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) 203– 229

Monotonicity preserving interpolatory subdivision schemes
Frans Kuijt ∗; 1 , Ruud van Damme
Faculty of Mathematical Sciences, University of Twente, P.O. Box 217, NL-7500 AE Enschede, The Netherlands
Received 30 April 1998; received in revised form 28 September 1998

Abstract
A class of local nonlinear stationary subdivision schemes that interpolate equidistant data and that preserve monotonicity
in the data is examined. The limit function obtained after repeated application of these schemes exists and is monotone for
arbitrary monotone initial data. Next a class of rational subdivision schemes is investigated. These schemes generate limit
functions that are continuously dierentiable for any strictly monotone data. The approximation order of the schemes is
four. Some generalisations, such as preservation of piecewise monotonicity and application to homogeneous grid renement,
c 1999 Elsevier Science B.V. All rights reserved.
are brie
y discussed.
AMS classication: 41A05; 41A29; 65D05; 65D17
Keywords: Subdivision; Interpolation; Monotonicity preservation; Shape preservation; Computer aided geometric design

1. Introduction
In this article, we examine four-point interpolatory monotonicity preserving subdivision schemes.

These schemes are used for interpolation of univariate data that are uniform and monotone increasing
(or decreasing).
Interpolatory subdivision schemes are based on iterative renement of a data set. The usual subdivision schemes roughly double the number of data points every iteration. Overviews on subdivision
schemes can be found in, e.g., [3, 6, 7]. Many subdivision schemes however fail to preserve monotonicity in the data. Linear monotonicity preserving subdivision schemes are discussed in [19], but
the schemes discussed there are not interpolatory. Monotonicity preservation of the interpolatory linear four-point scheme [5] is discussed in [2]. The author determines ranges on the tension parameter
such that the scheme is monotonicity preserving. Since the tension parameter depends on the initial
data, the resulting subdivision scheme is stationary but data dependent.
∗
1

Corresponding author. E-mail: f.kuijt@math.utwente.nl.
Financially supported by the Dutch Technology Foundation STW.

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 8 ) 0 0 2 2 0 - 9

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F. Kuijt, R. van Damme / Journal of Computational and Applied Mathematics 101 (1999) 203– 229

We restrict ourselves to subdivision schemes that guarantee the preservation of monotonicity in
the data. It is stressed that there is an analogy with [16] (and [15]) in which convexity preserving
interpolatory subdivision schemes are examined. The same constructive approach is used, and so there
is a similarity in some of the proofs. The remark in [16] that convexity preserving interpolatory
subdivision schemes generating C 1 limit functions must necessarily be nonlinear also holds for
monotonicity preserving schemes. The schemes examined in this article are stationary and they do
not contain any data-dependent tension parameter.
The overview of this article is as follows. Section 2 states the problem and the class of schemes
under investigation. The condition for preservation of monotonicity is derived in Section 3, and in
Section 4 convergence to a limit function is examined. The analysis for convexity preservation in
[16] led to a scheme that is unique in some sense, if convergence to a C 1 function is required.
Requiring convergence to a C 1 -smooth monotone function leads to a larger class of subdivision
schemes however. Sucient conditions for convergence to a C 1 limit function are given in Section 5,
and since these conditions are too complex for constructing explicit subdivision schemes, we restrict
ourselves to a specic class of schemes, namely rational subdivision schemes, see Section 6. Section 7
shows that these schemes have the property that ratios of adjacent rst order dierences tend to 1 as k
tends to innity. For any initial monotone data these schemes converge to continuously dierentiable
limit functions (see Section 8) and have approximation order four (Section 9). Some generalisations
are brie

y discussed in Section 10: e.g., piecewise monotonicity and application to homogeneous
grid renement which is a useful property for subdivision schemes for functional nonuniform data.
2. Problem denition
First, we state the problem that is examined in this article.
Denition 1 (Problem denition). Given is a nite bounded data set {(ti(0) ; xi(0) ) ∈ R2 }Ni=0 , where the
data are uniform, i.e., ti(0) = ih, where h¿0 is the mesh size. The data are assumed to be monotone,
(0)
(0)
, ∀i, or xi(0) ¿ xi+1
, ∀i. Subdivision in t is dened as ti(k) = 2−k ih, i = 0; : : : ; 2k N . The aim
i.e., xi(0) 6 xi+1
is to characterise a class of subdivision schemes that are interpolatory and monotonicity preserving
if the data are monotone. The second goal is to restrict this class of subdivision schemes to schemes
that generate continuously dierentiable limit functions and are fourth order accurate.
First we make a remark how to treat the boundaries, see [16]. Every initial monotone data set
+2
{(ti(0) ; xi(0) )}Ni=0 can be extended in an arbitrary but monotonicity preserving way to {(ti(0) ; xi(0) )}Ni=−2
,
(0) (0)
such that the limit function is dened in the whole interval I = [t0 ; tN ]. This means that all relevant

properties on the kth iterate are easily shown to hold for the index set {0; : : : ; 2k N }. Thus, all relevant
properties are consistently proved on the original domain I = [t0(0) ; tN(0) ] = [t0(k) ; t2(k)
k N ].
A constructive approach is used to derive monotonicity preserving subdivision schemes. Without
loss of generality, we consider monotone increasing data, brie
y denoted as monotone data.
We restrict ourselves to the following class of schemes:
x2i(k+1) = xi(k) ;
(k)
(k)
(k)
(k)
(k+1)
) + G1 (xi−1
; xi(k) ; xi+1
; xi+2
);
x2i+1
= 12 (xi(k) + xi+1

(1)

F. Kuijt, R. van Damme / Journal of Computational and Applied Mathematics 101 (1999) 203– 229

205

for some function G1 . This implies that
1. the subdivision schemes are interpolatory,
2. the subdivision schemes are local, using four points.
First order dierences si(k) are dened by
(k)
− xi(k) :
si(k) := xi+1

(2)

Subdivision scheme (1) can then be rewritten in the following form:
x2i(k+1) = xi(k) ;
(k+1)
(k)

(k)
(k)
(k)
x2i+1
= 12 (xi(k) + xi+1
) + G2 ( 21 (xi(k) + xi+1
); si−1
; si(k) ; si+1
);

(3)

where G2 is another function representing the same class of subdivision schemes.
The third condition on the subdivision schemes deals with invariance under addition of constants:
3. The subdivision scheme is invariant under addition of constant functions, i.e., if the data
generate subdivision points (ti(k) ; xi(k) ), then the data (ti(0) ; xi(0) + ), with  ∈ R yield subdivision points (ti(k) ; xi(k) + ).

(ti(0) ; xi(0) )

Imposing this condition yields

(k)
(k)
(k)
(k)
(k+1)
x2i+1
) +  + G2 ( 21 (xi(k) + xi+1
) + ; si−1
; si(k) ; si+1
):
+  = 12 (xi(k) + xi+1

It follows that G2 cannot depend on its rst argument. Condition 3 therefore yields that subdivision
scheme (3) must be of the following form:
x2i(k+1) = xi(k) ;
(k)
(k)
(k)
(k+1)
x2i+1

) + G3 (si−1
; si(k) ; si+1
):
= 21 (xi(k) + xi+1

(4)

Next we add a natural requirement on the subdivision schemes:
4. The subdivision scheme is homogeneous, i.e., if initial data (ti(0) ; xi(0) ) give subdivision points
then initial data (ti(0) ; xi(0) ) yield points (ti(k) ; xi(k) ).

