Linear Algebra and its Applications

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1 Linear Algebra and its Applications 441 ( Contents lists available at SciVerse ScienceDirect Linear Algebra and its Applications journal homepage: wwwelseviercom/locate/laa Sparse polynomial interpolation in Chebyshev bases Daniel Potts a,, Manfred Tasche b a Chemnitz University of Technology, Department of Mathematics, D Chemnitz, Germany b University of Rostock, Institute of Mathematics, D Rostock, Germany ARTICLE INFO Article history: Received 19 July 2012 Accepted 9 February 2013 Available online 11 March 2013 Submitted by V Mehrmann AMS classification: 65D05 41A45 65F15 65F20 Keywords: Sparse interpolation Chebyshev basis Chebyshev polynomial Sparse polynomial Prony-like method ESPRIT Matrix pencil factorization Companion matrix Prony polynomial Eigenvalue problem Rectangular Toeplitz-plus-Hankel matrix ABSTRACT We study the problem of reconstructing a sparse polynomial in a basis of Chebyshev polynomials (Chebyshev basis in short from given samples on a Chebyshev grid of [ 1, 1] A polynomial is called M-sparse in a Chebyshev basis, if it can be represented by a linear combination of M Chebyshev polynomials For a polynomial with known and unknown Chebyshev sparsity, respectively, we present efficient reconstruction methods, where Prony-like methods are used The reconstruction results are mainly presented for bases of Chebyshev polynomials of first and second kind, respectively But similar issues can be obtained for bases of Chebyshev polynomials of third and fourth kind, respectively 2013 Elsevier Inc All rights reserved 1 Introduction The central issue of compressive sensing is the recovery of sparse signals from a rather small set of measurements, where a sparse signal can be represented in some basis by a linear combination with few nonzero coefficients For example, a 1-periodic trigonometric polynomial of degree at most N 1 with only M nonzero exponential terms can be recovered by O(M log 4 (N sampling points that are Corresponding author addresses: potts@mathematiktu-chemnitzde (D Potts, manfredtasche@uni-rostockde (M Tasche /$ - see front matter 2013 Elsevier Inc All rights reserved

2 62 D Potts, M Tasche / Linear Algebra and its Applications 441 ( { j randomly chosen from the equidistant grid ; N j = 0,,N } 1, where M N (see [23] Recently, Rauhut and Ward [21] have presented a recovery method of a polynomial of degree at most N 1 given in Legendre expansion with M nonzero terms, where O(M log 4 (N random samples are taken independently according to the Chebyshev probability measure of [ 1, 1] The recovery algorithms in compressive sensing are often based on l 1 -minimization Exact recovery of sparse signals or functions can be ensured only with a certain probability The method of [21] can extended to sparse polynomial interpolation in a basis of Chebyshev polynomials too In contrast to these random recovery methods, there exist also deterministic methods for the reconstruction of an exponential sum H(t := c j e if jt j=1 (t R with distinct frequencies f j [ π, π and complex coefficients Such methods are the Prony-like methods [19], such as the classical Prony method, annihilating filter method [5], ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques [22], matrix pencil method [10,9], and approximate Prony method [3,18] This approach allows the recovery of all parameters of H, ie M, f j and c j for j = 1,,M, from equidistant samples H(k(k = 0,,, where N M Prony-like methods can be applied also for the reconstruction of sparse trigonometric polynomials [19,Example 42] Note that the classical Prony method is equivalent to the annihilating filter method Unfortunately, the classical Prony method is very sensitive to noise in the sampled data Hence numerous modifications have been proposed in order to improve the numerical behavior of the Prony method Efficient Prony-like methods are important within many disciplines in sciences and engineering (see [15] For a survey of the most successful methods for the data fitting problem with linear combinations of complex exponentials, we refer to [14] Note that a variety of papers compare the statistical properties of the different algorithms, see eg [10,1,2,6] Similar results for our new suggested algorithms are of great interest, but are behind the scope of this paper In this paper, we present a new deterministic approach to sparse polynomial interpolation in a basis of Chebyshev polynomials, if relatively few samples of a Chebyshev grid of [ 1, 1] are given Note that Chebyshev grids are much better suited for the recovery of polynomials than uniform grids (see [4] For n N 0,thenth Chebyshev polynomial of first kind can be defined by T n (x := cos(n arccos x (x [ 1, 1] (see for example [13, p 2] These polynomials are orthogonal with respect to the weight (1 x 2 1/2 on ( 1, 1 (see [13, p 73] and form the Chebyshev-1 basis Let M be a positive integer A polynomial h(x = d b k T k (x of degree d M is called to be M-sparse in the Chebyshev-1 basis,ifm coefficients b k are nonzero and if the other d M + 1 coefficients vanish Then such a M-sparse polynomial h can be represented in the form h(x = c j T nj (x j=1 (11 with c j := b nj = 0 and 0 n 1 < n 2 < < n M = d TheintegerM is called the Chebyshev-1 sparsity of the polynomial (11 Recently the authors have presented a unified approach to Prony-like methods for the parameter estimation of an exponential sum [19], namely the classical Prony method, the matrix pencil method [9], and the ESPRIT method [22] The main idea is based on the evaluation of the eigenvalues of a matrix

3 D Potts, M Tasche / Linear Algebra and its Applications 441 ( which is similar to the companion matrix of the Prony polynomial To this end we have computed the singular value decomposition (SVD or the QR decomposition of a special Toeplitz-plus-Hankel matrix (T+H matrix The aim of this paper is to generalize this unified approach in order to obtain stable algorithms for an interpolation problem of a sparse polynomial (11 in the Chebyshev-1 basis Similar sparse interpolation problems are formerly explored in [12,11,7] and solved by Prony methods For known Chebyshev-1 sparsity, Theorem 26 shows that an M-sparse polynomial (11 inachebyshev basis can be reconstructed from only 2M samples on a special Chebyshev grid Our method can be considered as special case of a reconstruction of sparse sums of eigenfunctions of a Chebyshev-shift operator, for details see [17, Remark 46] A Prony-like method for sparse Legendre reconstruction was suggested in [16] This method can be also generalized to other polynomial systems, but one needs there high order derivatives of the sparse polynomial For the sparse interpolation of a multivariate polynomial, we refer to [8] The outline of this paper is as follows In Section 2, we collect some useful properties of T+H matrices and Vandermonde-like matrices Further we formulate the algorithms, if the Chebyshev-1 sparsity M of (11 is known and if only 2M sampleddataof(11 on a special Chebyshev grid are given In Section 3, we obtain corresponding results on sparse polynomial interpolation for unknown Chebyshev-1 sparsity M of (11 Furthermore one can improve the numerical stability of the algorithms by using more sampling values (see Section 5 In Section 4, we discuss the sparse interpolation in the basis of Chebyshev polynomials of second kind Finally we present some numerical experiments in Section 5, where we apply our methods to sparse polynomial interpolation In the following we use standard notations By N and N 0, respectively, we denote the set of all positive and nonnegative integers, respectively The Kronecker symbol is δ k The linear space of all column vectors with N real components is denoted by R N, where o is the corresponding zero vector The linear space of all real M-by-N matrices is denoted by R M N, where O M,N is the corresponding zero matrix For a matrix A M,N R M N, its transpose is denoted by A T M,N, and its Moore Penrose pseudoinverse by A M,N A square matrix A M,M is abbreviated to A M ByI M we denote the M-by-M identity matrix By null A M,N we denote the null space of a matrix A M,N Further we use the known submatrix notation Thus A M,M+1 (1 : M, 2 : M + 1 is the submatrix of A M,M+1 obtained by extracting rows 1 through M and columns 2 through M + 1, and A M,M+1 (1 : M, M + 1 means the last column vector of A M,M+1 Definitions are indicated by the symbol := Other notations are introduced when needed 2 Interpolation for known Chebyshev-1 sparsity This section has an introductory character Under the restricted assumption that the Chebyshev-1 sparsity M of the polynomial (11isaprioriknown, we introduce the problem (21 of sparse polynomial interpolation in the Chebyshev-1 basis and the related Prony polynomial (23 Then we collect some useful properties of square T+H matrices and square Vandermonde-like matrices We find a factorization (28 of the T+H matrix and prove an interesting relation between the Prony polynomial (23 and its companion matrix (see Lemma 25 Similar sparse interpolation problems in the Chebyshev-1 basis are formerly explored in [12,11,7] and solved by a Prony method (such as Algorithm 27 In [12,11], the grid {T k (a = cosh (k ( arcosh a; k = 0,,2M 1} with fixed a > 1isusedforthe interpolation In [7], the grid {T k cos 2π N = cos 2kπ ; N k = 0,,2M } 1 with N 2 n M is applied for interpolation The main results of Section 2 are the Algorithms 29 and 210 Let N N be sufficiently large such that N > M and is an upper bound of the degree of the π polynomial (11 For u N := cos 2N 1 we form the nonequidistant Chebyshev grid { u N,k := T k (u N = kπ cos ; 2N 1 k = 0,,2M } 1 of the interval [ 1, 1] Note that T 2N 1 (u N,k = ( 1 k (k = 0,,2M 1 We consider the following problem of sparse polynomial interpolation in the Chebyshev-1 basis: For given sampled data ( h k := h(u N,k = h cos kπ (k = 0,,2M 1 (21

