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Extra info for 1996 IEEE International Conference on Systems, Man and Cybernetics

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The requirement (2) reduces to ciS'(X) ;:::: 0 for x E [Xi-l,xd, i = 1, ... ,n, (23) M. Bastian-Waltber and J. W. Scbmidt 38 where Ci 2': 0, and = +1 for Ti ° ° = -1 for Ti ::; 0. Hence, in (2) we set Li = 0, 0, and Li = -00, Ui = for Ti ::; 0. For Ti = one should Ci Ui = +00 for Ti 2': decide on one of these two possibilities. In view of (20), a sufficient condition for (23) reads Thus, because of CiTi 2': 0, i = 1, ... ,n, this system is solved by Po = ... = Pn = 0, and even al = ... = an = is allowed.

Therefore, in Sel (~) we should define = S'(Xi-l + 0), {Li,2(S) = S'(Xi - 0), {Li,3(S) = (3 + ai)(S(Xi - 0) - S(Xi-l + O))/h i - S'(Xi-l + 0) - S'(Xi - 0). {Li,l(S) (16) Then the conditions {Li,k(S) ;:::: 0 for k = 1,2,3, i = 1, ... , n (17) = 1, ... ,n. (18) ensure for SESe 1 (~) that S'(X);:::: 0 for x E [Xi-I, Xi] , i Let the bounds L and U be given by (10). , :s; {Li,k(S) :s; {Li,k(U) for k = 1,2,3, i = 1, ... , n :s; Pi-l :s; Ui , Li :s; Pi :s; Ui , (1 + ai)Li :s; (3 + aih - Pi-l - Pi :s; (1 + ai)Ui , i = Li 1, .

A spline (5) with the derivatives (21) satisfies the restrictions (2) on the first-order derivatives if the requirements (22) on the nonnegative tension parameters aI, ... ,an are valid. We point out that monotone and even co-monotone interpolation are considerably easier to handle than the general problem (1), (2). The requirement (2) reduces to ciS'(X) ;:::: 0 for x E [Xi-l,xd, i = 1, ... ,n, (23) M. Bastian-Waltber and J. W. Scbmidt 38 where Ci 2': 0, and = +1 for Ti ° ° = -1 for Ti ::; 0.

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