Dedicated to Richard S. Varga on the occasion of his 80th birthday.

Transcription

1 MATRICES, MOMENTS, AND RATIONAL QUADRATURE G. LÓPEZ LAGOMASINO, L. REICHEL, AND L. WUNDERLICH Dedicated to Richard S. Varga on the occasion of his 80th birthday. Abstract. Many probles in science and engineering require the evaluation of functionals of the for F u(a) = u T f(a)u, where A is a large syetric atrix, u a vector, and f a nonlinear function. A popular and fairly inexpensive approach to deterining upper and lower bounds for such functionals is based on first carrying out a few steps of the Lanczos procedure applied to A with initial vector u, and then evaluating pairs of Gauss and Gauss-Radau quadrature rules associated with the tridiagonal atrix deterined by the Lanczos procedure. The present paper extends this approach to allow the use of rational Gauss quadrature rules. 1. Introduction. Richard Varga has ade any significant contributions to nuerical analysis, approxiation theory, linear algebra, and analysis. His work is concerned with iterative ethods, atrices, oents, polynoial and rational approxiation, as well as quadrature; see, e.g., [9, 11, 15, 27, 28, 29, 31, 32]. This paper cobines results fro these areas to develop a new ethod for deterining fairly easily coputable upper and lower bounds for functionals of the for (1.1) F u (A) = u T f(a)u, where A R n n is a large, sparse or structured, syetric atrix, u R n, and f is a nonlinear function. The need to evaluate this kind of functionals arises in any applications, such as inverse probles, lattice quantu croodynaics, fractals, error estiation, as well as paraeter deterination for iterative ethods; see, e.g., [2, 6, 7, 8, 17, 24, 25] and references therein. There are several approaches to the evaluation of functionals F u (A). When the atrix A is sall, it ay be easiest to copute the spectral factorization of A and use it to evaluate f(a) with forula (1.5) below. We then copute F u (A) via (1.1). If, oreover, f is a rational function, (1.2) f(t) = p(t) l (t t ), where p is a polynoial, the t are poles, and the atrices A t I are positive definite for all, then it can be attractive to instead copute the Cholesky factorizations of A t I, 1 l, and use the to solve linear systes of equations with these atrices. Here I denotes the identity atrix. For instance, let p be a constant. Then we solve the sequence of linear systes of equations (1.3) (A t I)y = y 1, = 1, 2,...,l, with y 0 = u and evaluate F u (A) = p u T y l. In case the poles are distinct, a partial fraction representation of f can be used; see [5] for a discussion on soe coputational aspects of the latter approach. Departaento de Mateáticas, Universidad Carlos III de Madrid, Leganés, Spain. E-ail: lago@ath.uc3.es. Research supported by grants CCG06-UC3M/ESP-0690 and MTM C Departent of Matheatical Sciences, Kent State University, Kent, OH 44242, USA. E-ail: reichel@ath.kent.edu. Research supported in part by an OBR Research Challenge Grant. Institut für Matheatik, MA 4-5, Technische Universität Berlin, D Berlin, Gerany. E- ail: wunder@ath.tu-berlin.de. 1

2 When the atrix A is large, sparse or structured, and syetric, ethods that require the coputation of the spectral factorization of A or the Cholesky factorizations of the atrices A t I, 1 l, are too coputationally deanding to be attractive. For rational functions f of the for (1.2), one ay deterine approxiate solutions of linear systes of equations of the for (1.3) by an iterative ethod. A non-rational function f can be approxiated by a rational function. This approach often is used to deterine approxiations of f(a)u when A is large and sparse. However, generally only estiates of the error in f(a)u can be coputed; see Froer and Sioncini [12] for a recent discussion, and this in turn gives estiates for the error in the coputed value of F u (A). Bounds for the approxiation error introduced by replacing a non-rational function f by a rational one are discussed in, e.g., [3, 10, 22]. The evaluation of these bounds requires knowledge of an interval that contains all eigenvalues of A. Generally, only estiates of the extree eigenvalues or fairly crude bounds, deterined, e.g., by using Geršgorin disks [32], are available in applications. Crude eigenvalue bounds ay give poor error bounds; see, e.g., [3, 10, 22]. This paper describes a quite siple approach to evaluate upper and lower bounds of F u (A) with fairly little coputational effort for large, sparse or structured, syetric atrices A and suitable functions f. Our approach generalizes a technique proposed by Golub and Meurant [17]. Introduce the spectral factorization (1.4) A = SΛS T, Λ = diag[λ 1, λ 2,...,λ n ] R n n, S R n n, S T S = I and define (1.5) f(a) = Sf(Λ)S T. The function f is required to be differentiable sufficiently any ties in an interval containing the spectru of A. The exact requireents on f are specified in Sections 2 and 3. For notational siplicity, we order the eigenvalues according to λ 1 λ 2 λ n and scale the vector u in (1.1) so that u = 1, where denotes the Euclidean vector nor. We will use the spectral factorization (1.4) to derive properties of our ethod; however, the factorization does not have to be coputed to deterine upper and lower bounds for F u (A). Golub and Meurant [17] describe an elegant technique for coputing upper and lower bounds for F u (A) based on the connection between the Lanczos procedure, orthogonal polynoials, and Gauss-type quadrature rules. The quality of the bounds obtained by application of steps of the Lanczos procedure to A depends on how well the function f can be approxiated by a polynoial of degree 2 1 on the spectru of A. We extend this technique to allow rational approxiation of f. This extension allows cancellation of poles of the integrand, which akes it possible to deterine upper and lower bounds for functionals that cannot be bounded using the ethod in [17]. Our approach also can be attractive when the technique in [17] requires ore Lanczos steps to give bounds of coparable accuracy. Our ethod requires the solution of linear systes of equations of the for (A + si)y = u for one or a few values of the scalar s. This paper is organized as follows. The reainder of this section reviews properties of Gauss quadrature rules. Section 2 discusses the connection between the 2

