C. Bischof and C. Van Loan. The WY representation for products of Householder matrices. SIAM J. Sci. Statist. Comput., 8:s2--s13, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....matrices on both sides of the product and thus belong to the second class. Many of these matrix factorizations have found their way into the LAPACK software library [2] In these implementations, e#ciency is attained by employing WY representations of the involved orthogonal transformations [7, 9, 10, 16]. The application of such representations can be formulated in terms of matrixmatrix multiplications leading to reduced memory tra#c which in turn means better performance. For example, on an average work station, computing the Hessenberg form of a 1000 1000 matrix would take more than thrice ....

....to call matrices of the form E j (x) E j (v, w, #, #, #) H j (v, #)G j (#)H j (w, #) 2. 1) elementary (orthogonal symplectic) Block algorithms for QR factorization, Hessenberg, bidiagonal or tridiagonal reduction implicitly rely on WY representations for products of Householder matrices [7, 10, 16]. Furthermore, the generation of the involved orthogonal transformation matrices can be implemented e#ciently by making explicit use of such representations, see for example the LAPACK routine DORGBR. Thus, in order to develop block algorithms for orthogonal symplectic factorizations we have to ....

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C. Bischof and C. Van Loan. The WY representation for products of Householder matrices. SIAM J. Sci. Statist. Comput., 8(1):S2--S13, 1987. Parallel processing for scientific computing (Norfolk, Va., 1985).

....that their application becomes a matrix multiplication. The idea is to represent the product Q r = P r P r Gamma1 : P 1 of r Householder transformations P i = I Gamma v i v i 2 IR (where v i v i = 2) in the form Q r = I W r Y r ; W r ; Y r 2 IR as suggested by Bischof and Van Loan [17]. This is achieved using the recurrence W 1 = Gammav 1 ; Y 1 = v 1 ; W i = W i Gamma1 Gamma v i ] Y i = Y i Gamma1 Q i Gamma1 v i ] 4.1) A partitioned QR factorization can be developed as follows. Partition A 2 IR (m n) as A = A 1 B ] A 1 2 IR ; 4.2) and compute the ....

Christian H. Bischof and Charles F. Van Loan. The WY representation for products of Householder matrices. SIAM J. Sci. Stat. Comput., 8(1): s2--s13, 1987.

....is used in this performance evaluation. 1 Introduction The orthogonalization process is one of the most important processes to perform several linear algebra computations, such as eigendecomposition and QR decomposition [3, 10, 11] Many researchers have paid more attention to QR decomposition [2, 5, 6, 12, 13]. Notwithstanding, we focus on the re orthogonalization process in this paper. This is because a lot of iterative methods for solving linear equations and eigenvector computations need the re orthogonalization process to maintain accuracy of results. For instance, the GMRES method, which is one of ....

C. Bischof and C. van Loan. The WY representation for products of Householder matrices. SIAM J. Sci. Stat. Comput., 8(1):s2{s13, 1987.

....1) Y; R; T ) if (j jb .LE. n) then compute A(j : m; j jb : n) I Gamma Y T T Y T )A(j : m; j jb : n) endif compute B (I Gamma Y T T Y T )B enddo B is now Q T B solve RX = B X overwrites B Fig. 2. Linear least squares routine. Q = I Gamma Y TY T is used [2, 7]. Here T is a matrix formed by calculations using Householder vectors and values. The recursion in the QR factorization leads to an automatic variable blocking and as a consequence the Level 2 computations in a standard block algorithm are replaced by Level 3 operations. However, there are ....

C. Bischof and C. Van Loan. The WY representation for products of householder matrices. SIAM J. Scientific and Statistical Computing, 8(1):s2--s13, 1987.

....by R and the Householder vectors u i ; i = 1; n. The matrix Q is not explicitly formed, but available as the product of n Householder transformations H i = I Gamma i u i u T i . For the updates of B or the trailing part of A, the compact WY representation Q = I Gamma Y TY T is used [3, 4, 5, 6, 11]. Here T is a matrix formed by calculations using Householder vectors and i values. RGEQR3 returns upper triangular R, lower trapezoidal Y and jb Theta jb upper NEW FASTER AND SIMPLER ALGORITHM FOR LAPACK DGELS 3 do j = 1; n; nb nb is the block size jb = min(n Gamma j 1; nb) call RGEQR3 ....

