AA96 6b5 WISCONSIN UNIV MADISON MATHEMATICS RESEARCH

CENTER FI 12/ION THE CHOICE OF THE EXTERIOR KNOTS IN THE B-SPLINE BASIS FOR A--ETC(U)DEC 80 J KOZAK OAAB29-BO-C-0041

UNCLASSIFIED MRC-TSR-2148 NL

sIopE-E]hEEIEIEEIT

MRC Technical Summary Report # 2148

ON THE CHOICE OF THE EXTERIOR KNOTS

IN THE B-SPLINE BASIS

FOR A SPLINE SPACE

~J. Kozak

CO \

Mathematics Research Center

University of Wisconsin-Madison

610 Walnut StreetMadison, Wisconsin 53706 AAR23 i9 8 i

December 1980

Received May 29, 1980

"' Approved for public releaseLLJ-.-3 Distribution unlimited

Sponsored by

U. S. Army Research Office Council for International

P. 0. Box 12211 Exchange of Scholars

Research Triangle Park Suite 300, Eleven DuPont Circle

North Carolina 27709 Washington, D. C. 20036

81 3 19 0824

S

ON THE CHOICE OF THE EXTERIOR KNOTS IN THE B-SPLINE BASIS TcT'

FOR A SPLINE SPACE

J. KOZAK __

Av:.I, -. ,. Codes

Technical Summary Report #2148 bist

Decemb.. 1980

ABSTRACT

The B-spline representation of a spline on some interval [a,b] requires

the introduction of additional knots outside ]a,b[ that have nothing in

common with the spline space itself. In this note, it is shown that locating

all these additional knots at the endpoints a and b of the interval

minimizes the matrix norm II.I1 of the matrix (X N )- where X are given00 1i j,k

linear interpolation conditions, and so is preferable when the B-spline

representation of a spline interpolant is to be constructed. Such a choice

usually also simplifies the algorithms. In particular, one is able to compute

stably the B-spline coefficients of a complete spline interpolant by Gauss

elimination without pivoting though the corresponding matrix fails to be

totally positive or diagonally dominant.

AMS(MOS) Subject Classification - 65DI0, 41A15

Key Words: B-spline basis, exterior knots, spline interpolation

Work Unit No. 3 - Numerical Analysis and Computer Science

Sponsored by the United States Army under Contract No. DAAG29-80-C-0041 andFulbright Grant No. 79-045-A.

SIGNIFICANCE AND EXPLANATION

The B-spline representation of a spline on some interval [a,b requires

the introduction of additional knots outside ]a,b[ that have nothing in

common with the spline space itself. Nevertheless, their choice influences

the numerical accuracy when the B-spline representation

of a spline approximant is to be constructed. In this note we prove that

locating all the additional knots at the endpoints a and b of the interval

minimizes the norm

i ,k, )

and is therefore preferable. Here X (Xi ) is an arbitrary sequence of

linear functionals with support in I except for the assumption that the

interpolation problem is correct, i.e. (A N nonsingular. The abovei j,k,t

choice usually also simplifies the algorithms. In particular, one is able to

compute in a stable way the B-spline coefficients of a complete spline

interpolant by Gauss elimination without pivoting though the corresponding

matrix fails to be totally positive or diagonally dominant.

The proof relies on the observation that

(N i,k,) = (N i,k,t)Qt on [a,b]

with T,t being two knot sequences that coincide in [a,b]. The matrix

Qt T for the particular choice of additional knots t described above turns

to be totally positive and its columns sum up to 1 , i.e. Qt T is composed

of (discrete) B-splines.

The responsibility for the wording and views expressed in this descriptive

summary lies with MRC, and not with the author of this report.

ON THE CHOICE OF THE EXTERIOR KNOTS IN THE B-SPLINE BASISFOR A SPLINE SPACE

J. KOZAK

1. The Result.

In a practical computation one is rarely able to make statements about the inverse of

a given matrix, particularly if the linear system to be solved depends on several free

parameters. This note is intended to demonstrate that in an important linear problem, that

of spline interpolation, the properties of a chosen basis allow us to draw an interesting

conclusion.

To start, let I -- [a,b] c R be a given real interval, partitioned by the sequence

a -: t k < tk+ 4 . t < tn : b, .)

k k+1 n n+1with ti i , all i and some integer 1 4 k 4 n. For the purpose of using B-splines,

the sequence is extended by

t 1 t2 (tk( tk tn+14 (tn+2 C .tn+k (1.2)_ n+k

To simplify the distinction between both parts of t :- (t ii. put

n+1 : k-1;n+kint(=) :- (ti) i-k' extt) :- (ti) i-1;i-n+2 * The collection of polynomial splines of

order k on I with knot sequence t is defined by

S k,tI) :- Ifftif t i+1 is a polynomial of degree < kJ mp t r) -0,r ( k - card(jlt, - ti}, sit i}

Any f e S k,(I) admits a unique B-spline representationn

t h p i (1.3)

that has proved very successful in practical computations. Here, (N i,k,) is the B-splinei~ktt

partition of the unity, i.e.

