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Published byWinfred Greer Modified over 9 years ago
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Interpolation produces a function that matches the given data exactly. The function then can be utilized to approximate the data values at intermediate points.
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Interpolation may also be used to produce a smooth graph of a function for which values are known only at discrete points, either from measurements or calculations.
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Given data points Obtain a function, P(x) P(x) goes through the data points Use P(x) To estimate values at intermediate points
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Given data points: At x 0 = 2, y 0 = 3 and at x 1 = 5, y 1 = 8 Find the following: At x = 4, y = ?
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P(x) should satisfy the following conditions: P(x = 2) = 3 and P(x = 5) = 8 P(x) can satisfy the above conditions if at x = x 0 = 2, L 0 (x) = 1 and L 1 (x) = 0 and at x = x 1 = 5, L 0 (x) = 0 and L 1 (x) = 1
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The conditions can be satisfied if L 0 (x) and L 1 (x) are defined in the following way. At x = x 0 = 2, L 0 (x) = 1 and L 1 (x) = 0 and at x = x 1 = 5, L 0 (x) = 0 and L 1 (x) = 1
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Lagrange Interpolating Polynomial
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The Lagrange interpolating polynomial passing through three given points; (x 0, y 0 ), (x 1, y 1 ) and (x 2, y 2 ) is:
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At x 0, L 0 (x) becomes 1. At all other given data points L 0 (x) is 0.
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At x 1, L 1 (x) becomes 1. At all other given data points L 1 (x) is 0.
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At x 2, L 2 (x) becomes 1. At all other given data points L 2 (x) is 0.
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General form of the Lagrange Interpolating Polynomial
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Numerator of
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Denominator of
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Find the Lagrange Interpolating Polynomial using the three given points.
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The three given points were taken from the function
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An approximation can be obtained from the Lagrange Interpolating Polynomial as:
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Newton’s Interpolating Polynomials Newton’s equation of a function that passes through two points andis
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Set
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Newton’s equation of a function that passes through three points and is
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To find, set
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Newton’s equation of a function that passes through four points can be written by adding a fourth term.
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The fourth term will vanish at all three previous points and, therefore, leaving all three previous coefficients intact.
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Divided differences and the coefficients The divided difference of a function, with respect to is denoted as It is called as zeroth divided difference and is simply the value of the function, at
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The divided difference of a function, called as the first divided difference, is denoted with respect to and
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The divided difference of a function, called as the second divided difference, is denoted as with respect to and,
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The third divided difference with respect to, and,
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The coefficients of Newton’s interpolating polynomial are: and so on.
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Example Find Newton’s interpolating polynomial to approximate a function whose 5 data points are given below. 2.00.85467 2.30.75682 2.60.43126 2.90.22364 3.20.08567
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02.00.85467 -0.32617 12.30.75682-1.26505 -1.085202.13363 22.60.431260.65522-2.02642 -0.69207-0.29808 32.90.223640.38695 -0.45990 43.20.08567
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The 5 coefficients of the Newton’s interpolating polynomial are:
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P(x) can now be used to estimate the value of the function f(x) say at x = 2.8.
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