On the Analytical and Numerical Properties of the Truncated Laplace Transform II.

Transcription

1 The Laplace Transform is frequently encountered in mathematics, physics, engineering and other fields. However, the spectral properties of the Laplace Transform tend to complicate its numerical treatment; therefore, the closely related Truncated Laplace Transforms are often used in applications. We have constructed efficient algorithms for the evaluation of the left singular functions and singular values of the Truncated Laplace Transform. Together with the previously introduced algorithms for the evaluation of the right singular functions, these algorithms provide the Singular Value Decomposition of the Truncated Laplace Transform. The resulting algorithms are applicable to all environments likely to be encountered in applications, including the evaluation of singular functions corresponding to extremely small singular values e.g.. On the Analytical and Numerical Properties of the Truncated Laplace Transform II. Roy R. Lederman, Vladimir Rokhlin, Technical Report YALEU/DCS/TR-57 May 9, 5 Thisauthor sresearchwassupportedinpartbytheonrgrants#n4---78,#n and #N and the NSF grant # This author s research was supported in part by the ONR grants #N and #N Applied Mathematics Program, Yale University, New Haven CT 65 Approved for public release: distribution is unlimited. Keywords: Truncated Laplace Transform, SVD.

2 Introduction The Laplace Transform L is a linear mapping L [, L [, ; for a function f L [,, it is defined by the formula Lfω = e tω ftdt. As is well-known, L has a continuous spectrum, and L is not continuous see, for example, [3]. These and related properties tend to complicate the numerical treatment of L. In addressing these problems, we find it useful to draw an analogy between the numerical treatment of the Laplace Transform, and the numerical treatment of the Fourier Transform F; for a function f L R, the latter is defined by the formula Ffω = e itω ftdt, where ω R. In various applications in mathematics and engineering, it is useful to define the Truncated Fourier Transform F c : L [,] L [,]; for a given c >, F c of a function f L [,] is defined by the formula F c fω = e ictω ftdt. 3 The operator F c has been analyzed extensively; one of most notable observations, made by Slepian and Pollak around 96, was that the integral operator F c commutes with a second order differential operator see [8]. This property of F c was used in analytical and numerical investigations of the eigendecomposition of this operator; for example in [8, 9,, 6, 7, 9, 5]. For < a < b <, the linear mapping L a,b : L [a,b] L [, defined by the formula L a,b fω = b a e tω ftdt, 4 will be referred to as the Truncated Laplace Transform of f; obviously, L a,b is a compact operator see, for example, [3]. The Singular Value Decomposition SVD of L a,b has been analyzed, inter alia, in [3] and [5]; Bertero and Grünbaum observed that each of the symmetric operators L a,b L a,b and L a,b L a,b commutes with adifferential operator see [5]. Despite [5, 6, 3,, 4, 3, 8, 7], much more is known about the numerical and analytical properties of F c than about the properties of L a,b. We have constructed algorithms for the efficient evaluation of the SVD of L a,b. In [] we introduced algorithms for the efficient evaluation of the right singular functions and singular values of L a,b. In this paper, we introduce an efficient algorithm for the numerical evaluation

3 of the left singular functions, and an additional algorithm for computing the singular value. Additional properties of the Truncated Laplace transform will be discussed in upcoming papers. This paper is organized as follows. Section summarizes the various standard mathematical facts and simple derivations that are used later in this paper, as well as various properties of the Truncated Laplace Transform. Section 3 contains the derivation of various properties of the truncated Laplace transform which are related to the left singular functions; the properties are used in the algorithms presented in the following section. Section 4 describes the algorithms for the evaluation of the left singular functions and singular values of the Truncated Laplace Transform. Section 5 contains numerical results obtained using the algorithms. Section 6 contains generalizations and conclusions. Preliminaries. The Laguerre Functions In this section we summarize some of the properties of the Laguerre Polynomials, Generalized Laguerrepolynomials, andlaguerrefunctions. TheLaguerrepolynomialL k xofdegreek is defined by the formula L k x = k k m k m m! xm. 5 m= The Generalized Laguerre Polynomial of order α > and degree k, denoted by L α k x, is defined by the formula k k +α k x = m k m m! xm. 6 L α m= The Laguerre Function of degree k, which we denote by Φ k, is defined via the formula Φ k x = e x/ L k x. 7 The following well known properties of the Laguerre Polynomials and the Generalized Laguerre Polynomials can be found, inter alia, in []. L α k x = L α k x Lα k x 8 d dx L kx = L k 9 xl k x = k +L k+ x+k +L k x kl k x

