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ArticleTitle The “phase function” method to solve second-order asymptotically polynomial differential equations Article Sub-Title
Article CopyRight Springer-Verlag
(This will be the copyright line in the final PDF) Journal Name Numerische Mathematik
Corresponding Author Family Name Spigler
Particle
Given Name Renato
Suffix
Division Dipartimento di Matematica
Organization Universit à “Roma Tre”
Address 1, Largo S.L. Murialdo, 00146, Rome, Italy
Email marcov@math.unipd.it
Author Family Name Vianello
Particle
Given Name Marco
Suffix
Division Dipartimento di Matematica Pura e Applicata Organization Universit à di Padova
Address 63, Via Trieste, 35121, Padua, Italy Email
Schedule
Received 24 September 2010
Revised 29 November 2011
Accepted
Abstract The Liouville-Green (WKB) asymptotic theory is used along with the Borůvka’s transformation theory, to obtain asymptotic approximations of “phase functions” for second-order linear differential equations, whose coefficients are asymptotically polynomial. An efficient numerical method to compute zeros of solutions or even the solutions themselves in such highly oscillatory problems is then derived. Numerical examples, where symbolic manipulations are also used, are provided to illustrate the performance of the method.
Mathematics Subject Classification (2000) (separated by '-')
Primary: 65L99 - 34E20 - Secondary: 65D20
Footnote Information Work supported, in part, by the Italian GNFM-INdAM and GNIM-INdAM.
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we have followed jobsheet).
Journal: 211
Article: 441
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Numer. Math.
DOI 10.1007/s00211-011-0441-9 Mathematik
The “phase function” method to solve second-order asymptotically polynomial differential equations
Renato Spigler · Marco Vianello
Received: 24 September 2010 / Revised: 29 November 2011
© Springer-Verlag 2011
Abstract The Liouville-Green (WKB) asymptotic theory is used along with the
1
Bor˚uvka’s transformation theory, to obtain asymptotic approximations of “phase func-
2
tions” for second-order linear differential equations, whose coefficients are asymp-
3
totically polynomial. An efficient numerical method to compute zeros of solutions
4
or even the solutions themselves in such highly oscillatory problems is then derived.
5
Numerical examples, where symbolic manipulations are also used, are provided to
6
illustrate the performance of the method.
7
Mathematics Subject Classification (2000) Primary: 65L99 · 34E20; Secondary:
8
65D20
9
1 Introduction
10
An efficient asymptotic-numerical method, to approximate zeros of solutions of
11
second-order linear differential equations,
12
y ′′ + q(x)y = 0, (1)
13
on a half-line, was developed in [28,29 ] for the oscillatory case with q(x) = a +b/x +
14
O(x − p ), a > 0, b ∈ R, p > 1. Such result has been obtained connecting Bor˚uvka’s
15
Work supported, in part, by the Italian GNFM-INdAM and GNIM-INdAM.
R. Spigler ( B )
Dipartimento di Matematica, Università “Roma Tre”, 1, Largo S.L. Murialdo, 00146 Rome, Italy e-mail: marcov@math.unipd.it
M. Vianello
Dipartimento di Matematica Pura e Applicata, Università di Padova, 63, Via Trieste, 35121 Padua, Italy
Author Proof
uncorrected
proof
transformation theory ([3,4], [21, §§5.1, 9.1], [30]) to an Olver’s idea developed to
16
compute zeros of cylinder (Bessel) functions [22,23]. This approach actually allows
17
to compute also the solutions to (1). In this paper, we extend our method to the case
18
of an asymptotically polynomial coefficient,
19
q(x) ∼ cx m , x → +∞, c, m ∈ R + . (2)
20
Equivalently, q(x) = cx m [1 + o(1)] as x → +∞. More precisely, q(x) is required to
21
be the restriction to the real half-line x > ρ of a function, q(z), analytic in the annular
22
sector
23
S ρ,γ := {z ∈ C : |z| > ρ, | arg z| < γ }, (3)
24
with ρ ≥ 0, 0 < γ ≤ π/2, and
25
q(z) ∼ cz m , z → ∞ in S ρ,γ . (4)
26
The exponent m does not need to be an integer, in which case it is understood that
27
the principal branch of z m has to be considered. Without any loss of generality, we
28
stipulate that q(z) does not vanish in S ρ,γ , and thus q(x) > 0 for x > ρ.
29
We shall confine ourselves to the reduced form (1), to which every smooth second-
30
order linear differential equation, y ′′ +a(x)y ′ +b(x)y = 0, can be taken (see e.g. [ 4]),
31
with q(x) = b(x) − a 2 (x)/ 4 − a ′ (x)/ 2. In particular, when a(z) ∼ αz n , b(z) ∼ βz m ,
32
as z → ∞ in S ρ
0,γ
0, with m, n ∈ N, m > 2n, and β > 0, assuming that a ′ (z) ∼
33
αnz n −1 in an annular subsector S ρ,γ ⊆ S ρ
0,γ
0, we are led to an equation like (1)
34
with q(z) ∼ βz m in S ρ,γ , so that q(x) ∼ βx m → +∞ as x → +∞. Note that
35
the assumption concerning asymptotic differentiation of a(z) is satisfied, e.g., when
36
a(z) = αz n 1 + O(z − p ) , p > 0, see [23,31].
37
All the previous assumptions are satisfied, in particular, when a(x) := P a (x)/Q a
38
(x) and b(x) := P b (x)/Q b (x) , are rational functions, with b(x) → +∞ as x → +∞,
39
and deg (P b ) − deg (Q b ) > 2 (deg (P a ) − deg (Q a )). In this case, S ρ
0,γ
0= S ρ,γ is
40
an annular sector which leaves out the poles of a(x) and b(x), taking ρ sufficiently
41
large and/or γ sufficiently small. Note that, in fact, when a(x) and b(x) are rational,
42
so is q(x), and, in particular, when a(x) and b(x) are polynomials, so is q(x), with
43
deg (q) ≤ max{deg (b), 2 deg (a)}.
