1. Introduction. Many applications in science and engineering require the evaluation of expressions of the form (1.1)

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1 ERROR ESTIMATION AND EVALUATION OF MATRIX FUNCTIONS VIA THE FABER TRANSFORM BERNHARD BECKERMANN AND LOTHAR REICHEL Abstract. The need to evaluate expressions of the form fa or fab, where f is a nonlinear function, A is a large sparse n n matrix, and b is an n-vector, arises in many applications. This paper describes how the Faber transform applied to the field of values of A can be used to determine improved error bounds for popular polynomial approximation methods based on the Arnoldi process. Applications of the Faber transform to rational approximation methods and, in particular, to the rational Arnoldi process, also are discussed. Key words. matrix function, polynomial approximation, rational approximation, Arnoldi process, rational Arnoldi process, error bound AMS subject classifications. 30E0, 65E05, 65F30. Introduction. Many applications in science and engineering require the evaluation of expressions of the form. fa or fab, where A C n n, b C n, and f is a nonlinear function. The expressions. can be defined in terms of the Jordan canonical form of A, the minimal polynomial of A, or by a Cauchy-type integral. The latter definition requires f to be analytic in an open set containing the spectrum of A, with the path of integration in this set. Detailed discussions on these definitions and their requirements on f are provided by Golub and Van Loan [42, Chapter ], Higham [50], and Horn and Johnson [52, Chapter 6]. Of particular interest are the entire functions ft = expt, ft = expt/t, ft = cost, ft = sint, with applications to the solution of ordinary and partial differential equations [2, 2, 30, 32, 40, 46, 49, 5, 58, 59, 63, 70, 74, 76, 8] as well as to inverse problems [3, 4]. Other functions of interest include Markov functions, such as ft = t, which arises in the solution of systems of stochastic differential equations [3, 9, 29]. The function ft = logt is a modification of a Markov function and also can be treated with the methods of the present paper; see [5, 48, 50] for applications. When the matrix A is small to medium-sized, the expressions. can be evaluated by determining a suitable factorization of A, e.g., in combination with a rational approximation of f; algorithms that factor A are described and analyzed in several of the above references as well as in [9, 5, 2, 38, 6, 76]. The present paper is concerned with the approximation of the expressions. when f is an entire or Markov function and the matrix A is large, sparse, and nonnormal. The methods described also apply when A is a normal matrix and simplify in this case, in particular, when A is Hermitian or skew-hermitian. A convex compact set E, which contains the field of values of A, defined by { } Ay, y WA := y, y : y Cn \ {0} Laboratoire Painlevé UMR 8524 ANO-EDP, UFR Mathématiques M3, UST Lille, F Villeneuve d Ascq CEDEX, France. bbecker@math.univ-lille.fr. Department of Mathematical Sciences, Kent State University, P.O. Box 590, Kent, OH reichel@math.kent.edu.

2 2 B. Beckermann and L. Reichel is assumed to be explicitly known. Since the field of values is convex, it is natural to choose E to be convex as well. Here and throughout this paper, denotes the usual inner product in C n and is the induced Euclidean vector or spectral matrix norm; however, the results discussed extend to more abstract finite- or infinite-dimensional Hilbert spaces. For convenience, we sometimes will assume that, besides being convex and compact, the set E also is symmetric with respect to the real axis. Note that when A is transformed by multiplication by a scalar or by addition of a scalar multiple of the identity, the field of values, and thus E, are transformed in a similar fashion. This paper discusses polynomial and rational approximation methods. The polynomial methods are based on the Arnoldi process, which simplifies to the Hermitian Lanczos process when A is Hermitian. We would like to approximate fab by pab, where p is a polynomial of fairly low degree and therefore investigate how well f can be approximated by polynomials on E. In Section 2, we introduce the Faber transform and show that it often suffices to consider the situation when E is the closed unit disk D. The Faber transform is in Section 3 used to derive new error bounds for polynomial approximants determined via the Arnoldi process. Section 4 applies these results to the approximation of the exponential function and improves bounds reported by Druskin and Knizhnerman [26, 27, 28, 54], and Hochbruck and Lubich [5]. It is well known that some functions, such as the logarithm or fractional powers, can be approximated much better by rational functions than by polynomials on convex sets E close to the origin. For instance, Kenney and Laub [53] proposed to use Padé approximants at the origin for the computation of fa, when fz = log z, E = {z C : z A }, and A < ; see also Higham [48, 50] and Davies and Higham [2]. This paper discusses error bounds for rational approximation with preassigned poles. The Faber transform allows us to consider equivalent rational approximation problems on the unit disk, and obtain error bounds in this manner. Section 5 considers application of the rational Arnoldi process, first considered by Ruhe [68], for the determination of rational approximants. Section 6 is concerned with rational approximation of Markov functions. New upper and lower bounds for the approximation error are derived. The smallest error bounds are obtained for rational approximants with carefully chosen distinct poles. Each pole, z j, requires the solution of a linear system of equations with the matrix z j I A. If these systems are solved by LU-factorization, then the use of rational approximants with few distinct poles of fairly high multiplicity can be advantageous. We derive error bounds for this situation. The use of rational approximants with multiple poles at the origin and infinity, has been discussed by Druskin and Knizhnerman [29] for the situation when the matrix A is symmetric and positive definite, and by Knizhnerman and Simoncini [55] for more general matrices. Section 7 contains concluding remarks. Other approaches to derive error bounds for certain functions have recently been discussed by Diele et al. [23] and Moret [62]. A careful comparison of these methods with those of the present paper is presently being carried out. In the remainder of this section, we introduce notation used throughout the paper. Thus, E denotes a connected compact set in the complex plane C and is assumed to contain at least two points. The extended complex plane is denoted by C = C { } and φ is the Riemann mapping that maps C \ E conformally onto C \ D with the normalization φ = and φ > 0. The inverse map is denoted by ψ, i.e.,

