VARIABLE-STEP VARIABLE-ORDER 3-STAGE HERMITEBIRKHOFF ODE SOLVER OF ORDER 5 TO 15
TRUONG NGUYEN-BA, HEMZA YAGOUB, YI LI, AND REMI VAILLANCOURT To appear in the Canadian Applied Mathematics Quarterly Abstract. Variable-step variable-order 3-stage HermiteBirkho (HB) methods HB(p)3 of order p = 5 to 15 are constructed for solving non-sti dierential equations. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep and RungeKutta type order conditions which are reorganized into linear conuent Vandermonde-type systems of HB type. Fast algorithms are developed for solving these systems in O(p2 ) operations to obtain HB interpolation polynomials in terms of generalized Lagrange basis functions. The stability regions of the HB methods have a remarkably good shape. The order and stepsize of these methods are controlled by four local error estimators. When programmed in C++, HB(p)3 uses less CPU time than DormandPrince DP(8,7)13M in solving costly problems at stringent tolerance. Resume. On construit un solveur dHermiteBirkho (HB) ` 3 tages ` pas et a e a ordre variables, nomm HB(p)3, dordre p = 5 ` 15, pour syst`mes dquations e a e e diffrentielles non raides y = f (x, y), y(x0 ) = y0. En identiant les dveloppee e ments de Taylor tronqus des solutions exacte et numrique on obtient des cone e ditions dordre du type RungeKutta quon rorganise en syst`mes linaires de e e e Vandermonde conuents de type HB quon rsout en O(p2 ) oprations au moyen e e de nouveaux algorithmes rapides qui donnent lieu ` des polynmes dinterpolation a o HB sur une base de fonctions de Lagrange gnralises. La forme des domaines e e e de stabilit absolue des mthodes HB est remarquable. On contrle lordre et le e e o pas au moyen de 4 estimateurs de lerreur locale. Programm en C++, HB(p)3 e rsout des prol`mes coteux ` tolrance serre plus rapidement que Dormand e e u a e e Prince DP(8,7)13M.
1. Introduction There is a large variety of variable step variable order (VSVO) methods designed to solve nonsti and sti systems of rst-order dierential equations (ODEs). This introduction intends to put some perspective among the several approaches and methods. Gear advocated a quasi-constant step size implementation in DIFSUB
1991 Mathematics Subject Classication. Primary: 65L06; Secondary: 65D05, 65D30. Key words and phrases. general linear method for non-sti ODEs, HermiteBirkho method, maximum global error, number of function evaluations, non-sti DETEST problems, conuent Vandermonde-type systems, C++. This work was supported in part by the Natural Sciences ands Engineering Research Council of Canada and the Centre de recherches mathmatiques of the Universit de Montral. e e e
T. NGUYEN-BA, H. YAGOUB, YI LI, AND R. VAILLANCOURT
[19]. This software works with a constant step size until a change of step size is necessary or clearly advantageous. Then a continuous extension is used to get approximations to the solution at previous points in an equally spaced mesh. This was largely because constant mesh spacing is very helpful when solving sti problems. Another possibility is xed leading coecient, which is seen in Petzolds popular code DASSL [28]. Finally, the actual mesh can be chosen by the code as done in Matlabs ode113. This is the equivalent of a PECE Adams formula in contrast with the AdamsMoulton formulas of DIFSUB and DASSL. In this paper a fully variable step size implementation is used with actual mesh chosen by the code equivalent of a PECE Adams formula. A more basic point about the implementation of a method is the choice of the form. The present method uses a Lagrangian form and much of the paper is devoted to computing the coecients eciently. It might be acknowledged that there are pros and cons about the form; such matters are discussed by Gear with the Nordsieck form [27], Krogh with modied divided dierences [23], and Brayton et al. [6] with Lagrangian form. The Lagrangian form has the virtue of simpilicity. It should be pointed out that the cost depends on the order of the formula. Remark 1 in a later section connects the computation of coecients for various forms. Krogh [23] was concerned about the eects of roundo at stringent tolerances, which is an implicit assumption of this paper, and he concluded that divided dierences is a good way to minimize the eect of roundo. Working with this form does involve manipulation of vectors of the length of the number of equations. These manipulations can be largely vectorized nowadays. Sofroniou and Spaletta [34] were even more concerned about this for extrapolation because the solvers might be used to achieve extraordinary accuracies in Mathematica. The code DVDQ [22] and ode113 [32, 2] implement AdamsBashforthMoulton multistep methods in PECE modes. On the other hand DIFSUB and DASSL implement AdamsMoulton formulas. Although these codes predict with Adams Bashforth formulas, they iterate to completion so that, in principle, it does not matter what predictor was used. Extrapolation is another way of achieving high accuracy. Deuhard [13] is the person most responsible for drawing attention to the value of this approach but the codes of [20, Section II.9] are probably the most visible. Indeed, NDSolve of Mathematica is based on those codes. Hairer et al. [20] consider extrapolation to be the best way of solving problems very accurately. That, in fact, is why it is used in Mathematicathey want to provide users with more-or-less whatever accuracy they require. Extrapolation can be viewed as a variable order RungeKuttta scheme. Following an important comparison of Enright and Hull [16] on xed order RungeKutta codes and extrapolation codes, Shampine and Baca [30] argue that a xed order (7,8) pair is more eective than extrapolation at accuracies common in scientic computation, but high order always wins if one asks for enough accuracy. Finally, Taylor series can be very attractive in VSVO implementation. It has been an excellent choice in astronomical calculations [3]. For general problems one can see the work of Corliss and Chang [12]. Lastly, an interpolant [15, 11] for approximating the solution between mesh

VSVO 3-STAGE HB ODE SOLVERS
points is an important matter because two natural continuous extensions spring to mind, depending on the global smoothness desired. It is worth remarking that a principal reason for the Matlab ODE Suite was to provide solvers with an event location facility Indeed, this is the main reason why the Suite does not contain an extrapolation code nor a high order RungeKutta pair. In retrospect, one should acknowledge that Fehlbergs (7,8) pair is the one that drew attention to the value of high order RK pair. General linear methods for solving nonsti systems of rst-order ordinary dierential equations of the form (1) y = f (x, y), y(x0 ) = y0 ,
can be thought of as multistep methods with o-step points or as RungeKutta methods with backstep points. Like multistep methods, they use information prior to the last step and, like RungeKutta methods, they use derivative evaluations at points partway through the current step. The link between the two types of methods is that values at o-step points are obtained by means of predictors which use values at previous points. It has been noted in [10] that general linear methods incorporate function evaluations at o-step points in order to reduce the number of backsteps without lowering the order. In this paper, we construct new 3-stage VSVO general linear methods of order p = 5, 6,. , 15 which use the values yn , yn1 and fnj , j = 0, 1,. , p 4. Since these methods use HermiteBirkho (HB) interpolation polynomials they will be called HB(p)3 methods and the family of such methods will be designated by HB(5-15)3. It was found experimentally that increasing the number of backstep points is more ecient in increasing the accuracy of HB methods than increasing the number of o-step points. It was also found that increased speed is generally achieved by higher order HB methods. The performance of HB(5-15)3 and DP(8,7)13M [29] was compared on several problems often used to test higher order ODE solvers. It was seen that HB(5-15)3 uses lower CPU time in solving costly equations. Other HB methods of order 9, 10 and 11 have been studied in [25]. An ecient HB Obrechko 3-stage 6-step method of order 14 using (d/dx)f (x, y(x)) has been studied in [26]. In Section 2 we introduce a new family of general HB(p)3 methods of order p = 5, 6,. , 15. Order conditions are listed in Section 3. In Section 4 general HB(p)3 are represented in terms of Vandermonde-type systems. In Section 5 symbolic elementary matrices are constructed as functions of the parameters of the methods in view of factoring the coecient matrices of Vandermonde-type systems. In Section 6 a family of particular variable step HB(5-15)3 is dened by xing the o-step points and is constructed in Section 7. Section 8 considers the regions of absolute stability and principal local truncation coecients of constant step HB(5-15)3. Section 9 deals with the step and order control. In Section 10 three criteria are used to compare the numerical performance of the methods considered

in this paper. Appendix A lists the algorithms. Appendix B describes the Matlab programming for Matlab users. 2. General variable step HB(p)3 of order p The following terminology will be useful. An HB(p)3 method is said to be a general variable-step HB method if its backstep and o-step points are variable parameters. If the o-step points are xed, the method is said to be a particular variable-step method. If the stepsize is constant, and hence the backsteps and o-steps are xed parameters, the method is said to be a constant-step method. A general 3-stage HB(p)3 of order p = 5, 6,. , 15 requires the following four formulae to perform the integration step from xn to xn+1 , where, for simplicity, c1 = 0 is used in the summations. (P2 ) A HermiteBirkho polynomial of degree p 2 is used as predictor P2 to obtain yn+c2 to order p 2,

