Difference between revisions of "Givens method"
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1.1 General description of the algorithm | 1.1 General description of the algorithm | ||
| − | '''Givens' method''' (which is also called '''the rotation method''' in the Russian mathematical literature) is used to represent a matrix in the form ''A = QR'', where ''Q'' is a unitary and ''R'' is an upper triangular matrix. The matrix Q is not stored and used in its explicit form but rather as the product of rotations. Each (Givens) rotation can be specified by a pair of indices and a single parameter. In a conventional implementation of Givens' method, this fact makes it possible to avoid using additional arrays by storing the results of decomposition in the array originally occupied by ''A''. | + | '''Givens' method''' (which is also called '''the rotation method''' in the Russian mathematical literature) is used to represent a matrix in the form ''A = QR'', where ''Q'' is a unitary and ''R'' is an upper triangular matrix. The matrix Q is not stored and used in its explicit form but rather as the product of rotations. Each (Givens) rotation can be specified by a pair of indices and a single parameter. In a conventional implementation of Givens' method, this fact makes it possible to avoid using additional arrays by storing the results of decomposition in the array originally occupied by ''A''. Various uses are possible for the ''QR'' decomposition of ''A''. It can be used for solving a SLAE (System of Linear Algebraic Equations) ''Ax = b'' or as a step in the so-called ''QR'' algorithm for finding the eigenvalues of a matrix. |
Revision as of 19:10, 21 January 2016
Primary authors of this description: A.V.Frolov, Vad.V.Voevodin (Section 2.2)
1 Properties and structure of the algorithm[edit]
1.1 General description of the algorithm
Givens' method (which is also called the rotation method in the Russian mathematical literature) is used to represent a matrix in the form A = QR, where Q is a unitary and R is an upper triangular matrix. The matrix Q is not stored and used in its explicit form but rather as the product of rotations. Each (Givens) rotation can be specified by a pair of indices and a single parameter. In a conventional implementation of Givens' method, this fact makes it possible to avoid using additional arrays by storing the results of decomposition in the array originally occupied by A. Various uses are possible for the QR decomposition of A. It can be used for solving a SLAE (System of Linear Algebraic Equations) Ax = b or as a step in the so-called QR algorithm for finding the eigenvalues of a matrix.
