A generalized eigenvalue algorithm for tridiagonal pencils based on a nonautonomous discrete integrable system

Kazuki Maedaa, , Satoshi Tsujimotoa ∗ aDepartment of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan

Abstract A generalized eigenvalue algorithm for pencils is presented. The algorithm appears as the time evolution equation of a nonautonomous discrete integrable system associated with a polynomial sequence which has some orthogonality on the support set of the zeros of the characteristic polynomial for a tridiagonal matrix pencil. The convergence of the algorithm is discussed by using the solution to the initial value problem for the corresponding discrete integrable system.

Keywords: generalized eigenvalue problem, nonautonomous discrete integrable system, RII chain, dqds algorithm, orthogonal polynomials 2010 MSC: 37K10, 37K40, 42C05, 65F15

1. Introduction

Applications of discrete integrable systems to numerical algorithms are important and fascinating top- ics. Since the end of the twentieth century, a number of relationships between classical numerical algorithms and integrable systems have been studied (see the review papers [1–3]). On this basis, new algorithms based on discrete integrable systems have been developed: (i) singular value algorithms for bidiagonal matrices based on the discrete Lotka–Volterra equation [4, 5], (ii) Pade´ approximation algorithms based on the dis- crete relativistic Toda lattice [6] and the discrete Schur flow [7], (iii) eigenvalue algorithms for band matrices based on the discrete hungry Lotka–Volterra equation [8] and the nonautonomous discrete hungry Toda lattice [9], and (iv) algorithms for computing D-optimal designs based on the nonautonomous discrete Toda (nd-Toda) lattice [10] and the discrete modified KdV equation [11]. In this paper, we focus on a nonautonomous discrete integrable system called the RII chain [12], which is associated with the generalized eigenvalue problem for tridiagonal matrix pencils [13]. The relationship between the finite RII chain and the generalized eigenvalue problem can be understood to be an analogue of the connection between the finite nd-Toda lattice and the eigenvalue problem for tridiagonal matrices. In , the time evolution equation of the finite nd-Toda lattice is called the dqds (differential quotient difference with shifts) algorithm [14], which is well known as a fast and accurate iterative algorithm for computing eigenvalues or singular values. Therefore, it is worth to consider the application of the finite arXiv:1303.1035v1 [math.NA] 5 Mar 2013 RII chain to algorithms for computing generalized eigenvalues. The purpose of this paper is to construct a generalized eigenvalue algorithm based on the finite RII chain and to prove the convergence of the algorithm. Further improvements and comparisons with traditional methods will be studied in subsequent papers. The nd-Toda lattice on a semi-infinite lattice or a non-periodic finite lattice has a Hankel solution. In the background, there are monic orthogonal polynomials, which give rise to this solution; monic orthogonal polynomials have a determinant expression that relates to the Hankel determinant, and spectral

∗Corresponding author Email addresses: [email protected] (Kazuki Maeda), [email protected] (Satoshi Tsujimoto)

Preprint submitted to Elsevier September 18, 2018 transformations for monic orthogonal polynomials give the Lax pair of the nd-Toda lattice [15, 16]. Espe- cially, for the finite lattice case, we can easily solve the initial value problem for the nd-Toda lattice with the Gauss quadrature formula for monic finite orthogonal polynomials. This special property of the discrete integrable system allows us to analyze the behaviour of the system in detail and tells us how parameters should be chosen to accelerate the convergence of the dqds algorithm. We will give a review of this theory in Section 2. The theory above will be extended to the RII chain in Section 3. The three-term recurrence relation that monic orthogonal polynomials satisfy arises from a tridiagonal matrix. In a similar way, a tridiagonal matrix pencil defines a monic polynomial sequence. This polynomial sequence, called monic RII polynomials [17], possesses similar properties to monic orthogonal polynomials and their spectral transformations yield the monic type RII chain. A determinant expression of the monic RII polynomials gives a Hankel determinant solution and, in particular for the finite lattice case, a convergence theorem of the monic RII chain is shown under an assumption. This theorem enables us to design a generalized eigenvalue algorithm. The dqds algorithm is a subtraction-free algorithm, i.e., the recurrence equations of the dqds algorithm do not contain subtraction operations except origin shifts (see Subsection 2.3). The subtraction-free form is numerically effective to avoid the loss of significant digits. In addition, there is another application of the subtraction-free form: ultradiscretization [18] or tropicalization [19]; e.g., the ultradiscretization of the finite nd-Toda lattice in a subtraction-free form gives a time evolution equation of the box–ball system with a carrier [20]. In Section 4, for the monic type RII chain, we will present its subtraction-free form, which contains no subtractions except origin shifts under some conditions. It is considered that this form makes the computation of the proposed algorithm more accurate. At the end of the paper, numerical examples will be presented to confirm that the proposed algorithm computes the generalized eigenvalues of given tridiagonal matrix pencils fast and accurately.

2. Monic orthogonal polynomials, nd-Toda lattice, and dqds algorithm

First, we will review the connection between the theory of orthogonal polynomials and the nd-Toda lattice.

2.1. Infinite dimensional case Let us consider a tridiagonal semi-infinite matrix of the form

t ⎛u0 1 ⎞ ⎜ t t ⎟ ⎜w( ) u 1 ⎟ ⎜ 1 1 ⎟ t t t = ⎜ ( ) (t) t ⎟ ∈ ∈ − { } B ⎜ w u 1 ⎟ , un C, wn C 0 , ⎜ 2 2 ⎟ ( ) ⎜ ( ) (t) ⋱ ⋱⎟ ( ) ( ) w3 ⎝ ( ) ⋱ ⋱⎠ where t ∈ N is the discrete time, whose evolution will be introduced later. Let In denote the t t of order n and Bn the n-th order leading principal submatrix of B . We now introduce a polynomial { t ( )}( ) ( ) sequence φn x n 0: ( ) ∞ t ( ) ∶= = t ( ) ∶= ( − t ) = φ0 x 1, φn x det xIn Bn , n 1, 2, 3, . . . . ( ) ( ) ( ) t ( ) ( − t ) By definition, φn x is a monic polynomial of degree n. The Laplace expansion for det xIn 1 Bn 1 with respect to the last( ) row yields the three-term recurrence relation ( ) + + t ( ) = ( − t ) t ( ) − t t ( ) = φn 1 x x un φn x wn φn 1 x , n 0, 1, 2, . . . , (2.1) ( ) ( ) ( ) ( ) ( ) + t ∶= t ( ) ∶= − where we set w0 0 and φ 1 x 0. It is well known that the three-term recurrence relation of the form (2.1) gives the following( ) classical( ) theorem. − 2 { t ( )} Theorem 2.1 (Favard’s Theorem [21, Chapter I, Section 4]). For the polynomials φn x n 0 satisfying the three-term recurrence relation and any nonzero constant t , there exists a unique linear( ) functional∞ L t defined (2.1) h0 = on the space of all polynomials such that the orthogonality relation( ) ( )

t m t t L [x φn (x)] = hn δm,n, n = 0, 1, 2, . . . , m = 0, 1, . . . , n, (2.2) ( ) ( ) ( ) holds, where

t = t t t t = hn h0 w1 w2 ... wn , n 1, 2, 3, . . . , ( ) ( ) ( ) ( ) ( ) and δm,n is Kronecker delta. From the relation (2.2), we readily obtain the relation

t t t t L [φm (x)φn (x)] = hn δm,n, m, n = 0, 1, 2, . . . . ( ) ( ) ( ) ( ) { t ( )} L t Therefore, the polynomials φn x n 0 are the monic orthogonal polynomials with respect to . Let us define the moment of( order) ∞m by ( ) = t t m µm ∶= L [x ], m = 0, 1, 2, . . . , ( ) ( ) and its Hankel determinant of order n by R R R µ t µ t ... µ t R R 0 1 n 1 R R (t) (t) ( t) R t t t n 1 R µ µ ... µn R τ ∶= 1, τn ∶= Sµ S = R 1 2 − R , n = 1, 2, 3, . . . . 0 i j i,j 0 R (⋮) ( ⋮) ( ⋮ ) R ( ) ( ) ( ) − R R R t t t R + = R R Rµn 1 µn ... µ2n 2R ( ) ( ) ( ) Since the monic orthogonal polynomials− with respect to− L t are uniquely determined, we then find the t ( ) determinant expression of the polynomial φn (x): ( ) R R R µ t µ t ... µ t µ t R R 0 1 n 1 n R R (t) (t) ( t) (t ) R R µ µ ... µn µ R t 1 R 1 2 − n 1 R φ (x) = R (⋮) ( ⋮) ( ⋮) ( ⋮ ) R , n = 0, 1, 2, . . . . (2.3) n t R R ( ) τn R t t t t+ R Rµ µn ... µ µ R ( ) R n 1 2n 2 2n 1R R (1) x( ) ... x(n)1 (xn) R − − − − Next, we introduce the discrete time evolution into the monic orthogonal polynomials by the following { t ( )} { t 1 ( )} transformation from φn x n 0 to φn x n 0: ( ) ∞ ( + ) ∞ ( − t ) t 1 ( ) = t (=) + t t ( ) = = x s φn x φn 1 x qn φn x , n 0, 1, 2, . . . , (2.4) ( ) ( + ) ( ) ( ) ( ) where +