Ni,k,t(x) t (ti+k - t, t - x)+k-1

Sponsored by the United States Army under Contract No. DAAG29-80-C-0041 andFulbright Grant No. 79-045-A.

Consider now the following interpolation problem: for given interpolation conditions

X .:= (A) and prescribed numbers r := (r ) F Rn

find f E Sk,(I) such that

Xif = r i all i . (1.4)

There A is an arbitrary sequence of linear functionals with support in I except for the

assumption that the interpolation problem is correct, i.e.

f C S k,(I) and A = 0 , all i , implies f = 0

The particular representation (1.3) leads to the solution

n -;1:(a i), 1 A (1.5)

St

N. nwith At := (Xi Nj,k,)i,j=1 . We note that the solution f does not depend on ext(t)

but A and consequently a do. This fact can influence the numerical accuracy of thet

computed spline.

Cox [8] considered various ways of choosing the additional knots. As he concluded

from numerical evidence, in the case of general-purpose algorithms the coincident choice at

end points is preferable. In fact, he observed in [7] that the spectral condition number

of the matrix obtained in particular spline least-squares problems is considerably smaller

for

ext(t) = (t k t k... t k tn+1, tn+1 t n+) (1.6)

k-l k-l

compared with an equidistant and an average choice. We note also that in the book by

de Boor [5] the choice (1.6) is the rule.

Theorem 1.1. Let the knot sequences T,t satisfy

int(T) = int(t),

T.i i , i =

i ; t i = n+2,n+3,.oo,n+k

and let n ) 2(k-1). If the interpolation problem (1.4) is correct, then

UA I IA- 1 (1.7)

-2-

The inequality (1.7) displays the numerical advantage of using the ext(t) as defined in

(1.6). Also, following E2], consider the interpolation map

P: Rn - S ,t(I) :r Pr :- (X, Pr -r i , all i )

Then

n -

1PM :- sup (suplPr(x)l) = sup (eupl a i Ni,k,t(x)1) 4 IA 1 1

00E 41 xcI lirE 41 xcI i-I

(1.8)

and (1.6) gives the best bound of the form (1.8). It is also clear from (2] that A must

become singular, as T1+ - or T n - . We shall explicitly observe

1 k-1 tk+l-T r k- Tn+r-t n

A- Im Mx(. - , (1.9)- AtH. r-2 tk+i- T r-2 t n+r-t n

Our final observation concerns complete even order spline interpolation. Let k - 2m and

_I := (6l .6(1), ,6 (m-1) 6 ..*,* 6(m-1 1

,..,6n+ (1.10)

t 't t t t t tk k k k+1 n n+-1 n+1 I.

with 6 r)f - fr (t). It Is proved in (4) that the Gauss elimination without pivoting cant

be applied safely if the matrix is totally positive. Unfortunately for X given in (1.10)

the matrix is not totally positive. But (1.6) gives us At of the form

-3-

.... . .. .... ... fl - + +~ l + - -'i~' '

I~iII

m x x

xx xx 0

x x xx x

xx xx x

x x

xi

and we need to factor only the submatrix

rn-t, m+2, ",n-rn

tm4*, mn+2, ",n-rn

and this submatrix is totally positive.

-4-

Wf--:-

2. Its Proof.

Consider two knot sequences t , T that satisfy (1.1), (1.2). Assume that int ()

is strictly increasing. The general case will follow from continuity properties of divided

differences. Of course there exists a matrix Q such that

AT At Qt T* (2.1)

but it is not so obvious that we can find Qt T explicitly when int(T) - int(t). Recall

Marsden's identity [9]

kl n(y-x) k- = ,k,ty) N i,k,t(x) xy E 1. (2.2)

i=l _

Here

k-lt(2.3)

i,k,t t e+i

As observed in [1], (2.2) implies

kl nk-1 I

(y -1

x+

= (y- t +i + Ni,k,t (x) y , int(t) x I.

( 2.4 )

But (2.4) holds for an y 4 tk = a too, since then both sides vanish identically. Hence

k- n k-1 Kly Oyl o..,y k I(. - X) +

-I = [ [y0,y 1 , o ,,,y k 1 (°- xt)+i+Ni,k,t W on

(2.5)

under the assumption

ky (Y i C int(t) U ]-,tk I

Similarly, with y c int(t) u It n 1,_ n+1

X )+ = (yoy...,yk](X- k-1--ly .. ,k +I k +

n k-1

= O £kI (tx i + N i (x) on 1. (2.6)

One finds that (2.5), (2.6) are the identity [3, (5.10)], adjusted to the finite interval

I.

