James Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... n tridiagonal matrixT satisfies the recursion t(n) n 2t(n=2) which has the solution t(n) O(n ) In practice, because of deflation, it appears that the algorithm takes only O(n 2:3 ) flops on average and the cost can even be as low as O(n ) for some special cases (see [10]) 3 Parallelization Issues Divide and conquer algorithms have been successfully implemented on shared memory multiprocessors for solving the symmetric tridiagonal eigenvalue problem and for the computation of the singular value decomposition of bidiagonal matrices [14] 24] By contrast, the ....

James W. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1997.

....and a multigrid FMV cycle. Newton s method converges quadratically. However, since each step involves inverting a system, it tends to be very hile the use of the FMV solver speeds the method up somewhat, it still is slower than the techniques we present now. It has long been known ( 12] [13]) that on certain problems non linear analogs to the classical Jacobi or Gauss Seidel iteration methods could be employed with some success. Technically, one sweep of such a method means that for j = 1, 2, N i (or (N 1) 2 for the two dimensional problem) one solves, via the scalar Newton s ....

James M. Ortega. Numerical analysis, a second course. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1979.

....we wish to put forth, but the central concept of seeing how much data lies in some place remains. A more sophisticated methodology appears in 1625 in the calculations of John Graunt, who used church records to produce a table of the probabilities of dying in a particular ten year age bracket [TT90b] The results of Graunt s study were not presented graphically, but they nevertheless represent one of the earliest genuine estimates of density; in this case, the density associated with the record of age at death in early seventeenth century London. Nevertheless, the real development of density ....

....distribution function as a distribution in the partial di#erential equations sense. We can also give this correspondence a statistical meaning by introducing the likelihood functional L(g X 1 , Xn ) n # i=1 g(X i ) This is used as a standard measure of goodness of fit in statistics [TT90b] Essentially, we want g to take greater values where there is data, requiring it to have small values in sparse regions. Now if we let a sequence g i converge to #n we observe that the likelihood of g i increases to infinity. As before, this is not particularly useful, however, when we assume ....

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James R. Thompson and Richard A. Tapia. Nonparametric Function Estimation, Modeling, and Simulation. Society for Industrial and Applied Mathematics, Philadelphia, 1990.

....more sophisticated algorithms for the eigenvalue eigenvector problem than the Jacobi eigensolver, which was originally chosen for simplicity of implementation. Techniques based on QR iteration with shifting work well for medium size matrices, and for large matrices, divide and conquer works best [Demm97, Golu89]. Unfortunately, these algorithms are more efficient the larger the problem size, and our problem size is extremely small; matrices. For such a small problem size, it turns out that brute force Jacobi iteration is as fast as we can reasonably expect, and has the advantage of being more stable ....

....algorithms are more efficient the larger the problem size, and our problem size is extremely small; matrices. For such a small problem size, it turns out that brute force Jacobi iteration is as fast as we can reasonably expect, and has the advantage of being more stable than the other routines [Demm97]. This is illustrated in Table 7, which shows the average performance of several eigensolvers over a number of random box shaped covariance matrices on a Pentium II class processor. Jacobi takes 60 of the time of the QR shift based algorithm, where both are hard coded for matrices, and both are ....

James W. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Norfolk, VA, 1997.

....if such x exists and is no larger than the uncertainty in the input, we are happy with the computed result. Then, we call the algorithm backward stable. If there is more than one possible value of x, we use the smallest one. The following de nition is from [3] Similar de nitions are given in [1, 2, 4] and other books. 1 De nition 1 (Backward Stable) A method for computing y = g(x) is called backward stable if, for any x, it produces a computed b y with a small backward error, that is, b y = g(x x) for some small x. The above de nition implicitly assumes that such x exists and says ....

J.M. Demmel. Applied Numerical Linear Algebra. The Society for Industrial and Applied Mathematics, 1997.

....L is unit lower triangular and D 2 = diag(d 2 i ) is diagonal. The problem (1.1) is then reduced to the form Cy j D Gamma1 L Gamma1 Pi T A PiL GammaT D Gamma1 y = y; y = DL T Pi T x: 1. 3) Any method for solving the symmetric eigenvalue problem can now be applied to C [5], 18] In LAPACK s xSYGV driver, 1.1) is solved by applying the QR algorithm to (1.3) MATLAB 6 s eig function does likewise when it is given a symmetric definite generalized eigenproblem. As is well known, when B is ill conditioned numerical stability can be lost in the Cholesky based method. ....

