ELSEVIER Applied Numerical Mathematics 25 (1997) 443-459 MATHEMATICS

Numerical procedures based on Runge-Kutta methods for solving isospectral flows

L. Lopez a,*, T. Politib a Dipartimento di Matematica, Universitgt di Bari, Via E. Orabona 4, 1-70125 Bari, Italy b Dipartimento di Matematica, Politecnico di Bari, Via E. Orabona 4, 1-70125 Bari, Italy

Abstract

This paper deals with the numerical solution of the Lax system L' = [B(L), L], L(0) = L0 (*), where L0 is a constant symmetric matrix, B(.) maps symmetric matrices into skew-symmetric matrices, and [B(L), L] is the commutator of B ( L ) and L. Here two different procedures, based on the approach recently proposed by Calvo, Iserles and Zanna (the MGLRK methods), are suggested. Such an approach is a computational form for the Flaschka formulation of (*). Our numerical procedures consist in solving (*) by a Runge-Kutta method, then, a single step of a Gauss-Legendre Runge-Kutta (GLRK) method may be applied to the Flaschka formulation of (.). In the first procedure we compute the approximation of the Lax system by a continuous explicit RK method, instead, the second procedure computes the approximation of the Lax system by a GLRK method (the same method used for the Flaschka system). The computational costs have been derived and compared with the ones of the MGLRK methods. Finally, several numerical tests and computational comparisons will be shown. © 1997 Elsevier Science B.V.

1. Introduction

In this paper, for simplicity, we will use the following notations for three special subsets in R~x~:

s(~) : { x ~ R~xn I X : X~} ,

7~(~) = { X e R~Xn I X : - - s T } ,

u(~) = { x ~ ~ x n I X ~ X : I } .

Recently a certain effort has been devoted to the numerical solution of isospectral flows (see [2,3,7,12,14]). The usual form of an isospectral flow is the differential system:

L ' = [ B ( L ) , L ] , L ( O ) - - Lo, t ~ O, (1.1)

* Corresponding author. E-mail: lopezl@pascal.dm.uniba.it. Work supported, in part, by Italian C.N.R. Grant No. 96.00240.CT01 and Strategic Project "Modelli e Metodi per la Matematica e l'Ingegneria".

0168-9274/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S0168-9274(97)00051-2

444 L. Lopez, T. Politi / Applied Numerical Mathematics 25 (1997) 443-459

where Lo is a constant matrix in S(n) , B(.) maps S(n) into 7-/(n), and [., .] is the standard Lie-bracket commutator:

[B(L),L] = B ( L ) L - LB(L) . (1.2)

The solution L(t) E S(n) for t ~> 0, and the flow (1.1) is isospectral, that is, the matrix L(t) preserves the eigenvalues for every t (see [9,13,15]).

Special cases are the well-known Toda and double-bracket flows. In the first case we have

B(L) = L+ - (L+) T, (1.3)

where L+ is the strictly upper triangular part of L E S(n). When t --+ +oc, the flow L(t) tends to a diagonal matrix L(cc), thus the dynamics of (1.1) is strictly related to the QR algorithm for finding the eigenvalues of L0. In fact, the sequence defined by the solution of (1.1) at integer times (i.e., L(1), L ( 2 ) , . . . , L(k) , . . . ) is equivalent to the sequence obtained by the OR algorithm applied to exp(L0) (see [14]).

In the double-bracket flow, inlroduced by Brockett in [1], we have

B(L) = [D, L], (1.4)

where D is a constant symmetric matrix. When D is a diagonal matrix with distinct eigenvalues, the solution L(t) tends to a diagonal matrix L(zx~). If D = diag(n, n - 1 , . . . , 1) then the double-bracket flow becomes the Toda flow (1.3) (see [5]). The previous two flows are particular cases of the more general scaled Toda-like flow in which B(L) = T o L, where o denotes the Hadamard product on matrices and T is a skew-symmetric matrix characterizing the flow (see [5]).

In [2] Calvo et al. have shown that the classical numerical methods gives unsatisfactory results and that the isospectrality of L(t), generally, is lost. Furthermore, they have proposed a new approach, based on the Flaschka formulation, and a class of numerical procedures called modified Gauss-- Legendre Runge-Kutta methods (MGLRKs, the integer s denotes the stages of GLRK). In [3] the same authors proposed a cheaper but not symmetry preserving method and discussed the effect of symmetry-breaking on the solution.