4 64 D Potts, M Tasche / Linear Algebra and its Applications 441 ( determine all parameters n j and c j (j = 1,,M of the sparse polynomial (11 If we substitute x = cos t (t [0, π], then we see that the above interpolation problem is closely related to the interpolation problem of the sparse, even trigonometric polynomial g(t := h(cos t = c j cos(n j t (t [0, π], (22 j=1 ( where the sampled values g kπ 2N 1 = hk (k = 0,,2M 1 are given (see [7,20] We introduce the Prony polynomial P of degree M with the leading coefficient 2 M 1, whose roots are x j := T nj (u N = cos P(x = 2 M 1 M j=1 n jπ 2N 1 ( x cos (j = 1,,M, ie n j π (23 Then the Prony polynomial P can be represented in the Chebyshev-1 basis by P(x = p l T l (x (p M := 1 (24 l=0 The coefficients p j of the Prony polynomial (24 can be characterized as follows: Lemma 21 For all k = 0,, M 1,thesampleddatah k and the coefficients p l of the Prony polynomial (24 satisfy the equations M 1 j=0 (h j+k + h j k p j = (h k+m + h M k (25 Proof Using cos(α + β + cos(α β = 2 cos α cos β, we obtain by(22 that ( h j+k + h j k = 2 c l cos n l(j + kπ + cos n l(j kπ l=1 n l jπ = 2 c l cos cos n l kπ 2N 1 (26 l=1 Thus we conclude that ( n l kπ n l jπ hj+k + h j k pj = 2 c l cos p j cos j=0 l=1 j=0 ( n l kπ = 2 c l cos P n l π cos = 0 l=1 By p M = 1, this implies the assertion (25 Introducing the vectors h(k := (h j+k + h j k M 1 j=0 (k = 0,,M and the square T+H matrix

5 D Potts, M Tasche / Linear Algebra and its Applications 441 ( H M (0 := (h j+k + h j k M 1 = ( j, h(0 h(1 h(m 1 2 h 0 2 h 1 2 h M 1 2 h 1 h 2 + h 0 h M + h M 2 =, 2 h M 1 h M + h M 2 h 2M 2 + h 0 then by (25 the vector p := (p k M 1 H M (0 p = h(m is a solution of the linear system (27 Lemma 22 Let M and N be integers with 1 M N Further let h be an M-sparse polynomial of degree at most in the Chebyshev-1 basis If h(u N,j = 0 for j = 0,,M 1, then h is identically zero Further the Vandermonde-like matrix V M (x := ( T nj (u N,k M 1,M = (,j=1 T k (x j M 1,M = n j kπ M 1,M,j=1 (cos,j=1 with x := (x j M j=1 is nonsingular and the T+H matrix H M(0 can be factorized in the following form H M (0 = 2 V M (x(diag c V M (x T (28 and is nonsingular Proof 1 Assume that the Vandermonde-like matrix V M (x is singular Then there exists a vector d = (d l M 1 l=0 = o such that d T V M (x = o T We consider the even trigonometric polynomial D of order at most M 1givenby D(t = M 1 l=0 d l cos(lt (t R n jπ Hence d T V M (x = o T implies that t j = [0, π] (j = 1,,M are roots of D These 2N 1 M roots are distinct, because 0 n 1 < < n M < 2N But this is impossible, since the even trigonometric polynomial D = 0ofdegreeatmostM 1cannot have M distinct roots in [0,π] Therefore, V M (x is nonsingular If h(u N,j = 0forj = 0,,M 1, then V M (x c = osincev M (x is nonsingular, c is equal to o, such that h is identically zero 2 The factorization (28 of the T+H matrix H M (0 followsimmediatelyfrom(26 Since c j = 0(j = 1,,M, diagc is nonsingular Further the Vandermonde-like matrix V M (x is nonsingular, such that H M (0 is nonsingular too Introducing the matrix p p p 2 P M := R M M p M p M p M 1

6 66 D Potts, M Tasche / Linear Algebra and its Applications 441 ( and using the linear system (27, we see that H M (0 P M = H M (1 + ( oh(0 h(m 2 with the T+H matrix H M (1 := ( ( h(1 h(2 h(m = hj+k+1 + M 1 h j k 1 j, RM M This T+H matrix has the following properties: Lemma 23 The T+H matrix H M (1 can be factorized in the following form H M (1 = 2 V M (x(diag c V M (xt (29 with the Vandermonde-like matrix V (x := ( M T k (x j M k,j=1 Further the matrices H M(1 and V M (x are nonsingular Proof 1 By Lemma 21 we know that (h j+k + h j k p k = 0 (j = 0,,2N M 1 Consequently we obtain where H M (0(p k M 1 = h(m, H M(1(p k+1 M 1 = p 0 h(0, p 0 = 2 M 1 ( 1 M M n j π cos j=1 does not vanish This implies that h(m span {h(0,,h(m 1}, h(0 span {h(1,,h(m} Thus we obtain that rank H M (0 = rank H M (1 = M 2 The (j, kth element of the matrix product 2 V M (x(diag c V M (xt can be analogously computed as (26 such that 2 c l T nl (u N,j T nl (u N,k = h j+k+1 + h j k 1 l=1 Since H M (1, V M (x, and diag c are nonsingular, it follows from (29 that the Vandermonde-like matrix V M (x is nonsingular too In the following Lemmas 24 and 25 we show that the zeros of the Prony polynomial (24 canbe computed via solving an eigenvalue problem To this end, we represent the Chebyshev polynomial T M in the form of a determinant