(ti(k) ; xi(k) ),

A direct consequence of homogeneity of the subdivision schemes is that the function G3 is homogeneous:
G3 (a; b; c) = G3 (a; b; c);

∀:

(5)

Subdivision scheme (4) then necessarily reproduces constant functions, i.e., if xi(0) = 0 , then xi(k) = 0
(take  = 0 in (5)).
Further simplication of the representation of subdivision scheme (4) is obtained by using the
homogeneity of G3 in (5) as follows:
(k)
(k)
(k)
(k)
);
; si(k) ; si+1
) = si(k) G3 (ri(k) ; 1; Ri+1
) = 21 si(k) G(ri(k) ; Ri+1
G3 (si−1

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F. Kuijt, R. van Damme / Journal of Computational and Applied Mathematics 101 (1999) 203– 229

where ratios of adjacent rst dierences are dened by
(k)

ri(k) := si−1
=si(k)

and

Ri(k) = 1=ri(k) ;

(6)

and the function G is dened by G(r; R) := 2G3 (r; 1; R).
Since it follows in the next sections that G is a bounded function, this reformulation of G does
not cause problems in case si(k) = 0.
The class of subdivision schemes (1) is rewritten in the form:
x2i(k+1) = xi(k) ;
(k)
(k)
(k+1)
x2i+1
) + 21 si(k) G(ri(k) ; Ri+1
);

= 12 (xi(k) + xi+1

(7)

and it is this subdivision scheme that is examined in this article.
Remark 2. Note that the class of subdivision schemes (7) automatically satises conditions 1–4.
The next general assumption on the subdivision scheme concerns with invariance under ane transformations of the variable t:
5. The subdivision scheme is invariant under ane transformations of the variable t, i.e., if the
initial data (ti(0) ; xi(0) ) yield subdivision points (ti(k) ; xi(k) ), then the data (ti(0) + 0 ; xi(0) ), with 0 ∈ R
yield subdivision points (ti(k) + 0 ; xi(k) ) for ¿0, and (ti(k) + 0 ; −xi(k) ) for ¡0.
By taking  = −1 and 0 = 0 in Condition 5, it follows that G is anti-symmetric under interchanging
its arguments, i.e., it is obtained that
G(r; R) = − G(R; r);

∀r; R;

(8)

which directly implies G(r; r) = 0, ∀r. Under this condition on G, subdivision scheme (7) necessarily
reproduces linear functions, i.e., if xi(0) = 1 ti(0) + 1 , then xi(k) = 1 ti(k) + 1 (as si(k) = 1 h, ri(k) = 1, ∀i).
Remark 3. Invariance under addition of linear functions is not a natural condition in case of monotonicity preservation, in contrast with convexity preservation (see [16]).
In the following sections, we discuss conditions for monotonicity preservation and smoothness
properties of subdivision scheme (7) satisfying (8), which requires additional conditions on the
function G.

3. Monotonicity preservation
In this section, we examine monotonicity preservation of the class of four-point interpolatory
subdivision schemes (7) satisfying (8).

F. Kuijt, R. van Damme / Journal of Computational and Applied Mathematics 101 (1999) 203– 229

207

Theorem 4 (Monotonicity preservation). Subdivision scheme (7) satisfying condition (8) preserves
monotonicity if and only if the subdivision function G satises
|G(r; R)| 6 1;

∀r; R ¿ 0:

(9)

Proof. Monotonicity preservation is achieved if and only if the scheme generates dierences that
satisfy
si(k) ¿ 0;

∀i; ∀k:

Therefore assume that for some k, the data xi(k) satisfy si(k) ¿ 0, ∀i. Necessary and sucient for
monotonicity preservation is that the dierences in the data at level (k + 1) are also nonnegative,
i.e., si(k+1) ¿ 0, ∀i.
(k+1)
Two dierences, s2i(k+1) and s2i+1
, have to be analysed:
(k)
(k)
(k+1)
) + 12 si(k) G(ri(k) ; Ri+1
) − xi(k)
s2i(k+1) = x2i+1
− x2i(k+1) = 12 (xi(k) + xi+1
(k)
(k)
= 12 (xi+1
− xi(k) ) + 12 si(k) G(ri(k) ; Ri+1
)
(k)
));
= 12 si(k) (1 + G(ri(k) ; Ri+1
(k+1)
s2i+1

(k+1)
= x2i+2

=
=

−

(k+1)
x2i+1

(k)
= xi+1

−

(10)
1 (k)
(x
2 i

+

(k)
xi+1
)

−

(k)
1 (k)
s G(ri(k) ; Ri+1
)
2 i

(k)
1 (k)
(x − xi(k) ) − 21 si(k) G(ri(k) ; Ri+1
)
2 i+1
(k)
1 (k)
s (1 − G(ri(k) ; Ri+1
)):
2 i

(11)

Since si(k) ¿ 0, it is necessary for monotonicity preservation that
(k)
(k)
) ¿ 0 and 1 − G(ri(k) ; Ri+1
) ¿ 0;
1 + G(ri(k) ; Ri+1

which yields that condition (9) is sucient for monotonicity preservation of subdivision scheme (7).
Since we consider arbitrary monotone data, this condition is also necessary.
Remark 5 (Preservation of strict monotonicity). A sucient condition for preservation of strict
monotonicity is:
∃¡1 such that |G(r; R)| 6 ;

∀r; R¿0:

Remark 6 (Nonlinearity). Observe that the function G must be necessarily nonlinear for monotonicity preservation and C 1 -smoothness of subdivision scheme (7): the only scheme that is polynomial
in its arguments and that satises (9) is given by G ≡ 0. The resulting two-point subdivision scheme
however, generates the piecewise linear interpolant as limit function, which is obviously only C 0 .
Remark 7 (The linear four-point scheme). The well-known linear four-point scheme of Dyn,
Gregory and Levin [5] with w = 1=16 is given by the function
 R) = 1 (r − R):
G(r;
8
Since this function G is linear, it obviously cannot satisfy the monotonicity condition (9).

