2. Introduction:
Interpolation is the process of estimating intermediate values
between precisely defined data points.
For n+1 scattered data points there is a unique n-th order
polynomial fit function.
n+1=2
n+1=3
n+1=4
Polynomial interpolation is the process of determining the unique
nth-order polynomial that fits (n+1) data points.
One can define the n-th order polynomial in different formats, e.g.,
Newton polynomials
These formats are well-suited for
computational implementations
Lagrange polynomials
3. Newton’s Divided Differences
One of the most popular interpolating functions
Linear interpolation:
Simplest form of interpolation: connect two data points by a
underlying
straight line, and estimate the intermediate value.
f(x)
f1(x) fit function
Similarity of the triangles:
f1 ( x) f ( x0 )
x x0
f(x1)
f1(x)
f(x0)
f ( x1 ) f ( x0 )
x1 x0
finite divided difference
of first derivative
x0
x
x1
f1 ( x)
f ( x0 )
f ( x1 ) f ( x0 )
( x x0 )
x1 x0
for data points
f1 ( xi )
function
f ( xi )
represents the first
order interpolation
linear interpolation formula
4. Quadratic interpolation:
If you have three data points, you can introduce some curvature
for a better fitting.
A second-order polynomial (quadratic polynomial) of the form
f 2 ( x) b0 b1 ( x x0 ) b2 ( x x0 )( x x1 )
To determine the values of the coefficients;
x
x
x
x0
x1
x2
b0
f ( x0 )
b1
f ( x1 ) f ( x0 )
x1 x0
b2
If you expand the
terms, this is nothing
different a general
polynomial
Linear
interpolation
formula
f ( x2 ) f ( x1 ) f ( x1 ) f ( x0 )
x2 x1
x1 x0
( x2 x0 )
Quadratic
interpolation
formula
finite divided
difference of
second derivative
5. General form of Newton’s interpolating polynomials:
In general, to fit an n-the order Newton’s polynomial to (n+1) data
points:
f n ( x) b0 b1 ( x x0 ) b2 ( x x0 )( x x1 ) .. bn ( x x0 )( x x1 )..( x xn 1 )
where the coefficients:
b0
f [ x0 ]
b1
f [ x1 , x0 ]
b2
f [ x2 , x1 , x0 ]
data points
n-th finite divided difference:
…
bn
brackets represent the
function evaluations for
finite divided-differences
f [ xn , xn 1 ,.., x1 , x0 ]
f [ xn , xn 1 ,.., x1 , x0 ]
f [ xn , xn 1 ,.., x1 ] f [ xn 1 ,.., x1 , x0 ]
( xn x0 )
6. These differences can be evaluated for the coefficients and
substituted into the fitting function.
f n ( x)
f ( x0 ) ( x x0 ) f [ x1 , x0 ] ( x x0 )(x x1 ) f [ x2 , x1 , x0 ]
.. ( x x0 )(x x1 )..(x xn 1 ) f [ xn , xn 1 ,.., x0 ]
x values are not need to be equally
spaced.
x values are not necessarily in order.
Newton’s divideddifference interpolating
polynomial
7. Error for Newton’s interpolating polynomials:
Newton’s divided difference formula is similar to Taylor
expansion formula, adding higher order derivatives of the
underlying function.
A truncation error can be defined as in the case of Taylor series
approximation:
Rn
f ( n 1) ( )
( x x0 )( x x1 )..( x xn )
(n 1)!
where is somewhere in the
interval containing the unknown
and the data.
For Taylor series
approximation error
Rn
f ( n 1) ( )
( xi
(n 1)!
1
xi ) n
Above formulation requires prior knowledge of the underlying
function and its derivative, so cannot be evaluated.
1
8. An alternative formulation that does not require prior knowledge
of the underlying function:
Rn
f [ x, xn , xn 1 ,..., x0 ]( x x0 )( x x1 )..( x xn )
(n+1)th finite
divided difference
One more data point (xn+1) is needed to evaluate the equation.
Rn
f [ xn 1 , xn , xn 1 ,..., x0 ]( x x0 )( x x1 )..( x xn )
This relationship is equivalent to
Rn
f n 1 ( x)
f n 1 ( x)
f n ( x)
f n ( x) Rn
(next estimate) - (current estimate)
increment added to the (n)th order case
to calculate (n+1)th order case is equal to
the error for the n-th order case.
9. Lagrange Polynomial Interpolation
A Lagrange polynomial can be stated concisely as
f n ( x)
Li ( x) f ( xi )
Li ( x)
j 0
j i
i 0
For example:
n 1
n
2
x xj
n
n
x x0
f ( x1 )
x1 x0
xi
xj
In fact, Lagrange
polynomials is just a
different formulation of
Newton’s polynomials
f1 ( x)
x x1
f ( x0 )
x0 x1
f1 ( x)
( x x0 )(x x2 )
( x x1 )(x x2 )
f ( x0 )
f ( x1 )
( x0 x1 )(x0 x2 )
( x1 x0 )(x1 x2 )
( x x0 )(x x1 )
f ( x2 )
( x2 x0 )(x2 x1 )
10. In the formula, each term Li (x) will be equal to 1 for x=xi , and
zero for all other data points.