7 D Potts, M Tasche / Linear Algebra and its Applications 441 ( Lemma 24 Let M be a positive integer Further let E M := diag shift matrix S M := ( δ j k 1 + δ j k+1 M 1 j, = R M M ( 1 2, 1,, 1 T R M and the modified Then det (2x E M S M = T M (x (x R Proof We show this by induction For M = 1 and M = 2 it follows immediately the assertion For M 3 we compute the determinant x x x x x x using cofactors of the last row (cf [13, p 18] Then we obtain the known recursion of the Chebyshev polynomials T M (x = 2xT M 1 (x T M 2 (x (see [13, p 2] This completes the proof Now we show that 1 2 E 1 M P M is the companion matrix of the Prony polynomial (24 in the Chebyshev- 1basis Lemma 25 Let M be a positive integer Then 1 2 E 1 M P M is the companion matrix of the Prony polynomial (24 in the Chebyshev-1 basis, ie det ( ( 2 x E M P M = 2 M 1 det x I M 1 2 E 1 M P M = P(x (x R Proof Applying Lemma 24 and P M = S M ( o op, (210 we compute det (2 x E M P M using cofactors of the last column Then we obtain on the one hand

8 68 D Potts, M Tasche / Linear Algebra and its Applications 441 ( det ( M 1 2 x E M P M = TM (x + p l T l (x = P(x (x R On the other hand it follows that l=0 det ( ( 2 x E M P M = det (2 EM det x I M 1 2 E 1 M P M with det (2 E M = 2 M 1 This completes the proof Theorem 26 Let M and N be integers with 1 M < N Let h be a M-sparse polynomial of degree at most in the Chebyshev-1 basis Then the M coefficients c j R (j = 1,M and the M nonnegative integers n j (j = 1,M of ( (11 can be reconstructed from the 2M samples h k = h cos (k = 0,,2M 1 kπ 2N 1 Proof Using Lemma 21, we obtain the linear system (27 The matrix H M (0 is nonsingular by Lemma 22 By Lemma 25, the eigenvalues of the companion matrix 1 2 E 1 M P M of the Prony polynomial (24 in the Chebyshev-1 basis coincide with the zeros of (24 By (210, we compute the zeros of the Prony polynomial (24 via solving an eigenvalue problem such that we obtain the nonnegative integers n j (j = 1,M We form the Vandermonde-like matrix V M (x with x j = T nj (u N (j = 1,,M, which is nonsingular by Lemma 22, and obtain finally the coefficients c j R (j = 1,,M Thus we can summarize: Algorithm 27 (Prony method for sparse Chebyshev-1 interpolation Input: N N with N > M, h k = h(u N,k R (k = 0,,2M 1, M N Chebyshev-1 sparsity of the polynomial (11ofdegreeatmost 1 Solve the square system H M (0(p j M 1 j=0 = h(m 2 Determine the simple roots x j (j = 1,M of the Prony polynomial (24, where 1 x 1 > x 2 > > x M 1, and compute then n j := [ ] 2N 1 arccos x j (j = 1,,M, where [x] := x + 05 means rounding of x R to the nearest integer 3 Compute c j R (j = 1,,M as solution of the square Vandermonde-like system V M (x c = (h k M 1 π with c := (c j M j=1 Output: n j N 0 (0 n 1 < n 2 << n M < 2N, c j R (j = 1,,M Now we show that the matrix pencil method follows directly from the Prony method First we observe that H M (0 = 2 V M (x(diag c V M (x T Since c j = 0 (j = 1,,M, thematrixh M (0 has the rank M and is invertible Note that the Chebyshev-1 sparsity of the polynomial (11 coincides with the rank of H M (0 Hence we conclude that

9 D Potts, M Tasche / Linear Algebra and its Applications 441 ( det (2 x H M (0 E M H M (0 P M = det (H M (0 det (2 x E M P M = det (H M (0 P(x such that the eigenvalues of the square matrix pencil 2 x H M (0 E M H M (0 P M (x R (211 are exactly x j = cos n jπ [ 1, 1] 2N 1 (j = 1,,M Eacheigenvaluex j of the matrix pencil (211 is simple and has a right eigenvector v = (v k M 1 with M 1 v M 1 = T M (x j = p l T l (x j, l=0 since P(x j = 0 and P has the form (24 By this special choice of v M 1 onecaneasilydeterminethe other components v M 2,, v 0 which can be recursively computed from the linear system P M v = 2 x j E M v HenceweobtainH M (0 P M v = 2 x j H M (0 E M v, where the matrices can be represented in the following form H M (0 P M = H M (1 + ( oh(0 h(m 2, 2 H M (0 E M = H M (0 + ( oh(1 h(m 1 Example 28 In the case M = 3wehavetosolvethelinearsystem 01 p 0 v 0 x j v p 1 v 1 = 2x j v 1 01 p 2 v 2 2x j v 2 with v 2 = T 3 (x j = 2 p l T l (x j l=0 Then we determine the other components of the eigenvector v = (v l 2 l=0 as v 1 = p 1 T 0 (x j (2p 0 + p 2 T 1 (x j p 1 T 2 (x j, v 0 = (p 0 + p 2 T 0 (x j 2p 1 T 1 (x j 2p 0 T 2 (x j In the following, we factorize the square T+H matrices H M (s(s = 0, 1 simultaneously Therefore we introduce the rectangular T+H matrix H M,M+1 := ( ( H M (0 H M (1(1 : M, M = h(0 h(1 h(m (212

10 70 D Potts, M Tasche / Linear Algebra and its Applications 441 ( such that conversely H M (s = H M,M+1 (1 : M, 1 + s : M + s (s = 0, 1 (213 Then we compute the QR factorization of H M,M+1 with column pivoting and obtain H M,M+1 M+1 = Q M R M,M+1 with an orthogonal matrix Q M, a permutation matrix M+1, and a trapezoidal matrix R M,M+1, where R M,M+1 (1 : M, 1 : M is a nonsingular upper triangular matrix Note that the permutation matrix M+1 is chosen such that the diagonal entries of R M,M+1 (1 : M, 1 : M have nonincreasing absolute values Using the definition S M,M+1 := R M,M+1 T M+1, we infer that by (213 H M (s = Q M S M (s (s = 0, 1, where S M (s := S M,M+1 (1 : M, 1 + s : M + s (s = 0, 1 Hence we can factorize the matrices 2 H M (0 E M and H M (0 P M in the following form 2 H M (0 E M = H M (0 + ( oh(1 h(m 1 = Q M S (0, M H M (0 P M = H M (1 + ( oh(0 h(m 2 = Q M S (1, M where S M (0 := S M(0 + ( os M (1(1 : M, 1 : M 1, (214 S M (1 := S M(1 + ( os M (0(1 : M, 1 : M 1 (215 Since Q M is orthogonal, the generalized eigenvalue problem of the matrix pencil (211 isequivalent to the generalized eigenvalue problem of the matrix pencil x S M (0 S M (1 = S M (0 ( x I M ( S M (0 1 S M (1 (x R Since H M (0 is nonsingular by Lemma 22,thematrix2H M (0 E M is nonsingular too Hence S M (0 = 2 Q M H M(0 E M is invertible We summarize this method: Algorithm 29 (Matrix pencil factorization based on QR decomposition for sparse Chebyshev-1 interpolation Input: N N with N > M, h k = h(u N,k R (k = 0,,2M 1, M N Chebyshev-1 sparsity of the polynomial (11ofdegreeatmost