In the Flaschka formulation of (1.1), the solution L(t) may be written as

n(t) = U(t)LoUT(t), t >~ 0, (1.5)

where U(t) is the solution of the unitary differential system

U' = B(L)U, U(O) = I, t >~ O, (1.6)

and U(t) E bl(n) for t /> 0 (see [7,9,11,13]). For convenience, we assume that a fixed time step h > 0 is used, so that tk = kh. Then the

MGLRKs method consists of a numerical approximation of (1.5) by

Lk+l -: Vk+ILkVT+I , (1.7) where Lk+l ~ L(tk+l) and Vk+l is a unitary approximation of U(t) at tk+l, obtained by a GLRK method applied to

U'(t) = B(L(t ) )U(t) , V(tk) = I, t e [tk,tk+l], k >>. O. (1.8)

There are some numerical problems associated with this procedure, such as its high computational cost and the computation of L(t) at the internal Gauss points.

L. Lopez, T. Politi / Applied Numerical Mathematics 25 (1997) 443-459 445

The methods studied in this paper derive directly from MGLRKs and our major contribution deals with how to compute L(t) at the internal points of the GLRK method for (1.8). These procedures are an isospectral correction of the numerical solution of (1.1) obtained by RK methods. Both numerical procedures consist of solving (1.1), on [tk, tk+l], by an RK method, then, a single step of a GLRK method may be applied to the Flaschka system (1.8). Thus the unitary solution Vk+l of (1.8) may be used for converting Lk+l tO a symmetric isospectral solution by (1.7). Since the unitary system (1.8) is solved only once, our numerical procedures are, generally, cheaper than the iterative MGLRKs methods.

In more detail, we compute the approximation of the Lax equation (1.1), on [tk, tk+l], by continuous explicit RK methods (CERK) or by GLRK methods. The first procedure, that is the CERK method for (1.1) in tandem with GLRK for (1.8), seems to be advantageous when the solution is required with lower precision and cost. Instead, the second procedure computes the approximation of the Lax system (1.1) by a GLRK method, the same scheme used for the Flaschka system. This method provides an arbitrarily high order of accuracy and seems to be useful when the solution is required accurately on the whole time interval.

The paper is organized as follows: in Section 2 we describe the MGLRKs methods in more detail; in Section 3 we show the first procedure and some theoretical results; in Section 4 we introduce the second procedure and propose some numerical techniques to solve the nonlinear systems arising from the application of GLRK methods to (1.1); in Section 5 the computational costs of our procedures is compared with the one of MGLRKs; in Section 6 we will report several numerical results and comparisons with MGLRKs.

2. The MGLRKs methods

Here we briefly summarize the MGLRKs methods proposed in [2]. Let us consider the RK method defined by the Butcher tableau

c A

b v

with e = ( c l , . . . , cs) T, b ---- ( b l , . . . , bs) T, A = (aij). Let M be the following matrix:

M ---- (blalm + bmaml -- blb,,~)l~,,~.

We now recall a result stated in [8] and improved in [2].

(2.1)

Theorem 1. Let us consider the unitary system U' : F(U)U, U(O) = L with F:]~ n×n ~ 7-[(n). Let Vk be the numerical solution given by an RK scheme. I f the RK method is such that M : O, then Vk C bl(n) for each k.

All s-stage Gauss-Legendre RK methods (GLRKs) have M ---- 0, while no explicit RK method has M = 0 (see [6]). The choice of GLRKs for solving unitary differential systems has been motivated both in [2] and [8]. However, an alternative approach, based on projective techniques, exists (see [8,10]).

446 L. Lopez, T. Polit i / Appl ied Numer ica l Mathemat ics 25 (1997) 4 4 3 - 4 5 9

The MGLRKs method consists of solving (1.8) by GLRKs, that is

8

Vk+ 1 : I + h E biB(Lkt)Vkl, (2.2a) l = l

Vkz = I + h ~-~ a l jB(Lk j )Vk j , 1 = 1 , . . . , s , (2.2b) j = l

and by the correction

Lk+l = Vk+~ LkV~+ 1 (2.3)

which provides the numerical solution of the Lax equation at tk+l and preserves the eigenvalues of the initial matrix Lk.