4 e xt L k xdx = t k t k As is well-known, the Laguerre Polynomials L x,l x,... form an orthonormal basis in the Hilbert space induced by the weighted inner product f,g = e x fxgxdx. It follows that the Laguerre functions Φ x,φ x,... form an orthonormal basis in the standard L [, sense. By and 7, e xt Φ k xdx =. The Legendre Polynomials t k t+ k. 3 In this subsection we summarize some of the properties of the Normalized Shifted Legendre Polynomials, derived from the standard Legendre Polynomials. We define the Normalized Shifted Legendre Polynomial of degree k =,,..., which we will be denoting by Pk, by the formula P k x = P kx k +; 4 where P k is the Legendre Polynomial of degree k; the standard definition of the Legendre Polynomials can be found, inter alia, in []. As is well-known, the Legendre Polynomials form an orthogonal basis in L [,], but they are not normalized; a simple calculation shows that the Normalized Shifted Legendre Polynomials P,P,... form an orthonormal basis in L [,]. By 4, the first Normalized Shifted Legendre Polynomial is a constant, P x =. 5.3 Singular Value Decomposition of integral operators The SVD of integral operators and its key properties are summarized in the following theorem, which can be found, for example, in []. Theorem.. Suppose that the function K : [c,d] [a,b] R is square integrable, and T : L [a,b] L [c,d] is defined by the formula Tfx = b a Kx, tftdt. 6 3

5 Then, there exist two orthonormal sequences of functions u,u,..., where u n : [a,b] R and v,v,..., where v n : [c,d] R, and a sequence α,α,... R, where α α..., such that b Tfx = α n u n tftdt v n x 7 n= for any f L [a,b]. The sequence α,α,... is uniquely determined by K. a The functions u,u,... are referred to as the right singular functions, the functions v,v,... are referred to as the left singular functions, and the values α,α,... are referred to as the singular values of the operator T. Together, the right singular functions, the left singular functions and the singular values are referred to as the SVD of the operator T. It immediately follows from Theorem. that Tu n = α n v n, 8 T v n = α n u n. 9 Observation.. The right singular functions u,u,.. of T are eigenfunctions of the operator T T and the left singular functions v,v,... are eigenfunctions of the operator T T ; the singular values α,α,... of T are the square roots of the eigenvalues of T T and T T. In other words, for every n =,,..., and T Tu n τ = T T v n ξ = d c b a b Kx,τ Kx,tu n tdt dx = αnu n τ a d Kξ, t Kx,tv n xdx dt = αnv n ξ c Remark.3. The function K can be expressed using the singular functions as follows see [], Kx,t = v n xα n u n t n= and it can be approximated by truncation of small singular values also see []: Kx,t p v n xα n u n t 3 n= 4

6 .4 The Truncated Laplace Transform Definition.4. For any pair of real numbers a,b, such that < a < b <, the Truncated Laplace Transform L a,b is the linear mapping L [a,b] L [,, defined by the formula L a,b fω = b Obviously, the adjoint of L a,b is L a,b gt = a e tω ftdt, 4 e tω gωdω. 5 The operators L a,b and L a,b are compact, the range of L a,b is dense in L [a,b] and the range of L a,b is dense in L [, see, for example, [3]..5 The SVD of the Truncated Laplace Transform By Theorem., there exist an orthonormal sequence of right singular functions u,u,... L [a,b], an orthonormal sequence of left singular functions v,v,... L [, and a sequence of real numbers α,α,... R such that and for all n =,,..., L a,b fω = b α n n= a u n tftdt v n ω, 6 L a,b u n = α n v n, 7 and L a,b v n = α n u n, 8 α n α n+. 9 Remark.5. The multiplicity of the singular values of L a,b is one see [5]; in other words, for all n =,,... α n > α n+. 3 Remark.6. According to Observation., the right singular functions u,u,... of L a,b are eigenfunctions of the integral operator L a,b L a,b : L [a,b] L [a,b] given by the formula L a,b L a,b ft = e ωt b 5 a b e ωs fsds dω = a t+s fsds,