φ t (s t ) t ∶= − n 1 = qn ( ) , n 0, 1, 2, . . . , (2.5) t (t ) ( ) φn+ (s ) ( ) ( ) t t ( ) = { t ( )} and s is a parameter that is not a zero of φn x for all n 0, 1, 2, . . . . Suppose that φn x n 0 are the monic( orthogonal) polynomials with respect to( )L t and define a new linear functional L (t )1 by∞ = ( ) ( + ) L t 1 [P(x)] ∶= L t [(x − s t )P(x)] (2.6) ( + ) ( ) ( ) 3 ( ) { t 1 ( )} for all polynomials P x . Then, it is easily verified that φn x n 0 are monic orthogonal polynomials with respect to L t 1 again. Since it is shown that the monic( + orthogonal) ∞ polynomials satisfy the three-term = recurrence relation( + of) the form (2.1), another relation t ( ) = t 1 ( ) + t t 1 ( ) = φn x φn x en φn 1 x , n 0, 1, 2, . . . , (2.7) ( ) ( + ) ( ) ( + ) − t is derived for consistency. The variable en satisfies the compatibility condition ( ) t 1 = t 1 + t 1 + t 1 = t + t + t un qn en s qn en 1 s , (2.8a) ( + ) ( + ) ( + ) ( ) ( ) t 1 = t 1 t 1 = t( + t) ( ) wn qn 1 en qn en + (2.8b) ( + ) ( + ) ( + ) ( ) ( ) with the boundary− condition

t = ≥ e0 0 for all t 0. (2.8c) ( ) The transformations (2.4) and (2.7) are called the Christoffel transformation and the Geronimus transformation, respectively [22]. The discrete dynamical system (2.8) is the semi-infinite nd-Toda lattice. We have seen above the derivation of the nd-Toda lattice from the theory of orthogonal polynomials. Using this connection, we can give an explicit solution to the semi-infinite nd-Toda lattice (2.8); the solution is written in terms of the moments of the monic orthogonal polynomials. The time evolution of the linear functional (2.6) leads to t 1 = t − t t µm µm 1 s µm . (2.9) ( + ) ( ) ( ) ( ) By applying this+ relation to the determinant expression of the monic orthogonal polynomials (2.3), the definition of the variable (2.5) yields

t t 1 τn τ q t = n 1 . (2.10) n (t ) ( t+1) ( ) τn 1τn+ ( ) ( + ) Further, applying+ the orthogonality relation (2.2) to equation (2.7), we obtain

t t t 1 L t [xnφ (x)] h t τ τ t = n = n = n 1 n 1 en ( ) ( + ) . (2.11) L t 1( [) n 1 ( )t 1 ( )] t( )1 t t 1 ( ) x φn 1 x hn 1 τn+ τn − ( + ) − ( + ) ( + ) ( ) ( + ) t − − t If the moments µm , the elements of the Hankel determinant τn , are arbitrary functions satisfying the relation (2.9), then( these) (2.10) and (2.11) give particular solutions( to) the semi-infinite nd-Toda lattice (2.8). For instance,

t 1 t m j µm = S x M(x − s )ω(x) dx Ω − ( ) j 0 ( ) satisfies the relation= (2.9), where Ω is an interval of the real line and ω(x) is a weight function on Ω. If the integral of the right-hand side has a finite value for all m ∈ N, then this moment gives a solution.

2.2. Finite dimensional case In what follows, we shall reduce the size of the tridiagonal matrix B t to finite N: ( ) t ⎛u0 1 ⎞ ⎜ t t ⎟ ⎜w( ) u 1 ⎟ ⎜ 1 1 ⎟ t ∶= ⎜ ( ) (t) ⎟ B ⎜ w ⋱ ⋱ ⎟ . ⎜ 2 ⎟ ( ) ⎜ ( ) ⋱ ⋱ 1 ⎟ ⎝ t t ⎠ wN 1 uN 1 ( ) ( ) − − 4 The matrix B t determines a system of monic finite orthogonal polynomials. The corresponding nd-Toda lattice is also reduced( ) to the case of the non-periodic finite lattice of size N:

t 1 + t 1 + t 1 = t + t + t qn en s qn en 1 s , (2.12a) ( + ) ( + ) ( ) ( ) t 1 t 1 = t ( +t ) ( ) qn 1 en qn en , + (2.12b) ( + ) ( + ) ( ) ( ) t = t = ≥ e0 − eN 0 for all t 0. (2.12c) ( ) ( ) We can solve the initial value problem for the finite nd-Toda lattice (2.12) through the theory of finite or- thogonal polynomials. { t ( )}N The monic finite orthogonal polynomials φn x n 0 are defined in the same way as the infinite dimen- φ t ( ) ∶= ( − t ) ( ) φ t ( ) sional case: n x det xIn Bn . It should be remarked= that N x is the characteristic polynomial of t ( ) { t ( )}( )N (t ) B . For the polynomials φn x n 0 and any nonzero constant h0 , there exists a unique linear functional L(t) such that the orthogonality( ) relation ( ) = ( ) t m t t L [x φn (x)] = hn δm,n, n = 0, 1, . . . , N − 1, m = 0, 1, . . . , n, (2.13a) ( ) ( ) ( ) and the terminating condition

L t [ m t ( )] = = x φN x 0, m 0, 1, 2, . . . , (2.13b) ( ) ( ) hold. t ( ) Let x0, x1,..., xN 1 denote the zeros of the characteristic polynomial φN x , i.e., ( ) N 1 − t φ (x) = M (x − xi). (2.14) N − ( ) i 0

If for simplicity we= assume that these zeros are all simple, the linear functional L t is concretely given by the Gauss quadrature formula. ( )

Theorem 2.2 (Gauss quadrature formula [21, Chapter I, Section 6]). Let x0, x1,..., xN 1 be the simple zeros t ( ) L t of the characteristic polynomial φN x . For the linear functional of the monic finite orthogonal− polynomials { t ( )}N ( ) t t t ( ) φn x n 0, there exist some constants c0 , c1 ,..., cN 1 such that ( ) ( ) ( ) ( ) = N 1 − t t L [P(x)] = Q c P(xi) (2.15) − i ( ) i 0 ( )

= ( ) t t t t t t holds for all polynomials P x . Further, if w1 , w2 ,..., wN 1 are all real and positive, then c0 , c1 ,..., cN 1 are also all real and positive. ( ) ( ) ( ) ( ) ( ) ( ) − − { t ( )}N The formula (2.15) means that the monic finite orthogonal polynomials φn x n 0 with the termi- nating condition (2.13b) are orthogonal on the support set of the zeros x0, x1,...,( x) N 1 of the characteristic = t ( ) polynomial φN x . − ( ) t t t The constants c0 , c1 ,..., cN 1 are calculated as ( ) ( ) ( ) − h t c t = N 1 , i = 0, 1, . . . , N − 1, (2.16) i t ( ()) t ( ) ( ) φN 1 xi φ− N xi ( ) ′( ) − 5 t ( ) t ( ) where φ N x is the derivative of φN x . This formula is verified as follows. Due to the Gauss quadrature formula′ (2.15),( ) the moment is given( by)

N 1 t t m t m µm = L [x ] = Q c x , m = 0, 1, 2, . . . . − i i ( ) ( ) i 0 ( ) This yields the relation =

N 1 t t t m µ − xjµm = Q c (xi − xj)x , j = 0, 1, . . . , N − 1, m = 0, 1, 2, . . . . m 1 − i i ( ) ( ) i 0 ( ) + i j = This relation and the determinant≠ expression of the monic orthogonal polynomials (2.3) lead to

N 1 t 1 t 2 φ (xj) = M c (xj − xi) M (xν − xν ) , j = 0, 1, . . . , N − 1. N 1 t − i 1 0 ( ) τ i 0 ( ) 0 ν0 ν1 N 1 N 1 i j − ( ) ν0 j,ν1 j − = ≤ < ≤ − A similar calculation yields≠ ≠ ≠

N 1 t t 2 τ = M c M (xν − xν ) . N − i 1 0 ( ) i 0 ( ) 0 ν0 ν1 N 1 Further, we have= ≤ < ≤ −

N 1 t φ (xj) = M (xj − xi), j = 0, 1, . . . , N − 1, N − ′( ) i 0 i j = ≠ τ t h t = L t [xN 1φ t (x)] = N . N 1 N 1 (t ) ( ) ( ) − ( ) τN 1 − − ( ) These equations lead to the formula (2.16).− The spectral transformations (2.4) and (2.7) also work for the finite dimensional case except the Christof- fel transformation for n = N: t 1 ( ) = t ( ) φN x φN x , (2.17) ( + ) ( ) which is consistent with the Geronimus transformation for n = N. Equation (2.17) means that the char- t ( ) t acteristic polynomial φN x of the tridiagonal matrix B is invariant under the time evolution. In other words, the time evolution( ) of the monic finite orthogonal( polynomials) does not change the eigenvalues of the tridiagonal matrix B t . Since the time evolution of the moment is given by (2.9), we have the following expression for the moment:( )

N 1 ⎛ t 1 ⎞ t 0 m j µm = Q c x M(xi − s ) , (2.18) − ⎝ i i − ⎠ ( ) i 0 ( ) j 0 ( ) where we define= t = 0 as the= initial time. Substituting this expression of the moment (2.18) into the elements t of the Hankel determinant τn and applying the Binet–Cauchy formula and the Vandermonde determinant ( ) t formula, we obtain the expanded form of τn : ( ) ⎛n 1 ⎛ t 1 ⎞ ⎞ t 0 j 2 τn = Q M cr M(xr − s ) M (xrν − xrν ) . (2.19) ⎝ − ⎝ i − i ⎠ 1 0 ⎠ ( ) 0 r0 r1 rn−1 N 1 i 0 ( ) j 0 ( ) 0 ν0 ν1 n 1 ≤ < <⋅⋅⋅< ≤ − = = 6 ≤ < ≤ − Hence, we can conclude that the solution to the initial value problem for the finite nd-Toda lattice (2.12) is given by t t 1 t t 1 τn τ τ τ q t = n 1 , e t = n 1 n 1 , (2.20) n (t ) ( t+1) n ( t) (t +1) ( ) τn 1τn+ ( ) τn+ τn − ( ) ( + ) ( ) ( + ) t 0 with the expanded+ form of τn (2.19) and the expression of cn (2.16). In the rest of this subsection,( ) we will reformulate the matrix( ) forms of the finite nd-Toda lattice. Let L t t and R be bidiagonal matrices of order N: ( ) ( ) ⎛ 1 ⎞ ⎛q t 1 ⎞ ⎜ t ⎟ ⎜ 0 ⎟ ⎜e 1 ⎟ ⎜ ( ) t ⎟ ⎜ 1 ⎟ ⎜ q1 1 ⎟ t = ⎜ ( ) t ⎟ t = ⎜ ⎟ L ⎜ e ⋱ ⎟ , R ⎜ ( ) ⋱ ⋱ ⎟ , ⎜ 2 ⎟ ⎜ ⎟ ( ) ⎜ ( ) ⋱ ⋱ ⎟ ( ) ⎜ ⋱ 1 ⎟ ⎝ t ⎠ ⎝ t ⎠ eN 1 1 qN 1 ( ) ( ) t t ( ) ( ) − − and let φ x and φN x be the vectors of order N: ( ) ( ) ⎛ t ( ) ⎞ φ0 x ⎛ 0 ⎞ ⎜ t ⎟ ⎜ ( )( ) ⎟ t ⎜ ⋮ ⎟ t ( ) = ⎜ φ1 x ⎟ ( ) = ⎜ ⎟ φ x ⎜ ⎟ , φN x ⎜ ⎟ . ⎜ ( )⋮ ⎟ ⎜ 0 ⎟ ( ) ( ) t ⎝ t ( )⎠ ⎝φ (x)⎠ φN 1 x N ( ) ( ) Then, the three-term− recurrence relation (2.1) and the spectral transformations (2.4) and (2.7) are written as

