G. E. Forsythe and E. G. Straus (1955), `On best conditioned matrices', Proc. Amer. Math. Soc. 6, 340-345.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for A. We find that for matrices A such that the product FAF has Property A, the best conditioned circulant matrix is just the optimal circulant preconditioner C F . The following two Lemmas relate matrices with Property A to the best conditioned circulant matrix. Lemma 1 (Forsythe and Straus [8]) Let Q be a Hermitian positive definite matrix of the form Q = I p I q where I p and I q are identity matrices of order p and q respectively. Then (Q) Q) for any diagonal matrix . 4 Lemma 2 (Chan, Jin and Yeung [5] Let A be an n by n matrix and C F be the optimal circulant ....

G. Forsythe and E. Straus, On Best Conditioned Matrices, Proc. Amer. Math. Soc. Vol. 6 (1955), pp. 340-345.

....D in (1.1) to transform to a linear system involving the correlation matrix H. The motivation is the result (1.2) since one of the aims of preconditioning is to reduce the condition number 2 (A) For some matrices arising in PDE applications, D in (1. 1) is an optimal scaling: Forsythe and Straus [11] show that 2 (H) is minimal if A is symmetric positive definite with property A (that is, there exists a permutation matrix P such that PAP can be expressed as a block 2 Theta 2 matrix whose (1; 1) and (2; 2) blocks are diagonal) Thus, for example, any symmetric positive definite block ....

G. E. Forsythe and E. G. Straus. On best conditioned matrices. Proc. Amer. Math. Soc., 6:340--345, 1955.

....orthogonal polynomials , 12] functions of the measurements and the weighting) to represent respectively the numerator and denominator of the rational form, 22) the matrix left hand side of the normal equation system becomes: 23) with and unity matrices of the appropriate size. Reference [13] shows that (23) is best conditioned in the sense that no other rational function model can be found with a better conditioned set of normal equations. Once the appropriate basis is chosen, it must not be changed during the iterative minimization of the non linear cost function to guarantee ....

Forsythe G.E. and E.G. Straus, "On Best Conditioned Matrices.", Proceedings of the American Mathematical Society, Vol.6, pp.340345, 1955.

....simply a diagonal scaling of H 0 , and given an accurate estimate of #, this diagonal scaling will improve the condition number of H 0 , as we discuss next. 4.1. Diagonal scaling and improved condition number. Results on improvement of condition number of a matrix by diagonal scaling are scarce [2, 9, 16]. Rather than present a general proof that the diagonal scaling presented above improves the condition number of H 0 , we demonstrate this explicitly for a canonical case. Namely, let the vertices z j be the nth roots of unity: z j = exp(ij#) where # =2# n. The Vandermonde matrix V n is then ....

G. Forsythe and E. Straus, On best conditioned matrices, Proc. Amer. Math. Soc., 6 (1955), pp. 340--345.

....matrix will make all diagonal elements of the preconditioned Hessian equal to unity, one may assume that the condition number of the preconditioned Hessian will be greatly improved. This is especially true in the case when the Hessian matrix is strongly diagonally dominant. Besides, Forsythe et al. 1955) have shown that using the diagonal of the Hessian is optimal among all diagonal preconditioning methods. 3.3. Observations We assume that the cost function measuring misfit between forecast model solution and available observations is of the form given by Eq. 3.1.2) Introducing a bogus ....

Forsythe, G. E., and E. G. Strauss, 1955: On best conditioned matrices, Proc. Ameri. Math. Soc., 6, 340-345.

....in (1.1) to transform to a linear system involving the correlation matrix H . The motivation is the result (1.2) since one of the aims of preconditioning is to reduce the condition number 2 (A) For some matrices arising in PDE applications, D in (1. 1) is an optimal scaling: Forsythe and Straus [11] show that 2 (H) is minimal if A is symmetric positive definite with property A (that is, there exists a permutation matrix P such that PAP T can be expressed as a block 2 Theta 2 matrix whose (1; 1) and (2; 2) blocks are diagonal) Thus, for example, any symmetric positive definite block ....

G. E. Forsythe and E. G. Straus. On best conditioned matrices. Proc. Amer. Math. Soc., 6:340--345, 1955.

....is simply a diagonal scaling of H 0 , and given an accurate estimate of ae, this diagonal scaling will improve the condition number of H 0 ,aswe discuss next. 4.1. Diagonal scaling and improved condition number. Results on improvement of condition number of a matrix by diagonal scaling are scarce [16,9,2]. Rather than present a general proof that the diagonal scaling presented above improves the condition number of H 0 , we demonstrate this explicitly for a canonical 2 One may estimate ae using a variety of other techniques. For instance, the ratio k 1 = k converges to ae#orwe can approximate ....

G.Forsythe and E.Straus, On best conditioned matrices, Proc. Amer. Math. Soc., 6 (1955), pp.340--345.