(12)

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F. Kuijt, R. van Damme / Journal of Computational and Applied Mathematics 101 (1999) 203– 229

4. Convergence to a continuous function
In this section convergence of subdivision scheme (7) to continuous limit functions is investigated.
The proof follows the lines of the proof of existence and continuity of the limit curve generated by
the linear four-point scheme in [5].
Theorem 8 (C 0 -convergence). Given is a monotone data set {(ti(0) ; xi(0) ) ∈ R2 }i , where ti(0) = ih. The
kth stage data {(ti(k) ; xi(k) )}i are dened at values ti(k) = 2−k ih.
Repeated application of subdivision scheme (7) satisfying
∃¡1 such that |G(r; R)| 6 ;

∀r; R ¿ 0:

(13)

leads to a continuous function which is monotone and interpolates the initial data points (ti(0) ; xi(0) ).
Proof. The interpolatory property of the subdivision scheme is a direct consequence of the denition
x2i(k+1) = xi(k) . Preservation of monotonicity of the subdivision scheme was shown in the previous section. Therefore, the continuous function x(k) , dened as the linear interpolant to the data {(ti(k) ; xi(k) )}i ,
is monotone.
Remains to prove that the sequence of functions x(k) converges, i.e., the limit function
x(∞) := lim x(k)
k→∞

exists and is continuous. Sucient for convergence is that x(k) is a Cauchy sequence in k with limit
0, i.e., it suces to show that
kx(k+1) − x(k) k∞ 6 C0 0k

where C0 ¡∞ and 0 ¡1:

The distance kx(k+1) − x(k) k∞ is calculated by
n


 
 

o


(k+1)
(k)
− 12 (xi(k) + xi+1
kx(k+1) − x(k) k∞ = max max x2i(k+1) − xi(k)  ; x2i+1
) :
i

(k+1)
The maximal distance between the functions x(k) and x(k+1) occurs at a point t2i+1
= 2−k (i + 12 )h for
some i, which thus gives, using condition (9)
(k+1)
(k)
kx(k+1) − x(k) k∞ = max |x2i+1
− 12 (xi(k) + xi+1
)|
i

=

1
2

(k)
max |si(k) G(ri(k) ; Ri+1
)|
i

(k)
6 max |G(ri(k) ; Ri+1
)| max |si(k) | 6 12  max si(k) :
1
2

i

i

i

(14)

We now prove that maxi si(k) converges to 0. Using (10), (11) and monotonicity condition (9), it is
obtained that
(k)
(k+1)
max si(k+1) = max max{s2i(k+1) ; s2i+1
))}
} = 12 max{si(k) (1 ± G(ri(k) ; Ri+1
i

i

i

6

1
(1
2

+

) max si(k) :
i

F. Kuijt, R. van Damme / Journal of Computational and Applied Mathematics 101 (1999) 203– 229

209

Combining this result with (14) yields
kx(k+1) − x(k) k∞ 6 12  max si(0)
i



1+
2

k

:

As ¡1, this proves convergence of x(k) , and since all functions x(k) are continuous by construction,
the limit function x(∞) is continuous.

5. Convergence to a continuously dierentiable function
In the previous sections, we derived sucient conditions on the function G such that the subdivision scheme preserves monotonicity and that a continuous limit function exists. An additional
sucient condition on the scheme such that it generates continuously dierentiable functions is
presented in this section.
Theorem 9 (C 1 -convergence). Given is a strictly monotone data set {(ti(0) ; xi(0) ) ∈ R2 }i ; where ti(0) =
ih; with h¿0. The kth stage data {(ti(k) ; xi(k) )}i are dened at values ti(k) = 2−k ih.
Let the function G satisfy (13) and the Lipschitz condition
∃¿0; ∀x; y: |G(x + 1 ; y + 2 ) − G(x; y)| 6 B1 kk ;

B1 ¡∞:

(15)

Moreover; subdivision scheme (7) has the property that the ratios of adjacent rst order dierences;
dened in (6), obey
∃¡1:



)
(




(k) 1
max max ri ; (k) − 1 6 B2 k ;
i 

ri

B2 ¡∞:

(16)

Repeated application of such a subdivision scheme generates a continuously dierentiable function
which is monotone and interpolates the initial data points (ti(0) ; xi(0) ).
Proof. Starting from a strictly monotone data set {(ti(k) ; xi(k) )}i , rst order divided dierences yi(k) in
the data are dened by
yi(k) :=

(k)
xi+1
− xi(k) 2k (k)
2k (k)
(k)
(x
s :
−
x
)
=
=
i
i+1
(k)
h
h i
ti+1
− ti(k)

(17)

(k+1)
The function y(k) is dened as the linear interpolant of the data points (t2i+1
; yi(k) ). All functions
(k)
y are therefore continuous by construction.
It has to be proved that the functions y(k) converge to a function y(∞) , and secondly this y(∞)
must be the derivative of x(∞) , dened in the previous section.
Sucient for convergence of the sequence of functions y(k) is that they form a Cauchy sequence
in k with limit 0, i.e., there must exist a 1 ¡1 and C1 ∈ R such that

ky(k+1) − y(k) k∞ 6 C1 1 k :

(18)

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F. Kuijt, R. van Damme / Journal of Computational and Applied Mathematics 101 (1999) 203– 229

By construction, the maximal distance between the functions y(k+1) and y(k) occurs at a point
(k+2)
(k+1)
t4i+j
= 2−k (i + j=4)h, for some i and j, and these distances d4i+j
satisfy
(k+1)
(k+2)
(k+2)
d4i+j
= |y(k+1) (t4i+j
) − y(k) (t4i+j
)|:

(19)

Subsequent application of (19), (17), and the subdivision relations (10) and (11), for example yields
(k+1)
for the distance d4i+1
:
(k+1)
d4i+1






 (k+1)
1 (k)
3 (k)  2k  (k+1) 1 (k)
3 (k) 

= 2s2i − si−1 − si 
= y2i −
y + y
4 i−1 4 i 
h
4
4

k 
2 1
(k)
(k) 
=  (si(k) − si−1
) + si(k) G(ri(k) ; Ri+1
)
h 4


1
2k
(k) 
= si(k)  (1 − ri(k) ) + G(ri(k) ; Ri+1
) ;
h
4

and the other distances are determined similarly:


1 (k)
2k  1
(k)
(k)
) − si−1
G(ri−1
; Ri(k) ) ;
=  si(k) G(ri(k) ; Ri+1
h 2
2


k

1
2
(k+1)
(k) 
d4i+1
= si(k)  (1 − ri(k) ) + G(ri(k) ; Ri+1
) ;
h
4


d4i(k+1)



(k+1)
= 0;
d4i+2
(k+1)
d4i+3

1
2k
(k)
(k) 
= si(k)  (1 − Ri+1
) − G(ri(k) ; Ri+1
) :
h
4




The distance d4i(k+1) is easily estimated with
2k
(k)
)|;
max si(k) max|G(ri(k) ; Ri+1
i
h i
and thus, it is obtained that
d4i(k+1) 6

max dj(k+1) 6
j

1
2k
(k)
(k) 
max si(k) max max |G(ri(k) ; Ri+1
)|;  (1 − ri(k) ) + G(ri(k) ; Ri+1
) ;
i
h i
4


1

 (1 − R(k) ) − G(r (k) ; R(k) ) :
i
i+1
i+1 
4






(20)

Subsequently, we apply assumptions (15) and (16), and also use the fact that G(1; 1) = 0, see (8).
The function G can now be estimated as follows:
(k)
(k)
(k)
)| = |G(ri(k) ; Ri+1
) − G(1; 1)| 6 B1 k(ri(k) − 1; Ri+1
− 1)k
|G(ri(k) ; Ri+1

6 B3 k ;
where B3 ¡∞. According to (10) and (11), the rst part of (20) can be estimated as
max
i

si(k+1)

6

1
2



1+



(k)
max |G(ri(k) ; Ri+1
)|
i

max si(k) 6 21 (1 + B3 k ) max si(k) ;
i

i

F. Kuijt, R. van Damme / Journal of Computational and Applied Mathematics 101 (1999) 203– 229

211

which yields
max
i

si(k)

6

 k

1
2

max si(0)
i

k−1
Y

(1 + B3 ‘ ):

‘=0

Since 1 + x 6 e x , we obtain
k−1
Y

‘

(1 + B3  ) 6

‘=0

k−1
Y

‘

exp(B3  ) = exp B3

‘=0

k−1
X



‘

‘=0

1 − k
= exp B3
1 − 

!

