Convex cubic HERMITE-spline interpolation

Journal of Computational and Applied Mathematics 11 (1984) 377-378 North-Holland

377

Letter Section

Convex cubic HERMITE-spline interpolation Holger METTKE Sektion Mathematik, Technische UnioersitgTt Dresden, DDR-8027, Dresden, German Democratic Republic Received 4 July 1984 This letter is a corrigendum and addendum to the paper "Convex cubic HERMITE-spline interpolation", published in this journal, volume 9 (1983) 205-211.

The proof "(i) =~ (ii)" of Theorem 3.3 is incorrect. Therefore, we prove the equivalence of the three statements (i), (ii), (iii) by proving of "(i) =, (iii) =, (ii) =, (i)" where the part "(iii) =, (ii) ~ (i)" can be taken over from the original version. It remains to show the implication "(i) =, (iii)". To d o that we a s s u m e that (3.2) is solvable. T h e n we o b t a i n at first

i = 1, 2 . . . . . n.

m i _ 1 <~ "ri,

(.)

H e n c e , m . _ 1 is b o u n d e d b y r n . _ x ~< "r. a n d ½(3"r,,_ 1 - m . _ 2 ) ~< m . _ 1 ~< 3~,,_ 1 - 2 r n . _ 2. O n c e m o r e using ( * ) , (3.1) we h a v e !(3"r.-22

-

-

r n . _ 3) ~< ran-2 <~ 3'r.- 2 - 2 m , , - 3 ,

a } " - 2 ) = 3"r._ 1 -- 2'r,, ~< m,,_ 2 ~< 'rn_ I.

Finally, in the s a m e m a n n e r we get the following b o u n d s for m,,_ 3: ½(3.rn_ 3 - m,,_4) ~< ran_ 3 ~< 3"r,,_ 3 - 2 m . _ 4 , a } " - 3 ) - - 3~'.-2 - 2~,,-1 ~< m . - 3 ~< ~,,-2, m . _ 3 ~< ½(3"r._ 2 - 3"r._ 1 + 2"r,,) = b~"-3~ C o n s e q u e n t l y , the validity o f (a), (b) is s h o w n for i = n - 2 a n d i = n - 3. Let us a s s u m e n o w that the inequalities

a~i)<~rni~'ri+l,

k=1.2

....

m i ~ b ( k '',

k=l,2

.... ,[½(n-i-l)],

,[½(n-

i)1,

0377-0427/84/$3.00 © 1984, Elsevier Science Publishers B.V. (North-Holland)

(**)

378

H. Mettke

/

HERMITE-spfine

interpolation

are true for i = n - 2, n - 3 . . . . ,j + 1. F r o m this a n d with the help of (3.1) we obtain mj~<½(3~+l

- - a k(J+l)~=b~/), ]

m j > ~ 3 9 . + l - 2 b C , / + l ) = ' (Uj k) + l ~

mj >139+ Hence, ( . . )

1 -- 29.+2 =

k=l

,

2 . . . .• [ ½ ( n - j - l ) ]

k=l

,

2 . . . .•

[½(n-j-2)]

,

.

a~j).

is also valid for i = j .

[]

Remark. In a d d i t i o n to T h e o r e m 3.3 we have: If the system (3.2) is solvable then all solutions can be o b t a i n e d with A l g o r i t h m 1. Proof. Let m o . . . . . m n be a n y solution of (3.2). T h e n we get f r o m (* *) a~,°)~< mo~< r,,

k = 1 , 2 . . . . . [½n],

mo<~,..,t,(o), k = 1 , 2

. . . .. [ ½ ( n - I ) ] .

Hence, m o ~ [m o, ~o]- N o w let mi ~ [mi, ~ ] (3.1) and (* *)

be true for i = 0 . . . . ,j - 1. T h e n we obtain from

½(3'9 - m j _ , ) ~< mj ~< 3 " ~ - 2 m j _ , ,

a(.J)<~mj<~'rj+l, m i<~ b~,j), Therefore,

mj

~

[m j, m j]

is also true.

k=l,2,...,[½(n-j)], k = 1, 2 . . . . . [ ½ ( n - j []

Acknowledgment I am indebted to Prof. Dr. J.W. Schmidt w h o hinted me at the fault.

1)].