(2) (P3 )

yn+c2 = 20 yn + 21 yn1 + hn+1 a21 fn +

2j fnj

A HermiteBirkho polynomial of degree p 1 is used as predictor P3 to obtain yn+c3 to order p 2,
yn+c3 = 30 yn + 31 yn1 + hn+1 a31 fn+c1 + a32 fn+c2 +

3j fnj

A HermiteBirkho polynomial of degree p is used as integration formula IF to obtain yn+1 to order p:

(4) (P4 )

yn+1 = 10 yn + 11 yn1 + hn+1

b1j fn+cj + b13 fn+c3 +

1j fnj
An AdamsMoulton corrector of order p 2 is used as P4 to control the stepsize, hn+2 , and obtain yn+1 to order p 2,
yn+1 = yn + hn+1 a41 fn + a43 fn+1 +

4j fnj

For the 3-stage (p 3)-step methods considered in this paper, the o-step points satisfy the following RungeKutta type simplifying conditions:

(6) where (7)

aij + Bi (1),

i = 2, 3,

j Bi (j) = i+ j!

j1 +1 , i (j 1)!

j = 1, 2,. , p,

i = 2, 3.

and (8) j = 1 hn+1 (xn xn+1j ) = 1 hn+1

j = 2, 3,. , p 3.

In the sequel, j will be frequently used without explicit reference to (8). 3. Order conditions of general HB(p)3 By forcing a Taylor expansion of the numerical solution produced by formulae HB(5-15)3 to agree with an expansion of the true solution we obtain multistep and RungeKutta type order conditions that must be satised by general HB(p)3 methods of order p = 5,. , 15. As in similar search for ODE solvers, we impose the following simplifying assumptions:

j=0 i1

ij = 1, aij ck + k!Bi (k + 1) = j
i = 2, 3, i = 2, 3, k = 0, 1, 2,. , p 3.

1 ck+1 , k+1 i

There remain three sets of equations to be solved:
1i = 1, b1i ck + k!B1 (k + 1) = i

(12) (13)

1 , k+1

k = 0, 1,. , p 1,

cpj aij + Bi (p 1) + B1 (p) = , (p 2)! p! j=1
where the backstep parts, B1 (j), are dened by (14)
j 2 B1 (j) = 11 + j! j1 i+1 i , (j 1)!

j = 1,. , p + 1.

4. Vandermonde-type formulation of general HB(p)3 4.1. Integration formula IF. The (p + 1)-vector of reordered coecients of the integration formula IF in (4), u1 = [10 , b11 , b12 , b13 , 11 , 12 ,. , 1,p4 , 11 ]T , is the solution of the conuent Vandermonde-type system of order conditions (15) M 1 u1 = r 1 ,

M1 =

0 0. 0 0

c2! cp(p1)!

2 p3 2!

p(p1)!

p1 p3 (p1)!

2! 3! . .

p 2 p!
and r 1 = r1 (1 : p + 1) has components r1 (i) = 1/(i 1)!, The leading error term of IF is cp cp p+2 + b+ b+ (p + 1)! p! p!
p4 p j+hp+1 y p+1. 1j p! (p + 1)! n+1 n j=1

i = 1, 2,. , p + 1.

4.2. Predictor P2. The (p 1)-vector of reordered coecients of predictor P2 in (2), u2 = [20 , a21 , 21 , 22 ,. , 2,p4 , 21 ]T , is the solution of the system of order conditions (17) where 2 M = 0 0. M 2 u2 = r 2 , 2

p(p3)!

p3 p3 (p3)!

p(p2)!

and r 2 = r2 (1 : p 1) has components r2 (i) = ci1 /(i 1)!, 2
p+1 (j) S2 (j)hj yn n+1 j=0

i = 1, 2,. , p 1.

A truncated Taylor expansion of the right-hand side of (2) about xn gives
with coecients S2 (j) = M 2 (j + 1, 1 : p 1)u2 = r2 (j + 1) = j S2 (j) = + j!

cj 2 , j!

j = 0, 1,. , p 2,

j1 i+1 , (j 1)!

j = p 1, p, p + 1.
We note that P2 is of order p 2 since it satises the order conditions
2i = 1, S2 (j) = and its leading error term is cp1 p1 (p1) 2 hn+1 yn. S2 (p 1) (p 1)! 4.3. Predictor P3. The p-vector of reordered coecients of predictor P3 in (3), u3 = [30 , a31 , a32 , 31 , 32 ,. , 3,p4 , 31 ]T , is the solution of the system of order conditions (19) where 3 M = 0 0. c2
c2! cp(p2)! i=0 j c2 /j!,

j = 1,. , p 2,

M 3 u3 = r 3 , 2

p2 p3 (p2)!

The rst p 1 components of r 3 = r3 (1 : p) are r3 (i) = ci1 /(i 1)!, 3 and the pth component is r3 (p) = b12 S2 (p 1) B1 (p) , b13 p! i = 1, 2,. , p 1,
which corresponds to the RK order conditions (13). A truncated Taylor expansion of the right-hand side of (3) about xn gives
p+1 (j) S3 (j)hj yn n+1 j=0
with coecients S3 (j) = M 3 (j + 1, 1 : p)u3 = r3 (j + 1) = j S3 (j) = + a32 S2 (j 1) + j!

cj 3 , j!

j = 0, 1,. , p 2, j = p 1, p, p + 1.
4.4. Step control predictor P4. The (p 2)-vector of reordered coecients of P4 in (5), u4 = [a41 , a43 , 41 , 42 ,. , 4,p4 ]T , is the solution of the system of order conditions: (21) where M 4 u4 = r 4 ,

4 M = 0 . . . 0

cp(p3)!

p3 (p3)!

p3 2 p3 2!. . . p3
and r 4 = r4 (1 : p 2) has components r4 (i) = 1/i!, i = 1, 2,. , p 2.
The solutions u , = 1, 2, 3, 4, form generalized Lagrange basis functions for representing the HB interpolation polynomials. 5. Symbolic construction of elementary matrix functions Consider the matrices (23) M Rm

= 1, 2, 3, 4,

of the Vandermonde-type systems (15), (17), (19), and (21), where (24) m1 = p + 1, m2 = p 1, m3 = p, m4 = p 2,
and p is the order of the method. The purpose of this section is to construct elementary lower and upper bidiagonal matrices as symbolic functions of the parameters of HP(p). These matrices are most easily constructed by means of a symbolic software. These functions will be used in Section 7 to factor each M , = 1, 2, 3, into a diagonal+last-column matrix, W , which will be further diagonalized by a Gaussian elimination. This decomposition will lead to a fast solution of the systems M u = r in O(p2 ) operations. Since the Vandermonde-type matrices M can be decomposed into the product of a diagonal matrix containing reciprocals of factorials and a conuent Vandermonde matrix, the factorizations used in this paper hold following the approach of Bjrck o and Pereyra [5], Krogh [23], Galimberti and Pereyra [17] and Bjrck and Elfving o [4]. Pivoting is not needed in this decomposition because of the special structure of Vandermonde-type matrices.
5.1. Symbolic construction of lower bidiagonal matrices. We rst describe the zeroing process of a general vector x = [x1 , x2 ,. , xm ]T with no zero elements. The lower bidiagonal matrix Ik k+(25) Lk = . . . . m dened by the multipliers xi1 (26) i = = Lk (i, i), xi i = k + 1, k + 2,. , m,
zeros the last (m k) components, xk+1 ,. , xm , of x. This zeroing process will be applied recursively on M as follows. For k = 3, 4,. , m 1, left multiplying Tk = Lk1 L4 L3 M by Lk zeros the last (ml k) components of the kth column of Tk. Thus we obtain the upper triangular matrix (27) L M = Lm

