On optimal choice of nodes when approximating functions by interpolation splines

U.S.S.R. Comput.Maths.Math.Phys.,Vo1.24,No.5,pp.l-7,1984 Printed in Great Britain

0041-5553/84 $10.00+0.00 01985 Pergamon press Ltd.

ON OPTIMAL CHOICE OF NODES WHEN APPROXIMATING FUNCTIONS BY INTERPOLATIONSPLINES* A.A. LIGUN and A.A. SHUMEIKO

The asymptotically optimal choice of nodes when approximating concrete functions and their derivatives by parabolic interpolation splines of minimum deficiency and the derivatives of functions by cubic interpolation splines, is considered.

Introduction. Let 4,[a,bJ-(a=~f~~i... -%.=b) be a division

of the interval [a, b] and &~(&,[a, bl! the of all splines of order r and deficiency k with respect to division 4.[a, b], i.e., the set of functions having a continuous (r-k)-th derivative in [a, bl and coinciding in each interval (t,.,1,+,,"), i=O, l.....n-l, with an algebraic polynomial of degree not higher than r. Let the operator p(A.[a, b]) map C'[o, b) into S,,(A,[a, In particular, P(z, b]), par-k. (see &[a. bl) may be an Hermitian spline (see e.g., /l/), a spline of best approximation e.g., /2/), or an interpolation spline etc. For fixed {A.'[a,b]],"_,is called asympv, r, P and [a, 61, the sequence of divisions n-m, totically best for the function z(t) if, as set

=;inf {~~z(‘~-P~‘(z, ]]z"'-P'"'(z, &*[a, b])li,,.,t., In /3/ the minimization

with respect

2

A.[a,b])l~p~..bllAn[ar

to divisions

bl)[l+o(i) I-

A.[a.bl of functionals

of the type

(PM&) (l‘+L~--tc.)),

I_,

was considered, where 6in=[tl+0. ~1, f(t) is a positive function and the function cp(t) is convex. This problem is closely connected with the problem of choosing the sequence of asymptotically optimal divisions when approximating functions ret'+‘ such that z"+"(t)>O, &s [a, b]. by local splines. In /l/ the asymptotically best choice of nodes when approximating functions by Hermitian splines for p~[i, -1 was found; simultaneously, in /4/ the asymptotically best choice of nodes for Hermitian splines in p== was obtained, and for step-lines with p-2 this problem was solved in /5/. The problem was solved in /b-E/ for various generalizations of local splines. For the case when P(z, 4.[a, b]) is an operator of best approximation, the solution was published in /2/. Since splines of best approximation and interpolation splines of minimum deficiency are not local, the problem is more complicated in this case than in the previous cases. The problem of optimizing the nodes when interpolating by cubic splines was considered in a somewhat different statement in /9/. Those algorithms which provide equal principal terms of the asymptotic form of the approximation error in each interval were regarded as optimal in /9/. It is well known that, when approximating by splines, denser mesh nodes have to be taken where the (&I)-th derivative of the function is greater. For all the above problems, the asymptotically optimal choice of nodes A,'[a,b]={t,,')~' was determined from the equation I<". ,5('+l)(t) (~,~r+G-“+v~ &_ m

(jt. )z”+” (t) I uc.+l--v+p-*)

(1)

0

In the present paper we prove that this is correct for splines of minimum deficiency of orders 2 and 3. When finding the upper bound, we first solve the problem for a certain type of local splines, then we show that, for the nodes which are regarded as best, local and The lower bound is obtained by interpolation splines give asymptotically identical error. methods developed in the theory of quadrature formulae. At the end of the paper we give an algorithm for the asyptotically optimal choice of nodes, using only the values of the function at certain points, and calculations whereby the errors of deviation of the splines are equated at equidistant nodes and at asymptotically optimal nodes.

1. Defini,tions. Let C.']U, 61 be the space of functions ~&'[a, b) such that ]s("(t)]>O,t=[4, b]. The spline of minimum deficiency a,(~,,&[a, b])E&(An[a, b]) is said to be interpolating for the function x if, for r=l, 3,...

sFk A..]a,bl,&.)=z(t,,),

i--i, 2,. . .

