Computation of the Bernstein Coecients on Subdivided Triangles Ralf Hungerbuhler
Fakultat fur Mathematik und Informatik, Universitat Konstanz, Postfach 5560 D 197, D{78434 Konstanz, Germany
Jurgen Garlo (
[email protected])
Fachbereich Informatik, Fachhochschule Konstanz, Postfach 10 05 43, D{78405 Konstanz, Germany
Abstract. We present a procedure for computing the coecients of the expansion
of a bivariate polynomial into Bernstein polynomials over subtriangles. These triangles are generated by partitioning the standard simplex of IR2 . The coecients are computed directly from the coecients on the subdivided triangle from the preceding subdivision level. This allows a recursive computation of the coecients and facilitates the economical computation of bounds for the range of a bivariate polynomial over a given triangle.
Keywords: Range enclosure, Bernstein polynomials, subdivision
1. Introduction Finding a tight enclosure for the range of a function over a compact set D IRl is a basic problem of interval computations, cf. [5],[6] and the references therein. If the function is a multivariate polynomial and D is a box, there exist several methods, for references cf. [4]. Compared to boxes, simplices allow more exibility since far more general geometries can be treated. In [4] we present a method for computing bounds for the range of a bivariate polynomial
p(x; y) =
n X ; = 0
a x y with a 2 C
over a triangle. This method is based on the expansion of p into Bernstein polynomials. If p has only real coecients the minimum and the maximum of the coecients of this expansion, the so{called Bernstein coecients, provide lower and upper bounds for the range. All rounding errors appearing in the computation of the Bernstein coecients can be taken into account similarly as in [1] for the univariate case. In the case Author to whom all correspondence should be directed.
c 1999 Kluwer Academic Publishers.
Printed in the Netherlands.
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2 that p has complex coecients the convex hull of the Bernstein coecients encloses the range, for details see [4]. Without loss of generality we can assume that the given triangle is the unit triangle n
T = (x; y) 2 IR2 x; y 0 ^ x + y 1
o
since any nonempty triangle can be mapped by an ane transformation onto T . The bounds are improved by subdivision. In partitioning T we are led by the following useful fact [2]: When we subdivide a square by successively halving in both coordinate directions and calculate the Bernstein coecients on a generated subsquare then we obtain as a byproduct of the computation the Bernstein coecients on the neighbouring subsquares. This is one of the reasons why we partition T into squares and triangles, cf. Figures 2 and 3 in [4]. In this technical note we use the de nitions and notations from [4] and present a procedure for computing the Bernstein coecients on the subtriangles T2mt?+11 and T2mt +1 of the triangle Ttm , t = 1; : : :; 2m , m 1, which are subtriangles generated by subdivision; in the beginning these are the triangle T and the subtriangles with vertex sets f(0, 21 ), ( 21 , 12 ), (0,1)g and f( 12 ,0), (1,0), ( 12 , 12 )g. The Bernstein coecients are computed directly from the Bernstein coecients on the triangle Ttm . This allows a recursive computation of the Bernstein coecients on the subtriangles and facilitates the economical computation of the bounds for the range of the given polynomial over T . In passing we note that the procedure presented in [4] for computing the Bernstein coecients on the four subsquares generated by halving [0; 1] [0; 1] in both coordinate directions can be generalized to the l {dimensional case. The resulting algorithm is rather technical. The procedure is therefore omitted here and the interested reader is referred to [3].
2. Main Result In this section we give the procedure for computing the Bernstein coecients on the subtriangles T2mt?+11 and T2mt +1 of a triangle Ttm . In the following stands for T2mt?+11 or T2mt +1 . PROCEDURE Let m 1 and t 2 f1; : : :; 2mg.
Start by setting (0) ()
= b (Ttm ) for all (; ) 2 IT(2n) ;
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3
then for = 0; : ::; n ? 1
.
2 for (; ) 2 IT(2(n?)?1) ; (1) 9 ? m+1 ? m+1 . ( ) ( ) 2; > = T2t?1 + ; +1 T2t?1 > =
() () + () () = ; +1 +1;
() ()
(+1) T m+1 2t?1
?
?
?
?
.
> (+1) T m+1 = () T m+1 + () m+1 ; 2; > 2t 2t +1; T2t (; ) 2 IT(2(n??1)) , 1(; 1) () = ;()2(n?)?1? () ; = 0; : : :; 2(n ? ) ? 1 ; (2)
?
