Abstract In this paper, an generalization of a formula proposed by Van Loan in [1978, IEEE Trans. Automat. Control. 23, 395-404] for the computation of multiple integrals of exponential matrices is introduced. In this way, the numerical evaluation of such integrals is reduced to the use of a conventional algorithm to compute matrix exponentials. The formula is applied for evaluating some kinds of integrals that frequentely emerge in a number classical mathematical subjects in the framework of diﬀerential equations, numerical methods and control engineering applications.

AMC (2000): 15A15, 65F30 Key words: multiple integrals, matrix exponential.

1

Introduction

This note deals with the computation of multiple integrals involving matrix exponentials, which frequently emerge in a number of classical mathematical subjects in the framework of diﬀerential equations, numerical methods, control engineering applications, etc. Specifically, integral of the form Rt Rs1 sk−2 R A11 (t−s1 ) ... e A12 eA22 (s1 −s2 ) A23 ....eAkk sk−1 dsk−1 ...ds1 0 0

(1)

0

will be considered, where Aik are di × dk constant matrices, t > 0 and k = 1, 2, ... At glance, the analytical calculation of these integrals seems to be diﬃcult. Nevertheless, in [19] a simple explicit formula in terms of certain exponential matrix was given. In that way, a number of distinctive integrals such as Rt A11 s e A12 eA22 s ds 0

and

Rt Rs1 A11 s1 e A12 eA22 (s1 −s2 ) ds2 ds1 0 0

can easily computed as particular cases of the mentioned formula [19]. However, such formula is restricted to integrals with multiplicity k ≤ 4, which obviously limits its usefulness range. ∗

Instituto de Cibernética, Matemática y Física, Departamento de Matemática Interdisciplinaria, Calle 15, No. 551, e/ C y D, Vedado, La Habana 4, C.P. 10400, Cuba

1

The proposal of this note is generalizing the formula introduced in [19] for any positive value of k. This have been strongly motivated for the need of computing the integrals Rt

eF(t−s) G(s)ds

(2)

0

and

Rt

| (t−s)

eF(t−s) G(s)G| (s)eF

ds,

(3)

0

where F is an r × r constant matrix and G : R → Rr×d a smooth function. Integrals of the type (2) appear as a term in the analytical solution of linear time-depending control system [16], linear ordinary and delay diﬀerential equations [6]. Integrals like (3) are associated to the notions of controllability and observability Grammians of linear control systems [16]. They also appear as the covariance matrix of the solution of linear stochastic diﬀerential equations with time-depending additive noise [1] and, in the context of filtering theory, as the system noise covariance matrix of the extended Kalman filter for continuous-discrete state-space models with additive noise [10]. In addition, they arise in a number of numerical schemes for the integration of ordinary diﬀerential equations that are based on polynomial approximations for the remainder term of the variation of constant formula [9], [8], [7], [3], [4]. The paper has three sections. In the first one, the generalized formula is derived, whereas in the second one the formula is applied to the computation of the integrals (2) and (3). Last section deals with some computational aspects for implementing such formula.

2

Main result

Theorem 1 in [19] provides a simple way to compute single, double and triple integrals of the form B12 (t) ≡

Zt

eA11 (t−u) A12 eA22 u du,

0

B13 (t) ≡

Zt Zu 0

and B14 (t) ≡

Zt Zu Zr 0

0

eA11 (t−u) A12 eA22 (u−r) A23 eA33 r drdu

0

eA11 (t−u) A12 eA22 (u−r) A23 eA33 (r−w) A34 eA44 w dwdrdu,

0

in terms of a single exponential matrix, but not integrals of the form (1) with k ≥ 5. Next theorem overcomes this restriction. Theorem 1 Let d1 , d2 , ..., dn , be positive integers. If the n × n block triangular matrix A =[(Alj )]l,j=1:n is defined by ⎞ ⎛ A11 A12 ... A1n ⎜ 0 A22 ... A2n ⎟ ⎟ ⎜ A =⎜ . ⎟, ⎝ 0 0 . . . .. ⎠ 0

0

0 Ann

where (Alj ), l, j = 1, ..., n are dl × dj matrices such that dl = dj for l = j. Then for t > 0 ⎛ ⎞ B11 (t) B12 (t) ... B1n (t) ⎜ 0 B22 (t) ... B2n (t) ⎟ ⎜ ⎟ eAt = ⎜ .. ⎟ , . . ⎝ 0 . 0 . ⎠ 0