3 Error estimation and evaluation of matrix functions 3 ψ = φ. Both φ and ψ have similar Laurent expansions at infinity,.2 φz = dz + d 0 + d z +..., ψw = cw + c 0 + c w +..., with c > 0 and d = /c. The coefficient c is commonly referred to as the logarithmic capacity of E and is denoted by cap E. For any ρ, the set E ρ is defined via its complement E c ρ := {z C\E : φz > ρ}, i.e., E ρ = C \ E c ρ. In particular, E = E. The nth Faber polynomial F n = Fn E for the set E is defined as the polynomial part of the Laurent expansion at infinity of φ n for n = 0,,... ; cf..2. Faber polynomials are discussed further below; surveys of their properties are provided, e.g., by Gaier [39, Chapter ] and Suetin [78]. Example.. Let E be the closed unit disk. Then φz = z and the Faber polynomials are given by Fn E z = z n, n = 0,,.... Thus, the Fn E are Chebyshev polynomials for E. More generally, for E = {z C : z z 0 r}, we obtain the shifted monomials Fn E z = z z 0 n /r n. Example.2. Let E = [, ]. The mapping ψw = 2 w + w is known as the Joukowski map. The Faber polynomials F n [,], n =, 2,..., are twice the Chebyshev polynomials T n of the first kind, and F [,] 0 =. This follows from the property.3 w n F [,] n ψw = O/w, w, see, e.g., Gaier [39, p. 43], and the fact that T n 2 w + w = w n + w n. 2 More generally, when E is an ellipse, the Faber polynomials F E n are Chebyshev polynomials for E up to a scaling factor. When the foci coalesce, E becomes a disk, cf. Example.. Details when E is an ellipse, as well as further examples, can be found in [20, 78]. We are interested in polynomial approximation of entire functions and rational approximation of Markov functions. The latter are functions of the form.4 fz = β α dµx z x, where µ is a positive measure with suppµ [α, β], α < β <. Thus, f is analytic in C \ suppµ; in particular, f is analytic in C \ [α, β]. Example.3. The function has the representation fz = log + z z fz = /xdx z x,

4 4 B. Beckermann and L. Reichel and therefore is a Markov function. Moreover,.5 log + z = zfz is a simple modification of a Markov function. Example.4. Let < γ < 0 and z C \ R, where R = {z R : z 0}. Let C be an integration path in C \ R surrounding z. The principal branch of z γ can be represented by the Cauchy integral, z γ = 2πi Moving the path C towards R yields.6 C z γ = sinπγ π t γ t z dt, i =. 0 t γ t z dt, which shows that z γ is a Markov function. The integral in.6 exists because the integrand has a singularity of order γ > at the origin and a zero of order + γ > at infinity. Fractional powers z α, for 0 < α <, can be represented by multiplying z γ by z, similarly as in.5. It is possible to represent certain meromorphic functions as Markov functions with respect to a discrete measure. Example.5. We obtain from the product representation of the sine function, see, e.g., [64, Section 3.5], that dµx = z tanh z z x, µ = δ δ j 2 π 2, with δ x the Dirac measure at the point x. The error bounds of Section 6, however, are sharp only if the support of µ is the whole interval [α, β]. 2. The Faber transform. Let AE denote the Banach algebra of functions analytic in the interior, IntE, of E and continuous on E, equipped with the uniform norm L E on E. Moreover, let P k denote the set of polynomials of degree at most k, and P k E the set of polynomials on E of degree at most k equipped with the norm L E. The Faber transform F maps the polynomial to the polynomial j= pw = a 0 w 0 + a w a k w k, a j C, Fpz = a 0 F 0 z + a F z a k F k z, where F j = Fj E is the Faber polynomial of degree j for E. Thus, for pw := w j, we have Fpz = F j z. The mapping F is a bijection from P k D to P k E with inverse F pw = pψw dw 2. 2πi w w, w <, p P ke. D The representation 2. follows from the Cauchy formula and the observation that F n ψw = w n + O/w, w ;

5 Error estimation and evaluation of matrix functions 5 see, e.g., [39, p. 43] for a discussion of the latter; equation.3 is a special case. A set E is said to be a Faber set if there is a constant d, such that for all polynomials p, 2.2 Fp L E d p L D. For instance, sets E with a piecewise smooth boundary E without cusps are Faber sets; see, e.g., Gaier [39, Chapter ] or Ganelius [4]. The constant d depends on the total rotation of E. Let E be a rectifiable Jordan curve of bounded total rotation V. Then we may choose 2.3 d = + 2 V π ; see, e.g., Gaier [39, Theorem 2, pp ]. For convex sets E, we have V = 2π, i.e., F 5. In particular, finite intervals are Faber sets. A Faber set E is said to be an inverse Faber set if there is a constant d, such that for all polynomials p, 2.4 F p L D d p L E. Since the set of polynomials is dense in AE, it follows from 2.2 that if E is a Faber set, then F admits a unique extension that is continuous from AD to AE. We also denote this extension by F. Analogously, if E is an inverse Faber set, then the inequality 2.4 shows that F can be extended in a unique way to a continuous mapping from AE to AD. This extension is also denoted by F. Anderson and Clunie [4, Theorem 2] show that if E is the closure of a Jordan domain with nonempty interior, whose boundary E is rectifiable, of bounded boundary rotation, and has no cusps, then E is an inverse Faber set. Thus, in this situation, F is a bijection from AD to AE with bounded inverse F, and F d, F d, where d and d are the constants in 2.2 and 2.4, respectively. We note for future reference that for sets E with nonempty interior, Fpz = 2.5 pφζ dζ 2πi ζ z, z IntE. E Our interest in explicit bounds for the norms of F and F is motivated by our desire to bound the errors for best uniform polynomial and rational approximation with fixed poles of functions in AE. For some polynomial qz = m j= z z j, z j E, let 2.6 η q k f, E := min{ f p q L E : p P k }. The residue theorem and 2. show that, for any ŵ C \ D and z IntE, we have 2.7 Let 2.8 F w ŵ z = ψ ŵ z ψŵ. qw = w w w w 2... w w m, w j = φz j D.

