Iyad T. Abu-Jeib and Thomas S. Shores

Transcription

1 NEW ZEALAND JOURNAL OF MATHEMATICS Volume 3 (003, 9 ON PROPERTIES OF MATRIX I ( OF SINC METHODS Iyad T. Abu-Jeib and Thomas S. Shores (Received February 00 Abstract. In this paper, we study the determinant and eigen properties of I (, an important Toeplitz matri used in Sinc methods. Some Sinc method applications depend on the non-singularity of this matri and on the location of its eigenvalues. Among the theorems we prove is that I ( is nonsingular. We also show that if λ is a pure imaginary eigenvalue of I (, then λ >.. Introduction Our goal in this paper is to study properties of an important Toeplitz matri in the theory of Sinc indefinite integration and Sinc convolution. This matri is denoted as I (. Actually, this is an infinite family of Toeplitz matrices, generated by an infinite sequence of diagonal coefficients; definitions are given below. Although much is known about the global and asymptotic behavior of the eigenvalues of such families in general, as noted in [3, p. 87], much less is known about the behavior of individual eigenvalues. We are particularly interested in a conjecture of Frank Stenger: I ( is diagonalizable and its eigenvalues are located in the open right half plane (see [6]. Both properties are needed for the convergence of certain convolution algorithms and differential equation solvers based on Sinc theory developed in [7]. As noted in [7], it has been determined numerically that these properties hold for all such matrices of order at most 5. Although this is sufficient for most practical purposes, the general problem remains open. It follows rather easily from the definition of I ( that its eigenvalues lie in the closed right half-plane. One of the results we will prove in this paper is that if λ is a pure imaginary eigenvalue of I (, then λ >. It follows that I ( is always an invertible matri. Moreover, we will show that I ( is at least close to diagonalizable in the sense that it is a rank one update of a highly structured skew-symmetric matri with simple eigenvalues. We use the following notation: ( The square constant matri whose elements are all equal to ω is denoted by by [ω]. ( If (λ, is an eigenpair of the matri A, then we write λ evals(a and evecs(a. Also, we use the notation λ(a to mean that λ is an eigenvalue of A. 99 Mathematics Subject Classification 65F40, 65F5, 5A8, 5A4, 5A57. Key words and phrases: Sinc, eigenvalue, eigenvector, singular value, determinant, skew symmetric, Toeplitz.

2 IYAD T. ABU-JEIB AND THOMAS S. SHORES (3 The transpose (Hermitian transpose of the matri A is denoted by A t (A and the tuple notation = (,,... n represents the column vector [,,... n ] t in C n. Net, we introduce the sinc function, Sinc methods and I (. Definition.. The sinc function is defined as follows: { sin(π sinc( =, π for 0, for = 0. Sinc methods are a family of formulas based on the sinc function which provide accurate approimation of derivatives, definite and indefinite integrals and convolutions. These methods were developed etensively by Frank Stenger and his students (see [8]. Sinc methods can be used to numerically solve differential equations with boundary layers, integrals with infinite intervals or with singular integrands, and ordinary differential equations or partial differential equations that have coefficients with singularities. These are two numerical methods which depend on the sinc function defined above. For more information, see [4], [7], and [6]. For convolutions and integration, the following family of matrices is especially important. Definition.. I ( is the n n matri defined as follows: where η ij = e i j, e k = + s k, and s k = I ( = [η ij ] n i,j= k 0 sinc(d Thus, I ( can be epressed in the form [ ] I ( = + S, where S is the following skew-symmetric Toeplitz matri of dimension n: 0 s s... s n s 0 s... s n S = s s 0... s n s n s n s n Throughout this paper, S refers to the matri defined above. Observe that S is a real skew-symmetric Toeplitz matri. If we want to emphasize the dimension of S, we write S n in place of S. Also note that the matri [ ] = uu, with u = (,,...,, is a rank one matri. This decomposition of I ( is key to our analysis of its behavior.. Localization Results for I ( We have verified computationally (for a dimension up to 000 that the eigenvalues of I ( lie in the open right half plane. I (. The fact that in all cases they lie in the closed right half plane follows easily from the following elementary, but key result.