11 D Potts, M Tasche / Linear Algebra and its Applications 441 ( Compute the QR factorization with column pivoting of the rectangular T+H matrix (212 and form the matrices (214 and (215 2 Determine the eigenvalues x j [ 1, 1] (j = 1,,M of the square matrix ( S M (0 1 S M (1, where x j are ordered in the following way 1 x 1 > x 2 > > x M 1 Form n j := [ 2N 1 π ] arccos x j (j = 1,,M 3 Compute c j R (j = 1,,M as solution of the square Vandermonde-like system V M (x c = (h k M 1 with x := (x j M j=1 and c := (c j M j=1 Output: n j N 0 (0 n 1 < n 2 << n M < 2N, c j R (j = 1,,M In contrast to Algorithm 29, we use now the singular value decomposition (SVD of the rectangular Hankel matrix (212 and obtain a method which is known as the ESPRIT method Applying the SVD to H M,M+1,weobtain H M,M+1 = U M D M,M+1 W M+1 with orthogonal matrices U M, W M+1 and a diagonal matrix D M,M+1, whose diagonal entries are the ordered singular values σ 1 σ 2 σ M > 0ofH M,M+1 Introducing D M := D M,M+1 (1 : M, 1 : M, W M,M+1 := W M+1 (1 : M, 1 : M + 1, we can simplify the SVD of (212by H M,M+1 = U M D M W M,M+1 Note that W M,M+1 W T M,M+1 = I M Setting W M (s := W M,M+1 (1 : M, 1 + s : M + s (s = 0, 1, it follows from (213 that H M (s = U M D M W M (s(s = 0, 1 Hence we can factorize the matrices 2 H M (0 E M and H M (0 P M in the following form 2 H M (0 E M = H M (0 + ( oh(1 h(m 1 = UM D M W M (0, H M (0 P M = H M (1 + ( oh(0 h(m 2 = UM D M W M (1, where W M (0 := W M(0 + ( ow M (1(1 : M, 1 : M 1, (216 W M (1 := W M(1 + ( ow M (0(1 : M, 1 : M 1 (217

12 72 D Potts, M Tasche / Linear Algebra and its Applications 441 ( Clearly, W M (0 = 2 D 1 M UT M H M(0 E M is a nonsingular matrix by construction Then we infer that the generalized eigenvalue problem of the matrix pencil (211 is equivalent to the generalized eigenvalue problem of the matrix pencil x W M (0 W M (1 = W M (0 ( x I M ( W M (0 1 W M (1, since U M is orthogonal and D M is invertible Therefore we obtain that P M = ( H M (0 1 UM D M W M (1 Algorithm 210 (ESPRIT method for sparse Chebyshev-1 interpolation Input: N N with N > M, h k R (k = 0,,2M 1, M N Chebyshev-1 sparsity of the polynomial (11ofdegreeatmost 1 Compute the SVD of the Hankel matrix (212 and form the matrices (216 and (217 2 Determine the eigenvalues x j [ 1, 1] (j = 1,M of ( W M (0 1 W M (1, where x j are ordered in the following form 1 x 1 > x 2 > > x M 1 Form n j := [ ] 2N 1 π arccos x j (j = 1,,M 3 Compute the coefficients c j R (j = 1,,M as solution of the square Vandermonde-like system V M (x c = (h k M 1 with x := (x j M j=1 and c := (c j M j=1 Output: n j N 0 (0 n 1 < n 2 << n M < 2N, c j R (j = 1,,M Remark 211 The last step of the Algorithms can be replaced by the computation of the real coefficients c j (j = 1,,M as least squares solution of the overdetermined Vandermonde-like V 2M,M (x c = (h k 2M 1 with the rectangular Vandermonde-like matrix V 2M,M (x := ( T k (x j 2M 1,M,j=1 = (cos n j kπ 2M 1,M,j=1 In the case of sparse Chebyshev-1 interpolation of (11 with known Chebyshev-1 sparsity M, we have seen that each method determines the eigenvalues x j (j = 1,,M of the matrix pencil 2 x E M P M, where 1 2 E 1 M P M is the companion matrix of the Prony polynomial (24 inthe Chebyshev-1 basis 3 Interpolation for unknown Chebyshev-1 sparsity This section is the core of the paper Here we consider the problem of sparse polynomial interpolation in the important case of unknown Chebyshev-1 sparsity M of the polynomial (11 We assumeonly that an upper bound of the Chebyshev-1 sparsity is known Roughly spoken, we generalize the results of Section 2 to rectangular T+H matrices and rectangular Vandermonde-like matrices We show factorizations of rectangular T+H matrices and the interesting relation (38 between the modified Prony polynomial (36 and the T+H matrices (see Lemma 32 The zeros of the modified Prony polynomial can be computed via solving an eigenvalue problem of the related companion matrix The main results of Section 3 are the Algorithms Numerical examples in Section 5 show that the Algorithms 34 and 35 are numerically stable in the floating point arithmetic

13 D Potts, M Tasche / Linear Algebra and its Applications 441 ( Let L N be convenient upper bound of the unknown Chebyshev-1 sparsity M of the polynomial (11ofdegreeatmost, where N N is sufficiently large with M L N Inorderto improve the numerical stability, we allow to choose more sampling points Therefore we introduce an additional parameter K with L K N such that we use K + L sampling points of (11, more precisely we assume that noiseless sampled data h k = h(u N,k (k = 0,,L + K 1 are given With the L + K sampled data h k R (k = 0,,L + K 1 we form the rectangular T+H matrices H K,L+1 := ( h l+m + K 1,L h l m l,m=0 (31 H K,L (s := ( h l+m+s + K 1,L 1 h l m s l,m=0 (s = 0, 1 (32 Then H K,L (1 is a shifted version of the T+H matrix H K,L (0 and H K,L+1 = ( H K,L (0 H K,L (1(1 : K, L, H K,L (s = H K,L+1 (1 : K, 1 + s : L + s (s = 0, 1 (33 Note that in the special case M = L = K we obtain again the matrices (212 and (213 Using the coefficients p k (k = 0,,M 1 of the Prony polynomial (24, we form the vector p L := (p k L 1 with p M := 1, p M+1 = = p L 1 := 0 By S L := ( δ k l 1 + δ L 1 k l+1 k,l=0 we denote the sum of forward and backward shift matrix, where δ k is the Kronecker symbol Analogously, we introduce p L+1 := (p k L with p L := 0, if L > M, and S L+1 := ( δ k l 1 + δ L k l+1 k,l=0 Lemma 31 LetL,K,M,N N with M L K N be given Furthermore, let h k = h(u N,k (k = 0,,L + K 1 be noiseless sampled data of the sparse polynomial (11 of degree at most with coefficients c j R \{0} (j = 1,,MThen rank H K,L+1 = rank H K,L (s = M (s = 0, 1 (34 If L = M, then null H K,M+1 = span {p M+1 } and null H K,M (s ={o} for s = 0, 1 IfL> M, then and null H K,L+1 = span {p L+1, S L+1 p L+1,,S L M L+1 p L+1}, null H K,L (s = span {p L, S L p L,,S L M 1 L p L } (s = 0, 1 dim (null H K,L+1 = L M + 1, dim (null H K,L (s = L M (s = 0, 1 1 For x j = T nj (u N (j = 1,,M, we introduce the rectangular Vandermonde-like matri- Proof ces V K,M (x := ( T k 1 (x j ( K,M = cos n K,M j(k 1π, (35 k,j=1 k,j=1 V K,M (x := ( T k (x j K,M k,j=1 = (cos n j kπ K,M k,j=1, which have the rank M, sincev M (x and V M (x are nonsingular by Lemmas 22 and 23 Then the rectangular T+H matrices (31 and (32 can be factorized in the following form

