An algorithm for an eigenvalues problem in the Earth rotation
theory
Ana B. Gonzaleza;∗, Juan Getinob, Jose M. Fartoa
aGrupo de Mecanica Celeste, Depto. Matematica Aplicada a la Ingeniera, E.T.S. Ingenieros Industriales, Paseo del Cauce s=n, 47011 Valladolid. Spain
bGrupo de Mecanica Celeste, Depto. Matematica Aplicada Fundamental, Facultad de Ciencias, 47005 Valladolid. Spain
Received 10 March 1998; received in revised form 7 September 1998
Abstract
In this paper we present a new algorithm to parameterize some kind of hypersurfaces. Our technique extends the Newton– Puiseux algorithm for plane curves to several variables. It is based on the introduction of an order in the monomials of several variables compatible with the total degree and in a recursive construction of a sequence of convex-hulls together with a tree search. We have developed this algorithm to determine the free frequencies (appearing as eigenvalues of a certain matrix) of realistic non-rigid Earth rotation models. We have implemented the algorithm in a Maple V package calledPoliVar. c 1999 Elsevier Science B.V. All rights reserved.
AMS classication:41A58; 41A63; 4104; 8604; 8608 Keywords:Earth rotation theory
1. Introduction
In this work we shall consider an equation
f(”1; : : : ; ”s; z)≡f(”; z) = 0; (1)
where fis a polynomial. This relation describes implicit dependencies of z with respect to ”1; : : : ; ”s.
Our task will be to develop an algorithm to construct the solutions to this problem as
z≡z(”1; : : : ; ”s)≡z(”); (2)
truncated up to a given degree.
∗Corresponding author. E-mail: [email protected], [email protected], [email protected].
Our motivation is an eigenvalues problem arising in the Hamiltonian theory of the rotation of the non-rigid Earth. It is commonly accepted that the paper of Kinoshita [14] contains the best and most accurate theory for the rotation of the rigid Earth. The canonical formulation of this theory allows the application of a perturbation theory [13] and then, the facility of separating the secular and periodical perturbations, and the possibility of increasing the approximation of the solution as close as necessary. Nowadays, the precision of the measures of the Earth rotation require more realistic analytical models of the Earth to adequately describe them. Thus, non-rigid Earth models must be considered. The application of the Hamiltonian method to these models appears as a natural way of obtaining an analytical theory of the rotation of the Earth which extends to that of the rigid case, while being more appropriate to the real constitution of the planet. This task was undertaken by Getino and Ferrandiz some years ago.
Obviously, the elaboration of a complete new theory of the Earth’s rotation is so complex that it must be done by successive approximations: the consideration of an Earth with an elastic mantle [9], then the study of an Earth composed of an axis-symmetrical rigid mantle and a stratied liquid core [6, 7], and the improvement of this model by considering several new eects as the dissipative forces in the mantle–core boundary [10], the triaxiality of the Earth [12] and an Earth model composed of three layers: axis-symmetrical rigid mantle, uid outer core (FOC) and solid inner core (SIC) [11].
In all these models, we need to calculate the kinetic energy of the free motion of the Earth. To this end, a constant coecient linear system of ordinary dierential equations must be solved. Thus the eigenvalues of the system matrix (the free frequencies) must be calculated. In fact, it is sucient to obtain truncated expressions of the eigenvalues as functions of small parameters (ellipticities, coecients of the dissipation, : : :) of the motion. In this way, by considering the characteristic polynomial of the system matrix we have a problem as (1), where the variables ”1; : : : ; ”s play the role of the small parameters. As the coecients of the polynomial depend on
other formal parameters (whose actual values are experimentally given), an ecient algorithm is needed.
Clearly, our eigenvalues problem corresponds to the algebraic problem of parameterizing the hy-persurface given by an equation as Eq. (1) by constructing the solutions (2). For s= 1 we have plane algebraic curves and the problem can be solved by means of the well-known Newton–Puiseux polygon techniques [15], obtaining the parameterizations as series with rational exponents. Fors ¿1, not all hypersurfaces admit a solution with non-negative rational exponents. The Abhyankar–Jung Theorem [1] gives us sucient conditions for the existence of such a solution. A short and con-structive proof of this theorem can be found in Zurro [16], in which Hensel’s lemma is used but this is not the way that we shall take.