Q~‘(~, A"[a,b],i)-I'*'(i),

=I,,

lZh.vychis1.Mat.mat.Piz.,24,9,1283-1293,1984 V..”

z.*m-a

1

, n--l,

i-a, b,

2

and for

r-2,4,...

1.. . , n-i,

i-0,

x.(z, A.[a, b], I,.)==z(frn),

I-a,b,

zP'(=,An[arb],1)3=i*)(i),

v4,‘,

f,_-t,.+&,,/Z, h,,--t,+,,.-t,., i-0. I,..., n-i, h.-mar {hIi-0,l,...,n-1) and I,,I, are subsets where is equal to r-ki and such that the number of elements of of the set J-(O,l,...,r} MI, the number of elements of 1.'U!,' is equal to r, where, by z$")(t, A.(a, bl, a), s$“‘(z, A.[o,bl, .b)

s:Y’(t, L[a, bl, b-0)

we mean r? (z, A,[a. b], a+O) , henceforth, for brevity,

respectively. A., c',... instead of A.(O, ll,C’[O,

we write e..,(z),

-

inf(b”‘-s.“’

(t.

Let s(t, A,) be a parabolic interpolation

iI,...,

and we put

A.)I))1 A.1).

spline with boundary

s&r. A.,i)-=(l), i-0,

conditions (2)

1,

or

A.,O)-s:"(2, A.,l),

ss’*(t, s,(E, A.) the interpolation

and

v-=0,&

cubic spline with boundary SF)@,

(3)

conditions (4)

f,v-o,i,

A.,I)-z(.) (i),

or

2. Basic theorem. Let r-Z,3,~C'.+',~~[l,oo]; then, as Theorem 1. v--i,2.3

with r-3, the sequence of divisions

(5)

v=o,k2.

s:"(z,A.,O)-s:"'(z,AI,i),

n+ca

for

v-0, i,2

, with r-2,

or for

(A.').'L,-((t,.'):,).~,, given by the equation

11..

z.(t) is a sequence

where

of functions

i-0,

Jz.(t)I”dt,

j ,z.(t),W=+j 0

(6)

I,...,n,

I such that ]Z"_z"+')/,+o,

will be asymptotically

optimal

and we have the relation

D,+,(t) is the r-th l-periodic a=-(t-t-I-v+p-I)-' and D,(t) --t-0.5. [O,11, of the function The proof follows easily from the following theorems:

integral,

where

vanishing

Theorem 2. Let r=l, 2,. . . ( p=[i, =I, v-=1, Z,...,r when r is odd, and is even; then, given any function z&Y+', as zba=,

s,(z). Theorem

(t‘s*}LcJ

are

UD;;;;$

* -llZ”+“lI.a

t 0 ($--)

Let r-2, 3,pE[i,-1,z&L+', v-1,2,3 if r-3 3. chosen from conditions (6); then, as a--+=,

on average

v-0, I,...,r

when r

. and

v=O,i, 2

if

r-2, and nodes

IIDr+r-.b lls"'-r."'(z,A:) I/P’n’+l-‘l12 ~r*i~ll.+o(&). Note.

It can be shown that, if nodes

(tlm’):,,,

are

in

(7)

chosen from the condition

Ii.. [ Il(‘+‘l(t)

1+.r(z(‘+*),

n-‘)I’

dt - i

J

I

?I i I

[II”+“(t))

f z(t)-

t’+

dt,

(monosplines) of the type ,-L

~ac(t-tc.);-‘+~ t-L

d, i-0

whereu+"-(max(w O]]"~d ,,,-(zIzEM., :s~~~(O).-Z~~~(I),V-O, I,..., r-l], : tll), j, p,,“-(~)~~p.,

1,....r-i, t-0, 1). ~~,W$k+f.,:s~~ It was proved in /lo, ll/ that

II-‘)]=

rcC'+1; but the proof is rather more

where 1'[a(r+2)+2]-: then (7) will hold for any function complicated in this case.

3. Auxiliary propositions. Denoteby M., the set of all functions

+tlP(r(‘+‘),

m..O-(zIz~m,,

~,-(~lz~M..:t'*'(i)-O, V-O,

: zJ_i).

