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The following table gives the number of operations in the k-th step of the QR-decomposition by the Householder transformation, assuming a square matrix with size n. Operation Number of operations in the k -th step
An RRQR factorization or rank-revealing QR factorization is a matrix decomposition algorithm based on the QR factorization which can be used to determine the rank of a matrix. [1] The singular value decomposition can be used to generate an RRQR, but it is not an efficient method to do so. [2] An RRQR implementation is available in MATLAB. [3]
Instead, the QR algorithm works with a complete basis of vectors, using QR decomposition to renormalize (and orthogonalize). For a symmetric matrix A , upon convergence, AQ = QΛ , where Λ is the diagonal matrix of eigenvalues to which A converged, and where Q is a composite of all the orthogonal similarity transforms required to get there.
The QR factorization of a matrix is a matrix and a matrix so that A = QR, where Q is orthogonal and R is upper triangular. [1]: 50 [4]: 223 The two main algorithms for computing QR factorizations are the Gram–Schmidt process and the Householder transformation.
In the theory of Lie group decompositions, it is generalized by the Iwasawa decomposition. The application of the Gram–Schmidt process to the column vectors of a full column rank matrix yields the QR decomposition (it is decomposed into an orthogonal and a triangular matrix ).
QR decomposition should have a link to here, but they are different. QR iteration (this page) uses QR decomposition to find eigenvalues. If anything, this article could be expanded into a series of articles including multishift versions and the modern forms of the algorithm that do the QR decomposition only in a very hidden implicit form.--
In eigenvalue algorithms, the Hessenberg matrix can be further reduced to a triangular matrix through Shifted QR-factorization combined with deflation steps. Reducing a general matrix to a Hessenberg matrix and then reducing further to a triangular matrix, instead of directly reducing a general matrix to a triangular matrix, often economizes ...
In sparse direct solvers, pivoting may be needed, where ultimately the resulting matrix has 2x2 blocks on the diagonal, [3] rather than a working towards a completely pure LL H Cholesky decomposition for positive definite symmetric or Hermitian systems. Pivoting may result in unpredictable memory usage increases.