?
?
.
2 ; = 0; : : :; 2(n?)?1; 2(; 1) T2mt?+11 = 0() T2mt?+11 + 0(;)+1 T2mt?+11 . ? ? ? 2(; 1) T2mt +1 = (0) T2mt +1 + (+1) ;0 T2mt +1 2 ; = 0; : : :; 2(n?)?1;
nally for = 1; 2 , = 0; : : :; n ? 2 and = 1; : : :; 2(n ? ) ? 1 . (; ) () + (; ) () 2 ; = 0; : : :; 2(n?)? ?1: (; +1) () = ; +1 Theorem The Bernstein coecients on the subtriangles at subdivision level m +1 can be obtained by the Procedure from the following relations with (i; j ) 2 IT(2n): 8 (2n?i?j;j ?2(n?i)) ?T m+1 > if 2(n ? i) < j ; > 2t?1 10 > < ? ? bij T2mt?+11 = > 0(ij) T2mt?+11 if 2(n ? i) = j ; > ? > : (i;2(n?i)?j ) 2j T2mt?+11 if 2(n ? i) > j ; ?
m+1
bij T2t
=
8 (2n?i?j;i?2(n?j )) ?T m+1 > > 2t 1 > < i ? ( j ) m +1 i0 T2t > > > : (j;2(n?j )?i) ? m+1
2i
T2t
if 2(n ? j ) < i ; if 2(n ? j ) = i ; if 2(n ? j ) > i :
First one shows by induction on that ? ;(2() n?)?? T2mt?+11
= 4?
? 2((n) ?)??; T2mt +1
= 4?
X
u;v = 0
u
m) ; v b+u;2n?(+u)?(+v) (Tt (3)
X u;v = 0
u
m v b2n?(+u)?(+v);+v (Tt )
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for
4 holds true for all (; ) 2 IT(2(n?)) for = 0; : : :; n. Then one proves by induction on = 1; : : :; 2(n ? ) that the following identity is true for all = 1; 2 and = 0; : : :; n ? 2 X ?1 ? 1 ( ; ) ? +1 () = 2 (;1) () ; = 0; : : :; 2(n ? ) ? :
h=0
; +h
h
(4) We show the statements of the theorem only for the subtriangle T2mt?+11 and only in the case 2(n ? i) < j . The proof for the two remaining cases and for the subtriangle T2mt +1 is analogous. To simplify notation, we mark the dependency from T2mt?+11 and Ttm of the quantities considered in the sequel by writing m + 1 and m for short. From (2) and (1) it follows that for = 0; : : :; 2(n ? ) ? 1 1(; 1)(m + 1) = ;(2() n?)?1? (m + 1) . = ;(2() n?)? (m + 1) + (+1) ;2(n?)?1? (m + 1) 2 = 2?2?1
X
v b+u;2n?(+u)?v (m) X + b+u+1;2n?(+u+1)?v (m) u;v=0 u v
u;v=0
u
X +1 X +1 = 2?2?1 u u=0 v=0
b v +u;2n?(+u)?v (m) ;
where the last but one identity follows from (3). Now we apply identity (4) to conclude from this result that (2n?i?j;j ?2(n?i)) (m + 1) 10 = 2?(j ?2(n?i))+1 =
2j ?2n
j ?2(X n?i)?1
h=0 j ?2(X n?i)?1
j ? 2(n ? i) ? 1 h
h=0 2n?X i?j +1 2nX ?i?j u=0
= 2?i?(2n?i?j )
j ? 2(n ? i) ? 1 (2n?i?j;1) (m + 1) 1h h
2n ? i ? j + 1
v=0 2nX ?i?j
u
2n ? i ? j
v=0
v
2n ? i ? j b h+u;2n?(h+u)?v (m) v
X 2(n?i)?1g i minfu;j ?X
u=0
h=0
j ? 2(n ? i) ? 1 h
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5
2n ?ui??hj + 1 bu;2n?u?v (m) ?i?j i 2nX X i 2n ? i ? j b = 2?i?(2n?i?j ) u;2n?u?v (m) : u v u=0 v=0
To obtain the last but one identity we used the Vandermonde convolution formula, e.g., [7]. Now by Lemma 4.2 in [4] we can conclude (2n?i?j;j ?2(n?i)) (m + 1) = b (m + 1) : 10 ij
To compute the Bernstein coecients on the subtriangles T2mt?+11 and T2mt +1 by the Procedure requires 163 n3 + 4n2 + 83 n additions and the same amount of multiplications (binary shifts). Here we made use of the relations (0) ?T m+1 = (0) ?T m+1 for (; ) 2 I (2n?1)
2t 2t?1 T and ? ? 1(0; ) T2mt?+11 = 1(0; ) T2mt +1 for = 1; : : :; 2n and = 0; : : :; 2n ? : Then the cumulated operations count for the procedure for computing the Bernstein coecients on all subsquares and subtriangles lling the unit triangle T at subdivision level i for all i = 1; : : :; m sums up to O(4m n3 ) additions and the same amount of multiplications (binary shifts).