7 and the corresponding eigenvalues of L a,b L a,b are α,α,..., where α n is the singular value of L a,b associated with the right singular function u n. In other words, L a,b L a,b u n t = b a 3 t+s u nsds = α nu n t. 3 Similarly, the left singular functions v n of L a,b are eigenfunctions of the integral operator L a,b L a,b : L [, L [, given by the formula = b a L a,b L a,b gω = e ωt e ρt gρdρ dt = e aω+ρ e bω+ρ gρdρ, ω +ρ 33 and the corresponding eigenvalues L a,b L a,b are α,α,... In other words, L a,b L a,b v n ω = e aω+ρ e bω+ρ v n ρdρ = α ω +ρ nv n ω The differential operators D t and ˆD ω associated with the singular functions of L a,b In this subsection we summarize several properties related to the differential operator D t, defined by the formula Dt f t = d t a b t ddt dt ft t a ft, 35 where f C [a,b]; and properties related to the differential operator ˆD ω, defined by the formula ˆDω f ω = = ω d d dω dω fω +a +b d ω d dω dω fω + a b ω +a fω, 36 where f C 4 [, L [,. For a derivation of these properties, see [5]. Theorem.7. The differential operator D t, defined in 35, commutes with the integral operator L a,b L a,b, specified in 3 in L [a,b]. In other words, D t L a,b L a,b = L a,b L a,b D t 37 6

8 Theorem.8. The differential operator ˆD ω, defined in 36, commutes with the integral operator L a,b L a,b, specified in 33 in L [,. In other words, L a,b L a,b ˆD ω = ˆD ω L a,b L a,b. 38 Theorem.9. The right singular functions u,u,... defined in 7 of L a,b defined in 4 are also the eigenfunctions of D t. Theorem.. The left singular functions v,v,... defined in 7 of L a,b defined in 4 are also the eigenfunctions of ˆD ω. We denote the eigenvalues of the differential operator D t by χ, χ,..., and the eigenvalues of the differential operator ˆD ω by χ,χ,... By Theorem.9, the singular function u n is the solution to the differential equation d dt t a b t d dt u nt t a u n t = χ n u n t, 39 and by Theorem., the left singular function v n is the solution to the differential equation ω d d dω dω v kω +a +b d ω d dω dω v kω + a b ω +a v k ω = = χ k v kω. Remark.. The singular values α n defined in 7 of the integral operator L a,b are known to decay exponentially as n grows; consequently, the direct numerical computation of the singular functions of L a,b beyond the first few singular functions is impossible. The differential operators D t and ˆD ω are advantageous in the numerical treatment of the singular functions u n and v n because their eigenvalues are well-separated. 4.7 Properties of the right singular functions u n In this section we presents some of the numerical properties of the right singular functions u n, a more detailed discussion of these properties is found in []. We find it convenient to define the function ψ n x on the interval [,] by the formula ψ n x = b a u n a+b ax. 4 We introduce the notation h n = h n,hn,... for the vector of coefficients of the expansion of the function ψ n in the basis of Normalized Shifted Legendre Polynomials defined in 4; where the element h n k is defined by the formula h n k = ψ n xpk xdx, 4 7

9 so that ψ n x = k= h n k P k x. 43 Theorem.. The vector of coefficients h n is the n+-th eigenvector of the five-diagonal matrix M, defined by the formula where M k,k = k k 4 k 3k k+, M k,k = k3 +β k, k+ M k,k = 4 6β kβ+3β+k 7+β+β +k 3 +k 4 7+6β+8β k k+3, M k+,k = k+3 +β k+ k+3, M k+,k = β = a b a. k+ k+ 4 k+k+3 k Analytical apparatus 3. Expansion of v n in the basis of Laguerre functions Suppose that g is a smooth function in L [,. Then, g can be expressed in the basis of Laguerre functions Φ k defined in 7; let η = η,η,... be the a vector where η k = gωφ k ωdω, 46 then clearly η is the vector of coefficients in the expansion, gω = η k Φ k ω. 47 k= We introduce the notation η n = η n,ηn,... for the vector of coefficients of the expansion of the left singular function v k defined in 7 in the basis of Laguerre functions; where the element ηk n is defined by the formula so that ηk n = v n ωφ k ωdω, 48 v n ω = ηk n Φ kω. 49 k= 8