The set (2.2) represents a nonlinear system which may be solved by the simple iteration

8

-- t~ZjJ-'i~kj JVkj , l = 1, s, (2.4a) "k l " ' " j = l

vk - i + h +1), (2.4b) + 1 - -

/=1

L(m+l) ~T(m+l) L Tr(m+l)T k+l ---- vk+ 1 kvk+ 1 , (2.4C)

r,(m) for j = 1 , . . , s , are unknown matrices. Each iteration of (2.4) corresponds for m ~> 0, where ~kj , to the numerical solution, by GLRK, of the linear differential system

V'(t) : B ( L ( + ] ( t ) ) V ( t ) , V( tk ) : I , t E [tk,tk+,], (2.5)

(m) where Lk+ 1 (t) denotes, for each m ~> O, an approximation of L(t) on [tk, tk+l ] . r(m) The MGLRKs methods compute the matrices ~kj , at the internal points, by means of Hermite

interpolation of L(t) at tk and tk+l. Specifically, the methods set

L °) 1 (~) r(~) k+lJ : "k , r = 0 , . . . , 8 - 1,

where the upper index r denotes the rth derivative of the considered matrix function. Hermite interpo- (0) lation provides Lk+ 1 (t) using the values L~ r) and Wk+ljrr(0) ](r) for r = 0, . . . , s-- 1. Then, the approximate

problem

V'( t ) = B ( L (°) ( t ) )V( t ) , V( tk ) = I , k k + l

r(m) may be solved, in order to have ~k+l, r(1) and we may restart until the convergence of ~'k+l" However, the main problem of the MGLRKs methods seems to consist in their high computational

cost: although the approximate differential system (2.4) is linear, a certain number of iterations may be required to attain convergence. In the following we shall propose two procedures which overcome, in part, these difficulties. These procedures are different from the MGLRKs methods in the way of solving (2.2) and computing the matrices Lkj, j = 1 , . . . , s, at the intemal points.

L. Lopez, T. Politi / Applied Numerical Mathematics 25 (1997) 443-459 447

3. The first procedure

The first procedure may be considered as an isospectral correction of the numerical solution of the linearized system L' = [B(L), L], L(0) = L0, where L(t) is a symmetric approximation of L(t ) given by a continuous explicit RK scheme (CERK).

We compute the matrices L k j , j = 1 , . . . , 8, required in (2.2), by means of a CERK method applied tO

L' = [ B ( L ) , L ] , L( tk) = Lk, t e [tk, ta+l], k >~ O. (3.1)

In order to obtain a continuous approximation L(t) of the solution of (3.1), the natural continuous extension of a u-stage explicit Runge-Kutta (ERK) method will be considered (see [16]). Since B(.) is skew-symmetric for symmetric arguments, the matrix solution of V'( t ) = B ( L ( t ) ) V ( t ) , V( tk ) = I, on [tk, tk+l] is unitary (see [2,3]). Then, a single iteration of GLRKs will be applied to the linearized differential V' ( t ) = B ( L ( t ) ) V ( t ) , V( tk ) = I on [tk, he+l], in order to correct the numerical solution of L(t) to an isospectral one.

Let us suppose that the following tableau

C* A*

b*T

defines an explicit RK method, with c* = (c~ , . . . , C*) T, b* = (b~, . . . , b*) T, A* = (ai*j)- Then a CERK method, for solving (3.1), may be defined as follows:

V

-i(tk + O h ) = Lk + hy~b~(O)[B(-&~),-&z], 0 ~ 0 < 1, (3.2a) /=1

with

l--1

-Lkl = Lk + h Z a ~ j [ B ( L k j ) , - L k j ] , l = l , . . . , u . (3.2b) j=l

We suppose that b~(O), j = 1 , . . . ,u, are polynomials of degree d for which b~(0) = 0 and b~(1) = b~, with [±(p + 1)] ~< d ~< p; p is the order of the underlying ERK method and [l(p + 1)] is the integer 2 part of + as l (p 1). As far the accuracy is concerned we have

max IlL(tk + Oh) - ~(tk + 0h)ll = o(hd+~), k >1 O, (3.3) 0~<0~<1

where II f$ stands for every norm on matrices (see [16]). Note that, since Lk is symmetric, the matrices Lkl and the arguments of B(-) in (3.2b) are symmetric

for all I = 1 , . . . , u , and k ~> 0. We have to point out that every RK method has a continuous extension of minimal degree d =

[l(p + 1)] (see [16, Theorem 7]). For instance, the 1-stage RK process of order 1 (Euler method) has

448 L. Lopez, T. Politi / Applied Numerical Mathematics 25 (1997) 443--459

a continuous extension of degree 1 (d = 1) defined by bl(0) = 0, for 0 ~< 0 ~< 1. The well-known 4-stage ERK method (p = 4), given by

1

1 0

1 - 2 2

1 2 1 0 ~

(3.4)

has a continuous extension of degree 2 (d = 2) defined by

b~(O) = 3(2c* - 1)b~0 2 + 2 ( 2 - 2c*)b*O, i = 1 ,2 ,3 ,4 .