3 ON PROPERTIES OF MATRIX I ( OF SINC METHODS 3 Lemma.. Let (λ, be an eigenpair of A = [ω] + M, ω real and M skew- Hermitian, = (,,... n and =. Then Re(λ = n j and Im(λ = M j= Proof. With the given notation we have from A = λ that λ = [ω] + M. However, [ω] = ω n j= j is real and M is pure imaginary since M is skew-hermitian. The result follows. In particular, it follows from this lemma that the real part of an eigenvalue λ of I ( is nonnegative. The lemma can also be used to further localize the eigenvalues of I ( as follows: the real part of each eigenvalue of I ( is between 0 and n/. To see this, note that the eigenvalues of the rank matri [ ] are n/ and 0 (with multiplicity n. Since =, it follows that [ ] n/. Thus the issue of eigenvalues of I ( lying in the open half right plane is reduced to showing that no eigenvalues of I ( lie on the imaginary ais. The main result of this section shows that if there is a purely imaginary eigenvalue λ = bi of I (, then b > /π. Definition.. If v = [v 0, v,..., v n ] t is an eigenvector of A, then the corresponding eigenpolynomial is V (z = v 0 + v z v n z n. A function f( is said to be a generator for a matri M = [m ij ] of dimension n, if m ij = c i j, for all i, j n, where c k = π f(e ik d, k = 0,,..., n. In the following lemma, we prove that S is the Toeplitz matri generated by the comple function i/. In what follows, the integral sign denotes the Cauchy Principal Value and not the Lebesgue integral which has been used by other authors in dealing with generators. Lemma.3. i is a generator of S. Proof. We want f( = i to satisfy: Now we claim that r f(e ir sin(π d = d, r = 0,,..., n. π 0 π i π cos(r i sin(r r d = 0 sin(π d. ( π

4 4 IYAD T. ABU-JEIB AND THOMAS S. SHORES First notice that if r = 0, then the left hand side of ( is i π π to zero. And if r 0, then cos(r i π d, which is equal is odd. So the left hand side of ( is now i sin(r d, ( which is an ordinary integral since the integrand is continuous. In fact, the integrand is even. Thus, ( becomes sin(r d (3 π 0 Now substitute = π r y in (3, to get r sin(πy dy, πy which proves the lemma. 0 To study the determinant and eigen properties of S, it suffices to study the matri T is instead. T is generated by /. It is Hermitian and has the advantage of being generated by a real-valued function. These facts are helpful in the following theorem, the proof of which is modeled along the lines of Theorem. of [9]. Theorem.4. If µ is a pure imaginary eigenvalue of I (, then µ >. Proof. Let (µ, u be an eigenpair of I (, where µ = ib, b is real, and without loss of generality, assume that b is nonnegative. It follows from the identity I ( u = ibu that [ ] u u + u Su = ib u. Since the right hand side and u Su are purely imaginary, it follows that 0 = u [ ] u = n j= u j. Therefore [ ] u = 0, from which it follows that (µ, u is also an eigenpair of S and (0, u is an eigenpair of [ ]. This means that (b, u is an eigenpair of T. Now if U(z is the eigenpolynomial associated with the eigenvector u, then it follows that is a zero of U(z. For the fact that (0, u is an eigenpair of [ ] implies that the sum of the coordinates of u is zero, which means that the sum of the coefficients of U(z is zero. Thus, we can write U(z = (z Û(z, for some polynomial Û(z. Now let V (z be a polynomial corresponding to a vector v and which has - as a zero. Then V (z can be factored as V (z = (z + V (z. Thus, U(zV (z = (z (z + Û(z V (z. Now notice that if z = e i, then (z (z + = i sin(. Since f( = / is a generator of T, it follows that Thus, i π sin( Û(e i V (e i d = T u, v = λu, v = iλ π sin(û(ei V (e i d. ( sin( λ sin( Û(e i V (e i d = 0.

