Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol101.Issue1-2.1999:
Journal of Computational and Applied Mathematics 101 (1999) 243–254
An algorithm for an eigenvalues problem in the Earth rotation
theory
Ana B. Gonzalez a;∗ , Juan Getinob , Jose M. Fartoa
a
Grupo de Mecanica Celeste, Depto. Matematica Aplicada a la Ingeniera, E.T.S. Ingenieros Industriales,
Paseo del Cauce s=n, 47011 Valladolid. Spain
b
Grupo 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
c 1999 Elsevier Science B.V. All rights reserved.
called PoliVar.
AMS classi cation: 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 f is 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 (”);
truncated up to a given degree.
∗
Corresponding author. E-mail: [email protected], [email protected], [email protected].
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 7 ) 0 0 2 3 0 - 1
(2)
244
A.B. Gonzalez et al. / Journal of Computational and Applied Mathematics 101 (1999) 243–254
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 strati ed liquid
core [6, 7], and the improvement of this model by considering several new e ects 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 di erential 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 hypersurface 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. For s ¿ 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 constructive 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.
2. An ecient indeterminate coecients technique (based on the recursive construction of a sequence of convex-hulls and a tree search) to nd directly the exponents and the coecient of
each term.
A.B. Gonzalez et al. / Journal of Computational and Applied Mathematics 101 (1999) 243–254
245
Some examples from the Earth rotation theory are presented. In some (signi cative in the Earth
motion study) cases, negative exponents arise. Then the algorithm must be designed to deal with
them, and further modi cations 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
’(”) =
X
c ” ;
(4)
where = (1 ; : : : ; s ) ∈Q s and ” is ”1 1 · · · ”s s , is a solution of (3). Then the equality
f(”; ’(”)) = 0;
(5)
must be satis ed.
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
’=
X
c ” + 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 R s . For technical reasons, this order must be compatible
with the addition in R s , that is
1 62 ⇒ 1 + 62 + ; ∀1 ; 2 ; ∈R s :
(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 satis ed 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)
con guring its rst row. For every 1 ; 2 ∈R s , we shall say that 1 62 if and only if A1 6A2 for
the lexicographic order on R s .
Remark 2.1. In all this paper vectors are considered as row vectors, but if A is a matrix and is
a vector, by A we mean (At )t .
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A.B. Gonzalez et al. / Journal of Computational and Applied Mathematics 101 (1999) 243–254
In the following, we shall always consider the order for the monomials induced by the order
of R s de ned above, and we shall x the s × s matrix
1 1 1
0 1 0
A=0 0 1
. . .
. . .
. . .
0 0 0
···
···
···
..
.
1
0
0
:
..
.
··· 1
(9)
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=
X
d r ” z r :
We say that the set
D(f) := {( 1 ; : : : s ; r ) | d r 6= 0} ⊆ R s+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 c 6= 0. We make the formal substitution
f1 := f(”; (c ” + ’1 )) =
X
d r ” (c ” + ’1 )r :
By expanding the last expression we have
f1 :=
r
XX
j=0
d r
!
r j 1 +j1
· · · ” s s +js ’1r−j :
c ”
j 1
By the properties of the constructed order, given a term d r ” z r of f, the smallest monomial obtained
from it in f1 is
d r cr ” 11 +r1 · · · ”s s +rs :
(12)
Thus, for obtaining the smallest monomial in f1 , we must minimize the set
E (f; ) := { + r | ( ; r ) ∈D(f)} ⊆ R s :
Let
(f; ) = (1 ; : : : ; s ) := min E (f; );
which is well de ned by construction. Therefore, the smallest monomial of f1 is
X
( ; r) ∈ S(f; )
d r cr ” +r =
X
( ; r)∈S(f; )
d r cr ”(f; ) ;
A.B. Gonzalez et al. / Journal of Computational and Applied Mathematics 101 (1999) 243–254
247
where
S(f; ) := {( ; r ) ∈D(f) | + r = (f; )}
will be called generalized side for fand . By (5), f1 vanish. In particular, the smallest monomial
of f1 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
d r X r ;
( ; r) ∈ S(f; )
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 di erent monomials in (f; ) or equivalently there must be at least two di erent elements
in S(f; ) (and with the last components di erent). 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
Ẽ (f; ) := {A( + r) | ( ; r ) ∈D(f)}
for the lexicographic order. The following recurrence formulae characterize the value ˜ (f; ):
(0) S̃ 0 (f; ) := D̃(f).