C. Bischof and C. Van Loan. The WY representation for products of Householder matrices. SIAM J. Scientific and Statistical Computing, 8(1):s2--s13, 1987.

....1. Introduction LAPACK algorithm DGEQRF requires more floatingpoint operations than LAPACK algorithm DGEQR2; see [1] Yet DGEQRF outperforms DGEQR2 on an RS 6000 workstation by nearly a factor of 3 on large matrices. Dongarra, Kaufman, and Hammarling, in [2] later, Bischof and Van Loan, in [3], and still later, Schreiber and Van Loan, in [4] demonstrated why this is possible by aggregating the Householder transforms before applying #Copyright 2000 by International Business Machines Corporation. Copying in printed form for private use is permitted without payment of royalty provided ....

....by computer based and other information service systems. Permission to republish any other portion of this paper must be obtained from the Editor. 0018 8646 00 5.00 2000 IBM IBM J. RES. DEVELOP. VOL. 44 NO. 4 JULY 2000 E. ELMROTH AND F. G. GUSTAVSON 605 them to a matrix C. The result of [3] and [4] was the k way aggregating WY Householder transform and the k way aggregating storage efficient Householder transform. In the latter, the aggregated representation of Q # I # YTY T . Here, lower trapezoidal Y is m by k, consisting of k Householder vectors, and upper triangular T is k ....

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C. Bischof and C. Van Loan, "The WY Representation for Products of Householder Matrices," SIAM J. Scientif. & Statist. Computing 8, s2--s13 (1987).

....by R and the Householder vectors u i ; i = 1; n. The matrix Q is not explicitly formed, but available as the product of n Householder transformations H i = I Gamma i u i u T i . For the updates of B or the trailing part of A, the compact WY representation Q = I Gamma Y TY T is used [3, 10]. Here T is a matrix formed by calculations using Householder vectors and i values. NEW FASTER AND SIMPLER ALGORITHM FOR LAPACK DGELS 3 do j = 1; n; nb nb is the block size jb = min(n Gamma j 1; nb) call RGEQR3 A(j : m; j jb Gamma 1) Y; R; T ) if (j jb n) then compute A(j : m; ....

C. Bischof and C. Van Loan. The WY representation for products of Householder matrices. SIAM J. Scientific and Statistical Computing, 8(1):s2--s13, 1987. 12 E. ELMROTH AND F.G. GUSTAVSON

....Householder transformations [52] to reduce the generator matrix G of a scalar Toeplitz matrix to an upper triangular matrix. We extend their idea to block hyperbolic Householder transformations (required in the block Schur algorithm) using representations very similar to those proposed in [53] and [54] Let W be a diagonal matrix whose entries are either 1 or Gamma1. It is easy to verify that the matrix W satisfies the equalities W 2 = I and W T = W: 2.28) Any matrix U that satisfies the equation U T WU = W is called a W unitary matrix. Let x be a column vector such that x T ....

....could block these transformations together, and then apply this block to the appropriate matrices. This allows us to use BLAS3 primitives, rather than BLAS2 operations if we applied the transformations sequentially. Storage efficient ways to block regular Householder transformations are derived in [53] and [54] We extend these methods to hyperbolic Householder transforms. Suppose U (r) U r U r Gamma1 : U 2 U 1 is a product of r n Theta n hyperbolic Householder matrices. The matrix U can be written in two forms corresponding to the V Y form and the Y TY T form derived in [53] and ....

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C. Bischof and C. Van Loan, "The WY representation for products of Householder matrices, " SIAM J. Sci. Stat. Comput., vol. 8, pp. s2--s13, 1987.

....is given by A = QR where Q is unitary and R is upper triangular. For simplicity, we will assume that m = n. Matrix Q is usually computed and stored as a collection of Householder transformations. A blocked algorithm is derived by aggregating a number of Householder transforms into a WY transform [3] or Y TY T transform [10] The primary problem with creating an OOC version of the QR factorization is that on the surface it appears that columns of matrix A must be brought into memory simultaneously in order to compute Householder transforms from columns or apply Householder transforms to ....

Christian Bischof and Charles Van Loan. The WY representation for products of Householder matrices. SIAM J. Sci. Stat. Comput., 8(1):s2--s13, Jan. 1987.

