Interpolating Accuracy without underlying f (x)
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1 Example: Tabulated Data The following table x f (x) lists values of a function f at various points. The approximations to f (1.5) obtained by various Lagrange polynomials that use this data will be compared to try and determine the accuracy of the approximation. Numerical Analysis (Chapter 3) Data Approximation & Neville s Method R L Burden & J D Faires 5 / 46
2 x f (x) Solution (1/7) Numerical Analysis (Chapter 3) Data Approximation & Neville s Method R L Burden & J D Faires 6 / 46
3 x f (x) Solution (1/7) The most appropriate linear polynomial uses x 0 = 1.3 and x 1 = 1.6 because 1.5 is between 1.3 and 1.6. Numerical Analysis (Chapter 3) Data Approximation & Neville s Method R L Burden & J D Faires 6 / 46
4 x f (x) Solution (1/7) The most appropriate linear polynomial uses x 0 = 1.3 and x 1 = 1.6 because 1.5 is between 1.3 and 1.6. The value of the interpolating polynomial at 1.5 is P 1 (1.5) = ( ) 1.3) f (1.3)+(1.5 ( ) ( ) f (1.6) Numerical Analysis (Chapter 3) Data Approximation & Neville s Method R L Burden & J D Faires 6 / 46
5 x f (x) Solution (1/7) The most appropriate linear polynomial uses x 0 = 1.3 and x 1 = 1.6 because 1.5 is between 1.3 and 1.6. The value of the interpolating polynomial at 1.5 is P 1 (1.5) = = ( ) 1.3) f (1.3)+(1.5 ( ) ( ) f (1.6) ( ) 1.3) ( )+(1.5 ( ) ( ) ( ) Numerical Analysis (Chapter 3) Data Approximation & Neville s Method R L Burden & J D Faires 6 / 46
6 x f (x) Solution (1/7) The most appropriate linear polynomial uses x 0 = 1.3 and x 1 = 1.6 because 1.5 is between 1.3 and 1.6. The value of the interpolating polynomial at 1.5 is P 1 (1.5) = ( ) ( ) = ( ) ( ) = f (1.3)+( ) ( ) f (1.6) ( )+( ) ( ) ( ) Numerical Analysis (Chapter 3) Data Approximation & Neville s Method R L Burden & J D Faires 6 / 46
7 Solution (2/7) Numerical Analysis (Chapter 3) Data Approximation & Neville s Method R L Burden & J D Faires 7 / 46
8 Solution (2/7) Two polynomials of degree 2 can reasonably be used, one with x 0 = 1.3, x 1 = 1.6 and x 2 = 1.9, which gives P 2 (1.5) = ( )( ) ( )( ) ( ) Numerical Analysis (Chapter 3) Data Approximation & Neville s Method R L Burden & J D Faires 7 / 46
9 Solution (2/7) Two polynomials of degree 2 can reasonably be used, one with x 0 = 1.3, x 1 = 1.6 and x 2 = 1.9, which gives P 2 (1.5) = ( )( ) ( )( ) ( ) ( )( ) + ( )( ) ( ) Numerical Analysis (Chapter 3) Data Approximation & Neville s Method R L Burden & J D Faires 7 / 46
10 Solution (2/7) Two polynomials of degree 2 can reasonably be used, one with x 0 = 1.3, x 1 = 1.6 and x 2 = 1.9, which gives P 2 (1.5) = ( )( ) ( )( ) ( ) ( )( ) + ( )( ) ( ) ( )( ) + ( ) = ( )( ) and one with x 0 = 1.0, x 1 = 1.3, and x 2 = 1.6, which gives ˆP 2 (1.5) = Numerical Analysis (Chapter 3) Data Approximation & Neville s Method R L Burden & J D Faires 7 / 46
11 x f (x) Solution (3/7) Numerical Analysis (Chapter 3) Data Approximation & Neville s Method R L Burden & J D Faires 8 / 46
12 x f (x) Solution (3/7) In the third-degree case, there are also two reasonable choices for the polynomial. Numerical Analysis (Chapter 3) Data Approximation & Neville s Method R L Burden & J D Faires 8 / 46
13 x f (x) Solution (3/7) In the third-degree case, there are also two reasonable choices for the polynomial. One with x 0 = 1.3, x 1 = 1.6, x 2 = 1.9 and x 3 = 2.2, which gives P 3 (1.5) = Numerical Analysis (Chapter 3) Data Approximation & Neville s Method R L Burden & J D Faires 8 / 46