10 3. A matrix representation of the differential operator ˆD ω in the basis of Φ k The purpose of this subsection is to express the differential operator ˆD ω defined in 36 in the basis of Laguerre Functions Φ k as the matrix ˆM described in Lemma 3.. Theorem 3. shows that the matrix ˆM is in fact a five-diagonal matrix; and Corollary 3.3 provides the relationship between the eigenvectors of ˆM and the left singular functions vn defined in 7. Lemma 3.. Let g be a smooth function with the expansion η = η,η,... specified in 47: gω = η k Φ k ω. 5 k= Suppose that ψ = ˆD ω g, with the expansion c = c,c,... such that ψω = c k Φ k ω. 5 Then, c = ˆMη, k= where the matrix elements ˆM jk of ˆM are defined via the formula ˆM jk = Φ j ωˆdω Φ k ωdω, 53 with j,k <. Proof. By the linearity of the differential operator ˆD ω defined in 36, ψω = ˆDω g ω = η k ˆDω Φ k ω. 54 Combining 5 and 54, c k Φ k ω = k= k= k= 5 η k ˆDω Φ k ω. 55 Now, by multiplying both sides of 55 by Φ j and integrating, we have c j = η k ˆDω Φ k ω Φ j ωdω. 56 By linearity, c j = k= η k k= Φ j ωˆdω Φ k ωdω. 57 9

11 Theorem 3.. For any k, ˆDω Φ k ω = 4a 4b k k Φ k ω 6 + k 6a b Φ k ω 4 + kk + 48a b 4a 4b 3 + 6a b +a 4b Φ k ω 8 + k + 6a b Φ k+ ω 4 4a 4b k +k + Φ k+ ω, 6 58 where Φ k is the Laguerre function defined in 7. In other words, ˆM is the symmetric five-diagonal matrix with non-zero entries defined by the formulae 4a ˆM 4b k k k,k =, 6 ˆM k,k = k 6a b, 4 ˆM k,k = kk + 48a b 4a 4b 3 + 6a b +a 4b, 8 ˆM k+,k = k + 6a b, 4 4a ˆM 4b k +k + k+,k = Proof. By the definition of ˆD ω in 36, ˆDω Φ k x = = d d dωωdω Φ kω+a +b d d dω ω dω Φ kx+ a b ω +a Φ k ω A somewhat tedious derivation from 6, using identities 8, 9 and, yields 58. 6

12 Corollary 3.3. Suppose that η n = η n,ηn,... is the vector of coefficients defined in 48, in the expansion of the left singular function v n defined in 7; then, η n is the n+-th eigenvector of ˆM : ˆMη n = χ nη n, 6 where ˆM is the five-diagonal matrix 59, χ n are the eigenvalues of the differential operator ˆD ω, and k =,,,... Proof. By 4, v n is an eigenfunction of ˆD ω, with the eigenvalue χ n: ˆDω v k ω = χ k v kω, 6 so that ˆD ω ηk n Φ k ω = k= ˆDω v k ω = χ k ηk n Φ kω. 63 k= Therefore, by Lemma 3. we obtain 3.3. Remark 3.4. If the operator ˆD ω is represented in the basis associated with Hermite polynomials, rather than in the basis of Laguerre functions, the resulting matrix ˆM in Lemma 3. and Theorem 3. is seven-diagonal and of certain analytical advantage. This approach is under investigation and will be reported at a later date. 3.. The special cases L /,γ/ and L /γ,/ An inspection of formula 59 immediately indicates that there are two special cases in which the second off-diagonal of the matrix ˆM defined in 59 vanishes, and ˆM becomes a tridiagonal band matrix. The first case occurs when a = /. The substitution of a = /,b = γ/ into 59 yields ˆM k,k = 4 γ k, ˆM k,k = 4 γ γ + k γ + k +, 64 ˆM k+,k = 4 γ k +, and zero in all other elements of ˆM. In other words, the differential operator associated with the Truncated Laplace Transform L /,γ/ via 38 is a tridiagonal band matrix in the basis of Laguerre Functions.