L4 L3 M

in (m 3) steps. We note that L does not change the rst two rows of M. Process 1. At the kth step, starting with k = 3, M (k1) = Lk1 Lk2 L3 M is an upper triangular matrix in columns 1 to k 1. The multipliers in Lk are obtained from M (k1) (k+1 : ml , k) since M (i, k) = 0 for i = k + 1, k + 2,. , ml. Algorithm 1 in Appendix A describes this process. 5.2. Symbolic construction of upper bidiagonal matrices. For each matrix L M , = 1, 2, 3, we construct recursively upper bidiagonal matrices U2 , U3. , Um 2 such that the upper triangular matrix U = U2 U3 Um 2 transforms L M into a matrix W = L M U with nonzero diagonal elements, W (i, i) = 0, i = 1, 2,. , m , and nonzero W (1 : m , m ) = 0, in the last column, and zero elsewhere. We call such a matrix a diagonal+last-column matrix. We describe the zeroing process of the upper bidiagonal matrix Uk on the tworow matrix (28) L M U2 U3 Uk1 (k : k + 1, 1 : m ) = The divisors (29) i = 1 = Uk (i, i), y2,i y2,i1 i = k + 1, k + 2,. , m 1, yk1 yk+1,1 yk,k1 yk+1,k1 yk+1,k yk+1,k+1 1 yk+1,m

yk,m yk+1,m

dene the upper bidiagonal matrix Ikk+0 k+1 k+2 0 . (30) Uk = . . 0 m 2 m m 0

0 0 1

Right-multiplying (28) by Uk zeros the 1s in position k,. , m 1 in the rst row and puts 1s in position k + 1,. , m 1 in the second row: (31) L M U2 U3 Uk1 Uk (k : k + 1, 1 : m ) = yk1 yk+1,1 yk,k0 yk+1,k1 yk+1,k 1 0 yk,m 1 yk+1,m.
Thus, U = U2 U3 Um 2 transforms the upper triangular matrix L M into the diagonal+last-column matrix (32) in (m 3) steps. Process 2. At the kth step, starting with k = 2, M (k1) = L M U2 U3 Uk1 is a diagonal+last-column matrix in rows 1 to k 1. The divisors in Uk are obtained from M (k1) (k + 1, k + 1 : m ) since M (k1) (k + 1, j) M (k1) (k + 1, j 1) = 0, j = k + 1, k + 2,. , m 1. Algorithm 2 in Appendix A describes this process. 6. Particular variable-step HB(p)3 The general HB(p)3 methods obtained in Section 3 contain one free coecient, c2 , and depends on hn+1 and the previous nodes, xn , xn1 ,. , xn(p4) , which determine 2 ,. , p3 in (8). For simplicity and to reduce the number of ostep points to one, thus reducing the cost per step in the implementation of a particular variable-step HB(p)3, for p = 5, 6,. , 15, the following three coecients were chosen 2 (33) c1 = 0, c2 = , c3 = 1. 3 This choice turned out to be better than other neighboring choices on several problems. The remaining of this paper is concerned with the family of particular VSVO with coecients cj given in (33) again denoted by HB(5-15)3. The procedure to advance integration from xn to xn+1 is as follows. (a) The order p is obtained by the procedure of Section 9. Then, the stepsize, hn+1 , is obtained by formula (38) of Section 9 with = p 1. W = L M U2 U3 Um

(b) The numbers 2 ,. , p3 , dened in (8), are calculated. (c) The coecients of integration formula IF, predictors P2 , P3 and step control predictor P4 are obtained successively as solutions of systems (15), (17), (19) and (21). (d) The values yn+c2 , yn+c3 , yn+1 , and yn+1 are obtained by formulae (2)(5). (e) The step is accepted if |yn+1 yn+1 | is smaller than the chosen tolerance and the program goes to (a) with n replaced by n + 1. Otherwise the program returns to (a) with the same order p and the smaller step 0.7 hn+1. 7. Fast solution of Vandermonde-type systems for particular HB(p)3 Symbolic elementary matrix functions Lk and Uk , = 1, 2, 3, are constructed once as functions of j , for j = 2, 3,. , p 3 by Algorithms 1 and 2 in Appendix A to produce diagonal+last-column matrices, which, in turn, are diagonalized by a Gaussian elimination expressed as the product of two elementary matrix functions. These elementary matrix functions are used by fast Algorithms 3 and 4, in Appendix A, to solve systems (15), (17), (19) and (21) at each integration step. 7.1. Solution of M u = r , m1 = p + 1, = 1, 2, 3. We let m2 = p 1, m3 = p, m4 = p 2,
as dened in (24). Firstly, the elimination procedure of subsection 5.1 is applied to M to construct m m lower bidiagonal matrices Lk , k = 3,. , m 1, with multipliers (34) i = i+1k = Lk (i, i), M (3, k) i = k + 1, k + 2,. , m.
The matrix L = Lm 1 L4 L3 transforms the coecient matrix M into the upper triangular matrix L M of the form (27). Secondly, the elimination procedure of subsection 5.2 is used to construct m m upper bidiagonal matrices Uk , k = 2,. , m 2, with multipliers (35) i = k1 = Uk (i, i), M (3, i) M (3, i k + 1) i = k + 1, k + 2,. , m 1.
The right-product of the Uk will transform L M into a diagonal+last-column matrix W of the form (32). Finally, a factored Gaussian elimination, Lm +1 Lm , diagonalizes W as follows. First, W (m , m ) is set to 1 by the diagonal matrix Lm : Lm (i, i) = 1, i = 1,. , m 1, Lm (m , m ) = 1/W (m , m ). Then the non-diagonal entries in the last column of Lm W are zeroed by the unit diagonal+last-column matrix Lm +1 whose last column has top m 1 entries Lm
: m 1, m ) = (Lm W )(1 : m 1, m ).
This procedure transforms M into the diagonal matrix D = Lm where D (i, i) = 1, and D (i, i) = (i 2)! , [M (3, 3)] [M (3, 4)] [M (3, i 1)] i = 4, 5,. , m 1. i = 1, 2, 3, m ,

L3 M U2 U3 Um

Thus we have the following factorization of M into the product of elementary matrices: , M = Lm +1 Lm L3 D U2 U3 Um 2 and the solution is (36) u = U2 U3 Um

L3 r ,

where fast computation goes from right to left. Procedure (36) is implemented in Algorithm 3 in Appendix A in O(m2 ) operations. The input is M = M ; m = m ; r = r ; Lk = Lk , k = 3, 4,. , m 1; Uk = Uk , k = 2, 3,. , m 2; and D = D. The output is u = u ; 7.2. Solution of M 4 u4 = r 4. The algorithm to solve the system M 4 u4 = r 4 in O(m2 ) operations is similar to the algorithm for the primal system of [5, p. 896] 4 and is described in Algorithm 4 in Appendix A. The input is M = M 4 ; m = m4 ; r = r 4 and the output is u = u4. Remark 1. Formulae (2)(5) can be put in matrix form. For instance, (4) can be written as yn+1 = F 1 u1. where F 1 = yn , hn+1 fn , hn+1 fn+c2 , hn+1 fn+c3 , hn+1 fn1 , hn+1 fn2 ,. , hn+1 fn(p4) , yn1 , and u1 = [10 , b11 , b12 , b13 , 11 , 12 ,. , 1,p4 , 11 ]T , It is interesting to note the three decomposition forms of the system F u: F (U D1 Lr) (F U D1 )Lr (F U D1 L)r (generalized Lagrange interpolation), (Kroghs modied divided dierences), (Nordsiecks formulation).

The rst form is used in this paper, the second form for Vandermonde systems is found in [23], and the third form is found in [27].
8. Regions of absolute stability and principal error term The region of absolute stability, R, of HB(p)3 is obtained by applying the predictors P2 , P3 and the integration formula IF with constant h to the linear test equation y = y, y0 = 1. This gives the following dierence equation and corresponding characteristic equation

j yn+j = 0,

j rj = 0,
respectively, where p 3 is the number of steps of the method. A complex number h is in R if the p 3 roots of the characteristic equation satisfy the root condition |rs | 1 and the multiple roots satisfy |rs | < 1. The method used to nd R is similar to the one used for k-step multistep methods (see [20, pp. 256257]). Let ABM(p, p 1) denote the ABM method with predictor of order p 1 and corrector of order p in PECE mode [31, p. 135140]. To have a fair comparison of the performance of HB(5-15)3 and ABM(p, p 1) their regions of absolute stability should be scaled by 1/3 and 1/2, respectively, to take the number of function evaluations into account at each step. The upper part of the unscaled regions of absolute stability, R, of HB(5-15)3 are shown in grey in Fig. 1. The region R is symmetric with respect to the real axis. The good shape of the stability regions is remarkable. The scaled intervals of absolute stability (/3, 0) of HB(p)3 and (/2, 0) of ABM(p, p 1) are listed in the left part of Table 1. It is seen that HB methods have larger scaled intervals of absolute stability than ABM methods of comparable order for p > 7. The principal error term of HB(5-15)3 is of the form 1 {f p } + 2 (p){{f p2 }f } + 3 {2 f p1 }2 + 4 {3 f p2 }3 hp+1. where {f p }, {{f p2 }f }, {2 f p1 }2 , {3 f p2 }3 are elementary dierentials dened in [9, 20]. The principal local truncation coecients (PLTC), 1 , 2 , 3 and 4 , of the principal error term are listed in Table 2. The PLTC of ABM(p, p 1) are [k Cp , Cp+1 ] [24, p. 107]. The scaled norms 3 PLTC 2 of HB(5-15)3 and 2 PLTC 2 of ABM(p, p1) of order p = 5,. , 13 are listed in the right part of Table 1. It is observed that the scaled norm of HB(p) is smaller than the scaled norm of ABM(p, p 1). 9. Controlling stepsize and order A variant of the procedure described in [31] is used to control the stepsize and order of our VSVO HB methods. The program computes the maximum norm E = yn+1 yn+1,q