Theorem A. Let n,r--l,Z, ...and pE[1, -a];then there is a unique, apart from a shift lperiodic monospline, deviating least from zero in the metric of L,, where

inf(ll~ll,I~~,)-n-'inf (~lD,-~l~P~A~R'}.

3 Lemma 1. Let R, r-i, 2,... and pE[l, monospline t*(t)-n-'D.(M)such that

~1; then there

is a unique,apartfroma shift l-periodic

inf{ll~ll~l~~.O~~ll~‘ll~-~~‘lI~.lI~. For r odd, Lermna1 follows at once from Theorem A. If r is even, the proof is similar to the proof of Theorem A, except that, when proving the existence, we have to observe that, on all passage to the limit of monosplines, on average equal to zero, monosplines are again obtained, on average equal to zero; on proving the uniqueness, instead of the number of sign changes, we consider the number of alternations of rises and falls (i.e., changes of sign of the derivative), and instead of the functions G(t) and G’(t), appearing in /ll/, we have to take c(t);=

G(+&W--t,). ‘-I

n-rD.(nt).

We introduce the notation

E'(s,S,,(A.[n', bI)),,.,w -inf(llz-dll,r..r,lSES,I(A”[Q. e’k

&,(A.[Q,

~I))PI..~I-inf{112-sHp,..s’l~ES,,(A.[~,

2--(11i, _+‘(Q)-,(“‘(Q)=,(“‘(b)-z”‘(b), CO(Z, S,,(A.[a,

z-sli,

bl), t-sii]*

V=O.

b]))n..~,-inf(ll~-sll~~..~rlSES~~(~~[a~ p(i)-z’“‘(i),

v-0,

b]),

1,. . , r-i),

b]),

i,. . . , T, i=a, b)

and ~,.*(z).l..b'~inf{~(trSI1(A.[~, bl))~~~,s,lAJ~,bl1; the quantitities s..*(l).'..e' , e.r'(z)P'r.~' are defined in a similar way. Lemma

For all

2.

r-0,

l,...,

n51, 2,...,

pe[f.

ml , zW?[U,

s:+r(r+r'.r(r)P'..~' G &,'(r),'.,e' G enra(t)Pt.,~l

b] we have

G c~~~(~)~w.

The last two inequalities are obvious. Consider the proof of the first. Let s(t)=&,,(AJa, b]) be a spline such that Z--s_L~. Then, for every function ~&?[a, b] divisions &+,[s,a+s], A,+,[b-e, b], A.+,[B,r],[B,r]c(n+e, b-e) exist, and splines defined in them Proof.

respectively r.(t), r..(t), s...(t) such that z-S_Li and z(t)--S(t)lI;Jz(t)--s(t)J, %[a, b], where I--s(t)+r.(t)+~..(t)+r...(t), S(t)ES~,(A.+r(.+,,[a, b]); here, An+,(,+,)[a. bl=A,+,[a, a+elUA.,[a, blU Hence, for any s~&,(A.[a, b]) such that Z-~11, we have A.+,[B, ‘ylUA.+,[b-e, bl.

ll~--bIlrP.r, >llz-all,r..s’ 3 d++,v+d4 PI0.L’. From this and the arbitrarinessof the division 4. N,

Proofs

of

baais

&[a,

b],

we

obtain the lemma.

theorems.

Theorem 2. For any N>i we denote by 6X the division of [O,l] the spline of S,+,.,(pl,,) such that, for odd r, and by tl

by points i/N,i=O,i,...,

and for even r z,,(f)-z(f),

+0,1,...,N,

Then (see e.g., /12, p.132/) as

s~‘(i)=z”‘(i),

v-O,1

,...,

+,

i-0,1.

N-+m

]]z("'-zJl "'U_-O(~-'-'O(Z(~",N-')),v-0 , 1I..., r+i, where ~(z,t) is the modulus of continuity of the function x. We choose N=N(r,r,n) as the least integer satisfying the inequality NL'-1L,(2('+",N-')(n"-,-'o%(=(,+", n-l), From this and the above, as n-cm,

we have

lb (‘+“-*;+l’[j_,O(l). From /13/, in particular, for any xd”\S,,,+t(A.) 3M>O:VA,,

Vm!L,v+,(AJ

we have IId” - ~“‘11, * ;.

