<< Ïðåäûäóùàÿ ñòð. 14(èç 83 ñòð.)ÎÃËÀÂËÅÍÈÅ Ñëåäóþùàÿ >>
2 t âˆ’ 2i(t âˆ’ i âˆ’ i )2
0

where w = x âˆ’ y; r = |w|, = w2 /r and H0(1) (z) is a Hankel function of the Ã¿rst kind, as speciÃ¿ed
more fully in [10,11]. By taking x2 = 0 in (3.9), we see from (3.13) that u(x1 ; 0) satisÃ¿es an integral
equation with Toeplitz structure, and the fast Fourier transform yields its iterative solution e ciently.
Without loss of generality, we use the scale determined by k =1 in (3.9) â€“ (3.13). For this example,
the impedance is taken to be Ã¿ = 1:4ei =4 on the interval = {x: âˆ’ 40 Â¡ x1 Â¡ 40 ; x2 = 0}. At
two sample points (x1 â‰ˆ âˆ’20 and 20 ) taken from a 400-point discretisation of , we found the
following results with the VEA using 16 decimal place (MATLAB) arithmetic
(12)
= [ : : ; âˆ’0:36843 + 0:44072i; : : : ; âˆ’0:14507 + 0:55796i; : :];
0
(10)
= [ : : ; âˆ’0:36333 + 0:45614i; : : : ; âˆ’0:14565 + 0:56342i; : :];
2
(8)
= [ : : ; âˆ’0:36341 + 0:45582i; : : : ; âˆ’0:14568 + 0:56312i; : :];
4
(6)
= [ : : ; âˆ’0:36341 + 0:45583i; : : : ; âˆ’0:14569 + 0:56311i; : :];
6
(4)
= [ : : ; âˆ’0:36341 + 0:45583i; : : : ; âˆ’0:14569 + 0:56311i; : :];
8

where the converged Ã¿gures are shown in bold face.
Each of these results, showing just two of the components of a particular Ã„j) in columns Ã„ =
(

0; 2; : : : ; 8 of the vector-epsilon table, needs 12 iterations of (3.9) for its construction. In this appli-
cation, these results show that the VEA converges reasonably steadily, in contrast to Lanczos type
methods, eventually yielding Ã¿ve decimal places of precision.

Example 3.2 was chosen partly to demonstrate the use of the vector epsilon algorithm for a weakly
convergent sequence of complex-valued data, and partly because the problem is one which lends
itself to iterative methods. In fact, the example also shows that the VEA has used up 11 of the 15
decimal places of accuracy of the data to extrapolate the sequence to its limit. If greater precision
is required, other methods such as stabilised Lanczos or multigrid methods should be considered.
The success of the VEA in examples such as those given above is usually attributed to the fact
( j)
that the entries { 2k ; j = 0; 1; 2; : : :} are the exact limit of a convergent sequence S if S is generated
by precisely k nontrivial geometric components. This result is an immediate and direct generalisation
of that for the scalar case given in Section 2. The given vector sequence is represented by
j
k k
i
wÃ„ (Ã‚Ã„ )j ;
sj = C0 + CÃ„ (Ã‚Ã„ ) = sâˆž âˆ’ j = 0; 1; 2; : : : ; (3.14)
Ã„=1 i=0 Ã„=1

where each CÃ„ ; wÃ„ âˆˆ Cd ; Ã‚Ã„ âˆˆ C; |Ã‚Ã„ | Â¡ 1, and all the Ã‚Ã„ are distinct. The two representations used
in (3.14) are consistent if
k k
âˆ’1
CÃ„ = sâˆž âˆ’ wÃ„ and CÃ„ = wÃ„ (Ã‚Ã„ âˆ’ 1):
Ã„=0 Ã„=1

To establish this convergence result, and its generalisations, we must set up a formalism which
allows vectors to be treated algebraically.
64 P.R. Graves-Morris et al. / Journal of Computational and Applied Mathematics 122 (2000) 51â€“80

From the given sequence S = (si ; i = 0; 1; 2; : : : ; : si âˆˆ Cd ), we form the series coe cients
c0 := s0 ; ci := si âˆ’ siâˆ’1 ; i = 1; 2; 3; : : : (3.15)
and the associated generating function
f (z) = c0 + c1 z + c2 z 2 + Â· Â· Â· âˆˆ Cd [[z]]: (3.16)
Our Ã¿rst aim is to Ã¿nd an analogue of (2.15) which allows construction, at least in principle, of
the denominator polynomials of a vector-valued PadÃƒ approximant for f (z). This generalisation is
e
possible if the vectors cj in (3.16) are put in oneâ€“one correspondence with operators cj in a Cli ord
algebra A. The details of how this is done using an explicit matrix representation were basically
set out by McLeod [37]. We use his approach [26,27,38] and square matrices Ei , i = 1; 2; : : : ; 2d + 1
of dimension 22d+1 which obey the anticommutation relations
Ei Ej + Ej Ei = 2 ij I; (3.17)
where I is an identity matrix. The special matrix J = E2d+1 is used to form the operator products
Fi = JEd+i ; i = 1; 2; : : : ; d: (3.18)
Then, to each vector w = x + iy âˆˆ Cd whose real and imaginary parts x; y âˆˆ Rd , we associate the
operator
d d
w= xi E i + yi Fi : (3.19)
i=1 i=1

