scholarly journals Two-piece Cubic Spline Functions

2003 ◽  
Vol 2 (1) ◽  
pp. 25-33
Author(s):  
Hannah Vijayakumar
Keyword(s):  
Cubic Spline ◽  
Oscillatory Behavior ◽  
Spline Functions ◽  
Entire Interval ◽  
Natural Spline ◽  
Cubic Polynomial ◽  
Natural Cubic Spline ◽  
Cubic Spline Function ◽  
High Degree ◽  
The Given

.M Prenter defines a cubic Spline function in an interval [a, b] as a piecewise cubic polynomial which is twice continuously differentiable in the entire interval [a, b]. The smooth cubic spline functions fitting the given data are the most popular spline functions and when used for interpolation, they do not have the oscillatory behavior which characterized high-degree polynomials. The natural spline has been shown to be unique function possessing the minimum curvature property of all functions interpolating the data and having square integrable second derivative. In this sense, the natural cubic spline is the smoothest function which interpolates the data. Here Two-piece Natural Cubic Spline functions have been defined. An approximation with no indication of its accuracy is utterly valueless. Where an approximation is intended for the general use, one must , of course, go for the trouble of estimating the error as precisely as possible. In this section, an attempt has been made to derive closed form expressions for the error-functions in the case of Two-piece Spline Functions.

Spline Fitting

2011 ◽  
pp. 166-171
Author(s):  
Michael Wodny
Keyword(s):  
Optimization Problems ◽  
Smoothing Splines ◽  
Cubic Spline ◽  
Smoothing Parameter ◽  
Spline Functions ◽  
System Of Equations ◽  
Linear System Of Equations ◽  
Spline Fitting ◽  
Natural Cubic Spline ◽  
The Given

Given are the m points (xi,yi), i=1,2,…,m. Spline functions are introduced, and it is noticed that the interpolation task in the case of natural splines has a unique solution. The interpolating natural cubic spline is constructed. For the construction of smoothing splines, different optimization problems are formulated. A selected problem is looked at in detail. The construction of the solution is carried out in two steps. In the first step the unknown Di=s(xi) are calculated via a linear system of equations. The second step is the construction of the interpolating natural cubic spline with respect to these (xi,Di), i=1,2,…,m. Every optimization problem contains a smoothing parameter. A method of estimation of the smoothing parameter from the given data is motivated briefly.

A Computer System to Derive Developable Hull Surfaces and Tables of Offsets

1981 ◽  
Vol 18 (03) ◽  
pp. 227-233
Author(s):  
John C. Clements
Keyword(s):  
Computer System ◽  
Tangent Plane ◽  
Cubic Spline ◽  
Ruled Surface ◽  
Spline Functions ◽  
Simple Method ◽  
The Given

The fairbody, chine and sheer lines of a proposed vessel are represented by cubic spline functions. Between each pair of chine lines that ruled surface is generated which has the same tangent plane at all points of each generator or ruling line. A procedure based on the multiconic development of a surface is used to modify the given chine lines to ensure that no ruling lines intersect at a point within the surface. The result is a developable hull surface. A simple method is suggested for fairing the modified chine lines and the steps necessary to generate tables of offsets from ruling-line intercepts are outlined briefly.

scholarly journals NODE-INDEPENDENT METHOD FOR GASTROENTEROLOGICAL SIGNAL PROCESSING BASED ON CUBIC SPLINES

2021 ◽  
Keyword(s):  
Signal Processing ◽  
Spline Function ◽  
Cubic Spline ◽  
Cubic Splines ◽  
Independent Method ◽  
Spline Functions ◽  
Function Model ◽  
Basic Functions ◽  
Cubic Spline Function

This paper discusses a local cubic spline function built independently of node points using basic functions. the size of the calculations required to find the parameters to be determined during the construction of the spline function does not depend on the number of node points. Local-based splines are used to build such spline functions. Restoration of the gastroenterological signal was performed on the basis of the spline-function model discussed in the article. The result of a cubic spline-function error independent of the node points was compared with the result of the Lagrange classical polynomial error (Table 2).

