Editing Singular value decomposition (section)

=== Two-sided Jacobi algorithm ===
Two-sided Jacobi SVD algorithm—a generalization of the [[Jacobi eigenvalue algorithm]]—is an iterative algorithm where a square matrix is iteratively transformed into a diagonal matrix. If the matrix is not square the [[QR decomposition]] is performed first and then the algorithm is applied to the <math>R</math> matrix. The elementary iteration zeroes a pair of off-diagonal elements by first applying a [[Givens rotation]] to symmetrize the pair of elements and then applying a [[Jacobi transformation]] to zero them,

<math display=block>
M \leftarrow J^TGMJ
</math>

where <math>G</math> is the Givens rotation matrix with the angle chosen such that the given pair of off-diagonal elements become equal after the rotation, and where <math>J</math> is the Jacobi transformation matrix that zeroes these off-diagonal elements. The iterations proceeds exactly as in the Jacobi eigenvalue algorithm: by cyclic sweeps over all off-diagonal elements.

After the algorithm has converged the resulting diagonal matrix contains the singular values.
The matrices <math>U</math> and <math>V</math> are accumulated as follows:
<math>U\leftarrow UG^TJ</math>,
<math>V\leftarrow VJ</math>.