6 6 B. Beckermann and L. Reichel Then the operator F is a bijection from P k / q onto P k /q for k m ; see Ellacott [34], Ganelius [4], or Suetin [79, p. 324]. It follows that for all f AE and k m, we obtain the bounds 2.9 F η q k F f, D η q k f, E F η q k F f, D; see, e.g., [4, Theorem ] or [35] for details. The above inequalities show that it generally suffices to consider best uniform polynomial and rational approximation with fixed poles on D. In particular cases it is possible to improve the right-hand side bound in 2.9 by considering, instead of F, the modified Faber operators F ±, defined for g AD by 2.0 F gz := Fgz g0, F + gz := Fgz + g0. For convex E, it is known that F 2. This bound can be established, e.g., by modifying the proof of [39, Theorem 2, p. 49]. Moreover, it is shown implicitly by Kővari and Pommerenke [56] that F + 2. An explicit proof of the latter inequality is given in Theorem 2. below. Thus, for convex E and k m, we may replace the quantity F in 2.9 by 2. We remark that no simple explicit bound for F appears to be available. Theorem 2. below generalizes the bound F + g 2 g L D for the modified Faber transform to matrix arguments. This enables us to bound the error in matrix function approximations. This generalization is implicitly contained in the double layer potential representation of fa discussed by Badea et al. [5, Section 4], but these authors do not establish a connection to the modified Faber transform. Our proof follows fairly closely ideas of Crouzeix and his collaborators on norms of functions of matrices and operators [5, 6, 7, 8, 9, 22]. In particular, Crouzeix [7] shows that for any set E C and any matrix or Hilbert space operator A with WA E, the bound 2. fa K f L E, f AE, holds for the universal constant 2.2 K =.08. Crouzeix conjectures that the bound 2. holds for K = 2. Theorem 2.. Let the set E be convex and consist of more than one point. Then the operator F + defined by 2.0 satisfies 2.3 F + 2. Let WA E. Then, for f AD, we have 2.4 F + fa 2 f L D. Proof. We first show 2.3 under the assumption that IntE. Let z IntE. Then the function gw := f/w,

7 Error estimation and evaluation of matrix functions 7 where the bar denotes complex conjugation, is analytic in C \ D where, as usual, C denotes the extended complex plane and continuous on w, with gw = fw for w =. Thus, fw ψ wdw 2πi D ψw z = gw ψ wdw = g = f0, 2πi D ψw z where the second equality follows from the residue theorem applied in the closed complement of D. Adding the conjugate of the above equation to the relation F fz = fw ψ wdw 2πi ψw z, which is obtained by substituting ζ = ψw into 2.5, yields F + fz = 2.5 fwκw dw π with 2.6 κw := 2i ψ w ψw z D D dw dw ψ w ψw z dw. dw Let ζ := ψw E for w =. Then αζ := argψ w dw/ dw exists for almost all w =, with e iαζ being the the tangent to E at ζ. The convexity of E yields 2.7 e iαζ z ζ e 2i iαζ z ζ > 0. It follows that κw > 0 for all w =. We obtain from 2.5 that F + fz π fw κw dw f L D κw dw D π D 2.8 = f L DF + z = 2 f L D. This establishes 2.3 for z IntE. Furthermore, since the boundary can be neglected in the L -norm, the inequality 2.8 holds for all z E. Finally, if z E and E has no interior points, then E is an interval. In this situation E is traversed twice once in each direction as w traverses the unit circle. The tangent vectors vanish at the endpoints of the interval. The bound 2.8 also holds in this situation. This completes the proof of 2.3. We turn to the proof of 2.4, and first assume that WA is contained in the interior of E. In order to derive a matrix-valued analog of the expression 2.6, we observe that F fa = fwψwi A ψ wdw. 2πi D Moreover, since the matrix A and its transpose, A T, have the same eigenvalues, it follows that gwψwi A T ψ wdw = 2πi f0i. D

8 8 B. Beckermann and L. Reichel Adding the conjugate of the latter expression to the former yields F + fa = F fa + f0i = fwkw dw, π D where Kw := 2i ψ dw wψwi A dw ψ wψwi A dw dw and A denotes the conjugate transpose of A. We would like to show that the Hermitian matrix Kw is positive definite for all w =. This is equivalent to establishing that the matrix Gw := ψwi AKwψwI A is positive definite. With ζ and αζ defined as above, we have Gw = 2i e iαζ ζi A e iαζ ζi A. Let v C n be a unit vector. Then a := v Av lives in WA and we obtain v Gwv = 2i e iαζ ζ a e iαζ ζ a > 0, where the inequality follows similarly as 2.7. Application of the Cauchy-Schwarz inequality now yields F + fa sup u, v π fwv Kwu dw D f L D sup u u, v [ Kwu dw ] /2 v Kw v dw π D π D = f L D sup u, v [ /2 F + Au, uf + Av, v] = 2 f L D, in agreement with 2.4. We turn to the situation when WA is not contained in the interior of E. If E has no interior point, then both E and WA are intervals. In particular, the matrix A is normal, and it follows that F + fa max z σa F + fz max z E F + fz = F + f L E 2 f L D by 2.3, where σa denotes the spectrum of A. We therefore may assume that E has an interior point z 0, and let ɛ 0,. Then the field of values of the matrix A ɛ := ɛz 0 I + ɛa, given by WA ɛ = z 0 + ɛwa z 0, is in the interior of E. Hence, for all ɛ 0,, F + fa ɛ 2 f L D. The bound 2.4 follows by letting ɛ 0.

9 Error estimation and evaluation of matrix functions 9 Remark 2.2. Let fw := w n for n =, 2,.... Then F + f = F f = F E n, n =, 2,..., are Faber polynomials for E. We obtain from 2.4 that 2.9 F E n A 2, n =, 2,.... This inequality recently has been shown in [7, Theorem ] in a similar manner. We note that a for convex set E it follows from [56, Theorem 2] that Fn E L E 2. Consider the solution of the linear system of equations Ax = b by the GMRES iterative method, described, e.g., in [7, Chapter 6] and [72]. Let x 0 be an initial approximate solution and let, for k =, 2,..., x k denote the kth iterate generated by the method. Define the associated residual errors r k := b Ax k, k = 0,,.... The inequality 2.9 can be used to derive bounds for the r k in terms of the field of values of A when 0 E. Specifically, one can show that r k { r 0 min 2 φ0 k, 2 + φ0 } φ0 k ; see [7] for details. This inequality improves bounds reported in [8, 3, 33] and [44, Chapter 3]. The following result, which is a consequence of Theorem 2., is applied in the remainder of this paper. Corollary 2.3. Assume that E and WA satisfy the conditions of Theorem 2.. Let g AE, g := F g, r AD, and Then rz := F + rz + F F + gz = F + rz g0. ga ra 2 g r L D. 3. Polynomial approximation via the Arnoldi process. In this section, we assume that the matrix A C n n in. is large and sparse, and that the vector b C n is of unit length. The Arnoldi process applied to A with initial vector b yields, after m steps, the decomposition 3. AV m = V m H m + h m e T m, where V m = [v, v 2,..., v m ] C n m and h m C n satisfy v = b, V mv m = I, and V mh m = 0. Throughout this paper e j denotes the jth axis vector of appropriate dimension. The matrix H m C m m is of upper Hessenberg form, and where range V m = K m A, b, K m A, b := span {b, Ab,..., A m b} is a Krylov subspace. In particular, v j K j A, b, i.e., there is a polynomial p j P j, such that 3.2 v j = p j Ab, j =, 2,..., m;