3. Numerical Example It was shown in [4] that the convex hull of the Bernstein coecients on all subsquares and subtriangles generated at subdivision level m , denoted by Cm, provides an enclosure for the range of p over T . Obviously, a tighter enclosure for this range is given by taking the union of the convex hulls of the Bernstein coecients on each subregion at level m , i.e., [ [ S Cm0 = CTtm ; CSrsm where
r = 1;:::;2m ?1 s = 1;:::;2m ?r
n
t = 1;:::;2m o
CSrsm := conv bij (Srsm) (i; j ) 2 IS(n) ; r = 1; : : :; 2m ? 1 ;
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6
s = 1; : : :; 2m ? r
and
n
o
CTtm := conv bij (Ttm) (i; j ) 2 IT(2n) ; t = 1; : : :; 2m :
We consider now the polynomial p(x; y) = 3:1 + 1:2i + (?1:8 + 4i)y + (2:5 + i)x + (17 + 8i)xy : Figures 1 and 2 show Cm0 of C20 are sharp, viz. ? b00 S112 = ? b02 T12 = ? b20 T42 =
for m = 2; 4 . Already three extreme points
a00 = p(0; 0) = 3:1 + 1:2i ; a00 + a01 = p(0; 1) = 1:3 + 5:2i ; a00 + a10 = p(1; 0) = 5:6 + 2:2i :
y
6
1 0
-
1
x
Figure 1. Enclosure C20 for the range of p over T
The edges connecting 3:1+1:2i with the points 1:3+5:2i and 5:6+2:2i are part of the range of p and are therefore sharp. Yet the rest of the boundary of C20 provides a crude approximation of the image of the straight line connecting (0; 1) with (1; 0).
veroeff2b.tex; 21/09/1999; 17:37; p.6
7
y
6
1 0
-
1
x
Figure 2. Enclosure C40 for the range of p over T
The set C40 gives a much better approximation of the range of p . For illustration, the convex hulls of the Bernstein coecients which constitute C40 are displayed. The convex hulls of the Bernstein coecients on the subsquares are already identical to the respective ranges of p .
Acknowledgements Support from the Ministry of Science and Research Baden{Wurttemberg and from the Ministry of Education, Science, Research, and Technology of the Federal Republic of Germany under the contract no. 1706998 is gratefully acknowledged.
References 1. Fischer, H. C., \Range Computations and Applications," in \Contributions to Computer Arithmetic and Self{Validating Numerical Methods," C. Ullrich, Ed., J. C. Baltzer, Amsterdam, pp. 197{211, 1990. 2. Garlo , J., \The Bernstein Algorithm," Interval Computations, Vol. 2, 1993, pp. 154{168. 3. Hungerbuhler, R., \Bounds for the Range of a Multivariate Polynomial over Triangles (in German)," diploma thesis, Faculty for Mathematics and Computer Science, University of Constance, Constance, Germany, 1998.
veroeff2b.tex; 21/09/1999; 17:37; p.7
8 4. Hungerbuhler, R. and Garlo , J., \Bounds for the Range of a Bivariate Polynomial over a Triangle," Reliable Computing, Vol. 4, 1998, pp. 3{13. 5. Neumaier, A., \Interval Methods for Systems of Equations," Cambridge Univ. Press, Cambridge, 1990, Chapter 2. 6. Ratschek, H. and Rokne, J., \Computer Methods for the Range of Functions," Ellis Horwood Ltd., Chichester, 1984. 7. Riordan, R., \Combinatorial Identities," Wiley and Sons, New York, 1968, p. 8 and p. 12.
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