Polynomial Approximation

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Polynomial Approximation

• PSCI 702
• October 05, 2005

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What is a Polynomial?

• Functions of the form
• Polynomial of degree n, having n1 terms.
• Will take n(n1)n/2 multiplications and n
  additions. Can be re-written to take n additions
  and n multiplications.

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• Factored form
• N roots.
• N1 parameters.
• Both real and complex roots.
• No analytical solution for the polynomials of
  degree 5 or higher.

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Constraints on the roots of P(x)
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Synthetic Division
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Iterative Methods

• Searching with an initial guess the method finds
  the roots iteratively.
• Construct a tangent to the curve at the initial
  guess and then extend this to the x-axis.
• The new crossing point represent an improved
  value of the root.

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Iterative Methods
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Newton-Raphson Method
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Convergence
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Multiple Roots

• A multiple root (double, triple, etc.) occurs
  where the function is tangent to the x axis

two roots
single root
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Newton-Raphson Method
Newtons method - tangent line
root x
xi
xi1
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Newton Raphson Method

• Step 1 Start at the point (x1, f(x1)).
• Step 2 The intersection of the tangent of f(x)
  at this point and the x-axis.
• x2 x1 - f(x1)/f (x1)
  
• Step 3 Examine if f(x2) 0
• or abs(x2 - x1) lt tolerance,
• Step 4 If yes, solution xr x2
• If not, x1 x2, repeat the
  iteration.

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Newton-Raphson Method
Examples of poor convergence
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Secant Method
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Secant Method

• Use secant line instead of tangent line at f(xi)

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Convergence not Guaranteed
y ln x
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MATLAB Function fzero

• Bracketing methods reliable but slow
• Open methods fast but possibly unreliable
• MATLAB fzero fast and reliable
• fzero find real root of an equation (not
  suitable for double root!)

fzero(function, x0) fzero(function, x0 x1)
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Interpolation Methods
Interpolation uses the data to approximate a
function, which will fit all of the data points.
All of the data is used to approximate the values
of the function inside the bounds of the data.
All interpolation theory is based on polynomial
approximation.
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Lagrange Interpolation

• The problem find the (unique) polynomial f(x)
  of degree k-1 given a set of evaluation points
  xii1,k and a set of values yif(xi)
• Solution for each i1,...,k
• find a polynomial pi(x) that takes on the value
  yi at xi, and is zero for all other instances of
• x1, ...,xi-1,..xi1,..xk

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Lagrange Interpolation
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Cubic Lagrange Interpolation

• p(x) l1,4 f1 l2,4 f2 l3,4 f3 l4,4 f4
• where
• l1,4 ( x - x2 ) ( x - x3 ) ( x - x4 ) / (
  x1 - x2 ) ( x1 - x3 ) ( x1 - x4 )
• l2,4 ( x - x1 ) ( x - x3 ) ( x - x4 ) / (
  x2 - x1 ) ( x2 - x3 ) ( x2 - x4 )
• l3,4 ( x - x1 ) ( x - x2 ) ( x - x4 ) / (
  x3 - x1 ) ( x3 - x2 ) ( x3 - x4 )
• l4,4 ( x - x1 ) ( x - x2 ) ( x - x3 ) / (
  x4 - x1 ) ( x4 - x2 ) ( x4 - x3 )

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Cubic Lagrange Interpolation

