ChE 250 Numeric Methods

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ChE 250 Numeric Methods

• Lecture 18
• Chapra, Chapter 18 Interpolation
• Chapter 19 Fourier Approximation
• 20070319

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Today

• Interpolation
• Lagrange Interpolating Polynomial
• Spline
• Fourier Approximation
• Least Squares
• Continuous Fourier series
• Time and Frequency domain
• Fast Fourier Transform

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Lagrange Polynomials

• Calculating the finite differences required for
  Newtons Interpolation can sometimes be
  cumbersome
• It is often more convenient to use Lagrange
  Polynomials
• These are calculated by a product of the
  differences of the independent value for
  surrounding data points

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Lagrange Polynomials

• Terminology an nth order Lagrange Polynomial
  requires n1 data points
• The points used in the calculation should be
  those closest to the desired value, and throw
  away the rest of the given points
• Several polynomials can be solved and the best
  one chosen
• An r2 can also be calculated

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Lagrange Polynomials

• Equation The polynomials are Li which are
  calculated using the Pi, product
• Example 18.6
• Questions?

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Lagrange Polynomials

• Which is the best fit?
• (a) 4th order
• (b) 3rd order
• (c) 2nd order
• (d) 1st order

Time, s Measured Velocity, cm/s
1 800
3 2310
5 3090
7 3940
13 4755
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Lagrange Polynomials

• Summary
• Newtons polynomial may be better when attempting
  to determine the order of the polynomial to use
• Can calculate error easily
• Lagrange method is easier to implement when the
  order of the polynomial fit is known
• Both will have significant rounding error for
  higher orders

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Spline Interpolation

• Connect the dots method
• First order is exactly like a linear
  interpolation
• Second order (quadratic) matches the value and
  the first derivative
• Third order (cubic) matches value, first and
  second derivative at each interior point, or knot

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Spline Interpolation

• Cubic spline can be represented by a function in
  each interval of the form
• By setting the first derivatives at the interior
  knots equal on adjacent cubics
• And also the second derivatives

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Spline Interpolation

• Now, we have to solve for all the as, bs, cs
  and ds using some form of simultaneous solution,
  or matrix solution for larger sets

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Spline Interpolation

• For Scilab, The splin function with natural
  argument returns the derivative vector
• This can be used with the interp function to
  interpolate a value

Example 18.10, Cubic Spline on Scilab
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Spline Interpolation

• Splines have the distinct advantage of sitting a
  dataset very well, no matter how discontinuous
  the data is
• Even if the data changes regimes
• Example 18.10
• Questions?

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Fourier Approximation

• Polynomials fit many functions associated with
  physical properties
• But when functions of time are considered, they
  are often periodic and not well-modeled by
  polynomials

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Fourier Approximation

• Polynomials fit many functions associated with
  physical properties
• But when functions of time are considered, they
  are often periodic and not well-modeled by
  polynomials

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Fourier Approximation

• The Fourier series transforms a given periodic
  function into a superposition of sine and cosine
  waves
• The following equations are used

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Fourier Approximation

• For cases where the function is not known, but a
  data set with points equally spaced in time is
  available, the data can be fitted using a Fast
  Fourier Transform
• This is done routinely in electronic equipment
  including laboratory analysis
• e. g. spectroscopy
• FFT Example
• Questions?

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Preparation for 16Mar

• Reading
• Chapter 19 Fourier Approximation
• Homework Due Friday
• Chapter 17
• 4, 7,9,12, 17,25,29
• Chapter 18
• 5, 7, 11, 13
• Chapter 19
• 4, 6, 12, 14, 22
• Chapter 20
• 12, 14, 17