George E. Forsythe and Peter Henrici. The Cyclic Jacobi Method for Computing the Principal Values of a Complex Matrix. Transactions of the American Methematical Society, 94(1):1-23, January 1960.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....A being diagonal, which means that the columns of A are orthonormal with 2 norms equal to the singular values. The left singular vectors that make up the columns of U are then obtained by scaling the columns by these singular values. Algorithm 5. 1 does converge, as proved by Forsythe and Henrici [24], and the asymptotic rate of convergence is quadratic; see [29] 44, Chapter 9] for details. The overall cost of applying Algorithm 5.1 can be reduced by computing a QR factorization of A or A and then applying the algorithm to the square upper triangular factor. Table 5.1: Effect of ....

G. E. Forsythe and P. Henrici. The cyclic Jacobi method for computing the principal values of a complex matrix. Trans. Amer. Math. Soc., 94:1--23, 1960.

....Procrustes problem We will now present the left sided relaxation method. Without loss of generality let us assume that the planes (r; s) in which transformations operate are chosen in the row cyclic order, in the way analogous to that used in the cyclic Jacobi method for the SVD computation [7]. In this case N : D f1; 1 2 p(p Gamma 1)g, D = f(r; s) 1 p Gamma 1; r 1 s pg is given by N (r; s) s Gamma r (r Gamma 1) p Gamma r 2 ) and Q in (3.6) has the following form Q = p Gamma1 Y r=1 r Y s=1 J p Gammar;p Gammas 1 Q ; 4.1) where J r;s 2 J r;s (p) ....

G.E. Forsythe and P. Henrici, The cyclic Jacobi method for computing the principal values of a complex matrix, Trans. AMS 94 (1960), 1--23.

....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 ....

....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 [82] Hansen [39] discusses the convergence properties associated with various orderings for the serial Jacobi method. He ....

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G. E. Forsythe and P. Henrici. The Cyclic Jacobi Method for Computing the Principal Values of a Complex Matrix. Transactions of the American Mathematical Society, 94(1):1--23, January 1960. 107

....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] ....

....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 cyclic Jacobi method for SVD and conditions on the parameters of the two sided transformations ....

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G. E. Forsythe and P. Henrici. The Cyclic Jacobi Method for Computing the Principal Values of a Complex Matrix. Transactions of the American Mathematical Society, 94(1):1--23, January 1960.

....for finding the eigenvalues and eigenvectors of a real symmetric matrix is to construct a matrix sequence fA l g by means of A l 1 = Q l A l Q t l ; l = 1; 2; 1) where A 1 = A, and Q l is a plane rotation (Jacobi rotation) in the (p; q) plane with 1 p; q n. Under certain conditions [10], sequence fA l g converges to a diagonal matrix D, D = Q k Q k Gamma1 : Q 2 Q 1 AQ t 1 Q t 2 : Q t k Gamma1 Q t k (2) whose diagonal elements are then the eigenvalues of A. Each product Q l A l Q t l represents a similarity transformation that annihilates a pair of nondiagonal ....

.... bound, where off(A) q P n j=1;j 6=i P n i=1 a 2 ij . A sweep consists of successively nullifying the n(n Gamma 1) 2 nondiagonal elements in the lower triangular part of the matrix (and the corresponding symmetrical part) The convergence of the cyclic byrow method is analyzed in [10]. More recently, 18, 19] demonstrate that the Jacobi method converges with other ordering schemes when the cyclic by row method is convergent. These orderings are intended to simplify parallelization of the method. In this paper the odd even order will be used [20] Again, this order was ....

G. E. Forsythe and P. Henrici. The cyclic Jacobi method for computing the principal values of a complex matrix. Transactions of American Mathematical Society, 94:1--23, 1960. 36

....matrix A can be obtained by means of Jacobi like methods. The basic idea is to construct a matrix sequence fA l g by means of A l 1 = Q l A l Q t l ; l = 1; 2; 1) where A 1 = A, and Q l is a plane rotation (Jacobi rotation) in the (p; q) plane with 1 p; q n. Under certain conditions [10], sequence fA l g converges to a diagonal matrix D, D = Q k Q k Gamma1 : Q 2 Q 1 AQ t 1 Q t 2 : Q t k Gamma1 Q t k (2) whose diagonal elements are then the eigenvalues of A. Each product Q l A l Q t l represents a similarity transformation that annihilates a pair of nondiagonal ....

