Efficient Solution of Parameter Dependent Quasiseparable Systems and Computation of Meromorphic Matrix Functions

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1 Efficient Solution of Parameter Dependent Quasiseparable Systems and Computation of Meromorphic Matrix Functions Paola Boito, Yuli Eidelman, Luca Gemignani To cite this version: Paola Boito, Yuli Eidelman, Luca Gemignani. Efficient Solution of Parameter Dependent Quasiseparable Systems and Computation of Meromorphic Matrix Functions. Numerical Linear Algebra with Applications, Wiley, In press. <hal > HAL Id: hal Submitted on 2 Dec 206 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Efficient Solution of Parameter Dependent Quasiseparable Systems and Computation of Meromorphic Matrix Functions P. Boito a,b, Y. Eidelman c, and L. Gemignani d a XLIM DMI, UMR 7252 CNRS Université de Limoges, 23 avenue Albert Thomas, Limoges Cedex, France. paola.boito@unilim.fr b CNRS, Université de Lyon, Laboratoire LIP (CNRS, ENS Lyon, Inria, UCBL), 46 allée d Italie, Lyon Cedex 07, France. c School of Mathematical Sciences, Raymond and Beverly Sacler Faculty of Exact Sciences, Tel-Aviv University, Ramat-Aviv, 69978, Israel. eideyu@post.tau.ac.il d Dipartimento di Informatica, Università di Pisa, Largo Bruno Pontecorvo, Pisa, Italy. l.gemignani@di.unipi.it Abstract In this paper we focus on the solution of shifted quasiseparable systems and of more general parameter dependent matrix equations with quasiseparable representations. We propose an efficient algorithm exploiting the invariance of the quasiseparable structure under diagonal shifting and inversion. This algorithm is applied to compute various functions of matrices. Numerical experiments show the effectiveness of the approach. Keywords: Quasiseparable matrices, shifted linear system, QR factorization, matrix function, matrix equation. 200 MSC: 65F5 Introduction In this paper we propose a novel method for computing the solution of shifted quasiseparable systems and of more general parameter dependent linear matrix equations with quasiseparable representations. We show that our approach has also a noticeable potential for effectively solving some large-scale algebraic problems that reduce to evaluating the action of a quasiseparable matrix function to a vector.

3 Quasiseparable matrices found their application in several branches of applied mathematics and engineering and, therefore, there has been major interest in developing fast algorithms for woring with them in the last decade [6, 7, 7, 6]. The quasiseparable structure arises in the discretization of continuous operators due either to the local properties of the discretization schemes and/or the decaying properties of the operator or its finite approximations. It is a celebrated fact that operations with quasiseparable matrices can be performed in linear time with respect to their size. In particular the QR factorization algorithm presented in [5] computes in linear time a QR decomposition of a quasiseparable matrix A C N N of the form A = V U R, where R is upper triangular whereas U and V are banded unitary matrices and V only depends on the generators of the strictly lower triangular part of A. This implies that any shifted linear system A + σi N, σ C, can also be factored as A + σi N = V U σ R σ for suitable U σ and R σ. Relying upon this fact, in this paper we design an efficient algorithm for solving a sequence of shifted quasiseparable linear systems. The invariance of the factor V can be exploited to halve the overall computational cost. Section 2 illustrates the potential of our approach using motivating examples and applications. In Section 3 we describe the novel algorithm by proving its correctness. Finally, in Section 4 we show the results of numerical experiments which confirm the effectiveness and the robustness of our method. 2 Motivating Examples In this section we describe two motivating examples from applied fields that lead to the solution of several shifted or parameter dependent quasiseparable linear systems. 2. A Model Problem for Boundary Value ODEs Consider the non-local boundary value problem in a linear finite dimensional normed space X = R N : dv dt = Av, 0 < t < τ, (2.) τ v(t) dt = g (2.2) τ 0 where A is a linear operator in R N and g R N is a given vector. Under the assumption that all the numbers µ = 2πi/τ, = ±, ±2, ±3,... are regular points of the operator A the problem (2.), (2.2) has a unique 2