Thus, before applying (2.1) to solve (1.8), we compute the approximated solution L(t) of L(t), on the subinterval [tk, tk+l], by means of (3.2). In particular we set

v

-L(tk + c j h ) = Lk + h~-~b~(cj)[B(-Lkz),-Lkt], j = l , . . . , s , l = l

where the values cy, j = 1 , 2 , . . . , s, are the abscissae of GLRK. The numerical methods proposed will be denoted by EGLRK(p, p, d, s) and the following result

holds.

Theorem 2. Let L(t) be sufficiently smooth and B(L) given by (1.4). Let EGLRK(u,p, d, s) be the procedure in which CERK is a continuous extension of degree d of a u-stage ERK method (of order p) in tandem with the s-stage GLRK method. The order of accuracy of EGLRK(u, p, d, s), as h ---* O, is given by q = min{~, 2s} with ~ = max{d + 1,p}.

Proof. Without loss of generality we consider the solutions uT(t) and vT( t ) of the adjoint unitary differential systems

( u T ) ' = -UTB(L), (VT) ' = - V ~ B ( Z ) ,

uT(0) ---= I , t E [0, h],

V ¢(o) = I, t e [0, h],

the second of which is a perturbation of the first, with L(t) given by (3.2). The Alekseev-Grobner lemma yields

h

VT(h) - UT(h) = f eB(T)[(vT) '(T) + vT(T)B(L(T))] dT

0

h

= / ¢B(T)VT(T)[B(L('r)) - B ~ ( T ) ) ] d'r, (3.5)

0

L. Lopez, T. Politi / Applied Numerical Mathematics 25 (1997) 443-459 449

hence, being the fundamental matrix OB(t) unitary for all t, we have

IIVT(h) - UT(h)II ~ h max IIB(-L(t)) - B(L(t))][, (3.6) tE[0,h]

where II" II is the Euclidean norm on matrices. Then from (3.6), using (1.4) and (3.3), we get

V w(h) - U T(h) = O(hd+2). (3.7)

Furthermore, from (3.5), using (1.4), it follows also that

h VT(h) UT(h) j CB('r)vY('r)[D, L('r) - L('r)] aT.

0

Thus exploiting the asymptotic orthogonality conditions (13) given in [16], we obtain that

V T(h) -- U T(h) : O(hP+l) . (3.8)

Then, from (3.7) and (3.8), we have VW(h) - UT(h) = O(h ~+1) and

V(h)-U(h) :o(h +1) with p = max{d + 1,p}. Thus if we approximate (1.5), at t = h, by means of

nl = VILoV~

we commit an error of O(h ~+1) in replacing U(h) by V(h) and an error of O(h 2s+l) in replacing V(h) by V1, that is

L(h) - L1 = O(hq+l),

where q = min{p, 28}. []

The continuous extension of the Euler method in tandem with the implicit midpoint rule furnishes a second order numerical procedure denoted by E1GLRK1. If s ~> 2 the method will be the one in which CERK is the continuous extension of minimal degree of the ERK method of order p = 2s, in tandem with the s-stage GLRK method. These procedures will be denoted by EuGLRKs and, because of Theorem 2, will be of order 28. For instance, E4GLRK2 will be a fourth order method in which CERK is the continuous extension of degree 2 of (3.4); and E7GLRK3 is a sixth order method in which CERK is the continuous extension of degree 3 of a 7-stage ERK method (given in [16]) in tandem with GLRK3.

4. The second procedure

In this section we suggest to derive the matrices L k j , for j = 1 , . . . , s, applying GLRKs to (3.1). The same GLRKs will be used for solving (1.8). This procedure is implicit in the computation of Lkj, j ---- 1 , . . . , 8, but provides an arbitrarily high order of accuracy. More specifically, the application of GLRKs to (3.1) yields the following nonlinear set of equations:

8

= Lk + h l = 1 , . . . ,8, j = l

450 L. Lopez, T. Politi / Applied Numerical Mathematics 25 (1997) 443--459

which furnishes the numerical solution of the Lax system (3.1) at the internal points of [tk, tk+l]. Then (2.2) may be applied in tandem with the correction (2.3). As far as the accuracy is concerned, it is easy to prove that

I l L ( h ) - Llll = o(hZs+l) .

Now we observe that the main computational difficulty is the numerical solution of the previous nonlinear set, which in compact form may be written as

Lk = Mk + h(A ® I )[B(Lk) ,Lk] , (4.1)

with M k = e ® Lk and

e ---- (1, 1 , . . . , 1) T, Lk = ( L k l , . . . , Lks) T,

[B(Lk),Lk] = ([B(Lk, ) ,Lk ,] , . . . , [B(Lks),rks]) T.