James W. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1997. xi+419 pp. ISBN 0-89871-389-7.

....However, this algorithm goes through the array twice and requires more computation than the straightforward version. Especially, division is usually many times more expensive than either addition or multiplication. It is a challenge to develop an algorithm satisfying the following criteria [1]: a) Reliability: It must compute the answer accurately, i.e. nearly all the computed digits must be correct, unless the answer is outside the range of normalized oating point numbers. CS760, S. Qiao Part 2 Page 7 (b) Eciency: It must be nearly as fast as the straightforward algorithm in most ....

James W. Demmel. Applied Numerica Linear Algebra. Society for Industrial and Applied Mathematics, 1997.

....an n Theta n matrix A and a vector b 2 R n , find x 2 R n such that Ax = b. Its condition number (with respect to the 2 norm) is defined as (A) k A k 2 fl fl A Gamma1 fl fl 2 . Comprehensive treatment of the perturbation theory for this problem can be found in the literature, such as [3] Section 2.2, 4] Chapter 7, 14] Lecture 12, etc. Theorem 1. Let A be an n Theta n matrix with integer coefficients. If A is invertible, then (A) n n 2 1 max i;j jA ij j n . No originality is claimed for Theorem 1. This result is included for completeness and because its ....

....second problem in the list is minimal squares fitting. Let A be an m Theta n matrix, m n, with full rank, and let b 2 R m . One has to find x to minimize k Ax Gamma b k 2 2 . Let r = Ax Gamma b be the residual, we are minimizing k r k 2 2 . Let sin = k r k 2 k b k 2 . According to [3] p. 117 (Compare to [14] Lecture 18 and [4] Section 19.1) the condition number of the linear least squares problem is LS (A; b) 2(A) cos tan (A) 2 . Since we do not assume A to be square, we need to give a new definition for (A) Let oe MAX (A) and oe MIN (A) be respectively the ....

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James W. Demmel. Applied numerical linear algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997.

.... , while IEEE double precision numbers used in most modern computers cannot contain oating point values more than 2 1024 , since the exponent is represented by 11 bits (sign included) 8] As a matter of fact, the representation is a little more complicated, as it allows for subnormal numbers [5]) Therefore, we would have an over ow when computing the 8 th Grae e iterate of f . Example 2. On the example above, assume that f would have an additional root 1:01. Namely, f(x) x 1) x 1:01) x 2) x 3) x 4) 24:24 74:5x 85:35x 2 45:1x 3 11:01x 4 x 5 We will show that 8 Grae e ....

J. W. Demmel, Applied numerical linear algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997.

....solving: given an n n matrix A and a vector b 2 R n , nd x 2 R n such that Ax = b. Its condition number (with respect to the 2 norm) is de ned as (A) k A k 2 A 1 2 . Comprehensive treatment of the perturbation theory for this problem can be found in the literature, such as [3] Section 2.2, 4] Chapter 7, 14] Lecture 12, etc. Theorem 1. Let A be an n n matrix with integer coe cients. If A is invertible, then (A) n n 2 1 max i;j jA ij j n . No originality is claimed for Theorem 1. This result is included for completeness and because its proof is ....

....Minimal squares. The second problem in the list is minimal squares tting. Let A be an m n matrix, m n, with full rank, and let b 2 R m . One has to nd x to minimize k Ax b k 2 2 . Let r = Ax b be the residual, we are minimizing k r k 2 2 . Let sin = k r k 2 k b k 2 . According to [3] p. 117 (Compare to [14] Lecture 18 and [4] Section 19.1) the condition number of the linear least squares problem is LS (A; b) 2 (A) cos tan (A) 2 . Since we do not assume A to be square, we need to give a new de nition for (A) Let MAX (A) and MIN (A) be respectively the ....

[Article contains additional citation context not shown here]

James W. Demmel. Applied numerical linear algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997.