....operations and data accesses to hierarchal memories. A block or blocked algorithm at BLAS Level 3 usually renders a better ratio between data process and data access [16] The approach of WY form, as well as its variants for the product of Householder transformations by Bischof and Van Loan [2], is to aggregate a sequence of Householder transformations, say, nb 1 of them, and then to apply the aggregated transformation to A at once, turning nb matrix vector multiplications and k rank 1 updates into one matrix matrix multiplication and one rank nb update, both at BLAS Level 3. The YTY ....

C. H. Bischof and C. Van Loan. The WY representation for products of Householder matrices. SIAM J. Sci. Stat. Comp., 8:12--13, 1987.

....k # = # #(Y k 1 , y k )R 1 k Y H k 1 y H k #. From the illustrated relation between R k and R k 1 and by the induction hypothesis, the theorem is true for G k as well. The above theorem is a generalization of the compact WY representation for products of real Householder matrices [22,23,24]. If all transformations are Hermitian, then D = I 2, and #G can be determined by Y only; otherwise, # j j=1:k are needed. The theory here can also lead to an extension of the basis kernel representation for orthogonal matrices [17] We are now in a position to promote an alternative and ....

C. H. Bischof and C. Van Loan. The WY representation for products of householder matrices. SIAM J. Sci. Stat. Comp., 8:12--13, 1987. 14

....On the other hand, we know that there are algorithms which try to access data from high speed memory, like registers or caches, instead of accessing them from the low speed main memory. Such algorithms are generally known as blocked algorithms. A blocked algorithm with the WY representation[7] (a form of blocked Householder algorithm) which was proposed by Bischof and Loan, is known as one of the most efficient blocked algorithms for the Householder method. In the following we discuss parallel blocked algorithms. Early important parallel blocked algorithms for Householder method have ....

.... machine[6] An original parallel algorithm using blocked algorithm for the similarly transformation has been proposed by Dongarra and van der Geijn[3] Subsequently, Choi et al. 8] have reported performances for a parallel blocked algorithm by the WY representation proposed by van Loan et al.[7]. However, as we already pointed out in this thesis, these parallel 78 algorithms do not discuss the relations between blocks in sequential block algorithms and blocks in the data distribution. Accordingly, their evaluations may be improper from the point of view of parallel execution time and ....

Bischof, C. and van Loan, C.: The WY Representation for Products of Householder Matrices, SIAM J. Sci. Stat. Comput., Vol. 8, No. 1, pp. s2--s13 (1987).

....is a scalar (or point) algorithm in which the operations have been grouped and reordered into matrix operations. The partitioned form may involve some extra operations over the scalar form (as is the case with algorithms that aggregate Householder transformations using the WY technique of [4]) A block algorithm is a generalization of a scalar algorithm in which the basic scalar operations become matrix operations (ff fi, fffi, ff=fi become A B, AB and AB Gamma1 ) and a matrix property based on the nonzero structure becomes the corresponding property blockwise (in particular, ....

....is the block algorithms whose stability is most in question. We know of no examples of an unstable partitioned algorithm. Those partitioned algorithms based on the aggregation of Householder transformations that do slightly different arithmetic than the point versions have been shown to be stable [4, 7]) In Section 2 we derive backward error bounds for block LU factorization and for the solution of a linear system Ax = b using the block LU factors. In Section 3 we show that block LU factorization is stable if A is block diagonally dominant by columns; this generalizes the known results that ....

Christian H. Bischof and Charles F. Van Loan, The WY representation for products of Householder matrices, SIAM J. Sci. Stat. Comput., 8 (1987), pp. s2--s13.

....The techniques used in LAPACK for constructing block versions of these algorithms are based on the aggregation of Householder transformations. Our aim is therefore to analyse the stability of these aggregation techniques. One form of aggregation is the WY representation of Bischof and Van Loan [5]. This involves representing the product Q r = P r P r Gamma1 : P 1 of r Householder transformations P i = I Gamma u i u T i 2 IR n Thetan (u T i u i = 2) in the form Q r = I W r Y T r ; W r ; Y r 2 IR n Thetar : This can be done using the recurrence W 1 = Gammau 1 ; Y 1 = u 1 ....

....to the alternative method of aggregation (3. 1) as well as to the more storage efficient compact WY representation of Schreiber and Van Loan [30] First, we note that the construction of the W and Y matrices is done in a stable manner (indeed it does not involve the BLAS3) Bischof and Van Loan [5] show that the computed b Q = I c W b Y T is such that k b Q T b Q Gamma Ik = O(u) 3.2) k c Wk = O(1) k b Y k = O(1) 3.3) The condition (3.2) implies that b Q = U DeltaU; U T U = I; k DeltaU k = O(u) 3.4) that is, b Q is close to an exactly orthogonal matrix. ....