t t ( ) + t ( ) = t ( ) B φ x φN x xφ x , (2.21a) ( ) ( ( −) (t )) t 1 ( ) = t (t)( ) + t ( ) x s φ x R φ x φN x , (2.21b) ( ) ( + ) ( ) ( ) ( ) φ t (x) = L t φ t 1 (x). (2.21c) Note that,( ) from (2.14)( ) ( and+ ) (2.21a), t t t B φ (xi) = xiφ (xi), i = 0, 1, . . . , N − 1, ( ) ( ) ( ) t ( ) holds. This corresponds to the fact that the zeros of the characteristic polynomials φN x are the eigenvalues t t ( ) of B . Moreover, it indicates that φ (xi) is the eigenvector corresponding to the eigenvalue xi. From (2.21), we have( ) ( ) t 1 ( ) = t 1 t 1 ( ) + t 1 ( ) xφ x B φ x φN x ( + ) ( + ) = ( ( +t )1 ( +t 1) + t 1 ) t 1 ( ) + t 1 ( ) L R s IN φ x φN x ( + ) = ( ( t+ ) t (++ )t ()+ ) t 1 ( ()++) t 1 ( ) R L s IN φ x φN x . ( + ) This yields the matrix( form) ( ) of the( ) finite( + nd-Toda) lattice (2.12): t 1 t 1 t 1 t 1 t t t B = L R + s IN = R L + s IN. Since L(t+ is) always( + ) regular,( + ) it( is+ shown) that( ) ( the) tridiagonal( ) matrices B t and B t 1 are similar: ( ) 1 ( )1 ( + ) t 1 t t t t t t t t t t t B = R L + s IN = ŠL  ŠL R + s IN L = ŠL  B L . − − ( + ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Therefore, the eigenvalues of B t are conserved under the time evolution. This corresponds to the fact (2.17) ( ) t ( ) that the characteristic polynomial φN x is invariant under the time evolution. From the result, we can see that the spectral transformations (2.21b)( ) and (2.21c) correspond to the LU decomposition of the tridiagonal matrix B t with the shift s t . ( ) ( ) 7 2.3. The dqds algorithm For the finite nd-Toda lattice (2.12), let us introduce an auxiliary variable

t 1 ∶= t 1 − t = − dn qn en 1, n 0, 1, . . . , N 1. (2.22) ( + ) ( + ) ( ) Then, equations (2.12) are+ rewritten as

t 1 = t − ( t 1 − t ) d0 q0 s s , (2.23a) ( + ) ( ) ( + ) ( ) q t d t 1 = d t 1 n − (s t 1 − s t ), n = 1, 2, . . . , N − 1, (2.23b) n n 1 t( )1 ( + ) ( + ) qn 1 ( + ) ( ) − ( + ) t 1 = t + t 1 = − qn en 1 dn− , n 0, 1, . . . , N 1, (2.23c) ( + ) ( ) t ( + ) + q e t 1 = e t n , n = 1, 2, . . . , N − 1, (2.23d) n n t( )1 ( + ) ( ) qn 1 ( + ) t = t = ≥ e0 eN 0 − for all t 0. (2.23e) ( ) ( ) These recurrence equations are called the dqds algorithm. The spectral transformations (2.4) and (2.7) yields

t 1 t t φ (x) + dn φn (x) φ t 1 (x) = n 1 , n = 0, 1, . . . , N − 1. n ( − ) − (t ) ( ) ( + ) + x s Hence, we obtain ( )

φ t 1 (s t ) τ t σ t t = − n 1 = n n 1 dn ( − ) ( ) ( ) , (2.24) t ( t( ) t 1 t 1 ( ) φ+n s τn 1 τn+ ( ) ( ) ( − ) ( + ) with +

t ∶= S t 1 − t t 1 S σn µi j 1 s µi j 0 i,j n 1. ( ) ( − ) ( ) ( − ) ≤ ≤ − + + + t By a calculation similar to the derivation of the expanded form (2.19), we obtain the expanded form of σn ( ) ⎛n 1 ⎛ t 2 ⎞ ⎞ t 0 t j 2 σn = Q M cr (xr − s ) M(xr − s ) M (xrν − xrν ) . (2.25) ⎝ − ⎝ i i − i ⎠ 1 0 ⎠ ( ) 0 r0 r1 rn−1 N 1 i 0 ( ) ( ) j 0 ( ) 0 ν0 ν1 n 1 ≤ < <⋅⋅⋅< ≤ − = = ≤ < ≤ − We henceforth assume that, for the elements of the initial tridiagonal matrix B 0 , the following condi- 0 0 0 0 0 0 ( ) tions are satisfied: u0 , u1 ,..., uN 1 are all real and w1 , w2 ,..., wN 1 are all real and positive. Then, the ( 0) ( ) ( ) ( ) ( ) ( ) tridiagonal matrix B is similar to a real symmetric tridiagonal matrix. The eigenvalues x0, x1,..., xN 1 of − − 0 ( ) 0 0 0 B are thus all real and simple. In addition, the constants c0 , c1 ,..., cN 1 are all real and positive− by Theorem( ) 2.2. Accordingly, the solution (2.20) and (2.24) with the( ) expanded( ) forms( ) (2.19) and (2.25) gives the next theorem. − 0 0 0 0 0 0 Theorem 2.3. Suppose that u0 , u1 ,..., uN 1 are all real, and w1 , w2 ,..., wN 1 are all real and positive. Choose the parameter s t as ( ) ( ) ( ) ( ) ( ) ( ) − − ( ) t s < min{x0, x1,..., xN 1} for all t ≥ 0. (2.26) ( ) t t − t Then, the variables qn , en and dn of the dqds algorithm (2.23) are positive for all n and t ≥ 0. ( ) ( ) ( ) 8 This theorem guarantees that, under the assumption, the dqds algorithm does not contain subtraction operations except the parameter terms −(s t 1 − s t ) in equations (2.23a) and (2.23b). Namely, equations (2.23) are the subtraction-free form of the finite( + ) nd-Toda( ) lattice (2.12). It is known that this form improves the accuracy of the numerical computation. The asymptotic analysis of the dqds algorithm proves convergence of the algorithm and provides a method for accelerating the convergence. The solution derived in Subsection 2.2 is a key tool for the analysis. 0 Arrange the eigenvalues x0, x1,..., xN 1 of the initial tridiagonal matrix B in descending order: x0 > x1 > ⋅ ⋅ ⋅ > x . If the parameter s t chosen as (2.26), namely s t < x , then the( ) inequality x − s t > x − s t > N 1 − N 1 0 1 ⋅ ⋅ ⋅ > x − s t > 0 holds for( all) t ≥ 0. Under this assumption,( ) by the solution (2.20) with the expanded( ) form( ) N−1 − (2.19), we obtain( ) the asymptotic behaviour for t ≫ 0: − ⎧ ⎫ 1 ⎛ ⎪ ∏t (x − s j ) ∏t (x − s j )⎪⎞ ⎛ ∏t (x − s j ) ⎞ t t ⎪ j 0 n j 0 n 1 ⎪ t j 0 n q = xn − s + O max ⎨ , ⎬ , e = O . n ⎝ ⎪ t 1( − ( )j ) t 1( − j( ) ⎪⎠ n ⎝∏t −( − ( )j )⎠ ( ) ( ) ⎩∏j 0= xn 1 s ∏=j 0 x+n s ⎭ ( ) j =0 xn 1 s − ( ) − ( ) ( ) t t = − t = = − This shows that qn and en converge to xn − s and 0 as t → +∞, respectively. Hence, it is shown that the dqds algorithm (2.23)( ) with( appropriate) parameters( ) s t computes the eigenvalues of a given real symmetric (t) ( ) xn s tridiagonal matrix. It is clear that the convergence speed depends on (t) . Therefore, we should choose xn−1 s t < − the parameter s xN 1 as close as possible to the minimum eigenvalue− xN 1 for fast computation. The acceleration parameter( ) s t is called the origin shift. − − ( )

3. RII polynomials, RII chain, and generalized eigenvalue algorithm We shall extend the discussion in Section 2 for tridiagonal matrix pencils and its associated nonau- tonomous discrete integrable system.

3.1. Infinite dimensional case Let us consider two tridiagonal semi-infinite matrices in the following forms:

t ⎛ v0 κt ⎞ ⎜ ( )t t ⎟ ⎜λ1w v κt 1 ⎟ ⎜ 1 1 ⎟ t t t = ⎜ ( ) ( )t t ⎟ ∈ C ∈ C − { } A ⎜ λ2w v κ 2 ⎟ , vn , κt n, λn , wn 0 , ⎜ 2 2+ t ⎟ ( ) ⎜ ( ) ( )t ⋱ ⋱⎟ ( ) ( ) λ3w3 + + ⎝ ( ) ⋱ ⋱⎠ t ⎛u0 1 ⎞ ⎜ t t ⎟ ⎜w( ) u 1 ⎟ ⎜ 1 1 ⎟ t t = ⎜ ( ) (t) t ⎟ ∈ B ⎜ w u 1 ⎟ , un C. ⎜ 2 2 ⎟ ( ) ⎜ ( ) (t) ⋱ ⋱⎟ ( ) w3 ⎝ ( ) ⋱ ⋱⎠

t t t t Let An and Bn denote the n-th order leading principal submatrices of A and B , respectively. We now ( ) ( ) { t ( )} ( ) ( ) define a polynomial sequence ϕn x n 0 by ( ) ∞ t ( ) ∶= t ( ) ∶= ( t −= t ) = ϕ0 x 1, ϕn x det xBn An , n 1, 2, 3, . . . . ( ) ( ) ( ) ( ) t The polynomial ϕn (x) is a monic polynomial of degree n. In the same manner as in the case of monic orthogonal polynomials( ) in Section 2, we obtain the three-term recurrence relation