4 solution. Moreover this solution is given by the formula v(t) = q t (A)g, q t (z) = τzezt e zτ. (2.3) Without loss of generality one can assume that τ = 2π. Expanding the function q t (z) in the Fourier series of t we obtain q t (z) = + Z/{0} ze it z i, 0 < t < 2π. We consider the equivalent representation with the real series given by q t (z) = + 2 z(z cos t sin t)(z ), 0 < t < 2π. Using the formula we find that z = z 2 ( 2 + z 2 ) q t (z) = +2 z(z cos t+ z2 sin t)(z ) 2z sin t, 0 < t < 2π and since 2 sin t = π t, 0 we arrive at the following formula < t < 2π q t (z) = + 2 (z cos t + z2 sin t)(z ) z(π t), 0 < t < 2π. Thus we obtain the sought expansion of the solution v(t) of the problem (2.), (2.2): v(t) = g (π t)ag+2 Under the assumptions (A cos t+ A2 sin t)(a I N ) Ag, 0 t 2π. (2.4) (A ii N ) C, = ±, ±2,... (2.5) using the Abel transform one can chec that the series in (2.4) converges uniformly in t [0, 2π]. Hence, by continuity we extend here the formula 3

5 (2.4) for the solution on all the segment [0, 2π]. The rate of convergence of the series in (2.4) is the same as for the series. The last may be 2 improved using standard techniques for series acceleration but by including higher degrees of A. Indeed set Using the identity we get C (t)(a I N ) Ag = C (t) = A cos t + A2 sin t. (A I) = 2 I 2 A2 (A I) 2 C (t)ag The first entry here has the form ( cos t 2 C (t)ag = 2 A 2 g + Using the formulas cos t 2 Thus we get = π2 6 π 2 t + t2 4, sin t 3 v(t) = V 0 (t)g+v (t)ag+v 2 (t)a 2 g+v 3 (t)a 3 g 2 with 2 C (t)(a I) A 3 g. ) sin t 3 A 3 g. = π2 6 t π 4 t2 + t C (t)(a I) A 3 g. (2.6) V 0 (t) =, V (t) = t π, V 2 (t) = π2 3 πt + t2 2, V 3(t) = π2 3 t π 2 t2 + t3 6. Summing up, our proposal consists in approximating the solution v(t) of the problem (2.), (2.2) by the finite sum v l (t) = g (π t)ag+2 l (A cos t+ A2 sin t)(a I N ) Ag, 0 t 2π, or using (2.6) by the sum 2 l ˆv l (t) = 3 V j (t)a j g j=0 (2.7) 2 (A cos t + A2 sin t)(a I N ) A 3 g, 0 t 2π, (2.8) 4

6 where l is suitably chosen by checing the convergence of the expansion. The computation of v l (t i ) v(t i ), 0 i M +, requires the solution of a possibly large set of the shifted systems of the form (A + σ i I N )x i = y, i =,..., l. (2.9) The same conclusion applies to the problem of computing the function of a quasiseparable matrix whenever the function can be represented as a series of partial fractions. The classes of meromorphic functions admitting such a representation were investigated for instance in [3]. Other partial fraction approximations of certain analytic functions can be found in []. In the next section we describe an effective algorithm for this tas. A numerical approximation of the solution can be obtained by using the discretization of the boundary value problem on a grid and the subsequent application of the cyclic reduction approach discussed in []. In this approach the saving of an approximate quasiseparable structure in recursive LU-based solvers may be used as it was done in the recent papers [2, 4, 9]. The approximate quasiseparable structure of functions of quasiseparable matrices has also been investigated in [2]. 2.2 Sylvester-type Matrix Equations As a natural extension of the problem (2.9), the right-hand side y could also depend on the parameter, that is, y = y(σ) and y i = y(σ i ), i =,..., l. This situation is common in many applications such as control theory, structural dynamics and time-dependent PDEs [0]. In this case, the systems to be solved taes the form AX + XD = Y, A R N N, D = diag [σ,..., σ l ], Y = [y,..., y l ]. Using the Kronecer product this matrix equation can be rewritten as a bigger linear system A vec(x) = vec(y ), where A = I l A + D T I N R Nl Nl, vec(x) = [ x T,..., ] T [ xt l, vec(y ) = y T,..., y T ] T l. The extension to the case where D is replaced by a lower triangular matrix L = (l i,j ) R l l is based on the bacward substitution technique which amounts to solve (A + l i,i I N )x i = y i l j=i+ l i,j x j, i = l: :. (2.0) Such approach has been used in different related contexts where the considered Sylvester equation is occasionally called sparse-dense equation [4]. Then the classical reduction proposed by Bartels and Stewart [3] maes possible to deal with a general matrix F by first computing its Schur decomposition F = ULU H and then solving A(XU) + (XU)L = Y U. The 5