Now we examine some different techniques for solving (4.1). The simplest way is the functional iteration

8

L(~ +1) - Lk + h E alj[B(L(7)) (m) - - , Lkj ], 1 = 1 . . . . , s, (4.2a)

j = l

with ~kj r(°) = Lk, for j = 1, . . ., s, which in compact form becomes

L(m+l) I) [B (L(m)), L ('~)] k = M k + h ( A ® k J, m>~0 , (4.2b)

where

L~ m) (L (m) L(sm)) T, z \ kl ' ' ' ' ~

[B(L(m)) ,L~)] (m) (m) L~?)])T. ~ - ( [ B ( Z k l ) , L k l ] [ B { L (m)'~ ~ ' ' '~ \ ks 1'

It is easy to show that:

L e m m a 1. The matrices L (m) for l = 1, s, m >~ 1, given by the functional iteration (4.2a), kl " ' ' ' preserve the symmetry of Lo but not its banded structure.

The following preliminary result, necessary for the convergence of (4.2), may be shown.

L e m m a 2. Consider B(L) given by (1.4) and II II any norm on matrices. Then there exist constants cl and e2 such that

(a) VL 6 S(n): [IB(L)II ~< e, IILll, (b) VL, M E S(n) : lIB(L) - B(M)II <~ ClllZ - MII, (c) VL, M 6 S(n) : II[B(L),L] - [B(M),M]I I <~ c21lL- MII ,

where depends on IILII, IIMII.

Proof. The points (a) and (b) are a trivial consequence of the definition (1.4) of B(L). We now observe that, for every pair of symmetric matrices (L, M) we obtain

L. Lopez, T. Politi /Applied Numerical Mathematics 25 (1997) 443-459 451

[ B ( L ) , L ] - [ B ( M ) , M ]

= B ( L ) L - LB(L) - B ( M ) M + M B ( M )

= B ( L ) L - LB(L) - B ( L ) M + B ( L ) M - B ( M ) M + M B ( M )

= B ( L ) ( L - M) - LB(L) + (B(L) - B ( M ) ) M + M B ( M ) - L B ( M ) + L B ( M )

= B ( L ) ( L - M) + ( B ( L ) - B ( M ) ) M - L ( B ( L ) - B(M)) + ( M - L)B(M) ,

therefore, using (a) and (b), we get

][[B(L),L] - [B(M),M][ I <~ 2c,(llLl[ + ]IMll) I l L - M H

and the thesis follows. []

Now we consider the vector of s square matrices given by W = (W1, . . . , Ws) T and define

IlWll = max IlWjll, (4.3) l<~j~s

then the following result may be derived.

T h e o r e m 3. Let us consider the iterative method (4.2). Under the same hypotheses of Lemma 2, for sufficiently small h, it follows that

IIL~ r~+l) - L~m)ll ~< belIAl[ IIL~ "~) - L~m-1)]], m/> l,

with C suitable constant.

(,~) Proof. In the first part of the theorem we show that, for sufficiently small h, the n o r m s IILkj II are uniformly bounded by a constant depending on IIL01l. Let I1" II be the Frobenius norm on matrices. Since

II[B(L£7)),L£7)311 mc, llL£7)ll by (4.2a) we get

IIL 7 +')II ~< IILk[[ + 2~lh ~_,lagjll[L~7)ll 2, l= 1,...,s, j=l

therefore for m = 0 we obtain

]]L(I/)[I ~< IlLkll + 2e, hllmll~llLkll 2, l = 1,. . . ,s.

Set K = IIL0[l(1 + ~311L011) with c3 = 2clHAII~, then, since IIL0]l = IIL~II for k > 1, we derive

IlL,It ) 1[ < / £ and the proof of the first part of the theorem can proceed by using induction.

Assume ItL(k~ ) [[ < K , for m > 1, then it follows that

I[L(7+')II ~< IILkll + 2e, h ~ I%lltL~7)tf: ~< IILkll + hcsK z, Z = 1,.. . ,s, j=l

L(m+ 1) thus if h ~< 1/(1 + c311Loll) 2, we have k, [I < K, which completes the first part.