Thus, each product Li (x) fi(x) takes on the value of fi(x) at the
data point.
11. Error is defined same as before
Rn
f [ x, xn , xn 1 ,..., x0 ]( x x0 )( x x1 )..( x xn )
(An additional point (xn+1) is needed for evaluation)
In summary:
Newton’s method is preferable for exploratory computations (n
is not known a priori).
> Newton method has advantages because of the insight for the
behavior between different orders (consider Taylor series).
> Error estimate in Newton method can easily be implemented
as it employs a finite difference.
Lagrange method is preferable when only one interpolation is
performed (order n is known a priori),
> It is easier for computational implementation.
12. Polynomial coefficients:
Newton and Lagrange methods do not provide the coefficients
of the conventional form
f ( x)
a0
a1 x a2 x 2 ... an x n
With (n+1) data points, all the (n+1) coefficients can be
determined by using elimination techniques. For example for
n=2:
2
f ( x)
a0
a1 x a2 x
satisfies the following linear equations
f ( x0 )
a0
a1 x0
2
a2 x0
f ( x1 )
a0
a1 x1
a2 x12
f ( x2 )
a0
a1 x2
2
a 2 x2
The process is notoriously illconditioned and susceptible to round-off
errors:
keep the order (n) small.
use Lagrange or Newton interpolation.
13. Inverse Interpolation
dependant variable
independent variable
Values of x are usually evenly spaced.
Normally interpolation concerns finding an approximate f (x) for
a given intermediate value of x.
What if reverse is needed, that is value of f(x) is given and need
to find the corresponding x value (inverse interpolation).
Two possible solutions:
Switch x by f(x) and apply Lagrange/Newton interpolation.
(this method is not suitable because there is no guarantee that
the new abscissa values will be evenly distributed,-in fact,
usually highly uneven.)
Apply normal interpolation, and find the x value that satisfies
the given f(x) value
a root finding problem.
f (x)
x
14. Equally spaced data:
Newton/Lagrange methods are compatible for arbitrarily spaced
data
Before the computer era, equally spaced data had to be
used, but computer implementation of these methods do not
require it anymore.
Evenly spaced data is required for other applications
too, e.g., numerical differentiation and integration.
Extrapolation:
The process of f(x) for a point outside of
the range of x values.
If the extrapolated x value is not near
the evaluation points, the error can be
very large. So, extreme caution is
fit curve
required during extrapolation.
true curve
extrapolation
15. Spline Interpolation
Sometimes fitting higher order polynomials results in erroneous
results, especially at sharp changes.
Spline interpolation provide s smoother transition between data
points.
Apply a different lower order polynomial to each interval of the
data points.
Continuity is maintained by constraining the derivatives at the
knots.
Here the polynomial
interpolation overshoots
between data points.
Spline offer a smoother
and a meaningful
transition
knot
interval
interval
A different spline
function is defined
for each interval
16. Linear Splines:
Each interval is connected by a straight line.
For each interval
f ( x)
f ( xi ) mi ( x xi )
where mi
f ( xi 1 ) f ( xi )
( xi 1 xi )
Linear splines is identical to the first order polynomial fit.
Linear spline function is discontinuous at the knots. So, we need
to use higher order polynomials to maintain continuity.
In general, for m-th derivative to be continuous, an order (m+1)
spline fit must be used.
17. Quadratic Splines:
In quadratic splines each interval is represented by a different
quadratic polynomial.
f ( x)
ai x 2
bi x ci
For n+1 data points there are n intervals. This makes a total of 3n
unknowns to be solved.
Conditions:
1. At the interior knots adjacent functions
must meet the data: 2n-2 equations
2. First and last function must pass through
end points: 2 equations
3. First derivatives at the interior knots
must be equal: n-1 equations
4. The final constrain is chosen arbitrarily: 1
equation
Note that continuity of the second
derivative is not ensured at the knots.
These simultaneous linear
equations are solved to obtain
all the coefficients.
18. Cubic Splines:
For cubic splines a different third order (cubic) polynomial is
defined for each interval.
f ( x)
ai x 3 bi x 2
ci x d i
For n+1 data points there are n intervals: 4n unknowns to be
solved.
Fifth condition is chosen arbitrarily
(called natural spline)
Conditions:
1. The function values must meet
interior knots: 2n-2 equations
2. The first and last function must pass
through end points: 2 equations
3. First derivatives at the knots must be
equal: n-1 equations
4. Second derivatives at the interior
knots must be equal: n-1 equations
5. The second derivatives at the end
knots are zero: 2 equations
