|
Interpolation is any
procedure for fitting a function to a set of points in such a manner that
the function intercepts each of the points. Consider m points
where
 ,
 , and
the
are distinct. We wish to construct a function f:
such
that
= f()
for all k. There are various solutions to this problem. We consider
an approach called ordinary
interpolation. This is applicable to a wide range of interpolation
functions.
Linear interpolation uses
a linear polynomial as the
interpolation function. Between two points, (x[1], y[1]) and (x[2], y[2]) with
x[1] < x[2], we interpolate a linear polynomial of the form
|
 |
[1] |
Consider points (2,1) and (5,3). These and an interpolated
linear polynomial are graphed below.
|
|
 |
 |
|
A linear polynomial interpolated between
the two points (2,1) and (5,3). |
To solve for the interpolated polynomial, we must
determine constants
 and
 . If the linear polynomial is to intercept
(2,1) and (5,3), it must satisfy
|
f(2) =
2 +
= 1
|
[2] |
|
f(5) =
5 +
= 3 |
[3] |
This is a system of two linear equations in two unknowns.
We express it in matrix form:
|
|
[4] |
Solving, we obtain
= –1/3 and
= 2/3, so our
interpolated polynomial is
Quadratic interpolation uses
a quadratic polynomial as its
interpolation function.
Suppose we want to interpolate a quadratic polynomial between the five
points
of Exhibit 2
|
|
 |
|
Five points through which to interpolate a
quadratic polynomial. |
The general form for a quadratic polynomial from
2 to
is
|
 |
[6] |
Solving for the six constants
based upon the five points
of Exhibit 2 would entail a system of five equations
in six unknowns. Such a system is likely to have infinitely many
solutions. To obtain a unique solution, we may consider a less general
form of quadratic polynomial than [6]. We might require
 or set
= 0, etc. For this example, let’s interpolate a quadratic polynomial
with zero cross term,
 = 0. Our polynomial then has form
|
 |
[7] |
We require that this polynomial intercept each of our five
points. This renders a system of five equations in five unknowns, which we
express in matrix notation as
|
 |
[8] |
Solving, we obtain
|
 |
[9] |
This and the five points are graphed in Exhibit 3.
|
|
|
|
 |
|
A quadratic polynomial interpolated
between five points. |
We now formalize the interpolation procedure illustrated
in our two examples. Those examples used polynomials as interpolation
functions f. The following discussion extends the procedure to a broader
range of interpolating functions.
Consider m points
where
,
, and
the
are distinct. We wish to construct a function f:
of
the form
|
 |
[10] |
in such a manner that f intercepts each of the points.
Functions
 :
can take any form. In our above quadratic example,
they were:
but exponentials, roots, logarithms and other functions
are permissible. Let’s express our problem with matrices. Define
|
 |
[11] |
This is unknown. It is what we want to solve for. Define
f
as the m m matrix comprising values of each function
evaluated at each
point
:
|
 |
[12] |
Define the vector
|
 |
[13] |
Both the matrix f and vector y are constants. They are
known. Our requirement that f intercept each point
yields the
equation
|
 |
[14] |
If the matrix f is invertible, this has the unique
solution
|
 |
[15] |
|
Consider three points (x[k],
y[k]) = (1,2), (4,2), (5,3).
Interpolate a quadratic polynomial of the form
.
[
of the form
such that f (1, 0) = 1 and f (1, 1) = 1.
[