0

2

0 Bnn (t)

with Bll (t) = eAll t , l = 1, ..., n j−l−1 Rt P Rt Rs1 Rsk Blj (t) = M(l,j) (t, s1 )ds1 + ... k=1 0 0

0

(4) P

M(l,i1 ,...,ik ,j) (t, s1 , ..., sk+1 )dsk+1 ...ds1 ,

(5)

0 l

l = 1, ..., n − 1, j = l + 1, ..., n,

where for any multi-index (i1 , ..., ik ) ∈ Nk and vector (s1 , ..., sk ) ∈ Rk the matrices M(i1 ,...,ik ) (s1 , ..., sk ) are defined by ¶ µk−1 Q Ai i (sr −sr+1 ) e rr Air ir+1 eAik ik sk . M(i1 ,...,ik ) (s1 , ..., sk ) = r=1

Proof. This proof shall be based on complete induction techniques. The identities (4-5) hold for n = 1, 2, 3, 4 by Theorem 1 in [19]. Suppose that they also hold for n = m > 4. Then, let’s show that (4-5) hold for any (m + 1) × (m + 1) block triangular matrix A = [(Alj )]l,j=1:m+1 . e = [(A e lj )]i,j=1:2 defined by Let us rewrite the matrix A as the 2 × 2 block triangular matrix A ¶ µ [(Alj )]l,j=1:m [(Al,m+1 )]l=1:m e , A= 0 Am+1,m+1 and let B(t) = [(Blj (t))]l,j=1:m+1 be the (m + 1) × (m + 1) block triangular matrix given by h

B(t) = eAt = eAt . e it is obtained Then, by applying the identities (4-5) with n = 2 to the matrix A h

[(Bl,j (t))]l,j=1:m = eA11 t ,

Am+1,m+1 t

, Bm+1,m+1 (t) = e t R h e 12 eAm+1,m+1 s1 ds1 . [(Bl,m+1 (t))]l=1:m = eA11 (t−s1 ) A

(6) (7) (8)

0

e 11 = [(Alj )]l,j=1:m then (6) and (7) imply that Since A Bll (t) = eAll t , l = 1, ..., m + 1 j−l−1 Rt P Rt Rs1 Rsk Blj (t) = M(l,j) (t, s1 )ds1 + ... k=1 0 0

0

P

(9) M(l,i1 ,...,ik ,j) (t, s1 , ..., sk+1 )dsk+1 ...ds1 ,

(10)

0 l

l = 1, ..., m − 1, j = l + 1, ..., m,

h e 12 = [(Al,m+1 )]l=1:m , the identity (8) On the other hand, since eA11 (t−s1 ) = [(Bl,j (t − s1 ))]l,j=1:m and A implies the following expression for each block (Bl,m+1 (t)), l = 1, ..., m :

Bl,m+1 (t) =

m Rt P

j=l 0

which by (9) and (10) gives

Bl,j (t − s1 )Aj,m+1 eAm+1,m+1 s1 ds1 ,

m Rt t−s Rt R 1 (l,j,m+1) P Bl,m+1 (t) = M(l,m+1) (t, s1 )ds1 + M (t, s1 + s2 , s1 )ds2 ds1 + j=l+1 0

0

m P

j−l−1 R 1Rs2 P Rt t−s

j=l+2 k=1 0

...

0

0

sk+1 R 0

(11)

0

P

M(l,i1 ,...,ik ,j,m+1) (t, s1 + s2 , ..., s1 + sk+2 , s1 )dsk+2 ...ds1 .

l

3

For each k = 0, 1, .. the change of variables u1 = s1 + s2 , u2 = s1 + s3 , ..., uk+1 = s1 + sk+2 , uk+2 = s1 in each of the multiple integrals that appear in (11) yields to m−l Rt P Rt Ru1 uRk Bl,m+1 (t) = M(l,m+1) (t, u1 )du1 + ... k=1 0 0

0

P

M(l,i1 ,...,ik ,j) (t, u1 , ..., uk+1 )duk+1 ...du1 ,

0 l

which combined with (9) and (10) shows that the identities (4) and (5) hold for n = m + 1, and the proof concludes. Now we are able to evaluate the integrals B12 (t), B13 (t), B14 (t) and the integrals B1k (t) =