B3
6 exp
1 − 


!



=: B4 ¡∞;

and hence
max
i

si(k)

6 B4

 k

1
2

max si(0) :
i

The second part of (20) is estimated as
(k)
(k)
(k)
(k)
max max{|G(ri(k) ; Ri+1
)|; | 41 (1 − ri(k) ) + G(ri(k) ; Ri+1
)|; | 14 (1 − Ri+1
) − G(ri(k) ; Ri+1
)|} 6 B5 ˆk ;
i

where
ˆ := max{;  }¡1

and

B5 := max{2B1 ; 12 B2 ; B3 }:

We complete the proof of (18) with
ky

(k+1)

2k
− y k∞ =
h
(k)

 k

1
2

B4 B5 max si(0) ˆk = B6 ˆk
i

with
B6 :=

B4 B5
max si(0) ¡∞;
i
h
′

as ¡1.
ˆ
The remaining part of the proof is to show that also y(∞) = x(∞) , i.e., y(k) converges to the
derivative of the limit function x(∞) . This can be done by the standard approach using the uniform
convergence of Bernstein polynomials: the derivative of the Bernstein polynomial determined by the
data {xi(k) }i on the interval [x0(0) ; xN(0) ] is the Bernstein polynomial of the data {yi(k) }i on the same
interval. This is described in [5]. The limit derivative y(∞) is C 0 , so the limit function x(∞) is C 1 ,
which completes the proof.
Remark 10 (The linear four-point scheme). Note that minimisation of the second part of (20)
yields the linear four-point scheme [5] given in (12), i.e., the linear four-point scheme is “as smooth
as possible”.

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F. Kuijt, R. van Damme / Journal of Computational and Applied Mathematics 101 (1999) 203– 229

6. Construction of rational subdivision schemes
In this section we restrict the class of subdivision schemes to schemes that generate continuously
dierentiable limit functions.
Since G cannot be polynomial, see Remark 6, a relatively simple restriction is achieved by choosing the function G of a specic nonlinear form: a rational function. Observe that the convexity
preserving subdivision scheme in [16] is also rational.
Theorem 11. Let the data xi(0) be drawn from a strictly monotone and four times continuously
dierentiable function g; as follows:
xi(0) = g(ih):
Assume that the function G is rational and bilinear in the numerator and the denominator; and
additionally require that G satises monotonicity condition (9).
Then; subdivision scheme (7) can only have approximation order four if G is of the form
G(r; R) =

r−R
‘1 + (1 + ‘2 )(r + R) + ‘3 rR

(‘1 ; ‘2 ; ‘3 ) ∈
;

(21)

where the triangular domain
is dened as

= {(‘1 ; ‘2 ; ‘3 ); such that ‘1 ; ‘2 ; ‘3 ¿ 0 and ‘1 + 2‘2 + ‘3 = 6}:

(22)

Proof. The class of rational functions G where the numerator and the denominator are bilinear
functions in r and R is denoted by
G(r; R) =

b1 + b2 r + b3 R + b4 rR
b5 + b6 r + b7 R + b8 rR

where bj ∈ R:

(23)

First, we impose conditions on the parameters bj that are necessary for fourth order accuracy of the
subdivision scheme. Necessary conditions are achieved by application to initial data that by denition
(0)
satisfy x(0)
and therefore ri(0) = ri(0) , using the rational function G as in (23) compared with
i = xi
the linear function G as in (12). It is easily checked that the following condition is necessary for
fourth-order accuracy (see also Section 9):
(0)
(0)
(1)
4
1 (0)
 (0) (0)
|x2i+1
− x(1)
2i+1 | = 2 si |G(ri ; Ri+1 ) − G(ri ; Ri+1 )| = O(h );

where
ri(0) =

g(ih) − g((i − 1)h)
g((i + 1)h) − g(ih)

and

R(0)
i+1 =

g((i + 2)h) − g((i + 1)h)
:
g((i + 1)h) − g(ih)

Since si(0) = O(h), it must hold that
3
 (0) (0)
G(ri(0) ; R(0)
i+1 ) − G(ri ; Ri+1 ) = O(h ):

Combining these constraints on the parameters bj with condition (8), one easily obtains that the
function G can be written in the form
r−R
G(r; R) =
:
(24)
‘1 + ‘0 (r + R) + rR(8 − 2‘0 − ‘1 )

F. Kuijt, R. van Damme / Journal of Computational and Applied Mathematics 101 (1999) 203– 229

213

Additional necessary conditions on the values of ‘0 and ‘1 in (24) are determined by condition
(9) and the requirement that G may not contain poles for positive values of r and R. A simple
calculation shows that necessarily
‘0 ¿ 1;

‘1 ¿ 0;

8 − 2‘0 − ‘1 ¿ 0:

Dening ‘2 = ‘0 − 1 and ‘3 = 6 − ‘1 − 2‘2 yields that the function G can be written as (21).
Remark 12. A simple calculation shows that the subdivision scheme with G in (21) reproduces
quadratic polynomials if ‘3 = 0, i.e., ‘1 + 2‘2 = 6; ‘1 ; ‘2 ¿ 0.
Remark 13. It is easily checked that G in (21) satises condition 5 from Section 2.
In addition, note that G in (21) automatically satises the following natural property:
• G(r; R∗ ) is strict monotone increasing in r, at xed R∗ ¿ 0.
• G(r ∗ ; R) is strict monotone decreasing in R, at xed r ∗ ¿ 0.
Applying (8) yields that such a function G also satises the natural condition
G(r; R)¿0;

∀r¿R¿0:

Remark 14. In the special case ‘1 = 2, ‘2 = 1 and ‘3 = 2, the function G in (21) reduces to
1
GC (r; R) =
2



1
1
−
:
1+R 1+r


(25)

In this case, the subdivision function (21) can be factorised as a dierence of two univariate functions
in r and in R respectively. This factorisation is only possible for this specic choice of the parameters
‘1 , ‘2 and ‘3 .
In this section, we showed that the class of subdivision schemes (7) with (21) satises necessary conditions for approximation order four. It is proved in Section 9 that this class of rational
monotonicity preserving interpolatory subdivision schemes has indeed approximation order four.
To be able to prove the smoothness properties and approximation order four, we use some additional properties on subdivision scheme (7) with (21). These properties are discussed in the next
section.
7. Ratios of rst-order dierences
In this section we investigate the behaviour of ratios of rst-order dierences obtained after application of subdivision scheme (7) with G as in (21).
7.1. Boundedness of dierence ratios
In this section we prove that the ratios of adjacent dierences in iteration step k + 1 are bounded
by the maximum of the ratios of dierences in iteration k.