7 resulting approach is well suited especially when the size of A is large w.r.t. the number of shifts. If A is quasiseparable then (2.0) again reduces to computing a sequence of shifted systems having the same structure in the lower triangular part and the method presented in the next section can be used. 3 The basic algorithm To solve the systems (2.9), (2.0) we rely upon the QR factorization algorithm described in [5] (see also Chapter 20 of [6]). On the first step we compute the factorization A + σi = V T σ (3.) with a unitary matrix V and a lower banded, or a bloc upper triangular, matrix T σ. It turns out that the matrix V σ does not depend on σ at all and moreover essential part of the quasiseparable generators of the matrix does not depend on σ also. So a relevant part of computations for all the values of σ may be performed only once. Thus the problem is reduced to the solution of the set of the systems T σ x σ = V H y i, σ = σ i, i =,..., l. (3.2) The inversion of every matrix T σ as well as the solution of the corresponding linear system is basically simpler than for the original matrix. We compute the factorization T σ = U σ R σ with a bloc upper triangular unitary matrix U σ and upper triangular R σ and solve the systems R σ x σ = U H σ V H y i. Thus we obtain the following algorithm. Recall that a bloc matrix A = (A i,j ) N i,j=, A i,j R m i m j is said to have lower quasiseparable generators p(i) R m i ri L (i = 2,..., N), q(j) R rj L m j (j =,..., N ), a() R rl rl ( = 2,..., N ) of orders r L ( =,..., N ) and upper quasiseparable generators g(i) R m i ri U (i =,..., N ), h(j) R ru j m j (j = 2,..., N), b() R ru ru ; ( = 2,..., N ) of orders r U ( =,..., N ) if { p(i)a > i,j q(j) if j < i N; A i,j = g(i)b < i,jh(j) if i < j N where a > i,j = a(i ) a(j +) for i > j + and a j+,j = I r L j, and, similarly, b < i,j = b(i + ) b(j ) for j > i + and b i,i+ = I r U i. 6

8 Theorem 3.. Let A = (A i,j ) N i,j= be a bloc matrix with entries of sizes m i m j, lower quasiseparable generators p(i) (i = 2,..., N), q(j) (j =,..., N ), a() ( = 2,..., N ) of orders r L ( =,..., N ), upper quasiseparable generators g(i) (i =,..., N ), h(j) (j = 2,..., N), b() ( = 2,..., N ) of orders r U ( =,..., N ) and diagonal entries d() ( =,..., N). Let y i = (y i ()) N be a vector column with m - dimensional coordinates y() and let σ i, i =,..., l be complex numbers. Then the set of solutions x (i) = x σi, i =,..., l of the systems (3.2) is obtained as follows.. Set ρ N = 0, ρ = min{m + ρ, r L }, = N,..., 2, ρ 0 = 0, ρ = ρ + r U, =,..., N, ν = m + ρ ρ, =,..., N. Compute the quasiseparable representation of the matrix V and the basic elements of the representation of the matrix T σ as well as the vector w i = V H y i as follows... Using QR factorization or another algorithm compute the factorization ( ) X N p(n) = V N, (3.3) 0 νn r L N where V N is a unitary matrix of sizes m N m N, X N is a matrix of sizes ρ N r L N. Determine the matrices p V (N), d V (N) of sizes m N ρ N, m N ν N from the partition V N = ( p V (N) d V (N) ). (3.4) Compute ( ) ( ) ht (N) h(n) = d T (N) VN Hd(N), (3.5) ( ) cn,i = VN H y w i (N) i (N), i l (3.6) with the matrices h T (N), d T (N), c N,i, w i (N) of sizes ρ N m N, ν N m N, ρ N, ν N respectively. Compute ( ) 0r U γ N = N m N p H V (N). (3.7).2. For = N,..., 2 perform the following..2.. Using QR factorization or another algorithm compute the factorization ( ) ( ) p() X = V X + a(), (3.8) 0 ν r L 7

