CSE 541 Numerical Methods

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CSE 541Numerical Methods

• Divided Differences, Uniqueness in Polynomial
  Interpolation, and Chebyshev Nodes

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Example ln(2)

• Given ln(1), ln(4), and ln(6)
• (x0, f0), (x1, f1), (x2, f2) Þ (1, 0), (4,
  1.3863), (6, 1.79176)

Note the divergence for values outside ofthe
data range
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Linear Interpolation

• p1(x) b0 b1(x x0)
• 0 (1.3863-0)/(4-1)(x-1)
• 0.4621(x - 1)
• p1(2) 0.4621

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Quadratic Interpolation

• p2(x) b0 b1(x x0) b2(x x0)(x x1)
• 0 0.4621(x-1)(((1.7918 1.3863)/(6-4)-
• (1.3863-0)/(4-1))/(6-1))(x-1)(x-4)
• 0.4621(x-1) - 0.051874(x-1)(x-4)
• p2(2) 0.5658

Same approximation as Lagrange Well that was a
quadratic also! Oh yes, uniqueness!
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Example ln(x)

• Observation
• Quadratic interpolation catches some of the
  curvature
• Linear plus higher order correction
• Improves the result somewhat
• Does that mean that the higher the degree the
  better our results?
• Not necessarily tune in later

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Calculating The Divided-Differences

• Incrementally construct a divided-difference
  table
• Consider eight samples and the function ln(x)
• Compute bi starting from b0

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Calculating The Divided-Differences
Calculate b1
Calculate needed values for the next step
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Calculating The Divided-Differences
Calculate b2
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Calculating The Divided-Differences
Calculate b3
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Calculating The Divided-Differences
Calculate b4
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Calculating The Divided-Differences
Calculate b5
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Calculating The Divided-Differences
Calculate b6
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Calculating The Divided-Differences

• Finally, we can calculate the last coefficient

Calculate b7
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Calculating The Divided-Differences

• Remember, all of the coefficients for the
  resulting polynomial are in bold
• Can use Excel to construct this table very easily!

b0
b4
b7
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Polynomial Form

• Insert the divided-differences into the final
  polynomial

Divided Differences diagonal values of the table
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Many polynomials

• The order of the numbers (xi, yi)s only matters
  when writing the polynomial down
• The first column in the table represents pairs of
  adjacent points connected linearly
• i.e. linear splines
• The second column gives us quadratics thru three
  adjacent points
• Etc.

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Adding an Additional Data Point

• Easy to add another row to the table
• Hence, only n additional divided-differences need
  to be calculated for the n 1st data points

b8
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Uniqueness

• Let pn(x) and qn(x) be two different polynomials
  of degree n (or less) that both interpolate n 1
  data points
• Let rn(x) be the subtraction of these two
  polynomials
• Þ Polynomial rn(x) has at most degree n

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Uniqueness

• The n 1 data points are roots of rn(x)
• Since both pn(x) and qn(x) interpolate the n 1
  data points
• A polynomial of degree n can only have at most n
  roots
• But, we established that rn(x) can have degree at
  most n
• A Contradiction!!!!!!
• Therefore, rn(x) ? 0

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Error

• Define the error term
• If f (x) is an nth order polynomial pn(x) is of
  course exact
• Otherwise, this function has at least n 1 roots
  at the interpolation points
• Since there is a perfect match at x0, x1,,xn

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Interpolation Errors

• We already have a feel for what these divided
  differences represent
• Corollary 1 in book If f (x) is a polynomial of
  degree m lt n, then all (m 1)th divided
  differences and higher are zero

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Interpolation Errors

• Intuitively, the first n 1 terms of the Taylor
  Series is also an nth degree polynomial
• Proof is in the book.

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Interpolation Errors

• Use the point x to expand the polynomial
• Point is, we can take an arbitrary point x, and
  create an (n 1)th polynomial that goes thru the
  point x

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Choosing Data Points

• Given an interval a, b of interest, how do we
  choose data points?
• Divide into equal sized subintervals
• First two are the endpoints, the next point is
  the mid-point, followed by successive mid-points
  of the half-intervals
• How many data points?
• More means higher order approximation
• Does this translate to better?

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Problems with Polynomial Interpolation

• Is it always a good idea to use higher and higher
  order polynomials?
• No Usually 3-4 points is good and 5-6 points ok
• Too high a degree causes wiggles
• Particularly for sharp edges

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Sample Point DistributionChebyshev Nodes

• Equally distributed points is the easy way
• This may not be the optimal solution
• If you could select the xis unevenly, what would
  they be?
• Want to minimize the term
• Chebyshev nodes
• For x -1 to 1, define

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Chebyshev nodes

• Lets look at these for n 4
• Spreads the points out inthe center