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F. Kuijt, R. van Damme / Journal of Computational and Applied Mathematics 101 (1999) 203– 229

Theorem 15. Dene numbers qi(k) and q(k) by
qi(k)

:= max

(

ri(k) ;

1
ri(k)

)

and q(k) := max qi(k) :

(26)

i

Then; application of subdivision scheme (7) with (21) yields
q(k+1) 6 q(k) :

(27)

(k+1)
Proof. Let the data xi(k) be given and its ratios of dierences ri(k) as dened in (6). Since r2i+1
can
(k)
(k)
(k+1)
(k)
(k)
(k)
be written as a function of ri and ri+1 , and r2i
as a function of ri−1 , ri and ri+1 , we prove here
that ratios of rst-order dierences at level k + 1 are bounded as follows:

max

(

max

(

(k+1)
r2i+1
;

)

1
(k+1)
r2i+1

6 max

(

6 max

(

ri(k) ;

1

1
; r (k) ; (k)
(k) i+1
ri
ri+1

)

and
r2i(k+1) ;

)

1
r2i(k+1)

(k)
ri−1
;

1

1 (k) 1
; (k)
; r (k) ; (k) ; ri+1
(k) i
ri−1
ri
ri+1

)

:

(k+1)
We illustrate the proof by the treatment of r2i+1
. Since the properties that must be proved contain
maximum functions, the proof has to enumerate several situations depending on the size of the rj(k) .
We, therefore, order the ratios rj(k) in size. The proof is based on treating all partitions separately.
(k)
is maximal, i.e., one of the following two partitions is valid:
Consider the case that 1=ri+1
(k)
ri+1
6 ri(k) 6 1 6

1
1
6 (k)
rik
ri+1

(28)

or
(k)
6
ri+1

1
ri(k)

6 1 6 ri(k) 6

1
(k)
ri+1

(29)

:

Then it must be proved that
1
(k)
ri+1

(k+1)
− r2i+1
¿0

and

1
(k)
ri+1

−

1
(k+1)
r2i+1

¿ 0:

(30)

As an example we give the construction for the second partition (29). A convenient transformation
of variables,
(k)
ri+1
=

1
1+x

and

ri(k) = 1 + x

1
1+y

where x; y ¿ 0;

is substituted in (30). Inequalities (30) then result in rational expressions that must hold for all
x; y ¿ 0.
By requiring that both the numerator and the denominator of such an expression is positive (or
negative), it is sucient for positiveness of the rational expression that the coecients in both the
numerator and the denominator have the same sign.

F. Kuijt, R. van Damme / Journal of Computational and Applied Mathematics 101 (1999) 203– 229

215

The construction for the even ratios r2i(k+1) requires a more sophisticated substitution for three
(k)
(k)
variables (ri−1
, ri(k) and ri+1
) in each partition. As an example for the partition
(k)
6
ri+1

1
(k)
ri−1

6 ri(k) 6 1 6

1
ri(k)

(k)
6 ri−1
6

1
(k)
ri+1

;

the substitution
(k)
ri+1
=

1
1
1
; r (k) = 1 + x
; r (k) =
1
1 + x i−1
1+y i
1 + x 1+z

where x; y; z ¿ 0;

has been used. Again it is required that the coecient in the numerator and the denominator have
the same sign.
The coecients however depend on the parameters ‘1 and ‘2 (note that ‘3 = 6 − ‘1 − 2‘2 ). By
enumeration over all dierent expressions to be proved and all dierent partitions, a large set of
constraints cj (‘1 ; ‘2 ) ¿ 0 is constructed in this way. Since we have to prove the validity of many
constraints, the calculations are performed using algebraic manipulation software. We used Maple [4]
to generate all equations and to solve the constraints. It is algebraically checked that all constraints
lie outside the domain
dened in (22) or on its boundary. As an illustration, all constraints
cj (‘1 ; ‘2 ) = 0 are shown in Fig. 1. Since in addition cj (2; 1)¿0, ∀j, it is thus proved that cj (‘1 ; ‘2 )
¿0, ∀(‘1 ; ‘2 ) ∈
\@
. The conclusion also holds for
including its boundaries, since all rational
expressions are continuous.
It is thus shown in this proof that
(k+1)
(k)
q2i+1
6 max{qi(k) ; qi+1
};

∀i;

(k)
(k)
; qi(k) ; qi+1
};
q2i(k+1) 6 max{qi−1

∀i

which completes the proof.
This result is also maximal in a single step relation between two subsequent subdivision iterations:
Theorem 16. Let the numbers q(k) be dened as in Theorem 15. Then; in general; there does not
exist a ¡1 such that
q(k+1) − 16(q(k) − 1):
Proof. Consider the data xi(0) with dierences si(0) satisfying
(0)
s−1
= a;

s0(0) = b;

s1(0) = a

and

The maximum ratio q(0) is equal to
q(0) =

s1(0) a
= ¿1;
s0(0) b

s2(0) = b where a¿b¿0:

216

F. Kuijt, R. van Damme / Journal of Computational and Applied Mathematics 101 (1999) 203– 229

Fig. 1. The constraints cj (‘1 ; ‘2 ) = 0.

and because of the symmetry of the scheme, one subdivision yields
b
s1(1) = ;
2

s2(1) =

a
s(1) a
⇒ q(1) = 2(1) = = q(0) :
2
b
s1

This counterexample shows that the maximum ratio in general does not become smaller in a single
subdivision iteration.
Next it is proved, using a double step strategy, that the ratios ri(k) converge to 1.
7.2. Strict convergence of dierence ratios
In the previous subsection, Theorem 16 indicates that we cannot establish convergence of the
dierence ratios to 1 using a single step strategy. In this section, we prove that ratios converge to
1 using a double step strategy.
Theorem 17. Let the numbers q(k) be dened as in Theorem 15. Then
q(k+2) − 16 34 (q(k) − 1):

(31)

F. Kuijt, R. van Damme / Journal of Computational and Applied Mathematics 101 (1999) 203– 229

217

Proof. It is proved that the qj(k+2) satisfy
(k+2)
(k)
(k)
q4i−1
− 16 165 (max{qi−1
; qi(k) ; qi+1
} − 1);
(k)
(k)
; qi(k) ; qi+1
} − 1);
q4i(k+2) − 16 34 (max{qi−1
(k+2)
(k)
(k)
q4i+1
− 16 165 (max{qi−1
; qi(k) ; qi+1
} − 1);
(k)
(k+2)
(k)
(k)
q4i+2
; qi(k) ; qi+1
− 16 14 (max{qi−1
; qi+2
} − 1):

To illustrate the proof, we examine the rst pair of inequalities:
(k+2)
r4i−1
−1
5
−
¿0
(k)
(k) (k)
16 max{qi−1 ; qi ; qi+1 } − 1

and

(k+2)
r4i−1
−1
5
+
¿0:
(k)
(k) (k)
16 max{qi−1 ; qi ; qi+1 } − 1

Using the same approach as in the proof of Theorem 15, constraints on ‘1 and ‘2 have been
generated. Again, it is algebraically checked that all constraints lie outside the domain
, see (22),
or on its boundary. This proves that the theorem holds for all (‘1 ; ‘2 ) ∈
. The result can be written
as
max
j