3 Lanczos procedure, orthogonal polynoials, and Gauss-type quadrature rules, and describes how these quadrature rules can be applied to copute upper and lower bounds for the functional (1.1). Further details on these connections can be found in the survey by Golub and Meurant [17]. Section 3 presents an extension that allows rational approxiation of f. In particular, properties of rational Gauss quadrature rules are discussed. Section 4 describes a few coputed exaples and Section 5 contains concluding rearks. Define the vector [µ 1, µ 2,..., µ n ] = u T S and, using (1.5), express the functional (1.1) in the for (1.6) n F u (A) = u T Sf(Λ)S T u = f(λ )µ 2. The right-hand side of (1.6) is a Stieltes integral If = f(s)dµ(s) with a nonnegative easure dµ, such that µ is a nondecreasing step function defined on R with ups at the eigenvalues λ. It follows fro u = 1 that the easure dµ has total ass one. The -point Gauss quadrature rule associated with dµ, (1.7) is characterized by G f = f(θ )γ 2, (1.8) If = G f f P 2 1, where P 2 1 denotes the set of polynoials of degree at ost 2 1. The nodes θ of the quadrature rule are the zeros of the th degree orthonoral polynoial with respect to the inner product (1.9) (f, g) = I(fg). It is well known that for a 2 ties continuously differentiable function f in the interval Ω = [λ 1, λ n ], the error of the quadrature rule (1.7) can be expressed as (1.10) E f = (I G )f = f(2) (θ G ) (2)! (s θ ) 2 dµ(s) for soe θ G in the interior of Ω; see, e.g., Gautschi [13, p. 24]. It follows that if the derivative f (2) is of known constant sign in the interior of Ω, then the sign of the error E f can be deterined without evaluating the right-hand side of (1.10). For instance if f (2) is nonnegative on Ω, then so is E f. Let ˆθ R satisfy ˆθ λ 1 or ˆθ λ n, and let ˆΩ denote the convex hull of the set {λ 1, λ n, ˆθ}. The ( + 1)-point Gauss-Radau quadrature rule associated with the easure dµ and with a prescribed node at ˆθ is an expression of the for (1.11) Ĝ +1 f = f(ˆθ )ˆγ 2 + f(ˆθ)ˆγ 2. 3

4 Properties of the nodes ˆθ and weights ˆγ 2 are reviewed in Section 2. The Gauss-Radau rule (1.11) satisfies (1.12) If = Ĝ+1f f P 2. Let the function f be ties continuously differentiable in ˆΩ. Then the error in the quadrature rule (1.11) is given by Ê +1 f = (I Ĝ+1)f = f(2+1) (θĝ) (s (2 + 1)! ˆθ) (1.13) (s ˆθ ) 2 dµ(s) for soe θĝ in the interior of ˆΩ; see, e.g., Gautschi [13, p. 26]. We note for future reference that if the derivative f (2+1) is of known constant sign in ˆΩ, then the sign of the error Ê+1f can be deterined fro (1.13) without explicit evaluation of the right-hand side expression. 2. Bounds via the Lanczos procedure. The discussion of the present section reviews results by Golub and Meurant [17] and Hanke [21]. More details can be found in [17]. Gauss quadrature rules with respect to the easure dµ can be deterined conveniently by the Lanczos procedure. Application of steps of the Lanczos procedure to the atrix A with initial vector v 1 = u yields the decoposition (2.1) AV = V T + β v +1 e T, where V = [v 1, v 2,..., v ] R n and v +1 R n satisfy V TV = I, v +1 = 1, Vv T +1 = 0, and β 0. Moreover, e denotes the th axis vector and α 1 β 1 0 β 1 α 2 β 2 β 2 α 3 (2.2) T =... R β 1 0 β 1 α is a syetric tridiagonal atrix with positive subdiagonal entries; see, e.g., [18, Chapter 9] for a detailed discussion on the Lanczos procedure. We tacitly assue that is sufficiently sall so that the decoposition (2.1) with the stated properties exists. If β = 0, then we set v +1 = 0, and obtain fro (2.1) that F u (A) = e T 1 f(t )e 1. Thus, F u (A) can be evaluated exactly. Henceforth, we assue that β > 0. The relation (2.1) between the coluns v of V shows that (2.3) v = p 1 (A)u, 1 + 1, for certain polynoials p 1 of degree 1. It follows fro the orthonorality of the vectors v that (p 1, p k 1 ) = p 1(s)p k 1 (s)dµ(s) = u T Sp 1 (Λ)p k 1 (Λ)S T u = u T p 1 (A)p { k 1 (A)u = v1 Tp 1(A)p k 1 (A)v 1 0, k, = v Tv k = 1, = k. 4

