University of Portsmouth

1

• University of Portsmouth
• Department of Mathematics
• Project Presentation
• Barycentric representation of some interpolants
  Theory and numerics.
• By Maria Apostolou
• Supervisor Dr. A. Makroglou
• 2nd Assessor Dr. A. Osbaldestin

2
Aim

• To study some numerical methods in their
  classical and barycentric form such as
• Lagrange
• Rational
• To implement some of them numerically.
• The MATLAB package has been used for programming.

3
Outline of the Talk

• 1. Interpolation problem.
• 2. Lagrange interpolation.
• 3. Barycentric representation
• 3.1 Lagrange type
• 3.2 Linear rational type
• 4. Numerical results
• 5. Conclusions.
• 6. Further Research.

4
References

• The main references are
• Berrut, J. P. and Mittelmann, H. D., Lebesgue
  Constant Minimizing Liner Rational Interpolation
  of Continuous Functions over the Interval,comp.
  Maths applic. , 33 (1997), 77-86.
• 2) Salzer, H. E., Lagrangian interpolation at the
  Chebyshev points Xn,v ? cos (v?/n), v 0(1)n
  some unnoted advantages, The Computer Journal, 15
  (1972), 156-159.
• 3) Werner, W., Polynomial interpolation
  Lagrange versus Newton, Mathematics of
  Computation, vol. 43 (1984), 205-217.
• 4) Hahn, B. D. Essential MATLAB for Scientists
  and Engineers, Arnold, 1997.

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1. Interpolation problem

• One of the types of approximation methods is the
  interpolation.
• The problem of interpolation for one dimensional
  data can be stated as follows given a set of
  data of the form (xi, yi), i 0, 1, . . ., n
  where the xi are distinct, find a function say
  p(x) such that p(xi) yi, i 0, 1, . . ., n.
• Quite often the interpolation function is a
  polynomial such as Lagrange, Newton and Neville.
• A well known problem of polynomial interpolation
  is that when used with a large number of
  equidistant points xi, the errors at points close
  to the end points of the interval of
  consideration grow catastrophically.

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2. Lagrange interpolation

• -One approach to the interpolation problem is the
  Lagrange method.
• The Lagrange form is
• 
  (2.1)
• where

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3. Barycentric representation3.1 Lagrange type

• The Lagrange type barycentric formula is
• ,
• Proof.
• Since

8

• the Lagrange interpolation polynomial can be
  written
• Let
• So we rewrite lj as

9

• Thus
• 
  (4.1)
• We divide both nominator and denominator by

10

• So (4.1) became
• 
  (4.2)
• So (4.2) became

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Advantage of the Lagrange barycentric form

• The main advantage of the barycentric form
  is related to improvements in the numerical
  stability of the Lagrange interpolation process.
  The main reason for this improvement is reported
  in Werner (1984, p.210) to be the fact that even
  if the weights Aj are not computed very
  accurately, the barycentric formula continues to
  be an interpolating formula .

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3.2 Linear rational type

• The form of the linear rational interpolant of
  barycentric form is
• The constants uk are chosen so that the Lebesgue
• constant
• is minimized under the constrains

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4. Numerical results - Table1

• The results for the function yexp(-x2) using
  classical
• Lagrange with equidistant points fon n 21
• are xval function values comp. value abs errors
• -0.950 0.40555451 0.40555278
  1.723e-006
• -0.750 0.56978282 0.56978272
  1.002e-007
• -0.550 0.73896849 0.73896849
  3.123e-009
• -0.350 0.88470590 0.88470590
  3.307e-011
• -0.150 0.97775124 0.97775124
  2.698e-014
• 0.150 0.97775124 0.97775124
  1.110e-016
• 0.350 0.88470590 0.88470590
  3.531e-014
• 0.550 0.73896849 0.73896849
  1.641e-012
• 0.750 0.56978282 0.56978282
  1.758e-010
• 0.950 0.40555451 0.40555451
  5.392e-009

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Numerical results Table2

• The results for the function yexp(-x2) using
  classical
• Lagrange with chebyshev points fon n 21
• are xval function values comp. value abs errors
• -0.950 0.40555451 0.40555457
  6.070e-008
• -0.750 0.56978282 0.56978283
  1.544e-009
• -0.550 0.73896849 0.73896849
  2.786e-011
• -0.350 0.88470590 0.88470590
  1.312e-012
• -0.150 0.97775124 0.97775124
  1.033e-014
• 0.150 0.97775124 0.97775124
  8.882e-016
• 0.350 0.88470590 0.88470590
  6.661e-016
• 0.550 0.73896849 0.73896849
  2.849e-013
• 0.750 0.56978282 0.56978282
  5.472e-012
• 0.950 0.40555451 0.40555450
  1.016e-010

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Numerical results Table3

• The results for the function yexp(-x2) using
  barycentric with
• equidistant points for n 21
• are xval function values comp. value abs errors
• -0.950 0.40555451 0.40555451
  2.262e-012
• -0.750 0.56978282 0.56978282
  2.220e-014
• -0.550 0.73896849 0.73896849
  8.882e-016
• -0.350 0.88470590 0.88470590
  3.331e-016
• -0.150 0.97775124 0.97775124
  0.000e000
• 0.150 0.97775124 0.97775124
  0.000e000
• 0.350 0.88470590 0.88470590
  2.220e-016
• 0.550 0.73896849 0.73896849
  7.772e-016
• 0.750 0.56978282 0.56978282
  1.910e-014
• 0.950 0.40555451 0.40555451
  2.593e-012

