CSE 541 Numerical Methods

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CSE 541Numerical Methods

• Polynomial Interpolation
• Lagrange and Newtons

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Data Interpolation

• Problem
• Given A list of data points xi, fi
• Goal Evaluate the function in between the xis
• Solutions???
• Nearest value
• Average between neighboring values
• Find a function that models or fits
  (approximates?) the data points
• Local vs global information

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Interpolation

• Data from experimental/observational measurements
• Classic gravity drop location changes with time
• Pressure varies with depth
• Wind speed varies with time
• Temperature varies with location
• Census data

What was the population at 1964?

• The relationship looks linear
• Global information (trends)

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Science, Engineering, Statistics, etc.

• Scientific method (Also Engineering)
• Identify the underlying relationship in your data
  (patterns/trends)
• Deal with noise in the data
• Make predictions about the behavior of a process
• Not extrapolation!
• Remember distributions in statistics
• Tools used in social science, political science,
  etc.

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Regression vs Interpolation

• Pretty confident
• There is a polynomial relationship
• Little/no scatter
• Find an expression that passes exactly
• through all the points
• This function is an interpolant

• Unsure about the relationship
• Data looks scattered
• Maybe noisy data?
• Find an expression that captures the trend
• Minimize some measure of the error
• over all the points
• For example least squares fitting

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Interpolation

• Assumptions
• No error in the data
• Round-off is assumed to be negligible
• We dont know the underlying function f (x)
• Given n1 points xi, fi, x0,x1xi,xn xj gt
  xj-1
• Spacing may or may not be even
• All xi are distinct
• Interpolation
• Develop a simple function g(x) that
• Approximates f (x)
• Passes through all the points xi (i.e., an
  interpolant)
• Evaluate f (xt) where x0 lt xt lt xn

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Taylors Series

• Why not use a Taylor Series for interpolation?
• Interpolates at the point of expansion c
• Approximation gets worse further away from c
• It may not interpolate at other points
• Need f (c), f (c), f (c),
• We would want an interpolant at several f (c)s

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Modeling Choices

• How do we choose the simple function g(x)???
• Polynomials
• Splines
• Trigonometric functions (Data looks periodic)
• Fourier Transform representations
• Least-squares
• Error minimization

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Polynomial Interpolation

• Why are polynomials nice functions?
• Defines a line very easily (a linear
  relationship)
• Easily definable derivatives

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Existence Theorem

• Wierstrass Approximation Theorem
• Does not say how well the polynomial approximates
  a function
• What degree polynomial do we use?
• Weaker Theorem
• ò f(x) p(x)dx lt e
• But, we have existence of a polynomial

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Existence and Uniqueness

• Consider our data set of n 1 points yif(xi) at
  distinct points x0,x1xi,xn
• In general, there is a unique polynomial
  interpolant gn(x) of at most degree n
• What about uniqueness?
• Assume uniqueness for now and well prove it later

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Polynomial Interpolation

• How do we find gn(x)?
• Take n 3, so x0, y0, x1, y1, x2, y2, x3,
  y3
• Just solve, right?
• a0 a1x0 a2x02 a3x03 y0
• a0 a1x1 a2x12 a3x13 y1
• a0 a1x2 a2x22 a3x23 y2
• a0 a1x3 a2x32 a3x33 y3
• Express this as a matrix problem Ax b? Solve
  for A.
• This may not be easy to solve if it is an
  ill-conditioned system
• Large degree polynomials when n is large

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Lagrange Polynomials

• Construct a polynomial with the following form
• A linear combination of the sample points

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Constant Interpolation

• Take n 0,

x0
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Linear Interpolation

• Take n 1

Blending functions
x0
x1
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Lagrange Polynomials

• Take n 2, quadratic polynomial
• L0(x), L1(x), and L2(x)

• The third quadratic has roots at x0 and x1 and a
  value equal to the function data at x2.
• P(x0) 0
• P(x1) 0
• P(x2) f2

• The first quadratic has roots at x1 and x2 and a
  value equal to the function data
• at x0.
• P(x0) f0
• P(x1) 0
• P(x2) 0

• The second quadratic has roots at x0 and x2 and a
  value equal to the function data at x1.
• P(x0) 0
• P(x1) f1
• P(x2) 0

• Adding them all together, we get the
  interpolating quadratic polynomial, such that
• P(x0) f0
• P(x1) f1
• P(x2) f2

x0
x2
x1
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Newton Interpolation

• Consider our data set of n 1 points yif (xi)
  at distinct points x0,x1xi,xn
• Note, the notation for the divided difference is
  reverse from what is presented in the textbook

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Interpolation

• Constant interpolation
• pn(x) f(x0)
• Linear interpolation

x0
x1
slope
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Quadratic Interpolation

• Three data points

roots
Basis (support) for the interval
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Newton Interpolation
We have a recursive definition
Divided Differences
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Newton Interpolation

• The recursion formula
• The quadratic term

Divided Differences
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Invariance Theorem

• The order of the data points does not matter
• Though, the data points must be distinct
• Hence, the divided difference f x0, x1, , xk
  is invariant under all permutations of the xis