(

rj(k+2) ;

1
rj(k+2)

)

(

3
1
− 16
max rj(k) ; (k)
j
4
rj

)

−1

!

and this completes the proof.
The factors 5=16, 3=4 and 1=4 have been conjectured by an asymptotic analysis on arbitrarily
chosen data xi(0) with ri(0) = 1 + i , where 0¡ ≪ 1 and i ∈ [−1; 1]. The proofs however, are given
for general data.
Remark 18. Numerical experiments show that the factor 34 cannot be improved by optimising the
parameters ‘1 and ‘2 : all parameter choices within the triangle
give the same contraction factor.
8. Convergence of rational subdivision schemes
It is proved in this section that subdivision scheme (7) with (21) preserves monotonicity and
generates continuously dierentiable limit functions.
Theorem 19 (Monotonicity preservation). Subdivision scheme (7) with G given in (21) preserves
monotonicity.
Proof. The function G satises (9), which is a direct result from the construction in Section 6.
With respect to convergence, the following theorem holds:
Theorem 20 (C 0 -convergence). Let the same conditions hold as in Theorem 8.

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F. Kuijt, R. van Damme / Journal of Computational and Applied Mathematics 101 (1999) 203– 229

Then; repeated application of subdivision scheme (7) with (21) leads to a continuous function
which is monotone and interpolates the initial data points (ti(0) ; xi(0) ) if ‘2 ¿0 or the initial data are
strictly monotone.
Proof. First, if ‘2 ¿0, it is shown that G satises

r+R
r−R
6
‘1 + (1 + ‘2 )(r + R) + ‘3 rR  ‘1 + (1 + ‘2 )(r + R) + ‘3 rR
1
r+R
=
= ¡1;
6
(1 + ‘2 )(r + R) 1 + ‘2





|G(r; R)| = 

(32)

i.e., G satises condition (13). If the initial data are strictly monotone, we remark that the ratios of
rst order dierences ri(k) can be estimated using Theorem 15:
16 max max
i

(

ri(k) ;

1
ri(k)

)

6q(0) ¡∞:

The function G dened in (21) is monotone in both arguments, see Remark 13, and hence is maximal
in the case
ri(k) = q(0)

and

(k)
Ri+1
= 1=q(0) ;

which yields that G can be estimated as
(k)
|G(ri(k) ; Ri+1
)|6

(q(0) )2 − 1
= ¡1;
(‘1 + ‘3 )q(0) + (1 + ‘2 )((q(0) )2 + 1)

(33)

which proves (13) for all (‘1 ; ‘2 ; ‘3 ) ∈
, as q(0) ¡∞.
Concerning C 1 -convergence, we can formulate the following result:
Theorem 21 (C 1 -convergence). Let the same conditions hold as in Theorem 9, and let the data be
strictly monotone.
Then repeated application of subdivision scheme (7) with (21) leads to a continuously dierentiable function which is monotone and interpolates the initial data points (ti(0) ; xi(0) ).
Proof. As the function G is continuously dierentiable in r and R, for all r; R¿0, it satises Lipschitz
condition (15). It is shown in Section 7 that this function G yields that ratios of adjacent rst-order
dierences converge to 1, i.e., the C 1 -requirement (16) is satised.
The analysis of the subdivision scheme for monotone, but not strictly monotone data is more
dicult to examine. At any diadic point the left and right derivative can be proved to be equal,
but this is not sucient for convergence to a continuously dierentiable limit function. Numerical
experiments however show that the subdivision scheme yields limit functions that are continuously
dierentiable even in such cases:
Conjecture 22 (Always C 1 ). Let the same conditions hold as in Theorem 9, but let the data be
monotone but not necessarily strictly monotone.

F. Kuijt, R. van Damme / Journal of Computational and Applied Mathematics 101 (1999) 203– 229

219

Table 1
The data set used in the numerical example
ti(0)
xi(0)

−2
−2

−1
−1

0
0

1
1=2

2
1

3
6

4
6

5
7

6
8

7
9

8
10

Fig. 2. The limit function obtained by the monotonicity preserving scheme (with ‘1 = 2, ‘2 = 1, ‘3 = 2) and the linear
four-point scheme.

Repeated application of subdivision scheme (7) with (21) leads to a continuously dierentiable
function which is monotone and interpolates the initial data points (ti(0) ; xi(0) ).
In the following example we show the graphical capabilities of the subdivision scheme and illustrate tension control that is provided by the parameters ‘1 , ‘2 and ‘3 .
Example 23 (Numerical example). Consider the uniform data set dened in Table 23. Some visual
results are shown after repeated application of subdivision scheme (7) with G in (21).
First, the limit function is shown in the interval [t0(0) ; t6(0) ] for the monotonicity preserving scheme
with the parameter choice ‘1 = 2, ‘2 = 1 and ‘3 = 2, see Remark 14, which has proved to generate
results that are visually pleasing. This result is compared with the graphical performance of the
linear four-point scheme [5], see (12), in Fig. 2, which clearly does not preserve monotonicity for
this data set.
In the next plots, see Fig. 3, we again display the limit function of the monotonicity preserving
subdivision scheme with ‘1 = 2, ‘2 = 1 and ‘3 = 2, together with its derivative, which is nonnegative
in the whole interval.
Finally, it is shown in Fig. 4 that the parameters ‘j act as tension parameters. Two extreme choices
are compared: ‘1 = 0, ‘2 = 0, ‘3 = 6 and ‘1 = 3, ‘2 = 3=2, ‘3 = 0 respectively. The rst case leads to
a limit function that is almost piecewise constant in dicult areas, whereas the second choice of the
tension parameters leads to an almost piecewise linear function.

220

F. Kuijt, R. van Damme / Journal of Computational and Applied Mathematics 101 (1999) 203– 229

Fig. 3. The limit function x(∞) and its derivative y(∞) obtained by the monotonicity preserving scheme (with ‘1 = 2,
‘2 = 1, ‘3 = 2).

Fig. 4. Tension control illustrated by the limit function obtained by the monotonicity preserving scheme with ‘1 = 0,
‘2 = 0, ‘3 = 6 and ‘1 = 3, ‘2 = 3=2, ‘3 = 0 respectively.

9. Approximation order
In this section, the approximation properties of the monotonicity preserving subdivision scheme
from the previous sections are examined. Although a simple calculation shows that the scheme
only reproduces linear functions (and quadratics if ‘3 = 0), it can be proved that the scheme has
approximation order four.

