W. Kahan. Accurate eigenvalues of a symmetric tridiagonal matrix. Technical Report CS41, Computer Science Department, Stanford University, Stanford, CA, 1966.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....FloatingCount(x) is also monotonic. A su#cient condition for floating point to be monotonic is that it be correctly rounded or correctly chopped; thus IEEE floating point arithmetic is monotonic. This result was first proven but not published by Kahan in 1966 for symmetric tridiagonal matrices [18]; in this paper we extend this result to symmetric acyclic matrices, a larger class including tridiagonal matrices, arrow matrices, and exponentially many others [10] see section 6. Our third result is to formalize the notion of a correct implementation of a bracketing algorithm, and use this ....

.... and also gives some serial and parallel algorithms that are provably correct subject to some assumptions about the computer arithmetic and FloatingCount(x) Section 5 reviews the roundo# error analysis of FloatingCount(x) and how to account for over underflow; this material may also be found in [10, 18]. Section 6 illustrates how monotonicity can fail, and proves that a natural serial implementation of FloatingCount(x) must be monotonic if the arithmetic is. Section 7 gives formal proofs for the correctness of the bracketing algorithms given in section 4. Section 8 discusses some practical ....

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W. Kahan, Accurate eigenvalues of a symmetric tridiagonal matrix, Computer Science Dept. Technical Report CS41, Stanford University, Stanford, CA, July 1966.

....Decomposition The standard method for computing the singular value decomposition is the Golub Reinsch algorithm, which reduces a full matrix to bidiagonal form B and then applies the QR algorithm implicitly to the tridiagonal matrix B B. In an unpublished technical report in the 1960s, Kahan [96] showed that the singular values of a bidiagonal matrix are determined to approximately the same relative accuracy as the elements of the matrix. Demmel and Kahan [42] use this result, together with rounding error analysis, to show that a zero shift version of the QR algorithm computes the ....

W. Kahan. Accurate eigenvalues of a symmetric tri-diagonal matrix. Technical Report No. CS41, Department of Computer Science, Stanford University,

....I . In bisection algorithms for computing eigenvalues, the inertia of J Gamma I is determined via the LDU factorisation if the matrix J is Hermitian (or can be transformed to a Hermitian matrix via diagonal similarity transformations) See Section 8.4. 1 of Golub and Van Loan [12] and Kahan [17]. We show that the inertia of J Gamma I can be obtained via the BABE factorisations; thus they can be used to compute eigenvalues via bisection by counting the signs of the pivots. This factorisation is also a vehicle for obtaining the norms of the inverse of J Gamma I which could be used in ....

....as given by the factorisations in floating point arithmetic assuming that certain reasonable axioms hold in floating point arithmetic. These axioms are always satisfied by hardware software which conform to the IEEE binary arithmetic standard [16] This is an extension of the results of Kahan [17] who studied the problem in the context of the pivots d i of the LDL t factorisation. The elements of the computed eigenvectors are determined by the pivots of the BABE factorisation; thus the accurate determination of the magnitudes and the signs of the pivots are paramount. 2 2 ....

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W. Kahan. Accurate eigenvalues of a symmetric tri-diagonal matrix. Technical Report CS 41, Computer Science Department, Stanford University, 1966.

....to overflow and other numerical problems. More recent implementations depend upon the application of Sylvester s inertia theorem to the LDL t factorisation of the shifted matrix. There are inexpensive ways to overcome overflow and other numerical problems in the factorisation method. See Kahan [15] and Barth et al. [3] Demmel and Gragg have extended the classes of matrices which are easily amenable to bisection type algorithms. See [5] Problems associated with bisection on parallel computers have recently been studied by Demmel et al. [7] In this report, we study the counting mechanism to ....

....Violation of monotonicity can lead to spurious and missing singular values. There are ways to overcome this problem on serial computers. See the routine SSTEBZ in LAPACK [1] However, it is rather difficult to cope with nonmonotonic algorithms on parallel computers. Following the work of Kahan [15], we show that the algorithm for the Golub Kahan form retains the monotonicity of the count if monotonic (e.g. IEEE) arithmetic is used. Unfortunately, the orthodox differential qd algorithms do not possess this property although the modified differential qd algorithms do give monotonic counts. ....

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W. Kahan. Accurate eigenvalues of a symmetric tri-diagonal matrix. Technical Report CS 41, Computer Science Department, Stanford University, 1966.

....of positive (negative) d i gives the number of eigenvalues of J which are greater (less) than . Thus the recursion (3) can be found in the inner loop of most bisection algorithms for finding eigenvalues of symmetric tridiagonal matrices. See Section 8.4. 1 of Golub and Van Loan [10] and Kahan [13]. Let the number of negative d i (the inertia count) be ( If the UDL factorisation of J Gamma I is defined as U( Delta( L( then the pivots ffi i ( the diagonal elements of Delta( are given by ffi i ( a i Gamma Gamma b i c i =ffi i 1 ( i = n Gamma 1; ....

....be derived if ffi i 1 = 0 for a particular i, i k 1. Remark 4 In floating point arithmetic, the d i and similarly the ffi i should be thresholded to if these quantities are tiny: if d i Gamma then d i ; j where j is the smallest representable number in the machine. See Kahan [13] for further details. If j k j is tiny, we could expect very good approximations to eigenvectors from Theorem 5. However, if j k j is not tiny then we are computing the eigenvectors of the perturbed matrix J Gamma k ( e k e k and in that case the computed eigenvectors will not closely ....

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W. Kahan. Accurate eigenvalues of a symmetric tri-diagonal matrix. Technical Report CS 41, Computer Science Department, Stanford University, 1966.

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W. Kahan, Accurate eigenvalues of a symmetric tridiagonal matrix, Computer Science Dept. Techn. Report CS41, Stanford University, Stanford, CA (1966.

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W. Kahan. Accurate eigenvalues of a symmetric tridiagonal matrix. Computer Science Dept. Technical Report CS41, Stanford University, Stanford, CA, July 1966.

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W. Kahan. Accurate eigenvalues of a symmetric tridiagonal matrix. Computer Science Dept. Technical Report CS41, Stanford University, Stanford, CA, July 1966.

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W. Kahan. Accurate eigenvalues of a symmetric tridiagonal matrix. Technical Report CS41, Computer Science Department, Stanford University, Stanford, CA, 1966.

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W. Kahan. Accurate eigenvalues of a symmetric tridiagonal matrix. Computer Science Dept. Technical Report CS41, Stanford University, Stanford, CA, July 1966.

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W. Kahan. Accurate eigenvalues of a symmetric tridiagonal matrix. Computer Science Dept. Technical Report CS41, Stanford University, Stanford, CA, July 1966.

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W. Kahan, Accurate eigenvalues of a symmetric tridiagonal matrix, Tech. Rep. CS 41, Stanford University, Computer Science Department, Stanford, California, 1966.

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W. Kahan , Accurate Eigenvalues of a Symmetric Tridiagonal Matrix, Technical Report CS41,

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