4 5 2 L. Lopez, T. Politi / Applied Numerical Mathematics 25 (1997) 443-459

Now, let I1 II be any norm on matrices, then from (4.2b) we derive that

_ = _ r . ( m - 1 ) ] % ~kr'(m+l) L(k m) h(a ® I){ [B(L~ m)), L(k m)] [B(L(m-I)),,.% j j , m / > 1,

hence

B L (m) , L (m) IIL(m+l)-L~m)ll<~hliA®IHIl[ ( k ) k ]-[B(L~m-1)),L(km-1)]ll

= L ( m ) a - hllA®/ll l l t~ ' ,~k. ,, k . , ~<h2~lllmll(ll (m) r(m-i) L( .~) r(m-1)

<~ h2mllAIl(ll (m) r(m-1) (m-l) zkr II + II)IIL m) -Lk I' (4.4) where r denotes the index for which

FB(L(m)~ L(m)] ( m - l ) ( m - l ) , , ~ J - [ B ( L k ) ,Lk ]11=11[ kr ,,L(m)lk~ ~ [B(L~'~-i)),L~7-1)]II • Since L (m) k~ II is bounded by a constant depending on K, by (4.4) it follows that there exists a constant C, depending on Cl, IIAII and ILL011, for which the thesis holds. []

Then, for sufficiently small h, the iterative method (4.2a) converges although a great number of iterations could be needed. To improve the convergence of (4.2a), avoiding the computation of Jacobian matrices, we exploit the particular form of [B(L), L] in deriving the iterative procedure

[ I - hauB(L(7>)] r(m+l)- Lk + h ~ FB(L(m)~ L(m)l (m) (m) ~kl - - alj[ t kj I, kj j--haltLkl B(Lkt ), (4.5) j= l,js&l

for 1 = 1 , . . . , s, which in compact form becomes

[ I - h(AD ® I)B(L~'~))]L(km+I) = Mk + h(ao® I)[B(L~m)),L~ m)] - h(aD ® I)diag(L~m))B(L(m)), (4.6)

where AD = diag(A), Ao = A - AD, and

B ( L (m)) = (B(L~I)),...,B(L(km)))T , diag(L~ m)) = d i a g ( L ( 1 ) , . . . , L~m)).

The following is a cheaper iterative scheme, based on (4.6):

Is - h(AD ® I)B(L~°))]L (re+l)

=- Mk + h(Ao ® I)[B(L~m)),L (m)] - h(AD ® I)diag(L(°))B(L(°)), (4.7)

where the coefficient matrix remains constant with m. Under the same hypotheses of Theorem 3, and for sufficiently small h, we can show that

(re+l) hCIIAoll IIL~ m)-L~m-1)ll, .~/> 1 .

ILk - L(km)H <" 1 - hclNADl] ILL011 Finally, a more expensive iterative method, with a smaller convergence constant, may be derived in the following way:

8

r('~+') h E [B(L~°)~L(m+I)-L(°)B[L (~)~1 l= l . s, m ~ O , ~kl = L k -4- al j ] k j kj \ kj ] J , , • " ,

j = l

L. Lopez, T. Politi / Applied Numerical Mathematics 25 (1997) 443-459 453

which may be written as

r a . ( I". (°)~1 r.(m+') I)diag( L (°)) B (L(m)). [ I - h ( A ® ~ J ' - ' ~ ' k ]J ~ k = L k -- h ( A ® (4.8)

Then, under the same hypotheses of Theorem 3 and for sufficiently small h, we can show that

hCl IIAI[ IIL0[[ r'(m+') - L~m) ll <~ IlL (m)- L~-I) I[ m ~> I.

Given the tolerance c, the iteration may be stopped when we obtain

re+l ) - L~) ] [ ~< e. (4.9)

5. Computational costs

Here, fixed a time level tk, the computational cost, in terms of flops, of the studied procedures will be derived and compared with the one of MGLRKs. For simplicity we neglect lower computational costs such as the cost to multiply a scalar by a matrix and that due to add matrices. Instead, the main computational costs to be considered will be:

• c~(n) the cost for computing the Lie-bracket operator; • ~(sn) the cost for solving n linear systems with the same coefficient matrix of dimension sn and

of the form (2.4a). Let

m = re(k, e) the smallest integer satisfying the stopping criterion (4.9) for the iterative methods MGLRKs or IGLRKs (suppose that m is the same for both the procedures);

• v(s) and o-(s) constants depending on s, with v(1) = 1, ~(2) = 4, v(3) = 7, and or(s) = 2 s - i - l, f o r s / > 1.

Furthermore, note that if L E S(n) and B :S(n) --~ 7-/(n), then

[B(L),L] = B ( L ) L + (B(L)L) T,

thus the flops for computing [B(L), L] are the same of B(L)L. For instance, in the case of Toda and Brockett double-bracket flows, we get, respectively, a(n) = O(2n 3) and a(n) = O(4n 3) flops. Since Lk+l is symmetric, the flops to multiply the matrices in (2.3) (or in (2.4c)) are O(3n3).

