Givens rotation

In numerical linear algebra, a Givens rotation is a rotation in the plane spanned by two coordinates axes. Givens rotations are named after Wallace Givens, who introduced them to numerical analysts in the 1950s while he was working at Argonne National Laboratory. In numerical linear algebra, a Givens rotation is a rotation in the plane spanned by two coordinates axes. Givens rotations are named after Wallace Givens, who introduced them to numerical analysts in the 1950s while he was working at Argonne National Laboratory. A Givens rotation is represented by a matrix of the form where c = cos θ and s = sin θ appear at the intersections ith and jth rows and columns. That is, for fixed i > j, the non-zero elements of Givens matrix are given by: The product G(i, j, θ)x represents a counterclockwise rotation of the vector x in the (i, j) plane of θ radians, hence the name Givens rotation. The main use of Givens rotations in numerical linear algebra is to introduce zeros in vectors or matrices.This effect can, for example, be employed for computing the QR decomposition of a matrix. One advantage over Householder transformations is that they can easily be parallelised, and another is that often for very sparse matrices they have a lower operation count. When a Givens rotation matrix, G(i, j, θ), multiplies another matrix, A, from the left, G A, only rows i and j of A are affected. Thus we restrict attention to the following counterclockwise problem. Given a and b, find c = cos θ and s = sin θ such that where r = a 2 + b 2 {displaystyle r={sqrt {a^{2}+b^{2}}}} is the length of the vector ( a , b ) {displaystyle (a,b)} .Explicit calculation of θ is rarely necessary or desirable. Instead we directly seek c and s. An obvious solution would be However, the computation for r may overflow or underflow. An alternative formulation avoiding this problem (Golub & Van Loan 1996, §5.1.8) is implemented as the hypot function in many programming languages. The following fortran code is a minimalistic implementation of Givens rotation for real numbers. If the input values 'a' or 'b' are frequently zero, the code may be optimized to handle these cases as presented here.