13 The second case occurs when b = /. The substitution of a = γ,b = / into 59 yields γ ˆM k k,k = 4γ, γ ˆM + k + γ + k +γ k,k = 4γ, γ ˆM k + k+,k = 4γ, 65 and zero in all other elements of ˆM. Again, the differential operator associated with the Truncated Laplace Transform L /γ,/ via 38 is a tridiagonal band matrix in the basis of Laguerre Functions. 3.. The special case: L γ, γ An additional special case occurs when ab = /, i.e. in the case of the Truncated Laplace Transform L γ, γ. Substituting a = γ,b = γ into 59 eliminates the first off-diagonal of ˆM and yields the matrix ˆM s : ˆM s k,k =γ k k, 6γ ˆM s γ k,k = 6γ kk + γ γ +3, 8γ k+,k =γ k +k +, 6γ ˆM s 66 with zero in all other elements of ˆMs. 3.3 The standard form of the Truncated Laplace Transform As shown in [3], the behavior of the Truncated Laplace Transform L a,b is determined by the ratio γ = b/a. 67 The following lemma summarizes the connections between the SVD of the Truncated Laplace Transform L a,b and the SVD of the Truncated Laplace Transform L ; we will refer to the latter as the standard form of the Truncated Laplace Transform. γ, γ Lemma 3.5. Suppose that < a < b <, and that u n, v n and α n are the n + -th right singular function, left singular function and singular value of L a,b, respectively. Suppose that

14 ũ n, ṽ n and α n are the n+-th right singular function, left singular function and singular value of the standard form L, respectively, with γ = b/a. and Then, γ, γ u n t = γ a ũnt/ γa, 68 v n ω = γ a ṽ n ωa γ, 69 α n = α n. 7 Proof. The identities are readily obtained by the change of variables t = t/a γ in 3, the change of variables ω = a γω in 34, and normalization of the singular functions. We denote the differential operator associated with L γ, γ differential operator is specified by the formula ˆDω f ω = = ω d d dω dω fω + γ + 4γ d dω via Theorem.8 by ˆDω ; this ω d dω fω + 6 ω + fω, γ 7 which is a special case of formula 36. The following lemma specifies the relation between the eigenvalues of ˆD ω associated with L γ, γ and the eigenvalues of ˆDω defined in 36 associated with L a,b. Lemma 3.6. Suppose that χ n is the n+-th eigenvalue of the differential operator ˆD ω associated with L a,b, and that χ n is the n+-th eigenvalue of the differential operator ˆDω associated with L, where γ = b/a. Then, γ, γ χ n = 4γa χ n = 4ab χ n. 7 Proof. The relation is obtained from 36 by the change of variables ω = a γω. 3.4 The operator C γ [ In this discussion, we find it useful to introduce the transform C γ : L ] γ, γ L [,]; we define C γ of a function f by the formula C γ fs = γ s/4 f γ s/ /. 73 3

15 Remark 3.7. A simple calculation shows that C γ fs C γ gsds = 4 logγ γ γ ftgtdt. 74 The following lemma provides an expression for the function C γ L γ, γ Φ k the Laguerre function defined in 7. Lemma 3.8. For any < γ <, C γ L Φ k γ, γ Furthermore, the function C γ L γ, γ Φ k C γ L Φ k s = C k γ γ, γ, with Φ k s = γ s/4 k k. γ s/ γ + s/ 75 is even or odd, depending on the value of k: L γ, γ Φ k s. 76 Proof. The identity 75 is obtained by substituting 3 into 73. The symmetry property 76 follows immediately from 75. Corollary 3.9. For < γ <, the functions C γ L γ, γ Φ k s decay exponentially as k grows, in the following sense C γ L Φ k s γ s/ k γ s/ = + k γ s/ γ, γ Furthermore, if s, then, C γ L Φ k s γ / + Proof. By 75, C γ γ, γ L γ, γ Φ k s = γs/4 γ s/ γ s/ + γ s/ + k. 78 k, 79 so that for all < γ < and s R, γ s/ γ s/ <. 8 + The inequality 77 follows immediately from 79 and 8. Equation 78 follows immediately from 77. 4