t ( ) = ( t − t ) t ( ) − t ( − )( − ) t ( ) = ϕn 1 x un x vn ϕn x wn x κt n 1 x λn ϕn 1 x , n 0, 1, 2, . . . , (3.1) ( ) ( ) ( ) ( ) ( ) ( ) + + − 9 − t ∶= t ( ) ∶= where we set w0 0 and ϕ 1 x 0. We will assume in what follows that all the parameters κt k and λk, = 0, 1, 2, . . . , are( ) not zeros of( the) polynomial ϕ t ( ) for all ∈ N. The polynomials {ϕ t ( )} are called k − n x n n x n 0 + t ( ) ( ) the RII polynomials with respect to L , introduced by Ismail and Masson [17]. ∞ = We introduce the notations ( ) k 1 k 1 l t 0 K (x) ∶= M(x − κt j), Kk(x) ∶= K (x) = M(x − κj), Ll(x) ∶= M(x − λj), k − k − ( ) j 0 ( ) j 0 j 1 + t = = xm = and D(L ) a linear space spanned by the rational functions ( ) , k, l = 0, 1, 2, . . . ; m = 0, 1, . . . , k + l. K t x L x ( ) k l The following Favard type theorem is proved. ( ) ( ) { t ( )} Theorem 3.1 (Favard type theorem for the RII polynomials [17]). For the RII polynomials ϕn x n 0 and any nonzero constants t and t , which satisfy t ≠ t , there exists a unique linear functional defined( ) ∞ on D(L t ) h0 h1 h0 h1 = such that the orthogonality( ) relation( ) ( ) ( ) ( ) ⎡ ⎤ ⎢ m t ( ) ⎥ t ⎢ x ϕn x ⎥ t L ⎢ ⎥ = h δm,n, n = 0, 1, 2, . . . , m = 0, 1, 2, . . . , n, ⎢ t ( ) ⎥ n ( ) ⎣ Kn (x)Ln(x)⎦ ( ) ( ) t holds, where hn , n = 2, 3, . . . , are some nonzero constants. ( ) t t In the rest of this paper, we consider the monic RII polynomials, i.e., the case where un = 1 + wn holds t ( ) t ( ) for all n = 0, 1, 2, . . . . For general tridiagonal semi-infinite matrices of the form B , if det(Bn ) ≠ 0 holds (t) ( ) ϕn x ( ) for all n = 1, 2, 3, . . . , then › ( ) are the monic RII polynomials. Therefore, the following argument det B t ∞ (n ) n 0 is valid for such matrices. ( ) = t The moment of the RII linear functional L is introduced by ⎡ ⎤ ( ) ⎢ m ⎥ k,l,t t ⎢ x ⎥ µm ∶= L ⎢ ⎥ , k, l = 0, 1, 2, . . . , m = 0, 1, . . . , k + l, (3.2) ⎢ t ( ) ( )⎥ ( ) ⎣ Kk x Ll x ⎦ ( ) and its Hankel determinant by

R k,l,t k,l,t k,l,t R Rµ0 µ1 ... µn 1 R R k,l,t k,l,t k,l,t R k,l,t ∶= k,l,t ∶= S k,l,tS = Rµ1 µ2 ... µn R = τ0 1, τn µi j 0 i,j n 1 R − R , n 1, 2, 3, . . . . R ⋮ ⋮ ⋮ R R k,l,t k,l,t k,l,t R + ≤ ≤ − R R Rµn 1 µn ... µ2n 2R − − { t ( )} Then, the determinant expression of the monic RII polynomials ϕn x n 0 is presented: ( ) ∞ Rµn,n,t µn,n,t ... µn,n,t µn,n,t R = R 0 1 n 1 n R R n,n,t n,n,t n,n,t n,n,t R Rµ µ ... µn µ R t 1 R 1 2 − n 1 R ϕ (x) = R ⋮ ⋮ ⋮ ⋮ R , n = 0, 1, 2, . . . . (3.3) n n,n,t R R ( ) τn Rµn,n,t µn,n,t ... µn,n,t µn,+n,t R R n 1 n 2n 2 2n 1R R 1 x ... xn 1 xn R − − − − The discrete time evolution for the monic RII polynomials is introduced by an analogue of the spectral transformations for monic orthogonal polynomials:

( − t )( + t ) t 1 ( ) = t ( ) + t ( − ) t ( ) x s 1 qn ϕn x ϕn 1 x qn x κt n ϕn x , (3.4a) (t) t ( ) ( +t 1) (t) ( ) t 1 ( ) ( + ) ( ) = ( ) + ( − ) ( +) 1 en ϕn x ϕn x en+ x λn ϕn 1 x (3.4b) ( ) ( ) ( + ) ( ) ( + ) − 10 for n = 0, 1, 2, . . . , where

ϕ t (s t ) t ∶= −( t − ) n 1 = qn s κt n ( ) , n 0, 1, 2, . . . , (3.5) t (t ) ( ) ( ) ϕn+ (s ) + ( ) ( ) t and en is the variable determined by the compatibility condition: ( ) t 1 t 1 t 1 t 1 1 + qn t 1 t 1 1 + wn = −qn − en + (1 + qn )(1 + en ) + (t+1) ( + ) ( + ) ( + ) 1 qn 1 ( + ) ( + ) t ( + ) 1 + e − = −q t n 1 − e t + (1 + q t )(1 + e t ), (3.6a) n ( t) n 1 n n 1 ( ) 1 + en+ ( ) ( ) ( ) ( ) + t 1 + t 1 t 1 t 1 1 + qn t 1 t 1 t 1 vn = −κt n 1qn − λnen + s (1 + qn )(1 + en ) + (t+1) ( + ) ( + ) ( + ) 1 qn 1 ( + ) ( + ) ( + ) + + t ( + ) 1 + e − = −κ q t n 1 − λ e t + s t (1 + q t )(1 + e t ), (3.6b) t n n ( t) n 1 n 1 n n 1 ( ) 1 + en+ ( ) ( ) ( ) ( ) + + ( ) t 1 + + t + t 1 t 1 t 1 1 + qn t t 1 en 1 wn = q en = qn en (3.6c) n 1 + (t+1) + ( t) ( + ) ( + ) ( + ) 1 qn 1 ( ) ( ) 1 en+ − ( + ) ( ) with the boundary condition −

t = ≥ e0 0 for all t 0. (3.6d) ( ) This is the semi-infinite monic type RII chain. Note that, since (3.6a) and (3.6c) are identical, there are the two t t independent equations that determine the time evolution of the two variables qn and en . It is readily veri- { t ( )} L t ( ) ( ) { t 1 ( )} fied that if ϕn x n 0 are the monic RII polynomials with respect to , then polynomials ϕn x n 0 defined by the( spectral) ∞ transformation (3.4a) are again the monic RII polynomials,( ) where the corresponding( + ) ∞ = = RII linear functional is defined by

x − s t L t 1 [R(x)] ∶= L t  R(x) (3.7) x − κ( ) ( + ) ( ) t for all R(x) ∈ D(L t 1 ). Let us derive a( solution+ ) to the monic type RII chain. By the definition of the moment (3.2) and the time evolution of the linear functional (3.7), we obtain the relations

k,l,t = k 1,l,t − k 1,l,t = k,l 1,t − k,l 1,t µm µm 1 κt kµm µm 1 λl 1µm , (3.8a) k,l,t 1 = +k 1,l,t − t +k 1,l,t + + µm µ+m 1 +s µm . + + (3.8b) + + ( ) + The relation (3.8b),+ the determinant expression of the monic RII polynomials (3.3), and the definition of the variable (3.5) lead to

n,n,t n,n 1,t 1 τn τ t = ( t − κ ) 1 n 1 . (3.9) qn s t n n 1,n,t 1 n+ 1,n+ 1,t ( ) ( ) − τn τ+n 1 + − + + + Next, the relation (3.8a) and the spectral+ transformation (3.4a) yield

t n,n 1,t n,n,t 1 ϕ (κt n) τ τn 1 + q t = (κ − s t ) 1 n 1 = (s t − κ ) 1 n 1 . n t n (t )1 t n n 1,n+1,t n 1,n+,t 1 ( ) ( ) − ϕn + (κ+t n) ( ) − τn 1+ τn + ( + ) + + + − + + 11 + Further, the Jacobi identity for [23, Section 2.6] proves the bilinear equation

k 1,l 1,t k,l,t − k 1,l,t k,l 1,t − k 1,l 1,t k,l,t = τn τn τn τn τn 1 τn 1 0. − − − − − − By using this bilinear equation and the− three-term+ recurrence relation (3.1), we obtain

( ) xn+1 ϕ t x n (t) L t  n+1 − x ϕn x (t) (t) n 1,n 1,t n,n 1,t n 1,n,t K x Ln+1 x K x Ln x τ τ τ w t = ( ) n+1 ( ) n ( ) = n 1 n 1 n 1 . n ( ) (t) n−1,n,−t n,n 1,+t n 1,+n 1,t n ϕ( t) ( ) xn−1 ϕ( ) x( ) ( ) t x n x n−1 τn− τn + τn 1+ L  ( ) − ( ) t t − − + + Kn x Ln x Kn−1 x Ln−(1 )x ( ) ( ) + Hence, from equation( (3.6c)) ( ) and these( ) formulae,( ) we find a solution

t + t n 1,n 1,t 1 n 1,n,t t wn 1 qn 1 t τn 1 τn 1 en = = (s − κt n) . (3.10) (t ) + ( t) τ−n,n−1,t+τn,n,t+ 1 ( ) qn 1 1 qn− ( ) −n n + ( ) ( ) + − + − k,l,t If the moments µm are arbitrary functions satisfying the relations (3.8), e.g.,

m t 1( − j ) x ∏j 0 x s µk,l,t = S ω(x) dx, m −( ) ( ()) Ω Kt n= x Ln x + then (3.9) and (3.10) give a solution to the monic type RII chain (3.6) expressed by the Hankel determinant k,l,t τn . The reason why the Hankel determinant appears can be explained from the point of view of the dis- crete two-dimensional Toda hierarchy [24]. Note that there is another determinant expression of the RII polynomials and a solution to the RII chain: the Casorati-type determinant solution [25, 26].

