Garbow, B. S., Boyle, J. M., Dongarra, J. J., and Moler, C. B. 1977. Matrix Eigensystem Routines { EISPACK Guide Extension, Volume 51 of Lecture notes in computer science. Springer-Verlag, New York.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....computers which made it possible to write interactive programs with question and answer dialog interfaces that prompted the user for input and displayed the results on the screen. In the mid seventies, several Fortran numerical subroutine libraries were developed, whereby Linpack [26] and Eispack [95, 32] became the basis for more specialized libraries for solving control problems. These libraries had common data structures and integrated the different numerical routines that were available at the time. Nonetheless, a main program had to be written in which these routines were called. This was ....

B.S. Garbow, J.M. Boyle, J.J. Dongarra, and C.B. Moler. Matrix Eigensystem Routines -- Eispack Guide Extension. Springer-Verlag, Berlin, 1977.

....strategy. Note that the deflation window size is likely to be substantial. In the numerical examples reported in section 3, window sizes ranged from k = 48 to k = 450. Hence, windownorm stable deflation resembles norm stable deflation and su#ers many of the same drawbacks. EISPACK [28, 37] and LAPACK [1] use small subdiagonal deflation. Subdiagonal entries are considered small enough to deflate only if they are tiny compared to nearby matrix entries. EISPACK [28, 37] subroutines HQR HQR2 and LAPACK [1] subroutines DHSEQR DLAHQR ordinarily set a subdiagonal entry h i 1,i to zero ....

....Hence, windownorm stable deflation resembles norm stable deflation and su#ers many of the same drawbacks. EISPACK [28, 37] and LAPACK [1] use small subdiagonal deflation. Subdiagonal entries are considered small enough to deflate only if they are tiny compared to nearby matrix entries. EISPACK [28, 37] subroutines HQR HQR2 and LAPACK [1] subroutines DHSEQR DLAHQR ordinarily set a subdiagonal entry h i 1,i to zero only if ( h ii ) where is the unit roundo#. If both h ii = 0 and h i 1,i 1 = 0, then EISPACK falls back on norm stable deflation. LAPACK falls back on a modified ....

B. S. Garbow, J. M. Boyle, J. J. Dongarra, and C. B. Moler, Matrix Eigensystem Routines: EISPACK Guide Extension, Springer-Verlag, New York, 1972.

....[35] 2.4 LAPACK and ScaLAPACK LAPACK, first released in 1992 and regularly updated, is a collection of Fortran 77 programs for solving various linear equation, linear least squares and eigenvalue problems. It can be regarded as a successor to the 1970s packages LINPACK [44] and EISPACK [117] [66]. It has virtually all the capabilities of these two packages and much more besides. LAPACK improves on LINPACK and EISPACK in four main respects: speed, accuracy, robustness and functionality. While LINPACK and EISPACK are based on level 1 BLAS, LAPACK was designed at the outset to use ....

B. S. Garbow, J. M. Boyle, J. J. Dongarra, and C. B. Moler. Matrix Eigensystem Routines---EISPACK Guide Extension, volume 51 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1977. viii+343 pp. ISBN 3-540-08254-9.

....types in C classes. 1 Using C in the Matrix package for R Numerical linear algebra is a cornerstone of any numerical or statistical computing system, such as S. The di#erent implementations of S S PLUS and R use extended versions of the Fortran code in the Linpack[2] and Eispack[4, 3] packages for numerical linear algebra. S PLUS versions 3.4 and later have a separate Matrix library based on Lapack[1] which is an extended and enhanced Fortran package combining capabilities of both Linpack and Eispack. Linpack and Lapack both use the Basic Linear Algebra Subroutines (BLAS) for ....

B.S. Garbow, J.M Boyle, J.J. Dongarra, and C.B. Moler. Matrix Eigensystem Routines - EISPACK Guide Extension. Springer-Verlag, New York, 1977. 1

....whilst in the case of medium sized matrices an application may require knowledge of the complete eigensystem. No one algorithm is effective in all circumstances, but there do now exist comprehensive suites of FORTRAN programs in both the NAG [2] and EISPACK (see Smith et al. 4] and Garbow et al. [1]) software libraries, and these may be relied upon to solve accurately and efficiently a very wide range of eigensystem problems involving matrices of modest size (with matrix order not exceeding 500) Larger scale problems arise abundantly; the matrix order is frequently as much as 10 4 , and ....