....convergence criterion is fulfilled, normally until off(A) bound. A sweep consists of successively nullifying the n(n Gamma 1) 2 nondiagonal elements in the lower triangular part of the matrix (and the corresponding symmetrical part) The convergence of the cyclic by row method is analyzed in [10]. More recently, 8, 9, 13, 14, 15] demonstrate that the Jacobi method converges with other ordering schemes when the cyclic by row method is convergent. These orderings are intended to simplify parallelization of the method. In this paper the odd even order will be used [16] Again, this order ....

Forsythe, G. E; Henrici, P: The cyclic Jacobi method for computing the principal values of a complex matrix. Trans. Amer. Math. Soc., Vol 94, pp 1-23. 1960.

....matrix A will become lower triangular. The second sweep will restore upper triangular form, and so on [20, 19] There is a literature on the different sweep orders for sequential and parallel computations besides the conventional row and column order, for example [24] Forsythe and Henrici [16] considered the convergence of the row cyclic Kogbetliantz algorithm. Fernando [14] proved a global convergence theorem under the assumption that one of the rotation angles fOE k ; k g at each (i; j) transformation lies in a closed interval J ae ( Gamma=2; 2) i.e. OE k 2 J or k 2 J; k = 1; ....

G. E. Forsythe and P. Henrici, The cyclic Jacobi method for computing the principal values of a complex matrix. Trans. Amer. Math. Soc. 94:1--23(1960).

....fashion. Even though codes based on the method were running on the ILLIAC and UNIVAC 1103 A, the proof of convergence of row and column cyclic Jacobi, and its rate of convergence under mild restriction on the angle of rotation, did not appear until several years later, in Forsythe and Henrici [55], Henrici [77] Wilkinson [146] and others. It is interesting to note that the code for the IBM 704, for example, could compute all eigenvalues and eigenvectors of a real symmetric matrix of order 68. Typical execution time varied from about 1 minute, for an 8 8 matrix, to 25 minutes, for an ....

Forsythe G.E., Henrici P., The cyclic Jacobi method for computing the principal value of matrix, Trans. AMS, Vol94, 1960.

....V k , respectively. Then let A k 1 = U T k A k V k ; 5.12) where A 0 = A. After the first sweep through all the superdiagonal (i; j) in row cyclic order, the upper triangular matrix A becomes lower triangular. Then the second sweep will restore upper triangular form, and so on [31, 30] In [27, 26, 44, 15, 4], it has been proved that if one of the rotation angles fOE k ; k g at each transformation lies in a closed interval in ( Gamma =2; 2) then the sequence fA k g from (5.12) converges initially linearly, and ultimately quadratically, to a diagonal matrix, and gives the SVD of A. For literature ....

G. E. Forsythe and P. Henrici, The cyclic Jacobi method for computing the principal values of a complex matrix. Trans. Amer. Math. Soc. 94, (1960), pp.1-23.

....exactly once per sweep. The iterative procedure terminates if one complete sweep occurs in which all columns are orthogonal to working accuracy and no columns are interchanged. If the rotations in a sweep are chosen in a reasonable, systematic order, the convergence rate is ultimately quadratic [9, 11]. Exceptional cases in which cycling occurs are easily avoided by the use of a threshold strategy [23] It is known that one Jacobi plane rotation operation only involves two columns. Therefore, there are disjoint operations which can be executed simultaneously. In a parallel implementation, we ....

....SVD. The first is the method of ordering, i.e. how to order the n(n Gamma1) 2 rotations in one sweep of computation. Various orderings have been introduced in the literature. In sequential computation, the most commonly used is the cyclic Jacobi ordering (cyclic ordering by rows or by columns) [9, 12]. When discussing sequential Jacobi algorithms in this paper, we assume that the cyclic ordering by rows is applied. The second important detail is the method for generating the plane rotation parameters c and s in each iteration. For the one sided Jacobi method there are three main rotation ....