From this and the previous inequality, ]]&'- 8,'" (GA.) I], +c'

-sl"(zx, A..) lip+ o

and hence to prove the theorem, it suffices to show that, as n--c-,

(8)

(9)

We fix

v, V-l,Z,...,r, and for every

t-i,

Z,...,N we choose an interval [~br]~[(i-l)'N*

"(rn,A;)ii. Let ~,-b,-ac , 6.,-A.fl[~,bcl-(~t,.,}$ i/N] such that in it 'SF'-r, ~,!‘+~(t)-const,

p=[i,-)

te[(i-1)/N,i/N],i-l, Z,...,N,and putting cc-zj;+n (i/N-O), we obtain

Noting that

i-l,Z,...,N,

for

where a=(r+l-vip-‘)--1, Using Lemmas 1 and 2, we obtain

(10)

Theproblem

zn

--Ac I-,0:

inf.

APO,

i-t,Z,...,N,

under the conditions

has a unique solution, and its extremal value is

From this and (10) we obtain

IIz:.’ -8:’ (a,,

A,)

ID.++.I16 ,t,!b [n+(‘&+~)N]“+i-“~

(&d1c’1a) “-’ L-L

Then

Points a, and b, can be selected in such a way that the intervals [(i-i)/N, a,] and [br,i/N] contain not more than r nodes each of division A,,;moreover, from /13/,

p/m.

IEdf]

e...(zx),P> (2Nr)f~,_.)p [ j b~+‘)w lo.ll\rY

On the other hand (see e.g., /12, p.132/), there always exists a division A. for which there is LlmR' such that llz?-& '") (z,,, A,,)l(;G Zk~"+'-"~r, i.e., M

WW’+‘-’ and since scE[ U--1)lN,~.,,l,

s (i&+“(t)

[ Io,,,rx

("dt]i'a-

O(-$-),

bc~[~~~-r,~,, 1/N],N--o(n), it now follows that [

s I&-+“(t)

I'dt]"'-o(l).

5

With

v-0

and r even, using Theorem

We can prove

A and the above arguments,

in the same way as (10) that

IIz:) -s.“’6z.v.A,) 11..,110 ,+,_.11_Xinf(max(~li51,2 Noting

we obtain

,...,

N}J&vh+w+3)N)

that the problem

i=i,2,...,N

A,>O,

- inf,

i-i,2

,...,

N,

under conditions

has a unique

solution

and that its extremal

value is

B-’ (gA:jA) I-1

‘,

and repeating almost exactly the above arguments, this proves Theorem 2. -1 ;

we find that relation

Theorem 3. Consider the interpolation parabolic spline (2). Then, for t=[ti., ti+,. “1, i=O, 1, . . . , n-i,

s,(z,A.)

(9) holds

for

with boundary

pE[1,

con-

ditions

s*(z, A,, t)=2m,,(r*-0.5~)-~c,(t-O.25)+2m+~,.(~+O.5~), f,.-z(L), i=l, 2,...,n, m,“-s,(z, A., t,.), i-0, l,...,n, where v=(f-tt.)hk-‘, and the numbers mc, are found from the conditions for continuity of the derivative of an interpolation parabolic spline. If we put c,.-m,---ts,i=O,l, ....n, the system from which the rn(.can be found can be written as CO.-0, hmc,-,, “+3c,“+p‘J,+,,“-

clh,

i--l, 2,...,

n-l,

L&=0, where

z*=z(tc"),

i-0, l,..., n,

hi.=h,,(h,-,,.+hr.)-‘,

Irm=l-hc.,

~n~~‘~h+~~,,“~-hrZ‘_*,n-3kr-

pt”Z‘+,,“. i==i,2,...,n-i. The following

is well known

(see e.g., /9, p.334/:

Proposition. If the matrix of the system

Az=d has diagonal

la,,l-r( ]a*ll=r,ZO,

i,j=l,2,...,n,

i*, then

maxIz,~f;maxId,Ir,-'.