5 This shows that the polynoials p are orthonoral with respect to the inner product (1.9). Cobining (2.1) and (2.3) yields a recurrence relation for the polynoials, (2.4) β 1 p 1 (s) = (s α 1 )p 0 (s), p 0 (s) = 1, β p (s) = (s α )p 1 (s) β 1 p 2 (s), 2. Thus, the Lanczos procedure is equivalent to the Stieltes procedure for generating orthonoral polynoials. It follows fro the recurrence relation (2.4) that only the coluns v and v 1 have to be available in order to deterine the next colun, v +1. The recurrence relation (2.4) can be expressed as (2.5) [p 0 (s), p 1 (s),..., p 1 (s)]t = s[p 0 (s), p 1 (s),..., p 1 (s)] β [0,...,0, p (s)], which shows that the zeros of p are the eigenvalues of T. Introduce the spectral decoposition T = Q D Q T, D = diag[θ 1, θ 2,..., θ ], Q T Q = I. The weights of the Gauss rule (1.7) are given by γ 2 = (et 1 Q e ) 2, 1, see, e.g., Gautschi [13, Theore 3.1] or Golub and Meurant [17], and it follows that the Gauss rule (1.7) can be expressed as (2.6) G f = e T 1 Q f(d )Q T e 1 = e T 1 f(t )e 1. Hence, G f can be deterined by first coputing the Lanczos decoposition (2.1) and then evaluating one of the expressions (2.6). The following result is an iediate consequence of the above discussion. The atrix T 1 in Theore 2.1 is the leading principal subatrix of order 1 of T. Theore 2.1. Let the function f be 2 ties continuously differentiable in the interval Ω = [λ 1, λ n ]. Assue that β > 0 in (2.1). Then (2.7) (2.8) e T 1 f(t 1 )e 1 < e T 1 f(t )e 1 < u T f(a)u, if f (2) > 0 in Ω, e T 1 f(t 1)e 1 > e T 1 f(t )e 1 > u T f(a)u, if f (2) < 0 in Ω. Proof. It follows fro β > 0 that the integral on the right-hand side of (1.10) does not vanish. Therefore, the right-hand side inequality of (2.7) follows fro (1.10). The nodes and weights of the -point Gauss quadrature rule G associated with the easure dµ, given by (2.6), defines a discrete easure on the real axis. The ( 1)-point Gauss rule G 1 f = e T 1 f(t 1)e 1 associated with the easure dµ also is a Gauss rule associated with the discrete easure deterined by the nodes and weights of G. This shows the left-hand side inequality of (2.7). The inequalities (2.8) can be shown siilarly. We turn to the coputation of Gauss-Radau quadrature rules associated with the easure dµ with a preassigned node ˆθ, and follow the approach in [16]; see also 5

6 [13, Theore 3.2]. Introduce the syetric tridiagonal atrix (2.9) ˆT +1 = α 1 β 1 0 β 1 α 2 β 2 β 2 α β 1 β 1 α β 0 β ˆα +1 R (+1) (+1) with the leading principal subatrix (2.2). The last subdiagonal entry, β, is defined by (2.1) and the last diagonal entry, ˆα +1, is chosen so that ˆT +1 is seidefinite and has the eigenvalue ˆθ as follows. Introduce the polynoial (2.10) ˆp +1 (s) = (s ˆα +1 )p (s) β p 1 (s). Then analogously to (2.5), [p 0 (s), p 1 (s),..., p (s)] ˆT +1 = s[p 0 (s), p 1 (s),..., p (s)] [0,...,0, ˆp +1 (s)], which shows that ˆα +1 should be chosen so that ˆθ is a zero of ˆp +1. Substituting s = ˆθ into (2.10) yields (2.11) ˆα +1 = ˆθ β p 1 (ˆθ) p (ˆθ). This deterines the Gauss-Radau atrix (2.9) and we obtain siilarly to (2.6) that (2.12) Ĝ +1 f = e T 1 f( ˆT +1 )e 1. Hence, Ĝ +1 f can be coputed by applying steps of the Lanczos procedure and then evaluating the expressions (2.11) and (2.12). The following result is analogous to Theore 2.1. The syetric tridiagonal atrix ˆT R in Theore 2.2 is associated with the -point Gauss-Radau rule for the easure dµ with a prescribed node at ˆθ. Theore 2.2. Let the function f be ties continuously differentiable in ˆΩ, the convex hull of the set {λ 1, λ n, ˆθ}, and assue that the step function µ has at least + 2 points of increase. If ˆθ λ 1, then (2.13) (2.14) e T 1 f( ˆT )e 1 < e T 1 f( ˆT +1 )e 1 < u T f(a)u, e T 1 f( ˆT )e 1 > e T 1 f( ˆT +1 )e 1 > u T f(a)u, if f (2+1) > 0 in ˆΩ, if f (2+1) < 0 in ˆΩ. If instead ˆθ λ n, then (2.15) (2.16) e T 1 f( ˆT )e 1 < e T 1 f( ˆT +1 )e 1 < u T f(a)u, e T 1 f( ˆT )e 1 > e T 1 f( ˆT +1 )e 1 > u T f(a)u, if f (2+1) < 0 in ˆΩ, if f (2+1) > 0 in ˆΩ. 6