(1) ˜j 6 j + r ˜ j ; ∀ ( ; r ) ∈ S̃j−1 (f; ), and
S̃j (f; ) := {( ; r ) ∈ S̃j−1 (f; ) | j + r ˜ j = ˜j }; j = 1; : : : ; s,
where ˜ = (˜ 1 ; : : : ; ˜ s ) := A, and D̃(f) := {(A ; r ) | ( ; r ) ∈D(f)}.
Obviously,
S(f; ) = {(A−1 ; r ) | ( ; r ) ∈ S̃ s (f; )}
and
(f; ) = A−1 ˜ (f; ):
Since
S̃j (f; ) ⊆ S̃ j−1 (f; ); j = 1; : : : ; s;
if a value for is valid, then #S̃j (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
D1 (f) := {( 1 ; r ) | ∃( 1 ; : : : ; s ; r ) ∈ D̃(f)} ⊆ R2 :
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A.B. Gonzalez et al. / Journal of Computational and Applied Mathematics 101 (1999) 243–254
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, di ers 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 di erent points.
Once a valid ˜ 1 and the corresponding ˜ 1 are chosen, we iterate the construction for j = 2; : : : ; s,
by considering
∃( ; r ) ∈ D̃(f) with
Dj (f) := ( j ; r )
( ; r ) ∈S (f; ˜ ; : : : ; ˜ );
k
k
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
way that Nj (f; ˜ 1 ; : : : ; ˜ j−1 ) has a side of slope −1= ˜ j . Then ˜j is the abscise of the intersection of
A.B. Gonzalez et al. / Journal of Computational and Applied Mathematics 101 (1999) 243–254
249
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 di erent 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 ˜ ∈Q s 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
0
(C − Am )=Am
M :=
0
Cc =Am
(Bm − C )= (Bm )
0
C=Bm
0
−C=Am
0
;
−Cc =Bm
0
BCc = (Bm Bc )
0
−ACc = (Ac Am )
0
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A.B. Gonzalez et al. / Journal of Computational and Applied Mathematics 101 (1999) 243–254
Fig. 2. Resolution tree of Example 2.3.
where
Am = A − Ac ; Bm = B − Bc ; C = A(1 + e); Cc = Ac (1 + ec );
B=
A(1 + e)
;
1 + ê
Bc =
Ac (1 + ec )
:
1 + êc
Let f(e; ê; ec ; êc ; z ) be the numerator of det(M − zI4 ). We shall apply the algorithm to the equation
f = 0, to express z as certain truncated parameterizations up to total degree two in the variables
(e; ê; ec ; êc ), with the order matrix as in (9). In the rst stage, we can see that we have four di erent
sets of exponents and coecients, corresponding to all the eigenvalues of the matrix. For two of
them, we must apply nine stages to reach the total degree three. For the others we need to perform
only four stages. The results of the four resolutions process can be represented as the resolution
tree (see [4]) of Fig. 2, where
p11 := (0; 0; 0; 0);
p21 := (0; 0; 0; 1);
p31 := (0; 0; 1; 0);
p41 := (1; 0; 0; 1);
p51 := (0; 0; 0; 2);
p61 := (1; 0; 1; 0);
p71 := (0; 0; 1; 1);
p81 := (0; 0; 2; 0);
p91 := (2; 0; 0; 1);
p13 := ( 12 ; 12 ; 0; 0);
p23 :=( 23 ; 12 ; 0; 0);
p33 :=( 21 ; 21 ; 0; 1);
p43 := ( 21 ; 32 ; 0; 0);
p53 :=( 25 ; 12 ; 0; 0);
c11 := i; c21 :=
iA
−iA2
iA
−iAAc
;
c
:=
;
; c31 :=
; c41 :=
51
Am
Am
2A2m
8A2m
c61 :=
iA(A − 2Ac )
−iA(A − 4Ac )
−iAAc
; c71 :=
; c81 :=
;
2A2m
4A2m
8A2m
c91 :=
iA2 Ac
iA
−iAAc
iAAc
−iAAc
; c13 :=
; c33 :=
; c43 := 2 ;
; c23 :=
3
2
2
2Am
Am
2Am
Am
2Am
c53 :=
iAAc (4A − Ac )
8A3m