,

‘

From this and the above we obtain

It is easily

seen that d,.=

where

j D(u)z"'(u)du, k-1..

dominance,

i.e.,

6

t,+,,“1,

i=i,

2,. . , n-l,

such

that

h ,_,.I h ‘*

[z,,“‘(h,.‘-h:-,,.)+(h,.‘+h:-,,.)o(z”,L)

12(h,-,,.+hi.)

I.

The case of periodic boundary conditions is treated in a Similar way. Let p,(z, A,) be a parabolic spline of deficiency 2 with nodes of division interpolating function x at nodes t,. and f;.. Then, for S[f‘., k+,.“I

A.,

and

p%(z, A.. t)=22~n(~'-0.5~)-~cn(~~-0.25)+2z,+,,.(7*+0.5~). We have:

Lemma 3.

Let

v=o, 1, 2 , and let the division

and ~~=(i~)(allt~ll~+(2+41)o(z~', (2) and (3) we have the relation

llsp(z,A.)--Ps~ C,=,/,er, C,=‘i,,

where

Consider the arguments

A. be such that

k)); then, for the spline

sl(z,A.)

Iht,-h,-,,.(
I~l
with boundary

conditions

(2, AJ ti=,t ,_,,n., ,~~~‘%-“B~~

C,=‘/,.

the cubic interpolation splines with boundary of /g, p.W/). Then, for rE[t,.,&+,..],

conditions

(4) (here we follow

s,(r,A,,t)--2,.(~-~)*(1+2r)+~~+,,~~*(3-2r)+m,,h,.(l-r)*~-m,+,,.~~~~*(~-~)~ where r-(t-t,")&,,, m,"-=sa'(z, A., t,,),i-0, 1,.. . I n. Numbers mr, are found from the continuity conditions for the spline second derivative; if we put c,.=m,.-&,,I, i-o,& . . ..n. then the system from which m,. are found can be written as ca.=o, A,.c,-,,n+2c,n+~mc,+,. .=d,.,

i--1,2,...,n-l,

c..-0, where i ,...,n.

d,. - ~(P,~(z,+,,~ -I,.) hi.-, + I,.(zcm - z,-,.J Using

further

arguments

similar

h,:‘,,.) - k:-,.l

v-0,

h,_,,h,.[

The case of periodic If p,(t, A,)E&,(&) then we have i, Larva

a.)a(t,“, we have

4.

1, 2, 3.

h,.‘t,“(rl,.)

1h,_,,m-h,. 1 p(t,.)

~‘~2

(k) q +09

I + (h,-,,,+h,,)ub(“,

16 maxb-,..7

h,.)) 1.

boundary conditions is treated in a similar way. is an Hermitian cubic spline, i.e., PI(*‘(+,~,~~-Z(“‘(~),

Let the division

h,.)) , v-0,

and

to the above, we obtain

d ,” -;h,,,h,_,.(h,_,,,,+h,&‘[hf-,,.z(”(b.)6

- 2x,.’ - p,.t:+,..

Ih,,-hr-,,

A,, be such that

Then, for the spline

~~(2,A.)

./-=a&,..:

(aJ
with boundary

$I,-=(i-her) conditions

+@,

it...,%

(a&“‘11~+(2+ (4), (S),

U~~'(~,h,)-p~'(zrAr)Uc,~,_,,~.',~~_C C.h::'B., where

C,-'/,,,, C,--l/7,, C,-'I,,,C,-'1,. Clearly,

with

r-2,3 /JzW-r."'(z,A.)u,
with

(s,A.n)II,+IJp.“‘(z, A.)- ~:'(z,An)lti

It was shown in particular in /6/ that, when approximating z," by the @ins p~“‘(& A.1 optimal choice of nodes is given by (1) and r-2, v-=0, and r-3, V--I the asymptotically

b“-pi”’

_!++qz,~+"~~~ &An') Ijp

+ o(nv-r-,).