F. Kuijt, R. van Damme / Journal of Computational and Applied Mathematics 101 (1999) 203– 229

221

Theorem 24 (Approximation order). Consider the data set {(ti(0) ; xi(0) )}Ni=0 drawn from a strictly
monotone function g ∈ C 4 (I ); where I = [t0(0) ; tN(0) ] = [0; 1]; such that
xi(0) = g(ti(0) ) where ti(0) = ih and

Nh = 1:

Dene xh(∞) as the limit function obtained by repeated application of the monotonicity preserving
subdivision scheme (7) with (21) on the data xi(0) .
Then; subdivision scheme (7) with (21) is fourth-order accurate; i.e. there exists a constant C;
not depending on h; such that
kxh(∞) − gkI; ∞ 6Ch4 ;
provided the boundaries are treated properly.
Proof. The boundaries are treated rst. Note that, using Taylor series of g in t0(0) and tN(0) , any
strictly monotone function g ∈ C 4 (I ) can be extended to a strictly monotone function g̃ ∈ C 4 (I˜),
where I˜ = [−; 1 + ]), such that g̃(x) = g(x), ∀x ∈ I . Provided h is small enough, such an ¿0
(0)
(0)
always exists. The boundary data points x−2
, x−1
, xN(0)+1 , and xN(0)+2 , which are necessary to determine
the limit function x(∞) in I , are now drawn from this extended function g̃.
To prove fourth-order accuracy, we compare the monotonicity preserving scheme with the linear
four-point scheme [5] with w = 1=16, see (12). This scheme reproduces cubic polynomials. The data
generated by the linear scheme are denoted by x i(k) , and therefore
x2i(k+1) = xi(k) ;
(k+1)
=
x2i+1

9
(x (k)
16 i

(k)
+ xi+1
)−

1
(x (k)
16 i−1

(k)
+ xi+2
):

(34)

The linear four-point scheme with w = 161 has proved to be fourth-order accurate in [5]: there exists
a B ∈ R such that
kxh(∞) − gkI;∞ 6Bh4 :

(35)

Starting from the initial data x i(0) = x(0)
i ; ∀i, it will be proved that
∀k

∃ Ck ¡∞

such that max |xi(k) − xi (k) |6Ck h4 ;
i

(36)

and
lim Ck ¡∞:

k→∞

The proof of (36) is based on induction with respect to k. Clearly, C0 = 0, since the initial data
satisfy xi(0) = x (0)
i ; ∀i. Suppose that (36) is valid for some k, it will be proved that
(k+1)
(k+1)
max |xi(k+1) − x i(k+1) | = max max{|x2i(k+1) − x 2i(k+1) |; |x2i+1
− x 2i+1
|}6Ck+1 h4 ;
i

i

where a relation between Ck+1 and Ck will be given. The rst term gives
max |x2i(k+1) − x 2i(k+1) | = max |xi(k) − xi (k) |6Ck h4 ;
i

i

222

F. Kuijt, R. van Damme / Journal of Computational and Applied Mathematics 101 (1999) 203– 229

so it remains to estimate
(k+1)
(k+1)
|x2i+1
− x 2i+1
|
(k)
(k)
= |{ 21 (xi(k) + xi+1
) + 12 si(k) G(ri(k) ; Ri+1
)} − { 21 (xi (k) + xi+1 (k) ) + 21 si (k) G(ri (k) ; ri+1 (k) )}|
(k)
(k)
6 12 |xi(k) − xi (k) | + 12 |xi+1
− xi+1 (k) | + | 21 si(k) G(ri(k) ; Ri+1
) − 21 si (k) G(ri (k) ; ri+1 (k) )|:

(37)

Using the triangle inequality, the last contribution is written as
(k)
(k) 
(k)
) − s(k)
|si(k) G(ri(k) ; Ri+1
i G(ri ; Ri+1 )|
(k)
(k)
(k)
(k)
= |si(k) G(ri(k) ; Ri+1
) − si (k) G(ri(k) ; Ri+1
) + si (k) G(ri(k) ; Ri+1
) − si (k) G(ri(k) ; Ri+1
)
(k)
+si (k) G(ri(k) ; Ri+1
) − si (k) G(ri (k) ; R i+1 (k) )|
(k)
(k)
)| + |si (k) ||G(ri(k) ; Ri+1
)
6|si(k) − si (k) ||G(ri(k) ; Ri+1
(k)
(k)
−G(ri(k) ; Ri+1
)| + |si (k) ||G(ri(k) ; Ri+1
) − G(ri (k) ; R i+1 (k) )|:

In the appendix, we show convergence of the following contributions:
1
2

(k)
max | si(k) − si (k) | max |G(ri(k) ; Ri+1
)|6A1 1k h4 ;

(38)

1
2

(k)
(k)
max |si (k) | max |G(ri(k) ; Ri+1
) − G(ri(k) ; Ri+1
)|6A2 2k h4 ;

(39)

1
2

(k)
max |si (k) | max |G(ri(k) ; Ri+1
) − G(ri (k) ; R i+1 (k) )|6A3 3k h4 ;

(40)

i

i

i

i

i

i

where
j ¡1

and

Aj ¡∞;

j = 1; 2; 3;

are numbers depending on derivatives of the original function g. It is easily checked that these
inequalities are indeed sucient for conditions (38)–(40). The proofs of some of the estimates are
given by induction in k. Using Taylor series it is derived that the initial data indeed satisfy the
estimates.
Using induction hypothesis (36) together with the estimates (38)–(40), Eq. (37) yields
(k+1)
(k+1)
max |x2i+1
− x 2i+1
| 6 12 Ck h4 + 21 Ck h4 + A1 1k h4 + A2 2k h4 + A3 3k h4
i

6 Ck h4 + A  k h4 ;
where A = A1 + A2 + A3 and  = max{1 ; 2 ; 3 }. The coecient Ck+1 now satises
Ck+1 6max{Ck ; Ck + A  k } = Ck + A  k :
Since C0 = 0, this gives
1 −  k
1
⇒ lim Ck 6A
=: A¡∞:
Ck 6A
k→∞
1 − 
1 − 

F. Kuijt, R. van Damme / Journal of Computational and Applied Mathematics 101 (1999) 203– 229

223

Having proved the induction step, hypothesis (36) holds. Therefore
kxh(∞) − gkI; ∞ 6kxh(∞) − xh (∞) kI; ∞ + kxh (∞) − gkI; ∞ 6(A + B)h4 ;
which completes the proof, i.e., subdivision scheme (7) with (21) has approximation order four.
Remark 25. Observe that this analysis is only valid for strictly monotone data, i.e., data drawn from
a function g with g′ ()¿0, ∀ ∈ I . Numerical experiments show that if g′ () = 0 for some  ∈ I , the
approximation order in the max-norm decreases to 3.
Remark 26. More recently, see [12, 17], an alternative proof for the approximation order of subdivision schemes is provided. This approach uses the notion of stability of stationary subdivision
schemes.