The real linear space V is deÃ¿ned as the set of all elements of the form (3.19). If w1 ; w2 âˆˆV
C C
d
correspond to w1 ; w2 âˆˆ C and ; Ã¿ are real, then
w3 = w1 + Ã¿w2 âˆˆ V (3.20)
C

corresponds uniquely to w3 = w1 + Ã¿w2 âˆˆ Cd . Were ; Ã¿ complex, the correspondence would not be
oneâ€“one. We refer to the space V as the isomorphic image of Cd , where the isomorphism preserves
C
linearity only in respect of real multipliers as shown in (3.20). Thus the image of f (z) is
f(z) = c0 + c1 z + c2 z 2 + Â· Â· Â· âˆˆ V [[z]]: (3.21)
C

The elements Ei , i = 1; 2; : : : ; 2d + 1 are often called the basis vectors of A, and their linear combi-
nations are called the vectors of A. Notice that the Fi are not vectors of A and so the vectors of
A do not form the space V. Products of the nonnull vectors of A are said to form the Lipschitz
C
group [40]. The reversion operator, denoted by a tilde, is deÃ¿ned as the anti-automorphism which
reverses the order of the vectors constituting any element of the Lipschitz group and the operation
is extended to the whole algebra A by linearity. For example, if ; Ã¿ âˆˆ R and
D = E1 + Ã¿E4 E5 E6 ;
then
Ëœ
D = E1 + Ã¿E6 E5 E4 :
Hence (3.18) and (3.19) imply that
d d
w=
Ëœ xi E i âˆ’ yi Fi : (3.22)
i=1 i=1
P.R. Graves-Morris et al. / Journal of Computational and Applied Mathematics 122 (2000) 51â€“80 65

We notice that w corresponds to wâˆ— , the complex conjugate of w, and that
Ëœ
d
(xi2 + yi2 )I = ||w||2 I
ww =
Ëœ (3.23)
2
i=1

is a real scalar in A. The linear space of real scalars in A is deÃ¿ned as S := { I; âˆˆ R}. Using
(3.23) we can form reciprocals, and
wâˆ’1 = w=|w|2 ;
Ëœ (3.24)
where
|w| := ||w||; (3.25)
so that wâˆ’1 is the image of wâˆ’1 as deÃ¿ned by (3.5). Thus (3.19) speciÃ¿es an isomorphism between
(i) the space Cd , having representative element
wâˆ’1 = wâˆ— =||w||2 ;
w = x + iy and an inverse
(ii) the real linear space V with a representative element
C

d d
wâˆ’1 = w=|w|2 :
w= xi E i + yi Fi and its inverse given by Ëœ
i=1 i=1

The isomorphism preserves inverses and linearity with respect to real multipliers, as shown in (3.20).
Using this formalism, we proceed to form the polynomial q2j+1 (z) analogously to (2.15). The equa-
tions for its coe cients are
ï£¹ ï£® (2j+1) ï£¹ ï£®
ï£® ï£¹
qj+1
c0 Â·Â·Â· cj âˆ’cj+1
ï£ºï£¯ . ï£º ï£¯
ï£¯. . .ï£º
ï£»ï£¯ . ï£º = ï£°
ï£°. . .ï£» (3.26)
. . .
.ï£»
ï£°
(2j+1)
cj Â·Â·Â· c2j âˆ’c2j+1
q1
(2j+1)
which represent the accuracy-through-order conditions; we assume that q0 = q2j+1 (0) = I . In
(2j+1) (2j+1) (2j+1) (2j+1)
principle, we can eliminate the variables qj+1 ; qj ; : : : ; q2 sequentially, Ã¿nd q1 and then
(2j+1)
the rest of the variables of (3.26) by back-substitution. However, the resulting qi turn out to be
higher grade quantities in the Cli ord algebra, meaning that they involve higher-order outer products
of the fundamental vectors. Numerical representation of these quantities uses up computer storage
and is undesirable. For practical purposes, we prefer to work with low-grade quantities such as
scalars and vectors [42].
The previous remarks re ect the fact that, in general, the product w1 ; w2 ; w3 âˆˆ V when C
w1 ; w2 ; w3 âˆˆ V. However, there is an important exception to this rule, which we formulate as
C
follows [26], see Eqs. (6:3) and (6:4) in [40].