44
It is an immediate consequence of the Sturm comparison theorem (see e.g. [13,
45
Ch. 11]), that all solutions of (1)–(2) exhibit an extremely rapid oscillatory behavior,
46
with frequency increasing unboundedly. The numerical treatment of such problems
47
is known to represent a difficult task, which calls for special discretization methods
48
(see e.g. [14,24,27], and references therein). Since time-stepping methods, however,
49
require exceedingly small time-steps, and the global errors become very large due
50
to the large values taken by the higher derivatives of solutions [16,17], recently, a
51
number of alternatives have been proposed [6,18]. Some of these are, e.g., Magnus,
52
Cayley, and Neumann methods, which are related to Lie group techniques [18].
53
Others, like Filon-type methods, rest on suitable interpolations [6,7]. They are all
54
based on the reformulation of the differential equations in terms of highly oscillatory
55
Author Proof
uncorrected
proof
integrals. All the algorithms of the first group come from the Magnus series expansion
56
of solutions or variants of it, those of the second group involve suitable polynomial
57
(Hermite) interpolation of part of the integrand, namely the slowly varying amplitude
58
factor. In a recent application to forced oscillator equations arising from electronic
59
circuit simulation, Filon-type quadratures have been used very successfully [6,7]. An
60
extension of such problems was treated efficiently by still another method, based on
61
modulated Fourier series [8]. Throughout, extensive exploitation of asymptotics and,
62
when possible, of symbolic manipulations, have proved to be quite useful.
63
In this paper, we consider still another method to solve oscillator equations as (1),
64
whose coefficient q(x) is asymptotically polynomial. In some cases treated in [16,17],
65
q(x) was required to diverge to +∞ as x → +∞, while its derivatives should remain
66
small, in the same limit. The algorithm proposed in the present paper is not affected
67
by such limitation, even though here we do not consider such cases, e.g. when q(x)
68
diverges exponentially. Moreover, also the present algorithm avoids time-stepping and
69
seems to perform even better when oscillations are faster, as in the aforementioned
70
methods.
71
The inherent difficulties in handling rapidly oscillatory solutions strongly affects,
72
in particular, the numerical approximation of zeros of solutions, whenever one
73
computes them via the approximation of the solutions themselves. As in [29], instead
74
of evaluating in advance the solutions to (1)–(2), our approach for computing zeros of
75
any given solution consists in obtaining an asymptotic-numerical approximation of a
76
related function, that is one of the so-called “phases” or “phase functions”. A phase
77
function, α(x), is any C 3 -solution of tan α(x) = u(x)/v(x), where (u, v) denotes a
78
basis for equation (1). By differentiation, we obtain
79
α ′ (x) = − W
u 2 (x) + v 2 (x) , (5)
80
where W := W [u, v] is the (constant) Wronskian of u, v. By further differentiations
81
and using (1), the following third-order nonlinear differential equation,
82
α ′2 (x) = q(x) − 1
2 {α, x}, {α, x} := α ′′′ (x) α ′ (x) − 3
2
α ′′ (x) α ′ (x)
2
, (6)
83
is obtained. In (6 ), {α, x} represents the so-called “Schwarzian derivative” of α,
84
and the close-form equation satisfied by α is a special instance of the “Kummer
85
differential equation”, which plays a key role in the transformation theory of linear
86
ordinary second-order differential equations [3,4,21,30]. Indeed, every solution of (6)
87
is the “kernel” of a transformation which takes equation (1) into Y ′′ + Y = 0. As a
88
consequence of this fact, it is easily shown that every solution to (6) is a phase function
89
related to the basis
90
u(x) = |α ′ (x) | −1/2 sin α(x), v(x) = |α ′ (x) | −1/2 cos α(x), (7)
91
[3,4]. Therefore, knowing a single phase function, it is possible to retrieve a basis
92
(and thus all solutions) to equation (1). In addition, zeros of solutions can be directly
93
Author Proof
uncorrected
proof
obtained by means of α(x), since it can be proved that
94
|α(x k ) − α(x k−1 ) | = π (8)
95
for any two consecutive zeros, x k −1 and x k , of any given solution of (1): From (8), in
96
fact, given α(x) and one of the two zeros, it is possible to evaluate the other, solving
97
a nonlinear equation like α(x) = const. (note that, by ( 5), all phase functions are
98
strictly monotone).
99
Even though, in principle, obtaining a phase function α(x) requires solving a third-
100
order nonlinear differential equation, we consider the iterative scheme
101
(α ′ 0 ) 2 (x) = q(x), (α n ′ +1 ) 2 (x) = q(x) − 1
2 {α n , x }, n = 0, 1, 2, . . . , (9)
102
see [29], and show that, under the asymptotic assumptions in (4), α n (x) converges (in
103
a suitable sense) to a special phase function, α(x) ≡ α L G (x), the so-called Liouville–
104
Green phase. Such a convergence is usually very fast, and thus the method becomes
105
very competitive when compared to methods based on the preliminary numerical
106
solution of the original differential equation, (1).
107
Here is the plan of the paper. In Sect. 2, we develop the basic asymptotic anal-
108
ysis, obtaining the representation α ′2 L G (z) = q(z) + O(z −2 ) in a suitable subsector
109
of S ρ,γ . This makes it reasonable the initial approximation in the iterative scheme in
110
(9), whose convergence is proved in Sect. 3. In Sect. 4, we show that our method can
111
be implemented efficiently, in particular when q(x) is a polynomial, and numerical
112
examples are given for the purpose of illustration.