Lemma 2.1. Let n > 2(k-1), int(T) = int(t). Then

(N )n= (N ) Q (2.7)i,k,x i1 i,k,t i1I t T

with

-5-

____ - - - ,. =:*--~-- - - - --- --- ~~- -- r

a nnQt T {iJ'.tI i,J-1

anCT T r ; ( t )/, (t T , 1V.1 j V. k-1,j+k J Lj) i,k,t lt+k j-l,k+2,Tl +k

i

q _ ] -T) .j C i,k,t (t)/S CJ-,k+2,T (T) n-k+2 4 j V i 4 n,

i otherwise.

Proof. Assume for a moment that int(t) is strictly increasing. Choose 1 4 j V k-1.

Since n+l ) 2(k-I) + I - 2k-I we can use (2.5) for any such j to obtain

n k-1

Nj,k,T ( j+k - j )T .j+ J+k t L+i + Ni,k,t on I

and further

k-i

( - ) , (. - t )+)J+k J J+ ' 'Tj+k =+i

k-i

j+k r -t+i)-C j+k - ) J+k

r=j T (Tr- T 9

Er

k-i

J+k (T r " 4i

= 0 + (Tj.I - T - all ij+k j -~ J+k i~

HI (Tr T)"

r T C r E

The proof for n-k + 2 V j V n follows inthe same way from (2.6), and (2.7) obviously holds

for the remaining range of j . The proof is completed for a strictly increasing

int(t), but int (T) = int (t) simplifies gj, say for the range

I < 1. j 4 k-i , tok-i k-1

qiJ't 1 (T J+k - TJ)[T i+k' Ti+k+ .,T j+kIC T . - t)/ T C. - T)) , (2.8)-- ik =ki1 k=.

and the general case follows. U

-6-

2. Its Proof.

Consider two knot sequences t T that satisfy (1.1), (1.2). Assume that int ()

is strictly increasing. The general case will follow from continuity properties of divided

differences. Of course there exists a matrix Q such that

AT =At Qt T (2.1)

but it is not so obvious that we can find Qt T explicitly when int(T) = int(t). Recall

Marsden's identity [91

k-i n(y-x) = . ikt Ni,k,t(x) xy £ Ie (2.2)

i-i ' - -

Herek-l

"oi,k,t n (- - t£+i ) (2.3)

As observed in El], (2.2) implies

n k-1

k-i(y -x)+

I'- f (y -t£+) Nkt(x) , y E int(t) ,x e I.•

(y X) +i +i,k,t_

(2.4)

But (2.4) holds for an y 4 tk = a too, since then both sides vanish identically. Hence

ki n k-i

[yoy 1 ,.. .,y k I(. - x)+ = [ [yo, 1 ,. ..,y kT J c. - + )Ni,k,t (x) on Ii=I =1--

(2.5)

under the assumption

k

Similarly, with y c int(t) u )tn1, [,[y0,Yl, ,yk] + xI k-

k-l k-ily 0 'y I .... y k= y 0 ,y , ... ,YkI(x

n k-i

i=Iy ly' '" 1 k=I (t +i + Nik't (x) on (2.6)i= g1 _

One finds that (2.5), (2.6) are the identity (3, (5.10)], adjusted to the finite interval

I.

Lemma 2.1. Let n ) 2(k-1), int(T) = int(t). Then

(N n. = (N n= Q 12.7)i,k,T)=1 (i,k,t i=1 t T

with

-5-

... ,.

In order to investigate Q further we compute, for 1 4 1 4 j 4 k-1

t T

q ijt!J+k - J' "i~k~tt C k"PJ-1,k+2, (TI~

i j+k it x j i,k,t(tL+k/ "j-1,k+2,T (+k

j-1 !li (rt+k/

i i,k,t t1+k V j-1,k+1,T T+k) + --ik4t+k)/ ; k+1,r (T k

and consequently

n k-1 k-1I q I q P -i,k,t1(t+k)/P'k ,k+1,T (T+k

) wJ=1 qiJ , J i qij ,tT_ £"i

Since additionally int(T) = int(t) , then

k-111 (T£ tr )

2k-I r+iW r=1

X-i+k 2k 2

r-kr

r*£

k-1

k-1 i+k-1 T tr+i[ k'rk, l'** ', 2k 1 ] T C. - t2k-) - X 2k-I

r= 1 £fI k k (T - Tr

r=kr*1

=1-0 1

The following lemma summarizes this observation.

Lemma 2.2. Let n ) 2(k-1), and int(T) - int(t) . Thenn n

I= = 1 , all i .

