On the complex Falk–Langemeyer method

Abstract

The known Falk-Langemeyer method for the simultaneous diagonalization of two positive definite symmetric matrices is generalized to work with complex matrices. It is shown that the derived method is well defined for the Hermitian matrices which make a definite pair. Special attention is paid to the stability of the formulas for the computational parameters when the pivot submatrices are close to being proportional. Numerical tests indicate the high relative accuracy of the method provided that both matrices are definite and well-behaved, i.e. if they can be well-scaled symmetrically.In 1960 Falk and Langemeyer [1] proposed a Jacobi-type method for solving the generalized eigenvalue problem (GEP) Ax = λBx, x = 0 with symmetric positive definite matrices A, B. Later Slapničar and Hari [8] proved the quadratic convergence of the method in the case of simple eigenvalues. They also proved that the method was well defined for a more general, definite pair of symmetric matrices. In 2015 Matejaš [6] proved sharp error estimates for the method. Our numerical tests indicate the high relative accuracy of the method in the case of positive definite matrices that can be wellscaled symmetrically, i.e. when the condition numbers of D A AD A and D B BD B are small for some diagonal matrices D A and D B . Since it is a Jacobi method for the GEP it is very efficient and highly accurate when both matrices are almost diagonal (cf. [5]). This typically happens in the course of modelling the parameters of a system. The described properties make the method an excellent choice for the kernel algorithm of the block-Jacobi methods which are nowadays the prime choice of the methods for solving the definite GEP on contemporary parallel CPU and GPU computing machines [7].In this short communication, we present the main formulas of the complex Falk-Langemeyer (CFL) algorithm that are derived in [3]. Although they are the proper generalization of those in the real method, their derivation is not trivial. The formulas for the transformation parameters become useless when the pivot submatrices are proportional or very close to being proportional. In that case, several stable formulas are proposed in [3]. It is proved that the new complex algorithm is well defined when some real linear combination of the complex Hermitian matrices A and B is positive definite. Numerical tests strongly indicate the high relative accuracy of the method when both matrices are positive definite and can be well-scaled symmetrically. The quadratic convergence of the method can be proved following the original proof from [8]. The proof of the global convergence requires more preliminary work (cf. [2,4]) while the proof of the high relative accuracy seems like a more demanding task.

On the convergence of complex Jacobi methods

“…There are several Jacobi methods for solving PGEP, such as the complex Falk-Langemeyer (FL) method [15], Hari-Zimmermann (HZ) method [8] and Cholesky-Jacobi (CJ) method [12]. Preliminary numerical tests indicate that they all have excellent accuracy properties.…”

Subspace algorithms for detecting a definite Hermitian matrix pair

“…d) Finally, we briefly mention two more methods. We can use some algorithm for detecting the definiteness of a small compress pair, and in the definite case use the real or complex Falk-Langemeyer (FL) method [44,45], which is a Jacobi-type method, to diagonalize the definite pair (A i , B i ) by working on both matrices…”