113
2 Preliminary asymptotic analysis
114
Setting
115
φ := (α ′ ) 2 , φ n := (α n ′ ) 2 , (10)
116
equation (6) transforms into
117
φ (x) = q(x) + [φ, x], [φ, x] := − 1 4
φ ′′ (x) φ (x) + 5
16
φ ′ (x) φ (x)
2
, (11)
118
and the iterative scheme in (9) becomes
119
φ 0 (x) = q(x), φ n+1 (x) = q(x) + [φ n , x ], n = 0, 1, 2, . . . (12)
120
Note that the third-order problem in (6) has been reduced to a second-order problem
121
(in (11)), and α can be recovered by a quadrature. Moreover, the scheme in (12) has
122
the advantage, with respect to that in (9), of avoiding the evaluation of square roots.
123
This circumvents the difficulties due to the complexification of the problem. In fact,
124
Author Proof
uncorrected
proof
both, asymptotic estimates and proof of convergence, will be performed in suitable
125
sectors of C, since we need to control derivatives.
126
In the next section, we shall prove the convergence (in a suitable sense) of φ n to a
127
particular solution of equation (11 ), φ ≡ φ L G , which is the square of the derivative
128
of any “Liouville-Green phase”, that is any phase function related to a real-valued
129
Liouville-Green (WKB) basis of equation (1).
130
Recall that a complex-valued Liouville-Green (L G) basis is, according to Olver’s
131
theorem ([23, Ch. 6, Thm. 11.1], see also [32]),
132
U ± (z) = q −1/4 (z) e ±iξ(z) [1 + ε ± (z) ], (13)
133
where
134
ξ(z) =
z
q 1/2 (t ) dt, (14)
135
and
136
|ε ± (z) | ≤ exp V z, ± ∞ (F ) − 1, |ε ± ′ (z) | ≤ |q 1/2 (z) | exp V z, ± ∞ (F ) − 1 , (15)
137
the z-function V z,∞ ± (F ) denoting the variation of
138
F ≡ F(z) :=
z
q −1/4 D 2 q −1/4
dt (16)
139
along the path ℓ ± , see [23]. The paths ℓ ± ≡ ℓ ± (z) connect z to ∞ remaining inside a
140
suitable annular subsector of S ρ,γ , and are required to be ξ -progressive, i.e., Im ξ(z)
141
must be nonincreasing along ℓ − and nondecreasing along ℓ + . In (14), (15), (16), the
142
branches of the fractional powers of q(z) must be continuous, that of q 1/2 (z) being
143
the square of q 1/4 (z). Note that in (13) a family of bases is actually defined, corre-
144
spondingly to the arbitrary choice of the primitive, ξ(z).
145
When q(z) is as in (4), we can show that the estimates
146
ε ± (z) = O(z −m/2−1 ), q −1/2 (z)ε ′ ± (z) = O(z −m/2−1 ), z ∈ S ρ
′,γ
′⊂ S ρ,γ ,
147
(17)
148
hold, for ρ ′ > ρ sufficiently large, and γ ′ < min {γ, π/(m + 2)}. Such estimates have
149
been obtained using as path ℓ + the circular arc η ≡ η z (θ ) = |z|e i θ , arg z ≤ θ ≤ γ ′ ,
150
followed by the ray η ≡ η z (r ) = re i γ
′, r ≥ |z|, whereas ℓ − is the circular arc η ≡
151
η z (θ ) = |z|e i θ , arg z ≥ θ ≥ −γ ′ , followed by the ray η ≡ η z (r ) = re −iγ
′, r ≥ |z|.
152
Let check that such paths are indeed ξ -progressive. In fact, on the circular arc of
153
ℓ + , we have
154
d
dθ Im ξ(η(θ )) = Im dξ dη
dη dθ
= Im
q 1/2 (η(θ ))η ′ (θ )
155
= √ c Im
η m/2 (θ )η ′ (θ ) ( 1 + ω)
, (18)
156
Author Proof
uncorrected
proof
where ω → 0 as z → ∞ in S ρ,γ . Now,
157
η m/2 η ′ = |z| m/ 2+1 e i [(m/2+1)θ+π/2] , (19)
158
hence
159
Im
η m/2 η ′ ( 1 + ω)
= |z| m/ 2+1
cos m 2 + 1
θ
( 1 + Re ω)
160
− sin m 2 + 1
θ Im ω
∼ |z| m/ 2+1 cos m 2 + 1
θ
(20)
161
as z → ∞ in S ρ,γ
′, where 0 < γ ′ < π/(m + 2). Therefore, the left-hand side is
162
strictly positive for |z| ≥ ρ 1 > ρ, with ρ 1 sufficiently large (and | arg z| ≤ γ ′ ). This
163
shows that Im ξ(η(θ )) increases on the circular arc in such subsector.
164
Similarly, on the ray following the circular arc in ℓ + ,
165
d
dr Im ξ(η(r )) = √ c r m/2
sin m 2 +1
γ ′
( 1+Re ω)+cos m 2 +1
γ ′ Im ω
166
∼ √
c r m/2 sin m 2 + 1
γ ′
(21)
167
as z → ∞ in S ρ,γ
′, ω being an infinitesimal. Again, the left-hand side is positive for
168
|z| ≥ ρ 2 > ρ, for ρ 2 sufficiently large.
169
An analogous procedure is adopted to show that the path ℓ − is also ξ -progressive
170
in S ρ,γ
′, with ρ sufficiently large. Below, we denote with ρ ′ the largest among the
171
previous lower bounds for ρ (i.e., ρ 1 , ρ 2 , and the corresponding radii concerning ℓ − ).