G. E. Forsythe and P. Henrici, "The cyclic Jacobi method for computing the principal values of a complex matrix", Trans. Amer. Math. Soc., 94, 1960, pp. 1-23.

....an off diagonal element a (k) pq , i.e. a (k 1) pq = 0, where k denotes the iteration index. If each off diagonal element is annihilated once, a so called sweep is completed. For cyclic Jacobi methods, the matrix elements a pq are processed in a fixed order, e.g. cyclic ordering scheme [9] One sweep of the cyclic Jacobi method can be implemented on an upper triangular array of processors with nearest neighbor interconnections [1] 2] 3] Fig. 1) Each processor contains a 2 Theta 2 block of the upper triangular part of A (k) The n=2 diagonal processors simultaneously ....

....of entries is not strictly justified, since in subsequent steps already generated zeros are destroyed. Another strategy does not keep to rigid rotation of a (k 1) pq to zero but demands only a reduction fi fi fia (k 1) pq fi fi fi fi fi fia (k) pq fi fi fi from step to step [9], 22] 3] 10] This modification is called approximate rotation scheme. Generally, for these schemes the number of sweeps increases but with a simple approximation the cost per sweep is smaller than for the exact algorithm. In this paper, it is shown that the binary data representation in the ....

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G.E. Forsythe and P. Henrici. The cyclic Jacobi method for computing the principal values of a complex matrix. Trans. Amer. Math. Soc. 94(1960), 1--23.

....when P and Q are appropriately selected. We present here an iterative algorithm to perform this nulling operation. Motivated by computational efficiency, our algorithm makes use of selective nulling by Givens rotations, similar to that used in the Jacobi method for eigenvalue computations 2 [15, 18, 70], which can lead to parallel implementation. Let us call the N Theta N lower block the eliminatee block and the N Theta N upper block the elminator block. A diagonal band of the eliminatee block Q is formally defined as the set of elements q j b;j for j 2 J b given a fixed integer b 2 [ GammaN ....

....In this example, the diagonal bands of Q are nulled by the following specific order: 1. The main diagonal, i.e. b = 0. 2. The upper (right) first off diagonal, b = 1. 3. The lower (left) first off diagonal, b = Gamma1. 3 named after a similar operation in the iterative Jacobi method [15, 18] 4. The upper second off diagonal, b = 2. 5. The lower second off diagonal, b = Gamma2. 6. The upper third off diagonal, b = 3. 7. The lower third off diagonal, b = Gamma3. 8. The upper fourth off diagonal, b = 4. 9. The lower fourth off diagonal, b = Gamma4. Step 1 above is pictorally ....

G. E. Forsythe and P. Henrici. The cyclic Jacobi method for computing the principal values of a complex matrix. Trans. Amer. Math. Soc., 94:1--23, 1960.

....= 1; n0 1. It is well known that if use is made of outer rotations (see [23] exactly n such sequences constitute a double sweep for a cyclic ordering (pipelined forward and backward sweep) Each rotation reduces the off norm in R [k] and R [k] eventually converges to a diagonal matrix [6], so that finally we will have R [k] 6 ffi [k] V [k] V ffi [k] With the above procedure, the diagonal structure of R can be restored after each update. With r = O(n) on the average, the operation count is O(n 3 ) per update. In practice however, mostly it suffices to keep R [k] close ....

....we would now perform a few sweeps of Kogbetliantz s SVD algorithm (without any QR updates ) and then again check the norm of the cross terms. Classical convergence results for Kogbetliantz s algorithm turn out to be useless in this respect. Linear convergence bounds are extremely conservative [5, 6, 11, 13], while ultimate quadratic convergence results do not apply to the initial convergence, where the off diagonal elements in Rs [k] can be very large [3, 19, 24] Jacobi type algorithms are considered extremely fast, but for the general case no estimates are available whatsoever as for the speed of ....

FORSYTHE G.E., HENRICI P., 1960. The cyclic Jacobi method for computing the principal values of a complex matrix. Trans. Am. Math. Soc., Vol. 94, pp 1-23.