7 Proof. The requireent that µ have at least +2 points of increase secures that the integral on the right-hand side of (1.13) is nonvanishing. Assue that ˆθ λ 1 and f (2+1) > 0 in ˆΩ. Then the right-hand side inequality of (2.13) follows fro (1.13). The nodes and weights of the ( + 1)-point Gauss-Radau rule Ĝ+1 associated with the easure dµ define a discrete easure on the real axis. The -point Gauss- Radau rule Ĝ f = e T 1 f( ˆT )e 1 associated with the sae easure dµ also is a Gauss-Radau rule associated with the discrete easure deterined by the nodes and weights of Ĝ+1. This shows the left-hand side inequality of (2.13). The inequalities (2.14)-(2.16) follow siilarly. 3. Rational Gauss rules. This section is concerned with an extension of Gauss quadrature rules that is exact for certain rational functions with preselected poles. These rules are known as rational Gauss rules. They were first discussed in [19, 23] and have subsequently received considerable attention; see, e.g., [4, 13, 14, 20, 30] for discussions on the rate of convergence, error bounds, and the selection of poles. Pairs of rational Gauss and Gauss-Radau rules can be used to bound certain functionals (1.1) for which pairs of standard Gauss and Gauss-Radau rules are not guaranteed to provide upper and lower bounds. This is illustrated in Section 4. Moreover, when the integrand f is analytic in a set that contains the interval [λ 1, λ n ] and has a singularity close to this interval, quadrature rules which are exact for rational functions with poles at or near the singularity of f ay yield significantly higher accuracy than standard Gauss quadrature rules with the sae nuber of nodes. Therefore, for soe integrands rational Gauss rules provide tighter bounds than standard Gauss rules using the sae nuber of nodes. We review properties of rational Gauss rules and discuss their application to the coputation of upper and lower bounds for functionals of the for (1.1). Let {z } k be a set of not necessarily distinct real or coplex nubers outside the interval [λ 1, λ n ], and assue that the set is syetric with respect to the real axis. The z will be poles of rational functions that are integrated exactly by the rational Gauss quadrature rules. Introduce the polynoial (3.1) k w(s) = σ (s z ), where we choose the scaling factor σ R\{0} so that the easure (3.2) dµ (s) = dµ(s) w(s) has total ass one. The -point Gauss quadrature rule associated with this easure, (3.3) G f = f(θ )(γ ) 2, is the basis of the rational Gauss quadrature rules. Theore 3.1. Let {θ, (γ ) 2 } be the node-weight pairs of the Gauss rule (3.3). Assue that 1 2 (k+1), where k is the degree of the polynoial w; cf. (3.1). 7

