An improvement of the Jacobi singular value decomposition algorithm is proposed. The matrix is first reduced to a triangular form. It is shown that the row-cyclic strategy preserves the triangularity. Further improvements lie in the convergence properties. It is shown that the method converges globally and a proof of the quadratic convergence is indicated as well. The numerical experiments confirm these theoretical predictions. Our method is about 2-3 times slower than the standard QR method but it almost reaches the latter if the matrix is diagonally dominant or of low rank.