2 HB(5)3 1

1.5 HB(6)3 1.0 0.5
0 -1.5 -1.3 -1.0 1.5 HB(7)3 1.0 0.-1.5 1.0 HB(9)3 0.-0.94 -0.8 1.0 0.-0.79 1.0 0.-0.8 -0.6 HB(11)3 -1.03

-1.5 1.5

-1.14 -1.0
HB(8)3 1.0 0.5 -0.0 -0.96 -0.8 HB(10)3 0.1.0 0.0 -0.8 -0.69 -0.6 1.0 0.-0.8 -0.52 -0.4 HB(14)3 -0.91-0.8 HB(12)3

HB(13)3

-0.4 1.0

-0.2 HB(15)3

0.-0.45
Figure 1. Unscaled regions of absolute stability, R, of HB(5-15)3. where yn+1,q := yn+1 is the value obtained by the step control predictor P4 of order q = p 2. The stepsize hn+1 is obtained by the formula (see [21]): (38) hn+1 = min hmax , hn tolerance E

, 4 hn

Table 1. For given order p, the table lists the scaled abscissa of absolute stability, , and the scaled norm 3 PLTC 2 for HB(p)3 and 2 PLTC 2 ABM(p, p 1), respectively. p 15 /3 /2 HB(p)3 ABM(p, p 1) 0.43 0.70 0.38 0.52 0.34 0.39 0.32 0.30 0.31 0.22 0.30 0.17 0.26 0.13 0.23 0.11 0.20 0.03 0.17 0.PLTC HB(p)3 3.93e-02 2.36e-02 1.61e-02 1.19e-02 9.27e-03 7.47e-03 6.18e-03 5.25e-03 4.50e-03 3.93e-03 3.48e-03
2 PLTC 2 ABM(p, p 1) 2.44e-01 2.18e-01 2.00e-01 1.86e-01 1.75e-01 1.65e-01 1.57e-01 1.51e-01 1.45e-01
Table 2. For each order p, the table lists the principal local truncation coecients for HB(5-15)3. p 15 1

53849 207525400761

448503 8228865958

82398 19247139919

778718 952540877
949978046398679 61509369910607
with = p 1 and safety factor = 0.81. The coecients of integration formula IF, predictors P2 , P3 and step control predictor P4 are obtained successively as solutions of the linear systems (15), (17), (19) and (21).
The step to xn+1 is accepted if E tolerance, else it is rejected and the program returns to the previous step with smaller step 0.7 hn+1. If the step to xn+1 is successful, besides P4 , three other AdamsMoulton step control predictors,
yn+1, = yn + hn+1 a41 fn + a43 fn+c3 +
of order = q 1 and q 2 are used to produce the three values yn+1, , respectively, to control the order and stepsize by means of the following three maximum norms, E1 = yn+1 yn+1,q1

E2 = yn+1 yn+1,q2

which estimate the local error at xn+1 had the step to xn+1 been taken at orders q 1 and q 2, respectively. These three quantities are formed with E so that much of the order and step size selection can be done by using the following rules. The lowest satisfactory order is used. Thus, the order is lowered if E1 min{E, E+1 } or E max{E1 , E2 }. The order is raised only if the following stronger conditions, E+1 < E < max{E1 , E2 }, are satised. When the order q of P4 is 13, E+1 is not available; Thus, the order is lowered if E max{E1 , E2 }. When q = 3, the order is raised only if E+1 < E. After selecting the order to be used, and E are reassigned according to the selected order. For example, if the order is to be lowered in the next step, n+1 = n 1 and E = E1. The stepsize hn+1 is then controlled by formula (38). 10. Numerical results 10.1. Test problems. The numerical performance of HB(5-15)3 and DP(8,7)13M has been compared on the following problems: Arenstorfs orbits [1], the Brusselator and the Pleiades [20, pp. 244249], Eulers equation and the restricted threebody problem [31, pp. 232259], the cubic wave equation [8] (pointed out to the authors by Philip W. Sharp), and the following nonsti DETEST problems [21]: the growth problem B1 of two conicting populations, two-body problems D1D5, Van der Pols equation E2, and three easier problems: the oscillatory problem A3, the integral surface of a torus B4, and Bessels equation of order 1/2 with the origin shifted one unit to the left E1. We report on the performance of the present

Figure 2. CPU (horizontal axis) versus log10 (|MGE|) (vertical axis) for the Brusselator (left) and the cubic wave (right). VSVO HB(5-15)3 and DP(8,7). methods only on the Brusselator and the cubic wave equation since DP(8,7)13M wins on the other problems. 10.2. Starting procedure. The necessary three starting values for HB(5-15)3 were obtained by DP(5,4)7FM (see [14]) with initial step size, h1 , chosen by a method similar to steps (a) and (b) of [20, p. 169]. 10.3. CPU against maximum global error. CPU time (CPU) has been plotted in Fig. 2 versus the Maximum Global Error (MGE) in HB(5-15)3 and DP(8,7)13M for the Brusselator and the cubic wave. The horizontal axis is CPU for a given tolerance and the vertical axis is the common logarithm of MGE: (40) log10 (|MGE|).
10.4. CPU percentage eciency gain. The CPU percentage eciency gain (CPU PEG) is dened by formula (cf. Sharp [33]), (41) (CPU PEG)i = 100 CPU2,ij 1 , j CPU1,ij
where CPU1,ij and CPU2,ij are the CPU of methods 1 and 2, respectively, associated with problem i, and j = log10 (|MGE|). The CPU PEG for the the Brusselator and the cubic wave is listed in Table 3. Table 3. CPU percentage eciency gain, CPU PEG, of HB(5-15)3 over DP(8,7)13M for the listed problems. Problems CPU PEG Brusselator -32% Cubic Wave 151%
Similar to test results in [16], it is seen from Fig. 2 and Table 3 that for the cubic wave problem whose derivative evaluations are relatively expensive, the new VSVO HB(5-15)3 wins over DP(8,7)13M. Computations were performed on a Mac with a dual 2.5 GHz PowerPC G5 and 4 GB DDR SSRAM running under Mac OS X Version 10.4.7 and Matlab Version 7.0.4.352 (R14) Service Pack 2. 11. Conclusion and Future Work A family of variable-step variable-order 3-stage HermiteBirkho (HB) methods of orders 5 to 15 was constructed by solving generalized conuent Vandermonde systems containing RungeKutta type order conditions. The stability regions of the HB methods have a remarkably good shape. The order and stepsize of these methods are controlled by four local error estimators. These methods, in their vectorized Lagrange form, were tested on several problems and were found to have larger scaled regions of absolute stability at higher order and lower scaled error norm than multistep methods. When programmed in C++, they use less CPU time and require fewer function evaluations than DP(8,7)13M also programmed in C++ for costly problems at stringent tolerance. Future work includes extrapolation, iteration to completion, and continuous extension for approximating the solution between mesh points. A desirable goal is to have the code in Fortran. It is expected that the shallow water problem in [7], which was pointed out to the authors by Philip W. Sharp, will be expensive to solve and prove to be an ideal problem for HB(5-15)3. Acknowledgment The anonymous referee is deeply thanked for pointing out important modications to this paper, communicating invaluable insights for future work and suggesting to use a compiler language. Philip W. Sharp is heartily thanked for his suggestions which considerably improved the content and format of a preliminary version of this paper. Appendix A. Algorithms Algorithm 1. This algorithm constructs lower bidiagonal matrices Lk (applied to IF, P2 and P3 ) as functions of c2 , c3 and j , j = 2, 3,. , p 3. For k = 3 : m 1, do the following iteration: For i = m : 1 : k + 1, do the following two steps: Step (1) Lk (i, i) = M (i 1, k)/M (i, k). Step (2) For j = k : m, compute: M (i, j) = M (i 1, j) + M (i, j)Lk (i, i). Algorithm 2. This algorithm constructs upper bidiagonal matrices Uk (applied to IF, P2 and P3 ) as functions of c2 , c3 and j , j = 2, 3,. , p 3. For k = 2 : m 2, do the following iteration:

For j = m 1 : 1 : k + 1, do the following two steps: Step (1) Uk (j, j) = 1/[M (k + 1, j) M (k + 1, j 1)]. Step (2) for i = k : j, compute M (i, j) = (M (i, j) M (i, j 1))Uk (j, j). Algorithm 3. This algorithm solves the systems for IF, P2 and P3 in O(m2 ) operations Given [2 , 3 ,. , p3 ] and r = r(1 : m), the following algorithm overwrites r with the solution u = u(1 : m) of the system M u = r. Step (1) The following iteration overwrites r = r(1 : m) with Lm1 Lm2 L3 r: for k = 3, 4,. , m 1, compute r(i) = r(i 1) + r(i)Lk (i, i), Step (2) First put G(1 : m) = M (1 : m, m). We obtain the coecients of the last two row transformations, Lm and Lm+1 , by means of the recursion: for k = 3, 4,. , m 1, compute G(i) = G(i 1) + G(i)Lk (i, i), i = m, m 1,. , k + 1. Step (3) The following computation overwrites the newly obtained r with Lm+1 Lm r: r(m) = r(m)/G(m), and for k = m 1, m 2,. , 1, compute r(k) = r(k) G(k)r(m). Step (4) The following iteration overwrites r = r(1 : m) with U2 U3 Um2 D1 r: r(i) = r(i)/D(i, i), For k = m 2, m 3,. , 2, compute r(i) = r(i)Uk (i, i), r(i) = r(i) r(i + 1), i = k + 1, k + 2,. , m 1, i = k, k + 1,. , m 2. i = 1, 2,. , m. i = m, m 1,. , k + 1.
Algorithm 4. This algorithm solves the system for the step control predictor P4 in O(m2 ) operations Given [2 , 3 ,. , p3 ] and r = r(1 : m), the following algorithm overwrites r with the solution u = u(1 : m) of the system M u = r. Step (1) for k = 2, 3,. , m 1, compute r(i) = r(i 1) r(i) i+1k , M 4 (2, k) i = m, m 1,. , k + 1.
Step (2) compute r(i) = r(i), r(i) = r(i) i = 1, 2.
[M (2, 2)] [M 4 (2, 3)] [M 4 (2, i 1)] , (i 1)!

i = 3, 4,. , m.

For k = m 1, m 2,. , 1, compute k , i = k + 1, k + 2,. , m, + 1) M 4 (2, i k + 1) r(i) = r(i) r(i + 1), i = k, k + 1,. , m 1. r(i) = r(i) M 4 (2, i ABM(p, p 1) Appendix B. Matlab programming This appendix is included for the benet of Matlab users. When programmed in Matlab, HB(5-15)3 turned out to be superior to Matlabs ode113 on all the problems listed in subsection 10.1. Algorithm 3 which solves systems IF, P2 and P3 were programmed as subroutines in C, e.g., IFsub, P2sub and, P3sub Algorithm 4 which solves the P4 system was programmed in C as subroutines in C, e.g., P4sub. A calling program in C, e.g., IFP which calls IFsub, P2sub, P3sub and, P4sub was compiled together with the above four subroutines by the Matlab mex command into mex les, e.g., IFP.macmex. At runtime, the data of dierential equations were input. Then, IFP.macmex was called and run to calculate the values of the coecients of IF, P2 , P3 and P4 at each integration step until completion of the integration. As an option, CPU time and NFE of function f (x, y) in (1) at the runtime of Algorithms 3 and 4 can be recorded. As another option, MGE can also be run. Matlabs ode113 can be run with appropriate tolerance for comparison with HB(p)3. The elementary matrices Lk and Uk , = 1, 2, 3, 4, are constructed by Algorithms 1 and 2 as functions of j , for j = 2, 3,. , p 3. These algorithms are not needed at runtime since these matrix functions are already implemented in the four subroutines IFsub, P2sub, P3sub and, P4sub which are compiled together with the calling program into Matlab mex le IFP.macmex. References

[1] R. F. Arenstorf, Periodic solutions of the restricted three-body problem representing analytic continuations of Keplerian elliptic motions, Amer. J. Math., LXXXV (1963), pp. 27 35. [2] R. Ashino, M. Nagase, and R. Vaillancourt, Behind and beyond the Matlab ODE Suite, Comput. & Math. with Applics., 40 (2000) pp. 491512. [3] R. Barrio, F. Blesa and M. Lara, VSVO formulation of the Taylor method for the numerical solution of ODEs, Comput. Math. Applic., 50, pp. 93111.
[4] A. Bjrck and T. Elfving, Algorithms for conuent Vandermonde systems, Numer. Math., o 21 (1973), pp. 130137. [5] A. Bjrck and V. Pereyra, Solution of Vandermonde systems of equations, Math. Comp., o 24 (1970), pp. 893903. [6] R. K. Brayton, F. G. Gustavson and G.D. Hachtel, A new ecient algorithm for solving dierential-algebraic systems using implicit backward dierentiation formulas, Proc. IEEE, 60 (1972), pp. 98108. [7] P. J. Bryant, Periodic waves in shallow water, J. Fluid Mech, 59. part 4, (1973), 625644. [8] P. J. Bryant, Nonlinear wave groups in deep water, manuscript. [9] J. C. Butcher, Coecients for the study of RungeKutta integration processes, J. Aust. Math. Soc., 3 (1963), pp. 185201. [10] J. C. Butcher, A modied multistep method for the numerical integration of ordinary dierential equations, J. Assoc. Comput. Mach., 12 (1965), pp. 124135. [11] M. Calv and R. Vaillancourt, Interpolants for RungeKutta pairs of order four and ve, e Computing, 45 (1990), pp. 383388. [12] G. F. Corliss and Y. F. Chang, Solving ordinary dierential equations using Taylor series, ACM Trans. Math. Software, 8(2) (1982), pp. 114144. [13] P. Deuhard, Recent progress in extrapolation methods for ordinary dierential equations, SIAM Rev., 27 (1985), pp. 505535. [14] J. R. Dormand and P. J. Prince, A reconsideration of some embedded RungeKutta formulae, J. Comput. Appl. Math., 15 (1986), pp. 203211. [15] W. H. Enright, Continuous numerical methods for ODEs with defect control, J. Comput. Appl. Math., 125 (2000), pp. 159-170. [16] W. H. Enright and T. E. Hull, The test results on initial value methods for non-sti ordinary dierential equations, SIAM J. Numr. Anal., 13 (1976) pp. 944961. [17] G. Galimberti and V. Pereyra, Solving conuent Vandermonde systems of Hermite type, Numer. Math, 18 (1971), pp. 4460. [18] C. W. Gear, The numerical integration of ordinary dierential equations, Math. Comp., 21 (1967), pp. 146156. [19] C. W. Gear, Numerical Initial Value Problems in Ordinary Dierential Equations, Prentice-Hall, Englewood Clis, NJ, 1971. [20] E. Hairer, S. P. Nrsett and G. Wanner, Solving Ordinary Dierential Equations I. Nonsti Problems, Section III.8, Springer-Verlag. Berlin, 1987. [21] T. E. Hull, W. H. Enright, B. M. Fellen, and A. E. Sedgwick, Comparing numerical methods for ordinary dierential equations, SIAM J. Numer. Anal., 9 (1972), pp. 603 637. [22] F. T. Krogh, VODQ/SVDQ/DVDQ-variable order integrators for the numerical solution of ordinary dierential equations, TU Doc. No. CP-2308, NPO-11643, May 1969, Jet Propulsion Laboratory, Pasadena, CA. [23] F. T. Krogh, Changing stepsize in the integration of dierential equations using modied divided dierences, in Proc. Conf. on the Numerical Solution of Ordinary Dierential Equations, University of Texas at Austin 1972 (Ed. D.G. Bettis), Lecture Notes in Mathematics No. 362, Springer-Verlag, Berlin, 2271, (1974) [24] J. D. Lambert, Numerical Methods for Ordinary Dierential Systems, Wiley, Chichester UK, 1991. [25] T. Nguyen-Ba and R. Vaillancourt, HermiteBirkho dierential equation solvers, Scientic Proceedings of Riga Technical University, 5-th series: Computer Science, 46-th thematic issue, 21 (2004), pp. 4764. [26] T. Nguyen-Ba and R. Vaillancourt, HermiteBirkhoObrechko 3-stage 6-step ODE solver of order 14, Can. Appl. Math. Quarterly, 13 (Summer 2005), pp. 151181. [27] A. Nordsieck, On numerical integration of ordinary dierential equations, Math. Comp., 16 (1962), 2249.