9 where V is a unitary matrix of sizes (m + ρ ) (m + ρ ), X is a matrix of sizes ρ r L. Determine the matrices p V (), q V (), a V (), d V () of sizes m ρ, ρ ν, ρ ρ, m ν from the partition ( ) pv () d V = V (). (3.9) a V () q V ().2.2. Compute ( ) ht () b T () = d T () g T () ( Ir U 0 0 V H ) h() b() 0 d() g() 0 X + q() 0 I ρ. (3.20) with the matrices h T (), b T (), d T (), g T () of sizes ρ m, ρ ρ, ν m, ν ρ respectively. Compute ( ) 0r U γ = m p H V () (3.2) and ( c,i w i () ) = V H ( yi () c +,i ), i l (3.22) with the vector columns c,i, w i () of sizes ρ, ν respectively..3. Set V = I ν and ( ) ( ) ( ) d() g() 0 Im d T () =, g X 2 q() T () =, γ 0 I =, ρ 0 ρ m (3.23) ( ) w i () = V H yi (), i l. (3.24) c 2,i 2. For i =,..., l perform the following: 2.. Compute the factorization T σi = U σi R σi and the vector v (i) = v σi = Uσ H i w i, w i = V H y i Compute the QR factorization d T () + σ i γ = U (i) ( d (i) R () 0 ρ m ), (3.25) where U (i) is a ν ν unitary matrix and d (i) R () is an upper triangular m m matrix. Compute ( ) g (i) R () Y (i) = (U (i) )H g T (), (3.26) 8

10 ( v (i) () α (i) ) = (U (i) )H w i () (3.27) with the matrices g (i) R (), v(i) (), Y (i), α(i) of sizes m ρ, m, ρ ρ, ρ For = 2,..., N perform the following. Compute the QR factorization ( ) ( ) Y (i) (h T () + σ i γ ) d T () + σ i d H V () = U (i) d (i) R (), (3.28) 0 ρ m where U (i) is an (m + ρ ) (m + ρ ) unitary matrix and d (i) R () is an m m upper triangular matrix. Compute ( ) ( ) g (i) R () ( Y (i) v (i) () α (i) = (U (i) )H ) = (U (i) )H Y (i) b T () g T () ( α (i) w i () ), (3.29) (3.30) with the matrices g (i) R (), v(i) (), Y (i), α(i) of sizes m ρ, m, ρ ρ, ρ Compute the QR factorization ( ) Y (i) N h T (N) + σ i γ N d T (N) + σ i d H V (N) = U (i) N d(i) R (N), (3.3) where U (i) N is a unitary matrix of sizes (ν N +ρ N ) (ν N +ρ N ) and d (i) R (N) is an upper triangular matrix of sizes m N m N. Compute ( ) v (i) (N) = (U (i) α (i) N )H N. (3.32) w i (N) 2.2. Solve the system R σi x (i) = v (i) Compute x (i) (N) = (d (i) R (N)) v (i) (N), For = N,..., 2 compute x (i) () = (d (i) R ()) (v (i) () g (i) R ()η(i) ), Compute η (i) N = (h T (N) + σ i γ N )v (i) (N) x (i) () = (d (i) R ()) (v (i) () g (i) R ()η(i) ) 9 η(i) = b T ()η (i) +(h T ()+σ i γ )x (i) ().