A.B. Gonzalez et al. / Journal of Computational and Applied Mathematics 101 (1999) 243–254
251
and
N11 := (p11 ; c11 );
N31 := (p31 ; c31 );
N51 := (p51 ; c51 );
N71 := (p71 ; c71 );
N91 := (p91 ; c91 );
N23 := (p23 ; c23 );
N43 := (p43 ; c43 );
N12 := (p11 ; −c11 );
N32 := (p31 ; −c31 );
N52 := (p51 ; −c51 );
N72 := (p71 ; −c71 );
N92 := (p91 ; −c91 );
N24 := (p23 ; −c23 );
N44 := (p43 ; −c43 );
N21 := (p21 ; c21 );
N41 := (p41 ; c41 );
N61 := (p61 ; c61 );
N81 := (p81 ; c81 );
N13 := (p13 ; c13 );
N33 := (p33 ; c33 );
N53 := (p53 ; c53 );
N22 := (p21 ; −c21 );
N42 := (p41 ; −c41 );
N62 := (p61 ; −c61 );
N82 := (p81 ; −c81 );
N14 := (p13 ; −c13 );
N34 := (p33 ; −c33 );
N54 := (p53 ; −c53 ):
Thus, the four solutions for f up to degree two are:
’1 := i 1 +
AAc
A2
A
AAc
A
êc +
ec − 2 eêc − 2 ê2c − 2 eec
Am
Am
2Am
8Am
2Am
!
A(A − 2Ac )
A(A − 4Ac ) 2
+
ec êc −
ec ;
2
4Am
8A2m
’2 := −’1 ;
A 1=2 1=2 AAc 3=2 1=2 AAc 1=2 1=2
AAc 1=2 3=2
;
’3 := i
e ê − 2 e ê − 2 e ê êc + 2 e ê
Am
2Am
Am
2Am
’4 := −’3 :
3. Modi cations 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 − e12 − 3e22 e1 + 2e23 + e12 e2 = 0. The solution is
’(e1 ; e2 ) := 1 + e1 + e1 e2 = (e1 − e2 ) + 2e22 − e1 e2 , but if we directly apply our algorithm, we obtain an
in nite number of terms of degree one:
1 + e1 + e2 + e1−1 e22 + e1−2 e23 + e1−3 e24 + · · ·;
corresponding to an expansion of the term e1 e2 = (e1 − e2 ) of the solution, which mask 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
eA + Al + As
A(1 + e)
=
A(1 + e)
0
−Al (1 + el )
( As − A)(1 + el )
−Al (1 + el ) +
−As (1 + es )
−As (1 + es )
Al − A − es As −
−
0
0
e s Am +
;
252
A.B. Gonzalez et al. / Journal of Computational and Applied Mathematics 101 (1999) 243–254
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 := c11 e + c21 es + c31 eel + c41 el + c51 e−1 el 2 + c61 e−2 el 3 + · · ·;
where the coecients cij depend on A; As ; Al . We obtain an in nite 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 in nite 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=Am e − As =Am es − AAl =A2m eel :
If we continue, we obtain an in nite number of terms of degree two
−1 −1 2
−2
−2 3
−3 2 −3 4
As el ( A−1
m + A e + A Am e + A Am e + · · ·);
which correspond to the term
∞
X
Am
As
el
Am
Ae
j=0
j
=
AAs eel
=: ’11 :
(
Am Ae − Am )
Let f1 be the numerator of f0 (e; el ; es ; ; ’01 + ’11 + z ). By applying the algorithm to f1 we rstly
obtain one term
Al As =A2m el es := ’21
and then an in nite number of terms of total degree two
(A − Al )As A2m −4
(A − Al )As Am −3
(A − Al)As −2
4
2
e
e
e
+
3
e el es 4 + · · ·;
e
e
e
+
2
l
s
l
s
A2
A3
A4
which can be collected in
∞
X
(A − Al )As
Am
2
e
e
j
l s
2
2
Ae
Ae
j=1
j−1
=
(A − Al )As el es 2
=: ’31 :
(Ae − Am )2
Let f2 be the numerator of f1 (e; el ; es ; ; ’21 + ’31 + z ). By applying the algorithm, we obtain once
again an in nite number of terms of total degree two:
2
(A − Al )
As el es2
1 −3 2
Am −4 3
A2m −5 4
A3m −6 5
e
+
3
e
+
6
e
+
10
e +···
A3
A4
A5
A6
terms that we can write as
∞
j ( j − 1) Am j−2 (A − Al )2 As el es2 2
(A − Al )2 Al el es2 2 X
=: ’41 :
=
A3 e 3
2
Ae
(Ae − Am )3
j=2
!
