EARS1160

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EARS1160 Numerical Methodsnotes by G. Houseman

• Lecture 4 Interpolation.
• Interpolation requirements
• Polynomial interpolation local vs global
• Linear and bi-linear interpolation
• Local quadratic interpolation
• Lagrange polynomial interpolation
• Continuity of function and derivatives
• Cubic spline interpolation
• Cubic spline theory
• Taut splines

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Interpolation of Discrete Functions

• Suppose we know the function values at a set of
  points, e.g. measurement points, how do we find
  the value of the function at other intermediate
  (e.g. regularly spaced) points?
• Ideally we would like to obtain estimates for
  interpolated points that are consistent with the
  idea of first (and probably second) derivatives
  of the interpolated function being continuous.
  We would also like to be able to extend the
  method to 2 or more dimensions f(x,y) or f(x,y,z)

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Polynomial Interpolation

• Example Magnetic or gravity measurements may be
  collected at approximately constant spacing along
  a road or roads. Estimate the interpolated
  values on a set of regularly spaced points, for
  display and further processing.
• Most interpolation methods are based on fitting a
  polynomial to the data points, and using that
  polynomial to provide function values at
  arbitrary points.
• Global methods attempt to fit a single function
  to all of the data points. A polynomial of order
  N-1 may be fit to N data points. Accuracy can be
  a problem with these methods because they may
  show large minima and maxima between the points.
  In general they are not recommended.
• Local interpolation methods attempt to fit low
  order polynomials to small subsets of the data.
  These methods are preferred because they are
  conceptually simple and they are local, i.e.
  values of the interpolation function are only
  influenced by the nearby measurements.

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Linear Interpolation

• The linear interpolation method simply divides
  the line into segments defined by pairs of
  adjacent data points.
• If
• Then
• If we write the function in this form, you can
  simply verify that (i) it is linear in x, and
  (ii) it gives the required values at the segment
  end points xk and xk1.
• Difficulties with linear interpolation
• (i) The slope of the function is discontinuous at
  all sample points. Higher derivatives are
  unavailable
• (ii) Maxima and minima that fall between sample
  points are poorly represented.

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Bi-Linear Interpolation

• In 2-dimensions the linear interpolation method
  may be extended for data points that fall
  approximately on a grid.
• If, for the point (x, y), and
• Then
• This function is linear in x (if y is fixed) or
  vice versa, but note the presence of a term in xy
  - which allows us to fit a simple surface to 4
  points. A true linear function (representing a
  plane in 3D) could fit at most 3 points.
• This method does not necessarily require that the
  points fall on a regular grid, and may be
  extended to higher dimensions if necessary.

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Local Quadratic Interpolation

• To improve on linear interpolation, we might
  consider using local quadratic functions.
• The quadratic function has 3 unknown
  coefficients, so two sample points don't provide
  enough information to constrain the 3 unknowns.
• If we use 3 adjacent points, we can exactly fit a
  parabola which covers 2 sampling intervals
• Disadvantages
• (i) gradient of the function is still
  discontinuous at every second grid point
• (ii) odd-numbered points are treated differently
  from even-numbered points

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Polynomial Interpolation formula

• The equation given above for quadratic
  interpolation can be generalised to a polynomial
  of any order
• This is the Lagrange polynomial interpolation
  formula. You can verify by substitution that
  PN(xj) f(xj) for each sample point.
• You will use the routine POLINT from Numerical
  Recipes (Press et al., 1992) in the next
  practical exercise.
• POLINT implements the Lagrange formula using
  Neville's algorithm which builds the polynomial
  coefficients recursively.

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Continuity

• A function is continuous if there are no sudden
  steps in its value as we move along the
  independent variable. Continuity is usually an
  essential requirement of an interpolation
  function.
• With linear interpolation, sudden changes in
  slope of the interpolated function are observed
  at each sample point. A plot of the derivative
  of the interpolated function looks like a
  staircase flat between sample points, then
  stepping up or down at each sample point. The
  second derivative is undefined at these points.
• It is often useful to have continuous first and
  second derivatives for an interpolated function,
  since we may want to evaluate these quantities.
  If 1st derivative is continuous, then
  discontinuous 2nd derivative is less obvious - it
  implies a step-like change in curvature of the
  interpolated function.
• We refer to C0, C1, and C2 continuity, to
  indicate continuity of the function, its
  gradient, and its curvature, respectively.

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Cubic Spline Interpolation
With local cubic interpolation we can obtain C2
continuity without the stability and accuracy
problems associated with the Lagrange
Polynomials. A cubic polynomial has 4
coefficients. If we require that the polynomial
fits the points at xj and xj1, then two of the
coefficients are specified. If we also require
continuity of 1st and 2nd derivatives with the
cubic function on the adjoining segments, then we
have enough constraints to uniquely define all
the coefficients for all the cubic polynomial
segments. The resulting function is called a
Cubic Spline. The spline functions may be
quickly calculated, are accurate and stable, and
now are widely used for interpolation in many
different applications. We have not yet specified
the behaviour of first and second derivatives at
the end points of the sample range. There is
some choice possible, but setting the 2nd
derivative to zero is the natural choice, leading
to the Natural Cubic Spline.
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Cubic Spline Method 1

• The development of a spline method builds on the
  linear interpolation.
• with
• Suppose we also know values of the 2nd
  derivatives y"j at sample points xj, then we can
  add to the interpolation function a cubic
  component whose second derivative varies
  linearly
• where
• Exercise verify that the new function satisfies
  the data constraints and has continuous 2nd
  derivative.

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Cubic Spline Method 2

• So far we have assumed that we know the 2nd
  derivatives. In fact we don't, and in order to
  specify the values of the 2nd derivatives we need
  more constraints.
• We require that first derivatives of the spline
  function
• are continuous also, which gives
• This set of equations can now be solved for the
  set of unknown 2nd derivatives. This system of
  simultaneous equations is referred to as a
  tridiagonal system (the matrix has non-zero terms
  only on the 3 central diagonals).

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Splines further considerations

• The splines may be generalised to represent
  functions of 2 or more variables, which are C2
  continuous in all variables.
• Splines may be adapted to a parametric
  representation, which permits multi-valued
  functions (e.g. folded surfaces) to be
  represented.
• The requirement that the spline interpolants fit
  exactly to the samples points may be relaxed, by
  the use of taut splines used as a means of
  smoothing out data noise.
• Taut Splines use a tension parameter. Imagine
  that the spline surface is like a rubber sheet,
  which is stretched in the horizontal direction
  and the amount of tension in the horizontal
  directions is adjustable.
• If the tension is zero, the sheet can be forced
  to fit every sample point. As the tension is
  increased, we see some smoothing occurring as the
  sheet is pulled away from data points where there
  is extreme variation.