An algorithm for Ax=λBx with symmetric and positive-definite A and B

An algorithm is given for computing the solution of the eigenvalues of $Ax = \lambda Bx$ with symmetric and positive-definite $A$ and $B$. It reduces $Ax = \lambda Bx$ to the generalized singular value problem $LL^T x = \lambda ( L_B L_B^T )x$ by the Cholesky decompositions $A = LL^T $ and $B = L_B L_B^T $, and then reduces the generalized singular value decomposition of $L^T $ and $L_B^T $ to the CS decomposition of Q by the QR decomposition $( L,L_B )^T = QR$. Finally, it reduces $A$ and $B$ to diagonal forms by singular value decompositions. The algorithm provided is stable and, what is more, faster than the QZ algorithm. Numerical examples are also presented.