172
In order to establish the asymptotic estimate in (17), we compute first (by Ritt’s
173
theorem [23, Thm.4.2, p. 9], [31])
174
q −1/4 D 2 q −1/4 = O
z −m/2−2
, (22)
175
valid in any fixed subsector of S ρ,γ , for instance in S ρ
′,γ
′, and thus, after a little algebra,
176
we obtain
177
V z, ± ∞ (F ) :=
ℓ
1±q −1/4 D 2 q −1/4
| d z | +
ℓ
2±q −1/4 D 2 q −1/4
| d z |
178
≤ 2K
2
M + 1 + γ ′
|z| −m/2−1 , z ∈ S ρ
′,γ
′, (23)
179
where the superscript appended to ℓ ± refers to a circular arc (index 1) or to a ray
180
(index 2), and K is the constant implied by the O-symbol in (22). From this, (17)
181
follows, since e V − 1 = O(V ) as V → 0.
182
Author Proof
uncorrected
proof
From any complex L G basis in (13), we obtain a real L G basis,
183
u(x) := Re U + (x) = U + (x) + U − (x)
2 , v(x) := Im U + (x) = U + (x) − U − (x)
2i ,
184
(24)
185
on the real half-line x > ρ ′ , since U − (x) = U + (x).
186
We stipulate that a primitive ξ(z), real for z real, has been chosen, which is possible
187
since q(z) has such property, q(x) > 0, and the appropriate branch of the square root
188
has been selected. Then, we define
189
φ L G (x) := W 2 [u, v]
[u 2 (x) + v 2 (x) ] 2 ≡ 1
[u 2 (x) + v 2 (x) ] 2 , (25)
190
which is the square of the derivative of every phase defined by the family of bases in
191
(24), see (10). Recall, once for all, that φ L G is invariant in the family of real L G bases
192
(obtained varying the real constant of integration in (14)). In fact, all such bases are
193
related to each other by a unimodal orthogonal transformation, and the corresponding
194
phases are related to each other by an additive constant. The Wronskian in (25), which
195
is clearly a constant (see (1)), turns out to be equal to 1 for any L G basis. In fact,
196
simple direct calculations, using (24) and (13), yield
197
W [u, v] = 1 2i
2i (1 + ε + )( 1 + ε − ) + q −1/2 ( 1 + ε − )ε ′ + − q −1/2 ( 1 + ε + )ε − ′ ,
198
(26)
199
and thus, by (17 ), we obtain W [u, v] → 1 as x → +∞, and hence W [u, v] ≡ 1.
200
Extending analytically φ L G to the complex sector S ρ
′,γ
′, by using (24) and (25),
201
we obtain the asymptotic representation for φ L G (z)
202
φ L G (z) = 1
q −1/2 (z) [1 + ε + (z) ][1 + ε − (z) ] 2 = q(z) 1 + O
z −m/2−1 ,
203
(27)
204
valid in S ρ
′′,γ
′, where |ε ± (z) | < 1 for |z| > ρ ′′ . Therefore, by (4), φ L G (z) =
205
O (z m ) + O z m/ 2−1 = O (z m ) in S ρ
′,γ
′, while, by Ritt’s theorem ([23, Thm.4.2,
206
p. 9]; see also [31]),
207
φ L G ′ (z) = O z m −1
, φ ′′ L G (z) = O z m −2
, (28)
208
in any fixed annular subsector of S ρ
′′,γ
′, say S ρ
∗,γ
∗. Inserting these in the right-hand
209
side of the complexified differential equation (11) (recalling that φ L G (z) = 0 in
210
Author Proof
uncorrected
proof
S ρ
∗,γ
∗), we obtain
211
[φ L G , z ] = − 1 4
O z m −2
q(z) 1 + O z −m/2−1 + 5 16
O z m −1 q(z) 1 + O z −m/2−1
2
212
= O z −2
, in S ρ
∗,γ
∗, (29)
213
and hence
214
φ L G (z) = q(z) + O z −2
, z ∈ S ρ
∗,γ
∗. (30)
215
Note that this represents a refinement of the asymptotic estimate in (27).
216
3 An iterative method for approximating L G phases
217
In order to prove the convergence of the iterative scheme in (12), we embed the
218
problem in the complex domain, that is we consider the analytic extension of both,
219
equation (11 ) for φ ≡ φ L G , and equation (12) for φ n +1 ,
220
φ L G (z) = q(z) + [φ L G (z), z ], (31)
221
φ 0 (z) = q(z), φ n+1 (z) = q(z) + [φ n (z), z ], n = 0, 1, 2, . . . , (32)
222
where z ∈ S ρ
∗,γ
∗(the annular sector in (30)). Such a procedure is suggested by the
223
need of “controlling” the derivatives in the iterative scheme (32). We start with a
224
technical lemma.
225
Lemma 3.1 Consider the complex-valued function
226
G(w 1 , w 2 , w 3 ) := q(z) − 1 4
w 3 w 1 + 5
16
w 2 w 1
2
(33)
227
of the three complex variables w 1 , w 2 , w 3 , defined in the closed polydisc of C 3
228
P = D 1 × D 2 × D 3 , where D k := B
φ (k L G −1) (z) ; ε k (z)
, k = 1, 2, 3, z ∈ S ρ
∗,γ
∗.
229
Here B(a ; r) ⊂ C denotes the open disc centered at a with radius r, and choose
230
ε 1 (z) = |φ L G (z) |/2 and ε k (z) = |φ (k L G −1) (z) |, k = 2, 3. Then, the following Lipschitz
231
estimate holds,
232
|G(w) − G(v)| = χ(z)||w − v|| ∞ , ∀w, v ∈ P, (34)
233
for every fixed z ∈ S ρ
∗,γ
∗, where we set w = (w 1 , w 2 , w 3 ), v = (v 1 , v 2 , v 3 ), and the
234
estimate
235
0 ≤ χ(z) ≤ C|z| −m (35)
236
holds, C being a suitable constant.
