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UNIVERSITY OF NOTRE DAME
Department of Civil Engineering
and Geological Sciences
CE 30125
September 24, 2010
J.J. Westerink
Due: October 7, 2010
Homework Set #3
Backgound:
Lagrange
interpolation fits a polynomial
function exactly through a number of interpolation
points. The degree of this polynomial is N when we are fitting N+1 functional values. We have discussed three
ways of deriving
Lagrange
interpolation:
Method 1:
Power series
defines the polynomial, implements the constraints (i.e.
), and solves for
the unknown polynomial coefficients by solving a system of simultaneous equations. This method is conceptu
ally simple but is expensive when the order of the interpolator is high or when dealing with multidimensional
interpolators. The method may also be inaccurate due to the inherent illconditioning of the matrix that must be
solved.
Method 2:
Lagrange
Basis Functions
rewrites the interpolator as a product of functional values at the nodes
and the Lagrange basis function associated with each node:
Each basis function
equals zero at all nodes except at the node it is associated with where it equals 1, i.e.
. This makes each basis function very simple to formulate since we know that the roots are at all
nodes except at the node
that specific basis function is associated with. These definitions always ensure that
, exactly what we want. Note that we do not solve any system of equations and that each basis func
tion is developed as a sequence of products, simply implemented in do loops. Note that the interpolator that we
derive using the Basis Function approach is identical to that derived using the power series approach.
Method 3:
Newton Forward Interpolation
reformulates the interpolation function in terms of a sequence of
polynomial terms mutiplied by increasingly higher order forward differences. This method has the advantage
that each higher order polynomial term has the previous term embeded in it, making it quite inexpensive. In
addition, to increase the order of the interpolant, simply requires adding the next term in the sequence, making
the methodology hierarchal. Finally, the error is simply estimated by considering the subsequent term in the
sequence. Again this method gives the identical interpolation function as the power series and Lagrange Basis
function methods.
Problem 1
Consider the function:
on the interval [1, 4].
a) Develop a power series based interpolator for 4 equispaced points for the given function.
i)
Evaluate
,
,
and
using the specified function
.
ii) Define the appropriate polynomial expansion,
g(x)
.
iii) Define the constraints needed to solve for the coefficients in the generic expansion.
iv) Solve for the coefficients and plot the interpolation formula.
v) Plot
and
over the interval [1, 4] using increments of 0.01. Provide labels for the axes and
the curves.
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This note was uploaded on 03/02/2012 for the course CE 30125 taught by Professor Westerink,j during the Fall '08 term at Notre Dame.
 Fall '08
 Westerink,J
 Civil Engineering

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