....D in (1.1) to transform to a linear system involving the correlation matrix H. The motivation is the result (1.2) since one of the aims of preconditioning is to reduce the condition number 2 (A) For some matrices arising in PDE applications, D in (1. 1) is an optimal scaling: Forsythe and Straus [11] show that 2 (H) is minimal if A is symmetric positive definite with property A (that is, there exists a permutation matrix P such that PAP T can be expressed as a block 2 Theta 2 matrix whose (1; 1) and (2; 2) blocks are diagonal) Thus, for example, any symmetric positive definite block ....

G. E. Forsythe and E. G. Straus. On best conditioned matrices. Proc. Amer. Math. Soc., 6:340--345, 1955.

.... Gamma1 A) and min (C Gamma1 A) respectively, denote the largest and smallest eigenvalues of C Gamma1 A. This problem has been studied extensively in the literature, since it is important for several applications, such as the eigenvalue problem [9, 13] scaling problems in linear systems [17, 5, 23, 12] and the minimal factorization problem in control systems [4, 25] In fact, the problem is related to that of best conditioning of a matrix if its columns are constrained to span certain subspaces [25, 9] The following theorem of Demmel [9] shows that if we take the standard block Jacobi ....

G. E. Forsythe and E. G. Strauss. On best conditioned matrices. Proc. Amer. Math. Soc., 6:340--345, 1955.

....simply a diagonal scaling of H 0 , and given an accurate estimate of ae, this diagonal scaling will improve the condition number of H 0 , as we discuss next. 4.1. Diagonal scaling and improved condition number. Results on improvement of condition number of a matrix by diagonal scaling are scarce [16,9,2]. Rather than present a general proof that the diagonal scaling presented above improves the condition number of H 0 , we demonstrate this explicitly for a canonical case. Namely, let the vertices z j be the n th roots of unity: z j = exp(ij ) where = 2 =n. The Vandermonde matrix Vn is then ....

G.Forsythe and E.Straus, On best conditioned matrices, Proc. Amer. Math. Soc., 6 (1955), pp.340--345.

....the many preconditioners have been proposed or used in practice, there are not as many theoretic results known for these types of problems. In the context of minimizing the condition number (C Gamma1 A) where C is a preconditioning matrix, one classical result due to Forsythe and Strauss [58, 74] is worth mentioning. Theorem 2.3. Assume that a matrix A is symmetric positive definite and can be symmetrically permuted into the form D 1 B B T D 2 where D 1 and D 2 are diagonal matrices. Let D denote the diagonal of A. Then (D Gamma1=2 AD Gamma1=2 ) b DA b D) 32) for ....

G. E. FORSYTHE and E. G. STRAUSS, On best conditioned matrices, Proc. Amer. Math. Soc., 6 (1955), pp. 340-345.

....have been made for the construction of vectorizable preconditioners. The simplest is diagonal scaling, where the matrix A is scaled symmetrically so that the diagonal of the scaled matrix has unit entries. This is known to be quite effective, since it helps to reduce the condition number (Forsythe and Strauss 1955, van der Sluis 1969) and often has a beneficial influence on the convergence behavior. On some vector computers, the computational speed of the resulting iterative method (without any further preconditioning) is so high that it is often competitive with many of the approaches that have been ....

Forsythe, G. E. and Strauss, E. G. (1955), `On best conditioned matrices', Proc.Amer.Math.Soc. 6, 340--345.

....of solving problem (5.1) we will formulate our iterative method for the problem Ax = b (5.2) with the diagonally scaled matrix A = D Gamma1=2 A D Gamma1=2 ; where D = diag(A ) is the diagonal of A . This diagonal scaling reduces the spectral condition number of the problem (cf. [40]) and nondimensionalizes the system of equations [48] This last aspect is especially useful when solving problems mixing degrees of freedom having different physical interpretation, such as the problems of plates and shells (cf. 56] As the new matrix A is independent of the scaling of basis ....

G. E. Forsythe and E. G. Strauss, On the best conditioned matrices, Proc. Amer. Math. Soc., 6 (1955), pp. 340--345.

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G. E. Forsythe and E. G. Straus (1955), `On best conditioned matrices', Proc. Amer. Math. Soc. 6, 340-345.

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Forsythe, G. E., Straus, E.G.: On Best Conditioned Matrices. Proceedings of the American Mathematical Society, 6 (1955) 340-345

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Forsythe, G. E. and Straus, E. G., On best conditioned matrices, Proc. Amer. Math. Soc., 6 (1955), pp. 340--345.

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Forsythe ,G. E. and E. G. Strauss,1955: On best conditioned matrices.Proc. Amer. Math. Soc., 6, 340-345.

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Forsythe ,G.E. and E.G. Strauss,1955: On best conditioned matrices.Proc. Amer. Math. Soc., 6, 340-345.

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