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University of Portsmouth Department of Mathematics Project Presentation

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

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University of Portsmouth Department of Mathematics Project Presentation

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  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.

  5. 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.

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

  7. 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 denominatorby

  10. `` So (4.1) became (4.2) So (4.2) became

  11. 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 .

  12. 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

  13. 4. Numerical results - Table1 The results for the function: y=exp(-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

  14. Numerical results – Table2 The results for the function: y=exp(-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

  15. Numerical results –Table3 The results for the function : y=exp(-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 0.000e+000 0.150 0.97775124 0.97775124 0.000e+000 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

  16. Numerical results – Table4 The results for the function : y=exp(-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

  17. Numerical results - Table5 The results for the function : y=exp(-x2)using classical Lagrange with Chebychev points fon n = 101 are: xval function values comp. value abs errors -0.450 0.81668648 -8.91078343 9.727e+000 -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 1.140e+004 0.350 0.88470590 -1057124943.27642430 1.057e+009

  18. Numerical results – Table6 The results for the function : y=exp(-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 0.000e+000

  19. Numerical results – Table7 The results for the function : y=1/(1+25x2)using Classical Lagrange with Chebychev points fon n = 101 are: xval function values comp. value abs errors -0.350 0.24615385 -653263076.28299761 6.533e+008 -0.250 0.39024390 -5003.80824613 5.004e+003 -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 4.024e+000 0.550 0.11678832 -68734.99358674 6.874e+004

  20. Numerical results – Table8 The results for the function : y=1/(1+25x2)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

  21. 5. Conclusions • Evaluating the results of the Tables 1, 2, 3 and 4 for the y=exp(-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 n=101 and Chebyshev nodes are all of the order E-16, while the errors in Table 5 (classical Lagrange, n=101, Chebyshev nodes) increase catastrophically at points outside the interval [-0.15, 0.15].

  22. The errors in Table 8 with n=101 and Chebyshev nodes are all of the order E-10, E-09, while the errors in Table 7 (classical Lagrange, n=101, 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 (n+1 the number of points) and thus it should be used in all applications where Lagrange interpolation is used (Approximation theory, differential equations etc).

  23. 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.

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