14 74 D Potts, M Tasche / Linear Algebra and its Applications 441 ( H K,L+1 = 2 V K,M (x(diag c V L+1,M (x T, H K,L (0 = 2 V K,M (x(diag c V L,M (x T, H K,L (1 = 2 V K,M (x(diag c V L,M (xt with x = (x j M j=1 and c = (c j M j=1 This can be shown in similar way as in the proof of Lemma 22 Sincec j = 0 and since x j [ 1, 1] are distinct, we obtain (34 Using rank estimation, we can determine the rank and thus the Chebyshev-1 sparsity of the sparse polynomial (11 By (34 and H K,L+1 p M+1 = o (see (25, the 1-dimensional null space of H K,L+1 is spanned by p M+1 Furthermore, the null spaces of H K,L (s are trivial for s = 0, 1 2 Assume that L > MFrom ( p m hl+m+s + h l m s = 0 (l = 0,, 2N M s 1; s = 0, 1 m=0 it follows that H K,L+1 (S j L+1 p L+1 = o (j = 0,, L M and analogously H K,L (s(s j L p L = o (j = 0,, L M 1; s = 0, 1, where o denotes the corresponding zero vector By p M = 1, we see that the vectors S j L+1 p L+1 (j = 0,, L M and S j L p L (j = 0,, L M 1 are linearly independent and located in null H K,L+1, and null H K,L (s, respectively 3 Let again L > M Now we prove that null H K,L+1 is contained in the linear span of the vectors S j L+1 p L+1 (j = 0,,L M Letu = (u l L l=0 RL+1 be an arbitrary right eigenvector of H K,L+1 related to the eigenvalue 0 and let U be the corresponding polynomial U(x = L u l T l (x (x R l=0 Using the noiseless sampled data h k = h(u N,k (k = 0,,, weobtain 0 = L (h l+m + h l m u m = m=0 L u m m=0 j=1 [ c j Tnj (u N,l+m + T nj (u N, l m ] Thus by T nj (u N,l+m + T nj (u N, l m = T l+m (x j + T l m (x j = 2 T l (x j T m (x j it follows that 0 = 2 c j T l (x j U(x j (l = 0,,2N L 1 j=1 and hence by (35 V K,M (x ( c j U(x j M j=1 = o

15 D Potts, M Tasche / Linear Algebra and its Applications 441 ( Since x j [ 1, 1] (j = 1,,M are distinct by assumption, the square Vandermonde-like matrix V M (x is nonsingular by Lemma 22 Hence we obtain U(x j = 0 (j = 1,,M by c j = 0 Thus it follows that U(x = P(x R(x with certain polynomial R(x = L M r k T k (x (x R; r k R But this means for the coefficients of the polynomials P, R, and U that u = r 0 p L r 1 S L+1 p L r L M S L M L+1 p L+1 Hence the vectors S j L+1 p L+1 (j = 0,,L M form a basis of null H K,L+1 such that dim(null H K,L+1 = L M + 1 Similarly, one can show the results for the other T+H matrices (32 This completes the proof The Prony method for sparse Chebyshev-1 interpolation (with unknown Chebyshev-1 sparsity M is based on the following result Lemma 32 LetL,K,M,N N with M L K N be given Let h k = h(u N,k (k = 0,,L+K 1 be noiseless sampled data of the sparse polynomial (11 of degree at most with coefficients c j R \{0} Then following assertions are equivalent: (i The polynomial Q(x := L q k T k (x (x R; q L := 1 (36 with real coefficients q k has M distinct zeros x j [ 1, 1] (j = 1,,M (ii The vector q = (q k L 1 is a solution of the linear system H K,L (0 q = h(l (h(l := ( h L+m + K 1 h L m m=0 (37 (iii The matrix Q L := S L ( o oq R L L has the property H K,L (0 Q L = H K,L (1 + ( oh(0 h(l 2 (38 Further the eigenvalues of 1 2 E 1 L Q L coincide with the zeros of the polynomial (36 Proof 1 From (i it follows (ii: Assume that Q(x j = 0 (j = 1,,M Form = 0,,K 1, we compute the sums s m := L (h k+m + h k m q k Using h k = h(u N,k (k = 0,,L + K 1, (11, and the known identities (see eg [13,p17 and p 31] 2 T j (x T k (x = T j+k (x + T j k (x, T j ( Tk (x = T j+k (x (j, k N 0,

16 76 D Potts, M Tasche / Linear Algebra and its Applications 441 ( we obtain s m = = L q k [ h ( T k+m (u N + h ( T k m (u N ] L c l l=1 By q L = 1 this implies that q k [T k+m (x l + T k m (x l ] = 2 c l T m (x l Q(x l = 0 l=1 L 1 (h k+m + h k m q k = 1 (h L+m + h L m (m = 0,,K 1 Hence we get (37 2 From (ii it follows (iii: Assume that q = (q l L 1 l=0 is a solution of the linear system (37 Then by H K,L (0(δ k j L 1 = h(j = ( h k+j + h k j K 1 (j = 1,,L 1, H K,L (0 q = h(l = ( h k+l + h k L K 1, we obtain (38 column by column 3 From (iii it follows (i: By (38weobtain(37, since the last column of Q L reads (δ L 2 j L 1 j=0 q and since the last column of H K,L (1 + ( oh(0 h(l 2 is equal to h(l + h(l 2 Then(37implies L (h k+m + h k m q k = 0 (m = 0,,K 1 As shown in the first step, we obtain c l T m (x l Q(x l = 0 (m = 0,,K 1, l=1 ie by (35 finally V K,M (x ( c l Q(x l M l=1 = o Especially we conclude that V M (x ( c l Q(x l M l=1 = o Since x j [ 1, 1] (j = 1,,M are distinct, the square Vandermonde-like matrix V M (x is nonsingular by Lemma 22 such that Q(x j = 0 (j = 1,,M 4 From Lemma 25, itfollowsthat det ( 2x E L Q L = Q(x (x R Hence the eigenvalues of the square matrix 1 2 E 1 L Q L coincide with the zeros of the polynomial (36 This completes the proof

