L. Fox, H. D. Huskey and J. H. Wilkinson, Notes on the solution of algebraic linear simultaneous equations, J. Mech. Appl. Math., 1(1948), 149-173.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....lin eaires. Il semble que la m ethode de Cholesky fut ensuite exhum ee par John Todd qui l exposait dans son cours d analyse num erique au King s College a Londres d es 1946 [10] Todd la porta a l attention de Leslie Fox, Harry D. Huskey et James H. Wilkinson qui en firent la premi ere analyse [8]. Sa stabilit e num erique fut simultan ement etudi ee par Alan Turing [11] voir [9] pour un travail plus r ecent et plus complet sur cette question) Les travaux de Cholesky ont et e analys es en d etail dans [7, pp.347 351] et il n y a donc pas lieu de reprendre ici leur discussion. 2 La ....

L. Fox, H.D. Huskey, J.H. Wilkinson, Notes on the solution of algebraic linear simultaneous equations, Quart. J. Mech. Appl. Math., 1 (1948) 149--173.

....gradients and projection methods. More recently, those familiar with null space methods will also see a connection. To give an idea of the frequency with which these ideas have been discovered and rediscovered, some the earlier papers are itemized in the following table Fox, Huskey and Wilkinson [9] 1948 Hestenes and Stiefel [12] 1952 Purcell [16] 1953 Householder [13] 1955 Pietrzykowski [15] 1960 Faddeev and Faddeeva [7] 1963 Stewart [19] 1973 Enderson and Wassyng [6] 1978 Sloboda [18] 1978 Wassyng [23] 1982 Abaffy, Broyden and Spedicato [1] 1984 Hegedus [11] 1986 Most of the papers develop ....

Fox L., Huskey H.D. and Wilkinson J.H. (1948), Notes on the solution of algebraic linear simultaneous equations, Quart. J. Mech. Appl. Math., 1, 149-173.

....making a trivial change in the analysis Turing s bound can be made proportional only to kA Gamma1 k1 . Turing also showed that the factor 4 n Gamma1 in Hotelling s bound can be improved to 2 n Gamma1 and that still the bound is attained only in exceptional cases. Fox, Huskey and Wilkinson [18] presented empirical evidence in support of GE, commenting that in our practical experience on matrices of orders up to the twentieth, some of them very ill conditioned, the errors were in fact quite small . A major breakthrough in the error analysis of GE came with Wilkinson s pioneering ....

L. Fox, H.D. Huskey and J.H. Wilkinson, Notes on the solution of algebraic linear simultaneous equations, Quart. J. Mech. and Applied Math., 1 (1948), pp. 149--173.

.... [15] Pietrzykowski [14] Householder [12] 13] Faddeev and Faddeeva [9] Stewart [17] Sloboda [16] and Aba#y, Broyden, and Spedicato [1] For symmetric positive definite matrices, the algorithm essentially becomes a conjugate direction method already considered by Fox, Huskey, and Wilkinson [10] and by Hestenes and Stiefel [11] Other references together with a brief historical overview of this general class of projection methods can be found in [2] It was recognized early on that DPM can be interpreted in terms of an implicit triangular factorization of A . If the null vector ....

....positive definite case DPM becomes an orthogonalization method; the null vectors are obtained from the unit vectors e 1 , e 2 , e n by the Gram Schmidt orthogonalization process with inner product #x, y# = x T Ay. This procedure was first proposed by Fox, Huskey, and Wilkinson [10] and also studied by Hestenes and Stiefel [11] It is worthwhile observing that DPM can also be related to orthogonalization schemes in the following way. Let Z be any nonsingular matrix such that AZ = L where L is a lower triangular matrix. Then the columns of Z form a set of null vectors for A ....

L. Fox, H. D. Huskey, and J. H. Wilkinson, Notes on the solution of algebraic linear simultaneous equations, Quart. J. Mech. Appl. Math., 1 (1948), pp. 149--173.

....: z n ] W = w 1 ; w 2 ; w n ] and D = diag(p 1 ; p 2 ; p n ) This algorithm can be interpreted as a (two sided) generalized Gram Schmidt orthogonalization process with respect to the bilinear form associated with A. Some references on this kind of algorithm are [20] 30] 44] [45]. If A is SPD, only the process for Z need to be carried out (since in this case W = Z) and the algorithm is just a conjugate Gram Schmidt process, i.e. orthogonalization of the unit vectors with respect to the energy inner product hx; yi : x T Ay. It is easy to see that, in exact ....