10 0 B. Beckermann and L. Reichel see, e.g., [42, Chapter 9] for details on the Arnoldi process. We refer to 3. as an Arnoldi decomposition. We remark that if h m = 0, then range V m is an invariant subspace, and it follows that fab = V m fh m e. We therefore henceforth will assume that h m 0. When A is Hermitian, the Arnoldi process simplifies to the Hermitian Lanczos process and the matrix H m in the decomposition 3. is Hermitian and tridiagonal. The columns v j of V m are generated for increasing values of j; the computation of v j requires the evaluation of j matrix-vector product with A and orthogonalization against all the already computed columns v, v 2,..., v j. One would like to keep m in Arnoldi decompositions 3. used in applications fairly small, because the computational effort and storage required to generate the Arnoldi decomposition increases with m. Moreover, instead of computing fab, we will evaluate fh m e, and the computational effort required for the latter typically grows rapidly with m. We note for future reference that since H m = VmAV m, { } Ax, x 3.3 WH m = x, x : x = V my, y C m \ {0} WA E. One easily verifies by induction that for any p P m, we have 3.4 pab = V m ph m V mb = V m ph m e ; see, e.g., [26, 70]. This motivates the use of the polynomial approximation 3.5 V m fh m e fab, where in view of 3.2, the left-hand side can be written as pab for some p P m. The Crouzeix bound 2. with the constant 2.2 yields an immediate bound for the approximation error in 3.5 in terms best polynomial approximation of f on E; cf. 2.6 with q. Proposition 3.. Let E be a convex compact set, such that WA E. Assume that f AE. Then, for all m, 3.6 fab V m fh m e 23 η m f, E. Proof. It follows from 3.4 that for any p P m, we have fab V m fh m e = f pab V m f ph m e f pa + f ph m, where we have used that b =. The inequality 3.6 now is a consequence of 2., 2.2, 3.3, and the fact that WA E. The following theorem connects polynomial approximation of fab with polynomial approximation of F f on D. Theorem 3.2. Let E be a convex and compact set, such that WA E. Assume that f AE. Then, for all m, 3.7 fab V m fh m e 4 η m F f, D. More generally, for gz = q z + q 2 zfz with q P m+s, q 2 P s, there holds 3.8 gab V m+s gh m+s e 4 q 2 Ab η m F f, D.

11 Finally, we have the bounds Error estimation and evaluation of matrix functions 3.9 f m ηm F f, D f j j=m in terms of the coefficients in the Faber series expansion of f, f j := 3.0 fψw dw, j = 0,,.... 2πi wj+ D Proof. Since b =, the bound 3.7 follows from 3.8 by taking q z = 0, s = 0, and q 2 z =. In order to show the latter bound, we choose an extremal polynomial p P m, such that F f p L D = η m F f, D. Similarly as in Corollary 2.3, we define fw := F fw, pz := F + pz f0, qz := q z + q 2 zpz. Then q P m+s and, according to 3.4, gab V m+s gh m+s e = g qab V m+s g qh m+s e f paq 2 Ab + V m+s f ph m+s q 2 H m+s e fa pa q 2 Ab + fh m+s ph m+s q 2 H m+s e, where, again by 3.4, we have q 2 H m+s e = q 2 Ab. Corollary 2.3 yields fa ˆpA 2 f p L D = 2 η m F f, D, and 3.3 combined with Corollary 2.3 gives fh m ˆpH m 2 f p L D = 2 η m F f, D. This establishes the inequalities 3.7 and 3.8. Comparing 3.0 to 2., we observe that f m is the mth coefficient in the Taylor expansion of F f at the origin. Hence, with the extremal p P m as above, f m = 2πi w = F fw w m+ dw = 2πi w = F fw pw w m+ dw, the absolute value being bounded above by F f p L D = η m F f, D. Finally, if j=m f j <, then fz = f j F j z, F fw = f j w j, j=0 because both series are absolutely convergent. The upper bound 3.9 now follows by approximating F f by its Taylor sum m j=0 f jw j. Remark 3.3. Let f AE ρ for some ρ > and change the path of integration from D to {w C : w = ρ} in the definition of the Faber coefficients 3.0. Then one easily verifies that j=0 ρ m f j f L E ρ ρ, j=m

12 2 B. Beckermann and L. Reichel where the factor ρ m corresponds to the classical rate of best polynomial approximation on E of functions in AE ρ ; see, e.g., [83, Theorem IV.5]. In particular, the lower and upper bounds in 3.9 differ only by a term that decreases geometrically, or even faster when f is an entire function, such as the exponential function; see below. In order to compare Proposition 3. and Theorem 3.2, one may either use 2.9, or apply the bounds 3. f m ηm f, E 2 f j. j=m The lower bound can be shown similarly as in the proof of 3.9, and the upper bound by using a partial Faber sum, as well as the fact that F E j L E 2; see Remark 2.2. Remark 3.4. Let us compare Theorem 3.2 with bounds reported by Druskin and Knizhnerman [26, 27, 28, 54]. Knizhnerman [54, Theorem ] shows that there are positive constants C and α, which depend on the shape of E := WA, such that fab V m fh m e C f k k α. When E = [, ], the Faber polynomials Fj E, for j, are twice the Chebyshev polynomials of the first kind; cf. Example.2. The observation that in this case fab V m fh m e 4 k=m f k is at least implicitly included in [27, Proof of Theorem ]. For the exponential function and E = [, ] further improvements and more explicit bounds are derived in [26, 28] by using the fact that the Faber coefficients are explicitly known in terms of Bessel functions. Remark 3.5. Hochbruck and Lubich [5] derive error bounds for analytic functions f in terms of integral formulas and exploit the latter to obtain bounds for the error in polynomial approximations of expτ Ab, τ > 0, determined by the Arnoldi process with WA contained in various convex compact sets E. Let E be a convex compact set containing WA in its interior, and let E be a bounded set that contains E. The boundary Γ of E is assumed to be a piecewise smooth Jordan curve. Let the function f be analytic in the interior of Γ and continuous on the closure of E. Then Hochbruck and Lubich [5, Lemma ] show that dz 3.2 fab V m fh m e C min fz pz p P m 2π φz m for C := k=m length E dist E, WA distγ, WA. We would like to compare this bound to Theorem 3.2 and will use the inequalities Γ 3.3 distz, E φ z φz + φz 2 φz, z C \ E,