17 D Potts, M Tasche / Linear Algebra and its Applications 441 ( In the following, we denote a polynomial (36 asamodified Prony polynomial of degree L (M L N, if the corresponding coefficient vector q = (q k L 1 is a solution of the linear system (37 Then (36 has the same zeros x j [ 1, 1] (j = 1,,M as the Prony polynomial (24, but (36 has L M additional zeros, if L > M The eigenvalues of 1 2 E 1 L Q L coincide with the zeros of the polynomial (36 Now we formulate Lemma 32 as an algorithm Since the unknown coefficients c j (j = 1,,M do not vanish, we can assume that c j >εfor convenient bound ε(0 <ε 1 Algorithm 33 (Prony method for sparse Chebyshev-1 interpolation Input: L, K, N N (N 1, 3 L K N, L is upper bound of the Chebyshev-1 sparsity M of (11ofdegreeatmost, h k = h(u N,k R (k = 0,,L + K 1, 0<ε 1 1 Compute the least squares solution q = (q k L 1 of the rectangular linear system (37 2 Determine the simple roots x j [ 1, 1] (j = 1,, M of the modified Prony polynomial (36, ie, compute all eigenvalues x j [ 1, 1] (j = 1,, M of the companion matrix 1 2 E 1 L Q L Assume that x j are ordered in the following form 1 x 1 > x 2 > > x M 1 Note that rank H K,L (0 = M M 3 Compute c j R (j = 1,, M as least squares solution of the overdetermined linear Vandermonde-like system V L+K, M( x( c j M j=1 = (h k L+K 1 with x := ( x j M j=1 and V L+K, M( x := ( T k ( x j L+K 1, M,j=1 4 Delete all the x l (l {1,, M} with cl εand denote the remaining values by x j (j = 1,,M with M M Calculatenj := [ ] 2N 1 π arccos x j (j = 1,,M 5 Repeat step 3 and compute c = (c j M j=1 RM as least squares solution of the overdetermined linear Vandermonde-like system V L+K,M (x c = (h k L+K 1 with x := (x j M j=1 and V L+K,M(x := ( T k (x j L+K 1,M,j=1 = ( cos n jkπ L+K 1,M 2N 1,j=1 Output: M N, n j N 0 (0 n 1 < n 2 << n M < 2N, c j R (j = 1,,M Now we show that the Prony method for sparse Chebyshev-1 interpolation can be improved to a matrix pencil method As known, a rectangular matrix pencil may not have eigenvalues in general But this is not the case for our rectangular matrix pencil 2x H K,L (0 E L H K,L (0 Q L, (39 which has x j [ 1, 1] (j = 1,,M as eigenvalues Note that by (38 bothmatricesh K,L (0 E L and H K,L (0 Q L are known by the given sampled data h k (k = 0,, The matrix pencil (39 has at least the eigenvalues x j [ 1, 1] (j = 1,,MIfv C L is a right eigenvector related to x j, then by ( 2xj H K,L (0 E L H K,L (0 Q L v = HK,L (0 ( 2x j E L Q L v and det ( 2x j E L Q L = Q(xj = 0

18 78 D Potts, M Tasche / Linear Algebra and its Applications 441 ( we see that v = (v k L 1 is a right eigenvector of the square eigenvalue problem 1 2 E 1 L Q L v = x j v A right eigenvector can be determined by L 1 v L 1 = T L (x j = q l T l (x j, l=0 whereas the other components v L 2,, v 0 can be computed recursively from the linear system Q L v = 2x j E L v Now we factorize the rectangular T+H matrices (32 simultaneously For this reason, we compute the QR decomposition of the rectangular T+H matrix (31 By (34, the rank of the T+H matrix H K,L+1 is equal to MHenceH K,L+1 is rank deficient Therefore we apply QR factorization with column pivoting and obtain H K,L+1 L+1 = U K R K,L+1 with an orthogonal matrix U K, a permutation matrix L+1, and a trapezoidal matrix R K,L+1 = R K,L+1(1 : M, 1 : L + 1, O K M,L+1 where R K,L+1 (1 : M, 1 : M is a nonsingular upper triangular matrix By the QR decomposition we can determine the rank M of the T+H matrix (31 and hence the Chebyshev-1 sparsity of the sparse polynomial (11 Note that the permutation matrix L+1 is chosen such that the diagonal entries of R K,L+1 (1 : M, 1 : M have nonincreasing absolute values We denote the diagonal matrix containing these diagonal entries by D M With S K,L+1 := R K,L+1 T = S K,L+1(1 : M, 1 : L + 1 L+1, (310 we infer that by (33 with H K,L (s = U K S K,L (s (s = 0, 1 O K M,L+1 S K,L (s := S K,L+1 (1 : K, 1 + s : L + s (s = 0, 1 Hence we can factorize the matrices 2 H K,L (0 E L and H K,L (0 Q L in the following form 2 H K,L (0 E L = H K,L (0 + ( oh(1 h(l 1 = UK S K,L (0, H K,L (0 Q L = H K,L (1 + ( oh(0 h(l 2 = UK S K,L (1,

19 D Potts, M Tasche / Linear Algebra and its Applications 441 ( where S K,L (0 := S K,L(0 + ( os K,L (1(1 : K, 1 : L 1, S K,L (1 := S K,L(1 + ( os K,L (0(1 : K, 1 : L 1 Since U K is orthogonal, the generalized eigenvalue problem of the matrix pencil (39isequivalentto the generalized eigenvalue problem of the matrix pencil x S (0 K,L S K,L (1 (x R Using the special structure of (310, we can simplify the matrix pencil with x T M,L (0 T M,L (1 (x R (311 T M,L (s := S K,L (1 : M, 1 + s : L + s (s = 0, 1 (312 Here one can use the matrix D M as diagonal preconditioner and proceed with T M,L (s := D 1 M T M,L(s (313 Then the generalized eigenvalue problem of the transposed matrix pencil x T M,L (0T T M,L (1T has the same eigenvalues as the matrix pencil (311 except for the zero eigenvalues and it can be solved as eigenvalue problem of the M-by-M matrix F QR M := ( T M,L (0T T M,L (1T (314 Finally we obtain the nodes x j [ 1, 1] (j = 1,,M as the eigenvalues of (314 Algorithm 34 (Matrix pencil factorization based on QR decomposition for sparse Chebyshev-1 interpolation Input: L, K, N N (N 1, 3 L K N, L is upper bound of the Chebyshev-1 sparsity M of (11ofdegreeatmost, h k = h(u N,k R (k = 0,,L + K 1 1 Compute QR factorization of the rectangular T+H matrix (31 Determine the rank of (31 and form the matrices (312 and (313 2 Determine the eigenvalues x j [ 1, 1] (j = 1,,M of the square matrix (314 Assume that x j are ordered in the following form 1 x 1 > x 2 >> x M 1 Calculate n j := [ 2N 1 π arccos x j ] (j = 1,,M 3 Compute the coefficients c j R (j = 1,,M as least squares solution of the overdetermined linear Vandermonde-like system V L+K,M (x(c j M j=1 = (h k L+K 1 with x := (x j M j=1 and V L+K,M(x := ( T k (x j L+K 1,M,j=1 = ( cos n jkπ L+K 1,M 2N 1,j=1 Output: M N, n j N 0 (0 n 1 < n 2 << n M < 2N, c j R (j = 1,,M