8 Then the -point rational Gauss quadrature rule (3.4) satisfies (3.5) where (3.6) with Q 0 =. Moreover, (3.7) R f = f(θ )w(θ )(γ ) 2 If = R f f Q k P 2 1 k, Q k = span{ E f = (I R )f l ( z ) 1, 1 l k} = d2 ds 2 (fw) s=θ R 1 (2)! (s θ ) 2 dµ (s), where θ R is in the interior of Ω = [λ 1, λ n ]. Proof. Rational Gauss rules of the for (3.4) are discussed, e.g., by Gautschi [13, Section 3.1.4], where also (3.5) is shown. The proof follows by choosing suitable polynoials f in (1.8) with dµ replaced by (3.2). The reainder forula (3.7) is obtained by replacing dµ by (3.2) and f by fw in the reainder forula (1.10) for standard Gauss quadrature. The lower bound for secures that the quadrature rule integrates constants exactly. Rational Gauss-Radau rules can be defined analogously. Thus, let the prescribed node ˆθ satisfy ˆθ λ 1 or ˆθ λ n, and introduce the ( + 1)-point Gauss-Radau quadrature rule associated with the easure (3.2), (3.8) Ĝ +1 f = f(ˆθ f(ˆθ )(ˆγ ) 2 + f(ˆθ)(ˆγ ) 2. The following result, which is based on properties of this Gauss-Radau rule, is analogous to Theore 3.1. Theore 3.2. Let {ˆθ, (ˆγ ) 2 } {ˆθ, (ˆγ ) 2 } be the node-weight pairs of the Gauss-Radau rule (3.8). Assue that k/2, where k is the degree of the polynoial w; cf. (3.1). Then the ( + 1)-point rational Gauss-Radau quadrature rule ˆR +1 f = (3.9) )w(ˆθ )(ˆγ ) 2 + f(ˆθ)w(ˆθ)(ˆγ ) 2 satisfies (3.10) If = where Q k is defined by (3.6). Moreover, (3.11) Ê +1f = (I ˆR +1 )f = d2+1 ds 2+1 (fw) 1 s=θ ˆR (2 + 1)! ˆR +1 f f Q k P 2 k, 8 (s ˆθ) (s ˆθ ) 2 dµ (s),

9 where θ ˆR is in the interior of ˆΩ, the convex hull of the set {λ 1, λ n, ˆθ}. Proof. Rational Gauss-Radau rules (3.9) are discussed by Gautschi [13, Section ]. The theore can be shown siilarly as Theore 3.1. Thus, the property (3.10) is obtained by choosing suitable polynoials f in (1.12) with dµ replaced by (3.2). The reainder forula (3.11) follows by replacing dµ by (3.2) and f by fw in (1.13). Let T R be the syetric tridiagonal atrix associated with the Gauss are the eigenvalues of T and the quadrature rule (3.3), i.e., the nodes {θ weights {(γ ) 2 } are the square of the first coponent of the noralized eigenvectors. Thus, the atrix T relates to the Gauss rule (3.3) siilarly as the atrix (2.2) relates to the Gauss rule (1.7). Analogously to the right-hand side of (2.6), the rational Gauss rule (3.4) can be expressed as } (3.12) R f = e T 1 f(t )w(t )e 1. Substituting the function f 1 into (3.4) and (3.5) yields 1 = R f = which deterines the scaling factor σ = (e T 1 w(θ k (T )(γ ) 2 = e T 1 w(t )e 1, z I)e 1 ) 1 in (3.1). Whether it is preferable to deterine the spectral decoposition of T and evaluate (3.4) or to copute (3.12) depends on the function f, the degree k of the polynoial (3.1), the order of the quadrature rule, as well as on the nuber of ties the quadrature rule is to be evaluated. In the following analogue of Theore 2.1, T 1 subatrix of T point Gauss rule G 1 denotes the leading principal of order 1; the atrix T 1 is associated with the ( 1)- analogous to (3.3). The requireent in the theore below that the ( + 1)-node Gauss rule exists is equivalent to the requireent β > 0 in Theore 2.1. Theore 3.3. Let the function f be 2 ties continuously differentiable in the interval Ω = [λ 1, λ n ] and assue that the (+1)-point Gauss rule analogous to (3.3) exists. If d 2 (fw)/dt 2 > 0 in Ω, then (3.13) e T 1 f(t 1 )w(t 1 )e 1 < e T 1 f(t )w(t )e 1 < u T f(a)u. Siilarly, if d 2 (fw)/dt 2 < 0 in Ω, then (3.14) e T 1 f(t 1 )w(t 1 )e 1 > e T 1 f(t )w(t )e 1 > u T f(a)u. Proof. The theore follows fro the observation that the rational Gauss rule (3.4) is the Gauss rule (3.3) applied to the function f w. The inequalities (3.13) and (3.14) are a consequence of Theore