14 x f (x) Solution (3/7) In the third-degree case, there are also two reasonable choices for the polynomial. One with x 0 = 1.3, x 1 = 1.6, x 2 = 1.9 and x 3 = 2.2, which gives P 3 (1.5) = The second third-degree approximation is obtained with x 0 = 1.0, x 1 = 1.3, x 2 = 1.6 and x 3 = 1.9, which gives ˆP 3 (1.5) = Numerical Analysis (Chapter 3) Data Approximation & Neville s Method R L Burden & J D Faires 8 / 46
15 x f (x) Solution (4/7) The fourth-degree Lagrange polynomial uses all the entries in the table. Numerical Analysis (Chapter 3) Data Approximation & Neville s Method R L Burden & J D Faires 9 / 46
16 x f (x) Solution (4/7) The fourth-degree Lagrange polynomial uses all the entries in the table. With x 0 = 1.0, x 1 = 1.3, x 2 = 1.6, x 3 = 1.9, and x 4 = 2.2, the approximation is P 4 (1.5) = Numerical Analysis (Chapter 3) Data Approximation & Neville s Method R L Burden & J D Faires 9 / 46
17 P 3 (1.5) ˆP3 (1.5) P 4 (1.5) Solution (5/7) Numerical Analysis (Chapter 3) Data Approximation & Neville s Method R L Burden & J D Faires 10 / 46
18 P 3 (1.5) ˆP3 (1.5) P 4 (1.5) Solution (5/7) Because P 3 (1.5), ˆP 3 (1.5), and P 4 (1.5) all agree to within units, we expect this degree of accuracy for these approximations. Numerical Analysis (Chapter 3) Data Approximation & Neville s Method R L Burden & J D Faires 10 / 46
19 P 3 (1.5) ˆP3 (1.5) P 4 (1.5) Solution (5/7) Because P 3 (1.5), ˆP 3 (1.5), and P 4 (1.5) all agree to within units, we expect this degree of accuracy for these approximations. We also expect P 4 (1.5) to be the most accurate approximation, since it uses more of the given data. Numerical Analysis (Chapter 3) Data Approximation & Neville s Method R L Burden & J D Faires 10 / 46
20 P 3 (1.5) ˆP3 (1.5) P 4 (1.5) Solution (5/7) Because P 3 (1.5), ˆP 3 (1.5), and P 4 (1.5) all agree to within units, we expect this degree of accuracy for these approximations. We also expect P 4 (1.5) to be the most accurate approximation, since it uses more of the given data. The function we are approximating is actually the Bessel function of the first kind of order zero, whose value at 1.5 is known to be Numerical Analysis (Chapter 3) Data Approximation & Neville s Method R L Burden & J D Faires 10 / 46
21 Solution (6/7) Therefore, the true accuracies of the approximations are as follows: P 1 (1.5) f (1.5) P 2 (1.5) f (1.5) ˆP 2 (1.5) f (1.5) P 3 (1.5) f (1.5) ˆP 3 (1.5) f (1.5) P 4 (1.5) f (1.5) Numerical Analysis (Chapter 3) Data Approximation & Neville s Method R L Burden & J D Faires 11 / 46
22 Concluding Remarks (7/7)
23 Concluding Remarks (7/7) Although P 3 (1.5) is the most accurate approximation, if we had no knowledge of the actual value of f (1.5), we would accept P 4 (1.5) as the best approximation since it includes the most data about the function.
24 Concluding Remarks (7/7) Although P 3 (1.5) is the most accurate approximation, if we had no knowledge of the actual value of f (1.5), we would accept P 4 (1.5) as the best approximation since it includes the most data about the function. The theoretical Lagrange error term Theorem cannot be applied here because we have no knowledge of the fourth derivative of f.
25 Concluding Remarks (7/7) Although P 3 (1.5) is the most accurate approximation, if we had no knowledge of the actual value of f (1.5), we would accept P 4 (1.5) as the best approximation since it includes the most data about the function. The theoretical Lagrange error term Theorem cannot be applied here because we have no knowledge of the fourth derivative of f. Unfortunately, this is generally the case.
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