ACHILLES INFLATABLE CRAFT
A Division of Achilles USA, Inc.
1407 80th Street, SW Everett, WA 98203 Email: boats@achillesusa.com www.achillesboats.com

EAST COAST SALES OFFICE:

(201)-438-6400

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(951)-587-8101

MANUFACTURED BY:

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EUROPEAN AGENT:

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AUTHORIZED DEALER

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AUSTRALIAN DISTRIBUTOR:

Photography in this catalog depicts scenes using professional models. Achilles recommends the use of a personal flotation device or life jacket when boating.
Specifications of Achilles boats may change without notice. Always follow manufacturers recommendations for boats and engines.
BURSILL SPORTSGEAR (Holdings) Pty. Ltd. 157 Cleveland Street Chippendale N.S.W. 2008 Australia Phone: (61) 8022 Fax: (61) 5708 Email: grahambursill@bigpond.com www.inflatableboats.com.au

DINGHY SERIES

Pages 3 - 4

RIGID HULL

Pages 10 - 12
Any inflatable boat is only as good as its fabric. For over 40 years Achilles has been setting the standard for inflatable boat fabric using our customized formula combining tough and airtight elastomer coatings over a heavy duty Nylon or Polyester core fabric. Today, Achilles Fabric continues to set the standard by using an exterior coating of chlorosulphonated polyethylene, or CSM, for its unsurpassed durability and two interior layers of chloroprene for its remarkable air tightness. With so many fabric claims made by inflatable boat manufacturers, it is important to recognize that Achilles Fabric has been proven where it counts most - on the water. Many types of fabric have come and gone under many different names, but none have ever been proven more effective than Achilles Fabric. Trust your next inflatable boat to the proven name in fabric reliability, durability and toughness - Achilles.
CSM exterior coating for toughness Heavy-duty Nylon or Polyester core. Two layers of Chloroprene for unsurpassed air retention

for 2011

Series

SPORT TENDERS

Pages 5 - 9
FULL FIVE-YEAR WARRANTY ON FABRIC AND SEAMS
We back our claim to having superior fabric and construction with a strong warranty commitment.
Performax tube design is standard on all HB, LSI, LSR and SPD models!
Better performance with heavier 4-stroke motors Increased buoyancy and load capacity Increased waterline for faster, smoother planing

LSR-290

A SMOOTH, SIMPLE OAR SYSTEM
We invented the fold-down, locking oar system that makes rowing a breeze while keeping oars secured and out of the way when not in use.
UNSURPASSED SEAM CONSTRUCTION
Outside Seam Tape Our seams are overlapped a full inch and reinforced with 1 Fabric Overlay seam tape both inside and out. No one else takes so Inside Seam Tape many steps to ensure seams will last. Side View of Achilles Seam
NON-CORROSIVE CHECK VALVES
ACHILLES WORK & RESCUE BOATS
Achilles produces heavy-duty workboats and rapid deployment rescue boats. Call or email for our Commercial Boat catalog.
All Achilles valves are non-corrosive and double-sealed for safety with no moving parts that might break. They are engineered for easy "one-way" inflation and instant deflation.

SPORT & UTILITY

Pages 12 - 16
Introducing the new Achilles HB-300FX

Add the convenience of a folding transom to the great features and design found in Achilles RIBs and you have the new Achilles HB-300FX. See page 10.

HB-300-FX

ACHILLES RIVERBOATS
Achilles produces a complete line of riverboats and kayaks. Call or email us for our riverboat brochure.
Achilles boats are certified by the National Marine Manufacturers Association as adhering to federal regulations and rigid industry standards established by the U.S. Coast Guard and the American Boat & Yacht Council. Achilles boats must pass rigorous annual inspections from independent third party inspectors to maintain this prestigious certification that guarantees our boats meet the highest safety and production standards.
These two and four-person boats go from their handy carry bags to fully equipped dinghies in just minutes. Compact enough to be stored virtually anywhere, they are safe, durable and dependable dinghies capable of carrying substantial loads. They come ready to work with fold-down oar system, wood seat, fold-up floor and motor mount. Even though they are our most economical boats, they are made with the same durable Achilles CSM fabric and backed by the same five-year warranty as our large sport boats.

D I N G H Y S E R I E S

A lot of quality in a small package. That describes our LT Series dinghies.

D I N G H I E S

Value packed, roll-up option for boaters who need the space-saving R O L L U P convenience of a lightweight easyT E N D E R S E R I E S to-stow tender.
Both our LS-RU rollup tenders can be stowed in a space just 10" high. With a roll-up wood/Achilles CSM fabric floor and solid transom, they can be inflated and ready to go in minutes. When they are, they deliver surprisingly large load and person capacities. Easy to carry, easy to set up, easy to store - the LS-RU.

T E N D E R S

LS4-RU
LS-RU STANDARD FEATURES INCLUDE: The LT4 has a heavy-duty removable motor mount that fits quickly and securely into molded rubber mounting attachments.
Achilles CSM fabric; wood/CSM roll-up floor; full-length rubbing strake; fold-down locking oar system and oar holders; two-piece breakdown aluminum oars and holders; protective motor clamp plate; removable wood rowing seat; towing D-rings; foot pump; carry bag; and maintenance kit.

* See pages 17-18

Available in Europe only
LT STANDARD FEATURES INCLUDE: Achilles CSM fabric; wood/Achilles CSM roll-up floor (LT4); hinged wood floor (LT2); full-length rubbing strake; fold-down locking oar system with two-piece breakdown aluminum oars and holders; bow D-ring (LT2); towing bridle D-rings (LT4); life lines with hand loops; removable wood rowing seat; motor mount; foot pump; carry bag; and maintenance kit.
LS-4 STANDARD FEATURES INCLUDE:
Achilles CSM fabric; sectional wood floor; full-length rubbing strake; fold-down locking oar system and oar holders; two-piece breakdown aluminum oars; inflatable thwart seat; towing D-rings; foot pump; carry bag; and maintenance kit.
S P O R T T E N D E R S E R I E S
These timeless tenders still offer boaters the classic inflatable look and functionality that has made them so popular.

LEX-88

S P O R T T E N D E R S

T E N D E R S E R I E S

At last, inflatable tenders that combine the convenience of a roll-up floor with the S P O R T rigidity and strength of floorboards.
Achilles has developed a unique aluminum floor that makes it a snap to quickly inflate and break down these tenders. This new floor design combined with an innovative hinge system makes this roll-up floor stronger and firmer than others while keeping it lightweight. This is combined with an inflatable keel and V-shaped fiberglass-encased transom to enhance performance. Large 17 tubes allow the LSR tenders to carry heavy loads. Add in the improved buoyancy, extra load capacity and smoother planing provided by the Performaxtube design, and you have a roll-up tender that is tough to beat.
They represent the culmination of many subtle refinements over the years that have kept them technologically current while maintaining their traditional appearance. A sleek bow design and ample interior space come together in this classic tender. A teardrop rub strake deflects spray and improves protection from the inevitable collisions with docks and boat hulls. Other subtle quality features range from an adjustable range wood seat to molded rubber oar holders. They all add up to a solid, reliable tender that combines practicality and performance in an economical package.
Painted marine grade floorboards, seat and transom

Performax tubes perform

better with todays heavier 4-stroke motors.
LEX STANDARD FEATURES INCLUDE: Achilles CSM fabric; air keel; self-locking wood
floorboards; fold-down, locking oar system; two-piece breakdown aluminum oars and oar holders; full-length tear drop rubbing strake; removable wood rowing seat; carry bag; foot pump; towing bridle D-rings; protective transom motor clamp plate; life lines; self bailer
LSR STANDARD FEATURES INCLUDE: Achilles CSM fabric; air keel; aluminum roll-up floor; fiberglass

encased transom; full length tear drop rubbing strake; bow carry handle; internal bow lifting D-ring; fold down locking oar system; two-piece break down aluminum oars & oar holders; towing bridle Drings; life lines; removable wood rowing seat; extra seat attachment patches; rear carry handles; protective
valve and maintenance kit.
motor clamp plate; self-bailer valve; foot pump; carry bag; and maintenance kit.
The worlds best high-pressure air floor tenders are better than ever thanks to their Performax tube design. Our Performax tube design provides more buoyancy and a greater load capacity essential for todays heavier 4-stroke motors. This tube design also results in a longer waterline enabling our LSI series dinghies to plane even quicker and stay on plane at slower speeds. Our LSI series tenders also have Achilles CSM S P O R T Drop Stitch High Pressure Floors. When you consider the abuse floors take from scrapes, the sun, salt-water T E N D E R and even oil and gasoline, it makes sense to have the toughest material available to protect them-and S E R I E S that is Achilles CSM. So our remarkable AirFirm floors are not only incredibly lightweight and rigid, they are also the most durable you will find. Thousands of polyester threads make our air floors super firm. The additional support of an air keel increases floor rigidity and along with a fiberglass-encased transom** enhances overall performance. With large tube diameters, LSI tenders carry heavy loads despite their light weight. There are six different LSI model sizes to choose from. No other manufacturers air floor tenders offer more convenience, more quality or more choices.
Our LSI boats were the first inflatable tenders made with Achilles CSM - reinforced high pressure air floors.