13 Error estimation and evaluation of matrix functions 3 which follow from [80, Theorem 3.] and its proof. Let p P m minimize the righthand side of 3.2. Then, for all j m, f j = fz φ z dz 2πi φz j+ = [fz pz] φ z dz 2πi φz j+, Γ and, using 3.3, the right-hand side of 3.7 can be bounded according to 4 f j 2 fz pz π Γ 4 fz pz π Γ C fz pz 2π j=m Γ Γ φ z dz φz m+ / φz dz distz, E φz m dz φz m, C := 8 distγ, E, where we note that the bound in each step may be quite crude. Nevertheless, the ratio C /C can be made arbitrarily small by choosing E close to WA. We would expect the bound of Theorem 3.2 to be most accurate in this situation. Thus, Theorem 3.2 may provide useful bounds when 3.2 does not. The next section discusses applications of Theorem Approximation of the exponential function. The following result provides bounds for the Faber coefficients of the exponential function, and thereby also for the approximation error achieved by the Arnoldi process, via Theorem 3.2, and for best polynomial approximation, via 3.. The bounds of Corollary 4. below only depend on the logarithmic capacity of E for large values of m, whereas Corollary 4.2 discusses the dependence on the outer angle at the right-most boundary point of E for small values of m. Corollary 4.. Let fz = expτz, where τ > 0 is an arbitrary parameter. Let the set E be compact, convex, and symmetric with respect to the real axis, with capacity c = cap E = ψ. Then, for r, the Faber coefficients satisfy 4. and 4.2 f m eτψr r m η m F f, D e τψr r m r. The minimum of the right-hand side of 4. is attained for r = if τψ m, and otherwise at the unique solution of the equation rψ r = m/τ. In particular, if m 2cτ, then 4.3 f m 7 2 τcm eτψ, η m! m F f, D 7e τψ τcm m!. Proof. According to 3.0, we obtain for any r the simple upper bound f m π e τψreit 2π r m dt = r m exp τ max t [ π,π] Rψreit, π

14 4 B. Beckermann and L. Reichel where by symmetry of E, the maximum of the right-hand side is attained at the rightmost point of E, i.e., at t = 0. This shows 4.. The bound 4.2 is obtained by using the fact that ηm F f, D f j. It remains to be shown that the equation τrψ r = m has at most one solution. First notice that, + r ψ r is real by symmetry of E, does not change sign, and tends to c > 0 as r. Hence, ψ r is strictly positive in, +. It follows from convexity that R + wψ w ψ w > 0 for all w >. Therefore, r rψ r is increasing. The choice r = m ct in 4. and 4.2 leads to 4.3. However, there is a missing factor / m, which requires refinement of our bounds. We first show that for w, ψw ψr cw r c w 4.4 r. By convexity of E, we have the inequality j=m ψ w c c, w, w 2 due to Grötzsch and Golusin; see [56, Section 2]. Hence, ψw ψr cw r w R ψ ζ c dζ c w R dζ ζ 2, and we obtain the inequality 4.4 by taking as path of integration the circular arc [0, ] t /r+t/w /r, staying outside of the unit disk. Notice that 4.4 for w = implies that ψr ψ cr + c = c r 4.5. r r Applying our inequality for w = re it, t π, and using again the symmetry of E, we obtain Rψre it R ψr + cw r + c w r which yields f m 2πr m = ψr + crcost + c r e it = ψr 2cr sin 2 t/2 + 2c r sin t/2 ψr 2 cr π 2 t2 + c 2 cr t = ψr 2 t r π 2 π2 4r 2 + cπ2 8r 3, π π τcπ 2 eψr+ 8r 3 = π πr m exp τψr 2 τcr 2 t π 2 π2 4r 2 + τcπ2 8r 3 dt π exp 2 τcr 2 t π 2 π2 4r 2 dt 0 2 τcπ τc expτψr + 8r expτψ + τcr 3 r m r π 2τcr r m 2πτcr + τcπ2 8r 3,

15 Error estimation and evaluation of matrix functions 5 where, in the last inequality, we applied 4.5. Now the choice r = m τc 2 gives τc r + τcπ2 8r 3 0 and τc m f m π expτψ 7 2mm/e m 2 expτψτcm, m! as claimed in the first inequality of 4.3. The second inequality of 4.3 follows by observing that / /r 2. Let E = {z C : z z 0 c} for some constants z 0 R and c > 0. Then τz0 τcm ψw = cw + z 0 and the Faber coefficients are given by f m = e m!. Hence, the bound 4.3 for f m is sharp up to the factor 7 2 expτc, independently of m, whereas the bound 4. with optimal parameter r = m τc is sharp up to a factor 2πm. Further, when E is an interval on the real or imaginary axis, explicit formulas for the Faber coefficients f m can be given in terms of Bessel functions [2, 26]. These formulas show the bound 4.3 for f m to be asymptotically sharp as m up to a constant independent of m. Hochbruck and Lubich [5, Theorems 2,4-6] apply the bound 3.2 to the exponential function when WA E for i E = [ 4c, 0] an interval on the negative real axis and ψw = cw + /w 2, ii E = [ 2ic, 2ic] an interval on the imaginary axis and ψw = cw /w, iii E a disk, such that ψw = cw + c 0, and iv E a drop-shaped region, for which ψw = cw /w α with α >. The latter set has an outer angle απ at the vertex ψ = 0. The bounds for large m given in [5, Theorems 2,4-6] essentially coincide with 4. for r = m τc, though the absolute constants in [5] are somewhat larger. When E is the interval [ 4c, 0] or the drop-shaped region, Hochbruck and Lubich [5] also provide upper bounds for the situation when m 2τc. These sets E have an outer angle απ > π at the right-most boundary point, which is the pre-image of w = under ψ. Therefore, ψ = 0 and inequality 4. indicates that f m may be smaller than e τψ also for m 2τc. It is possible to give a bound depending only on this outer angle. Corollary 4.2. Under the assumptions of Corollary 4., suppose in addition that E has an outer angle απ with α > at its right-most boundary point ψ. Then, for m 2τc, we have f m exp τψ α 7 m 3τc η m F f, D 3 m m 3τc α α, exp τψ α m 7 m α. 3τc 4.8 Proof. In the first part of the proof we show the improvement of 4.5, ψr ψ cr α + 2 α. r r Introduce the generating function for the Faber polynomials, rψ r ψr ψ = + n= F n ψ r n,