11 Proof. The shifted matrix A + σi N has the same lower and upper quasiseparable generators as the matrix A and diagonal entries d()+σi m ( =,..., N). To compute the factorization (3.) we apply Theorem 20.5 from [6]. Directly from Theorem 20.5 we obtain the representation of the unitary matrix V in the form where V = ṼNṼN Ṽ2Ṽ, (3.33) Ṽ = V I φ, Ṽ = I η V I φ, = 2,..., N, Ṽ N = I ηn V N with η = j= m j, φ = N j=+ and (m + ρ ) (m + ρ ) unitary matrices V, as well as the formulas (3.3), (3.4), (3.8), (3.9), V = I ν. Here we see that the matrix V does not depend of σ. Moreover the representation (3.33) yields the formulas (3.6), (3.22), (3.24) to compute the vectors w i = V H y i, i l. Next we apply the corresponding formulas from the same theorem to compute diagonal entries d σ T () ( =,..., N) and upper quasiseparable generators gt σ (i) (i =,..., N ), hσ T (j) (j = 2,..., N), bσ T () ( = 2,..., N ) of the matrix T σ. Hence, we obtain that ( ) ( ) h σ ( ) T (N) Ir U 0 h(n) d σ T (N) = N 0 VN H, d(n) + σi mn ( d σ T () gt σ () ) ( ) d() + σim g() 0 =, X 2 q() 0 I ρ and for = N,..., 2, ( h σ T () b σ T () d σ T () gσ T () ) = ( Ir U 0 0 V H From here we obtain the formulas ) h σ T () = h T () + σγ, = N,..., 2, h() b() 0 d() + σi m g() 0 X + q() 0 I ρ d σ T () = d T () + σd V (), = N,..., 2, d σ T () = d T () + σγ, gt σ () = g T (), =,..., N, b σ T () = b T (), = 2,..., N (3.34) with h T (), d T (), g T (), b T () and γ as in (3.5), (3.7), (3.20), (3.2) and (3.23). Now by applying Theorem 20.7 from [6] to the matrices T σi, i =, 2,..., M with the generators determined in (3.34) we obtain the formulas (3.25), (3.26), (3.28), (3.29), (3.3) to compute unitary matrices U (i) and quasiseparable generators of the upper triangular R σi such that T σi = U σi R σi, where. U σi = Ũ (i) Ũ (i) 2 Ũ N Ũ (i) (i) N (3.35) 0

12 with Ũ (i) = U (i) I φ, Ũ (i) = I η U (i) I φ, = 2,..., N, Ũ (i) N = I η N U (i) N. Moreover the representation (3.35) yields the formulas (3.27), (3.30), (3.32) to compute the vector v (i) = Uσ H i w i, i l. Finally applying Theorem 3.3 from [6] we obtain Step 2.2 to compute the solutions of the systems R σi x (i) = v (i). Concerning the complexity of the previous algorithm we observe that under the simplified assumptions of r L = ru = r, m i = m j = m and r << m the cost of step is of order (6r 2 m+2m 2 )(Nm) whereas the cost of step 2 can be estimated as (2m 2 l)(nm) arithmetic operations. Therefore, the proposed algorithm saves at least half of the overall cost of solving l shifted quasiseparable linear systems. 4 Numerical Experiments The proposed fast algorithm has been implemented in MATLAB. All experiments were performed on a MacBooPro equipped with MATLAB R206b. Example 4.. Let us test the computation of functions of quasiseparable matrices via series expansion, as outlined in Section 2. We choose A as the one-dimensional discretized Laplacian with zero boundary conditions, and g as a random vector with entries taen from a uniform distribution over [0, ]. Define v ex = 2πAe At (e 2πA I) g, as exact solution (computed in multiprecision) to the problem (2.), (2.2). Let v l and ˆv l be the approximate solutions obtained from (2.7) and (2.8), respectively, with l expansion terms. Figures and 2 show logarithmic plots of the absolute normwise errors v ex v l 2 and v ex ˆv l 2 for t = π/2 and t = π/2, and values of l ranging from 0 to 500. The results clearly confirm that the formulation (2.8) has improved convergence properties with respect to (2.7). Note that the decreasing behavior of the errors is not always monotone. Example 4.2. This example is taen from [5], Example 3.3. We consider here the matrix A R stemming from the centered finite difference discretization of the differential operator u + 0u x on the unit square with homogeneous Dirichlet boundary condition. Note that A has (scalar) quasiseparability order 50, but it can also be seen as bloc -quasiseparable with bloc size 50. The code is available at software.html.