From Lemma 2.1 we conclude

and to complete now the proof of the theorem it is enough to show that, for the

-7--

particular t and T as specified there,

Q ) 0 := (q ,t ) 0 , all i,j)

since then by Lemma 2.2 and invertible A -

RA-1t t

Also (1.9) is confirmed just by changing At and Qt T in the previous argument. It was

pointed out to me by de Boor (6] that Qt T is known to be totally positive if t is a

refinement of T on I, i.e. i ,t(t) - int (t), T 4 t , t ) tn. The following lemma_-- k 1, n42 > n+k*

indicates that this fact holds in the more general situation of the Theorem 1.1, and the

columns of Qt T deserve to be called discrete B-splines as in [3].

Lemma 2.3. Under assumptions of the Theorem 1.1

q j-iq ij, T t > 0 , all ij . (2.9)qij ,t 7 ij , t

Proof. Choose again 1 4 i 4 j 4 k-1. The assumption on T,t reads

Ti 4 ti r t k < Ti+k , 1 4 i < k-1 . (2.10)

From (2.8) we observe that (2.9) holds for j - i . Let j > i+1. The elements

qij,t Tare linear in any t. , i+11 4 Z k-1. Thus from (2.10)

sgn(q ) orij't T

sgn(qi., ) = tk

or after repeating the argument k-i-1 times

-8-

sgn(q ) " sgn(qij....it+1 tk-1

ti+1 or ,otk-1 or

tk tk

m m+1j sgn((Tj+k I j ETi+k Ti+k+1 ,o.*Tj+k - k ) / 1 (o - T V

for some m , J-i-1 4 m 4 k-i-i, and some j 4 Vj < VJ+1 ( -1 Vi+M < k-1 . For

m - j-i-1 the argument of sgn(*) in (2.11) vanishes, and it is enough to consider

m J-i . Then

[T i+ku T i+k~l '...T J+k]U t k )C l - )

m[T i+k, Ti+k+1,*'' T J+k

( tk / (*T I - 1w (k))

[TT ,T T..T+

i+k Ti+k+l' J+k z m+-+i-j(Z -t )

r-m-i+j-1 m-r

(T i+k' Ti+k+ 1 ,*' T J+kI z W) (W t k)r-O

( itj (J-i (m- 1 - r-I-m m-r + 0

~ ( I )(ii - v^)(J-i) 1 r (m+i-j-r)I k

for some w,z with Tj 4 7 V w V m+i ( Tk- I I TI+ k 4 , j+k . Thus the first

inequality in (2.91 is confirmed. The other follows similarly. U

Acknowledgement.

I thank Professor Carl de Boor for his careful reading and questioning of the

manuscript. His suggestions helped me very much in developing the final draft of the

paper.

-9-

References

1. C. de Boor, On calculating with B-splines, J. Approximation Theory 6 (1972), 50-62.

2. C. de Boor, On bounding spline interpolation, J. Approximation Th.ory 14 (1975),

191-203.

3. C. de Boor, Splines as linear combinatins of .-splines. A survey, MRC Technical

Summary Report No. 1667, 1976.

4. C. de Boor, A. Pinkus, Backward error analysis for totally positive linear systems,

Numer. Math. 27 (1977), 485-490.

5. C. de Boor, A Practical Guide to Splines, Springer-Verlag, 1978.

6. C. de Boor, Private communication.

7. M. G. Cox, Numerical methods for the interpolation and approximation of data by

spline functions, Ph. D. Thesis, 1975.

8. M. G. Cox, The incorporation of boundary conditions in spline approximation problems,

NPL NAC Report No. 80, 1977.

9. M. Marsden, An identity for spline functions and its application to variation

diminishing spline approximations, J. Approximation Theory 3 (1970), 7-49.

JK/db

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U. S. Army Research Office Council for International

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19. KEY WORDS (Continue on reverse side it necessary ind identifv by block nhumber)

B-spline basis, exterior knots, spline interpolation

20 AUDST itACT (Continue on reveres side if necessary and identify by block number)

The B-spline representation of a spline on some interval [a,b] requires the in-troduction of additional knots outside ]a,b[ that have nothing in common with thespline space itself. In this note, it is shown that locating all these additionalknots at the endpoints 1 a and b of the interval minimizes the matrix norm II .II

). the matrix (XiN ) where i. are given linear interpolation conditions, andso is preferable wen the B-spline representation of a spline interpolant is to beconstructed. Such a choice usually also simplifies the algorithms. In particulalone is able to compute stably the B-spline coefficients of a complete spline

FORM i (continued)DO I 1473 EDITION OF I NOV Of15 OBSOLETE UNCLASSIFIED

SECURITY CLASSIFICATION OF THIS PAGE (When Data SEn rta

'- -___... ........ "- .., .,-. -- -o . .

20. Abstract (continued)

interpolant by Gauss elimination without pivoting though the corresponding matrixfails to be totally positive or diagonally dominant.

Ii

,DA7FE

,ILMED,