;
A.B. Gonzalez et al. / Journal of Computational and Applied Mathematics 101 (1999) 243–254
253
Now if we observe ’31 + ’41 , an in nite number of terms appear:
As el
2
∞
X
(A − Al ) j+1 esj+1
j=0
(Ae − Am ) j+2
∞
(A − Al )es j
(A − Al )As es el 2 X
=
(Ae − Am )2 j=0 Ae − Am
=
(A − Al )As el es 2
=: ’51 :
(Ae − Am )(Ae − Am − (A − Al )es )
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 ce2 el , 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 substituted (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 substituted 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 di erential 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 e ort. We have implemented
c . This package is experimental at this moment,
a software package in Maple V called PoliVar
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
c (intended for the management of di erential polynomials), and we hope that they will
PoliDif
be uni ed (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.
References
[1] S.S. Abhyankar, On the rami cation of algebroid functions, Amer. J. Math. 77 (1955) 575 – 592.
[2] J.M. Farto, Multiplicity of the solutions of a di erential polynomial, J. Pure Appl. Algebra 108 (1996) 203 – 218.
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[3] J.M. Farto, Dicritic Di erential Polynomials, Annales de la Faculte des Sciences de Toulouse, in press.
[4] J.M. Farto, A.B. Gonzalez, D.J. Lopez, Solving strategies for an ODE at a singular point, J. Pure Appl. Algebra,
1997, submitted.
[5] J.M. Farto, A.B. Gonzalez, P. Martn, An algorithm for the systematic construction of solutions to perturbed problems,
Comput. Phys. Commun. 111 (1998) 110 –132.
[6] J. Getino, An interpretation of the core-mantle interaction problem, Geophys. J. Int. 120 (1995) 693 – 705.
[7] J. Getino, Forced nutations of a rigid mantle-liquid core Earth model in canonical formulation, Geophys. J. Int. 122
(1995) 803 – 814.
[8] J. Getino, J.M. Farto, J.M. Ferrandiz, Obtaining the free frequencies of the non-rigid Earth, Celes. Mech. Dyn.
Astrom., 1998, in press.
[9] J. Getino, J.M. Ferrandiz, On the e ect of the elastic mantle on the Earth rotation, Celes. Mech. Dyn. Astrom. 61
(1995) 117–180.
[10] J. Getino, J.M. Ferrandiz, Hamiltonian approach to dissipative phenomena between Earth mantle and core, and e ects
on free nutation, Geophys. J. Int. 130 (1997) 326–334.
[11] J. Getino, J.M. Ferrandiz, The e ect of the solid inner core on the free nutations of the Earth, preprint (1997).
[12] A.B. Gonzalez, J. Getino, The rotation of non-rigid, non-symmetrical Earth I: Free nutations, Celes. Mech. 68 (1997)
139–149.
[13] G. Hori, Theory of general perturbations with unspeci ed canonical variables, Publ. Astronom. Soc. Japan 18
(1966) 287.
[14] H. Kinoshita, Theory of the rotation of the rigid Earth, Celes. Mech. Dyn. Astrom. 15 (1977) 277– 326.
[15] R.J. Walker, Algebraic Curves, Dover Publications, New York, 1962.
[16] M.A. Zurro, The Abhyankar–Jung theorem revisited, J. Pure Appl. Algebra 90 (1993) 275 – 282.
An algorithm for an eigenvalues problem in the Earth rotation
theory
Ana B. Gonzalez a;∗ , Juan Getinob , Jose M. Fartoa
a
Grupo de Mecanica Celeste, Depto. Matematica Aplicada a la Ingeniera, E.T.S. Ingenieros Industriales,
Paseo del Cauce s=n, 47011 Valladolid. Spain
b
Grupo 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
c 1999 Elsevier Science B.V. All rights reserved.
called PoliVar.
AMS classi cation: 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 f is 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 (”);
truncated up to a given degree.
∗
Corresponding author. E-mail: [email protected], [email protected], [email protected].
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 7 ) 0 0 2 3 0 - 1
(2)
244
A.B. Gonzalez et al. / Journal of Computational and Applied Mathematics 101 (1999) 243–254
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 strati ed liquid
core [6, 7], and the improvement of this model by considering several new e ects 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 di erential 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 hypersurface 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. For s ¿ 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 constructive 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.