3.2. Finite dimensional case In this subsection, we will derive the solution to the initial value problem and the convergence theorem for the monic type finite RII chain. Let us start with a pair of tridiagonal matrices of order N:

t 1 1 ⎛ v0 κt ⎞ ⎛ ⎞ ⎜ t t ⎟ ⎜ t t ⎟ ⎜λ w( ) v κ ⎟ ⎜w 1 + w 1 ⎟ ⎜ 1 1 1 t 1 ⎟ ⎜ 1 1 ⎟ A t = ⎜ ( ) ( )t ⋱ ⋱ ⎟ , B t = ⎜ ( ) w t( ) ⋱ ⋱ ⎟ . (3.11) ⎜ λ2w2 + ⎟ ⎜ 2 ⎟ ( ) ⎜ ⎟ ( ) ⎜ ( ) ⋱ ⋱ ⎟ ⎜ ( ) ⋱ ⋱ κt N 2⎟ ⎜ 1 ⎟ ⎝ t t ⎠ ⎝ w t 1 + w t ⎠ λN 1wN 1 v+N −1 N 1 N 1 ( ) ( ) ( ) ( ) − − − The corresponding monic type finite RII−chain is−

t 1 t 1 t 1 1 + qn t 1 t 1 t 1 κt n 1qn + λnen − s (1 + qn )(1 + en ) + (t+1) ( + ) ( + ) 1 qn 1 ( + ) ( + ) ( + ) + + t ( + ) 1 + e − = κ q t n 1 + λ e t − s t (1 + q t )(1 + e t ), (3.12a) t n n ( t) n 1 n 1 n n 1 ( ) 1 + en+ ( ) ( ) ( ) ( ) + + t 1 ( ) + t + + t 1 t 1 1 + qn t t 1 en 1 q en = qn en , (3.12b) n 1 + (t+1) + ( t) ( + ) ( + ) 1 qn 1 ( ) ( ) 1 en+ − ( + ) ( ) t = t = ≥ e0 eN 0 for− all t 0. (3.12c) ( ) ( ) 12 To derive the solution to the initial value problem for the monic type finite RII chain (3.12), we consider { t ( )} t ( ) ∶= ( t − t ) the monic finite RII polynomials ϕn x n 0 defined by ϕn x det xBn An . We should remark ϕ t ( ) ( ) ∞ ( ) ( )( t ( ) t ) that N x is the characteristic polynomial= of the tridiagonal matrix pencil A , B ; the zeros of the ( ) t ( ) ( t t ()) ( ) polynomial ϕN x are the generalized eigenvalues of the matrix pencil A , B , i.e., the solutions of the equation ( ) ( ) ( )

A t Φ = xB t Φ, x ∈ C, Φ ∈ CN − {0}.

( ) ( ) m D(L t ) x = Let be a linear space spanned by the rational functions (t) , m 0, 1, 2, . . . . For the monic K x LN x ( ) N { t ( )}N t finite RII polynomials ϕn x n 0 and any nonzero constant H , there( ) exists( ) a unique linear functional defined on D(L t ) such that( ) the orthogonality relation ( ) = ⎡ ( ) ⎤ ⎢ m t ( ) ⎥ t ⎢ x ϕn x ⎥ t L ⎢ ⎥ = h δm,n, n = 0, 1, . . . , N − 1, m = 0, 1, . . . , n, (3.13a) ⎢ t ( ) ⎥ n ( ) ⎣ Kn (x)Ln(x)⎦ ( ) ( ) and the terminating condition ⎡ ⎤ ⎢ m t ( ) ⎥ t ⎢ x ϕN x ⎥ L ⎢ ⎥ = 0, k, l = 0, 1, . . . , N, m = 0, 1, 2, . . . , (3.13b) ⎢ t ( ()) ( )⎥ ( ) ⎣ Kk x Ll x ⎦ ( ) t t t hold, where the constants h0 , h1 ,..., hN 1 are given by solving the following linear equation ( ) ( ) ( ) ⎛ −1 1 − ⎞ ⎛ t ⎞ t ⎜ t −( + t ) ⎟ h0 ⎛−H ⎞ ⎜w1 1 w1 1 ⎟ ⎜ t ⎟ ⎜ ⎟ ⎜ h( ) ⎟ ⎜ 0( )⎟ ⎜ ( ) w t ( ) ⋱ ⋱ ⎟ ⎜ 1 ⎟ = ⎜ ⎟ , ⎜ 2 ⎟ ⎜ (⋮ ) ⎟ ⎜ ⋮ ⎟ ⎜ ( ) ⋱ ⋱ ⎟ ⎜ ⎟ ⎜ 1 ⎟ t ⎝ ⎠ t t ⎝h ⎠ 0 ⎝ w −(1 + w )⎠ N 1 N 1 N 1 ( ) ( ) ( ) − i.e., − − t = t ( + t + t t + ⋅ ⋅ ⋅ + t t t ) h0 H 1 w1 w1 w2 w1 w2 ... wN 1 , ( ) ( ) ( ) ( ) ( ) ( ) ( ) t = (t)( t + t t + ⋅ ⋅ ⋅ + t t t ) h1 H w1 w1 w2 w1 w2 ... wN 1 ,− ⋮ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) − t = t t t t hN 1 H w1 w2 ... wN 1. ( ) ( ) ( ) ( ) ( ) Note that,− for the infinite dimensional− case, there are two degrees of freedom: the choice of the two constants t t h0 and h1 (see Theorem 3.1). For the finite dimensional case, however, there is only one degree of freedom: the( ) choice( of) the constant H t . The cause of this is the terminating condition (3.13b). t To derive a realization of(L) , we give a quadrature formula for the RII linear functional. Suppose that ( ) t ( ) all the zeros x0, x1,..., xN 1 of the characteristic polynomial ϕN x are simple. ( ) Theorem 3.2 (The quadrature− formula for the RII linear functional). Let x0, x1,..., xN 1 be the simple zeros of t ( ) L t { t ( )}N the characteristic polynomial ϕN x . For the linear functional of the monic finite RII polynomials− ϕn x n 0, t ( )t t ( ) ( ) there exist some constants c0 , c1 ,..., cN 1 such that = ( ) ( ) ( ) N 1 − t t L [R(x)] = Q c R(xi) (3.14) − i ( ) i 0 ( ) holds for all R(x) ∈ D=(L t ). ( ) 13 Proof. This proof is an analogue of the proof to the Gauss quadrature formula (Theorem 2.2). For the given rational function R(x), consider the following interpolation rational function

N 1 t Λ(x) ∶= Q ` (x)R(xi), − i i 0 ( ) where = t t ϕ (x)K (xi)LN(xi) ` t (x) ∶= N N , i = 0, 1, . . . , N − 1. i ( − ( )) t (( )) ( ) t ( ) ( ) x xi KN x LN x ϕ N xi ( ) ′( ) It is readily shown that

` t ( ) = = − i xj δi,j, i, j 0, 1, . . . , N 1, ( ) holds. Let

Q(x) ∶= R(x) − Λ(x).

t Then, the numerator of Q(x) is a polynomial that has zeros at x0, x1,..., xN 1. Since R(x) ∈ D(L ), there exists a polynomial P(x) such that ( ) − P(x)ϕ t (x) Q(x) = N . t ( )( ) ( ) KN x LN x ( ) By the terminating condition (3.13b), we obtain

L t [R(x)] = L t [Λ(x)] + L t [Q(x)] ( ) ( ) ( ) ⎡ ⎤ N 1 ⎢ ( ) t ( ) ⎥ t t t ⎢ P x ϕN x ⎥ = Q L [` (x)]R(xi) + L ⎢ ⎥ − i ⎢ t ( )( ) ( )⎥ i 0 ( ) ( ) ( ) ⎣ KN x LN x ⎦ N 1 ( ) = t t = Q L [` (x)]R(xi). − i i 0 ( ) ( ) t ∶= L t [` t ( =)] = − Set ci i x , i 0, 1, . . . , N 1, then the proof is completed. ( ) ( ) ( ) t t t Zhedanov [13] derived a formula to calculate the constants c0 , c1 ,..., cN 1. He used the second kind polynomials to derive it. Here, we give a direct calculation to check( ) his( ) result.( From) the quadrature formula (3.14), the moment is written as − ⎡ ⎤ ⎢ m ⎥ N 1 t m k,l,t t ⎢ x ⎥ ci xi µm = L ⎢ ⎥ = Q . ⎢ t ( ) ( )⎥ − t (( )) ( ) ( ) ⎣ Kk x Ll x ⎦ i 0 Kk xi Ll xi ( ) ( ) In the same manner as in Subsection= 2.2, we thus obtain the following formulae for j = 0, 1, . . . , N − 1:

N 1 t ( − ) t 1 ci xj xi 2 ϕ (xj) = M M (xν − xν ) , N 1 N 1,N 1,t − t ( ) 1 0 ( ) τN 1 i 0 K (xi)LN 1(xi) 0 ν0 ν1 N 1 i j N 1 − − − ( ) ν0 j,ν1 j = ≤ < ≤ − N 1− − − t ≠ ≠ ≠ ϕ (xj) = M (xj − xi), N − ′( ) i 0 i j = ≠ 14 and

N 1 t N 1,N 1,t ci 2 τ = M M (xν − xν ) , N − t ( )( ) ( ) 1 0 − − i 0 KN 1 xi LN 1 xi 0 ν0 ν1 N 1 ( ) ν0 j,ν1 j = ≤ < ≤ − N 1,N 1,t − − τ ≠ ≠ t = N . hN 1 N−1,N−1,t ( ) τN 1 − − − Hence, we find the− formula

t t h K (xi)LN 1(xi) c t = N 1 N 1 , i = 0, 1, . . . , N − 1. i ( ) t ( () ) t ( ) ( ) −ϕN 1 −xi ϕ N −xi ( ) ′( ) For the finite dimensional− case, in the same manner as for the monic finite orthogonal polynomials (see Subsection 2.2), the characteristic polynomial is invariant under the time evolution:

t 1 ( ) = t ( ) ϕN x ϕN x . ( + ) ( ) From the results in Subsection 3.1, we can thus see that the solution to the initial value problem for the monic type finite RII chain is given by

n,n,t n,n 1,t 1 n 1,n 1,t 1 n 1,n,t τn τ τ τ t = ( t − κ ) 1 n 1 , t = ( t − κ ) n 1 n 1 , qn s t n n 1,n,t 1 n+ 1,n+ 1,t en s t n −n,n−1,t+ n,n,t+ 1 ( ) ( ) − τn τ+n 1 ( ) ( ) τ−n τn + + − + + + + − + where, because the moment is concretely+ given by