10 We turn to Gauss-Radau quadrature rules (3.8) associated with the easure (3.2) with a preassigned node ˆθ. Let ˆT +1 R(+1) (+1) be the syetric tridiagonal atrix associated with the Gauss-Radau rule (3.8). Then the rational Gauss-Radau rule (3.9) can be evaluated as In the following theore, ˆT with the -point Gauss-Radau rule Ĝ ˆR +1 f = et 1 f( ˆT +1 )w( ˆT +1 )e 1. denotes the syetric tridiagonal atrix associated analogous to the ( + 1)-point rule Ĝ +1. Theore 3.4. Let the function f be ties continuously differentiable in ˆΩ, the convex hull of the set {λ 1, λ n, ˆθ}. Assue that the ( +2)-point Gauss-Radau rule analogous to (3.9) exists. If d 2+1 (fw)/dt 2+1 > 0 in ˆΩ, then (3.15) e T 1 f( ˆT )w( ˆT )e 1 < e T 1 f( ˆT +1 )w( ˆT +1 )e 1 < u T f(a)u. Siilarly, if d 2+1 (fw)/dt 2+1 < 0 in ˆΩ, then (3.16) e T 1 f( ˆT )w( ˆT )e 1 > e T 1 f( ˆT +1 )w( ˆT +1 )e 1 > u T f(a)u. Proof. The theore follows fro the observation that the rational Gauss-Radau rule (3.9) is the Gauss-Radau rule (3.8) applied to the function fw. The inequalities (3.15) and (3.16) are a consequence of Theore 2.2. We turn to the coputation of the -point rational Gauss rule (3.4) when the easure dµ is defined by (1.6). In view of Theore 3.1, we need to deterine the syetric tridiagonal atrix T R associated with the Gauss rule (3.3). The nontrivial entries of this atrix are recurrence coefficients for orthonoral polynoials with respect to the inner product (3.17) (f, g) = u T f(a)g(a)(w(a)) 1 u. The atrix T can be coputed in several ways. First assue that the polynoial (3.1) can be factored according to (3.18) w(s) = ( w(s)) 2, where w is a polynoial of degree k/2, say, k/2 w(s) = (s z ). Then steps of the standard Lanczos procedure with initial vector ( w(a)) 1 u yields the atrix T. We note that the first k/2 steps of the Lanczos procedure can be carried out without evaluating atrix-vector products with A if the interediate vectors w = (A z I) 1 w 1, = 1, 2,..., k 2 1, are stored with w 0 = u. 10

11 If the polynoial w cannot be factored according to (3.18), then a Lanczos-type procedure that generates two biorthogonal vector sequences with respect to the inner product (3.17), such as v = p 1 (A)u, w = p 1 (A)(w(A)) 1 u, = 1, 2, 3,..., can be used to copute T. Such a procedure requires the evaluation of two atrix-vector products with the atrix A in each step. The need to deterine two biorthogonal sequences arises when it is infeasible or ipractical to copute the vector (w(a)) 1/2 u. Alternatively, we ay first generate the syetric tridiagonal atrix T associated with the standard Gauss quadrature rule G for the easure dµ by applying steps of the Lanczos procedure to A with initial vector u, as described in Section 2, and then odify this atrix to obtain T as follows. Assue that the atrix T and the next subdiagonal eleent, β, already have been coputed, cf. (2.1), and let z 1 be a real zero of the polynoial (3.1). We copute the oent µ 1 = u T (A z 1 I) 1 u, e.g., by solving the linear syste of equations (3.19) (A z 1 I)y (1) = u. Algorith 2.8 in Gautschi [13, p. 129], with T, β, and µ 1 as input, yields the syetric tridiagonal atrix T (1), associated with the -point Gauss quadrature rule for the easure dµ (1) (s) = σ (1) (s z 1 ) 1 dµ(s), and the next subdiagonal entry β (1). Here σ (1) is a scaling factor chosen to give the easure dµ (1) total ass one. If the polynoial (3.1) has another real zero, say z 2, then we update T (1) and β (1) siilarly as T and β. Thus, we first copute the oent µ 2, e.g., by solving the linear syste of equations (A z 2 I)y (2) = y (1), where y (1) satisfies (3.19), and then use T (1), β (1), and µ 2 as input for Algorith 2.8 in [13]. The algorith deterines the syetric tridiagonal atrix T (2), associated with the -point Gauss quadrature rule for the easure dµ (2) (s) = σ (2) (s z 1 ) 1 (s z 2 ) 1 dµ(s), and the next subdiagonal entry β (2). The coefficient σ (2) is a scaling factor. When the polynoial (3.1) has a pair of coplex conugate zeros, say z 3 and z 4, Algorith 2.9 in [13, p. 131] can be used to copute the tridiagonal atrix T (4), associated with the -point Gauss quadrature rule for the easure (3.20) 4 dµ (4) (s) = σ (4) (s z ) 1 dµ(s), and the next subdiagonal entry β (4). The algorith coputes the tridiagonal atrix associated with the poles z 3 and z 4 without using coplex arithetic. It requires the atrix T (2), the next subdiagonal entry β (2), and the oent µ 3 = u T (A z 3 I) 1 y (2) as input. The output fro Algoriths 2.8 and 2.9 in [13] allows the coputation of the atrix associated with the ( + 1)-point Gauss-Radau ˆT (4) +1 quadrature rule for the easure (3.20) and a specified node, ˆθ, as described in Section 2. Given T, β, and the oents µ, 1 3, the coputation of the tridiagonal atrices T (4) (4) and ˆT +1 by Algoriths 2.8 and 2.9 in [13] requires only O() arithetic floating point operations. In the applications of the present paper, k typically is sall and is not large. Algoriths 2.8 and 2.9 in [13] perfor well in this situation. 11