....and a multigrid FMV cycle. Newton s method converges quadratically. However, since each step involves inverting a system, it tends to be very hile the use of the FMV solver speeds the method up somewhat, it still is slower than the techniques we present now. It has long been known ( 12] [13]) that on certain problems non linear analogs to the classical Jacobi or Gauss Seidel iteration methods could be employed with some success. Technically, one sweep of such a method means that for j = 1; 2; N 0 1 (or (N 0 1) 2 for the two dimensional problem) one solves, via the ....

James M. Ortega. Numerical analysis, a second course. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1979.

.... matrixT satisfies the recursion t(n) n 3 2t(n=2) which has the solution t(n) 4 3 n 3 O(n 2 ) In practice, because of deflation, it appears that the algorithm takes only O(n 2:3 ) flops on average and the cost can even be as low as O(n 2 ) for some special cases (see [10]) 3 Parallelization Issues Divide and conquer algorithms have been successfully implemented on shared memory multiprocessors for solving the symmetric tridiagonal eigenvalue problem and for the computation of the singular value decomposition of bidiagonal matrices [14] 24] By contrast, the ....

James W. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1997.

....1 shows some of the common trends for those kernels. Computational Algorithmic Software kernel choices (implementation) Hardware xGEMM triple nested loop, based on Level 1 vector processor, Strassen [14] BLAS, Level 2 superscalar RISC, Winograd [15] BLAS, or Level 3 VLIW processor BLAS [16] Solving a linear explicit inverse, left looking, right sequential, SMP, system of equa alecompositional looking, Crout [17] MPP, constellations method (e.g. LU, recursive [18,19] tions [2] QR, or LL T) Common trends in strategies for computational kernels. Another aspect that ....

J. W. Demmel, Applied Numerical Linear Algebra, Society for Industrial and Applied Mathematics, Philadelphia, 1997.

....interest since the 1980s because of its suitability for parallel implementation and its high accuracy properties. We describe Jacobi s method for the SVD, concentrating on its accuracy and stability. The perturbation theory and error analysis in this chapter is based on that in the book of Demmel [18] and gives a relatively accessible explanation of the accuracy properties of Jacobi methods. The research literature should be consulted for further details, of which there are many. The best starting points are Demmel and Veseli c [19] and Mathias [39] Recall that the SVD of A 2 R , where m ....

James W. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1997. xi+419 pp. ISBN 0-89871-389-7.

....chain must have started from its stationary distribution. Full rigor is postponed until Section 2.1. 2 This and other italicized terms are de ned in Section 2.1. 3 By traditional eigenvalue algorithms we refer to those found, for example, in Golub and Van Loan[5] See also the book by Demmel [1] for a more recent discussion. 4 space of the Markov process. This section and Chapter 3 develop the context in which we formulate the new ideas of the paper. In the last section of Chapter 3, Section 3.3, we present the familiar Krylov subspace and explain why this represents our best ....

....as the slowest modes of the process) However, this subspace spans only two dimensions of the entire d dimensional space, and it is more likely that 1 , at best, only comes close to lying in the subspace of interest. Now, given 1 , a judicious choice for the second dimension 2 , and hence 2 [ 1 2 ], would be that which makes max a6=0 j ( 2 a)j is as large as possible. To establish that this is indeed the pertinent objective, note the following: max a6=0 j ( 2 a; M)j max a6=0 a t t 2 M 2 a a t t 2 2 a = max x2ran( 2 ) j (x; M)j max j (x; M)j ....

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J. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, 1997.

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James Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, 1997.

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J. W. Demmel, Applied Numerical Linear Algebra, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1997.

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James W. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, 1997.

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J. W. Demmel, Applied Numerical Linear Algebra, Society for Industrial and Applied Mathematics, 1997.

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Demmel J.W., Applied Numerical Linear Algebra, Society of Industrial and Applied Mathematics, Philadelphia (1997).

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James W. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1997. xi+419 pp. ISBN 0-89871-3897.

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James W. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1997. xi+419 pp. ISBN 0-89871-3897.

No context found.

J. Demmel, Applied Numerical Linear Algebra, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1997.

No context found.

James W. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1997. xi+419 pp. ISBN 0-89871-3897.

No context found.

James W. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1997. xi+419 pp. ISBN 0-89871-3897.

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