16 3.4. The function C γ u n We introduce the notation U n for the C γ of the n+ right singular function of L defined by the formula γ, γ ; U n is U n s = C γ u n s. 8 where u n is the n+-th right singular function of the operator L γ, γ and C γ is defined in 73. The following lemma summarizes some of the observations found in [5] and reformulates them using the notation used in this paper. Lemma 3.. The function U n defined in 8 is an eigenfunction of the differential operator D s, defined by the formula Ds f s =log γ d γ + γcoshslog γ d ds 3 γcoshslog γ+ 4 γ 7 4 fs. ds fs Since the differential operator Ds is symmetric around, the function U n is even or odd in the sense that 8 U n s = n U n s Decay of the coefficients Since the left singular function v n defined in 7 is a smooth solutions of a differential equation specified in 4, we expect the coefficients ηk n defined in 48 in the expansion of v n to decay rapidly. In this section we provide an estimate for the actual decay. Lemma 3.. Suppose that v n is the n+-th left singular function of the operator L γ, γ defined in 4; then η n k α n logγ + γ k. 84 where η n k is defined in 48 and α n is the n+-th singular value of L Proof. By 48 and 7, η n k = α n b a γ, γ e ωt u n tdt Φ k ωdω

17 Changing the order of integration and using 5, η n k = α n γ γ A simple calculation using 74 and 8 shows that η n k = α n By the Cauchy-Schwarz inequality, η n k α n u n t L γ, γ Φ k tdt. 86 U n s C γ L γ, γ Φ k sds. 87 U n s ds C γ L γ, γ Φ k sds. 88 Now, since the right singular function u n is normalized, and using 74 and 8, η n k α n Finally, using inequality 78, η n k α n logγ logγ + γ C γ L γ, γ Φ k sds. 89 k. 9 Remark 3.. Lemma 3. can be generalized to the following bound for the coefficients of the expansion of the left singular functions of the Truncated Laplace Transform L a,b in the nonstandard form; the proof is found in our report []. The coefficients ηk n in the expansion of the left singular function v n of L a,b are bounded in the sense that η n k α n logγ γ smax/ γ smax/ + k, 9 where s max = max loga logγ, logb logγ. 9 6

18 3.6 Integral of the function u n The following lemma describes the relation between the integral b a u ntdt, where u n is the right singular function defined in 7, and the expansion specified in 43. The expansion 43 is associated with the right singular function u n via 4 and 4 and it is studied in []. Lemma 3.3. Suppose that u n be the n+-th right singular function of L a,b, then b a u n tdt = b ah n. 93 where h n is the first coefficient in the expansion 4 of the function ψ n defined in 43 in the basis of Legendre Polynomials. Proof. By 4, h n = Substituting 5 into 94 yields h n = ψ n xp xdx, 94 ψ n xdx. 95 Now, substituting 4 into 94 with the change of variable t = a+b ax yield A relation between u n, v n, and the singular value α n The following lemma provides the relation between the coefficient h n function v n, and the singular value α n. see 4, the left singular Theorem 3.4. Suppose that u n, v n and α n are the n + -th right singular function, left singular function and singular value of L a,b and suppose that h n is the first coefficient in the expansion defined in 43. Then, α n = b a hn v n 96 Proof. Evaluating both side of 7 at yields α n v n = L a,b u n = α n v n = Substituting 93 into 97, b a u n tdt. 97 α n v n = b a h n. 98 7

19 4 Algorithms 4. Computing the left singular function v n In this section we introduce an algorithm for the numerical evaluation of v n ω, the n+-th left singular function defined in 7 of L a,b the operator defined in 4. By Theorem., the function v n ω is an eigenfunction of the differential operator ˆD ω defined in 36; by Corollary 3.3, the expansion η n = η n,ηn,... defined in 49 of v n in the basis of Laguerre functions is an eigenvector of the matrix ˆM defined in 59. Based on Lemma 3.5 and Lemma 3.6, it is sufficient to compute ṽ n ω, the n + -th left singular function of the Truncated Laplace Transform in the standard form L, where γ, γ γ = b/a; v n t is computed from ṽ n t using 69. In this special case of the Truncated Laplace Transform, the expansion η n = η n,ηn,... of ṽ n ω is an eigenvector of the matrix ˆM s specified in 66. Since the matrix ˆM s is five-diagonal, and is non-zero only in even rows of even columns, and on odd rows of odd columns, an eigenvector of ˆMs must vanish at either the even or odd positions. Therefore, for any even n = m where m, equation 49 is reformulated as ṽ m ω = where η even,m = η even,m for all integer k : k=,η even,m η even,m k Φ k ω, 99,... is simply the even numbered elements of the vector η m ; η m k = ηeven,m k. In other words, η even,m is the m+-th eigenvector of the tridiagonal matrix ˆM s,even, obtained by removing the odd numbered rows and columns of ˆMs ; the non-zero elements of ˆMs,even are specified by the formula ˆM s,even k,k ˆM s,even = γ k k, 8γ k,k = γ 6γ kk + γ γ +3, 8γ k+,k = γ k +k +. 8γ ˆM s,even Similarly, for any odd n = m+ where m, equation 49 is reformulated as ṽ m+ ω = k= η odd,m k Φ k+ ω, 8

