Mechanical Engineering in Context 1
ENERGY/PROJECTILE
Name: Ahmed Gouda
Registration No.: H00231949
Username:asg6
Subject Name and code: Mechanical in engineering in context 1
Supervisor: Mounif, Abdallah
Briefly summarise in your own words the purpose of
Unit 5: Cubic Splines
Let K = cfw_x0, . . . , xm be a set of given knots with
a = x0 < x1 < < xm = b
Denition. [11.2] A function s C 2[a, b] is called a cubic spline on [a, b],
if s is a cubic polynomial si in each interval [xi, xi+1].
It is called a cubi
Lecture 9 Cubic Spline Examples
9
Cubic Spline Examples
9.1
Mon 08/02/10
22
Solution:
From (8.12) we see we are looking for S(x)=
Recall.
During Wednesdays lecture we saw that each cubic in
the spline interpolant can be written as
si (x) = i (x xi1 ) + i
.
Spline interpolation
Given a tabulated function fk = f(xk ), k = , . . . N, a spline is a polynomial between each pair of tabulated
points, but one whose coefficients are determined slightly non-locally. The non-locality is designed to
guarantee global
SPLINE INTERPOLATION
Spline Background
Problem: high degree interpolating polynomials often
have extra oscillations.
1
Example: Runge function f (x) = 1+4x2 , x [1, 1].
1/(1+4x2) and P8(x) and P16(x)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1
0.8
0.6
0.4
0.
Natural Cubic Interpolation
Jingjing Huang
10/24/2012
Interpolation
Construct a function crosses known points
Predict the value of unknown points
Interpolation in modeling
3
Interpolation
Polynomial Interpolation
Same polynomial for all points
Vander
CUBIC SPLINE INTERPOLATION: A REVIEW
George Walberg
Department of Computer Science
Columbia University
New York, NY 10027
[email protected]
September 1988
Technical Repon CUCS-389-88
ABSTRACT
The purpose of this paper is to review the fundamentals o
c19-b.qxd
9/16/05
810
19.4
6:24 PM
Page 810
CHAP. 19
Numerics in General
Spline Interpolation
Given data (function values, points in the xy-plane) (x0, 0), (x1, 1), , (xn, n) can be
interpolated by a polynomial Pn(x) of degree n or less so that the curve
x
$
s w t H t TX h T H q wX g vX h ` vX d g
u1Q1rfWir$QSfQpi1
wX y h T ` g d g y r R d g v g g vX h ` vX d g T H q g w ` w ` y hX h ` y h y h H H V w H w wX T g y
r1pcfw_1Qu13uWWfeepuw1QuuhQ$epfQUYpp6Ssmu1ph
yWrpi1SW1Sp~furupi1Ark1pDWurW11fSUQut
T HX h
CUBIC SPLINE INTERPOLATION: A REVIEW
George Walberg
Department of Computer Science
Columbia University
New York, NY 10027
[email protected]
September 1988
Technical Repon CUCS-389-88
ABSTRACT
The purpose of this paper is to review the fundamentals o
Lecture 19
Polynomial and Spline Interpolation
A Chemical Reaction
In a chemical reaction the concentration level y of the product at time t was measured every half hour. The
following results were found:
t 0 .5 1.0 1.5 2.0
y 0 .19 .26 .29 .31
We can inpu
Chapter 05.10
Shortest Path of a Robot
After reading this chapter, you should be able to:
1. find the shortest smooth path through consecutive points, and
2. compare the lengths of different paths.
Example
Peter: Dr. Kaw, I am taking a course in manufactu
:. L, m 4: $ $- $9 $19
Alzheimers
He stands at. the door, a crazy old man
Back from the hospital, his mind rattling
Like the suitcase, swinging from his hand,
I hat contains shaving cream, a piggy bank,
1. Scarlett O'Haras maid in Margaret Mitchells novel
Basis Basics
Selected from presentations by
Jim Ramsay, McGill University,
Hongliang Fei, and Brian Quanz
1. Introduction
Basis: In Linear Algebra, a basis is a
set of vectors satisfying:
Linear combination of the basis can
represent every vector in a giv
Cubic Spline Interpolation
Sky McKinley and Megan Levine
Math 45: Linear Algebra
Abstract. An introduction into the theory and application of cubic splines with accompanying Matlab
m-file cspline.m
Introduction
Real world numerical data is usually difficu
Spline Functions An Elegant View of Interpolation
Bruce Cohen
[email protected]
http:/www.cgl.ucsf.edu/home/bic
David Sklar
[email protected]
Start with xv intensity controls
Some Goals
To present a concrete introduction to a widely used class of
methods
Introduction to Numerical Analysis
Doron Levy
Department of Mathematics
and
Center for Scientic Computation and Mathematical Modeling (CSCAMM)
University of Maryland
September 21, 2010
D. Levy
Preface
i
D. Levy
CONTENTS
Contents
Preface
i
1 Introduction
1
Splines
Let a = x0 < x1 < . . . < xn1 < xn = b. A spline of degree m is a function
S(x) which satises the following conditions:
1. For x [xi , xi+1 ], S(x) = Si (x): polynomial of degree m
2. S (m1) exists and continuous at the interior points x1 , . . .
CHAPTER 5
Spline Approximation of Functions
and Data
This chapter introduces a number of methods for obtaining spline approximations to given
functions, or more precisely, to data obtained by sampling a function. In Section 5.1, we
focus on local methods
A Note On Cubic Splines, AMATH 352, March 4, 2002
We would like to use a spline to approximate a function represented by the points 0 0 1 0 3 2
and 4 2 . The rst task is to determine the spacing between the points h k , the slopes dk and then
(though the
Cubic Spline Interpolation
Introduction
In the last lecture we saw that linear splines are a cheap way to interpolate our n + 1 points (x 0 , y0 ) to
(xn , yn ).
In general, the linear spline function is not continuously dierentiable. Its slope changes ab
Math 128A Spring 2002
Sergey Fomel
Handout # 17
March 14, 2002
Answers to Homework 6: Interpolation: Spline Interpolation
1. In class, we interpolated the function f (x) =
satised the natural boundary conditions
1
x
at the points x = 2, 4, 5 with the cubi
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