Rt Rs1 sk−2 R A11 (t−s1 ) ... e A12 eA22 (s1 −s2 ) A23 ....eAkk sk−1 dsk−1 ...ds1 0 0

0

for k ≥ 5 as well, all of them computed by just a single exponential matrix. This is, by applying the theorem above follows that the blocks B12 (t), B13 (t),..., B1k (t) are easily obtained from etA , with ⎡ ⎤ A11 A12 0 0 0 ⎢ 0 ⎥ 0 A22 A23 0 ⎢ ⎥ ⎢ ⎥ .. ⎢ ⎥. . A=⎢ 0 0 0 A33 ⎥ ⎢ ⎥ .. ⎣ 0 . Ak−1,k ⎦ 0 0 0 0 0 0 Akk

3

Application of the generalized formula

This section is devoted to compute integrals of the form (2) and (3) by means of the theorem stated in the previous section. Firstly, suppose that G(s) is a r × d polynomial matrix function of degree p. That is, G(s) =

p P

Gi si ,

i=0

or equivalently,

u1 uR p Rs R i−1 Rs P G(s) = G0 + G1 du + ... (i!Gi )dui dui−1 ...du1 . i=2 0 0

0

0

Then, by Theorem 1 it is easy to check that Rt

eF(t−s) G(s)ds = B1,p+2 (t)

0

where

B(t) = [(Blj (t))]l,j=1:p+2 = eAt , and the block triangular matrix A =[(Alj )]l,j=1:p+2 is given by ⎞ ⎛ F p!Gp (p − 1)!Gp−1 · · · G1 G0 ⎜ 0 0d×d Id ··· 0 0 ⎟ ⎟ ⎜ ⎜ . .. ⎟ . . . ⎜0 0 . . . ⎟ 0d×d ⎟. A =⎜ ⎟ ⎜ .. .. .. .. ⎜. . Id 0 ⎟ . . ⎟ ⎜ ⎝0 0 ··· 0 0d×d Id ⎠ 0 0 ··· 0 0 0d×d 4

(12)

In a similar way, it is obtained that Rt

| (t−s)

eF(t−s) G(s)G| (s)eF

0

where B(t) =eAt and the matrix A in this case ⎛ F H2p ⎜ 0 −F| ⎜ ⎜ ⎜0 0 A=⎜ ⎜ .. .. ⎜. . ⎜ ⎝0 0 0 0 with

Hi =

P

ds = B1,2p+2 (t)B|11 (t),

(13)

is given by ⎞ H2p−1 · · · H1 H0 Id · · · 0 0 ⎟ ⎟ .. .. ⎟ .. | . −F . . ⎟ ⎟, ⎟ .. .. . Id 0 ⎟ . ⎟ ··· 0 −F| Id ⎠ ··· 0 0 −F|

(l!j!)Gl G|j , i = 0, ..., 2p.

l+j=i

In case that G :R → Rr×d be a function with p continuous derivatives it can be approximated by means its truncated Taylor expansion. That is, G(s) ≈

p G P i i s, i=0 i!

where the coeﬃcient Gi is the i-th derivative of G at s = 0. In this case, the above procedure to compute the integrals (12) and (13) for polynomial G will give a convenient way to approximate integrals like (2) and (3). In [11], such approximation for (3) was early considered and an upper bound for it was also given. However, no closed formula in term of a simple exponential matrix was given for this approximation. Such is the main improvement of this paper in comparison with [11]. At this point it worths to remark that such approximations have been successfully applied to the computation of the predictions provided by the Local Linearization filters for non-linear continuous-discrete state space models [12, 14], as well as in [2, 15] for computing other types of multiple integrals involving matrix exponentials. This evidence the practical usefulness of the result achieved in this paper.