Suppose also to use the functional iteration (4.2b) for solving (4.1) in the iterative procedure IGLRKs. Then Table 1 may be derived, where a(s )a(n)+ma(s )a(n) are the flops, in MGLRKs, for computing the sequence of Hermite interpolants at the internal Gauss points; u(s)a(n) are the flops, in Et/GLRKs, for computing the continuous extension at the internal Gauss points; rosa(n) are the

flops, in IGLRKs, for evaluating the sequence r(m) ~kj at the internal Gauss points.

Table 1

Method Flop counts

MGLRKs

EvGLRKs

IGLRKs

o(s)o~(n) + m[o(s)o~(n) q- ~(sn) q- (2s + 3)n 3]

v(s)o~(n) 4- ~(sn) q- (2s + 3)n 3

rosa(n) +/3(sn) + (2s + 3)n 3

454 L. Lopez, T. Politi / Applied Numerical Mathematics 25 (1997) 443-459

Table 2

q m > l m = l

2 E l GLRK 1 MGLRK 1

4 E4GLRK2 IGLRK2

6 E7GLRK3 IGLRK3

Given q, in order to derive the optimal method we need to distinguish the case of m > 1 from that of m = 1. By Table 1, we can derive the results presented in Table 2, from which it follows that MGLRK1 is better than our procedures of the same order only when q = 2 and m = 1. We note that m = 1 means that the iterative procedure considered (i.e., MGLRKs or IGLRKs) needs one iteration to converge, that is Lk is close to the limit constant matrix L(ec). Thus, only against a large time interval, MGLRK1 requires less flops than both E1GLRK1 and IGLRK1. Notice that this kind of cost analysis is enough partial because to evaluate the performance of a numerical method the flop count (or the CPU time) would be related to the global error at a grid point. In the sequel comparisons among the performances of the studied methods applied to several differential problems will be reported.

6. Numerical tests

Here the proposed methods have been applied to the scaled Toda-like flow

L' = IT o L, L] (6.1)

for different choices of the scaling matrix T. All the numerical results have been obtained by Matlab codes implemented on a scalar computer Alpha 200 4/233 with 128 Mb RAM. Comparisons with MGLRKs, in terms of flops against global errors, are derived by using Matlab's f l o p s function. We used the following list of scaled Toda flows (see [5]).

Example 1 (Toda flow). Choose T = [tij] such that

- 1 , i f i > j , t~j = 1, if j > i, (6.2)

0, otherwise,

and

L 0 =

, 0) 0.5 1 .

1 0.5

Thus the flow represents the Toda lattice which describes the motion of 3 particles on a line interacting pairwise with exponential forces (see [2,15]).

L. Lopez, T. Politi / Applied Numerical Mathematics 25 (1997) 443-459 455

Example 2 (Toda flow). Choose T = [tij] given by (6.2) and

/ ' 0.6489 0.3286 -0 .1870 0.4121 -1 .3056 '~ [ 0.3286 2.0112 -0 .4956 3.4960 0.4198

Lo = / - 0 . 1 8 7 0 -0 .4956 2.1534 -0 .1380 -1 .8459 . 0.4121 3.4960 -0 .1380 0.5415 -0 .5440

\ - 1.3056 0.4198 - 1.8459 -0 .5440 - 1.3550

(6.3)

Example 3 (Continuous power method). Choose T -- [tij] such that

- 1 , if j = 1 a n d i > 3 , t i j = 1, if i = 1 and j > 1,

0, otherwise,

and Lo given by (6.3).

Example 4 (Double-bracket flow). Choose T = [tij] such that

t~j = di - dj

with dl = 1, d• = 0.8, d3 = 0.6, d4 = 0.4, d5 = 0.2 and Lo given by (6.3).

Example 5 (Simultaneous iteration). Choose T = [tij] such that

tij = di - dj

with dl = l, de = 0.8, d3 = da = d5 = 0.4 and Lo given by (6.3).

Example 6. Let A the set of integer pairs given by A = { ( i , j ) I 1 <<. j < i <~ n} . Let A the subset of A given by

A = {(2, 1), (4, 1), (3, 2), (5, 2), (5, 4)}.

Choose T = [tij] such that

- 1 , if ( i , j ) C A, tij ----- 1, if (j, i) E A,

0, otherwise,

and L0 given by (6.3).

Table 3 regards estimates, at time t = 2.5, obtained by using different time steps. These results confirm the theoretical order of each method: the global error reduction factors for the second, fourth, and sixth order methods, approach 4, 16 and 64, respectively.