20 where η odd,m = η odd,m for all integer k :,η odd,m,... is simply the odd numbered elements of the vector η m+ ; η m+ k+ = ηodd,m k. 3 In other words, η odd,m is the m+-th eigenvector of the tridiagonal matrix ˆM s,odd, obtained by removing the even numbered rows and columns of ˆMs ; the non-zero elements of ˆMs,odd are specified by the formula ˆM s,odd k,k =γ kk +, 8γ ˆM s,odd k,k = γ 6γ k +k + γ γ +3, 8γ k+,k =γ k +k +3. 8γ ˆM s,odd 4 Therefore, the algorithm for computing the left singular function v n of the Truncated Laplace Transform L a,b, where n = m is an even number: Step : Compute η even,n, the n+-th eigenvector of the matrix ˆM s,even, defined in. Step : Compute the function ṽ n from η even,n, using the expansion specified in 99. Step 3: Obtain v n from ṽ n using 4. Similarly, the algorithm for computing the left singular function v n of the Truncated Laplace Transform L a,b, where n = m+ is an odd number: Step : Compute η odd,n, the n+-th eigenvector of the matrix ˆM s,odd, defined in 4. Step : Compute the function ṽ n from η odd,n, using the expansion specified in. Step 3: Obtain v n from ṽ n using 4. Remark 4.. For computations to precision ǫ, the vector η n = η n,ηn,... is truncated at K, such that ηk n ǫ for all k > K. By lemma 3. the coefficients ηn k decay rapidly as k grows; since the vectors η even,n and η odd,n, which contain only the even and odd elements of ηk n, respectively, these vectors are truncated at approximately K/ elements. Theactualpositionofthelastsignificantcoefficientofη even,n andη odd,n, largerinmagnitude than ǫ, is given in Figure 6 and Table 4 in Section 5, for several combinations of γ and n. 4. Computing the singular value α n Theorem 3.4 provides formula 96 for computing the singular value α n using the function v n and the first coefficient of the vector h n the vector associated with the right singular function u n via 4 and 4. The function v n is computed using the algorithm in Section 4., and the vector h n is the n+-th eigenvector of the matrix M defined in 44, see []. 9

21 Remark 4.. The numerical difficulty in this algorithm is that the first element h n of the vector h n see 4 is of the order of α n, so that is decays exponentially as n grows. It has been shown in [4] that in some band matrices, such as M, the first element of the eigenvector h n can be computed to relative precision, not not just to absolute precision. Consequently, if h n is computed as in [4], formula 96 yields α n with high relative precision. 5 Numerical results In this section we present results of several numerical experiments. The algorithms for computing the left singular functions v n and singular values α n of L a,b the operator defined in 4 were implemented in FORTRAN 77, using double precision arithmetic, and compiled using GFORTRAN. In Figures, and 3 we present examples of left singular functions of the operator L a,b, where a = and b =., b = and b = respectively. In Figure 4 and Table we present the singular values α n of the operator L a,b, for several ratios γ = b/a; α n depends only on γ and n see Lemma 3.5. In table we present several singular values smaller than ; the Fujitsu compiler with quadruple precision was used in this experiment. In Figure 5 and Table 3 we present the eigenvalues of the matrix ˆM s defined in 66. In Figure 6 and Table 4 we present for several combinations of γ and n the position of the last significant coefficient η n in the expansion defined in 49, that is larger in magnitude than ǫ = 6. In numerical computations, the vectors η even,m defined in and η odd,m defined in 3 are truncated at about half this number see Remark 4.. In figure 7 we present the CPU time required for the computation of the expansion of the -st right singular function v of L,γ, for varying γ; The experiment was performed on a ThinkPad X3 laptop with Intel Core i7-35 CPU and 6GB RAM n= n= n= 5 n=.5 n= n= n= 5 n=. v n ω v n ω ω ω Figure : Left singular functions of L,.. Figure : Left singular functions of L,.