13 log 0 (error) 0 - formula (2.7) formula (2.8) number of terms Figure : Results for Example 4. for t = π/2. This is a logarithmic plot of the errors given by the application of formulas (2.7) (blue circles) and (2.8) (red diamonds) for the computation of a matrix function times a vector. We want to compute the matrix function A 2 b l κ (ω 2 I A) Ab, where b is the vector of all ones. We apply the rational approximations presented in [] as method2 and method3 (the latter is designed for the square root function and it is nown to have better convergence properties). Table shows 2-norm relative errors with respect to the result computed by the MATLAB command sqrtm(a)*b, for several values of l (number of terms in the expansion or number of integration nodes). We have tested three approaches to solve the shifted linear systems involved in the expansion: classical bacslash solver, fast structured solver with blocs of size and quasiseparability order 50, and fast structured solver with blocs of size 50 and quasiseparability order. For each value of l, the errors are roughly the same for all three approaches: in particular, the fast algorithms appear to be as accurate as standard solvers. Example 4.3. The motivation for this example comes from the classical problem of solving the Poisson equation on a rectangular domain with uni- 2

14 log 0 (error) 0 formula (2.8) formula (2.7) number of terms Figure 2: Results for Example 4. for t = π/2. This is a logarithmic plot of the errors given by the application of formulas (2.7) (blue circles) and (2.8) (red diamonds) for the computation of a matrix function times a vector. In this example the errors do not decrease monotonically. Table : Relative errors for Example 4.2. l rel. error (method 2) rel. error (method 3) e e e e e e e 8.6e e 9.56e e 2.6e e e e 3 2.4e 3 3

15 Table 2: Relative errors for Example 4.3. N a N b 0 2.8e 5 3.5e e e 5.8e e e 5.34e e e e 6.98e e e 4 2.4e e 5 2.4e e 4.93e 3 4.5e e e e 4 2.e 3 5.6e 3 form zero boundary conditions. The equation taes the form u(x, y) = f(x, y), with 0 < x < a, 0 < y < b, and its finite difference discretization yields a matrix equation AX + XB = F with X, F R N b N a, (4.36) where N a is the number of grid points taen along the x direction and N b is the number of grid points along the y direction. Here A and B are matrices of sizes N b N b and N a N a, respectively, and both are Toeplitz symmetric tridiagonal with nonzero entries {, 2, }. See e.g., [8] for a discussion of this problem. A widespread approach consists in reformulating the matrix equation (4.36) as a larger linear system of size N a N b N a N b via Kronecer products. Here we apply instead the idea outlined in Section 2.2: compute the (well-nown) Schur decomposition of B, that is, B = UDU H, and solve the equation A(XU) + (XU)D = F U. Note that, since D is diagonal, this equation can be rewritten as a collection of shifted linear systems, where the right-hand side vector may depend on the shift. Table 2 shows relative errors on the solution matrix X, computed w.r.t. the solution given by a standard solver applied to the Kronecer linear system. Here we tae F as the matrix of all ones. In the next examples we test experimentally the complexity of our algorithm. Example 4.4. We consider matrices A n R n n defined by random quasiseparable generators of order 3. The first column of Table 3 shows the running times of our structured algorithm applied to randomly shifted systems (A n + σ i I n )x i = y, for i =,..., 50 and growing values of n. The same data are plotted in Figure 3: the growth of the running times loos linear with n, as predicted by theoretical complexity estimates. 4

16 Table 3: Running times in seconds for Example 4.4. system size n fast algorithm sequential algorithm ratio The second column of Table 3 shows timings for the structured algorithm applied sequentially (i.e., without re-using the factorization (3.)) to the same set of shifted systems. The gain obtained by the fast algorithm of Section 3 w.r.t. a sequential structured approach amounts to a factor of about 2, which is consistent with the discussion at the end of Section 3. Experiments with a different number of shifts yield similar results. Example 4.5. In this example we study the complexity of our algorithm w.r.t. bloc size (that is, the parameter m at the end of Section 3). We choose N = 2, l = 2, r L = r U =, with random quasiseparable generators, and focus on large values of m. Running times, each of them averaged over ten trials, are shown in Table 4. A log-log plot is given in Figure 4, together with a linear fit, which shows that the experimental growth of these running times is consistent with theoretical complexity estimates. 5 Conclusion In this paper we have presented an effective algorithm based on QR decomposition for solving a possibly large number of shifted quasiseparable systems. Two main motivations are the computation of a meromorphic function of a quasiseparable matrix and the solution of linear matrix equations with quasiseparable matrix coefficients. The experiments performed suggest that our algorithm has good numerical properties. Future wor includes the analysis of methods based on the Mittag- Leffler s theorem for computing meromorphic functions of quasiseparable matrices. Approximate expansions of Mittag-Leffler type can be obtained by using the 5

