vecTomat

International Journal of Computer Mathematics

Communications of The ACM, 1973

This algorithm uses a rational variant of the QR transformation with explicit shift for the computation of all of the eigenvalues of a real, symmetric, and tridiagonal matrix. Details are described in Ill. Procedures tredl or tred3 published in [2] may be used to reduce any real, symmetric matrix to tridiagonal form. Turn the matrix end-for-end if necessary to bring very large entries to the bottom right-hand corner. References h := g--h;f:=f+h; for i := k d-1 step 1 until n do d[i] := d[i] --h; comment Rational QL transformation, rows k through m; g := d[m]; if g = 0.0 then g := b; h := g; s2 := 0.0; fori:= m --1 step --1 untilkdo begin

This article introduces two vectors intended to formalize some triadic transformations, considering specially the Chromatic Transformational System by David KOPP (2002). The numeric content of vector K describes concisely the processes associated to a given operation that must be applied for transforming a referential perfect triad onto a derived one. Vector G informs the spatial position of an operation considering its geometric projection on a referential two-dimensional plan (a Tonnetz). A practical application concerning analysis by computational means is presented in the last section of the study.

Per Musi, 2018

This article introduces two vectors intended to formalize some triadic transformations, considering specially the Chromatic Transformational System by David KOPP (2002). The numeric content of vector K describes concisely the processes associated to a given operation that must be applied for transforming a referential perfect triad onto a derived one. Vector G informs the spatial position of an operation considering its geometric projection on a referential two-dimensional plan (a Tonnetz). A practical application concerning analysis by computational means is presented in the last section of the study.

American Journal of Computational and Applied Mathematics, 2019

This article presents a special case of symmetric matrices, matrices of transpositions (Tr matrices) that are created from the elements of given n-dimensional vector XR n , n=2 m , mN. Has been proposed algorithm for obtaining matrices of transpositions with mutually orthogonal rows (Trs matrices) of dimensions 2, 4, and 8 as Hadamard product of Tr matrix and matrix of Hadamard and has been investigated their application for QR decomposition and n-dimensional rotation matrix generation. Tests and analysis of the algorithm show that obtaining an orthogonal Trs matrix of sizes 4 and 8 that rotates a given vector to the direction of one of the coordinate axes requires less processing time than obtaining a Housholder matrix of the same size.

Linear Algebra and its Applications, 2004

Many numerical methods produce sequences of vectors converging to the solution of a problem. When the convergence is slow, the sequence can be transformed into a new vector sequence which, under some assumptions, converges faster to the same limit. The construction of a sequence transformation is based on its kernel, that is the set of sequences which are transformed into a constant sequence. In this paper, new vector sequence transformations are built from kernels which extent those of the most general transformations known so far.

Linear and Multilinear Algebra, 1981

The vec-permutation matrix I m, n is defined by the equation vec A m?? n= I m, n vecA???, Where vec is the vec operator such that vecA is the vector of columns of A stacked one under the other. The variety of definitions, names and notations for I m, n are discussed, ...

Kalman Filtering, 2002

The scalars a ij are called the elements of A. We will use upper case letters to denote matrices and the corresponding lowercase letters to denote scalar elements of the associated matrices.

Linear Algebra and its Applications, 2000

We present a method for the multiplication of an arbitrary vector by a symmetric centrosymmetric matrix, requiring 5 4 n 2 + O(n) floating-point operations, rather than the 2n 2 operations needed in the case of an arbitrary matrix. Combining this method with Trench's algorithm for Toeplitz matrix inversion yields a method for solving Toeplitz systems with the same complexity as Levinson's algorithm.

Advances in Linear Algebra & Matrix Theory, 2014

This paper presents a new technique for expressing rhotrices in a generalize form. The method involves using multiple array indexes as analogous to matrix expressions, unlike the earlier method in the literature, which can only be functional in a single array computational environment. The new rhotrix look will encourage the study of rhotrix algebra and analysis from a better perspective. In addition, computing efficiency and accuracy will also be improved, particularly when the operations in rhotrix space over the new expression are algorithmatized for computing machines.