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Differential Qd Algorithm for Totally Nonnegative Hessenberg Matrices

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Differential Qd Algorithm for Totally Nonnegative Hessenberg Matrices JSIAM Letters Vol.2 (2010) pp.69–72 c 2010 Japan Society for Industrial and Applied Mathematics Differential qd algorithm for totally nonnegative Hessenberg matrices: introduction of origin shifts and relationship with the discrete hungry Lotka-Volterra system Yusaku Yamamoto1 and Takeshi Fukaya2 1 Department of Computer Science and Systems Engineering, Kobe University, 1-1 Rokkodai- cho, Nada-ku, Kobe 657-8501, Japan 2 Department of Computational Science and Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan E-mail yamamoto cs.kobe-u.ac.jp Received February 27, 2010, Accepted April 21, 2010 Abstract We propose an approach for introducing the origin shift into the multiple dqd algorithm for computing the eigenvalues of a totally nonnegative matrix. Numerical experiments show that the shift speeds up the convergence while retaining the accuracy of the computed eigenvalue. Keywords eigenvalue, totally nonnegative, Hessenberg matrix, qd algorithm, origin shift Research Activity Group Algorithms for Matrix / Eigenvalue Problems and their Applications 1. Introduction theoretical proof of high relative accuracy has yet to be Let A be an m × m matrix. A is called totally non- established. We also point out the close relationship be- negative (TN) if all of its minors are nonnegative. TN tween our shifted multiple dqd algorithm and the dis- matrices have many applications in areas such as com- crete hungry Lotka-Volterra system [5]. binatorics and statistics [1]. Recently, there is a growing interest in numerical algorithms for TN matrices, and 2. The multiple dqd algorithm several algorithms for eigenvalue computation, singular In this section, we define our target problem and re- value decomposition and linear equation solving with view the unshifted multiple dqd algorithm. TN coefficient matrices have been developed [1–3]. Let L and Ri (i = 1,...,M) be m × m lower and In [2], we proposed an algorithm for computing the upper bidiagonal matrices defined by eigenvalues of a totally nonnegative band matrix. The q algorithm, called the multiple dqd (differential quotient- 1 1 q difference) algorithm, is a natural extension of the dqd 2 1 q3 algorithm for computing the eigenvalues of a symmetric L = and . positive-definite tridiagonal matrix [4]. It has two fea- .. .. tures: first, it exploits the fact that TN matrices can be 1 qm represented as a product of positive bidiagonal factors 1 e and works directly on these bidiagonal factors. Second, i1 e it preserves the total nonnegativity throughout the iter- 1 i2 .. ations. These features enable us to show that the algo- Ri = 1 . , (1) rithm can compute all the eigenvalues of a TN matrix .. . ei,m−1 to high relative accuracy [2]. Unfortunately, due to the structure of the algorithm, it seemed difficult to intro- 1 duce the origin shift to accelerate the convergence into respectively, where qk (1 ≤ k ≤ m) and eik (1 ≤ i ≤ M, the multiple dqd algorithm. For this reason, the order of 1 ≤ k ≤ m − 1) are some positive numbers. We consider convergence of this algorithm remained only linear. the problem of computing the eigenvalues of a matrix In this paper, we consider the case where A is a TN defined as the product of these bidiagonal factors: Hessenberg matrix. For this type of matrices, we show that we can introduce the origin shift while retaining the A = LR1R2 · · · RM . (2) above two features. Our preliminary numerical experi- A is a Hessenberg matrix with upper bandwidth M. Fur- ment shows that the resulting algorithm exhibits faster thermore, A is totally nonnegative because it is a prod- convergence and can compute the smallest eigenvalue to uct of positive bidiagonal factors [1]. Then, from the gen- the same accuracy as the shiftless algorithm, although eral theory of the TN matrix, we know that all of the – 69 – JSIAM Letters Vol. 2 (2010) pp.69–72 Yusaku Yamamoto et al. eigenvalues of A are simple, real and positive. 