237
Author Proof
uncorrected
proof
Proof By Hartog’s theorem [26, Ch. 1], G is analytic in P, being separately analytic
238
in each of its arguments, w k ’s. In fact, w 1 ∈ B(φ L G (z) ; ε 1 (z)) implies that
239
|w 1 | ≥ | |φ L G (z) | − |φ L G (z) − w 1 | | ≥ |φ L G (z) | − ε 1 (z) = |φ L G (z) |
2 > 0. (36)
240
By the (generalized) mean value theorem,
241
|G(w) − G(v)| ≤ max
s ∈[w,v] {|grad G(s)|} ||w − v|| ∞ . (37)
242
This property holds in general for C 1 -mappings, G : (U ⊂ B 1 ) → B 2 , where U is
243
a convex subset of B 1 , B 1 and B 2 are Banach spaces, see e.g. [19], and the triple bar
244
denotes the norm of the Frêchét derivative of G. In (37 ), [w, v] denotes the segment
245
in C 3 , joining w and v, which is a subset of P, for any w, v ∈ P, P being a convex
246
set in C 3 . To estimate grad G, using (36) and observing that, by (27)–(28 ), |w 1 | −1 =
247
O z −m , |w 2 | ≤ 2
φ ′ L G (z)
= O z m −1 , |w 3 | ≤ 2
φ L G ′′ (z)
= O z m −2 in P, we
248
first evaluate
249
∂G
∂w 1
≤
w 3 4w 2 1
+ 5
8
w 2 2 w 3 1
= O z −2m
+ O z −m−2
,
250
∂G
∂w 2
= 5 8
w 2 w 1 2
= O z −2m
+ O z −m−1
, (38)
251
∂G
∂w 3
= 1
4|w 1 | = O z −m ,
252
valid for every m > 0. Then
253
sup
s ∈P |grad G(s)| ≤ sup
s ∈P 3
k =1
∂ G
∂w k
=: χ(z) = O z −m , (39)
254
for every z ∈ S ρ
∗,γ
∗. Therefore, the estimate in (34)–(35) is proved, in view of (37),
255
C being the constant implied by the O-symbol in (39). ⊓ ⊔
256
We are now ready to state the main convergence theorem.
257
Theorem 3.2 Assume that equation (1) be given, with q(x) asymptotically polyno-
258
mial as in (2), (3), (4). Then, there exists x 0 ∈ (ρ, +∞), depending on all parameters
259
of the problem, such that φ n (x) in the scheme (12) converges to the function φ L G (x)
260
defined in (25), in the sense that
261
|φ L G (x) − φ n (x) | ≤ C h n (x) x − n √
2 −2
, x > x n , (40)
262
Author Proof
uncorrected
proof
where
263
h 0 (x) ≡ 1, h n (x) := C n
n −1
j =0
x − j √
2 −m
, n = 1, 2, 3, . . . , (41)
264
being x n := x 0 + n √
2/ sin γ ∗ , x 0 ≥ ρ ∗ and n = 0, 1, 2, . . .; γ ∗ and ρ ∗ are, respec-
265
tively, the semi-angle and the radius of the annular sector in (30), and C the constant
266
in (35).
267
Remark 3.3 While the convergence of φ n to φ L G in the case of an asymptotically
268
constant coefficient is linear (see [28,29 ], with m = 0), the inequality in ( 40) as well
269
as the numerical results in the examples below suggest that now the convergence might
270
be superlinear,
271
|φ L G (x) − φ n (x) | ≤ C n +1 x 0 −2
n−1
j =0
x 0 + (n − j) √ 2 −m
272
= C x 0 2
C 2 m/2
n ⎡
⎣ Ŵ
x
0√ 2 + 1 Ŵ
x
0√ 2 + 1 + n
⎤
⎦
m
, uniformly for x > x 0 + n √ 2 sin γ ∗ , (42)
273
where Ŵ(·) is the Euler gamma function. Note that the convergence rate increases
274
with m, which is an essential feature of the present method. Large values of m imply
275
extremely rapid oscillations of solutions to equation (1).
276
Remark 3.4 As it will be clear in the proof below, the convergence results in (40)–(41)
277
are also valid in the complex domain and, more precisely, for z replacing x, z ∈ S n ≡
278
S(0, γ ∗ ) + x 0 + n √
2/ sin γ ∗ . Note that the geometrical structure of such a decreasing
279
sequence of sectors parallels the similar one introduced in [29].