We shall propose an algorithm which extends the Newton–Puiseux polygon based algorithm for plane curves (s= 1), to the case s ¿1. It will be useful to solve, not only quasi-ordinary problems, but also any appearing in the Earth rotation theory which are not. This new technique is mainly based on:
1. The introduction of an order in the monomials in ”1; : : : ; ”s compatible with the order of the
terms of a Taylor series expansion.
Some examples from the Earth rotation theory are presented. In some (signicative in the Earth motion study) cases, negative exponents arise. Then the algorithm must be designed to deal with them, and further modications must be included. Some geophysical consequences of the application of our algorithm can be found in [8].
2. The Algorithm
Let
f(”; z) = 0 (3)
be an equation where ”= (”1; : : : ; ”s) and f is a power series in ”i and a polynomial in z. Suppose
that
’(”) =Xc”; (4)
where = (1; : : : ; s)∈Qs and ” is ”11· · ·”ss, is a solution of (3). Then the equality
f(”; ’(”)) = 0; (5)
must be satised.
To calculate the solutions (4) of Eq. (3), we can carry out the following stage-by-stage strategy: 1. Supposing that ’ can be expressed as
’=Xc”+ g:t:; (6)
where g.t. means greater terms, calculate and c by imposing (5).
2. Construct
f1=f(”; c”+z): (7)
3. Obtain the following term of ’ from f1, make a similar change of variable as in (7) and so on. We must study the various problems arising from this strategy to develop a valid algorithm.
The rst one is to get that the words greater terms and following make sense for each s¿1. To this end, an order for the monomials in the variables ”i, must be given, i.e., an order in their
exponents. This can be done by ordering Rs. For technical reasons, this order must be compatible
with the addition in Rs, that is
162⇒1+62+; ∀1;2;∈Rs: (8)
On the other hand, the order must agree with the natural order of the monomials in a Taylor series expansion, i.e., all the terms with a given total degree must be smaller than the terms with greater total degree. This must be satised because we try to calculate truncations of the solutions up to a given total degree. Then, let A be an invertible s×s matrix with the vector (1;1; : : : ;1) conguring its rst row. For every 1;2∈Rs, we shall say that
162 if and only if A16A2 for the lexicographic order on Rs.
In the following, we shall always consider the order for the monomials induced by the order
The second problem related to the previous strategy is to obtain every term c” (by determining
and c) in an ecient way. Assume that
f=Xdr”zr:
We say that the set
D(f) :={(1; : : : s; r)|dr6= 0} ⊆Rs+1
is the diagram of f. There is a bijective map
D(f)↔ {Monomials of f}: (10)
We shall write (6) as
’=c”+’1; (11)
where ’1 contains only terms greater than c”, and c6= 0. We make the formal substitution
f1:=f(”;(c”+’1)) = X
dr”(c”+’1)r:
By expanding the last expression we have
f1:=
By the properties of the constructed order, given a termdr”zr off, the smallest monomial obtained
from it in f1 is
Thus, for obtaining the smallest monomial in f1, we must minimize the set
E(f;) :={+r|(; r)∈D(f)} ⊆Rs:
Let
(f;) = (1; : : : ; s) := minE(f;);
which is well dened by construction. Therefore, the smallest monomial of f1 is
where
S(f;) :={(; r)∈D(f)|+r=(f;)}
will be called generalized side for fand . By (5), f1 vanish. In particular, the smallest monomial off1 calculated before must be equal to 0, and therefore its coecient cancels out. As a consequence, c must be a root of the polynomial
(f;)(X) = X (; r)∈S(f;)
drXr;
that will be called characteristic polynomial for f and .
This last result imposes a restriction not only on the coecient c but also in the exponents. In
fact, since (f;)6= 0 (due to property (10)) and must have a non-zero root, c, there are at least
two dierent monomials in (f;) or equivalently there must be at least two dierent elements in S(f;) (and with the last components dierent). Thus if #S(f;)¡2, then is not a valid
exponent for a solution.
Note that to nd (f;) we can calculate the smallest value,
˜
(f;) = (˜1; : : : ;˜s)
of the set
˜
E(f;) :={A(+r)|(; r)∈D(f)}
for the lexicographic order. The following recurrence formulae characterize the value ˜(f;): (0) ˜S0(f;) := ˜D(f).