16 6 B. Beckermann and L. Reichel where we use the representation of the Faber polynomials from [66, Lemma ], here for convex E, for n, F n ψ = π π π e ins d s arg ψe is ψ, π π d s arg ψe is ψ 2π. Since the Stieltjes integral has a jump of πα at s = 0 and elsewhere the argument is increasing, we obtain, by taking out the jump and using the symmetry, that and, therefore, F n ψ = α + π π 0+ rψ r ψr ψ α r = 2 π π e ins + e ins d s arg ψe is ψ, 0+ π Integrating this inequality from r to gives 4.8. Since α >, we may choose r, such that 4.9 e is R r e is d s arg ψe is ψ = 2 r coss π 0+ r 2 + 2r coss d s arg ψe is ψ 2 π d s arg ψe is ψ = 2 α r + π r r α = m r 3τc. Hence, r r = + 5/2, and we obtain from 4.8 that r τψr ψ m logr τc m r + r + Since y y α is concave, we deduce that α r α r + 2 α + m log r r α r + + m r 3 r. r τψr ψ m logr m α + m r + r r = mα r r r + 2 r r mα r + 2 mα r r. 7 Inserting 4.9 gives 4.6. The bound 4.7 follows by observing that / /r r + /r /r. We conclude this section with three further illustrations/extensions of Corollary 4.. Example 4.. If the matrix A has a negative semi-definite real part, then a simple set containing WA is given by } A + A E = ρd {w C : Rw λ max, 2

17 Error estimation and evaluation of matrix functions 7 provided that ρ > 0 is large enough. For instance, ρ can be chosen to be the norm of A or the numerical radius, max{ z : z WA}. Define the angle β [0, π/2 by cosβ = λ max A+A 2 0. ρ In order to apply the bounds 4.3, we only require the value ψ = ρ cosβ = λ max A+A 2 and the capacity of E, which given by c = ρ π 2π β cos sinβ β 4 2β/π. The latter can be seen by constructing the conformal mapping φ, cf..2, which can be expressed as the composition φ := ρ T 3 T 2 T, where iβ/2 z + e iβ T z := e z + e iβ, T 2z := z π 2π β, T3 z := γ i z + γ, γ := exp i z γ β 4 2β/π see, e.g., [47, 60] for discussions on the construction of conformal mappings. The symmetry of E with respect to the real axis is not essential for showing bounds of the form 4.3. A bound valid for nonsymmetric sets E can be obtained by replacing ψ by Rψ in 4.3. The essential ingredient in the proof is the property that fz expτ Rz. Similar properties hold for hyperbolic and trigonometric functions. Example 4.2. Let fz = sinhτz or fz = coshτz with τ > 0. Then fz expτ Rz and { } max 2 f m, ηm F f, D 7 e τrψ + e τrψ τc m, m 2cτ. m! In order to show this bound, it suffices to slightly modify the proof of 4.3: Let π θ θ 2 π be such that Rψre it 0 if and only if t [θ, θ 2 ]. In this interval, we obtain as above Rψre it Rψ + c r + crcost + c e it, r r as required for our conclusion. For t [θ 2, 2π +θ ] we have from 4.4, with r replaced by r, that Rψre it = Rψre it ψ r crcost + + c r Rψ + c r r e it + crcost π + c r 2 sin t π 2 and the second part of the integral can be bounded as before. In particular, replacing A by ia yields for fz = sinτz or fz = cosτz that { } τc max 2 f m, ηm F m f, D 4 exp τ max Iz, m 2τc. z E m!, ;

18 8 B. Beckermann and L. Reichel Example 4.3. Define, for integers l, the functions φ l z := l e z z j z l = j! j=0 e z u 0 u l l! du, which are of interest in connection with exponential integrators. Let fz = φ l τz for some τ > 0 and fixed integer l, and let E be a subset of the left half-plane. Then, for m 2τc, f m 0 2π 7 τc m 2 m! 7 τc m 2 m! w = m uτc 0 0 e τ uψw w m+ dw ul u l l! du e τ urψ u m l! du u m ul l! du = 7 τc m 2 m + l!. The same upper bound holds for ηm F f, D/2. Druskin et al. [30] provide a nice discussion on rational approximation of the matrix exponential for symmetric matrices. We consider rational approximation in the following sections. 5. Rational approximation and the rational Arnoldi process. We consider the approximation of f on E by a rational function r, determined by approximating f := F f on D by a rational function r = p/ q, where p, q P m, and the polynomial q is monic with zeros w j = ψz j D. Let r be such a rational function. Then 5. rz = F + p/ qz f0 is a rational function of the form r = p/q with p, q P m. The monic polynomial q has the zeros z j = φw j of the same multiplicity as the corresponding zeros w j of q, i.e., q and q are related as in 2.8. It follows from Corollary 2.3 that 5.2 fa ra 2 f p/ q L D. We therefore are interested in results on the approximation of f on D by rational functions with prescribed poles. The case when f is a Markov function is discussed in Section 6, where we also consider the choice of suitable poles w j. In this section, we are concerned with the evaluation of rab, either for a given rational function r, or by using the rational Arnoldi process. The latter approach determines the numerator p for a user-specified denominator q. Here and in the remainder of this paper, we assume the set E to be symmetric with respect to the real axis, and that f satisfies fz = fz. The Faber pre-image f of f also has the latter property. Therefore, it suffices to consider rational approximants r with real or complex conjugate poles and residues. In order to fix ideas, suppose that r has m simple finite poles, rw = r + m j= c j w j w.