280
Proof Theorem 3.2 has been stated in R, but we shall prove it in C, for technical
281
reasons. The proof will be inductive on n. By (30),
282
|φ L G (z) − φ 0 (z) | = |φ L G (z) − q(z)| ≤ C |z| −2 , z ∈ S 0 := S(0, γ ∗ ) + x 0 , (43)
283
and assume
284
|φ L G (z) − φ n (z) | ≤ C h n ( |z|)(|z| − n √
2) −2 , z ∈ S n := S 0 + n √ 2
sin γ ∗ , (44)
285
for a fixed n, n = 1, 2, 3, . . ., where h n ( |z|) is defined in ( 41), and x 0 ≥ ρ ∗ will be
286
given below. Then,
287
|φ L G (z) − φ n +1 (z) | = | [φ L G (z), z ] − [φ n (z), z ] |
288
≡ | G(φ L G (z), φ ′ L G (z), φ ′′ L G (z)) − G(φ n (z), φ n ′ (z), φ n ′′ (z)) |
289
≤ sup
s ∈[w,w
n] {|grad G(s)|} ||w − w n || ∞ , (45)
290
Author Proof
uncorrected
proof
where w := (φ L G (z), φ ′ L G (z), φ L G ′′ (z)), w n := (φ n (z), φ ′ n (z), φ ′′ n (z)), and, by
291
Lemma 3.1,
292
|φ L G (z) − φ n +1 (z) | ≤ C |z| −m ||w − w n || ∞
293
≤ C |z| −m max |φ L G (z) − φ n (z) |, |φ ′ L G (z) − φ n ′ (z) |, |φ ′′ L G (z) − φ n ′′ (z) | , (46)
294
provided that w n ∈ P (the polydisc in Lemma 3.1). By Cauchy integral formula,
295
φ L G (k) (z) − φ n (k) (z)
=
k ! 2π i
Ŵ
φ L G (u) − φ n (u) (u − z) k +1 du
296
≤ k ! ( √
2) k sup
u∈Ŵ |φ L G (u) − φ n (u) |, z ∈ S n +1 , k = 0, 1, 2, (47)
297
where Ŵ denotes the circle centered at z, with radius √
2, and thus u = z + √ 2e i θ
298
on Ŵ. Note that S n+1 ⊂ S n , with dist (S n+1 , S n ) = √
2, see (44). Using the inductive
299
assumption in (44), we are able to estimate the sup term in (47), obtaining
300
φ (k) L G (z) − φ n (k) (z)
≤ k ! ( √
2) k C sup
u ∈Ŵ
h n ( |u|)
|u| − n √ 2 −2
301
≤ k ! ( √
2) k C n +1
⎛
⎝
n −1
j =0
|z| − ( j + 1) √ 2 −m
⎞
⎠
|z| − (n + 1) √ 2 −2
302
≤ C n +1
⎛
⎝
n
j =1
|z| − j √ 2 −m
⎞
⎠
|z| − (n + 1) √ 2 −2
, z ∈ S n +1 , k = 0, 1, 2.
303
(48)
304
Now, we should show that w n ∈ P, for every z ∈ S n +1 , which is true by (48) whenever
305
C n +1
⎛
⎝
n
j =1
|z| − j √ 2 −m
⎞
⎠
|z| − (n + 1) √ 2 −2
< min
k ε k (z) = O(z m −2 ),
306
(49)
307
see Lemma 3.1. This can be seen to be true, for x 0 sufficiently large, uniformly in n,
308
estimating the left-hand side of (49) by the right-hand side of (42) (with n replaced by
309
n + 1), observing that the right-hand side of ( 49) is of order O x 0 m−2
, uniformly in
310
z ∈ S n+1 . Finally, from (46),
311
Author Proof
uncorrected
proof
|φ L G (z) − φ n +1 (z) | ≤ C n +2 |z| −m
⎛
⎝
n
j=1
|z| − j √ 2 −m
⎞
⎠
|z| − (n + 1) √ 2 −2
312
= C n+2
⎛
⎝
n
j =0
|z| − j √ 2 −m
⎞
⎠
|z| − (n + 1) √ 2 −2
, z ∈ S n +1 , (50)
313
that is inequality (44 ) with n + 1 replacing n. ⊓ ⊔
314
4 Numerical implementation and examples
315
In order to compute a given zero (x ≡ x k ) of any given solution to equation (1), know-
316
ing (an approximation of) the successive zero, x k +1 , we solve numerically the equation
317
α L G (x k +1 ) − α L G (x) =
x
k+1x
φ 1/2 L G (t ) dt = π, x < x k +1 , (51)
318
see equation (8). In fact, any phase, α(x), is by (5) strictly monotone, and we can
319
choose it to be monotonic increasing selecting W ≡ const. < 0 in ( 5) (the two basis
320
functions, u and v, can be interchanged). Successive approximations to x k , say x k (n) ,
321
can be obtained from (51), replacing φ L G with φ n , see Theorem 3.2, and solving the
322
ensuing equation,
323
f k,n (x) :=
x
k+1x
φ n 1/2 (t ) dt − π = 0. (52)
324
This can be done by Newton’s method, provided that f k,n ′ (x) ≡ −φ n 1/2 (x) is
325
evaluated efficiently, and a suitable quadrature rule is adopted to compute f k,n (x).
326
It is worth noting that the familiar hypotheses ensuring global convergence of
327
Newton’s method hold, choosing x k+1 as an initial guess ( f k,n (x k+1 ) = −π),
328
f k,n ′′ (x) ∼ − 1 2 q −1/2 (x)q ′ (x) < 0 as x → +∞, in view of the asymptotic results
329
obtained in the complex domain within the proof of Theorem 3.2 (see Remark 3.4).
330
In the following examples, Simpson’s rule with the usual a posteriori error estimate
331
has been used successfully throughout to compute f k,n (x).
332
The key point is, however, evaluating φ n (x) in practice, by means of the itera-
333
tions in the algorithm (12). As already observed in [29], numerical differentiation to
334
compute [φ n , x ] is impractical because of its inherent instability, which propagates
335
destructively as the iterations proceed. Alternatively, one may consider using sym-
336
bolic differentiation tools, but the simple example of the Airy equation, i.e. equation
337
(1 ) with q(x) = x, and thus, by ( 12),
338
φ 0 (x) = x, φ 1 (x) = x + 5 16 x 2 ,
339
φ 2 (x) = −25 − 780 x 3 + 960 x 6 + 1024 x 9 4 x 2 ( 5 + 16 x 3 ) 2 ,
340
Author Proof
uncorrected
proof
φ 3 (x) = N 3 (x)
D 3 (x) , where
341
N 3 (x) := −15625 − 4500000 x 3 + 55192500 x 6 + 833304000 x 9
342
+ 1367424000 x 12 + 6930432000 x 15 + 8945664000 x 18 (53)
343
+ 377487360 x 21 + 3019898880 x 24 + 1073741824 x 27 ,
344
D 3 (x) := 4 x 2 ( 5 + 16 x 3 ) 2 ( −25 − 780 x 3 + 960 x 6 + 1024 x 9 ) 2 ,
345
suggests a blow-up of the numerical coefficients of the (here rational) functions φ n (x),
346
as well as an (exponentially) increasing computational complexity. Note that in (54)
347
φ n (x) ∼ x as x → +∞, for n = 1, 2, 3, as it should be expected; for instance, the
348
coefficients of the leading terms in N 3 (x) and D 3 (x) both coincide with 1024 3 .
