are explicitly updated. SVD Updating, 1st Version An SVD updating algorithm (for infinite precision arithmetic) is readily constructed by combining QR updating with a Jacobi type SVD diagonalization procedure (Kogbetliantz s algorithm, modified for triangular matrices [14, 15]) Suppose that at a certain time step k01, the A [k01] matrix is reduced to R [k01] upper triangular and almost diagonal with corresponding matrices U [k01] and V [k01] A [k01] U [k01] 1 R [k01] 1 V T [k01] After appending a new row a T [k] one has a decomposition of the type ....

....does not alter the V matrix. The U matrix does change, but it does not have to be stored anyway, as we are only interested in R and V . 3. SVD steps in order to obtain a diagonal matrix. This diagonalization procedure consists in applying a sequence of plane rotations as follows (see [14, 15] for details) R [k] R [k] V [k] V [k01] for j = 1; r for i = 1; n 0 1 R [k] 2 T [i;j;k] 1 R [k] 1 8 [i;j;k] V [k] V [k] 1 8 [i;j;k] end end The parameter i is called the pivot index. The matrices 2 [i;j;k] and 8 [i;j;k] represent rotations in the (i; i ....

KOGBETLIANTZ E., 1955. Solution of linear equations by diagonalization of coefficient matrices. Quart. Appl. Math. 13, 123-132.

....for the case where it is time varying. Finally, in most cases the U (k) matrices (of growing size ) need not be computed explicitly, and only V (k) and 6 (k) are explicitly updated. An adaptive algorithm can be constructed by interlacing a Jacobi type SVD procedure (Kogbetliantz s algorithm [9], modified for triangular matrices [8, 10] with repeated QR updates. See [12] for further details. Initialization V (0) I n2n R (0) O n2n Loop for k = 1; 1 input new measurement vector a (k) a (k) T ( a (k) T 1 V (k01) R (k) 1 R (k01) QR updating R (k) ....

KOGBETLIANTZ E., 1955. Solution of linear equations by diagonalization of coefficient matrices. Quart. Appl. Math., Vol. 13, pp 123-132.

....data matrix, when new observations (rows) are added continuously. The algorithm may be organized such that it provides at each time a certain approximation for the exact singular value decomposition. It combines a Jacobi type diagonalization scheme, based on two sided orthogonal transformations [10], with QR updates. A systolic implementation for this algorithm is described in [15] Here, we improve upon these results. An alternative algorithm is described, which employs only one sided transformations. Row and column transformations are applied in an alternating fashion. The algorithm is ....

....in a finite number of time steps, e.g. with a sequence of Givens transformations [8] Then an iterative procedure is applied to the triangular factor RA , transforming it into a diagonal matrix. This diagonalization procedure consists in applying a sequence of plane transformations as follows, see [10, 11] for details: R ( RA U ( I V ( I for k = 1; 1 for i = 1; m Gamma 1 2 6 6 4 R ( U T [i;k] Delta R Delta V [i;k] U ( U Delta U [i;k] V ( V Delta V [i;k] end end The parameter i is called the pivot index. The matrices U [i;k] ....

E. Kogbetliantz, `Solution of linear equations by diagonalization of coefficient matrices'. Quart. Appl. Math., 13 (1955), pp 123-132.

.... developed analogous to the Jacobi like method of Eberlein [15] for the computation of the Schur form of a general matrix (with which the reader is assumed to already be familiar) When used for matrices which are Hermitian and Hamiltonian, the algorithm is equivalent to the Kogbetliantz algorithm [27] for n dimensional matrices; the Kogbetliantz algorithm is globally convergent (under mild assumptions) and converges asymptotically quadratic. Section 2 describes the basic idea of the method and the connection to Kogbetliantz s algorithm. In each step of the algorithm a 4 Theta 4 ....

....try to achieve a form as close as possible to the Hamiltonian Schur form. That is, we want to choose a transformation such that in the almost Hamiltonian form (5) the norm of K 22 is as small as possible. A different motivation for the proposed algorithm can be given via the Kogbetliantz algorithm [27]. This algorithm is a generalization of the Jacobi method for solving the following problem: Let C 2 C n Thetan . Determine unitary matrices Z; W 2 C n Thetan such that P : WCZ H is a diagonal matrix (upto scaling this is the SVD of C) An elementary step of the Kogbetliantz method ....

E.G.Kogbetliantz. Solutions of linear equations by diagonalization of coeffients matrix. Quart.Appl.Math., 13(1955),123--132

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E. G. Kogbetliantz. Solution of linear equations by diagonalization of coef- cient matrix. Quart. Appl. Math., 13:123-132, 1955. 29

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E. Kogbetliantz, Solution of linear equations by diagonalization of coecients matrix, Quarterly of Applied Mathematics, 13 (1955), pp. 123-132.

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E. G. Kogbetliantz, "Solution of Linear Equations by Diagonalization of Coefficients Matrix," Quart. Applied Mathematics, vol. 13, 1955.

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