19 Then by 5. and 2.7, we obtain rz = r + Error estimation and evaluation of matrix functions 9 m j= ψ w j c j z j z, r = r + r0 f0, z j = ψw j. The evaluation of rab requires the solution of m shifted linear systems of equations 5.3 z j I Ax j = b. The approximation of fab by rab is meaningful when A is a large sparse matrix, such that the shifted systems 5.3 can be solved efficiently by a sparse direct method, but solution by Krylov subspace methods or by Schur reduction to triangular form are impractical. For example, discretization of the two-dimensional Laplace operator on a square, using the standard 5-point finite difference stencil, gives rise to such a matrix. Remark 5.. The matrices in 5.3 generate the same Krylov subspaces K j A, b, j =, 2,..., m. This makes it possible to solve the m linear systems of equations simultaneously by an iterate method that uses the same Krylov subspace; see, e.g., [37, 82]. However, solving these shifted systems in this manner, e.g., by the GMRES iterative method, implies that we determine a polynomial approximant of f. It may be possible to compute more accurate polynomial approximants of f for the same computational effort by using the approach described in Section 3. There are situations when it suffices to solve fewer than m shifted systems of equations. For instance, when all poles are distinct and the poles and coefficients c j appear in complex conjugate pairs, say, z m+ j = z j, and c m+ j = c j, we obtain m/2 rab = r + 2R ψ w j c j z j I A b, j= where we have taken into account that A and b have real entries. Thus, only m/2 shifted systems of equations have to be solved. In case of multiple poles, one has to solve several linear systems of equations with the same matrix z j I A, but with different right-hand sides. The number of LU-factorizations required is the number of distinct poles with nonnegative imaginary part. For an efficient implementation of our approach, we need to compute the inverse Faber image of f and the Faber image of a rational function. This poses no difficulty if ψ is known in closed form or is a Schwarz-Christoffel mapping; see, e.g., Driscoll and Trefethen [25] or Henrici [47, Chapter 5] for discussions of the latter; software for computing Schwarz-Christoffel mappings is made available by Driscoll [24]. The above approach requires knowledge of a suitable rational approximant r, not only its poles. The rational Krylov method, introduced by Ruhe [68, 69], only requires the poles to be specified, and gives an error, which similarly to 3.7, is bounded by 4η q m f, D; see Theorem 5.2 below. Thus, in view of 5.2, the rational Krylov method is quasi-optimal up to a factor 2. For the sake of completeness, we shortly describe this method. The introduction of an artificial pole z m+ := leads to a slight simplifications compared to the presentation in [68, 69]. Given complex poles z, z 2,..., z m, including the case of a pole z j =, we let q be the product of the linear factors corresponding to finite poles. Let z 0 C be sufficiently far away from the poles z j, but otherwise arbitrary. We compute by an Arnoldi-type process an orthonormal

20 20 B. Beckermann and L. Reichel basis {v j } m+ j= of the rational Krylov subspace qa span{b, Ab,..., A m b} in the following manner. Let v = b and determine v j+ by orthogonalizing z j z 0 z j I A A z 0 Iv j against the available vectors v, v 2,..., v j, followed by normalization. If z j =, then we orthogonalize A z 0 Iv j. The vectors v j satisfy for suitable scalars h k,j the recursion formula h j+,j v j+ = z j z 0 z j I A A z 0 Iv j h,j v... h j,j v j, j =, 2,..., m, with v = b. Let V m+ = [v, v 2,..., v m+ ] and define the upper Hessenberg matrix H m+ = [h j,k ] j,k=,...,m+. The formula for the projection A m+ := Vm+AV m+ is more complicated than for the standard Arnoldi process. Introduce [ ] D m+ = diag,...,. z z 0 z m+ z 0 Then, for j m, 5.4 j+ A z 0 Iv j = A z 0 IV m+ e j = I A z 0 I h k,j v k z j z 0 = V m+ H m+ A z 0 IV m+ H m+ D m+ e j. When j = m +, we have to include the additional term h m+2,m+ I A z 0 I v m+2 = h m+2,m+ v m+2 z m+ z 0 in the right-hand side of 5.4, where the equality follows from the choice z m+ =. We obtain from 5.4 that A z 0 IV m+ I + H m+ D m+ = V m+ H m+ + h m+2,m+ v m+2 e T m+. In view of that V m+v m+2 = 0 and V m+v m+ = I, this leads to the formula k= 5.5 A m+ = V m+av m+ = z 0 I + H m+ I + H m+ D m+. Notice that the choices z 0 = 0 and z =... = z m = yield the standard Arnoldi process, with A m+ = H m+ determined by 3. with m replaced by m+. A bound analogous to 3.7 for the standard Arnoldi method also holds for the rational Arnoldi method. Theorem 5.2. Let E be a compact convex set, such that WA E. Assume that f AE, and let z, z 2,..., z m E, z m+ =. Then, for all m, 5.6 fab V m+ fa m+ e 4 η q mf f, D with q as in 2.8. More generally, for gz = q z + q 2 zfz with q P m+s and q 2 P s, there holds with z m+ = z m+2 =... = z m+s+ =, 5.7 gab V m+s+ ga m+s+ e 4 q 2 Ab η q mf f, D.

21 Error estimation and evaluation of matrix functions 2 Proof. Since v j qa K m+ A, b, there exists p j P m, such that v j = qa p j Ab, j =, 2,..., m +. With v, v 2,..., v m+ being a basis of the rational Krylov subspace, the polynomials p 0, p,..., p m form a basis of P m. Since v = b, we have p 0 = q. We now show that 5.8 e j = ṽ j := qa m+ p j A m+ e, j =, 2,..., m +. This is trivially true for j =, and the general case follows by induction. By definition of v j and p j, the vectors ṽ j satisfy h j+,j ṽ j+ = z j z 0 z j I A m+ A m+ z 0 Iṽ j h,j ṽ... h j,j ṽ j. It only remains to observe that, by 5.5, z j z 0 z j I A m+ A m+ z 0 Ie j = z j z 0 A m+ z 0 I z j z 0 I A m+ z 0 I e j = z j z 0 H m+ z j z 0 I + H m+ z j z 0 D m+ I e j = H m+ e j, since z j z 0 D m+ Ie j = 0. This shows 5.8. Any p P m may be written as p = c p c m+ p m. Therefore, the Arnoldi approximation is exact for g = p/q, qa pab = = m+ j= m+ j= c j qa p j Ab = m+ j= m+ c j v j = V m+ j= c j e j c j qa m+ p j A m+ b = V m+ qa m+ pa m+ e. As a consequence, we obtain similarly as in the proof of Theorem 3.2, that fab V m+ fa m+ e min p P m f p q Ab + f p q A m+e 4 min p P m f p q = 4η q m f, D, L D since WA m+ WA E. This yields 5.6. The bound 5.7 can be shown in a similar way as in Theorem 3.2. We therefore omit the details. A bound similar to 5.6 for the case when the matrix A is symmetric recently has been shown independently by Druskin et al. [30]. Concerning the implementation of the rational Arnoldi process, we have to solve shifted linear systems of equations z j I Ax j = v j. Complex conjugation of z j does not correspond to complex conjugation of v j. In situations when it is feasible to compute LU-factorizations of the matrices z j I A, only factorizations for distinct finite nonnegative z j have to be determined. In particular, we just need to compute one LU-factorization of A if z 2j = 0 and z 2j =, j =, 2,.... This kind of rational approximant is discussed in [29, 55]. The derivation of an analogue of Theorem 5.2 for the approximation of entire functions, such as the exponential function, and the application of 5.2 to such functions, is beyond the scope of this paper; see, e.g., Ganelius [39] for a discussion on the rate of convergence of rational approximants of such functions.