12 4. Coputed exaples. The nuerical exaples of this section illustrate the application of rational Gauss quadrature rules to copute bounds for functionals of the for (1.1). All coputations are carried out in MATLAB with approxiately 16 significant decial digits. F u (A) R f F u (A) ˆR +1 f Table 4.1 Exaple 4.1: f(s) = (s + 1) 1 exp(s/2), w(s) = s + 1, F u(a) = , and A is a syetric positive definite Toeplitz atrix. Exaple 4.1. Let A R n n be the syetric Toeplitz atrix with first row [1, 1/2,..., 1/n] and n = Its sallest and largest eigenvalues are λ in (A) = and λ ax (A) = , respectively. Let u = n 1/2 [1, 1,...,1] T R n. We seek to deterine upper and lower bounds for the functional (1.1) with f(s) = 1 s + 1 exp(s 2 ). The polynoial w(s) = s+1 deterines the rational Gauss and Gauss-Radau quadrature rules R f and ˆR +1 f, respectively. The latter have the fixed node ˆθ = 13. We copute the tridiagonal atrices for the quadrature rules by Algorith 2.8 in [13]. The coputations require the solution of a linear syste of equations (3.19) with the syetric positive definite Toeplitz atrix A + I. Fast algoriths are available for this purpose; they require only O(n log 2 2 n) arithetic floating point operations, see, e.g., [1]. This is not uch ore coputational work than the O(n log 2 n) arithetic floating point operations needed for the evaluation of a atrix-vector product with the atrix A; see, e.g., [26, Section 3.4] for a discussion of the latter. Table 4.1 displays the errors in the coputed rational Gauss and Gauss-Radau quadrature rules. The errors are seen to decrease quickly as increases. The table illustrates that R f < F u (A) < ˆR +1 f. We reark that the standard Gauss and Gauss-Radau rules of Section 2 are not guaranteed to deterine approxiations that bracket F u (A). F u (A) R f F u (A) ˆR +1 f Table 4.2 Exaple 4.2: f(s) = (s ) 1 log( s), w(s) = s , Fu(A) = , and A is a syetric positive definite Toeplitz atrix. Exaple 4.2. Let A R be the syetric Toeplitz atrix with the first row 1 10 [1, 1/2,..., 1/1024]. Its extree eigenvalues are λ in(a) =

13 and λ ax (A) = The vector u is the sae as in Exaple 4.1. We would like to copute upper and lower bounds for the functional (1.1) with f(s) = 1 s log( s). 4 The polynoial w(s) = s deterines the rational Gauss and Gauss-Radau quadrature rules R f and ˆR +1 f, respectively. The latter have a fixed node at the origin. We copute the tridiagonal atrices for the quadrature rules by Algorith 2.9 in [13]. Table 4.2 displays the errors in the coputed rational Gauss and Gauss-Radau quadrature rules. The table illustrates that R f > F u (A) > ˆR +1 f. The values deterined by the standard Gauss and Gauss-Radau rules of Section 2 are not guaranteed to bracket F u (A). t F u (A) G 6 f F u (A) Ĝ7f F u (A) R 6 f F u (A) ˆR 7 f Table 4.3 Exaple 4.3: f(s) = (s + t) 9/10, w(s) = s + 1, Fu(A) 0.6 for the tabulated values of t, and 2 A is a syetric positive definite Toeplitz atrix. Exaple 4.3. Let the atrix A and vector u be the sae as in Exaple 4.2. We wish to deterine upper and lower bounds for the functional (1.1) with f(s) = (s + t) 9/10 for a few values of the paraeter t 1/2. Pairs of Gauss and Gauss- Radau rules of Section 2 yield such bounds, and so do pairs of rational Gauss and Gauss-Radau rules of Section 3 with w(s) = s The Gauss-Radau and rational Gauss-Radau rules have a fixed node at the origin. The so deterined quadrature rules satisfy G f < F u (A) < Ĝ+1f, ˆR+1 f < F u (A) < R f. Table 4.3 shows the rational Gauss-Radau rule to give saller errors of the sae sign than the standard Gauss rule, and a uch saller error for t = 1/2. The rational Gauss rule yield saller errors of the sae sign than the standard Gauss-Radau rule. The exact values of F u (A) are (after rounding) for t = 0.5, for t = 0.6, and for t = 0.7. The accuracy of the standard Gauss rules can be iproved by increasing the sizes of the corresponding tridiagonal atrices. The availability of an accurate approxiation of the functional (1.1) with a sall atrix can be iportant if the approxiant is to be evaluated for any values of the paraeter t. 5. Conclusion. Rational Gauss rules can be used to bound functionals of the for (1.1) in situations when standard Gauss rules cannot. Moreover, when both standard and rational Gauss rules provide bounds, the latter ay give higher accuracy with the sae nuber of nodes. 13