A Cubic Spline Algorithm on the Rotation Group Using Cayley Parameters

1996 ◽  
Author(s):  
F. C. Park ◽  
I. G. Kang
Keyword(s):  
Ordered Set ◽  
Angular Acceleration ◽  
Rotation Group ◽  
Cubic Spline ◽  
Computationally Efficient ◽  
Rotation Matrices ◽  
Cubic Polynomial ◽  
Transcendental Functions ◽  
The Given ◽  
Dense Set

Abstract This article presents a cubic spline algorithm for interpolation on the rotation group SO(3). Given an ordered set of rotation matrices and knot times, the algorithm generates a twice-differentiable curve on SO(3) that interpolates the given rotation matrices at their specified times. In our approach SO(3) is locally parametrized by the Cayley parameters, and the generated curve is cubic in the sense that the Cayley parameter representation is a cubic polynomial. The resulting algorithm is a computationally efficient way of generating bi-invariant (i.e., invariant with respect to choice of both inertial and body-fixed frames) trajectories on the rotation group that does not require the evaluation of transcendental functions, and can also be viewed as an approximation to a minimum angular acceleration trajectory. Because the Cayley parameters provide a one-to-one correspondence between R3 and a dense set of SO(3), the resulting trajectories do not have the “multiple winding” effect that occurs in several existing methods.

scholarly journals Numerical Method for Solving Nonlinear Equations Arising in Astrophysics

2019 ◽  
Vol 8 (4) ◽  
pp. 1075-1078
Keyword(s):  
Singular Point ◽  
Spline Function ◽  
Linear Problem ◽  
Cubic Spline ◽  
Linear Problems ◽  
Isothermal Gas ◽  
Cubic Spline Function ◽  
Mathematica Software ◽  
The Given ◽  
Parametric Cubic Spline

In this paper, a parametric cubic spline function was used to get the solution of a non-linear problem for an isothermal gas sphere. The quasi-linearization procedure was used to reduce the given problem to a sequence of linear problems, the resulting equations are modified at the singular point and are handled by using parametric cubic spline for determining the numerical results. All computations have been carried out by the Mathematica software program package. The findings of computational outcomes on those astrophysics problems confirmed that the technique is legitimate for the solution of these kinds of equations.

Use of Cubic Spline Functions in Solving Calibration Problems

1985 ◽  
pp. 167-181 ◽  
Author(s):  
WOLFHARD WEGSCHEIDER
Keyword(s):  
Cubic Spline ◽  
Spline Functions

scholarly journals Elements of Bi-cubic Polynomial Natural Spline Interpolation for Scattered Data: Boundary Conditions Meet Partition of Unity Technique

2020 ◽  
Vol 8 (4) ◽  
pp. 994-1010
Author(s):  
Weizhi Xu
Keyword(s):  
Boundary Conditions ◽  
Spline Interpolation ◽  
Partition Of Unity ◽  
Scattered Data ◽  
Rectangular Domain ◽  
Natural Boundary ◽  
Natural Spline ◽  
Cubic Polynomial ◽  
Natural Boundary Conditions ◽  
Cubic Polynomials

This paper investigates one kind of interpolation for scattered data by bi-cubic polynomial natural spline, in which the integral of square of partial derivative of two orders to x and to y for the interpolating function is minimal (with natural boundary conditions). Firstly, bi-cubic polynomial natural spline interpolations with four kinds of boundary conditions are studied. By the spline function methods of Hilbert space, their solutions are constructed as the sum of bi-linear polynomials and piecewise bi-cubic polynomials. Some properties of the solutions are also studied. In fact, bi-cubic natural spline interpolation on a rectangular domain is a generalization of the cubic natural spline interpolation on an interval. Secondly, based on bi-cubic polynomial natural spline interpolations of four kinds of boundary conditions, and using partition of unity technique, a Partition of Unity Interpolation Element Method (PUIEM) for fitting scattered data is proposed. Numerical experiments show that the PUIEM is adaptive and outperforms state-of-the-art competitions, such as the thin plate spline interpolation and the bi-cubic polynomial natural spline interpolations for scattered data.

Sign in / Sign up

Export Citation Format

Share Document