In some applications it may be important that the extrema of the interpolating function are within the extrema of the given data.

Transcription

1 Shape-preserving piecewise poly. interpolation In some applications it may be important that the extrema of the interpolating function are within the extrema of the given data. For example: If you the data represents a concentration then you may want an interpolant that can only take values between and. Also, one may one to preserve the shape of the data, which usually translates into meaning that the interpolating function should not introduce any local maxima or minima. Piecewise linear interpolation satisfies the above criteria but it does not result in a smooth interpolating function. So, we instead look for a piecewise cubic interpolating function. The resulting method is called PCHIP, which stands for piecewise cubic Hermite interpolating polynomial. This is not a great name.

2 For each interval [x k,x k+ ], k =,...,n, fit a cubic Hermite polynomial to the data H k (x) =b k (x x k ) 3 + c k (x x k ) 2 + d k (x x k )+e k = x k+ k = f k+ x k f k c k = 3 k 2d k d k+ b k = d k 2 k + d k+ h 2 k e k = f k d k = f k

3 For each interval [x k,x k+ ], k =,...,n, fit a cubic Hermite polynomial to the data = x k+ k = f k+ x k f k H k (x) =b k (x x k ) 3 + c k (x x k ) 2 + d k (x x k )+e k c k = 3 k 2d k d k+ b k = d k 2 k + d k+ h 2 k e k = f k d k = f k But, we are not given the derivatives

4 Idea: Start with the piecewise linear interpolant and approximate the derivative at each data point by averaging the slopes of piecewise linear interpolant from the left and right of the point. <latexit sha_base64="br3zudncye6vbywyrksehioaj7s=">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</latexit> <latexit sha_base64="br3zudncye6vbywyrksehioaj7s=">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</latexit> <latexit sha_base64="br3zudncye6vbywyrksehioaj7s=">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</latexit> <latexit sha_base64="br3zudncye6vbywyrksehioaj7s=">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</latexit> How do we average the slopes?

5 Two ideas: Weighted arithmetic averaging: d k = w k k + w k k w k + w k Weighted harmonic averaging: d k = w k w k k + w k + w k k w k =2 + and w k = +2

6 Two ideas: Weighted arithmetic averaging: d k = w k k + w k k w k + w k Gives a shape preserving interpolant Weighted harmonic averaging: d k = w k w k k + w k + w k k w k =2 + and w k = +2

7 <latexit sha_base64="55neak4vo4+xwmpswxfo9aw3f7e=">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</latexit> <latexit sha_base64="55neak4vo4+xwmpswxfo9aw3f7e=">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</latexit> <latexit sha_base64="55neak4vo4+xwmpswxfo9aw3f7e=">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</latexit> <latexit sha_base64="55neak4vo4+xwmpswxfo9aw3f7e=">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</latexit> How do we make maximum/minimum preserving? If the sign of the slopes from the left and right are different then set the derivative to zero: If k k < then set d k =