(1) ˜j6j+r˜j; ∀(; r)∈S˜j−1(f;), and ˜
Sj(f;) :={(; r)∈S˜j−1(f;)|j+r˜j= ˜j}; j= 1; : : : ; s,
where ˜= ( ˜1; : : : ;˜s) :=A, and ˜D(f) :={(A; r)|(; r)∈D(f)}. Obviously,
S(f;) ={(A−1; r)|(; r)∈S˜
s(f;)}
and
(f;) =A−1˜(f;):
Since
˜
Sj(f;)⊆S˜j−1(f;); j= 1; : : : ; s;
if a value for is valid, then # ˜Sj(f;)¿2; j= 1; : : : ; s.
Now, by taking into consideration these facts, we can successively construct the components of valid values of ˜ (hence ) and simultaneously the components of ˜(f;) (hence (f;)), and therefore (f;) and thus, the valid values for c. This construction, based on Newton polygons,
is as follows: let
Fig. 1. A typical newton polygon.
The border of the convex-hull of the set
{(; r) + (a;0)|(; r)∈D1(f) and a¿0}
will be called the Newton polygon of D1(f). We denote it by N1(f). A typical Newton polygon is a set like the one presented in Fig. 1. Every vertex of N1(f) corresponds to a point of D1(f). Thus, every straight segment of slope not equal to 0 (called in the following side) of this gure has at least two points of D1(f). Furthermore, the valid values of ˜1 are those in which there exists a side of N1(f) with slope −1=˜1, and ˜1 is the abscissa of the intersection of the prolongation of this side with the OX axis.
Remark 2.2. The construction of a Newton polygon seen here, diers slightly of the conventional construction (see for example [2–4]), but we do it in this manner to allow all the possible negative exponents.
Let
S1(f;˜1) :={(; r)∈D1(f)|+r˜1= ˜1}:
This set has at least two dierent points.
Once a valid ˜1 and the corresponding ˜1 are chosen, we iterate the construction for j= 2; : : : ; s, by considering
Dj(f) :=
(j; r)
∃(; r)∈D(f) with˜
(k; r)∈Sk(f;˜1; : : : ;˜k); k= 1; : : : ; j−1
and constructing Nj(f;˜1; : : : ;˜j−1) as the Newton polygon of Dj(f), and then taking ˜j in such a
such a side with the OX axis. Let
Sj(f;˜1; : : : ;˜j) :={(; r)∈Dj(f)|+r˜j= ˜j}:
It has at least two points.
If for any j we cannot nd a valid ˜j, this is a closed way and we must start from the beginning. By exploring all the possibilities, we can obtain all the possible values for ˜= ( ˜1; : : : ;˜s) and thus for =A−1˜, with the corresponding ˜(f;) = (˜
1; : : : ;˜s), and thus (f;) =A−1˜(f;).
The dierent valid values for and, for every , all the possible solutions c of (f;), give us
all the possibilities of starting a solution of the form
c”+ g.t.
Note that ˜∈Qs and all the constructions carried out above can be implemented in a computer
algebra system.
Once we have obtained values for and c, we will say that we have completed a stage of the
algorithm. Thus, we can write now the strategy presented above in the following manner: 1. f0:=f.
2. For j= 0;1; : : ::
Calculate a valid j and cj from fj with the Newton polygons above mentioned techniques and
construct
fj+1:=fj(”; cj”
j+z)
with the condition j¿j−1 if j ¿0.
Thus, we can calculate as many terms of a solution of f as we need. If we should want to make a complete theoretical study of this algorithm, we should need a complete analysis of continuation
conditions, that is to say, to study the possibility of constructing j and cj once fj is constructed.
But, since we shall apply these techniques to physical problems, and we already know that they have solutions, we will be able to calculate them in any way, and our techniques will be valid.
This algorithm is A-dependent, and in particular the order of the variables”j has great importance.
Note that we obtain solutions ordered as in a Taylor expansion: if a term obtained in a given stage has total degree n, all the subsequent terms will have total degree greater than or equal to n.
We will present an example from the Earth rotation theory.
Example 2.3. Now, we shall consider the case of an Earth model composed of a rigid mantle enclosing a liquid core. Triaxiality of the Earth is also considered (see [12]). The free frequencies can be obtained from the eigenvalues of the matrix
M:=
0 (Bm−C)=(Bm) 0 C=Bm (C−Am)=Am 0 −C=Am 0
0 −Cc=Bm 0 BCc=(BmBc) Cc=Am 0 −ACc=(AcAm) 0
and
Thus, the four solutions for f up to degree two are:
’1:=i 1 + A
3. Modications of the Algorithm
The introduction of negative exponents is needed to solve some non-quasi-ordinary problems. However, some diculties appear in many cases. We shall try to illustrate these diculties with a very simple example.