5 ON PROPERTIES OF MATRIX I ( OF SINC METHODS 5 Observe that V (z can be chosen to be any polynomial of degree at most n. Make the choice V (z = Û(z to obtain ( sin( b sin( Û(ei d = 0. Since u is an eigenvector, the polynomial Û(z is nonzero. It follows that the integrand above is positive on the open interval (, 0. Moreover, the factor sin( Û(ei is also positive on the interval (0, π. It follows that b < 0 for some in the interval (0, π. This implies that b > /π and completes the proof. It follows from Theorem.4 that zero is ecluded as a possible eigenvalue of I (. Corollary.5. For all dimensions n the matri I ( is nonsingular. 3. Diagonalizability of S n The principal theorem of this section is an analogue to Theorem. of [9]. Trench s theorem is valid only for real-valued Lebesgue-integrable functions on (,π, which are not constant on a set of measure π, so it cannot be applied to f( = / and hence, to the matri generated by if(. In the following theorem, we shall use the notation S n instead of S to emphasize the dimension (which is n, and use T n to represent is n. Theorem 3.. The eigenvalues of is n = T n (t r s n r,s= are simple, where t k = π e ik d, k = 0,,..., n. (4 Proof. Let f( = /. First, note that f( is a generator of T n. We will show that if λ r is an eigenvalue of T n of multiplicity greater than or equal to, then f( λ r must change sign at least 3 times in (, π. Clearly λ r does not change sign more than once in (, π for any choice of λ r. This will imply that all eigenvalues of T n are simple. Note that all of the following integrals are finite, because the Cauchy principal part of any integral whose integrand is a product of a polynomial and / is finite. Now for each vector v = [v, v,..., v n ] t in C n, associate the polynomial n V (z = [, z, z,..., z n ]v = v j z j. If u and v are in C n, then the standard inner product on C n satisfies u, v = π where z = e i. Also from (4, we obtain that T n u, v = π j= U(zV (zd, (5 U(zV (zd. (6

6 6 IYAD T. ABU-JEIB AND THOMAS S. SHORES Now let λ λ... λ n be the eigenvalues of the Hermitian matri T n, with corresponding eigenvectors,,..., n, and corresponding eigenpolynomials X i (z = [, z,..., z n ] i, i n. From (5 we have X i (zx j (zd = δ ij, i, j n. (7 π From (6 we also have π X i(zx j (zd = δ ij λ i, i, j n, (8 where δ ij is the Kronecker delta. Thus, if λ r is an eigenvalue of multiplicity one, then we claim that f( λ r must change sign at some point. This point must be /λ r (if λ r 0 or 0 (if λ r = 0. Notice that multiplying (7 by λ r, and taking i = j = r, yields λ r X r (z d = λ r. (9 π And by taking i = j = r in (8, we get π Now subtract (9 from (0 to obtain that ( π λ r X r (z d = 0. X r(z d = λ r. (0 Now if f( λ r is of constant sign in (, π, say it is positive, then the integral ( λ r X r (z d should be positive. This is impossible, which proves our assertion about the sign change of f( λ r. Net suppose that m. The epression ( λ r / changes sign only at and, where = 0 and = λ r. We will show that this assumption (that m leads to a contradiction. Notice that it suffices to handle the case when λ r 0, since nonzero eigenvalues of S n are of the form ia, a 0, a R, and they occur in pairs. Therefore, if a (assume a > 0 is an eigenvalue of T n of multiplicity greater than or equal to, then a is an eigenvalue of multiplicity greater than or equal to. So assume that λ r 0, and define g( = π (f( λ r. ( We assert that the function h( defined by ( ( h( = g( sin sin ( does not change sign in (, π. To see why this is true, notice first that since and i, i =,, are all in (, π, then i is also in (, π. Thus, sin( i defined on (, π can change sign

7 ON PROPERTIES OF MATRIX I ( OF SINC METHODS 7 only when i = 0; i.e. when = i, for i =,. But, g( changes sign only at these points. Consequently, h( can change sign only at these points also. To show that h does not change sign at these two points, it suffices to prove that the sign of h( does not change in a small neighborhood of i, for i =,. Notice that for each i, we can choose this neighborhood not to include j, for j i. Now write h as h( = π h (h (. where h ( = sin(/ ( and h ( = ( λ r sin (. Recall that = 0 and = λ r. First, let check the sign of h about. But, h does not change sign at, nor does h. Thus, h does not change sign at. It remains to check the sign of h about. Since h does not change sign at, we need only to check the sign of h about. But, when <, we have λ r > 0 and sin (( / < 0. Thus when <, h( is negative. On the other hand, when >, λ r < 0, and sin(( / > 0. Thus, when >, h( is negative. Therefore, h does not change sign at, and hence h does not change sign in (, π. Now suppose that λ r = λ r+ is of multiplicity greater than or equal to. For i = r, r + and j n, we have that g(x i(zx j (zd = 0, because g(x i (zx j (zd = π = λ i δ ij λ r δ ij = (λ i λ r δ ij. f(x i (zx j (zd λ r π X i (zx j (zd Now if i = r, then the above integral is zero. It is also zero if i = r +, because λ r+ = λ r. Therefore, where g(p(zx j (zd = 0, j n, p(z = c 0 X r (z + c X r+ (z and c 0 and c are constants, which are not both zero, and which satisfy c 0 X r ( + c X r+ ( = 0. This means that p(z is a nonzero polynomial. Now set Q(z = p(z(z ei, which is a polynomial of degree less than or equal z to n, and obtain that Thus, g(p(zq(zd = 0. (3 ( z e g( p(z i d = 0. z