B. S. Garbow, J. M. Boyle, J. J. Dongarra, and C. B. Moler. Matrix Eigensystem Routines: EISPACK Guide Extensions, volume 51 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1977.

....to Ma directly or indirectly. In our example we have chosen a tree topology which is shown in Figure 3. In step 2 of the parallel concept the condensed problem and the slave eigenproblems may be solved (in parallel but each of them sequentially) by appropriate library routines from EISPACK [5], e.g. It turns out however, that these eigenproblems in general are of extremely different size and sparseness (cf. 16] and the example in section 6) A dominating condensed problem (7) which usually is dense leads to an unbalanced load of the processors in this phase and consequently the ....

B.S. Garbow, J.M. Boyle, J.J Dongarra, and C.B. Moler, Matrix eigensystem routines -- EISPACK Guide Extension. Lecture notes in computer science 51 (Springer, New York, 1977).

....and engineering problems and more generally for those areas where significant numeric computations have to be performed. It is based on software produced in the LINPACK and EISPACK projects which provides firmly established, tried and tested numerical software for matrix computations, see [1,2]. There is a scripting system available which allows the user to develop and modify the software for their own needs and even add complete tool boxes for their area of work. To give a brief overview of MATLAB s facilities the major facilities are listed below: 1) a vast number of powerful ....

Garbow, B.S., Boyle, J.M., Dongarra, J.J., Moler, C. B. Matrix Eigensystem Routines --- EISPACK Guide Extension, Lecture Notes in ComputerScience,Volume51,SpringerVerlag 1977.

....2 j 1 = r j r j ; which is also a real number. 5 Therefore, the one vector approach used in our implementation yields a real symmetric tridiagonal matrix as de ned in (7) For the solution of the associated reduced eigenproblem a QL method based subroutine, IMTQL, available in EISPACK [7], has been used. Actually, that subroutine has been modi ed to compute only the bottom entries of each eigenvector of T j s l = s l l when the residuals are computed by means of relation (11) The full eigenvectors s l are evaluated only for the computation of the aproximations x. On the ....

B. S. Garbow, J. M. Boyle, J. J. Dongarra, and C. B. Moler. Matrix Eigensystem Routines: Eispack Guide Extension, volume 51 of Lecture Notes in Computer Science. Springer Verlag, Berlin, Germany, 1977.

....(Single Instruction Multiple Data) 31] computer with good inter processor 1 three left and three right rotation angles per step 2 Connection Machine TM is a registered trademark of Thinking Machines Corporation, Cambridge, MA. 5 communication capability, are compared with LINPACK EISPACK [26,33] routines for validation. The simulation also shows the convergence behavior of the complex SVD scheme. 1.4 Overview of the Thesis The next twochapters review previous work in the context of systolic arrays for SVD, CORDIC algorithms and Jacobi type methods. In the following chapter, the serial ....

....of algorithms available. They range from serial algorithms to parallel Jacobi type methods. On a conventional uniprocessor system, the most commonly used procedure is the Golub Kahan Reinsch[35, 36] SVD algorithm. The Golub Kahan Reinsch algorithm is implemented both in LINPACK [26] and EISPACK [33]. The first step in the Golub Kahan Reinsch SVD algorithm is the bidiagonalization of the matrix. This is followed by an iterative diagonalization of the bi diagonal matrix to complete the SVD. The time complexity of the Golub KahanReinsch algorithm is O(mn 2 ) An improvement to the ....

B. S. Garbow, J. M. Boyle, J. J. Dongarra, and C. B. Moler. Matrix Eigensystem Routines -- EISPACK Guide Extension. Springer-Verlag, Berlin, 1977.

....uniprocessor system, the most commonly used procedure is the Golub Kahan Reinsch [46, 47] SVD algorithm. Algorithms for the computation of the SVD, specific to some parallel architectures are also known [39, 85] The Golub Kahan Reinsch algorithm is implemented both in LINPACK [33] and EISPACK [43]. The first step in the Golub Kahan Reinsch SVD algorithm is the bidiagonalization of the matrix. This is followed byaniterative diagonalization of the bi diagonal matrix to complete the SVD. The time complexity of the Golub KahanReinsch algorithm is O(mn 2 ) An improvement to the ....

