Runge's Phenomenon
5:53
YouTubeWhite Crane Education
Runge's Phenomenon
At first glance, the remainder formula for Lagrange polynomials seems to say that more points are better. In a lot of situations this is the case, but there are some scenarios where something called Runge's phenomenon occurs and you get an oscillating effect at the end points of an interval that makes the results from a polynomial more points ...
2.6K viewsApr 17, 2024
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Runge's phenomenon: when equally spaced nodes betray you. Increase the degree of your interpolating polynomial, and instead of converging, it explodes near the endpoints of [-1,1]. The culprit isn't the function's real-line behaviour — it's complex singularities hiding just off the axis, silently poisoning the convergence that equally spaced nodes can never achieve. Thankfully chebyshev interpolation points save the day!
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Runge's phenomenon: when equally spaced nodes betray you. Increase the degree of your interpolating polynomial, and instead of converging, it explodes near the endpoints of [-1,1]. The culprit isn't the function's real-line behaviour — it's complex singularities hiding just off the axis, silently poisoning the convergence that equally spaced nodes can never achieve. Thankfully chebyshev interpolation points save the day!
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Runge's phenomenon: when equally spaced nodes betray you. Increase the degree of your interpolating polynomial, and instead of converging, it explodes near the endpoints of [-1,1]. The culprit isn't the function's real-line behaviour — it's complex singularities hiding just off the axis, silently poisoning the convergence that equally spaced nodes can never achieve. Thankfully chebyshev interpolation points save the day!
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