....to 2n 2 complex numbers if Q is not required. As with the serial Jacobi algorithm, to promote rapid convergence in the case of multiple eigenvalues, it is a good idea to use a similarity transformation by a permutation matrix to put the diagonal entries of A and B in lexicographic order [20]. Such an eigenvalue ordering is required by our proof of local quadratic convergence in Section 3. In our experience, for randomly chosen examples with n 80, rarely does ffl = 10 Gamma14 make Algorithm 1 require more than six sweeps. In Section 3 we show that Algorithm 1 has local quadratic ....

....if 1=2 doesn t work. We have not been able to show that Algorithm 1 with the above modification is globally convergent. 3. Convergence Properties. Algorithm 1 shares many of the desirable properties of algorithms related to the serial Jacobi algorithm for the real symmetric eigenvalue problem [20, 23, 36, 38, 41, 46]. In our experience, Algorithm 1, with the above strategy for avoiding stagnation, converges globally and ultimately quadratically. In this section we establish the local quadratic convergence and numerical stability of Algorithm 1. In parenthetical remarks we give a rough sketch of how the ....

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G. E. Forsythe and P. Henrici, The cyclic Jacobi method for computing the principal values of a complex matrix, Transactions of the American Mathematical Society, 94 (1960), pp. 1-- 23.

....exactly once per sweep. The iterative procedure terminates if one complete sweep occurs in which all columns are orthogonal to working accuracy and no columns are interchanged. If the rotations in a sweep are chosen in a reasonable, systematic order, the convergence rate is ultimately quadratic [9, 11]. Exceptional cases in which cycling occurs are easily avoided by the use of a threshold strategy [24] It can be seen from equation (1) that one Jacobi plane rotation operation only involves two columns. Therefore, there are disjoint operations which can be executed simultaneously. In a parallel ....

....SVD. The first is the method of ordering, i.e. how to order the n(n Gamma 1) 2 rotations in one sweep of computation. Various orderings have been introduced in the literature. In sequential computation, the most commonly used is the cyclic Jacobi ordering (cyclic ordering by rows or by columns) [9, 12]. When discussing sequential Jacobi algorithms in this paper, we assume that the cyclic ordering is applied. The second important detail is the method for generating the plane rotation parameters c and s in each iteration. For the one sided Jacobi method there are three main rotation algorithms, ....

G. E. Forsythe and P. Henrici, "The cyclic Jacobi method for computing the principal values of a complex matrix", Trans. Amer. Math. Soc., 94, 1960, pp. 1-23.

....Procrustes problem We will now present the left sided relaxation method. Without loss of generality let us assume that the planes (r; s) in which transformations operate are chosen in the row cyclic order, in the way analogous to that used in the cyclic Jacobi method for the SVD computation, see [5]. In this case N : D f1; 1 2 p(p Gamma 1)g, D = f(r; s) 1 p Gamma 1; r 1 s pg is given by N (r; s) s Gamma r (r Gamma 1) p Gamma r 2 ) and Q in (3.6) has the following form Q = p Gamma1 Y r=1 r Y s=1 J p Gammar;p Gammas 1 Q ; 4.1) where J r;s 2 J r;s (p) ....

G.E. Forsythe and P. Henrici, The cyclic Jacobi method for computing the principal values of a complex matrix, Trans. AMS 94 (1960), 1--23. 18 A. W. BOJANCZYK AND A. LUTOBORSKI

....equivalent to oddeven one. The use of various orderings does make a difference in convergence rates, which in some instances is quite striking [ME93] But, quadric convergence is always observed in real symmetric cases. Convergence for symmetric matrices has been proven by Forsythe and Henrici [FH60] for column cycling ordering. Luk and Park [LP89b] proved the convergence for odd even Jacobi sets by proving it is equivalent to column cycling orderings. The odd even ordering, however, is not optimal for parallel computation in that it completes a sweep in n sweeps instead of (n Gamma 1) We ....

G. E. Forsythe and P. Henrici. The cyclic Jacobi method for computing the principal values of a complex matrix. Trans. Amer. Math. Soc., 94:1--23, 1960.

....It is desirable for a sweep of n(n Gamma 1) 2 rotations to include all pairs (i; j) with i j. On a serial machine a simple strategy is to choose the cyclic by rows ordering (1; 2) 1; 3) Delta Delta Delta ; 1; n) 2; 3) Delta Delta Delta ; n Gamma 1; n) Forsythe and Henrici [20] have shown that the cyclic by rows ordering and condition (6.3) ensure convergence of the Jacobi method applied to A T A, and convergence of the cyclic by rows Hestenes method follows. In practice only a small number of sweeps are required. The speed of convergence is discussed in [9] 6.2 ....