B. S. Garbow, J. M. Boyle, J. J. Dongarra, and C. B. Moler. Matrix Eigensystem Routines -- EISPACK Guide Extension. Springer-Verlag, Berlin, 1977.

....r ; v r ) This is used as an initial guess to converge to the turning point using inverse power iterations. Implementation, Performance and Applications The loop detection algorithm has been implemented and its performance was measured on a number of models. The algorithm uses existing EISPACK [GBDM77] and LAPACK [ABB 92] routines for some of the matrix computations. At each stage of the algorithm, we can compute bounds on the accuracy of the results obtained based on the accuracy, condition numbers and convergence of numerical methods used like eigenvalue computation, power iterations and ....

B.S. Garbow, J.M. Boyle, J. Dongarra, and C.B. Moler. Matrix Eigensystem Routines -- EISPACK Guide Extension, volume 51. Springer-Verlag, Berlin, 1977.

....; v r ) This is used as an initial guess to converge to the turning point using inverse power iterations. 4 Implementation, Performance and Applications The loop detection algorithm has been implemented and its performance was measured on a number of models. The algorithm uses existing EISPACK [GBDM77] and LAPACK [ABB 92] routines for some of the matrix computations. At each stage of the algorithm, we can compute bounds on the accuracy of the results obtained based on the accuracy, condition numbers and convergence of numerical methods used like eigenvalue computation, power iterations and ....

B.S. Garbow, J.M. Boyle, J. Dongarra, and C.B. Moler. Matrix Eigensystem Routines -- EISPACK Guide Extension, volume 51. Springer-Verlag, Berlin, 1977.

....that (A Gamma I)x = 0: Once again, the methods used depend on the type of matrix, its order and how many eigenvalues and or eigenvectors are required. More generally, given M ( a matrix being a function of , find ; x such that M ( x = 0: See also Wilkinson (1965) Golub and van Loan (1989) Garbow et al. 1977), Smith et al. 1976) and Anderson et al. 1995) iii) Computation of matrix expressions like projection operators (e.g. A(A t A) Gamma1 A t ) and updating formulae (e.g. A uv t ) Gamma1 in terms of A Gamma1 ) in optimization and linear programming. Very often the algebraic ....

....matrix (or upper triangular form) by the accumulated similarity transformations from the first step (and the second step) to get the eigenvectors of A. 10.9 EISPACK. EISPACK was written over a number of years around 1972 at a cost of some millions of dollars. As well as the published book, Garbow et al. 1977), Smith et al. 1976) there is machine readable documentation, a short description of the main driver subroutines and a large file with a description of all subroutines. Except for routines which supply specialized output most of the package can be described by the flow chart. If only eigenvalues ....

Garbow B.S., Boyle J.M., Dongarra J.J. and Moler C.B., Matrix Eigensystem Routines: EISPACK Guide Extension, Lecture Notes in Computer Science, Springer-Verlag, 1977.

....some of the matrix computations. At each stage of the algorithm, we can compute bounds on the accuracy of the results obtained based on the accuracy and convergence of numerical methods adopted like eigenvalue computation, power iteration and Gaussian elimination. Our implementation uses EISPACK [GBDM77] routines (in Fortran) to compute the eigenvalues of matrices. The algorithm was run on a SGI Onyx workstation with a R4400 CPU with 128 Mbytes of main memory. We have not implemented the randomized algorithm for performing the general hidden surface removal. Currently, our system takes a set of ....

B.S. Garbow, J.M. Boyle, J. Dongarra, and C.B. Moler. Matrix Eigensystem Routines -- EISPACK Guide Extension, volume 51. Springer-Verlag, Berlin, 1977.

....and split them into non intersecting faces. Figure 10: Partitioning of patch in Figure 6 based on visibility curves 6.1 Implementation and Performance The algorithm to compute non overlapping regions using silhouette and algebraic curves has been implemented. The algorithm uses existing EISPACK [GBDM77] routines for some of the matrix computations. At each stage of the algorithm, we can compute bounds on the accuracy of the results obtained based on the accuracy and convergence of numerical methods adopted like eigenvalue computation, power iteration and Gaussian elimination. Our implementation ....