20 80 D Potts, M Tasche / Linear Algebra and its Applications 441 ( In the following we derive the ESPRIT method by similar ideas as above, but now we use the SVD of the T+H matrix (31, which is rank deficient by (34 Therefore we use the factorization H K,L+1 = U K D K,L+1 W L+1, where U K and W L+1 are orthogonal matrices and where D K,L+1 is a rectangular diagonal matrix The diagonal entries of D K,L+1 are the singular values of (31 arranged in nonincreasing order σ 1 σ 2 σ M >σ M+1 = = σ L+1 = 0 Thus we can determine the rank M of the Hankel matrix (31 which coincides with the Chebyshev-1 sparsity of the polynomial (11 Introducing the matrices D K,M := D K,L+1 (1 : K, 1 : M = diag (σ j M j=1, O K M,M W M,L+1 := W L+1 (1 : M, 1 : L + 1, we can simplify the SVD of the Hankel matrix (31 as follows H K,L+1 = U K D K,M W M,L+1 Note that W M,L+1 W T M,L+1 = I M Setting W M,L (s = W M,L+1 (1 : M, 1 + s : L + s (s = 0, 1, (315 it follows from (33 that H K,L (s = U K D K,M W M,L (s(s = 0, 1 Hence we can factorize the matrices 2 H K,L (0 E L and H K,L (0 Q L in the following form 2 H K,L (0 E L = H K,L (0 + ( oh(1 h(l 1 = UK D K,M W K,L (0, H K,L (0 Q L = H K,L (1 + ( oh(0 h(l 2 = UK D K,M W K,L (1, where W K,L (0 := W K,L(0 + ( ow K,L (1(1 : K, 1 : L 1, W K,L (1 := W K,L(1 + ( ow K,L (0(1 : K, 1 : L 1 Since U K is orthogonal, the generalized eigenvalue problem of the rectangular matrix pencil (39 is equivalent to the generalized eigenvalue problem of the matrix pencil x D K,M W (0 M,L D K,M W M,L (1 (316 If we multiply the transposed matrix pencil (316 from the right side with 1 diag (σ j M j=1, O K M,M we obtain the generalized eigenvalue problem of the matrix pencil

21 x W M,L (0T W M,L (1T, D Potts, M Tasche / Linear Algebra and its Applications 441 ( which has the same eigenvalues as the matrix pencil (316 except for the zero eigenvalues Finally we determine the nodes x j [ 1, 1] (j = 1,,M as eigenvalues of the matrix F SVD M := ( W M,L (0T W M,L (1T (317 Thus the ESPRIT algorithm reads as follows: Algorithm 35 (ESPRIT method for sparse Chebyshev-1 interpolation Input: L, K, N N (N 1, 3 L K N, L is upper bound of the Chebyshev-1 sparsity M of (11ofdegreeatmost, h k = h(u N,k R (k = 0,,L + K 1 1 Compute the SVD of the rectangular T+H matrix (31 Determine the rank M of (31 and form the matrices (315 2 Compute all eigenvalues x j [ 1, 1] (j = 1,,M of the square matrix (317 Assume that the eigenvalues are ordered in the following form 1 x 1 > x 2 > > x M 1 Calculate n j := [ 2N 1 π arccos x j ] (j = 1,,M 3 Compute the coefficients c j R (j = 1,,M as least squares solution of the overdetermined linear Vandermonde-like system V L+K,M (x c = (h k L+K 1 with x := (x j M j=1 and c := (c j M j=1 Output: M N, n j N 0 (0 n 1 < n 2 << n M < 2N, c j R (j = 1,,M 4 Sparse polynomial interpolation in Chebyshev-2 basis In this section, we discuss the sparse interpolation in the basis of Chebyshev polynomials of second kind Here we use analogous ideas as in Sections 2 and 3 Thus Lemma 41 corresponds to Lemma 21 Note that one can extend this approach to the Chebyshev polynomials of third and fourth kind, respectively For n N 0 and x ( 1, 1, thechebyshev polynomial of second kind is defined by U n (x := (1 x 2 1/2 sin ( (n + 1 arccos x (see for example [13, p 3] These polynomials are orthogonal with respect to the weight (1 x 2 1/2 on [ 1, 1] (see [13, p 74] and form the Chebyshev-2 basis For M, N N with M < N, we consider a polynomial h of degree at most 2N 1, which is M-sparse in the Chebyshev-2 basis, ie h(x = c j U nj (x j=1 (41 with 0 n 1 < n 2 < < n M The integer M is called Chebyshev-2 sparsity of (41 Note that the sparsity depends on the choice of Chebyshev basis Using T 0 = U 0, T 1 = U 1 /2 and T n = (U n U n 2 /2 forn 2(cf[13,p4],weobtainforN 1 U 2N 2 + U 2N 1 = T (T T 2N 1

22 82 D Potts, M Tasche / Linear Algebra and its Applications 441 ( Thus the 2-sparse polynomial U 2N 2 + U 2N 1 in the Chebyshev-2 basis is not a sparse polynomial in the Chebyshev-1 basis For sake of brevity, we restrict us on the discussion of the sparse polynomial interpolation in the Chebyshev-2 basis We present only the Prony method in the case of given Chebyshev-2 sparsity (see Algorithm 42 But we emphasize that one can extend this approach the Chebyshev polynomials of third and fourth kind (see [13, p 5], which are defined for n N 0 by (( V n (x := cos cos arccos x n ( 1, W n (x := 2 arccos x Substituting x = cos t, we obtain for all t [0, π] h(cos t sin t = By sampling at t = (( sin sin arccos x n ( (x 1 ( 1, 1 2 arccos x c j sin ( (n j + 1 t (42 j=1 πk (k = 0,,, itfollowsthat 2N 1 ( h k := h cos πk πk sin 2N = M c j sin 1 j=1 ( (n j + 1 πk (43 Further we set h k := hk (k = 1,, In this case, we introduce the Prony polynomial by P(x := 2 M 1 M ( j=1 x cos (n j + 1π which can be represented again in the Chebyshev-1 basis in the form, (44 P(x = p l T l (x (p M = 1 l=0 The coefficients p l of the Prony polynomial (44 can be characterized as follows: Lemma 41 For all k = 1,,M, the scaled sampled values (43 and the coefficients p l of the Prony polynomial (44 fulfill the equations M 1 j=0 ( hj+k hj k p j = ( hm+k hm k Proof Using sin(α + β sin(α β = 2sinα cos β, weobtainforj, k = 0,,M h j+k hj k = 2 l=1 c l sin (n l + 1πk cos (n l + 1πj (45 Note that the Eq (45 istrivialfork = 0 and therefore omitted From (45 it follows that

23 D Potts, M Tasche / Linear Algebra and its Applications 441 ( M ( hj+k hj k p j = 2 p j c l sin (n l + 1πk cos (n l + 1πj j=0 j=0 l=1 = 2 c l sin (n ( l + 1πk P cos (n l + 1πj = 0 l=1 By p M = 1, this implies the assertion If we introduce the T+H matrix H M (0 := ( hj+k hj k M,M 1 k=1,j=0 and the vector h(m := ( hm+k hm k M k=1, then by Lemma 41 the vector p := (p j M 1 j=0 is a solution of the linear system H M (0 p = h(m (46 By (45, the T+H matrix HM (0 can be factorized in the form H M (0 = 2 V s M (diag c ( V c M T (47 with the Vandermonde-like matrices ( V c := M cos (n M 1,M ( l + 1πj, V s 2N := M sin (n M l + 1πk 1 j=0,l=1 k,l=1 and the diagonal matrix of c = (c l M l=1 Both Vandermonde-like matrices are nonsingular Assume that V c M is singular Then there exists a vector d = (d l M 1 l=0 = o with d T V c = M ot Introducing D(x := M 1 l=0 d l cos(lx, this even trigonometric polynomial of order at most M 1 has M distinct zeros (n l+1π 2N 1 (0, π] (j = 1,,M But this can be only the case, if D vanishes identically Similarly, one can see that V s M is nonsingular too From (47 it follows that HM (0 is also nonsingular Thus we obtain: Algorithm 42 (Prony method for sparse Chebyshev-2 interpolation Input: N N with N > M, hk R (k = 0,,2M 1, M N Chebyshev-2 sparsity of the polynomial (41 ofdegreeatmost 1 Solve the square linear system (46 2 Determine the simple roots x j (j = 1,M of the Prony polynomial (44, where 1 x 1 > x 2 >> x M 1, and compute then n j := [ 2N 1 π arccos x j ] 1(j = 1,,M 3 Compute c j R (j = 1,,M as solution of the square Vandermonde-like system V s M c = ( hk M 1 with c := (c j M j=1 Output: n j N 0 (0 n 1 < n 2 << n M < 2N, c j R (j = 1,,M