....be used to compute the eigen decomposition of a symmetric n by n matrix in time O(n 3 S=P ) using P = O(n 2 ) processors. It is natural to ask if we can use more than Omega Gamma n) processors efficiently when computing the SVD. The answer is yes Kogbetliantz [40] and Forsythe Henrici [20] suggested an analogue of Jacobi s method, and this can be used to compute the SVD of a square matrix using a parallel algorithm very similar to the parallel implementation of Jacobi s method. The result is an algorithm which requires time O(n 3 S=P ) using P = O(n 2 ) processors. Details ....

G. E. Forsythe and P. Henrici, "The cyclic Jacobi method for computing the principal values of a complex matrix", Trans. Amer. Math. Soc. 94 (1960), 1-23.

....of approximate (but still orthogonal) rotation schemes and the use of factorized rotation schemes for gaining square root free [11, 12] or square root and division free [14] rotations. The possibility of using approximate rotation schemes has already been investigated for the sequential case in [29, 10, 33]. Since in a parallel environment the arithmetic is much more costly than other components (e.g. storage access) Modi and Pryce [25] and Charlier et al. 6] have shown, that the use of approximate rotations gives worthwile speedups on parallel computers. The second method for modifying rotations ....

....accuracy of the approximations for the TKA, i.e. jd K j, can be derived from the respective measure for the JA, i.e. jd J j, where jd K j = jrj Delta jd J j holds with jrj 1. 4. Convergence. The global convergence of the JA and the original KA has already been proved by Forsythe and Henrici [10] including approximate rotation schemes. The ultimate quadratic convergence of the JA has been proved by Wilkinson [32] and Paige and van Dooren [28] have extended this result to the original KA. However, for the original KA there are problems with the global and (in the case of clusters of ....

[Article contains additional citation context not shown here]

G. E. Forsythe and P.Henrici, The cyclic Jacobi method for computing the principal values of a complex matrix, Trans. Amer. Math. Soc., 94 (1960), pp. 1--23.

....and (l; k) entries are annihilated by a transformation with rotations W and Z in the k; l plane. W and Z differ from the identity only in the four matrix elements with indices (k; k) k; l) l; k) and (l; l) Convergence can be proved under certain conditions and can be shown to be quadratic [17, 34]. JACOBI METHOD FOR RICCATI EQUATIONS ON PARALLEL COMPUTERS 7 The following proposition gives the connection between the transformation of a Hermitian and Hamiltonian matrix to Hamiltonian Schur form and a singular value decomposition. Proposition 2.2. Let M = A B B GammaA be a 2n Theta ....

.... see [25] Hence, in the case of Hermitian and Hamiltonian matrices, the algorithm proposed here is equivalent to Kogbetliantz s algorithm and its convergence properties are the same, if we choose the pivot index sequence and the rotation angles according to the demands of the convergence proofs [17, 34]. The Hamiltonian Jacobi iteration proposed by Byers [12] performs a sequence of unitary symplectic similarity transformations each of which decreases the norm of Q plus twice the norm of the lower triangle of A. This quantity will be denoted by oe(H) oe( A G Q GammaA H ) n X ....

G.E.Forsythe,P.Henrici. The cyclic Jacobi method for computing the principal values of a complex matrix. Trans.Amer.Math.Soc., 94(1960),1--23

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George E. Forsythe and Peter Henrici. The Cyclic Jacobi Method for Computing the Principal Values of a Complex Matrix. Transactions of the American Methematical Society, 94(1):1-23, January 1960.

No context found.

G. E. Forsythe and P. Henrici. The cyclic Jacobi method for computing the principal values of a complex matrix. Transaction of the American Mathematical Society, 94:1-23, 1960.

No context found.

G. E. Forsythe and P. Henrici, The cyclic Jacobi method for computing the principal values of a complex matrix, Transactions of the American Mathematical Society, 94 (1960), pp. 1-- 23.

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