Example 3.1. Consider f:= (e1 −e2)z −e1 +e2 −e2
1 −3e22e1 + 2e32 +e12e2= 0. The solution is ’(e1; e2) := 1 +e1+e1e2=(e1−e2) + 2e22−e1e2, but if we directly apply our algorithm, we obtain an innite number of terms of degree one:
1 +e1+e2+e−1
corresponding to an expansion of the term e1e2=(e1−e2) of the solution, whichmask the subsequent terms of the solution and prevent us from calculating them in a nite number of stages.
We shall expose how to treat these diculties by means of a relevant example in the Earth rotation theory. If we consider an Earth model composed by an axis-symmetrical rigid mantle, FOC and SIC, we must calculate the eigenvalues of the matrix
expanded in the ordered variables (e; el; es; ), and where Am=A−Al−As. Let f= det(M −z ). To solve f= 0 we encounter the above-mentioned diculties.
If we apply directly the algorithm to f, we encounter the four solutions. We go on to focus our attention on one of them. We are interested in obtaining a truncation up to degree two. The xed solution has successive terms:
’1:=c11e+c21es+c31eel+c41el+c51e−1el2+c61e−2el3+· · ·;
where the coecientscij depend onA; As; Al. We obtain an innite number of terms of total degree 2,
which make the subsequent terms unreachable in a nite number of stages. We need to apply some extensions of the basic algorithm. With the aid of blowing-up techniques (see [15]) we can see that the solution contains terms of degree two greater than the innite sequence of terms presented above. Then it makes sense to attempt to calculate them.
In the rst three stages of the algorithm, negative exponents do not appear:
’01:=A=Ame−As=Ames−AAl=A2meel:
If we continue, we obtain an innite number of terms of degree two
Asel(A
which correspond to the term
As
and then an innite number of terms of total degree two
(A−Al)As
which can be collected in
(A−Al)As again an innite number of terms of total degree two:
(A−Al)2Asele2s
terms that we can write as
Now if we observe ’31+’41, an innite number of terms appear:
Asel2
∞ X
j=0
(A−Al)j+1esj+1 (Ae−Am)j+2 =
(A−Al)Asesel2 (Ae−Am)2
∞ X
j=0
(A−A
l)es Ae−Am
j
= (A−Al)Aseles 2
(Ae−Am)(Ae−Am−(A−Al)es)
=:’51:
Let f3 be the numerator of f(e; el; es; ; ’01+’11+’21+’51+z). If we apply the algorithm to f3, the next term is of the form ce2e
l, and has a total degree three. This fact ensures that
’1≈’01+’11+’21+’51
is the solution ’1 truncated up to degree two.
Finally, some numerical tests can be done. The actual values of the parameters can be substi-tuted (taking, for example, the PREM Earth model) rstly in the analytical calculated solution and secondly in f, solving it then numerically (using for example, the Newton–Raphson method). The agreement of the free frequencies obtained by means of the true numerical solution and the substi-tuted analytical solution agree correctly for the given degree of truncation, as can be seen in [8]. This fact, together with the nature of the expressions obtained, ensures that the calculations make geophysical sense.
4. Final considerations
Our techniques could be improved by introducing truncation techniques similar to those presented by Farto et al. [4, 5] for dierential equations and for perturbed problems. The computational complexity of the algorithm, converts this work into a very dicult task. However, these truncation techniques might be necessary in the future to solve more complicated problems (for example, more realistic models of the Earth rotation) with a reasonable computational eort. We have implemented a software package in Maple V called PoliVar c. This package is experimental at this moment,
but it works well for the problems treated with a reasonable computer resources consumption. All the calculations appearing in this paper have been performed with it. This program is based on
PoliDif c (intended for the management of dierential polynomials), and we hope that they will
be unied (together with other techniques) in a future computer package that we plan to develop in the future.
Acknowledgements
This work has been partially supported by Junta de Castilla y Leon, Projects No. VA47=95 and No. VA40=97, and DGES, Project No. PB95-0696.
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