24 84 D Potts, M Tasche / Linear Algebra and its Applications 441 ( Table 51 ResultsofExample51 N K L Alg 33 Alg 34 Alg 35 e(c e e e e e e e e e e e-15 Immediately we can see that the Algorithms 34 and 35 can be generalized in a straightforward manner, since the Prony polynomial P is represented in the Chebyshev-1 basis We will denote these generalizations by Algorithms 34 and 35, respectively 5 Numerical examples Now we illustrate the behavior and the limits of the suggested algorithms Using IEEE standard floating point arithmetic with double precision, we have implemented our algorithms in MATLAB In the Examples 51 53,anM-sparse polynomial is given in the form (11 with Chebyshev polynomials T nj of degree n j and real coefficients c j = 0 (j = 1,,M We compute the absolute error of the coefficients by e(c := max c j c j (c := (c j M, j=1 j=1,,m where c j are the coefficients computed by our algorithms In Example 54 we generalize the method to a sparse nonpolynomial interpolation Finally in Example 55, we present an example of sparse polynomial interpolation in the Chebyshev-2 basis In all examples we observe that the numerical stability of the Algorithms 34 and 35 can be improved by using more sampling values π Example 51 We start with the following example We choose M = 5, c j = j, u N := cos 2N 1 and (n 1, n 2, n 3, n 4, n 5 = (6, 12, 176, 178, 200 in (11 The symbols + and in the Table 51 mean that all degrees n j are correctly reconstructed and accordingly the reconstruction fails Since after a successful reconstruction the last step is the same in the Algorithms 33 35,wepresenttheerrore(c in the last column of the Table 51 Note that for the parameters N = 300 and K = L = 5theT+H matrix in step 1 of Algorithm 33, see(37, has a condition number cond(h 5 ( Due to roundoff errors, some eigenvalues x j are not contained in [ 1, 1] We can improve the stability by choosing more sampling values Further we remark that the stability of computing the eigenvalues x j depends on the stability of the different methods used in step 1 of the Algorithms 33, 34 and 35, respectively Example 52 It is difficult to reconstruct a sparse polynomial (11 in the case, if some degrees n j of the Chebyshev polynomials T nj differ only a little Therefore we consider the sparse polynomial (11 with (n 1, n 2, n 3, n 4, n 5 = (60, 120, 1760, 1780, 2000 and again c j = j (j = 1,,5 Theresults are shown in Table 52

25 D Potts, M Tasche / Linear Algebra and its Applications 441 ( Table 52 ResultsofExample52 N K L Alg 33 Alg 34 Alg 35 e(c e e e-16 Table 53 ResultsofExample53 N K L σ Alg 33 Alg 34 Alg 35 e(c e e e e-04 Example 53 Similarly as in Example 51, we choose M = 5, c j = j (j = 1,,5 and (n 1, n 2, n 3, n 4, n 5 = (6, 12, 176, 178, 200 We reconstruct the sparse polynomial (11 from samples of a random Chebyshev grid For this purpose, we choose a random integer σ [1, N 1] such that its inverse σ 1 modulo exists Assume that N fulfills the conditions n j By ( knj π T nj (u N,k = cos ( (σ k(σ cos 1 n j mod (2N 1π if σ 1 n 2N 1 j mod ( N, = ( (σ k(2n 1 (σ cos 1 n j mod (2N 1π if σ 1 n j mod ( >N 2N 1 T σ = 1 n j mod (2N 1(u N,σ k if σ 1 n j mod ( <N, T 2N 1 (σ 1 n j mod (2N 1 (u N,σ k if σ 1 n j mod ( N we are able to recover the degrees n j from the sampling set u N,σ k = cos σ kπ for k = 0,,K +L 1 2N 1 The main advantage is that the degrees σ 1 n j are much better separated than the original degrees n j The results are shown in the Table 53 Note that the Algorithm 33 determines the eigenvalues x j, which give the correct degrees n j after step 2, but the selection of these correct degrees fails in general in step 4 Example 54 This example shows a straightforward generalization to a sparse nonpolynomial interpolation We consider special functions the form h(x := c j cos(ν j arccos(x (x [ 1, 1], j=1 where ν j R with 0 ν 1 < < ν M < 2N are not necessarily integers Using t = arccos(x, we obtain g(t = c j cos(ν j t (t [0, π] j=1

26 86 D Potts, M Tasche / Linear Algebra and its Applications 441 ( Fig 51 The sparse polynomial (11ofExample53 for N = 300 and 100 samples with σ = 1(leftandσ = 251 (right Table 54 ResultsofExample54 N K L Alg 33 Alg 34 Alg 35 e(c e e e e e e e e-10 π As in Example 51 we choose M = 5, c j = j, u N := cos and(ν 2N 1 1,ν 2,ν 3,ν 4,ν 5 = (61, 122, 1763, 1784, 2005 We compute the error of the values ν j R by e(ν := max ν j ν j (ν := (ν j 5, j=1 j=1,,5 where ν j are the values computed by our algorithms This corresponding errors e(ν are shown in the kπ Table 54Wesamplethefunctiong at the nodes for k = 0,,L + K 1 and present the error 2N 1 e(c in the last column of Table 54 based on Algorithm 33 The results show that the Algorithms 34 and 35 can be used to find the entries ν j and the coefficients c j Example 55 Finally, we consider a sparse polynomial (41 in Chebyshev-2 basis To this end, we π choose M = 5, c j = j (j = 1,,5, u N := cos 2N 1 and(n 1, n 2, n 3, n 4, n 5 = (6, 12, 176, 178, 190 The symbols + and in the Table 55 mean that all degrees n j of the Chebyshev polynomials U nj are correctly reconstructed and accordingly the reconstruction fails Remember that the generalizations of Algorithms 34 and 35 for the Chebyshev-2 basis are denoted by Algorithms 34 and 35, respectively Since after a successful reconstruction the last step is the same in our algorithms, we present the error e(c in the last column of the Table 55 FromTable55 we observe that the algorithms for sparse polynomial interpolation in Chebyshev-2 basis behaves very similar as the algorithms for sparse polynomial interpolation in Chebyshev-1 basis Similar as in Example 54, we can deal with functions of the form h(t = M j=1 d j sin(μ j t by using the relation (42, and furthermore with functions of the form f (t = ( cj cos(ν j t + d j sin(μ j t (t [0, π] j=1