Lemma 3.3. Let w; t âˆˆ V be the images of w = x + iy; t = u + iC âˆˆ Cd . Then
C

(i) t w + wtËœ = 2 Re(wH t)I âˆˆ S;
Ëœ (3.27)

(ii) wtËœw = 2w Re(wH t) âˆ’ t||w||2 âˆˆ V: (3.28)
C
66 P.R. Graves-Morris et al. / Journal of Computational and Applied Mathematics 122 (2000) 51â€“80

Proof. Using (3.17), (3.18) and (3.22), we have
d d
t w + wtËœ =
Ëœ (ui Ei + vi Fi )(xj Ej âˆ’ yj Fj ) + (xj Ej + yj Fj )(ui Ei âˆ’ vi Fi )
i=1 j=1

= (uT x + CT y)I = 2 Re(wH t)I
because, for i; j = 1; 2; : : : ; d;
Fi Ej âˆ’ Ej Fi = 0; Fi Fj + Fj Fi = âˆ’2 ij I:
For part (ii), we simply note that
wtËœw = w(tËœw + wt) âˆ’ wwt:
Ëœ Ëœ
We have noted that, as j increases, the coe cients of q2j+1 (z) are increasingly di cult to store.
Economical approximations to q2j+1(z) are given in [42]. Here we proceed with
ï£® ï£¹ ï£® ï£¹
(2j+1)
qj+1 0
ï£® ï£¹
c0 Â·Â·Â· cj+1 ï£¯.ï£º ï£¯.ï£º
ï£ºï£¯ . ï£º ï£¯.ï£º
ï£¯. . .
ï£¯.ï£º
ï£°. . ï£»ï£¯ =ï£¯ (3.29)
ï£º
. . (2j+1) ï£º
ï£» ï£°0ï£»
ï£° q1
cj+1 Â·Â·Â· c2j+2
e2j+1
I
which are the accuracy-through-order conditions for a right-handed operator PadÃƒ approximant (OPA)
e
âˆ’1
p2j+1 (z)[q2j+1 (z)] for f(z) arising from
f(z)q2j+1 (z) = p2j+1 (z) + e2j+1 z 2j+2 + O(z 2j+3 ): (3.30)
The left-hand side of (3.29) contains a general square Hankel matrix with elements that are operators
from V. A remarkable fact, by no means obvious from (3.29) but proved in the next theorem, is
C
that
e2j+1 âˆˆ V: (3.31)
C

This result enables us to use OPAs of f(z) without constructing the denominator polynomials. A
quantity such as e2j+1 in (3.29) is called the left-designant of the operator matrix and it is denoted
by
c0 Â·Â·Â· cj+1
. .
. .
e2j+1 = : (3.31b)
. .
cj+1 Â·Â·Â· c2j+2 l

The subscript l (for left) distinguishes designants from determinants, which are very di erent con-
structs. Designants were introduced by Heyting [32] and in this context by Salam [43]. For present
purposes, we regard them as being deÃ¿ned by the elimination process following (3.26).

Example 3.4. The denominator of the OPA of type [0=1] is constructed using
(1)
c0 c1 0
q1
= :
c1 c2 e1
I
P.R. Graves-Morris et al. / Journal of Computational and Applied Mathematics 122 (2000) 51â€“80 67

(1)
We eliminate q1 as described above following (3.26) and Ã¿nd that
c2 c1 âˆ’1
e1 = = c2 âˆ’ c1 c0 c1 âˆˆ span{c0 ; c1 ; c2 }: (3.32)
c1 c0 l

Proceeding with the elimination in (3.29), we obtain
ï£® ï£¹ ï£® ï£¹
(2j+1)
qj+2 0
ï£® ï£¹
âˆ’1 âˆ’1
c2 âˆ’ c1 c0 c1 Â·Â·Â· cj+2 âˆ’ c1 c0 cj+1 ï£¯.ï£º ï£¯.ï£º
ï£ºï£¯ . ï£º ï£¯.ï£º
. .
ï£¯ .
ï£¯.ï£º
. . ï£» ï£¯ (2j+1) ï£º = ï£¯ ï£º: (3.33)
ï£° . . ï£°0ï£»
 << Ïðåäûäóùàÿ ñòð. 14(èç 83 ñòð.)ÎÃËÀÂËÅÍÈÅ Ñëåäóþùàÿ >>