Search results
Results From The WOW.Com Content Network
In linear algebra, a Householder transformation (also known as a Householder reflection or elementary reflector) is a linear transformation that describes a reflection about a plane or hyperplane containing the origin. The Householder transformation was used in a 1958 paper by Alston Scott Householder. [1]
In linear algebra, the Householder operator is defined as follows. [1] Let be a finite-dimensional inner product space with inner product , and unit ...
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.
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 GSL also offers an alternative method that uses a one-sided Jacobi orthogonalization in step 2 (GSL Team 2007). This method computes the SVD of the bidiagonal matrix by solving a sequence of 2 × 2 {\displaystyle 2\times 2} SVD problems, similar to how the Jacobi eigenvalue algorithm solves a sequence of 2 × 2 {\displaystyle 2 ...
In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace.Formally, starting with a linearly independent set of vectors {v 1, ... , v k} in an inner product space (most commonly the Euclidean space R n), orthogonalization results in a set of orthogonal vectors {u 1, ... , u k} that generate the same subspace as the vectors v 1 ...
Orthogonalization algorithms: Gram–Schmidt process; Householder transformation. Householder operator — analogue of Householder transformation for general inner product spaces; Givens rotation; Krylov subspace; Block matrix pseudoinverse; Bidiagonalization; Cuthill–McKee algorithm — permutes rows/columns in sparse matrix to yield a ...
A Householder reflection is typically used to simultaneously zero the lower part of a column. Any orthogonal matrix of size n × n can be constructed as a product of at most n such reflections. A Givens rotation acts on a two-dimensional (planar) subspace spanned by two coordinate axes, rotating by a chosen angle. It is typically used to zero a ...