10. Generalisations
In this section, we brie
y describe an extension and look forward to generalisations of the monotonicity preserving subdivision scheme discussed in this article.
10.1. Piecewise monotonicity
The subdivision scheme discussed in this article can be extended to a piecewise monotonicity
preserving subdivision scheme suited for interpolation of piecewise monotone data.
We rst observe that any monotonicity preserving subdivision scheme of the form (7) is also
directly applicable to monotone decreasing data. It is therefore only necessary to examine regions in
the initial data where the dierences si(0) change sign and to split the domain in monotone increasing
parts and monotone decreasing parts.
If one of the dierences in the initial data is zero, i.e., the case that the dierences satisfy s0(0) = 0,
(0)
s−1
¡0 and s1(0) ¿0, a simple and natural way to split the monotonicity regions is provided by the
data: the solution in the interval [t0(0) ; t1(0) ] becomes constant. The monotonicity preserving subdivision
scheme can be applied both on the left-hand side of t0(0) and on the right-hand side of t1(0) . The limit
function is then monotone decreasing left to t1(0) and monotone increasing right to t0(0) . In fact, the
scheme is piecewise monotonicity preserving for these data.
(0)
(0)
The general case is characterised by data with dierences satisfying s−2
¡0, s−1
¡0, s0(0) ¿0 and
s1(0) ¿0, since all other cases degenerate to this situation after one iteration. We dene the split point
as the point where the dierences change sign, i.e., in this case t0(0) .
A simple method to adapt the subdivision scheme to piecewise monotonicity preservation is as
follows:
• Apply the monotonicity preserving subdivision scheme to determine xj(k) for j¡0, where s0(0) is
replaced by 0.
(0)
• Apply the monotonicity preserving subdivision scheme to determine xj(k) for j¿0, where s−1
is
replaced by 0.

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F. Kuijt, R. van Damme / Journal of Computational and Applied Mathematics 101 (1999) 203– 229

The corresponding function GPM of this piecewise monotonicity preserving subdivision scheme can
be written as
GPM (r; R) = G (max{0; r}; max{0; R}) ;
with for example G dened as in (21). It is clear that the limit function is monotone decreasing for
t6t0(0) and monotone increasing for t¿t0(0) . The derivative of the limit function in t0(0) is equal to
zero. Convergence to a continuously dierentiable limit function is achieved if conjecture 22 is true.
10.2. Relation with monotonicity preserving splines
An alternative way to derive subdivision scheme (7) with G as in (25) originates from Hermite
interpolation using quadratic splines, see [18, 11].
Consider a strictly monotone data set {(2−k i; xi(k) )}i and dene Bezier points as follows
b2i(k) = xi(k) ;

(k)
b2i+1
= xi(k) +

2−k (k)
 ;
4 i

(k)
(k)
b2i+3
= xi+1
−

2−k (k)
 ;
4 i+1

(k)
(k)
b2i+4
= xi+1
;

(k+1)
where the i(k) are suitable derivative estimates. Dene now the subdivision points x2i+1
as follows:

2−k (k)
(k)
):
( − i+1
8 i

(k)
(k+1)
(k)
x2i+1
)+
= b2i+2
= 12 (b2i+1 + b2i+3 ) = 21 (xi(k) + xi+1

(41)

If the derivatives are estimated using Butland slopes, see [1, 9], i.e.,
i(k) = 2

(k) (k)
yi
yi−1
(k)
yi−1 + yi(k)

where yj(k) = 2k sj(k) ;

then subdivision scheme (41) preserves monotonicity and is written as
(k+1)
x2i+1

1
1
(k)
= (xi(k) + xi+1
) + si(k)
2
4

(k)
(k)
si−1
si+1
−
(k)
(k)
si−1
+ si(k) si(k) + si+1

!

1
1
(k)
(k)
= (xi(k) + xi+1
)d + si(k) GB (ri(k) ; Ri+1
);
2
2
with
(k)
GB (ri(k) ; Ri+1
)=

1
2

1
=
2

(k)
(k)
si−1
1
si+1
=
−
(k)
(k)
(k)
(k)
2
si−1 + si
si + si+1

!

1
(k) (k)
1 + si+1
=si

−

1
(k)
1 + si−1
=si(k)

!

si(k)
si(k)
−
(k)
(k)
si(k) + si+1
si−1
+ si(k)

!

1
1
−
(k)
1 + Ri+1 1 + ri(k)

!

1
=
2

;

which coincides with the special case GC in (25), see Remark 14.
In fact this approach does not only dene a subdivision scheme, but the construction using Bezier
points also provides an explicit interpolation method using quadratic (B-)splines that is monotonicity
preserving.

F. Kuijt, R. van Damme / Journal of Computational and Applied Mathematics 101 (1999) 203– 229

225

Remark 27. Note that the Butland slope is the harmonic average of two adjacent dierences, where
the convexity preserving subdivision scheme in [16] is a scheme that contains the harmonic average
of two adjacent second-order dierences.
A relation between rational interpolation and convexity preserving subdivision is observed in [8].
There exists also a connection between monotonicity preserving rational interpolation and subdivision scheme (7) with (21). In [10], a class of rational splines is dened on an interval [ti ; ti+1 ] as
follows:
xi (t) =

yi xi+1 2 + (xi i+1 + xi+1 i ) (1 − ) + yi xi (1 − ) 2
;
yi  2 + (i+1 + i )(1 − ) + yi (1 − )2

where
=

t − ti
ti+1 − ti

and yi =

xi+1 − xi
;
ti+1 − ti

and the j are suitable chosen derivative estimates at tj .
The construction is restricted to equidistant data, and a subdivision scheme is obtained by halfway
(k)
(k+1)
evaluation of xi (t), i.e., at t2i+1
). This results in the subdivision scheme
= 12 (ti(k) + ti+1
1
1
(k) − (k)
(k+1)
(k+1)
(k)
x2i+1
= x(t2i+1
) = (xi(k) + xi+1
) + hi(k) (k) i+1 (k) i
:
(k)
2
2
i + 2
xi + i+1
Two choices for the derivative estimates j(k) are presented: the arithmetic mean and the harmonic
mean:
(k)
j(k) = 12 (
xj−1
+
xj(k) );

j(k) =

or

(k)
2
xj−1

xj(k)
(k)

xj−1
+
xj(k)

;

and both choices yield a subdivision scheme as in (7), and the functions GL and GB are respectively
given by
GL (r; R) =

r−R
6+r+R

and

GB (r; R) =

r−R
:
1 + 2(r + R) + 3rR

Both functions are contained in class (21), as (‘1 ; ‘2 ; ‘3 ) = (6; 0; 0) ∈
, and (‘1 ; ‘2 ; ‘3 ) = (1; 1; 3) ∈
.
10.3. Application to grid renement
The application area of the subdivision scheme discussed in this article is not restricted to monotonicity preserving uniform subdivision. In this section we brie
y describe the importance of monotonicity preserving subdivision to grid renement.
We examined uniform subdivision schemes applied to data points (ti(k) ; xi(k) ) that preserve strict
monotonicity. This is exactly the same as subdivision scheme for a grid xi(k) that keeps the grid
ordered. In addition, the schemes discussed in this article have the property that ratios of rst-order
dierences converge to 1, see Section 7. Applying such a scheme to xi(k) yields that the grid becomes
homogeneous, i.e., ratios of the size of two adjacent cells tends to 1. This is important in applications
such as grid generation and nite element methods.

226

F. Kuijt, R. van Damme / Journal of Computational and Applied Mathematics 101 (1999) 203– 229

In case of functional nonuniform data (xi(k) ; fi(k) ), we propose to apply a suitable monotonicity
preserving scheme to the data xi(k) , e.g., scheme (7) with (21), which makes the grid homogeneous.
Another subdivision scheme is applied to the data fi(k) . This leads to stationary nonuniform subdivision schemes for functional nonuniform data, which are discussed in a separate report [14]. We
describe nonuniform extensions of the convexity preserving sch