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Interpolation and Error Analysis
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What are the interpolatory conditions of the piecewise rational cubic function?
S_i(xi) = fi, S_i(xi+1) = fi+1, S_i'(xi) = di, S_i'(xi+1) = di+1
How is the error analysis of the piecewise rational cubic function estimated?
Using the Peano Kernel Theorem to obtain the error analysis in each subinterval.
What is the formula for the Peano Kernel in the error analysis?
R(x) = f(x) - S_i(x) + (1/2)(x - xi)(f''(xi+1) - f''(xi))
What is the absolute value of the error in each subinterval?
|f(x) - S_i(x)| + (1/2)|f''(xi+1) - f''(xi)| * (x - xi)
What is the main focus of this work?
Developing a rational cubic spline function for shape-preserving convexity curves
What degree of smoothness does this work achieve?
C1 smoothness
What advantages does the developed scheme offer?
Local computational efficiency, time-saving, and visually pleasing interpolation
How does this work refine convex curves?
By adjusting free parameters in the rational cubic function interpolation
What is achieved in terms of error bound in this paper?
A good O(h^3) error bound
What are the major sections of this paper?
Section 2 defines the rational cubic function, Section 3 discusses the error of the interpolant, Section 4 explores shape-preserving convexity curves, Section 5 presents derivative approximation, Section 6 provides numerical results, and Section 7 concludes the work
How many free parameters are introduced by Abbas et al.?
Three
What are the parameters defined as for each subinterval?
S i (x), p i, q i
What are the values of p i, u i, f i, w i, h i, and d i?
1, 2, 3
What is the result when the values of u i, v i, and w i are 1, 1, and 3 respectively?
Reduces to standard cubic Hermite spline
Rational Cubic Spline Function
Study
What is the degree of smoothness in the developed scheme?
C1
What is the advantage of the developed scheme compared to existing schemes?
Local computationally economical, easy to compute, visually pleasant
What additional modification is required to refine the convex curve while preserving its shape?
Adjustment of free parameters in the rational cubic function interpolation
What error bound is achieved in this paper?
O(h^3)
How is the shape of convex data preserved in the developed scheme?
By imposing constraints on free parameters without any extra knots
What is the rational cubic function introduced by Abbas et al?
A cubic spline function with three free parameters
How is the rational cubic function defined in each subinterval?
S(x) = pi + qi(x - xi) + wi/(fi - ui) + vi/(fi - ui)^2 + wi/(fi - ui)(di - 1 - hi)
What are the positive free parameters in the function?
ui, vi, wi
What happens when the values of the free parameters are set to 1, 1, and 3?
The rational cubic function reduces to a standard cubic Hermite spline
What are some applications of rational cubic functions?
Curve design, shape-preserving interpolation
What is the purpose of the schemes developed by Costantini and Fontanella?
To preserve the convexity of data
What problem did Delbourgo and Gregory's scheme have?
The resulting surfaces were not visually pleasing and smooth
What parameter was introduced by Meng and Shi Long in their scheme?
Two free parameters
What alternative to the spline under tension was provided by Gregory, McAllister, Passow, and Roulier?
Rational splines represented in terms of first derivative values at the knots
What did Schumaker's scheme for quadratic spline interpolation involve?
Inserting an extra knot in each interval
What shape parameters did Sarfraz and Hussain use in their scheme?
Two shape parameters
What did Sarfraz develop in his scheme?
A piecewise rational cubic function with two families of parameters
What technique did Sarfraz use to preserve the physical shape properties of data?
Piecewise rational cubic interpolant in parametric context
What is the drawback of the local schemes used for shape preservation of curves?
They have no flexibility in preserving convexity
How many free parameters are used in the construction of the rational cubic function?
Three
Which parameter is used to preserve the convexity of convex data?
One of the free parameters
What are the attributes of the proposed scheme?
It achieves shape preservation of convex data, allows modification of convex curves, and has derived datadependent constraints
