C² Rational Quadratic Spline Interpolation to Monotonic Data

Shape Preserving Piecewise Rational Interpolation

“…T h e r a t i o n a l c u b i c h a s t h e f o l l o w i n g i n t e r p o l a t o r y p r o p e r t i e s s ( 6) w h e r e s ( 1 ) …”

“…This choice requires a knowledge of s ( 1 ) (x) and s ( 2 ) (x) which are given in the relevant sections below.…”

“…Theorem 2.1 Let f ∈ C 4 [x 1 , x n ] and let s be the piecewise rational c u b i c i n t e r p o l a n t s u c h t h a t s ( x i ) = f ( x i ) a n d s ( 1 ) …”

“…geometric approximations of subsection 3.4, (ii) the C 2 spline approximations of Delbourgo and Gregory [1], and (iii) the known exact derivatives. All curves are monotonic but (i) exhibits an inflexion.…”

Weighted Fifth Degree Polynomial Spline

“…Piecewise interpolation schemes may be classified as local (see, for example, [1][2][3][4][5]), which may be calculated in any subinterval independently of other subintervals, and global, which calculation involves solving of the system of I or I*k equations, I -number of interpolation knots. Recent spline belongs to the second class, which contains, for example, rational splines [6,7], discrete hyperbolic tension splines [8,9]. It is known that high accuracy interpolations are connected with possible producing of undesirable oscillations near points of interpolated function discontinuities or points of discontinuities of the function first derivative.…”