By J. L. Synge
Basic advent for starting pupil of absolute differential calculus and for these attracted to functions of tensor calculus to mathematical physics and engineering. issues contain areas and tensors; easy operations in Riemannian area, curvature of area, unique varieties of house, relative tensors, principles of quantity, more.
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Hence we define the level number li of vertex i in the elimination tree to be the number of vertices on the chain joining the root and vertex i. The level numbers will be useful in setting up a mechanism for directly accessing the nonzeros in a row of H. We now provide a 6 x 6 example in Figures 11 and 12 to illustrate these ideas. It is easy to verify that the structures of its Householder matrix H and its triangular X X X A= X X X X X X X X X X Figure 11: Structure of the matrix A. factor R are as shown in Figure 12.
Since P APT is also symmetric and positive definite for any permutation matrix P, this means we can choose to reorder A symmetrically to reduce fill without regard to numerical stability and before the actual numerical factorization begins. These options, which are not normally available to us when A is a general indefinite matrix, have enormous practical implications. Since the ordering can be determined before the factorization begins, the locations of the fill suffered during the factorization can also be determined.
With P = n, the calculation of Ax and Ad requires n parallel steps rTd, dTAd, rTr requires log n parallel steps. >E do 26 So one iteration costs 2n + 3 log n + 4 steps. If p = n 2 we can do all the multiplications in Ax and Ad in parallel so one iteration is only 3 log n + 6 steps. 6 Preconditioning In practice on a sequential machine both Jacobi and Conjugate Gradients are preconditioned before use. In the Jacobi method this involves the use of either a matrix C or a matrix M where the equation becomes CAx = Cb or M- 1Ax M- 1b.
Tensor calculus by J. L. Synge