It can be shown by an almost exact repetition of the arguments of /6/ that this relation also holds for the nodes given by (61, where, for all v
if nodes

P$~)) (z, A.), then t""'(t)-[s"'(t-o)+z,"'(t+0)]/2. this and Lemmas 3 and 4, it follows that, to prove Theorem 3, we have to show that, (61, then I&'--hT_, [' are chosen from conditions

For, if the nodes are chosen from conditions .A h ,,%-= , nA, Consequently,

A-j

la.(t)I”& 0

(61, then A, =lz,(t,.)

1..

I ht*‘-&-,,,I

+@-a,_,ld‘

‘

where C is a constant,

which

L

depends

on

0

nAtAc-t

tzn=,h,‘)

G Ch,.‘o (La, k.‘) .

z,.

Appendix Let us give an algorithm polation splines.

for the asymptotically

optimal

choice of the nodes for inter-

be the eecond divided Let A-P (t($?Pa be a division of the interval [O,l]and A% it0 . Denote by yr(z.An~.t) the piecewise constant function difference of xi at the point given by

Then (see e.g.,

/9/j,

as

f2+- , ll~(‘+~)-y,(r.

A.o)ll,-0.

We choose the nodes of division Ant from conditions etc.; then, as nodes of Anr from (6) with ~,(t)=x~(r,A~~.t). optimal. a..., will be asymptotically The results of ES-1022 computer computations of the uniform metric for the cubuc splines ~~(z.h~,..,Q are shown a uniform division with 50 nodes.

(6) with r,(t)=-y,(r,AmO,t) end the n-m , the nodes of divisions &,V=i, error of spline interpolation in a in Table 1, where as Aso,,we took Table

v

(I + 0.001)‘~’

etp(lOt)

1

[i+100(1--0_5P1-’

I

0 :

8.1728.10-= 1.831&10-3 1.64a3.10-2

l&53*10-* 2.0729.10-' 2.4Q42.10-‘

REFERENCES 1. XGUN A.A. and STORCBAI V.F., On the best choice of nodes when interpolating functions by Hermitian splines, Analys. math., 2, No.4, 267-275, 1976, 2. XGtJN A.A., Some extremal problems of approximation of functions, in: Proc. Conf. Approximation and Function Spaces 79, Gdansk, 218-225, 1981. problem, Numer. Methoden Approximations Theorie, Bd. 1, 3. X'ECBKIN S.B., An optimization Numer. Math., 16, 205-208, 1972. 4. ;REBENNIKOV A.I., On the choice of nodes when approximating functions by splines, Zh. vych. Mat. i mat. Fiz., 16, No-l, 219-223, 1976. 5. WARIN V.S. and BARMIN V.I., Approximation by piecewise linear functions, in: Matem. cb. (Mathematics Collection), Naukova duska, Kiev, 1976. 6. XGTJN A.A. and STORCBAI V.F., On the best choice of nodes in the approximation of functions by local Hermitian splines, Ukr. mat. zh., 32, No.6, 824-830, 1980. 7. BXEBENNIKOV A.I., On the choice of nodes of the interpolation of functions by L-splines, in: Computational methods and programing (Vychisl. metody i programmirovanie), 1~0.26, Ixv-vo MGU, 1977. of functions by Hermitian L-splines, in: Studies in modern 8. XUMBJKO A.A., Interpolation problems of summation and approximation of functions and applications, Izd-vo DGU, Dnepropetrovsk, 126-131. 9. ZAV'YALOV YU.S., KVASOV B.I. and MIROSHNICHENKO V.L., Methods of spline functions (Metody splain-funktsii), Nauka, Moscow, 1980. 10. ZBBWSYKBAEV A.A., Monosplines of minimum norm and best quadrature formulae,Usp. mat. nauk, 36, No.4, (220), 107-159, 1980. 11. ZBENSYKBAEV A.A., Best quadrature formula for some classes of periodic differentiable functions, Izv. Akad. Nauk SSSR, Ser. matem., 41, No.5, 1110-1124, 1977. 12. STEXHKIN S.B. and SUBBOTIN YlJ.N., Splines in computational mathematics (Splainy v vychislitel'noi matematike), Nauka, Moscow, 1976. 13. LIGUN A.A. and STORCBAI V-F., On the best choice of nodes when approximating by splines in a metric of LP, Matem. zametki, 20, N0.4, 611-618, 1976. Translated

by D.E.B.