2. An ecient indeterminate coecients technique (based on the recursive construction of a sequence of convex-hulls and a tree search) to nd directly the exponents and the coecient of
each term.
A.B. Gonzalez et al. / Journal of Computational and Applied Mathematics 101 (1999) 243–254
245
Some examples from the Earth rotation theory are presented. In some (signi cative in the Earth
motion study) cases, negative exponents arise. Then the algorithm must be designed to deal with
them, and further modi cations 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
’(”) =
X
c ” ;
(4)
where = (1 ; : : : ; s ) ∈Q s and ” is ”1 1 · · · ”s s , is a solution of (3). Then the equality
f(”; ’(”)) = 0;
(5)
must be satis ed.
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
’=
X
c ” + 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 R s . For technical reasons, this order must be compatible
with the addition in R s , that is
1 62 ⇒ 1 + 62 + ; ∀1 ; 2 ; ∈R s :
(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 satis ed 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)
con guring its rst row. For every 1 ; 2 ∈R s , we shall say that 1 62 if and only if A1 6A2 for
the lexicographic order on R s .
Remark 2.1. In all this paper vectors are considered as row vectors, but if A is a matrix and is
a vector, by A we mean (At )t .
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A.B. Gonzalez et al. / Journal of Computational and Applied Mathematics 101 (1999) 243–254
In the following, we shall always consider the order for the monomials induced by the order
of R s de ned above, and we shall x the s × s matrix
1 1 1
0 1 0
A=0 0 1
. . .
. . .
. . .
0 0 0
···
···
···
..
.
1
0
0
:
..
.
··· 1
(9)
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=
X
d r ” z r :
We say that the set
D(f) := {( 1 ; : : : s ; r ) | d r 6= 0} ⊆ R s+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 c 6= 0. We make the formal substitution
f1 := f(”; (c ” + ’1 )) =
X
d r ” (c ” + ’1 )r :
By expanding the last expression we have
f1 :=
r
XX
j=0
d r
!
r j 1 +j1
· · · ” s s +js ’1r−j :
c ”
j 1
By the properties of the constructed order, given a term d r ” z r of f, the smallest monomial obtained
from it in f1 is
d r cr ” 11 +r1 · · · ”s s +rs :
(12)
Thus, for obtaining the smallest monomial in f1 , we must minimize the set
E (f; ) := { + r | ( ; r ) ∈D(f)} ⊆ R s :
Let
(f; ) = (1 ; : : : ; s ) := min E (f; );
which is well de ned by construction. Therefore, the smallest monomial of f1 is
X
( ; r) ∈ S(f; )
d r cr ” +r =
X
( ; r)∈S(f; )
d r cr ”(f; ) ;
A.B. Gonzalez et al. / Journal of Computational and Applied Mathematics 101 (1999) 243–254
247
where
S(f; ) := {( ; r ) ∈D(f) | + r = (f; )}
will be called generalized side for fand . By (5), f1 vanish. In particular, the smallest monomial
of f1 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
d r X r ;
( ; r) ∈ S(f; )
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 di erent monomials in (f; ) or equivalently there must be at least two di erent elements
in S(f; ) (and with the last components di erent). 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
Ẽ (f; ) := {A( + r) | ( ; r ) ∈D(f)}
for the lexicographic order. The following recurrence formulae characterize the value ˜ (f; ):
(0) S̃ 0 (f; ) := D̃(f).
(1) ˜j 6 j + r ˜ j ; ∀ ( ; r ) ∈ S̃j−1 (f; ), and
S̃j (f; ) := {( ; r ) ∈ S̃j−1 (f; ) | j + r ˜ j = ˜j }; j = 1; : : : ; s,
where ˜ = (˜ 1 ; : : : ; ˜ s ) := A, and D̃(f) := {(A ; r ) | ( ; r ) ∈D(f)}.
Obviously,
S(f; ) = {(A−1 ; r ) | ( ; r ) ∈ S̃ s (f; )}
and
(f; ) = A−1 ˜ (f; ):
Since
S̃j (f; ) ⊆ S̃ j−1 (f; ); j = 1; : : : ; s;
if a value for is valid, then #S̃j (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
D1 (f) := {( 1 ; r ) | ∃( 1 ; : : : ; s ; r ) ∈ D̃(f)} ⊆ R2 :
248
A.B. Gonzalez et al. / Journal of Computational and Applied Mathematics 101 (1999) 243–254
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, di ers 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 di erent points.