L. Fox, H. D. Huskey, and J. H. Wilkinson. Notes on the solution of algebraic linear simultaneous equations. Quarterly Journal of Mechanics and Applied Mathematics, 1:149--173, 1948.

.... From the proof of the corollary the Cholesky factor is AU and not U with the latter being related to the inverse of the Cholesky factor of A. The biconjugation process here is equivalent to Gram Schmidt in the inner product x; y : y T Ax which was considered by Fox, Huskey and Wilkinson [6] and later by Hestenes and Stiefel [12] in their conjugate gradient paper. The A biconjugation process may be thought of as a function f acting on the space R n Thetak Theta R m Thetak by f(X; Y ) U; V ) 21) where U and V are a biconjugate pair of matrices that result from the ....

....of this paper. Dr. Michele Benzi and Professor Carl Meyer noticed the connection of our work with Stewart s [16] and we thank them for calling our attention to it. Dr. Benzi also pointed out the relation of the Cholesky factorization of Corollary 3. 5 to the work of Fox, Huskey and Wilkinson [6] and that of Hestenes and Stiefel [12] Related work by Dr. Benzi and Professor Meyer (associated with preconditioners among other issues) will appear in forthcoming publications. ....

L. Fox, H. D. Huskey and J. H. Wilkinson, Notes on the solution of algebraic linear simultaneous equations, J. Mech. Appl. Math., 1(1948), 149-173.

....and the p (i Gamma1) j . The columns of the resulting Z form a set of conjugate directions for A. If A is SPD, no breakdown can occur (in exact arithmetic) so that pivoting is not required and the algorithm computes the L T DL factorization of A Gamma1 . This method was first described in [26]. Geometrically, it amounts to Gram Schmidt orthogonalization with inner product hx; yi : x T Ay applied to the unit vectors e 1 ; e n . It is sometimes referred to as the conjugate Gram Schmidt process . The method is still impractical as a direct solver because it requires about twice ....

L. Fox, H. D. Huskey and J. H. Wilkinson. Notes on the solution of algebraic linear simultaneous equations. Quart. J. Mech. Appl. Math., 1:149--173, 1948.

....Lanczos, Paige, Rosser, Stein, and others. He specifically credited Forsythe and Rosser for the 3 term recurrence and L. J. Paige for the usual 2 term recurrence. The conjugate direction algorithm is somewhat older, discussed in a 1948 paper by Leslie Fox, H. D. Huskey, and Jim Wilkinson [22]. Meanwhile, Eduard Stiefel of E.T.H. Zurich, visited the INA, and presented a paper on the conjugate gradient algorithm at a workshop in August, 1951. His description of the n step iteration was published in 1952 and noted the connection with the Lanczos (1950) work [49] Hestenes and Stiefel ....

L. Fox, H. D. Huskey, and J. H. Wilkinson, Notes on the solution of algebraic linear simultaneous equations, Quart. J. of Mech. and Appl. Math., 1 (1948), pp. 149--173.

....for conjugate gradient 1137 feature for cases where the matrix is only implicitly given as an operator. Once Z and D are available, the solution of Ax = b can be computed as x # = A 1 b = ZD 1 Z T b = n X i=1 z T i b p i z i . A similar algorithm was first proposed in [14]; see also [13, 18] Further references and a few historical notes can be found in [3, 4] For a dense matrix this method requires roughly twice as much work as Cholesky. For a sparse matrix the cost can be substantially reduced, but the method is still impractical because the resulting Z tends ....

L. Fox, H. D. Huskey, and J. H. Wilkinson, Notes on the solution of algebraic linear simultaneous equations, Quart. J. Mech. Appl. Math., 1 (1948), pp. 149--173.

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L. Fox, H. D. Huskey and J. H. Wilkinson, Notes on the solution of algebraic linear simultaneous equations, J. Mech. Appl. Math., 1(1948), 149-173.

No context found.

L. Fox, H.D. Huskey, J.H. Wilkinson, Notes on the solution of algebraic linear simultaneous equations, Quart. J. Mech. Appl. Math., 1 (1948) 149--173.

No context found.

L. Fox, H.D. Huskey, J.H. Wilkinson, Notes on the solution of algebraic linear simultaneous equations, Quart. J. Mech. Appl. Math., 1 (1948) 149-173.

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