Example of Newtons Interpolating Polynomials

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Example of Newtons Interpolating Polynomials Dr.
Ferri
Try to interpolate the value of cos(0.3) using
computed data points
xi
fxi,xi1
fxi
fxi,xi1,xi2
fxi,xi3
fx0,x4
i
1.0000e000 -9.9667e-002 -4.8840e-001
4.9008e-002 3.8122e-002 9.8007e-001
-2.9503e-001 -4.5900e-001 7.9506e-002
0 9.2106e-001 -4.7863e-001 -4.1129e-001
0 0 8.2534e-001 -6.4314e-001
0 0 0 6.9671e-001
0 0 0 0
0.0 0.2 0.4 0.6 0.8
0 1 2 3 4
f x1 f x0
f x0 , x1
x1 x0
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Example of Newtons Interpolating Polynomials
Try to interpolate the value of cos(0.3) using
computed data points
xi
fxi,xi1
fxi
fxi,xi1,xi2
fxi,xi3
fx0,x4
i
1.0000e000 -9.9667e-002 -4.8840e-001
4.9008e-002 3.8122e-002 9.8007e-001
-2.9503e-001 -4.5900e-001 7.9506e-002
0 9.2106e-001 -4.7863e-001 -4.1129e-001
0 0 8.2534e-001 -6.4314e-001
0 0 0 6.9671e-001
0 0 0 0
0.0 0.2 0.4 0.6 0.8
0 1 2 3 4
f x1, x2 f x0, x1
f x0 , x1 , x2
x2 x0
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Example of Newtons Interpolating Polynomials
Try to interpolate the value of cos(0.3) using
computed data points
xi
fxi,xi1
fxi
fxi,xi1,xi2
fxi,xi3
fx0,x4
i
1.0000e000 -9.9667e-002 -4.8840e-001
4.9008e-002 3.8122e-002 9.8007e-001
-2.9503e-001 -4.5900e-001 7.9506e-002
0 9.2106e-001 -4.7863e-001 -4.1129e-001
0 0 8.2534e-001 -6.4314e-001
0 0 0 6.9671e-001
0 0 0 0
0.0 0.2 0.4 0.6 0.8
0 1 2 3 4
Terms in the top row of the table are used to
form the interpolating polynomial
P4(x) (1.0000e000) (-9.9667e-002)(x)
(-4.8840e-001)(x)(x-0.2)
(4.9008e-002)(x)(x-0.2)(x-0.4)
(3.8122e-002)(x)(x-0.2)(x-0.4)(x-0.6)
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Other Orders of Interpolating Polynomials
P0(x) (1.0000e000) P1(x) (1.0000e000)
(-9.9667e-002)(x) P2(x) (1.0000e000)
(-9.9667e-002)(x) (-4.8840e-001)(x)(x-0.2)
P3(x) (1.0000e000) (-9.9667e-002)(x)
(-4.8840e-001)(x)(x-0.2)
(4.9008e-002)(x)(x-0.2)(x-0.4) P4(x)
(1.0000e000) (-9.9667e-002)(x)
(-4.8840e-001)(x)(x-0.2)
(4.9008e-002)(x)(x-0.2)(x-0.4)
(3.8122e-002)(x)(x-0.2)(x-0.4)(x-0.6)
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Comparison Plot
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Error Analysis
Pi(x) Pi1(x) Pi(x)
exact Pi(x)
P0(0.3) 1.0000e000 P1(0.3) 9.7010e-001
P2(0.3) 9.5545e-001 P3(0.3) 9.5530e-001
P4(0.3) 9.5534e-001 Exact 9.5534e-001
-4.4664e-002 -1.4763e-002 -1.1132e-004
3.5706e-005 1.3957e-006
-2.9900e-002 -1.4652e-002 -1.4702e-004
3.4310e-005
See that there is a general agreement between the
magnitude of the interpolation error and the
difference between the predictions of the
successive polynomial approximations