What is the main question in polynomial interpolation?

How do we represent mathematical functions on a computer?

Why are polynomials easy to store?

Only need to store n + 1 coefficients

What is the Weierstrass Approximation Theorem?

What is interpolation?

Finding a p_{n}(x_{i}) = f(x_{i}) at a finite set of points x_{i }called nodes. Sometimes we also require the derivatives to match.

What is the simplest interpolating polynomial?

Truncated taylor series

Why is a truncated Taylor series the simplest interpoalting polynomial?

It uses only a single node x_{0}

What is Taylor's Theorem?

What is the following known as?

Taylor polynomial of degree n

What is the remainder in Taylor's theorem called?

Lagrange form of the remainder

What is the formula for the Lagrange form of the remainder?

What can we use to bound the error of an approximation using Taylor's theorem?

The Lagrange remainder

What is truncation error?

The error that arises from approxiamting f with a truncated series, rather than due to rounding

How does truncation error change as you use more terms?

It decreases

What is Rolle's Theorem?

Prove the Lagrane form of the remainder.

For general n, what are the polynomial interpolation conditions?

What is another way to write the following conditions?

What is the following called?

The Vandermonde matrix

What is the Vandermonde matrix?

What is the determinant of the Vandermonde matrix?

Prove that the determinant of the Vandermonde matrix is the following.

Problem Sheet

What is the existence/uniqueness theorem for interpolating polynomials?

Prove the unique part of the following theorem.

What is one way to solve the Vandermode matrix?

Gaussian elimination

What basis do we use in the Vandermode matrix?

{1, x, x^{2} .. }

When we use the basis of polynomials {l_{k}} what does p_{n}(x) equal?

For general n, what is the equation of the n+1 Lagrange polynomials?

What is a problem with the Lagrange form of the interpolating polynomial?

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- Expensive to evaluate since all the l
_{k} must be computed
- Chaning any nodes or adding a new node means all the l
_{k} must all be recomputed from scratch

_{k}must be computed_{k}must all be recomputed from scratchHow is the basis {l_{k}} picked?

So that we get the identity matrix meaning we get the following p_{n}(x). So we need the following.

What is the main idea for the Newton form of the basis?

Find a basis where p_{n+1} = p_{n} + g_{n+1}(x) so that adding new nodes is easier than the lagrange form os the basis

What is the formula for the {n_{k}} in the Newton basis.

What is the Newton form of the interpolating polynomial?

What is the the other notation which is commonly used for a_{k }in the Newton form of the interpolating polynomial?

a_{k} = f[x_{0},x_{1}, ..., x_{n}]

What is the theorem about the divided differences?

Prove the following theorem.

What do the diagrams for the Newton form of the interpolting polynomial look like?

Tree diagrams

What is the generic error for an interpolating formula?

| f(x) - p_{n}(x) |

What is Cauchy's theorem for the error in an interpolating formula?

Prove the following theorem.

How do you find what the maximum value of the error in Cauchy's formula?

The maximum of the node polynomial w(x) occurs at a turning point or at an end point, so find and use that value of x.

What is the formula for w(x), the node polynomial?

What is the Runge phenomenon?

When p_{n} converges to f in the middle, but diverges more and more near the ends.

What type of nodes do we choose to avoid the Runge phenomenon?

Chebyshev nodes

How do the spacing of chebyshev nodes change?

Places more nodes at the end of the interbal where most of the error is occuring at

What is the x_{j} formula to find the Chebyshev nodes?

What is the Lemma about the Chebyshev points?

Prove the following Lemma.

What is the theorem about Chebyshev interpolation?

Prove the following theorem.

What are the Chebshev nodes projections of?

Equally spaced points around a circle

What is the upper bound for the error in a Chebyshev polynomial?

What is the basic idea behind adding derivative conditions to interpolating polynomials?

Find polynomial that match one or more derivatives of f at the nodes, as well as function values

What two interpolating polynomials have derivative conditions?

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- Taylor polynomial
- Herminite polynomial

What is the theorem about Hermite interpolation?

Prove the following theorem.

What is the formula for the Hermite basis functions?

What is the formula for the Hermite interpolating polynomial?