0 N 1 c xm ∏t 1(x − s j ) k,l,t i i j 0 i µ = Q ( ) , m − ( −) ( )( ) i 0 Kt k x=i Ll xi the expanded= form of+ the Hankel determinant is

0 ⎛n 1 c ∏t 1(x − s j ) ⎞ k,l,t ri j 0 ri 2 τ = Q ⎜M ( ) M (xr − xr ) ⎟ . n − K (−x )L (x ()) ν1 ν0 0 r0 r1 rn−1 N 1 ⎝ i 0 t k = ri l ri 0 ν0 ν1 n 1 ⎠ + The solution≤ < derived<⋅⋅⋅< above≤ − yields= the following theorem.≤ < ≤ −

Theorem 3.3 (Convergence theorem for the monic type finite RII chain). Suppose that all the generalized eigen- 0 0 values x0, x1,..., xN 1 of the initial tridiagonal matrix pencil (A , B ) are real, simple and arranged in descending order as x > x > ⋅ ⋅ ⋅ > x . Choose the parameters s t and κ ( ) as( )x > s t and x ≫ κ for all t ≥ 0, 0 1 − N 1 t N 1 N 1 N 1 t N 1 respectively. Then, we have the asymptotics of the variables( ) for t ≫ 0: ( ) − + − − − + − t t xn − s qn = t − ( ) ( ) s κt n ( ) ⎧ t ( − j ) t n 1( − ) t ( − j ) t n 1( − ) ⎫ ⎛+ ⎪ ∏j 0 xn s ∏j 0 xn 1 κj ∏j 0 xn 1 s ∏j 0 xn κj ⎪⎞ + O max ⎨ , ⎬ , ⎝ ⎪ t 1( − ( )j ) +t −n 1( − ) t 1( − j( ) t +n −1( − )⎪⎠ ⎩⎪∏j 0= xn 1 s ∏=j 0 x−n κj ∏=j 0 x+n s ∏j =0 xn 1 κj ⎭⎪ t 1( − j−) t n 1( ) − + )− − ( ) + − ⎛ ∏j 0 xn s = ∏−j 0 xn 1 =κj ⎞ = = + e t = O . n ⎝ t −( − ( )j ) + t−n( − ) ⎠ ( ) ∏j =0 xn 1 s ∏= j 0 xn− κj ( ) + = − = (t) t t xn s → +∞ Hence, the variables qn and en converge to (t) and 0 as t , respectively. s κt+n ( ) ( ) − − 15 t t t This theorem implies that, from (3.6), the elements vn and wn of the tridiagonal matrices A and t t B converge to xn and 0 as t → +∞, respectively. Further,( ) we can( ) see that the parameters s and( ) κt n determine( ) the convergence speed; the parameter t works as the origin shift, which is the same( ) as for the s + dqds algorithm (see the end of Subsection 2.3). ( ) Next, we discuss the matrix form of the monic type finite RII chain. Introduce the rational functions defined by the following three-term recurrence relation:

t ( ) ∶= t ( ) ∶= Φ 1 x 0, Φ0 x 1, ( ) ( ) ( − ) t ( ) ∶= − Š( + t ) − t  t ( ) − t ( − ) t ( ) = − x− κt n Φn 1 x 1 wn x vn Φn x wn x λn Φn 1 x , n 0, 1, . . . , N 1. (3.15) ( ) ( ) ( ) ( ) ( ) ( ) By comparing+ to the+ three-term recurrence relation (3.1), the relation −

ϕ t (x) Φ t (x) = n , n = 0, 1, . . . , N, n (t) ( ) Kn (x) ( ) is verified. Let ⎛ t ( ) ⎞ Φ0 x ⎛ 0 ⎞ ⎜ t ⎟ ⎜ ( )( ) ⎟ t ⎜ ⋮ ⎟ t ( ) ∶= ⎜ Φ1 x ⎟ ( ) ∶= ⎜ ⎟ Φ x ⎜ ⎟ , ΦN x ⎜ ⎟ . ⎜ ( )⋮ ⎟ ⎜ 0 ⎟ ( ) ( ) t ⎝ t ( )⎠ ⎝Φ (x)⎠ ΦN 1 x N ( ) ( ) Then, the three-term− recurrence relation (3.15) is rewritten as

t t ( ) + t ( ) = Š t t ( ) + t ( ) A Φ x κt NΦN x x B Φ x ΦN x . (3.16a) ( ) ( ) ( ) ( ) ( ) ( ) + t t t Further, let LA , LB , and R be bidiagonal matrices: ( ) ( ) ( ) ⎛ κt ⎞ ⎛ 1 ⎞ ⎜ t ⎟ ⎜ t ⎟ ⎜−λ1e κt 1 ⎟ ⎜−e 1 ⎟ ⎜ 1 ⎟ ⎜ 1 ⎟ t ∶= ⎜ ( ) t ⎟ t ∶= ⎜ ( ) t ⎟ LA ⎜ −λ2e ⋱ ⎟ , LB ⎜ −e ⋱ ⎟ , ⎜ +2 ⎟ ⎜ 2 ⎟ ( ) ⎜ ( ) ⋱ ⋱ ⎟ ( ) ⎜ ( ) ⋱ ⋱ ⎟ ⎝ − t ⎠ ⎝ − t ⎠ λN 1eN 1 κt N 1 eN 1 1 t ( ) ( ) ⎛q −1 ⎞ − − + − − ⎜ 0 ⎟ ⎜ ( ) t − ⎟ ⎜ q1 1 ⎟ t ∶= ⎜ ⎟ R ⎜ ( ) ⋱ ⋱ ⎟ , ⎜ ⎟ ( ) ⎜ ⋱ −1 ⎟ ⎝ t ⎠ qN 1 ( ) t t t − and Dq , De and Dˆ e be diagonal matrices: ( ) ( ) ( ) t ∶= Š + t + t + t  Dq diag 1 q0 , 1 q1 , . . . , 1 qN 1 , ( ) ( ) ( ) ( ) t t t t t t ∶= Š + +  − ˆ ∶= Š + +  De diag 1, 1 e1 , . . . , 1 eN 1 , De diag 1 e1 , . . . , 1 eN 1, 1 . ( ) ( ) ( ) ( ) ( ) ( ) − − { t ( )}N Then, the spectral transformations (3.4) are written in terms of the rational functions Φn x n 0 as ( ) ( − t ) t t 1 ( ) = ( − ) Š t t ( ) − t ( ) = x s Dq Φ x x κt R Φ x ΦN x , (3.16b) ( ) ( ) ( + ) ( ) ( ) ( ) t t ( ) = ( − ) 1 Š t − t  t 1 ( ) De Φ x x κt xLB LA Φ x . (3.16c) ( ) ( ) − ( ) ( ) ( + ) 16 Equations (3.16) yield

Š t 1 t 1 ( ) + t 1 ( ) − t 1 t 1 ( ) − t 1 ( ) x B Φ x ΦN x A Φ x κt NΦN x ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) = ŠŠ− t 1 t 1 ( t 1 ) 1 t 1 + t 1 t 1  t+ 1 ( ) + t 1 ( ) x Dq LB Dq R Dq De Φ x ΦN x ( + ) ( + ) ( + ) − ( + ) ( + ) ( + ) ( + ) ( + ) − Š− t 1 t 1 ( t 1 ) 1 t 1 + t 1 t 1 t 1  t 1 ( ) − t 1 ( ) Dq LA Dq R s Dq De Φ x κt NΦN x ( + ) ( + ) ( + ) − ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) = ŠŠ− ˆ t t ( t ) 1 t + t ˆ t  t 1 ( ) + t 1 ( ) + x De R De LB Dq De Φ x ΦN x ( ) ( ) ( ) − ( ) ( ) ( ) ( + ) ( + ) − Š− ˆ t t ( t ) 1 t + t t ˆ t  t 1 ( ) − t 1 ( ) De R De LA s Dq De Φ x κt NΦN x ( ) ( ) ( ) − ( ) ( ) ( ) ( ) ( + ) ( + ) = 0. +

Hence, the compatibility condition for (3.16), i.e. the matrix form of the monic type finite RII chain, is given by

t 1 = − t 1 t 1 ( t 1 ) 1 t 1 + t 1 t 1 t 1 A Dq LA Dq R s Dq De ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) = − ˆ t t ( t ) 1 t −+ (t + ) t ˆ( +t ) De R De LA s Dq De , ( ) ( ) ( ) ( ) ( ) t 1 = − t 1 ( ) t 1 ( −t 1 ) 1 ( t) 1 + t 1 t 1 B Dq LB Dq R Dq De ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) = − ˆ t t ( t ) 1 t −+ ( t+ )ˆ t De R De LB Dq De . ( ) ( ) ( ) − ( ) ( ) ( ) This leads to

t 1 t t t t 1 t t 1 t A = Dˆ e R (Dq De ) A (R ) Dq , (t +1 ) (t ) (t ) (t ) (t ) −1 (t ) (t ) −1 (t ) B = Dˆ e R (Dq De ) B (R ) Dq , ( + ) ( ) ( ) ( ) ( ) − ( ) ( ) − ( ) and

t 1 t 1 t t t t 1 t t t 1 t xB − A = Dˆ e R (Dq De ) ŠxB − A  (R ) Dq . ( + ) ( + ) ( ) ( ) ( ) ( ) − ( ) ( ) ( ) − ( ) The last equation implies that the generalized eigenvalues of the tridiagonal matrix pencil (A t , B t ) are conserved under the time evolution. ( ) ( )

4. Generalized eigenvalue algorithm

4.1. Subtraction-free form of the monic type RII chain In Section 3, we have presented the convergence theorem for the monic type finite RII chain (Theo- rem 3.3). This theorem allows us to design a generalized eigenvalue algorithm for tridiagonal matrix pencils. Consider a pair of tridiagonal matrices of order N as input:

⎛a0,0 a0,1 ⎞ ⎛b0,0 b0,1 ⎞ ⎜a a a ⎟ ⎜b b b ⎟ ⎜ 1,0 1,1 1,2 ⎟ ⎜ 1,0 1,1 1,2 ⎟ = ⎜ ⋱ ⋱ ⎟ = ⎜ ⋱ ⋱ ⎟ A ⎜ a2,1 ⎟ , B ⎜ b2,1 ⎟ . (4.1) ⎜ ⋱ ⋱ a ⎟ ⎜ ⋱ ⋱ b ⎟ ⎝ N 2,N 1⎠ ⎝ N 2,N 1⎠ aN 1,N 2 aN 1,N 1 bN 1,N 2 bN 1,N 1 − − − − − − − − − − − − Suppose that all the subdiagonal elements b0,1, b1,2,..., bN 2,N 1 and b1,0, b2,0,..., bN 1,N 2 of the matrix B are nonzero, and all the leading principal minors of the matrix B are nonzero. Then, the transformation − − − − 0 1 0 1 A ∶= V1UAU V2, B ∶= V1UBU V2 ( ) − ( ) − 17 gives the initial matrix pencil of the form (3.11) for the monic type finite RII chain, where

U ∶= diag(1, b0,1, b0,1b1,2,..., b0,1b1,2 ... bN 2,N 1), 1 1 1 V1 ∶= diag ‰(det B1) , (det B2) ,..., (det−BN−) Ž , V2 ∶= diag(1, det B1, det B2, . . . , det BN 1), − − − −0 0 and Bn is the n-th order leading principal submatrix of the matrix B. Namely, the elements of A and B are computed by ( ) ( )

0 det Bn 0 det Bn 1 an,n 1 an,n 1 v ∶= a , , w ∶= b b , κ ∶= , λ ∶= . (4.2) n n n det n n 1,n n,n 1 det n n ( ) Bn 1 ( ) Bn−1 bn,n+1 bn,n−1 − − + + + − det Bn Note that, if n is large, an overflow may occur when one computes det Bn directly. The values and det Bn+1 det Bn−1 should be computed by the LU decomposition. Next, by the relation (3.6), “decompose” the matrix det Bn+1 0 0 pencil (A , B ) to the variables of the monic type finite RII chain: ( ) ( ) 0 ∶= 0 ∶= e0 0, eN 0, (4.3a) ( ) ( ) 0 + 0 0 wn 0 0 1 qn 1 e˜n ∶= , en ∶= e˜n , n = 1, 2, . . . , N − 1, (4.3b) (0 ) + (0) ( ) qn 1 ( ) ( ) 1 qn− (0) 0 0 ( 0) 0 0 vn− − s (1 + wn ) − (s − λn)e˜n qn ∶= , n = 0, 1, . . . , N − 1. (4.3c) ( ) ( ) 0( )− ( ) ( ) ( ) s κn ( ) 0 0 0 Notice that the initial matrix pencil (A , B ) does not fix the values of the parameters s and κN 1. We must choose the parameters s 0 and κ( ) appropriately.( ) We will discuss how to choose the( ) parameters in N 1 − the end of this subsection. After( ) that, compute the time evolution of the monic type finite RII chain by using − (3.12) iteratively; i.e., for each t ≥ 0, compute

t 1 ∶= t 1 ∶= e0 0, eN 0, (4.4a) ( + ) ( + ) t + t 1 + t t 1 t qn 1 qn 1 1 en 1 en ∶= en , n = 1, 2, . . . , N − 1, (4.4b) t( )1 + (t+1) + ( t) ( + ) ( ) qn 1 1 qn− 1 en+ ( + ) ( + ) ( ) ⎛ + t t 1 t 1 t 1− 1 t 1 t 1 en 1 t 1 t 1 1 + qn qn ∶= (s − κt n 1) (s − κt n)qn − (s − λn)en ⎝ + ( t) + (t+1) ( + ) ( + ) − ( + ) ( ) 1 en+ ( + ) ( + ) 1 qn 1 + + + ( ) ( + ) ⎞ + (s t 1 − λ )e t − (s t 1 − s t )(1 + q t )(1 + e−t ) , n 1 n 1 n n 1 ⎠ ( + ) ( ) ( + ) ( ) ( ) ( ) + + n = 0,+ 1, . . . , N − 1. (4.4c)

t 1 Here, we also have to choose the parameters s and κt N for computing the above recurrence equations. From the results in Subsection 3.2, we can see( + that) if the absolute values of all the subdiagonal elements + t t t t t λnwn and wn of the matrix pencil (A , B ) become sufficiently small at a time t, then the values (s − ( ) t t( ) ( ) ( ) ( ) κt n)qn + s give the generalized eigenvalues of the initial tridiagonal matrix pencil (A, B). In general, however,( ) equation( ) (4.4c) requires subtraction operations, which may degrade the accuracy by the loss of + significant digits. A subtraction-free form of the monic type finite RII chain may resolve the problem. Let us introduce an auxiliary variable

( t 1 − ) t 1 − ( t 1 − ) t t 1 s κt n 1 qn s λn 1 en 1 dn = , n = 0, 1, . . . , N − 1. ( + ) ( ++) t ( + ) ( ) ( + ) + + 1 en 1 + + ( ) + 18 This is an analogue of the auxiliary variable (2.22) introduced in the dqds algorithm. Then, the subtraction- free form is derived as

q t d t 1 ∶= (s t − κ )q t − (s t 1 − s t ), d t 1 ∶= d t 1 n − (s t 1 − s t )(1 + q t ), 0 t 0 n n 1 t( )1 n ( + ) ( ) ( ) ( + ) ( ) ( + ) ( + ) qn 1 ( + ) ( ) ( ) − ( + ) n = 1, 2, . . . , N − 1, (4.5a) − t 1 t t 1 t (s − λn 1)e + dn (1 + e ) q t 1 ∶= n 1 n 1 , n = 0, 1, . . . , N − 1, (4.5b) n ( + ) t 1( )− κ ( + ) ( ) ( + ) s+ + t n 1 + ( + ) t + t 1 + t t 1 t 1 t qn + 1+ qn 1 1 en 1 t 1 e ∶= 0, en ∶= en , n = 1, 2, . . . , N − 1, e ∶= 0. (4.5c) 0 t( )1 + (t+1) + ( t) N ( + ) ( + ) ( ) qn 1 1 qn− 1 en+ ( + ) ( + ) ( + ) ( ) From the spectral transformations− (3.4), we have

−( + t 1 ) t 1 ( t ) = Š( t − ) t − ( t − ) t 1  t ( t ) 1 en 1 ϕn 1 s s κt n qn s λn 1 en 1 ϕn s ( − ) ( − ) ( ) ( ) ( ) ( ) ( − ) ( ) ( ) t 1 n,n,t n,n 1,t + ϕ + (s t ) τ +σ + + ⇒ t = − n 1 = n n 1 dn ( − ) , t t( ) n 1,n 1,t 1 n+ 1,n,t 1 ( ) ϕ+n (s ) τn 1 τ+n ( ) ( ) + + − − + where +

k,l,t ∶= S k 1,l,t 1 − t k 1,l,t 1S σn µi j 1 s µi j 0 i,j n 1. + − ( ) + − ≤ ≤ − + + + t In addition, we already have the expression of qn (3.5). Hence, we obtain a sufficient condition for com- puting the recurrence equation (4.5) without subtraction( ) operations except the shift terms in (4.5a): for all n and t,

0 wn > 0, (4.6a) ( ) n t t t t t (−1) ϕn (s ) = det(An − s Bn ) > 0, (4.6b) n (t) (t) 1 ( ) t ( ) t (1) t (−1) ϕn (s ) = det(An − s Bn ) > 0, (4.6c) t ( ) ( + t) ( ) ( + ) ( ) s > κt n, s > λn. (4.6d) ( ) ( ) By (4.2), if the+ input tridiagonal matrix B is a real symmetric positive (or negative) definite matrix, then the condition (4.6a) is satisfied. Further, assume that the generalized eigenvalues x0, x1,..., xN 1 of the input tridiagonal matrix pencil (A, B) are all real and simple, the matrix A is a real matrix and the conditions t − wn > 0 and κt n 1 = λn are satisfied for n = 1, 2, . . . , N − 1 at some time t. Then, it is shown that if the t t parameter( ) s is chosen as s < min{x0, x1,..., xN 1}, the condition (4.6b) is satisfied. The condition (4.6c) + − is also satisfied( ) with s t 1 <( min) {x , x ,..., x }. From Theorem 3.3, if s t is chosen as close as possible 0 1 N 1 − to min{x0, x1,..., xN 1(}+under) the conditions (4.6), the convergence speed( is) accelerated. − By summarizing this subsection, Algorithm 1 is proposed as a new generalized eigenvalue algorithm − for tridiagonal matrix pencils based on the monic type finite RII chain.

4.2. Numerical examples We shall give numerical examples. To construct test problems with known generalized eigenvalues, let { ( )}N us consider the monic finite orthogonal polynomials pn x n 0 defined by N − 1 n(N − n) = p (x) ∶= ‹x −  p (x) − p (x), n = 0, 1, . . . , N − 1, n 1 2 n 4 n 1 + − 19 Algorithm 1 The proposed generalized eigenvalue algorithm based on the monic type finite RII chain 1: function GEVRII(A, B) ▷ A and B are tridiagonal matrices of the form (4.1) { 0 }N 1 { 0 }N 1 { }N 2 { }N 1 2: Compute vn n 0 , wn n 1 , κn n 0 , and λn n 1 by (4.2) ( ) 0 ( ) 3: Set the parameters− s and κ−N 1 appropriately− − ▷ See Theorem 3.3 and the condition (4.6) = = = = { 0 }N 1 ( ) { 0 }N 4: Compute qn n 0 and en −n 0 by (4.3) 5: t ∶= 0 ( ) − ( ) = = 6: repeat t 1 7: Set the parameters s and κt N appropriately ▷ See Theorem 3.3 and the condition (4.6) { t 1 }N (1 + ) { t 1 }N 8: Compute qn n 0 and en + n 0 by (4.5) 9: t ∶= t + 1 ( + ) − ( + ) = = 10: for n = 1, 2, . . . , N − 1 do (t) t t t 1 qn 11: wn ∶= q en ( ) n 1 1 q t ( ) ( ) ( ) + n−1 12: end for − + t t 13: until the absolute values of wn and λnwn are sufficiently small for all n = 1, 2, . . . , N − 1 {( t − ) t + t }(N) 1 ( ) 14: return s κt n qn s n 0 15: end function ( ) ( ) ( ) − + =

