KOVARIK’S FUNCTION ORTHOGONALIZATION ALGORITHM WITH APPROXIMATE INVERSION ∗

. Z. Kovarik proposed in 1970 a method for approximate orthogonal-ization of a ﬁnite set of linearly independent vectors from a Hilbert space. This method uses at each iteration a symmetric and positive deﬁnite matrix inversion. In this paper we describe an algorithm in which the above matrix inversion step is replaced by an arbitrary odd degree polynomial matrix expression. We prove that this new algorithm converges to the same orthonormal set of vectors as the original Kovarik’s method. Some numerical experiments presented in the last section of the paper show us that, even for small degree polynomial expressions the convergence properties of the new algorithm are comparable with those of the original one.

Let H be a (real) Hilbert space with the scalar product and the associated norm denoted by •, • and • , respectively.In some space R q the vectors will be considered as column vectors and the euclidean scalar product and norm will be denoted by •, • 2 and • 2 , respectively.For a q × q matrix A we shall denote by A t , (A) i , (A) ij , σ(A), ρ(A) the transpose, i-th row, (i, j)-th element, spectrum and spectral radius, respectively, and by A 2 , A ∞ the matrix norms defined by (see e.g.[1]) For a linearly independent system of vectors Φ = {φ 1 , . . ., φ n } ⊂ H the Gram matrix G(Φ), defined by (2) (G(Φ)) ij = φ j , φ i , i, j = 1, . . ., n exists and is also SPD.We define the number trace(G(Φ)) by If σ(G(Φ)) = {σ 1 , . . ., σ n }, then trace(G(Φ)) = σ 1 + • • • + σ n .For the above system Φ the author considered in [1] the norm ||| • ||| defined by (4) and proved the inequality where G(Φ) 2 is the spectral norm of the matrix G(Φ).Moreover, for an we can write (6) in the following "matrix by vector multiplication form", very useful in computations (see Section 2) and the following result holds.
In [2] Z. Kovarik proposed his approximate orthogonalization algorithm.It will be briefly presented in what follows, together with the corresponding convergence results.
Theorem 4. For any linearly independent system Φ, the sequence (Φ k ) k≥0 generated by (11) converges to Φ ∞ (in H).Moreover, the following estimate holds Remark 5.The above convergence of Φ k = {φ Now, if we denote by λ 1 , . . ., λ (0) n } ⊂ (0, ∞), then, from the second equality in (11), we obtain which tells us that the convergence in (12) is at least quadratic.Moreover, from the equalities (see [2]) k ), ∀k ≥ 0, which together with (9) tell us that the limit in (13) is equivalent with can be recursively generated by the following formulas with K k and S k computed as in (11).
Proof.From ( 11) and (6) we get (also using the symmetry of which completes the proof.

THE MODIFIED KOVARIK ALGORITHM
As already observed by Z. Kovarik in [2], for applying the algorithm (3) we need to compute at each iteration the inverse (I + G k ) −1 of the (SPD) matrix I + G k .We shall avoid this difficulty by replacing the matrix inverse with a polynomial expression with respect to G k .For this we shall first observe that after the scaling of the system Φ = Φ 0 = {φ Then, for a given sequence of integers q k ≥ 1, k ≥ 0, we shall approximate the inverse (I + G k ) −1 in (11) by the truncated Neumann series S(q k ; G k ) defined by ( 22) Then, the modified Kovarik algorithm is the following.
The next (main) result of the paper shows the convergence of the algorithm (7).
In order to prove it we need some auxiliary results which will be presented below.
For this, we consider the function and observe that, ∀x ∈ (0, 1), Then, from (28) and (36), it results (34) and the proof is complete.

The above inequality tells us that it exists an integer k
which contradicts (39)-(41).It results that (37) is true and the proof is complete.
Remark 11.In the other cases for the integers q k , i.e. q k = constant (even) or q k = arbitrary, the modified algorithm ( 22)-(23) does not always converge.

NUMERICAL EXPERIMENTS
We considered in our experiments a regular discretization of (0, 1):
We first applied the original Kovarik algorithm (11) (using ( 19)), for different values of N ≥ 2 and the stopping test The numbers of iterations for obtaining (53) are described in Table 1.
The last tests were made as those from above, but only for the first three (odd) values of q k and different values of N ≥ 2. The numbers of iterations are described in Table 3.

FINAL REMARKS AND COMMENTS
(1) We observe that for very small values of q k (which means less computational effort per iteration in ( 22)-( 23)), we got good enough results (Tables 2 and 3) by comparing them with those for the algorithm (11) (Table 1).(2) All the tests indicate a "mesh-independent" behaviour for both Kovarik, original and modified algorithms.(3) All the numerical experiments were performed with the numerical linear algebra software "OCTAVE", freely available on the Internet.q k = 1 q k = 3 q k = 5 q k = 7 q k = 9 q k = 11 28 22 19 17 16 15