B.S. Garbow, J.M. Boyle, J. Dongarra, and C.B. Moler. Matrix Eigensystem Routines -- EISPACK Guide Extension, volume 51. Springer-Verlag, Berlin, 1977.

....set before the text is moved. The band storage scheme used by the GB, HB, SB, and TB routines has columns of the matrix stored in columns of the array, and diagonals of the matrix stored in rows of the array. This is the storage scheme used by LINPACK. An alternative scheme (used in some EISPACK [8,11] routines) has rows of the matrix stored in rows of the array, and diagonals of the matrix stored in columns of the array. The latter scheme has the advantage that a band matrix vector product of the form y oAx 13y can be 14 computed using long vectors (the diagonals of the matrix) stored ....

B.S. Garbow, J.M. Boyle, J.J. Dongarra, C.B. Moler, Matrix Eigensystem Routines - EISPACK Guide Extension, Lecture Notes in Computer Science, Vol. 51, Springer-Verlag, Berlin, 1977.

....LAPACK can also be used satisfactorily on all types of scalar machine (PC s, workstations, mainframes) See Chapter 2 for some examples of the performance achieved by LAPACK routines. How does LAPACK compare with LINPACK and EISPACK LAPACK has been designed to supersede LINPACK [10] and EISPACK [25, 19], principally by restructuring the software in order to achieve much greater efficiency on modern high performance computers; also by adding extra functionality, by using some new or improved algorithms, and by integrating the two sets of algorithms into a unified package. Chapter 7 lists the ....

....Can we provide portable software for computations in dense linear algebra which is efficient on a wide range of modern high performance computers If so, how Answering these questions and providing the desired software has been the goal of the LAPACK project. LINPACK [10] and EISPACK [25, 19] have for many years provided high quality portable software for linear algebra; but on modern high performance computers they often achieve only a small fraction of the peak performance of the machines. Therefore, LAPACK has been designed to supersede LINPACK and EISPACK, principally by achieving ....

B. S. Garbow, J. M. Boyle, J. J. Dongarra, and C. B. Moler. Matrix Eigensystem Routines -- EISPACK Guide Extension. Springer-Verlag, New York, 1977.

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Garbow, B. S., Boyle, J. M., Dongarra, J. J., and Moler, C. B. 1977. Matrix Eigensystem Routines { EISPACK Guide Extension, Volume 51 of Lecture notes in computer science. Springer-Verlag, New York.

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B. S. Garbow, J. M. Boyle, J. J. Dongara, and C. B. Moler, Matrix Eigensystem Routines-EISPACK Guide Extension, in Lecture Notes in Computer Science, Springer Verlag, New York, 1977.

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B.S. Garbow, J.M. Boyle, J.J. Dongarra, and C.B. Moler, Matrix Eigensystem Routines { EISPACK Guide Extension, Berlin: Springer-Verlag, 1977.

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B.S. Garbow, J.M. Boyle, J.J. Dongarra, and C.B. Moler, Matrix Eigensystem Routines---EISPACK Guide Extension. Berlin: SpringerVelag, 1977.

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B. S. Garbow, J. M. Boyle, J. J. Dongarra, and C. B. Moler. Matrix Eigensystem Routines: EISPACK Guide Extension, volume 51 of Lecture Notes in Computer Science. Springer-Verlag, 1977.

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B. S. Garbow, J. M. Boyle, J. J. Dongarra, C. B. Moler, Matrix Eigensystems Routines: EISPACK Guide Extension, Springer-Verlag, New York, N. Y. (1972).

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Garbow, B.S.; Boyle, J.M.; Dongarra, J.J.; Moler, C.B.: Matrix eigensystem routines -- EISPACK Guide Extension. Lecture notes in computer science 51, Springer,New York, 1977.

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B.S. Garbow, J.M. Boyle, J. Dongarra, and C.B. Moler. Matrix Eigensystem Routines -- EISPACK Guide Extension, volume 51. Springer-Verlag, Berlin, 1977.

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