Once a valid ˜ 1 and the corresponding ˜ 1 are chosen, we iterate the construction for j = 2; : : : ; s,
by considering
∃( ; r ) ∈ D̃(f) with
Dj (f) := ( j ; r )
( ; r ) ∈S (f; ˜ ; : : : ; ˜ );
k
k
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
way that Nj (f; ˜ 1 ; : : : ; ˜ j−1 ) has a side of slope −1= ˜ j . Then ˜j is the abscise of the intersection of
A.B. Gonzalez et al. / Journal of Computational and Applied Mathematics 101 (1999) 243–254
249
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 di erent 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 ˜ ∈Q s 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
0
(C − Am )=Am
M :=
0
Cc =Am
(Bm − C )= (Bm )
0
C=Bm
0
−C=Am
0
;
−Cc =Bm
0
BCc = (Bm Bc )
0
−ACc = (Ac Am )
0
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A.B. Gonzalez et al. / Journal of Computational and Applied Mathematics 101 (1999) 243–254
Fig. 2. Resolution tree of Example 2.3.
where
Am = A − Ac ; Bm = B − Bc ; C = A(1 + e); Cc = Ac (1 + ec );
B=
A(1 + e)
;
1 + ê
Bc =
Ac (1 + ec )
:
1 + êc
Let f(e; ê; ec ; êc ; z ) be the numerator of det(M − zI4 ). We shall apply the algorithm to the equation
f = 0, to express z as certain truncated parameterizations up to total degree two in the variables
(e; ê; ec ; êc ), with the order matrix as in (9). In the rst stage, we can see that we have four di erent
sets of exponents and coecients, corresponding to all the eigenvalues of the matrix. For two of
them, we must apply nine stages to reach the total degree three. For the others we need to perform
only four stages. The results of the four resolutions process can be represented as the resolution
tree (see [4]) of Fig. 2, where
p11 := (0; 0; 0; 0);
p21 := (0; 0; 0; 1);
p31 := (0; 0; 1; 0);
p41 := (1; 0; 0; 1);
p51 := (0; 0; 0; 2);
p61 := (1; 0; 1; 0);
p71 := (0; 0; 1; 1);
p81 := (0; 0; 2; 0);
p91 := (2; 0; 0; 1);
p13 := ( 12 ; 12 ; 0; 0);
p23 :=( 23 ; 12 ; 0; 0);
p33 :=( 21 ; 21 ; 0; 1);
p43 := ( 21 ; 32 ; 0; 0);
p53 :=( 25 ; 12 ; 0; 0);
c11 := i; c21 :=
iA
−iA2
iA
−iAAc
;
c
:=
;
; c31 :=
; c41 :=
51
Am
Am
2A2m
8A2m
c61 :=
iA(A − 2Ac )
−iA(A − 4Ac )
−iAAc
; c71 :=
; c81 :=
;
2A2m
4A2m
8A2m
c91 :=
iA2 Ac
iA
−iAAc
iAAc
−iAAc
; c13 :=
; c33 :=
; c43 := 2 ;
; c23 :=
3
2
2
2Am
Am
2Am
Am
2Am
c53 :=
iAAc (4A − Ac )
8A3m
A.B. Gonzalez et al. / Journal of Computational and Applied Mathematics 101 (1999) 243–254
251
and
N11 := (p11 ; c11 );
N31 := (p31 ; c31 );
N51 := (p51 ; c51 );
N71 := (p71 ; c71 );
N91 := (p91 ; c91 );
N23 := (p23 ; c23 );
N43 := (p43 ; c43 );
N12 := (p11 ; −c11 );
N32 := (p31 ; −c31 );
N52 := (p51 ; −c51 );
N72 := (p71 ; −c71 );
N92 := (p91 ; −c91 );
N24 := (p23 ; −c23 );
N44 := (p43 ; −c43 );
N21 := (p21 ; c21 );
N41 := (p41 ; c41 );
N61 := (p61 ; c61 );
N81 := (p81 ; c81 );
N13 := (p13 ; c13 );
N33 := (p33 ; c33 );
N53 := (p53 ; c53 );
N22 := (p21 ; −c21 );
N42 := (p41 ; −c41 );
N62 := (p61 ; −c61 );
N82 := (p81 ; −c81 );
N14 := (p13 ; −c13 );
N34 := (p33 ; −c33 );
N54 := (p53 ; −c53 ):
Thus, the four solutions for f up to degree two are:
’1 := i 1 +
AAc
A2
A
AAc
A
êc +
ec − 2 eêc − 2 ê2c − 2 eec
Am
Am
2Am
8Am
2Am
!