22 5 v n ω α n 4 γ=. γ= γ= γ= γ= γ= γ= ω Figure 3: Left singular functions of L, n Figure 4: Singular values α n of L a,b, with γ = b/a. 5 8 χ n * γ=. γ= 5 γ= γ= γ= γ= γ= 3 n # of coefficients 6 4 γ=. γ= γ= γ= γ= γ= γ= γ= n Figure 5: Magnitude of the eigenvalues of the matrix ˆM s defined in 66. Figure 6: The position of the last significant coefficient larger in magnitude than 6.

23 CPU time s γ Figure 7: CPU time required for computing the expansion of the -st left singular function of L,γ, as a function of γ. The experiment was performed on a ThinkPad X3 laptop with an Intel Core i7-35 CPU and 6GB RAM. Table : Singular values α n of L a,b n γ =.E+ γ =.E+4 γ =.E+7 γ =.E+.356E E +.673E E E.88E +.437E E E E.487E E E E 8.95E.664E E E E E.9476E E.83E.96456E 3.85E.77967E E E E 4.694E E E E E.45E E E E E E E E 37.49E E E E E.384E E E E E E E E E E E.585E E E 88

24 Table : Examples of singular values α n smaller than γ n α n.e E.E E.E E.E E.E E.E E Table 3: Eigenvalues of ˆMs defined in 66 n γ =.E+ γ =.E+4 γ =.E+7 γ =.E+.378E E E E E +.439E E E + 8.7E +.394E E E E E E E E E E E + 9.E E E E E E E E E E E E +.848E + 4.4E E E E E E E E E E E E E E E +.37E E E E E E +.37E E E E E E E + 3 3

25 Table 4: The position of the last significant coefficient larger in magnitude than 6. n γ =.E+ γ =.E+4 γ =.E+7 γ =.E

26 6 Conclusions and generalizations In this paper we introduced effective algorithms for the evaluation of the left singular functions and singular values of the Truncated Laplace Transform L a,b. Together with the algorithms introduced in [] for the computation of the right singular functions, these algorithms conclude the construction of the SVD of the operators L a,b. As is evident from Remark.3 and the more detailed discussion in [], the left singular functions of L a,b are an efficient basis for representing combinations of decaying exponentials whose decay constants are in the interval [a, b]. References [] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs and Mathematical Tables, National Bureau of Standards, New York, 966. [] M. Bertero, P. Boccacci,, and E. Pike, On the recovery and resolution of exponential relaxation rates from experimental data. II. the optimum choice of experimental sampling points for Laplace transform inversion, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., , pp [3] M. Bertero, P. Boccacci, and E. R. Pike, On the recovery and resolution of exponential relaxation rates from experimental data: A singular-value analysis of the Laplace transform inversion in the presence of noise, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., , pp [4] M. Bertero, P. Brianzi, and E. R. Pike, On the recovery and resolution of exponential relaxation rates from experimental data. III. the effect of sampling and truncation of data on the Laplace transform inversion, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., , pp [5] M. Bertero and F. A. Grunbaum, Commuting differential operators for the finite Laplace transform, Inverse Problems, 985, pp [6] M. Bertero, F. A. Grunbaum, and L. Rebolia, Spectral properties of a differential operator related to the inversion of the finite Laplace transform, Inverse Problems, 986, pp [7] M. Bertero and E. Pike, Exponential-sampling method for Laplace and other dilationally invariant transforms: II. examples in photon correlation spectroscopy and fraunhofer diffraction, Inverse Problems, 7 99, pp. 4. [8] M. Bertero and E. R. Pike, Exponential-sampling method for Laplace and other dilationally invariant transforms: I. singular-system analysis, Inverse Problems, 7 99, pp.. 5

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