349
The method which turned out to be the most efficient one, exploits the analyticity
350
of the coefficient, q, and the ensuing analyticity of all functions φ n , reducing the com-
351
putational complexity to O(n 3 ) (see [29]). Indeed, knowing the Taylor expansion of
352
q(x) about a given point up to the order 2n suffices to obtain φ n at such a point, by
353
simple rational manipulations. From the iterative scheme in (12), it is in fact easily
354
seen that, having a number of coefficients, say k, for the Taylor expansion of φ n , results
355
in obtaining k − 2 coefficients for φ n +1 , the key-step being the trivial algorithm used
356
to compute ratios of (formal) power series (see for instance [23, Ch. 11, p. 20]). The
357
problem of computing the 2n Taylor coefficients needed to start such a local evaluation
358
algorithm, is straightforward when q(x) is a polynomial. Otherwise, when q(x) is a
359
rational function or, more generally, is an analytic function possessing the asymptotic
360
structure in (2), one can take advantage from the highly efficient methods for repeated
361
numerical differentiation of analytic functions developed, e.g., in [11,20] and even
362
better in the recent work in [2].
363
In the following subsections, we present first some examples of numerical
364
computation of zeros of solutions to asymptotically polynomial equations of the type
365
of equation (1), to illustrate the performance of our method. We then show how to solve
366
numerically, by the same “phase function” method, a given boundary-value problem
367
associated to equation (1). In fact, we stress that the present approach yields both a
368
way to compute zeros of solutions, without computing first the solutions themselves,
369
and also a procedure to approximate a certain basis of solutions, and thus the solution
370
to any given initial- or even boundary-value problem.
371
4.1 Approximating zeros
372
In this subsection we are concerned with the numerical approximation of zeros of
373
a given solution to equation (1) in four different cases. All examples have been
374
worked out within the Mathematica environment [34], with an extended precision of 18
375
significant figures. Part of these results were anticipated in [5,25,33].
376
Example 1 (Airy equation) Consider equation (1 ) with q(x) = x (Airy equation
377
[1,23]). The first 20 zeros of the solution J 1/3 (x) (Bessel function of the first kind
378
of order 1/3) have been computed starting from the 21th zero, which was obtained
379
independently with Mathematica.
380
Author Proof
uncorrected
proof
Fig. 1 Approximations of φ(x) by φ
n(x) with n = 0, . . . , 4, when q(x) = x
Fig. 2 log( ˜x −1
k
−x
k) for k = 1, 2, . . . , 20, using φ
n(x) with n = 0, . . . , 4, when q(x) = x
In practice, equation (52) has been solved as described at the beginning of Sect. 4,
381
replacing x k +1 with the approximated value x k (n) +1 , in turn approximated by the Newton
382
method with an absolute error smaller than 10 −16 .
383
In Fig. 1, few functions φ n (x) approximating φ(x) are plotted for this case. In
384
Fig. 2, the corresponding errors made are shown as functions of n and k. Here one
385
can see clearly the fast approach of the functions φ n (x) to φ(x) as n increases, and
386
the decay of the error on the k-th zero as k grows, for fixed n.
387
Example 2 (generalized Airy equation) In Table 1, we show only the (absolute) errors
388
on the first 20 positive zeros (computed as above) of the solution x 1/2 J 1/4 (x 2 /2) to
389
equation (1 ) with q(x) = x 2 (see [1]).
390
Example 3 (a perturbed generalized Airy equation) Here, the case of a perturbed
391
generalized Airy equation is considered, taking q(x) = x 2 − 3/(4 x 2 ) in equation (1).
392
Author Proof
uncorrected
proof
Fig. 3 Approximations of φ(x) = x 2 by φ
n(x) with n = 0, . . . , 3, when q(x) = x 2 − 4x 3
2Fig. 4 − log( ˜x 1
k
−x
k) for k = 1, . . . , 20 using φ
n(x) with n = 0, . . . , 4, when q(x) = x 2 − 4x 3
2This coefficient is obtained by equation (6 ) choosing the phase α(x) = x 2 /2, and thus
393
the zeros of the solution y(x) = |α ′ (x) | −1/2 sin α(x) = x −1/2 sin (x 2 /2) are known
394
exactly, x k = √
2kπ . We get from this the value of the 21th zero, and then compute
395
the first 20 zeros by our algorithm, for n = 0, 1, . . . , 4. In Fig. 3, few functions φ n (x),
396
approximating the function φ(x) relevant to this case are plotted. In in Fig. 4, the
397
corresponding errors are shown, as in the previous example.
398
Example 4 (a cubic polynomial coefficient) Finally, we consider, as coefficient in (1),
399
the cubic polynomial q(x) = x 3 − x + 1. In this case, we show the errors made in
400
computing the 10 consecutive zeros of the solution vanishing at x = 6, located on the
401
left of such a zero. The same is done for the solution vanishing at x = 20. In Fig. 5,
402
we plotted again few approximating functions φ n (x).
403
Recall that the choice of a given zero defines a solution to equation (1) up to a
404
multiplicative constant.