C.H. Bischof and C.F. Van Loan, The WY representation for products of Householder matrices, SIAM J. Sci. Stat. Comput., 8 (1987), pp. s2--s13.

....vector multiprocessors. Demmel and Higham investigated the stability of block algorithms in [36] and, with Schreiber, analyzed the stability of block LU factorization in [37] Van Loan, in [140] considered a block version of the Jacobi method for singular value decomposition. Bischof and Van Loan [15] introduced the WY representation for block Householder QR factorization algorithm. A memory efficient WY representation for the products of Housholder transformation was later introduced by Schreiber and Van Loan [132] see also the related results by Schreiber and Parlett [131] This is by no ....

Bischof C., Van Loan C., The WY representation for products of Householder matrices, SIAM J. Sci. Statist. Comput., 8, 1987.

....more matrix matrix products, than reducing the total number of floating point operations. To apply a Householder transformation to a matrix involves a rank 1 update. To obtain BLAS 3 operations, several rank one updates has to be blocked and a rank r update used instead. Bishof and Van Loan [5] introduce a way to block the Householder transformations. They show that r rank one updates of the form I uv T can be collected into one rank r update in the following way, Q = I Gamma u 1 v T 1 ) Delta Delta Delta (I Gamma u r v T r ) I W r Y T r W; Y 2 IR m Thetar . W and ....

C. H. Bischof and Charles F. Van Loan. The WY representation for products of Householder matrices. SIAM J. Sci. Statist. Comput., 8(1):2--13, 1987.

....similar to the RSand MHL algorithms for tridiagonalization. As in the MHL algorithm, Householder transformations are used to annihilate unwanted elements, but in the case b 1 we are able to aggregate n b b of the transformations into the WY or compact WY representation introduced by Bischof and Van Loan [1987] and Schreiber and Van Loan [1989] respectively. Thus, data locality is further improved. First, the d outmost subdiagonals are annihilated from the rst n b columns of A. This step can be done with a QR decomposition of an h n b , h = d n b , upper trapezoidal block of A, as shown in the ....

Bischof, C. and Van Loan, C. 1987. The WY representation for products of Householder matrices. SIAM J. Sci. Stat. Comput. 8, 1 (January), s2{s13.

....no update 5.25 DSBRDT (bandwidth b = 64 tridiagonal, with update 15.70 As pointed out by one of the referees, the workspace requirements of the reduction algorithms might be reduced by doing the updates in chunks . For an example, a blocked Householder transformation A A(I WY T ) see Bischof and Van Loan [1987] for the WY representation) involves two matrix matrix products, Z = AW and A A ZY T . If the second step is done by block rows then only single blocks of the block column Z are needed at a time. A similar technique has been described by Kaufman [1995] in the context of symmetric inde nite ....

Bischof, C. and Van Loan, C. 1987. The WY representation for products of Householder matrices. SIAM J. Sci. Stat. Comput. 8, 1 (January), s2{s13.

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C. Bischof and C. Van Loan. The WY representation for products of Householder matrices. SIAM J. Sci. Statist. Comput., 8:s2--s13, 1987.

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C. Bischof and C. Van Loan. The WY representation for products of Householder matrices. SIAM J. Sci. Stat. Comput., 8:s2--s13, 1987. 25, 75

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Christian Bischof and Charles Van Loan. The WY representation for products of Householder matrices. SIAM J. Sci. Stat. Comput., 8(1):s2{s13, Jan. 1987.

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C. Bischof and C. Van Loan, "The WY representation for products of Householder matrices," SIAM J. Sci. Stat. Comput., vol. 8, pp. s2--s13, 1987.

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C. H. Bischof and C. Van Loan. The WY representation for products of householder matrices. SIAM Journal on Scienti c and Statistical Computing, 8:s2{s13, 1987.

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C. H. Bischof and C. Van Loan. The WY representation for products of householder matrices. SIAM Journal on Scienti c and Statistical Computing, 8:s2{s13, 1987.

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C. Bischof and C. Van Loan. The WY representation for products of householder matrices. SIAM J. Scientific and Statistical Computing, 8(1):s2--s13, 1987.

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