N. J. Higham; Error analysis of the Bjorck-Pereyra algorithms for solving Vandermonde systems. Numer. Math. 50 (1987), 613--632.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....more accurate computations than those based on the use of general (structure ignoring) algorithms, say Gaussian elimination with pivoting. At the same time, such favorable numerical properties are much better understood for Vandermonde and related matrices (see, for example [BP70] TG81] CF88] [Hig87], Hig88] Hig90] RO91] CR92] CR93] V93] Ty94] as compared to the analysis of numerical issues related to Cauchy matrices (see [GK90] GK93] The Bj orck Pereyra algorithm for Vandermonde systems. In particular, most of the above mentioned papers were devoted to the analysis of ....

....the algorithm requires only O(n) locations of memory. Remarkable accuracy for monotonically ordered nodes. It turns out that along with dramatic speed up and savings in storage, the BP algorithm often produces a surprisingly high relative accuracy in the computed solution. N.J. Higham analyzed in [Hig87] the numerical performance of the BP algorithm and identified a class of Vandermonde matrices, viz. those for which the nodes can be strictly orderded, 0 x 1 x 2 : xn ; 1.4) with a favorable forward error bound, ja Gamma aj 5nuV (x 1:n ) jf j O(u ) 1.5) for the solution ....

N.J.Higham, Error analysis of the Bjorck--Pereyra algorithms for solving Vandermonde systems, Numer. Math., 50 (1987), 613 -- 632.

....matrix can be inverted by applying the Bjorck Pereyra algorithm to solve n linear systems, using the columns of identity matrix for the right hand sides. The latter O(n ) scheme is no longer fast. But, for the special case of positive and monotonically ordered points an error analysis of [Hig87] implies the following favorable bound : j j 5nujV (x) 0.4) Here stands for the inverse matrix computed by the Bjorck Pereyra algorithm, u is the machine precision, and the operation of taking the absolute value and comparison of matrices are understood in a componentwise ....

....the following favorable bound : j j 5nujV (x) 0.4) Here stands for the inverse matrix computed by the Bjorck Pereyra algorithm, u is the machine precision, and the operation of taking the absolute value and comparison of matrices are understood in a componentwise sense. In [Hig87] Higham used his pleasing bound (0.4) to argue that the fast O(n ) Traub algorithm is inaccurate, whereas the slow O(n ) Bjorck Pereyra algorithm is the other way around. To the best of our knowledge the possibility of simultaneously fast and accurate inversion of a Vandermonde matrix was ....

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N.Higham, Error analysis of the Bjorck--Pereyra algorithms for solving Vandermonde systems, Numerische mathematic, 50 (1987), 613 -- 632.

....solution of the least squares problem more efficient. One example is the matrix that arises from fitting polynomials using the power basis and equally spaced data points. The resulting matrix for the normal equations, a Vandermonde matrix, has beautiful structure but is quite ill conditioned [11, 9, 27, 47]. A second example is the band matrix structure that results from fitting functions whose support is local [80, 23] Wavelet [20] and Fourier bases often give matrices with small displacement rank [51] again leading to efficient solution algorithms [86, 24, 67, 44, 76] Some models give rise to ....

Nicholas J. Higham. Error analysis of the Bjorck-Pereyra algorithms for solving Vandermonde systems. Numer. Math., 50:613--632, 1987.

.... the system Ax = b: The method used is a direct method based on a sparse multifrontal variant of Gaussian elimination, discussed in detail by Duff and Reid [10] The method uses a parallel implementation of this multifrontal approach for shared memory computers which has been discussed in [7] [8], 2] 1] A multiprocessor version of the code is now available on the IBM 3090 6VF, the Alliant FX 80, the CRAY 2 and the CRAY Y MP. The multifrontal method belongs to a class of methods which separate the LU factorization into an analysis phase, a numerical factorization, and a solution step. ....