22 22 B. Beckermann and L. Reichel 6. Rational approximation of Markov functions. This section applies the error bounds of Theorems 3.2 and 5.2 for the standard and rational Arnoldi processes, respectively, to Markov functions f, given by.4, and to simple modifications of Markov functions, such as those discussed in Examples.3 and.4. As far as we know, only asymptotic results are known for rational interpolants with free poles, see, e.g., [77, Section 6] or [], and a posteriori error bounds are available for rational approximants obtained by balanced truncation and AAK theory; see, e.g., [0]. The present section derives explicit sharp upper and lower bounds for the error of best approximation η q mf f, D for rational approximants with prescribed denominator q of degree at most m. These bounds are believed to be new. We also construct nearly optimal approximants r, which can be used for explicit evaluation as explained in the previous section. Since, by 5. and 2.7, η q mf, E 2 η q mf f, D, we obtain explicit upper bounds for rational approximants of Markov functions on E. The following theorem establishes our main results for Markov functions. It discusses properties of the Blaschke product 6. Bw = wm q/w qw = m j= w j w w w j, whose poles w j are assumed to be real or occur in complex conjugate pairs and satisfy < w j. It follows that Bw has real coefficients when expressed in terms of positive and negative powers of w and, moreover, B/w = /Bw. Theorem 6.. Let the set E be compact, convex, and symmetric with respect to the real axis. Let f be a Markov function.4, and assume that α < β < γ := min{rz : z E}. a Then f = F f is a Markov function, 6.2 fw = β α φ x dµx w φx =: d µx w x. b Let R = P/ q with P P m be the rational interpolant of f with prescribed poles w j at the reflected points /w j for j =, 2,..., m counting multiplicities, and let Then r P m / q and f R rw = Rw + Bw 2B + f R. 2B η q mf f, D f r L D f L E φβ β α max y φ[α,β] φ x dµx Bφx φx 2 By. c If, in addition, the poles w j φα, φβ have even multiplicity, then 6.5 η q mf f, D β α φ x dµx Bφx φx 2 / Bφx,

23 Error estimation and evaluation of matrix functions 23 and for the approximant r of part b, we also have the a posteriori bound f r L D = f r = β We comment on the bounds before showing the theorem. Remark 6.2. For x [α, β], we have Bφx > and α φ x dµx Bφx φx 2. φx 2 φβ 2 φx 2 / Bφx, independently of the choice of poles w j and of their number m. Therefore, for all poles w j and m, the lower bound 6.5 is bounded below by the factor φβ 2 times the upper bound 6.3. The upper and lower bounds give a quite precise idea of the accuracy of the best approximation of f in P m / q on the unit circle. Concerning part b, we also should mention that our quasi-optimal approximant r is obtained by a simple modification of the interpolant R, where R is known to be the best approximant of f in P m / q with respect to the 2 norm on the unit circle. Remark 6.3. Consider the polynomial case when w =... = w m = and, hence, Bw = w m and qw =. From Theorem 6.a and its proof, we see that the Faber coefficients of f satisfy f j = β α φ x dµx φx j+. It is not difficult to verify that for this special case, the two bounds of Theorem 6.b and c take the form 6.6 f m+jm+ ηm f, D f m+2j f L E φβ m. j=0 j=0 These bounds improve on the inequalities 3.9. Moreover, the quantity ρ in Remark 3.3 is at most φβ ; ρ has to be chosen smaller if f is not continuous at β. Hence, 6.6 also is an improvement of the bound furnished by Remark 3.3. The proof of Theorem 6. is divided into three parts. Following the proof, we discuss some configurations of poles obtained by minimizing the bound 6.4. This enables us to compare our approach with the shifted Arnoldi process, see, e.g., [8], and the use of Talbot quadratures rules discussed in [45]. Proof. Theorem 6.a: The Faber coefficients of the Markov function f of.4 satisfy, by the Fubini theorem and the Cauchy formula, f j = 2πi = β α E φ zdz φz j+ dµx 2πi E β α dµx z x φ zdz φz j+ z x = β α φ x dµx φx j+. The last identity is obtained by deforming the path of integration E in C\E in order to obtain a circle around x with mathematically negative orientation. Recalling that F fw = f 0 + f w +..., we conclude that f = F f is of the form 6.2. By the symmetry of E, it follows that the function φx = φx is decreasing for

24 24 B. Beckermann and L. Reichel x, γ and that φ x is positive for x, γ. Thus, µ is a positive measure and f is a Markov function. Theorem 6.b: We first establish a well-known integral formula for rational interpolants with prescribed poles of Markov functions; see, e.g., [83, Theorem VIII.2]. The numerator P is the interpolation polynomial of q f at the points /w, /w 2,..., /w m, and, therefore, fw Rw = w /w... w /w m [/w, /w 2,..., /w m, w] q qw f = w /w... w /w m [/w, /w 2,..., /w m, w] t qx d µx qw t x = w /w... w /w m qx d µx = Bw qw d µx Bx w x. In particular, we find for the modified approximant that fw rw = Bw Bx w x /2 x /2 x = Bx w x + x x 2 d µx = x /w... x /w m w x d µx Bx wx w x d µx x 2. Taking into account that Blaschke factors are of unit modulus on the unit circle, and proceeding similarly as in the proof of 6.2, gives the upper bound 6.3. It follows from 3.3 that β α φ x dµx Bφx φx 2 β α dµx φx Bφx distx, E. The distance is achieved for γ E for all x [α, β], and x / φx is increasing in [α, β]. This shows 6.4. Theorem 6.c: We first recall that φ x > 0 for x [α, β]. Moreover, by assumption, Bw := w 2 Bw is a rational function with real coefficients, having all its m + 2 roots in the open unit disk, and its poles in the interval φ[α, β] have even multiplicity. Hence, B is of constant sign on φ[α, β]. Thus, ɛ Bφx > 0 for x [α, β] with ɛ 2 =. Let D m+2 = {x 0, x,..., x m+ } := {w C : ɛ Bw = }, with the x j being distinct points on the unit circle, ordered according to increasing argument. Theorem 6.c will follow by showing that 6.7 η q m f, D m+2 = min p P m f p/ q L D m+2 = β α φ x dµx ɛ Bφx =: δ. Let R P m+ / q be the rational interpolant of f at the points in D m+2, and with prescribed denominator q. Denote the coefficient for w m+ of Rw qw by a. Since wbw qw q0w m+ P m, we obtain that p w qw := Rw wbw q0 a P m/ q.