Polynomial Interpolation: Extending Function Values using Power Series - Prof. Ramani Dura, Study notes of Computer Science

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Computational Methods CMSC/AMSC/MAPL 460Polynomial Interpolation

Ramani Duraiswami,

Dept. of Computer Science

Interpolation

•^

Given a function at

N

points, find its value at other

point(s)– “within” the points

• Interpolation
  • “outside” the points
    • Extrapolation

•^

How do we extend the value from known points tounknown points?– Have to have prior knowledge about the function– Can do what is convenient

•^

Occam’s razor–

Parsimony: “One should not increase, beyond what is necessary, the numberof entities required to explain anything”–^ Key in all scientific modeling – Will return to this issue

Properties of Power Series

For any power series there exists

r

,,*^

such that the series

converges absolutely at |

y^

-^

x *

r

and diverges at |

y -

x *

r

,.*^

•^

The number

r

,*^

is called

the convergence radius

of the series,

•^

≤^

r*

•^

For any number

q

, such that 0 <

q

r

, the power series*

uniformly converges at |

y^

-^

x *

q.

2. Convergent power series can be summed, multiplied by a scalar, or

multiplied according to the Cauchy rule.

Cauchy’s rule

Taylor Series

•^

Let

f(y)

be a real function,

f(y)

C

n

[

x*

, x

+r*

–^

th n derivative

n f exists for

x

≤ *^

y

x

+r*

• Residual determines accuracy– Two evaluations of remainder
  • Cauchy evaluation• Lagrange evaluation

Polynomial Facts

•^

A polynomial of degree

k

has at most

k

distinct zeroes,

unless it is identically zero

•^

Sum of two polynomials of degree

k

is another

polynomial of degree at most

k.

•^

Polynomials can be expressed in many ways– Degree of basis functions is 2, 1 and 0 …– Basis 3 is the power basis or monomial basis– Any basis can be used … often “orthogonal polynomials” are

used

Uniqueness of interpolant

•^

We know that the polynomial exists

-^

Suppose that there are two different polynomials that caninterpolate the data

-^

Let them be

pn-

and

q

n-

•^

So we have

pn-

(x

) = yi

, i=1, …ni qn-

(x

) = yi

, i=1, …ni

•^

So

pn-

(x

) -qi

n-

(x

)= 0, i=1, …ni

•^

p^ n-

-qn-

is the difference of two polynomials of degree

n-1.

•^

It has

n

zeroes.

•^

Recall polynomial of degree

k

has at most

k

zeroes, or is the zero

polynomial.

-^

Here we have more zeroes than degree … so it is the zeropolynomial

-^

So interpolant is unique

Vandermonde matrices

•^

Matlab code

•^

In matlab Vandermondematrix is defined inflipped order using thefunction

vander

•^

Example …

•^

Also there is a function to fit polynomials to data, polyfit

•^

Vandermonde matrices are nonsingular if the points aredistinct

•^

However they can be very poorly conditioned

Computing interpolants by hand

•^

Suppose we have data at

n

points and we have fit a

polynomial

•^

How can we do this fit efficiently?

•^

Fit for one point is

y=y

1

•^

Fit for two points can be written as

y=a(x-x

)+b(x-x 1

•^

Fit for three points can be written as

y=a(x-x

)(x-x 1

)+b(x-x 2

)(x-x 1

)+c(x-x 3

)(x-x 2

•^

And so on …

•^

Advantage: each coefficient can be calculatedindependent of others– Why?– What is the form of the coefficient computed?