4

Computational Aspects

It is obvious that the main computational task on the practical application of Theorem 1 is the use of an appropiate numerical algorithm to compute matrix exponentials. For instance, those based on rational Padé approximations, the Schur decomposition or Krylov subspace methods (see [18] and [17] for excellent reviews on eﬀective methods to compute matrix exponential). The choice of one of them will mainly depends on the size and structure of the matrix A in Theorem 1. In many cases, it is enough to use the algorithms developed in [5], which take advantage of the special structure of the matrix A. For a high dimensional matrix A, Krylov subspace methods are strongly recommended. Nowadays, a number of professional mathematical softwares such as MATLAB 7.0 provide eﬃcient and precise codes for computing matrix exponentials. Therefore, the numerical evaluation of the integrals under consideration can be carried out in an eﬀective, accurate and simple way. In fact, some numerical experiments have been performed for an application of the classical result of Van Loan (i.e. integrals of type (1) for k ≤ 4). Specifically, in [13] were compared diﬀerent numerical algorithms for the evaluation of some integrals of type (1) with k ≤ 4. Those results could be easily extrapolated to our general setting since such comparision was mainly focused on the dimension of the block triangular matrix A. 5

Acknowledgment This work was partially supported by the Research Grant 03-059 RG/MATHS/LA from the Third World Academy of Science.

References [1] L. Arnold, Stochastic Diﬀerential Equations: Theory and Applications, Wiley Interscience Publications, New York, 1974. [2] Carbonell, F., Jimenez, J. C. and Biscay, R. Weak Local Linear discretizations for stochastic diﬀerential equations: convergence and numerical schemes. J. Comput. Appl. Math., 197 (2006) 578-596. [3] S. M. Cox and P. C. Matthews, Exponential time diﬀerencing for stiﬀ systems. J. Comput. Phys, 176 (2002) 430-455. [4] de la Cruz H., Biscay R.J., Carbonell F., Ozaki T., and Jimenez J.C. A higher order Local Linearization method for solving ordinary diﬀerential equations, to appear in Appl. Math. Comput., 2007 (published online August 3, 2006; doi: 10.1016/j.amc.2006.06.096). [5] L. Dieci, A. Papini, Pade approximations for the exponential of a block triangular matrix, Linear Algebra Appl. 308 (2000) 103-202. [6] R. D. Driver, Ordinary and delay diﬀerential equations, Springer-Verlag, 1977. [7] R. A. Friesner, L. S. Tuckerman, B. C. Dornblaser and T. V. Russo, A method for exponential propagation of large systems of stiﬀ nonlinear diﬀerential equatyions. J. Sci. Comput, 4 (1989) 327-354. [8] A. Iserles, Quadrature methods for stiﬀ ordinary diﬀerential systems. Math. Comp, 36 (1981) 171182. [9] R. K. Jain, Some A-stable methods for stiﬀ ordinary diﬀerential equations. Math. Comp, 26 (1972) 71-77. [10] A. H. Jazwinski, Stochastic processes and filtering theory, Academic Press, San Diego, 1970. [11] J. C. Jimenez, P. Valdes, L. M. Rodriguez, J. Riera, and R. J. Biscay, Computing the noise covariance matrix of the Local Linearization scheme for the numerical solution of stochastic diﬀerential equations, Appl. Math. Lett., 11 (1998) 19-23. [12] J.C. Jimenez and T. Ozaki, Linear estimation of continuous-discrete linear state space models with multiplicative noise, Syst. Control Letters, 47 (2002) 91-101. [13] J.C. Jimenez, A simple algebraic expression to evaluate the local linearization schemes for stochastic diﬀerential equations, Appl. Math. Lett., 15 (2002) 775-780. [14] J.C. Jimenez and T. Ozaki, Local Linearization filters for nonlinear continuous-discrete state space models with multiplicative noise. Int. J. Control, 76 (2003) 1159—1170. [15] J.C. Jimenez, L.M. Pedroso, F. Carbonell and V. Hernandez, Local linearization method for numerical integration of delay diﬀerential equations, SIAM J. Numer. Anal., to appear. [16] P. S. Maybeck, Stochastic models, estimation, and control, Academic Press, New York, 1982. [17] C.B. Moler, C.F. Van Loan., Nineteen dubious methods for computing the matrix exponential, twenty-five years later. SIAM Rev. 45, 1 (2003) 3-49 [18] Sidje, R. B., EXPOKIT: software package for computing matrix exponentials, AMC Trans. Math. Software, 1998, 24, 130-156. [19] C. F. Van Loan, Computing integrals involving the matrix exponential, IEEE Transactions on Automatic Control, 23 (1978) 395-404.

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