LSI-260

LSI-290 LSI-365
Achilles high-pressure, drop-stitch air floors provide superior rigidity.

LSI-290

Extra Seat Attachment Patches On LSI-290, 310, 335 and 365.
Seat storage bag shown is an optional accessory.
LSI STANDARD FEATURES INCLUDE: Achilles CSM fabric; air keel & fiberglass encased transom (all models except LSI-230); CSM drop stitch AirFirmtm Floors; full
length tear drop rubbing strake; bow carry handle and internal bow lifting D-ring; fold down locking oar system; two-piece break down aluminum oars & oar holders;
* See pages 17-18 ** LSI-230 has no air keel and a wood transom
towing bridle D-rings; life lines; removable wood rowing seat; protective motor clamp plate; self-bailer valve; large high pressure foot pump with pressure gauge; carry bag; rear carry handles; maintenance kit; and floor protection pad for gas tank.

These solid sport tenders have been a staple in the Achilles lineup for over 30 years.

SPD-290

And in all that time we have never stopped making them better. Their square bow configuration allows for more interior space and the V-shaped fiberglass transom and inflatable keel deliver better tracking and performance. These deluxe tenders also feature large tube diameters and Performax tubes for better stability, performance and load capacities. They also feature a host of deluxe features such as non-slip, foam-matted floorboards, aluminum floor joints and stringers for a more rigid floor system, and a foam matted wood seat that can be attached in two different locations. If you compare fabric, features, and price with competing brands youll see what a great value these SPD tenders really are.

H U L L S E R I E S

An innovative folding transom NEW makes this new lightweight RIB for 2011 R I G I D the easiest to store and transport.

HB-300FX

R I G I D H U L L
With it's unique folding transom and lightweight design, the new Achilles HB-300FX makes it easier than ever to transport and store a true rigid fiberglass hull inflatable. Despite weighing only 91 pounds, the HB-300FX offers a healthy 4-person, 1260 pound capacity. The "V" hull and Perfomax tubes insure the superior carrying load capacity and performance expected from an Achilles RIB. Equipped with a deluxe full-length, padded storage bag, you are now able to store a rigid hull Achilles inflatable in tighter places than ever before, transport it much more easily and launch it just about anywhere.

SPD-365

Easy to fold and store

Performax tubes

perform better with todays heavier 4-stroke motors.
Deluxe padded storage bag is standard.
SPD STANDARD FEATURES INCLUDE: Achilles CSM fabric; V-shaped fiberglass encased transom; foam-matted floorboards and seat; extra seat attachment patches; aluminum battens and stringers; bow carry handle; internal bow lifting D-ring; fold-down locking oar system with oar holders; rear carry handles; two-piece break-down aluminum oars; protective transom motor clamp plate; foot pump; carry bag; (2 bags for SPD-335 and SPD-365); full length tear drop rubbing strake and maintenance kit.
HB-FX STANDARD FEATURES INCLUDE: Pearl Gray Achilles CSM fabric; deep "V" fiberglass hull; folding fiberglass transom; deluxe padded full length storage bag; full length teardrop rubbing strake; D-rings for towing bridle; stainless steel bow eye; inside bow davit lifting eye; bow and stern carry handles; grab handles; folddown locking oar system with oar holders; interior oar straps; two-piece breakdown aluminum oars; gas tank tie down hardware; protective transom motor clamp plate; extra seat attachment patches; foot pump and maintenance kit.
The Achilles HB-LX boats provide the critically acclaimed performance of the Achilles DX deep "V" fiberglass hull design in a lighter RIB. While maintaining the same hull and tube design as DX boats, these models are on average 20% lighter. This means they can perform well with smaller engines and can be carried or transported more easily. Despite their lighter weight and lower cost compared to the HBDX series, they are still loaded with all the high quality standard features for which Achilles is known.

R I G I D H U L L S E R I E S
Our lightweight rigid hull line is packed with high performance features.

HB-240LX

These deluxe hard bottom inflatables offer boaters the best combination of style, performance and R I G I D functionality.
Sleek, Euro-style tubes give them a stylish edge over other RIBs. A deep V fiberglass hull delivers superior performance. The built-in bow locker provides both a seat and a place for gear and, combined with a large flat deck area, makes it easier to step in and out of the boat. Like every Achilles boat they are made with our own Achilles CSM fabric so they can handle the burning sun of the tropics or cold northern waters without fading or deteriorating. Choose a rigid-hulled inflatable that gives you a little extra. Choose an Achilles HB-DX.

HB-315DX

Eurohelm shown is optional equipment manufactured by St. Croix Davits.

HB-280LX

HB-350DX

HB-280DX

HB-350LX
Fiberglass seat with cushion Fiberglass console with windshield
A deep "V" fiberglass hull takes Achilles RIBs to an even higher level of performance.
HB-LX STANDARD FEATURES INCLUDE: Pearl Gray Achilles CSM fabric; deep V fiberglass hull; full length tear drop rubbing strake; removable rowing seat; Drings for towing bridle; stainless bow eye; lifting hardware for davits; grab handles; Bow and stern carry handles; fold down locking oar system with oar holders; interior oar straps; two piece breakdown aluminum oars; gas tank tie down hardware; protective transom motor clamp plate; extra seat attachment patches (except HB240-LX); foot pump and maintenance kit.
HB-DX STANDARD FEATURES INCLUDE: Pearl Gray Achilles CSM fabric; deep V fiberglass hull; bow locker; double heavy duty full length tear drop rub strake; extra seat attachment patches; helmsman grip; removable rowing seat; D-rings for towing bridle; stainless bow eye; lifting hardware for davits; lift and carry handles; grab handles; fold down locking oar system with oar holders; interior oar straps; two piece breakdown aluminum oars; gas tank tie down hardware; protective transom motor clamp plate; extra **15 H.P. Max. with Remote Steering; 10 H.P. with tiller steering seat attachment patches foot pump and maintenance kit.
S P O R T & U T I L I T Y S E R I E S
A versatile utility boat thats ready to go in minutes.

FRB-124

Designed originally as quick response boats for fire departments and rescue units, these aluminum floor roll ups can fill a variety of functions. Extremely lightweight, they are easy to store or carry and can be set up in minutes. Equipped with pressure relief valves, they can be inflated either by compressed air bottles or the standard equipment foot pump. In addition to their Achilles CSM fabric, their additional special features like abrasion wear patches and removable storage bags make them practical alternatives for tackling a range of jobs on the water.

S P O R T & U T I L I T Y
They are rugged and roomy so they can be put to the test as dive boats, heavy duty tenders or utility boats. Their sporty nature also makes them great runabouts that really perform thanks to their deep V inflatable keel and SGX-132 large tube diameters. A range of quality features include a fiberglass-encased transom and removable, dual position wood seat. Their Achilles CSM fabric pedigree makes these sport boats extremely tough.and tough to beat.
These versatile aluminum and wood-floored boats are the perfect solution for a range of uses.

SGX-122

Aluminum Floor and Fiberglass Encased Transom
Color Length Inside Length Beam Inside Beam Tube Diameter Weight Load Capacity Person Capacity Recommended H.P. Maximum H.P. Air Chambers FRB-104 Red 10'4" (315 cm.) 7'3" (220 cm.) 5'2" (158 cm.) 2'4 (72 cm.) 17" (43 cm.) 108 lbs. (49 kg.) 1230 lbs. (560 kg.) (4-15) Standard 15 Standard 2 + Keel + PRV FRB-124 Red 12'4" (375 cm.) 8'11" (272 cm.) 5'3" (161 cm.) 2'6" (75 cm.) 17" (43 cm.) 154 lbs. (70 kg.) 1500 lbs. (680 kg.) (6-25) Standard 25 Standard 4 + Keel + PRV
Courtesy Liz Clark Voyage of Swell
FRB STANDARD FEATURES INCLUDE: Achilles CSM fabric; aluminum roll-up floor with non-skid panels, fiberglass encased transom with protective motor clamp plate; the FRB-104 has two (2) pressure relief valves; the FRB-124 has two (4) pressure relief valves; lift and carry handles; two (2) paddles with tie downs; two removable storage bags; full length splash guard with life lines; abrasion patches; five (5) tie-down/davit rings; full-length rubbing strake; foot pump and maintenance kit.
SGX STANDARD FEATURES INCLUDE: Achilles CSM fabric; SGX boats have aluminum and wood self-locking floorboards and fiberglass encased transom; deep Vinflatable keel; towing bridle D-rings; fold-down locking oar system with two-piece breakdown aluminum oars and oar holders; bow handle; four lift & carry handles; helmsman grip; 2 self bailer valves; gas tank tie downs; lifelines; removable dual position wood seat; high-volume foot pump; two carry bags; and maintenance kit.
These top-of-the-line models continue to set the standard for inflatable sport-boats.
Built for high performance with deep V trigon air keels, Achilles SG boats offer a roomy, square bow configuration which provides room for extra gear, and an aluminum and fiberglass floor* system which delivers superior rigidity and durability. Achilles SG Sportboats bring performance and practicality together for better diving, fishing, or all-around sport-boating pleasure! They are also perennial rescue and emergency boat favorites of fire departments, first responders and others around the world.