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Numerical results Table4

• The results for the function yexp(-x2) using
  barycentric with
• Chebychev points fon n 21
• are xval function values comp. value abs errors
• -0.950 0.40555451 0.40555451
  1.249e-014
• -0.750 0.56978282 0.56978282
  9.437e-015
• -0.550 0.73896849 0.73896849
  2.998e-015
• -0.350 0.88470590 0.88470590
  4.885e-015
• -0.150 0.97775124 0.97775124
  5.551e-016
• 0.150 0.97775124 0.97775124
  1.110e-016
• 0.350 0.88470590 0.88470590
  4.552e-015
• 0.550 0.73896849 0.73896849
  2.665e-015
• 0.750 0.56978282 0.56978282
  9.437e-015
• 0.950 0.40555451 0.40555451
  1.255e-014

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Numerical results - Table5

• The results for the function yexp(-x2) using
  classical
• Lagrange with Chebychev points fon n 101
• are xval function values comp. value abs errors
• -0.450 0.81668648 -8.91078343
  9.727e000
• -0.350 0.88470590 0.88616143
  1.456e-003
• -0.250 0.93941306 0.93941296
  9.892e-008
• -0.150 0.97775124 0.97775124
  2.802e-012
• -0.050 0.99750312 0.99750312
  1.010e-014
• 0.050 0.99750312 0.99750312
  2.069e-011
• 0.150 0.97775124 0.97626064
  1.491e-003
• 0.250 0.93941306 -11403.06993416
  1.140e004
• 0.350 0.88470590 -1057124943.27642430
  1.057e009

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Numerical results Table6

• The results for the function yexp(-x2) using
  barycentric with
• Chebychev points fon n 101
• are xval function values comp. value abs errors
• -1.000 0.36787944 0.36787944
  1.665e-016
• -0.800 0.52729242 0.52729242
  2.220e-016
• -0.600 0.69767633 0.69767633
  5.551e-016
• -0.400 0.85214379 0.85214379
  1.110e-016
• -0.200 0.96078944 0.96078944
  4.441e-016
• 0.000 1.00000000 1.00000000
  4.441e-016
• 0.200 0.96078944 0.96078944
  1.110e-016
• 0.400 0.85214379 0.85214379
  5.551e-016
• 0.600 0.69767633 0.69767633
  2.220e-016
• 0.800 0.52729242 0.52729242
  2.220e-016
• 1.000 0.36787944 0.36787944
  0.000e000

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Numerical results Table7

• The results for the function y1/(125x2) using
  Classical
• Lagrange with Chebychev points fon n 101
• are xval function values comp. value abs errors
• -0.350 0.24615385 -653263076.28299761
  6.533e008
• -0.250 0.39024390 -5003.80824613
  5.004e003
• -0.150 0.64000000 0.64080180
  8.018e-004
• -0.050 0.94117647 0.94117647
  8.735e-010
• 0.050 0.94117647 0.94117647
  8.546e-010
• 0.150 0.64000000 0.64000000
  8.834e-010
• 0.250 0.39024390 0.39024400
  9.680e-008
• 0.350 0.24615385 0.24822379
  2.070e-003
• 0.450 0.16494845 4.18889959
  4.024e000
• 0.550 0.11678832 -68734.99358674
  6.874e004

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Numerical results Table8

• The results for the function y1/(125x2) using
• barycentric with Chebychev points fon n 101
• are xval function values comp. value abs errors
• -1.000 0.03846154 0.03846154
  7.409e-010
• -0.800 0.05882353 0.05882353
  5.050e-010
• -0.600 0.10000000 0.10000000
  9.597e-010
• -0.400 0.20000000 0.20000000
  1.019e-009
• -0.200 0.50000000 0.50000000
  1.920e-009
• 0.000 1.00000000 1.00000000
  4.441e-016
• 0.200 0.50000000 0.50000000
  1.920e-009
• 0.400 0.20000000 0.20000000
  1.019e-009
• 0.600 0.10000000 0.10000000
  9.597e-010
• 0.800 0.05882353 0.05882353
  5.050e-010
• 1.000 0.03846154 0.03846154
  7.409e-010

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5. Conclusions

• Evaluating the results of the Tables 1, 2, 3 and
  4 for the
• yexp(-x2) function when n 21 we notice that
  the errors with
• the barycentric form are a lot more accurate than
  the
• corresponding results with the Classical form.
• Also we notice that the results for Chebyshev
  nodes
• are more accurate that those with equidistant
  nodes as
• expected.
• The errors in Table 6 with n101 and Chebyshev
  nodes are
• all of the order E-16, while the errors in Table
  5 (classical
• Lagrange, n101, Chebyshev nodes) increase
  catastrophically
• at points outside the interval -0.15, 0.15.

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• The errors in Table 8 with n101 and Chebyshev
• nodes are all of the order E-10, E-09, while the
  errors
• in Table 7 (classical Lagrange, n101, Chebyshev
• nodes) increase catastrophically at points
  outside the
• interval -0.15, 0.15.
• So the general conclusion is that with respect to
  accuracy
• and numerical stability the barycentric form of
  the Lagrange
• interpolation method stays very accurate for
  large n (n1 the
• number of points) and thus it should be used in
  all
• applications where Lagrange interpolation is used
  
• (Approximation theory, differential equations
  etc).

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6. Further research

• Ideas for further research may include
• Exploring the application of the barycentric form
  to 2-dimensional data.
• The computation of other types of interpolation
  methods using their barycentric form.
• The use of barycentric approach when using
  interpolation in solving equations of various
  types.