14.14 Givens Rotations

A Givens rotation is a rotation in the plane acting on two elements of a given vector. It can be represented in matrix form as

where the \cos{\theta} and \sin{\theta} appear at the intersection of the ith and jth rows and columns. When acting on a vector x, G(i,j,\theta) x performs a rotation of the (i,j) elements of x. Givens rotations are typically used to introduce zeros in vectors, such as during the QR decomposition of a matrix. In this case, it is typically desired to find c and s such that

with r = \sqrt{a^2 + b^2}.

Function: void gsl_linalg_givens (const double a, const double b, double * c, double * s)

This function computes c = \cos{\theta} and s = \sin{\theta} so that the Givens matrix G(\theta) acting on the vector (a,b) produces (r, 0), with r = \sqrt{a^2 + b^2}.

Function: void gsl_linalg_givens_gv (gsl_vector * v, const size_t i, const size_t j, const double c, const double s)

This function applies the Givens rotation defined by c = \cos{\theta} and s = \sin{\theta} to the i and j elements of v. On output, (v(i),v(j)) \leftarrow G(\theta) (v(i),v(j)).