17 time (sec) running times linear fit matrix size Figure 3: Running times for Example 4.4. The linear fit appears to be a good approximation of the actual data. Its equation is y = x Table 4: Running times in seconds for Example 4.5. bloc size m running time (sec)

18 log0 ( time ) data linear fit log0 ( bloc size ) Figure 4: Log-log plot of running times versus bloc size m for Example 4.5. The equation of the linear fit is y = 2.9x 8.4, which confirms that the complexity of the algorithm grows as m 3. Carathéodory-Fejér approximation theory (see [8]). The application of these techniques for computing quasiseparable matrix functions is an ongoing research. References [] Pierluigi Amodio and Marcin Paprzyci. A cyclic reduction approach to the numerical solution of boundary value ODEs. SIAM J. Sci. Comput., 8():56 68, 997. Dedicated to C. William Gear on the occasion of his 60th birthday. [2] J. Ballani and D. Kressner. Matrices with hierarchical low-ran structure. In CIME Summer School on Exploiting Hidden Structure in Matrix Computations. [3] R. H. Bartels and G. W. Stewart. Algorithm 432: Solution of the Matrix Equation ax + xb = c. Comm. of the ACM, 5(9): , 972. [4] Dario A Bini, Stefano Massei, and Leonardo Robol. Efficient cyclic reduction for quasi-birth death problems with ran structured blocs. Applied Numerical Mathematics, 206. [5] Y. Eidelman and I. Gohberg. A modification of the Dewilde-van der Veen method for inversion of finite structured matrices. Linear Algebra 7

19 Appl., 343/344:49 450, Special issue on structured and infinite systems of linear equations. [6] Yuli Eidelman, Israel Gohberg, and Iulian Haimovici. Separable type representations of matrices and fast algorithms. Vol., volume 234 of Operator Theory: Advances and Applications. Birhäuser/Springer, Basel, 204. Basics. Completion problems. Multiplication and inversion algorithms. [7] Yuli Eidelman, Israel Gohberg, and Iulian Haimovici. Separable type representations of matrices and fast algorithms. Vol. 2, volume 235 of Operator Theory: Advances and Applications. Birhäuser/Springer Basel AG, Basel, 204. Eigenvalue method. [8] Roberto Garrappa and Marina Popolizio. On the use of matrix functions for fractional partial differential equations. Mathematics and Computers in Simulation, 8(5): , 20. [9] J. Gondzio and P. Zhlobich. Multilevel quasiseparable matrices in pdeconstrained optimization. Technical Report ERGO--02, School of Mathematics, The University of Edinburgh, 20. [0] G.-D. Gu and V. Simoncini. Numerical solution of parameter-dependent linear systems. Numer. Linear Algebra Appl., 2(9): , [] Nicholas Hale, Nicholas J. Higham, and Lloyd N. Trefethen. Computing A α, log(a), and related matrix functions by contour integrals. SIAM J. Numer. Anal., 46(5): , [2] Stefano Massei and Leonardo Robol. Decay bounds for the numerical quasiseparable preservation in matrix functions. arxiv preprint arxiv: , 206. [3] V. B. Sherstyuov. Expansion of the reciprocal of an entire function with zeros in a strip in the Kreĭn series. Mat. Sb., 202(2):37 56, 20. [4] V. Simoncini. Computational Methods for Linear Matrix Equations. SIAM Rev., 58(3):377 44, 206. [5] Valeria Simoncini. Extended Krylov subspace for parameter dependent systems. Applied numerical mathematics, 60(5): , 200. [6] Raf Vandebril, Marc Van Barel, and Nicola Mastronardi. Matrix computations and semiseparable matrices. Vol.. Johns Hopins University Press, Baltimore, MD, Linear systems. 8

20 [7] Raf Vandebril, Marc Van Barel, and Nicola Mastronardi. Matrix computations and semiseparable matrices. Vol. II. Johns Hopins University Press, Baltimore, MD, Eigenvalue and singular value methods. [8] Frederic Y.M. Wan. An in-core finite difference method for separable boundary value problems on a rectangle. Studies in Applied Mathematics, 52(2):03 3,