8 8 IYAD T. ABU-JEIB AND THOMAS S. SHORES If we let then the above equation becomes Now if z = e i, then Notice that l=0 g ( = g( c lx r+l (z, z g ((z (z e i d = 0. (4 (z (z e i = 4e i sin ( sin (. ( e i e iφ +φ i( = e φ i( e φ i( e +φ i( = e sin( φ. Therefore, (4 becomes Thus, 4 This implies that where Therefore, g ( ( ( e i sin sin g ( sin ( sin ( ( α(zg( sin sin ( ( ( α(z = p(z z. α(zh(d = 0. d = 0. (5 d = 0. (6 d = 0. Since h does not change sign in (, π and α(z 0, then it must be that p(z/(z 0. Therefore, p 0, a contradiction proving the multiplicity of any eigenvalue is. The equalities S = T and I ( = [ ] + S show that I ( is at most a rank one update away from a matri with simple eigenvalues, as stated in the introduction. Corollary 3.. The matri I ( can be epressed as the sum of a rank one matri and a skew-symmetric matri with simple eigenvalues. We conclude by relating the determinants of S and I (. The result that we need can be phrased in a somewhat more general contet. Theorem 3.3. Let M be a non-singular real skew-hermitian matri and A = [ω] + M, where ω is a nonzero real number. Then det(a = det(m.

9 ON PROPERTIES OF MATRIX I ( OF SINC METHODS 9 Proof. The rank one matri M [ω] has at most one nonzero eigenvalue. which is real. Hence all eigenvalues of M [ω] are real. Let (λ, be an eigenpair for this matri. From M [ω] = λ we see that [ω] = λ M. The right hand side of this identity is pure imaginary and the left hand side real, so each is zero. Thus [ω] = 0, so that 0 = [ω] = λm. Since M is non-singular, it follows that λ = 0. Therefore all eigenvalues of M [ω] are zero. Thus the only eigenvalues of I + M [ω] are and det(a = det(m(i + M [ω] = det(m. Corollary 3.4. Let n be even. Then det(i ( n = det(s n. Proof. All eigenvalues of S n occur in signed pairs ±ib, where b is real. If zero were an eigenvalue of S n, it would be repeated, which does not occur by Theorem 3.. Therefore, S n is nonsingular and Theorem 3.3 applied to the equality I ( = [ ]+S gives the desired result. References. Fabio Di Benedetto, Generalized updating problems and computation of the eigenvalues of rational toeplitz matrices, Linear Algebra and Its Applications, 67 (997, Douglas N. Clark, Perturbation and similarity of toeplitz operators, Operator Theory: Advances and Applications, 48 (990, T. Kailath and A. Sayed, Fast Reliable Algorithms for Matrices with Structure Equations, SIAM, Philadelphia ( J. Lund and K. Bowers, Sinc Methods for Quadrature and Differential Equations, SIAM, Philadelphia, John Makhoul, On the eigenvectors of symmetric toeplitz matrices, IEEE Transactions on Acoustics, Speach and Signal Processing, 9 (98, Frank Stenger, Matrices of sinc methods, Journal of Computational and Applied Mathematics, 86 (997, Frank Stenger, Collocating convolutions, Mathematics of Computation, 64 (995, Frank Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer, New York ( William F. Trench, Some Special Properties of Hermitian Toeplitz Matrices, Siam J. Matri Anal. Appl. 5 (994, Iyad T. Abu-Jeib Department of Mathematics and Computer Science SUNY College at Fredonia Fredonia, NY 4063 U.S.A abu-jeib@cs.fredonia.edu Thomas S. Shores Department of Mathematics and Statistics University of Nebraska Lincoln Lincoln, NE U.S.A tshores@math.unl.edu