Data Analysis-I. Interpolation. Soon-Hyung Yook. December 4, Soon-Hyung Yook Data Analysis-I December 4, / 1

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1 Data Analysis-I Interpolation Soon-Hyung Yook December 4, 2015 Soon-Hyung Yook Data Analysis-I December 4, / 1

2 Table of Contents Soon-Hyung Yook Data Analysis-I December 4, / 1

3 Introduction Interplation Estimate the value between data x x x x x Polynomial interpolation for a set of data {(x i, y i)} no knowledge of the relationship between y and x e.g.: estimate (x i+δ, y i+δ ) from the given dataset. Weierstrass theorem: In general a continuous function f(x) in a finite interval x [a, b] can be fitted by a polynomial P (x). Find a polynomial approximation for f(x) from N pairs of numbers {(x i, f(x i))}, for i = 0, 1,, N 1. Soon-Hyung Yook Data Analysis-I December 4, / 1

4 Lagrange Interpolation Lagrange Interpolation Construct a polynomial of the form P (x) = N 1 k=0 p k (x)f(x k ) (1) Lagrange Polynomial p k (x) = (x x 0 )(x x 1 ) (x x k 1 )(x x k+1 ) (x x N 1 ) (x k x 0 )(x k x 1 ) (x k x k 1 )(x k x k+1 ) (x k x N 1 ) (2) If we use the Lagrange polynomial then it is clear that P (x i ) = f(x i ) (3) Soon-Hyung Yook Data Analysis-I December 4, / 1

5 spline (in dictionary): a piece of flexible wood or plastic that can be bent into arbitrary smooth shapes in the days before computers, it was used to trace a smooth curve between points on a sheet of graph paper. Idea: use a simple function to approximate the relation between the dependent and independent variables use the third-degree polynomial : cubic -spline f(x) = a + bx + cx 2 + dx 3 (4) To find the coefficients in each interval use the condition for the continuity across each boundary Soon-Hyung Yook Data Analysis-I December 4, / 1

6 xi+1 f(x i ) f(x i+1) xi continuity condition: f (n) i (x i ) = f (n) i+1 (x i+1) for nth derivative Soon-Hyung Yook Data Analysis-I December 4, / 1

7 Conditions for cubic spline Given a dataset {(x i, y i )} in the interval [a, b] (a = x 0 < x 1 < < x n = b), a cubic spline interpolant, f, for {y i } is a function that satisfies the following condition: (a) f is a cubic polynomial, denoted by f i, on the subinterval [x i, x i+1 ]. (b) f(x i ) = y i for each i. (c) f i+1 (x i+1 ) = f i (x i+1 ) for each i. (d) f i+1 (x i+1) = f i (x i+1) for each i. (e) f i+1 (x i+1) = f i (x i+1) for each i. (f) One of the following sets of boundary condition is frequently used: (i) f (x 0) = f (x n) for natural or free boundary (ii) f (x 0) = y (x 0) and f (x n) = y (x n) for clamped boundary * Other boundary conditions are also possible. Soon-Hyung Yook Data Analysis-I December 4, / 1

8 To construct the cubic-spline interpolant for a ginven set of data {y i }, we assume a third-order polynomial for all i. f i (x) = a i + b i (x x i ) + c i (x x i ) 2 + d i (x x i ) 3 (5) from condition (b) and Eq. (5): from condition (c): a i+1 = f i+1 (x i+1 ) = f i (x i+1 ). and if we let h i x i+1 x i then we obtain f i (x i ) = a i = y i (6) a i+1 = a i + b ih i + c ih 2 i + d ih 3 i (7) From Eq. (5) b i = f i(x i ) (8) Soon-Hyung Yook Data Analysis-I December 4, / 1

9 From condition (d) and f i (x) = b i + 2c i (x x i ) + 3d i (x x i ) 2 : b i+1 = b i + 2c i h i + 3d i h 2 i (9) From condition (e): and c i = f (x i )/2 (10) c i+1 = c i + 3d j h j (11) Soon-Hyung Yook Data Analysis-I December 4, / 1

10 Solving for d i in Eq. (11) and substituting d i into Eq. (7) and Eq. (9) a i+i = a i + b i h i + h3 i 3 (2c i + c j+1 ) (12) and b i+i = b i + h i (c i + c i+1 ) (13) Note: Eq. (8) and Eq. (10) do not give us any numerical information to determine {b i } and {c i }. We cannot determine {b i } and {c i } from Eq. (8) and Eq. (10). Therefore, we have to do more things! Soon-Hyung Yook Data Analysis-I December 4, / 1

11 The final relationship between the coefficients is obtained by first solving the appropriate equation in the form of Eq. (12) for b i. From Eq. (12) b i = 1 h (a i+1 a i ) h i 3 (2c i + c i+1 ) (14) and then, with reduction of the index (i i 1): b i 1 = 1 h (a i a i 1 ) h i 1 3 (2c i 1 + c i ) (15) Substituting Eq. (14) and Eq. (15) into Eq. (13), when the index is reduced by 1 (i i 1), we obtain the linear system of equations: h i 1 c i 1 + 2(h i 1 + h i )c i + h i c i+1 = 3 h i (a i+1 a i ) 3 h i 1 (a i a i 1 ), (16) for each i = 1, 2,, n 1. Soon-Hyung Yook Data Analysis-I December 4, / 1

12 In Eq. (16), only {c i } is a set of unknown parameters. In Eq. (16), {h i } and {a i } are given by the spacing between each data point and the value of {y i }. Once the values of {c i } is obtained, we can easily find {b i } from Eq. (14). Then we can obtain {d i } from Eq. (11). Eq. (16) is an n-coupled linear equation. solve Eq. (16) using matrix. Soon-Hyung Yook Data Analysis-I December 4, / 1