• Find the cubic polynomial whose graph contains
  the four successive points (0,1), (1,2), (2,0),
  and (3,-2). Setting x1 0, f1 1, x2 1, f2
  2, x3 2, f3 0, and x4 3, f4 -2, we can
  form the values of the ls
• l1,4 ( x - x2 ) ( x - x3 ) ( x - x4 ) / (
  x1 - x2 ) ( x1 - x3 ) ( x1 - x4 ) ( x - 1 )
  ( x - 2 ) ( x - 3 ) / ( 0 - 1 ) ( 0 - 2 ) ( 0
  - 3 ) ( x - 1 ) ( x - 2 ) ( x - 3 ) / - 6
• l2,4 ( x - x1 ) ( x - x3 ) ( x - x4 ) / (
  x2 - x1 ) ( x2 - x3 ) ( x2 - x4 ) ( x - 0 )
  ( x - 2 ) ( x - 3 ) / ( 1 - 0 ) ( 1 - 2 ) ( 1
  - 3 ) x ( x - 2 ) ( x - 3 ) / 2
• l3,4 ( x - x1 ) ( x - x2 ) ( x - x4 ) / (
  x3 - x1 ) ( x3 - x2 ) ( x3 - x4 ) ( x - 0 )
  ( x - 1 ) ( x - 3 ) / ( 2 - 0 ) ( 2 - 1 ) ( 2
  - 3 ) x ( x - 1 ) ( x - 3 ) / - 2
• l4,4 ( x - x1 ) ( x - x2 ) ( x - x3 ) / (
  x4 - x1 ) ( x4 - x2 ) ( x4 - x3 ) ( x - 0 )
  ( x - 1 ) ( x - 2 ) / ( 3 - 0 ) ( 3 - 1 ) ( 3
  - 2 ) x ( x - 1 ) ( x - 2 ) / 6 .

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Cubic Lagrange Interpolation

• Inserting the values for the ls into Equation 2,
  we have
• f(x) l1,4 f1 l2,4 f2 l3,4 f3 l4,4 f4
  ( x - 1 ) ( x - 2 ) ( x - 3 ) / - 6 1
  x ( x - 2 ) ( x - 3 ) / 2 2 x ( x - 1 )
  ( x - 3 ) / - 2 0 x ( x - 1 ) ( x - 2 )
  / 6 - 2 (-1/6) ( x - 1 ) ( x - 2 ) ( x -
  3 ) x ( x - 2 ) ( x - 3 ) (-1/3) x (
  x - 1 ) ( x - 2 ) .
• Multiplying out the terms and collecting yields
  the desired polynomial
• f(x) 0.5x3 - 3x2 3.5x 1 .

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Hermite Interpolation

• The Advantages
• The segments of the piecewise Hermite polynomial
  have a continuous first derivative at support
  points (xis).
• The shape of the function being interpolated is
  better matched, because the tangent of this
  function and tangent of Hermite polynomial agree
  at the support points

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Cubic Spline Interpolation
Hermite Polynomials produce a smooth
interpolation, they have a disadvantage that the
slope of the input function must be specified at
each breakpoint. Cubic Splines interpolation use
only the data points used to maintaining the
desired smoothness of the function and is
piecewise continuous.
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Cubic Spline
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Cubic Spline
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Tangent vector
p(t)
p(t)
t
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Significant values

• p(0) startpoint segment
• p(0) tangent through startpoint
• p(1) endpoint segment
• p(1) tangent through endpoint

p(0)
p(1)
t
p(1)
p(0)
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Rational Function Interpolation
Polynomial are not always the best match of data.
A rational function can be used to represent the
steps. A rational function is a ratio of two
polynomials. This is useful when you deal with
fitting imaginary functions zx iy. The
Bulirsch-Stoer algorithm creates a function where
the numerator is of the same order as the
denominator or 1 less.
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Rational Function Interpolation
The Rational Function interpolation are required
for the location and function value need to be
known. or
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Legendre Polynomial

• The Legendre polynomials are a set of orthogonal
  functions, which can be used to represent a
  function as components of a function.

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Legendre Polynomial

• These function are orthogonal over a range -1, 1
  . This range can be scaled to fit the function.
  The orthogonal functions are defined as

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Legendre Polynomials

• Orthogonal Polynomials that covers the finite
  interval from -1 to 1

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Laguerre Polynomials

• Orthogonal Polynomials that covers the semi
  finite interval from 0 to infinity.

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Hermite polynomial

• Orthogonal Polynomials that covers the infinite
  interval from -infinity to infinity.

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Orthogonal Polynomials
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Orthogonal Polynomials