Fig. 1 plots the smallest integer m satisfying the stopping criterion (4.9), when e = 10 -12, re- spectively for the iteration (4.2a) (solid line), (4.7) (dashed line) and (4.8) (dash-dotted line). We can observe that, after a small initial time interval, the iterative procedures require the same m, thus the more effective method for solving the nonlinear system (4.1) seems to be (4.2a).

Fig. 2 shows flop counts, against the global errors at t = 10, of the second order methods E1GLRK1 (dashed line), IGLRK1 (solid line) and MGLRK1 (dotted line), applied to Example 1 for different time steps (i.e., h = l / T , for i = 2, 3,4, 5, 6, 7, 8). While Figs. 3 and 4 regard the numerical performances

456 L. Lopez, T. Politi / Applied Numerical Mathematics 25 (1997) 443-459

'-

0 0 5 10 15 20 25 30

T i m e

Fig. 1.

10 -5

10 -6

10 -7

.,_ 10 -8

10 -9

10 -1 o

10 -11

10 4

-+ . . . . . - I~ . 0. '~.

\ \ \ \®

\ \ \ \ \ ' { \ 'X

" \ \ . . \ X . . \ _ ~ \ " " X . . . . . . X . . . . . . . X . . . . . . . X

Ck ,, ~ , .

"" \ "" "0 \

, , i , i i i I , i ~ , h i , I

10 5 10 6

f lops

, , i , , i ,

10 7

Fig. 2. Second order methods.

L. Lopez, T. Politi / Applied Numerical Mathematics 25 (1997) 443-459 457

10-8

10 -7

10 -8

10 -9

~ 10 -Io

10 -11

10 -12

10 -13

10 -I 4 10 4

\

\ \ \ X

\Q \

\ \ \ );( ,,

\ \ .

X. , q \ - \1~ .

\ x \ x x ' ~ ' X-. • N X

\ \ +X

h i i , J i l l , i i , , , i l l . . . . . . .

10 5 10 6 10 7 f lops

Fig. 3. Fourth order methods.

10 .7

10-8

I 0 -9

10 -1 o

10 -11

10 -12

10 -13

10 -14 10 4

. . . . . . . . i Q

\ \ \ \

\

. . . . . . . . i

10 5

\ X \

\Q\

\ \ \

~ . . . . X . . . . . X . . . . . . . . i . . . . . . . . =

10 6 10 7

flops

Fig. 4. Sixth order methods.

10 8

458 L. Lopez, T. Politi /Applied Numerical Mathematics 25 (1997) 443-459

Table 3 Errors at t = 2.5

h = 1/4 h = l / 8 h = l / 1 6 h = 1 / 3 2 h = 1/64

EIGLRK1 9.2488e-03 1.9969e-03 4.6994e-04 1.1430e-04 2.8137e-05

IGLRK1 1.2840e-03 2.4250e-04 5.1056e-05 1.1652e-05 2.7566e-06

E4GLRK2 8.8868e-05 6.0746e-06 4.1423e-07 2.7304e-08 1.7563e-09

IGLRK2 9.7912e-06 8.5716e-07 6.1485e-08 2.6188e-10 2.7566e-06

E7GLRK3 9.6565e-07 2.6197e-07 6.4577e-09 1.3662e-10 2.5787e-12

IGLRK3 6.3365e-08 2.0050e-09 3.9240e- 11 6.7215e- 13 1.2283e- 14

at t = 6 respectively the fourth order methods E4GLRK2, IGLRK2, MGLRK2 and the sixth order methods E7GLRK3, IGLRK3, MGLRK3. Similar results have been obtained for the methods applied to Examples 2--6.

From these numerical comparisons it follows that Eu GLRKs and IGLRKs perform better than MGLRKs. Note that E u G L R K s seems to require less flops than IGLRKs while IGLRKs seems to give more accurate results than EuGLRKs. The E u G L R K s methods may be modified in order to exploit the convergence of the numerical solution to the limit matrix L(oc): if II (tk+ +cjh)-- (tk q-cjh)II < e, j -- 1 , . . . , s, one could assume the same continuous extension, at the internal Gauss points, for a certain number p of time steps. These new methods will be denoted by M p E u G L R K s and if p is a small integer its the errors are similar to the ones of the underlying E u G L R K s methods. In Fig. 2 we can see the performance of M2E1GLRK1 (see dash-dotted line).

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[12] J.B. Moore, R.E. Mahony and U. Helmke, Numerical gradient algorithms for eigenvalue and singular value calculations, SIAM J. Matrix Anal. Appl. 15 (1994) 881-903.

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