14 Acknowledgeent. LR would like to thank Guillero López Lagoasino and Volker Mehrann for enoyable visits to Leganés and Berlin, during which work on this paper was carried out. REFERENCES [1] G. S. Aar and W. B. Gragg, Superfast solution of real positive definite Toeplitz systes, SIAM J. Matrix Anal. Appl., 9 (1988), pp [2] Z. Bai, M. Fahey, and G. H. Golub, Soe large scale atrix coputation probles, J. Coput. Appl. Math. 74 (1996), pp [3] B. Beckerann and L. Reichel, Error estiation and evaluation of atrix functions via the Faber transfor, in preparation. [4] B. de la Calle Ysern, Error bounds for rational quadrature forulae of analytic functions, Nuer. Math., 101 (2005), pp [5] D. Calvetti, E. Gallopoulos, and L. Reichel, Incoplete partial fractions for parallel evaluation of rational atrix functions, J. Coput. Appl. Math., 59 (1995), pp [6] D. Calvetti, P. C. Hansen, and L. Reichel, L-curve curvature bounds via Lanczos bidiagonalization, Electron. Trans. Nuer. Anal., 14 (2002), pp [7] D. Calvetti, S. Morigi, L. Reichel, and F. Sgallari, An iterative ethod with error estiators, J. Coput. Appl. Math., 127 (2001), pp [8] D. Calvetti and L. Reichel, A hybrid iterative ethod for syetric indefinite linear systes, J. Coput. Appl. Math., 92 (1998), pp [9] G. Csordas and R. S. Varga, Moent inequalities and the Rieann hypothesis, Constr. Approx., 4 (1988), pp [10] V. L Druskin and L. A. Knizhneran, Two polynoial ethods for the coputation of functions of syetric atrices, USSR Coput. Maths. Math. Phys., 29 (1989) pp [11] M. Eierann, W. Niethaer, and R. S. Varga, A study of seiiterative ethods for nonsyetric systes of linear equations, Nuer. Math., 47 (1985), pp [12] A. Froer and V. Sioncini, Stopping criteria for rational atrix functions of Heritian and syetric atrices, SIAM J. Sci. Stat. Coput., 30 (2008), pp [13] W. Gautschi, Orthogonal Polynoials: Coputation and Approxiation, Oxford University Press, Oxford, [14] W. Gautschi, L. Gori, and M. L. Lo Cascio, Quadrature rules for rational functions, Nuer. Math., 86 (2000), pp [15] W. Gautschi and R. S. Varga, Error bounds for Gaussian quadrature of analytic functions, SIAM J. Nuer. Anal., 20 (1983), pp [16] G. H. Golub, Soe odified atrix eigenvalue probles, SIAM Review, 15 (1973), pp [17] G. H. Golub and G. Meurant, Matrices, oents and quadrature, in Nuerical Analysis 1993, eds. D. F. Griffiths and G. A. Watson, Longan, Essex, England, 1994, pp [18] G. H. Golub and C. F. Van Loan, Matrix Coputations, 3rd ed., Johns Hopkins University Press, Baltiore, [19] A. A. Gonchar and G. López Lagoasino, On Markov s theore for ultipoint Padé approxiants, Math. USSR Sb., 34 (1978), pp [20] P. González-Vera, M. Jiénez Paiz, G. López Lagoasino, and R. Orive, On the convergence of quadrature forulas connected with ultipoint Padé-type approxiation, J. Math. Anal. Appl., 202 (1996), pp [21] M. Hanke, A note on Tikhonov regularization of large linear probles, BIT, 43 (2003), pp [22] M. Hochbruck and C. Lubich, On Krylov subspace approxiations to the atrix exponential operator, SIAM J. Nuer. Anal., 34 (1997), pp [23] G. López Lagoasino, Conditions for the convergence of ultipoint Padé approxiants for Stieltes type functions, Math. USSR Sb., 35 (1979), pp [24] G. Meurant, The coputation of bounds for the nor of the error in the conugate gradient algorith, Nuer. Algoriths, 16 (1997), pp [25] S. Morigi, L. Reichel, and F. Sgallari, An iterative Lavrentiev regularization ethod, BIT, 46 (2006), pp [26] M. K. Ng, Iterative Methods for Toeplitz Systes, Oxford University Press, Oxford, [27] A. Ruttan and R. S. Varga, A unified theory for real vs. coplex rational Chebyshev approxiation on an interval, Trans. Aer. Math. Soc., 312 (1989), pp [28] E. B. Saff, A. Schönhage, and R. S. Varga, Geoetrical convergence to e z by rational functions with real poles, Nuer. Math., 25 (1976), pp

15 [29] E. B. Saff and R. S. Varga, The behavior of the Padé table for the exponential, in Approxiation Theory II, eds. G. G. Lorentz, C. K. Chui, and L. L. Schuaker, Acadeic Press, New York, 1976, pp [30] W. van Assche and I. Vanherwegen, Quadrature forulas based on rational interpolation, Math. Cop., 61 (1993), pp [31] R. S. Varga, Matrix Iterative Analysis, 2nd ed., Springer, Heidelberg, [32] R. S. Varga, Geršgorin and His Circles, Springer, Heidelberg,