A(A − 2Ac )
A(A − 4Ac ) 2
+
ec êc −
ec ;
2
4Am
8A2m
’2 := −’1 ;
A 1=2 1=2 AAc 3=2 1=2 AAc 1=2 1=2
AAc 1=2 3=2
;
’3 := i
e ê − 2 e ê − 2 e ê êc + 2 e ê
Am
2Am
Am
2Am
’4 := −’3 :
3. Modi cations 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 − e12 − 3e22 e1 + 2e23 + e12 e2 = 0. The solution is
’(e1 ; e2 ) := 1 + e1 + e1 e2 = (e1 − e2 ) + 2e22 − e1 e2 , but if we directly apply our algorithm, we obtain an
in nite number of terms of degree one:
1 + e1 + e2 + e1−1 e22 + e1−2 e23 + e1−3 e24 + · · ·;
corresponding to an expansion of the term e1 e2 = (e1 − e2 ) of the solution, which mask 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
eA + Al + As
A(1 + e)
=
A(1 + e)
0
−Al (1 + el )
( As − A)(1 + el )
−Al (1 + el ) +
−As (1 + es )
−As (1 + es )
Al − A − es As −
−
0
0
e s Am +
;
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A.B. Gonzalez et al. / Journal of Computational and Applied Mathematics 101 (1999) 243–254
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 := c11 e + c21 es + c31 eel + c41 el + c51 e−1 el 2 + c61 e−2 el 3 + · · ·;
where the coecients cij depend on A; As ; Al . We obtain an in nite 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 in nite 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=Am e − As =Am es − AAl =A2m eel :
If we continue, we obtain an in nite number of terms of degree two
−1 −1 2
−2
−2 3
−3 2 −3 4
As el ( A−1
m + A e + A Am e + A Am e + · · ·);
which correspond to the term
∞
X
Am
As
el
Am
Ae
j=0
j
=
AAs eel
=: ’11 :
(
Am Ae − Am )
Let f1 be the numerator of f0 (e; el ; es ; ; ’01 + ’11 + z ). By applying the algorithm to f1 we rstly
obtain one term
Al As =A2m el es := ’21
and then an in nite number of terms of total degree two
(A − Al )As A2m −4
(A − Al )As Am −3
(A − Al)As −2
4
2
e
e
e
+
3
e el es 4 + · · ·;
e
e
e
+
2
l
s
l
s
A2
A3
A4
which can be collected in
∞
X
(A − Al )As
Am
2
e
e
j
l s
2
2
Ae
Ae
j=1
j−1
=
(A − Al )As el es 2
=: ’31 :
(Ae − Am )2
Let f2 be the numerator of f1 (e; el ; es ; ; ’21 + ’31 + z ). By applying the algorithm, we obtain once
again an in nite number of terms of total degree two:
2
(A − Al )
As el es2
1 −3 2
Am −4 3
A2m −5 4
A3m −6 5
e
+
3
e
+
6
e
+
10
e +···
A3
A4
A5
A6
terms that we can write as
∞
j ( j − 1) Am j−2 (A − Al )2 As el es2 2
(A − Al )2 Al el es2 2 X
=: ’41 :
=
A3 e 3
2
Ae
(Ae − Am )3
j=2
!
;
A.B. Gonzalez et al. / Journal of Computational and Applied Mathematics 101 (1999) 243–254
253
Now if we observe ’31 + ’41 , an in nite number of terms appear:
As el
2
∞
X
(A − Al ) j+1 esj+1
j=0
(Ae − Am ) j+2
∞
(A − Al )es j
(A − Al )As es el 2 X
=
(Ae − Am )2 j=0 Ae − Am
=
(A − Al )As el es 2
=: ’51 :
(Ae − Am )(Ae − Am − (A − Al )es )
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 ce2 el , 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 substituted (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 substituted 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 di erential 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 e ort. We have implemented
c . This package is experimental at this moment,
a software package in Maple V called PoliVar
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
c (intended for the management of di erential polynomials), and we hope that they will
PoliDif
be uni ed (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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254
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