POLYNOMIAL INTERPOLATION Fitting polynomial to given data points

LINEAR INTERPOLATION • The linear interpolation shown in figure previous is given by •

Can three or more data points be fitted by a curve ?

LAGRANGE INTERPOLATION • Suppose N+1 data points are given. The Lagrange interpolation formula of

• The maximum error of Lagrange interpolation is expressed in the form •

NEWTON INTERPOLATION The drawback of the Lagrange interpolation: • The amount of computation needed

DIVIDED DIFFERENCE • To evaluate a Newton interpolation formula, a forward difference table is

• Therefore, the forward difference table is given by (for third order) i

• Hence, the Newton interpolation formula is written as follows where are obtained

Application Consider the data points given in the following table i 0 1 2

• Derive the Lagrange and Newton forward interpolation fitted to the data points

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POLYNOMIAL INTERPOLATION • Fitting polynomial to given data points • Most of numerical method schemes are based on polynomial interpolation, e. g. numerical integration and differentiation.

LINEAR INTERPOLATION • The linear interpolation shown in figure previous is given by • The maximum error of the linear interpolation is expressed in the form

Can three or more data points be fitted by a curve ?

LAGRANGE INTERPOLATION • Suppose N+1 data points are given. The Lagrange interpolation formula of order N-th is written as follows

• The maximum error of Lagrange interpolation is expressed in the form • There is no guarantee that the interpolation polynomial converges to the exact function when the number of data point is increased. In general, interpolation with a large-order polynomial should be avoided or used with extreme cautions

NEWTON INTERPOLATION The drawback of the Lagrange interpolation: • The amount of computation needed for one interpolation is large • No part of the previous application can be used to interpolate another value of x • When the number of data points has to be increased or decreased, the results of the previous computations cannot be used • Evaluation of error is not easy

DIVIDED DIFFERENCE • To evaluate a Newton interpolation formula, a forward difference table is necessary

• Therefore, the forward difference table is given by (for third order) i 0 1 2 3

• Hence, the Newton interpolation formula is written as follows where are obtained from forward difference table • The maximum error of Newton interpolation is in the form

Application Consider the data points given in the following table i 0 1 2 3 4 5 0. 1 0. 2 0. 3 0. 5 0. 7 0. 9975 0. 9776 0. 9384 0. 8812 0. 8075 0. 7196

• Derive the Lagrange and Newton forward interpolation fitted to the data points at a. i = 0, 1, 2 (evaluate for x = 0. 21) b. i = 1, 2, 3 (evaluate for x = 0. 21) c. i = 1, 2, 3, 4 (evaluate for x = 0. 21) • Estimate the maximum error for every evaluate of x