( ) ∶= ( ) ∶= { ( )}N with p 1 x 0 and p0 x 1. The polynomials pn x n 0 are the monic Krawtchouk polynomials with a special parameter and it is well known that the Krawtchouk polynomials are orthogonal on x = 0, 1, . . . , N − 1 − with respect to the binomial distribution [27]. This means= that the tridiagonal matrix of order N ⎛(N − 1)~2 1 ⎞ ⎜ ⎟ ⎜(N − 1)~4 (N − 1)~2 1 ⎟ ⎜ ⎟ ⎜ 2(N − 2)~4 (N − 1)~2 1 ⎟ N N K˜ ∶= ⎜ ⎟ ∈ R N ⎜ 3(N − 3)~4 ⋱ ⋱ ⎟ ⎜ ⎟ × ⎜ ⋱ ⋱ 1 ⎟ ⎝ (N − 1)~4 (N − 1)~2⎠ has the eigenvalues 0, 1, . . . , N − 1. The symmetric tridiagonal matrix » ⎛»(N − 1)~2 (N − 1)~4 » ⎞ ⎜ ⎟ ⎜ (N − 1)~4 (N − 1)~2 2(N − 2)~4 ⎟ ⎜ » » ⎟ ⎜ ( − )~ ( − )~ ( − )~ ⎟ ∶= ⎜ 2 N 2 4 » N 1 2 3 N 3 4 ⎟ ∈ N N KN ⎜ ⎟ R ⎜ 3(N − 3)~4 ⋱ ⋱ ⎟ ⎜ » ⎟ × ⎜ ⋱» ⋱ (N − 1)~4⎟ ⎝ (N − 1)~4 (N − 1)~2 ⎠ is similar to K˜ N. Hence, it is readily shown that the tridiagonal matrix pencil (KN + 2IN, KN + IN) has the generalized eigenvalues (n + 1)~n, n = 1, 2, . . . , N. The following experiments were run on a Linux PC with 3.7.4 and gcc 4.7.2 on Intel Core i5 760 2.80 GHz CPU and 4 GB memory. All the computations were performed in double precision and the t 20 t 20 stopping criterion (line 13 in Algorithm 1) was Swn S < 10 and Sλnwn S < 10 for all n = 1, 2, . . . , N − 1. ( ) − ( ) − Example 1. The first example is the case of N = 5:

⎛2 1 » ⎞ ⎜ ⎟ ⎜1» 2 3~2 » ⎟ ⎜ ⎟ K5 = ⎜ 3~2 2 3~2 ⎟ . ⎜ » ⎟ ⎜ 3~2 2 1⎟ ⎝ 1 2⎠ 20 100 t t q0 e1 1 q(t) 10 e(t) 2 1 2 10 ( ) − ( ) t 2 t q2 10 e3 (t) (t) q − e 3 10 3 4 q(t) ( ) 4 − ( ) 10 4 101.5 − 10 5 − 10 6 − 101 10 7 0 200 400 600 800 1,000 0 200 400 600 800 1,000 − t t

100 t t 2 v0 w1 (t) 10 1 (t) v1 w2 ( ) − ( ) v t 2 w t 1.5 2 10 3 ( ) (t) v t − w 3 3 4 (t) 10 ( ) v4 1 − ( ) 10 4 − 10 5 0.5 − 10 6 − 0 10 7 0 200 400 600 800 1,000 0 200 400 600 800 1,000 − t t

Figure 1: The behaviour of the variables of the monic type finite RII chain for the input tridiagonal matrix pencil (K5 + 2I5, K5 + I5) (t) with the parameters s = 1.01 for all t ≥ 0 and κn = 1 for all n ≥ 4.

The generalized eigenvalues of the matrix pencil (K5 + 2I5, K5 + I5) are 2, 3~2, 4~3, 5~4, and 6~5. By this example, we will observe the behaviour of the variables of the monic type finite RII chain and confirm that the proposed algorithm computes the generalized eigenvalues of a given matrix pencil and its convergence t speed depends on the parameters s and κt n. ( ) t = ≥ = ≥ t Figure 1 shows the result with the parameters+ s 1.01 for all t 0 and κn 1 for all n 4, where qn t ( ) t t t ( ) and en are the variables of the monic type RII chain, vn are the diagonal elements of A , and wn are the ( ) t ( ) t ( ) t ( ) subdiagonal elements of B (see equations (3.6b) and (3.6c)). We can confirm that vn and wn converge linearly to the eigenvalues and( ) zero, respectively. Since the shift parameter s t is not so( close) to the( ) minimal eigenvalue 6~5 = 1.2, the stopping criterion is satisfied at t = 4605. ( ) t Figure 2 shows the result with more suitable parameters: s = 1.19 for all t ≥ 0 and κn = −10000 for all n ≥ 4. The convergence speed is much faster than the former example;( ) the stopping criterion is satisfied at t = 48. Table 1 shows the computed eigenvalues. Example 2. Next, the test cases for N = 512, 1024, 2048, 4096, 8192 were computed by two methods. By these examples, we will compare the computation time and the accuracy of the proposed algorithm with a routine called DSYGV in LAPACK 3.4.2 [28]. DSYGV computes the generalized eigenvalues of a given matrix pencil (A, B) in double precision, where A is real symmetric and B is real symmetric and positive definite. Internally, DSYGV computes the Cholesky factorization B = LLT, where L is a lower , trans- 21 100 t t 0 q0 e1 10 (t) (t) q1 e2 10 1 (t) 10 12 (t) q2 e3 − − 2 (t) (t) 10 q3 e4 (t) ( ) − q 24 10 3 4 10 ( ) − − 10 4 − 10 36 10 5 − − 10 6 − 10 48 0 10 20 30 40 50 0 10 20 30 40 50 − t t

100 t t 2 v0 w1 (t) (t) v1 w2 12 v(t) 10 w(t) 1.5 2 3 (t) − (t) v3 w4 (t) ( ) v4 10 24 1 ( ) −

36 0.5 10 −

0 10 48 0 10 20 30 40 50 0 10 20 30 40 50 − t t

Figure 2: The behaviour of the variables of the monic type finite RII chain for the input tridiagonal matrix pencil (K5 + 2I5, K5 + I5) (t) with the parameters s = 1.19 for all t ≥ 0 and κn = −10000 for all n ≥ 4.

(t) Table 1: The eigenvalues computed by Algorithm 1. The parameters are s = 1.19 and κn = −10000 for all t ≥ 0 and n ≥ 4.

Computed eigenvalues True eigenvalues 1.9999999999999998 2.0000000000000000 1.4999999999999991 1.5000000000000000 1.3333333333333335 1.3333333333333333 1.2500000000000000 1.2500000000000000 1.2000000000000000 1.2000000000000000

22 Table 2: The results of the computation by Algorithm 1 for the generalized eigenvalue problems of (KN + 2IN, KN + IN).

Problem size (N) 512 1024 2048 4096 8192 Computation time [sec.] 0.0958 0.392 1.58 6.24 24.6 Maximum relative error 3.109 × 10 15 3.405 × 10 15 1.776 × 10 15 3.701 × 10 15 2.043 × 10 14 16 16 16 16 16 Average relative error 1.344 × 10− 1.211 × 10− 1.154 × 10− 1.072 × 10− 1.129 × 10− − − − − −

Table 3: The results of the computation by DSYGV in LAPACK for the generalized eigenvalue problems of (KN + 2IN, KN + IN).

Problem size (N) 512 1024 2048 4096 8192 Computation time [sec.] 0.162 1.92 30.3 307 2400 Maximum relative error 3.664 × 10 15 6.815 × 10 15 1.304 × 10 14 1.684 × 10 14 2.949 × 10 14 16 16 15 15 15 Average relative error 6.673 × 10− 8.469 × 10− 1.035 × 10− 1.276 × 10− 1.508 × 10− − − − − − forms the generalized eigenvalue problem Aϕ = xBϕ to the eigenvalue problem L 1 AL T(LTϕ) = x(LTϕ) 1 T and solves the eigenvalue problem. We should remark that, even if A and B are both− tridiagonal,− L AL is a dense matrix in general. Hence, we expect that DSYGV spends much time for large problems. On− the other− hand, the proposed algorithm preserves the tridiagonal form of the matrices A t and B t . The proposed algorithm will thus compute the generalized eigenvalues of tridiagonal matrix( pencils) fast( ) and accurately for large problems. Tables 2 and 3 show the results of the computation by the proposed algorithm and DSYGV, respectively. t The parameters for the proposed algorithm are s = (N + 2)~(N + 1) for all t ≥ 0 and κn = −10000 for all n ≥ N − 1. In all the cases, the proposed algorithm( is) faster and more accurate than DSYGV. In particular, the proposed algorithm has an advantage in computation time for large problems. Remark that the techniques t t called deflation and splitting (if Swn S and Sλnwn S become sufficiently small for some n at a time t, then the problem can be deflated or split( into) two problems)( ) were not implemented in the program used for the experiments. These techniques may further accelerate the proposed algorithm.

5. Conclusion

In this paper, we have studied the monic type RII chain in detail and proposed a generalized eigenvalue algorithm for tridiagonal matrix pencils based on a subtraction-free form of the monic type finite RII chain. t It has been shown that, similarly to the dqds algorithm, the parameter s in the monic type finite RII chain plays the role of the origin shifts to accelerate convergence and the proposed( ) algorithm computes the generalized eigenvalues of tridiagonal matrix pencils fast and accurately. In Example 2, the shift parameter s t is chosen ideally and all the conditions (4.6) are satisfied. However, it is difficult to make this situation in general.( ) Further improvements are thus required for practical use. First, in general, the condition for positivity (4.6) is not sufficient for applications; the condition does not provide concrete ways to choose the parameters for general cases. Second, for applying the proposed algorithm to general (not tridiagonal) matrix pencils, a preconditioning called simultaneous tridiagonalization (see, e.g., [29, 30]) is required. In addition to the improvements, comparisons with traditional methods should be discussed. These are left for future work.

Acknowledgments

The authors would like to thank Professor Yoshimasa Nakamura and Professor Alexei Zhedanov for valuable discussions and comments. This work was supported by JSPS KAKENHI Grant Numbers 11J04105 and 22540224. 23 References

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