6 May, 2004 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
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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
ReferencesThe main references are:1) 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
)()(...)(0
00 xlyxlylyxpn
kkknnn
n
kjj jk
jk xx
xxxl
,0 )(
)()(
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3. Barycentric representation3.1 Lagrange type
The Lagrange type barycentric formula is:
,
Proof. Since
1
1
1
1
])(
[
])(
[
)(n
j j
j
n
j j
jj
n
xx
Axx
yA
xp
n
jkkkj
j
xxA
,1
)(
1
1
1
1)(n
jj xl
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the Lagrange interpolation polynomial can be written
Let:
So we rewrite lj as:
1
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1
1
1
)(
)(
)()(n
jn
jj
n
jjj
jjn
xl
yxl
yxlxp
n
jkkkj
j
xxA
,1
)(
1
1
,1
)()(n
jkkkjj xxAxl
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Thus
(4.1)
We divide both nominator and denominator by
1
1
1
,1
1
1
1
,1
)(
)(
)(n
j
n
jkkkj
n
j
n
jkkkjj
n
xxA
xxyA
xp
1
,1
)(n
jkkkxx
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So (4.1) became
(4.2)
So (4.2) became
1
1
1
1
1
,1
1
1
1
1
1
,1
)(
),(
)(
)(
)(
n
kk
n
j
n
jkkkj
n
kk
n
j
n
jkkk
jj
n
xx
xxA
xx
xxyA
xp
1
1
1
1
])(
[
])(
[
)(n
j j
j
n
j j
jj
n
xx
Axx
yA
xp
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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 Lebesgueconstant
is minimized under the constrains
n
k k
k
n
k k
kk
un
xxuxxyu
xr
0
0)(
)(
)()(
n
kkk
n
kkk
bxa
n
k
uk
bxa
un
xxu
xxuxl
0
0)()(
))/((
)/()( supsup
ikMum k ,
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4. Numerical results - Table1The results for the function: y=exp(-x2) using classicalLagrange with equidistant points fon n = 21are: 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 – Table2The results for the function: y=exp(-x2) using classicalLagrange with chebyshev points fon n = 21are: 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 –Table3The results for the function : y=exp(-x2) using barycentric withequidistant points for n = 21are: 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
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Numerical results – Table4The results for the function : y=exp(-x2) using barycentric withChebychev points fon n = 21are: 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 - Table5The results for the function : y=exp(-x2) using classicalLagrange with Chebychev points fon n = 101are: 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
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Numerical results – Table6The results for the function : y=exp(-x2) using barycentric withChebychev points fon n = 101are: 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
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Numerical results – Table7The results for the function : y=1/(1+25x2) using ClassicalLagrange with Chebychev points fon n = 101are: 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
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Numerical results – Table8The results for the function : y=1/(1+25x2) usingbarycentric with Chebychev points fon n = 101are: 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 they=exp(-x2) function when n = 21 we notice that the errors withthe barycentric form are a lot more accurate than thecorresponding results with the Classical form.• Also we notice that the results for Chebyshev nodesare more accurate that those with equidistant nodes asexpected.• The errors in Table 6 with n=101 and Chebyshev nodes areall of the order E-16, while the errors in Table 5 (classicalLagrange, n=101, Chebyshev nodes) increase catastrophicallyat points outside the interval [-0.15, 0.15].
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• The errors in Table 8 with n=101 and Chebyshevnodes are all of the order E-10, E-09, while the errorsin Table 7 (classical Lagrange, n=101, Chebyshevnodes) increase catastrophically at points outside theinterval [-0.15, 0.15].
• So the general conclusion is that with respect to accuracyand 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).
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6. Further researchIdeas 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.
