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In linear algebra, a Householder transformation ... the given matrix is transformed to the similar tridiagonal matrix A 3 by using the Householder method. = ...
A tridiagonal matrix is a matrix that is both upper and lower Hessenberg matrix. [2] In particular, a tridiagonal matrix is a direct sum of p 1-by-1 and q 2-by-2 matrices such that p + q/2 = n — the dimension of the tridiagonal.
In fact, reduction of any matrix to a Hessenberg form can be achieved in a finite number of steps (for example, through Householder's transformation of unitary similarity transforms). Subsequent reduction of Hessenberg matrix to a triangular matrix can be achieved through iterative procedures, such as shifted QR-factorization.
A Householder reflection (or Householder transformation) is a transformation that takes a vector and reflects it about some plane or hyperplane. We can use this operation to calculate the QR factorization of an m-by-n matrix with m ≥ n. Q can be used to reflect a vector in such a way that all coordinates but one disappear.
The matrix is first brought to upper Hessenberg form = as in the explicit version; then, at each step, the first column of is transformed via a small-size Householder similarity transformation to the first column of () [clarification needed] (or ()), where (), of degree , is the polynomial that defines the shifting strategy (often ...
This new matrix A 3 is the upper triangular matrix needed to perform an iteration of the QR decomposition. Q is now formed using the transpose of the rotation matrices in the following manner: Q = G 1 T G 2 T . {\displaystyle Q=G_{1}^{T}\,G_{2}^{T}.}
the first has somehow, in some way, been my best year yet. So, as I often say to participants in the workshop, “If a school teacher from Nebraska can do it, so can you!”
In mathematics, and more specifically in numerical analysis, Householder's methods are a class of root-finding algorithms that are used for functions of one real variable with continuous derivatives up to some order d + 1. Each of these methods is characterized by the number d, which is known as the order of the method.