....Anal. and Applics. 10, 165 190. 6] Dongarra, J.J. Du Croz, J. Duff, I.S. and Hammarling, S. 1990) A set of level 3 Basic Linear Algebra Subprograms. ACM Trans. Math. Softw. 16, 1 17. 7] Duff, I.S. 1986) Parallel implementation of multifrontal schemes. Parallel computing 3, 193 204. [8] Duff, I.S. 1989) Multiprocessing a sparse matrix code on the Alliant FX 8. J. Comput. Appl. Math. 27, 229 239. 9] Duff, I.S. and Reid, J.K. 1983) The multifrontal solution of indefinite sparse symmetric linear systems. ACM Trans. Math. Softw. 9, 302 325. 10] Duff, I.S. and Reid, J.K. ....

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N. J. Higham. Error analysis of the Bjorck-Pereyra algorithm for solving Vandermonde systems. Numer. Math., 50:613-632, 1987.

....extrema of the Chebyshev polynomial T r 1 (x) shifted to [0; 1] Degrees 1 5 are typical of BDF and Runge Kutta codes. The higher degrees can occur in Adams codes. The condition numbers are uncomfortably large for the higher degrees. However, they are notoriously pessimistic, and the analysis of [7] shows that we should not expect the size of the condition number to be reflected in the errors in the coefficients a r in (4.1) as long as we use an appropriate algorithm for solving the van der Monde system. The general purpose package that we have written uses the nodes specified, except that ....

N.J. Higham, Error Analysis of the Bjorck-Pereyra Algorithms for Solving van der Monde Systems, Numer. Math., 50 (1987) 613--632.

....and Higham [12, 14] this method estimates kCk1 at the cost of forming a few matrix vector products Cx and C T y. We mention also two interesting nonlinear structures, those of Vandermonde matrices V = ff i Gamma1 j ) and Cauchy matrices H = Gamma (ff i fi j ) Gamma1 Delta . In [11] explicit expressions are derived for cond1 (V; x) and cond1 (V T ; x) in the case where f = 0 and g = 1; 1; 1) T . In [8] a structured condition number with respect to the inversion of H is derived. 4. Applications. In this section we look in detail at the structured componentwise ....

Nicholas J. Higham, Error analysis of the Bjorck-Pereyra algorithms for solving Vandermonde systems, Numer. Math., 50 (1987), pp. 613--632.

....the Vandermonde systems V x = b and V T a = f in O(n 2 ) operations, where V = ff i Gamma1 j ) 2 IR n Thetan . These algorithms have been used in various applications [1, 48, 56] have been generalized in several ways [6, 28, 31, 53] and their numerical stability has been investigated [26, 31, 58]. In [7] it was pointed out that the algorithms sometimes produce surprisingly accurate results, and an explanation for this was given in [26] However, analysis in [31] predicts that the algorithms can be moderately unstable, and an example of instability is given in [31] together with suggested ....

.... used in various applications [1, 48, 56] have been generalized in several ways [6, 28, 31, 53] and their numerical stability has been investigated [26, 31, 58] In [7] it was pointed out that the algorithms sometimes produce surprisingly accurate results, and an explanation for this was given in [26]. However, analysis in [31] predicts that the algorithms can be moderately unstable, and an example of instability is given in [31] together with suggested remedies. The example of [31] appears to be rare since no instances of instability of the Bjorck Pereyra algorithms were reported in the ....

Nicholas J. Higham, Error analysis of the Bjorck-Pereyra algorithms for solving Vandermonde systems, Numer. Math., 50 (1987), pp. 613--632.

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N. J. Higham; Error analysis of the Bjorck-Pereyra algorithms for solving Vandermonde systems. Numer. Math. 50 (1987), 613--632.

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Higham, N.J., Error analysis of the Bjorck--Pereyra algorithms for solving Vandermonde systems, Numer. Math. 50 (1987), 613--632.

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N. Higham, Error analysis of the Bjorck-Pereyra algorithms for solving Vandermonde systems, Numer. Math., 50 (1987), pp. 613--32.

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N. Higham, Error analysis of the Bjorck-Pereyra algorithms for solving Vandermonde systems, Numer. Math., 50 (1987), pp. 613--32.

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