SG-140

SU Boats are built tough to work hard for rugged private and commercial use.
They are made with an additional heavy-duty layer of 840-denier nylon fabric in a two-ply technique which results in a fabric even stronger than a single base fabric of equal thickness. A number of commercial grade features such as recessed valves and a full-length, protective wear patch combine to makes these boats the right choice for the toughest marine uses.
Wood seat & paddles shown are optional features.
Seat/console/arch shown are custom dealer options
HEAVY-DUTY SELF-BAILER AND ALUMINUM-PLATED TRANSOM.
SG STANDARD FEATURES INCLUDE: Achilles CSM fabric; trigon air keel; aluminum & fiblerglass* self-locking floorboards with aluminum stringer system; fiberglass encased transom with protective motor clamp plate; fuel tank tie-downs; 2 self-bailer valves; full-length teardrop rubbing strake; fold-down locking oar system with oar holders (except SG-156); bow carry handle; lift & carry handles; towing bridle D-rings; helmsman grip; full-length splash guard with life lines; removable wood seat (except SG-156); extra seat attachment patches; large, high-volume foot pump; double carry bag system; and maintenance kit.
*SG-124 floorboards are aluminum and wood.
SU STANDARD FEATURES INCLUDE: Achilles CSM fabric; extra durable double layer (840 denier each) nylon core fabric; trigon air keel; aluminum & fiberglass self-locking floorboards & heavy duty aluminum stringer system; fiberglass encased transom with aluminum plate & motor pad; fuel tank tie-downs; heavy duty self bailer sleeve and 2 self bailer valves; extra-wide, heavy-duty, full-length teardrop rubbing strake; tie-down straps for paddles; bow handle; towing bridle D-rings; lift & carry handles; splash guard with life lines; heavy-duty reinforced cargo wear patch all around; extra tie-down D-rings; heavy duty recessed valves; heavy duty lifting points (hull and transom) two (2) high-volume large foot pumps; and maintenance kit.

SPECIFICATIONS FOR

MODEL LENGTH INSIDE LENGTH BEAM INSIDE BEAM TUBE DIAMETER
15 (33 cm) (38 cm) (35 cm) (38 cm) (38 cm)

2011 ACHILLES BOATS

BOAT WEIGHT ACCESSORIES WEIGHT TOTAL WEIGHT LOAD CAPACITY

22 kg 33.5 kg 31 kg 35.5 kg 36 kg 640 lbs. 970 lbs. 600 lbs. 840 lbs. 860 lbs. 290 kg 440 kg 270 kg 380 kg 390 kg

HP RATING

RECOMMENDED/RANGE

MAX ENGINE WEIGHT

HULL WEIGHT

FLOOR WEIGHT

PERSON CAPACITY

2 4* 2 4* 2

NO. AIR CHAMBERS
33x16x9 35x110x10 35x110x10 35x110x10 35x110x10

STOWED DIMENSIONS

(100 x 46 x 24) (103 x 55 x 26) (103 x 55 x 26) (103 x 55 x 26) (103 x 55 x 26)
LT-2 LT-4 LS2-RU LS4-RU LS-4 LEX-77 LEX-88 LEX-96 LSR-290 LSR-310 LSI-230 LSI-260 LSI-290 LSI-310 LSI-335 LSI-365 SPD-290 SPD-310 SPD-335 SPD-365

NEW for 2011!

44 lbs. 44 lbs. 55 lbs. 66 lbs. 66 lbs.
20 kg 20 kg 25 kg 30 kg 30 kg
29 lbs. 37 lbs. 40 lbs. 46 lbs. 47 lbs.
13 kg 17 kg 18 kg 21 kg 21.5kg 8 lbs 20 lbs. 17 lbs. 20 lbs. 20 lbs.
3.5 kg 9 kg 7.5 kg 9 kg 9 kg
37 lbs. 57 lbs. 57 lbs. 66 lbs. 67 lbs.
16.5 kg 26 kg 25.5 kg 30 kg 30.5 kg
12 lbs. 17 lbs. 12 lbs. 12 lbs. 12 lbs.
5.5 kg 7.5 kg 5.5 kg 5.5 kg 5.5 kg
49 lbs. 74 lbs. 69 lbs. 78 lbs. 79 lbs.
(220 cm) (260 cm) (228 cm) (265 cm) (265 cm)
(153 cm) (188 cm) (156 cm) (185 cm) (185 cm)
(124 cm) 111 (142 cm) 22 (132 cm) 20 (142 cm) 22 (142 cm) 22
(58 cm) (66 cm) (62 cm) (66 cm) (66 cm)
3 (2 - 3) standard 3 (2 - 3) standard 4 (2 - 4) standard 4 (2 - 6) standard 4 (2 - 6) standard

2 & keel

122 (230 cm) (265 cm) (290 cm) (290 cm) (310 cm) (230 cm) (260 cm) (290 cm) (310 cm) (335 cm) (365 cm) (290 cm) (310 cm) (335 cm) (365 cm) 810 (149 cm) (180 cm) (205 cm) (205 cm) (220 cm) (155 cm) (185 cm) (205 cm) (220 cm) (226 cm) (268 cm) (205 cm) (220 cm) (226 cm) (268 cm) 56 (144 cm) 22 (144 cm) 22 (144 cm) 22 (158 cm) 24 (158 cm) 24 (144 cm) 22 (144 cm) 22 (158 cm) 24 (158 cm) 24 (168 cm) 27 (168 cm) 27 (158 cm) 24 (158 cm) 24 (168 cm) 27 (168 cm) 27 (66 cm) (66 cm) (66 cm) (72 cm) (72 cm) (66 cm) (66 cm) (72 cm) (72 cm) (78 cm) (78 cm) (72 cm) (72 cm) (78 cm) (78 cm) 15.5 15.5 15.17 15.5 15.18 (39 cm) (39 cm) ( 39 cm) (43 cm) (43 cm) (39 cm) (39 cm) (43 cm) (43 cm) (45 cm) (45 cm) (43 cm) (43 cm) (45 cm) (45 cm) 4 (2 - 4) standard 4 (2 - 6) standard 6 (4 - 8) standard 6 (4 - 8) standard 8 (4 - 10) standard 4 (2 - 4) standard 4 (2 - 6) standard 6 (4 - 8) standard 8 (4 - 10) standard 10 (6 - 15) standard 15 (6 - 25) standard 6 (4 - 8 ) standard 8 (4 - 10) standard 15 (6 - 20) standard 15 (6 - 25) standard 55 lbs. 66 lbs. 88 lbs. 88 lbs. 110 lbs. 55 lbs. 66 lbs. 88 lbs. 110 lbs. 110 lbs. 154 lbs. 88 lbs. 110 lbs. 121 lbs. 154 lbs. 25 kg 30 kg 40 kg 40 kg 50 kg 25 kg 30 kg 40 kg 50 kg 50 kg 70 kg 40 kg 50 kg 55 kg 70 kg

B = Boat F = Floor

10 (6 - 15) standard 15 (4 - 25) standard 20 (6 - 35) standard 30 (9.9 - 40) long 20 (6 - 35) standard 30 (9.9 - 50) long 35 (9.9 - 55) long 35 (9.9 -55) long 40 (20 - 75) long 50 (25 - 90) long
108 lbs. 154 lbs. 164 lbs. 187 lbs. 175 lbs. 226 lbs. 272 lbs. 280 lbs. 371 lbs. 433 lbs.
43.5 kg 51 kg 48 kg 55.5 kg 68.5 kg 78 kg 105 kg 120 kg

(230 cm) 37

The maximum HP Rating, Person Capacity and Load Capacity have been calculated according to the standards of ISO6185. *3 Person Capacity according to NMMA Certification Standards
Persons operating boats alone
should exercise caution when using maximum HP engines.
* *15 H.P. Max. with Remote Steering; 10 H.P. with tiller steering