3.2 Introduction of the origin shift into the multiple dqd The multiple dqd algorithm for A is a variant of the algorithm LR algorithm that performs the LR step solely on the Assume that A is given as in (2). In considering the bidiagonal factors. Let M = 3 and consider applying an shifted version of the multiple dqd algorithm, it would be LR step to A. Noting that (2) already gives the LR de- natural to require that the next iterate Aˆ is also given as composition of A, we can compute the next iterate Aˆ by a product of bidiagonal factors: making the product R R R L and computing its LR de- 1 2 3 Aˆ LˆRˆ Rˆ · · · Rˆ , composition. In the multiple dqd algorithm, we do this = 1 2 M (8) by repeating the LR decomposition of a product of upper where Lˆ is a lower bidiagonal matrix whose lower subdi- and lower bidiagonal factors as follows: agonal elements are all 1’s and Rˆi is a unit upper bidi- agonal matrix. We denote the diagonal elements of Lˆ by Aˆ = R1R2R3L qˆk and the upper subdiagonal elements of Rˆi bye ˆik. (1) = R1R2L Rˆ3 We want to design the shifted algorithm in such a way that the conditions (i) and (ii) explained in Section R L(2)Rˆ Rˆ = 1 2 3 2 are satisfied. This is because then we will be able to (3) perform relative error analysis of the algorithm as in the = L Rˆ1Rˆ2Rˆ3 ≡ LˆRˆ1Rˆ2Rˆ3. (3) unshifted case. In the present subsection, we concentrate Here, the products such as R1R2R3L are not computed on constructing the shifted algorithm so that the condi- explicitly and the LR transformations such as R3L = tion (i) is satisfied. Positivity of the variables will be dis- (1) ˆ L R3 are done with the dqd algorithm [4]. This algo- cussed in the next subsection. Also, in this subsection, rithm has the following two features: we assume that breakdown of the algorithm, such as di- (i) It works on the bidiagonal factors directly and never vision by zero, does not occur. A sufficient condition for forms their products explicitly. this will be given in the next subsection. Inserting (2) and (8) into (6) and (7), we have (ii) Positivity of the bidiagonal factors is preserved due (0) (0) to the characteristics of the dqd algorithm. LR1 · · · RM − sI = L R , (9) By combining these features with mixed error analysis of (0) −1 (0) LˆRˆ · · · RˆM = (L ) LR · · · RM L . (10) the dqd algorithm [4] and relative perturbation theory 1 1 on the eigenvalues of a TN matrix [1], one can show In (9), L(0) is a lower bidiagonal matrix whose subdi- that the multiple dqd algorithm can compute all the agonal elements are all 1’s. Now, suppose that we know (0) eigenvalues of a TN matrix to high relative accuracy [2]. L . Then we can compute L,ˆ Rˆ1,..., RˆM by applying the LR transformations repeatedly to the right-hand side 3. Introduction of the origin shift of (10) as follows: (0) (1) 3.1 The shifted LR algorithm for a general matrix RM L = L RˆM , (11) As a preparation for introducing the origin shift into (1) (2) ˆ the multiple dqd algorithm, we first explain the shifted RM−1L = L RM−1, (12) LR algorithm. Let s be some properly chosen shift and . denote the m × m identity matrix by I. Then, one step . of the shifted LR algorithm can be written as follows: (M−1) (M) R1L = L Rˆ1, (13) (0) (0) A − sI = L R , (4) LL(M) = L(0)L.ˆ (14) ˆ (0) (0) (0) −1 (0) A = R L + sI = (L ) AL . (5) Here, the last equation is obtained by rewriting the equa- tion Lˆ = (L(0))−1LL(M). Eqs. (11) through (14) show Here, we define the LU decomposition in (4) so that the that once we know L(0), all the subsequent computations (0) diagonals of R are 1’s, in accordance with (1). can be done by working only on the bidiagonal factors. Now, assume that A is given in the factored form as The remaining problem is how to compute L(0). From ˆ ˆ A = LR and we want A also in the factored form A = (9), it seems that one needs to form LR1 · · · RM − sI LˆRˆ. Then, the above formulae can be rewritten as explicitly and compute its LR decomposition. But this LR − sI = L(0)R(0), (6) approach would violate the condition (i). Fortunately, we can compute L(0) simultaneously with (0) −1 (0) LˆRˆ = (L ) LRL . (7) L,ˆ Rˆ1,..., RˆM by rearranging the computations. To see this, first compare the (1, 1) elements of both sides of (9) ˆ ˆ This shows that the computation of L and R from L and to obtain R can be done in the following two steps. q(0) = q − s. (15) • Obtain the lower triangular factor L(0) from the LR 1 1 decomposition of LR − sI. Next, by writing down (11) through (13) element by el- • Apply similarity transformation by L(0) to LR. ement, we have the following set of equalities: (0) Based on this observation, we introduce the origin shift qk eˆ − = e − (2 ≤ k ≤ m), (16) into the multiple dqd algorithm in the next subsection. M,k 1 M,k 1 (1) qk−1 – 70 – JSIAM Letters Vol. 2 (2010) pp.69–72 Yusaku Yamamoto et al. (1) (0) 1:k 1:k qk = eM,k + qk − eˆM,k−1 (1 ≤ k ≤ m), (17) values of C1:k = A1:k − sIk are positive and accordingly C1:k > 0 (1 ≤ k ≤ m). In this case, the left-hand side (1) 1:k qk e − − e − − ≤ k ≤ m , of (9) is LR-decomposable and the diagonal elements of ˆM 1,k 1 = M 1,k 1 (2) (2 ) (18) qk−1 the lower triangular factor are given as (2) (1) 1:k (0) C k qk = eM−1,k + qk − eˆM−1,k−1 (1 ≤ k ≤ m), (19) q = 1: > 0, (26) k 1:k− 1 .
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