405
Author Proof
uncorrected
proof
Fig. 5 Approximations of φ(x) by φ
n(x) with n = 0, . . . , 4, when q(x) = x 3 − x + 1
4.2 Numerical solution of a boundary-value problem
406
The “phase function” method described in the previous sections, besides providing the
407
direct computation of zeros of solutions (see Sect. 4.1), makes it possible to approxi-
408
mate a (Liouville-Green) basis, and thus any solution, to a given initial or boundary-
409
value (BV) problem for equation (1). In fact, knowing any phase, α(x), a basis can be
410
constructed as in (7). In practice, we obtain an approximate basis through the iterative
411
scheme in (32),
412
u n (x) := φ −1/4 n (x) sin
⎛
⎝
x
φ n 1/2 (t ) dt,
⎞
⎠ , v n (x) := φ −1/4 n cos
⎛
⎝
x
φ n 1/2 (t ) dt
⎞
⎠ ,
413
(54)
414
recalling that φ n (x) approximates φ L G (x) = (α ′ L G ) 2 , see Theorem 3.2. We also recall
415
that the choice of the primitive in (54) is irrelevant to the purpose of approximating a
416
basis, see [3,4,30].
417
Consider the following BV problem for equation (1): y ′′ + q(x)y = 0 on a < x <
418
b, y(a) = A, y(b) = B, with a coefficient q(x) belonging to the class described in
419
Sect. 1 . Then, its solution has the form y(x) = c 1 u(x) + c 2 v(x), (u(x), v(x)) being,
420
e.g., the L G basis in (7), and the constants c 1 , c 2 can be determined as functions of
421
a, b, A, B. We shall use, instead, the approximate solution, y n (x) := c 1 (n)u n (x) +
422
c 2 (n)v n (x), where u n (x), v n (x) are given in (54). The “constants” c 1 (n), c 2 (n) can
423
also be determined approximately as functions of the known data. It is lengthy but
424
elementary to establish that the error made in evaluating y(x) on the interval [a, b] by
425
means of y(x) can be estimated as
426
|y(x) − y n (x) | ≤ const. x n −m/2 sup
x>x
n|φ n (x) − φ L G (x) |, for b > x > a > x n ,
427
(55)
428
Author Proof
uncorrected
proof
where the sup term can be estimated as in Remark 3.3, and x n is defined in Theorem 3.2.
429
We give now an example of application of such a procedure.
430
Example 5 Consider the two-point BV problem for the perturbed generalized Airy
431
equation of Example 3,
432
y ′′ +
x 2 − 3
4x 2
y = 0, a < x < b, y(a) = A, y(b) = B. (56)
433
Tables 4 and5 refer to two cases for the exact solution y(x) = |α ′ (x) | −1/2 [sin α(x) −
434
cos α(x)] = (2/x) 1/2 sin(x 2 / 2 − 1/4), see Example 3. The two cases are obtained
435
choosing the intervals 6 < x < 8, and 9.5 < x < 11.5, respectively. Clearly, A
436
and B are evaluated from such a test solution. In Tables 4 and5, the effectiveness of
437
the method is shown, since the error turns out to be of order of 10 −11 and 10 −14 ,
438
respectively, already with n = 4.
439
5 Comparing the “phase function” method with a standard high-order
440
algorithm
441
The problems considered in the previous examples still concern solutions with an
442
oscillatory behavior not so hard, given the modest powers there, but they may become
443
more challenging on larger intervals. To test the effectiveness of the “phase func-
444
tion” method to solve highly oscillatory problems, we compared its performance with
445
that of a standard but highly accurate method. Instead of resorting to Runge–Kutta
446
methods combined with dense output, whose order seems to be not higher than 8 in
447
the existing literature, we prefer to use the MATLAB RKN12(10) code (also denoted
448
by RKN1210), which is a 12th/10th order Runge–Kutta–Nyström integrator with
449
automatic control of the step size. This is widely used in problems where extremely
450
stringent error tolerances are required [9,10,15].
451
Example 6 We considered two examples concerning a perturbed generalized Airy
452
equation, whose phase function α(x), and hence the coefficient q(x), can be
453
estimated, and the exact solutions are known.Thus, the accuracy of both, our method
454
and the RKN12(10) can be compared computing the discrepancy between the approx-
455
imate and the exact solution. In case (a) we chose the same equation of Example 5,
456
i.e., q(x) = (x 2 − 3/(4x 2 )) (for which α(x) = x 2 /2), in case (b) we chose
457
q(x) = (x 6 − 15/(4x 2 ) (corresponding to α(x) = x 4 /4); see Figs. 6 and 7. The
458
numerical solution based on RKN12(10) of course is faster, in this examples, since
459
our method exploited symbolic manipulations. On the other hand, our method works
460
for solutions with arbitrarily rapid oscillations, and has an infinite accuracy in those
461
parts of the code where symbolic manipulations are used.
462
Example 7 Moreover, in several cases the RKN12(10) method fails completely, for
463
instance when the step-size falls below the smallest acceptable value (t f i n −t i n )/10 12 ,
464
see [10]. An example of such occurrence is encountered solving the initial-value prob-
465
lem for the harmonic oscillator, y ′′ +ω 2 y = 0, x > 0, y(0) = 0, y ′ ( 0) = ω. Choosing
466
Author Proof
uncorrected
proof
Fig. 6 Absolute error, |y(x) − y
n(x) |, made solving the Cauchy problem of case (a) in Example 6, with RKN12(10), on the interval [9, 11]
Fig. 7 Absolute error, |y(x) − y
n(x) |, made solving the Cauchy problem of case (b) in Example 6, with RKN12(10), on the interval [9, 11]
increasingly larger values of ω, we found that when ω = 10 5 the algorithm RK12(10)
467
starts failing, while our method works, yielding a solution with an error about equal
468
to zero. This performance of our method was observed even for larger values of ω.
469
In closing, we remark that the method proposed by Glaser et al. [12] presents some
470
similarities with ours: Both methods need one starting zero accurately computed; both
471
try to compute zeros solving directly a suitable, related ODE, circumventing the need
472
