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- [/
- Copyright (c) 2019 Nick Thompson
- Use, modification and distribution are subject to the
- Boost Software License, Version 1.0. (See accompanying file
- LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
- ]
- [section:whittaker_shannon Whittaker-Shannon interpolation]
- [heading Synopsis]
- ``
- #include <boost/math/interpolators/whittaker_shannon.hpp>
- ``
- namespace boost { namespace math { namespace interpolators {
- template <class RandomAccessContainer>
- class whittaker_shannon
- {
- public:
- using Real = RandomAccessContainer::value_type;
- whittaker_shannon(RandomAccessContainer&& v, Real left_endpoint, Real step_size);
- Real operator()(Real x) const;
- Real prime(Real x) const;
- };
- }}} // namespaces
- [heading Whittaker-Shannon Interpolation]
- The Whittaker-Shannon interpolator takes equispaced data and interpolates between them via a sum of sinc functions.
- This interpolation is stable and infinitely smooth, but has linear complexity in the data,
- making it slow relative to compactly-supported b-splines.
- In addition, we cannot pass an infinite amount of data into the class,
- and must truncate the (perhaps) infinite sinc series to a finite number of terms.
- Since the sinc function has slow /1/x/ decay, the truncation of the series can incur large error.
- Hence this interpolator works best when operating on samples of compactly-supported functions.
- Here is an example of interpolating a smooth "bump function":
- auto bump = [](double x) { if (std::abs(x) >= 1) { return 0.0; } return std::exp(-1.0/(1.0-x*x)); };
- double t0 = -1;
- size_t n = 2049;
- double h = 2.0/(n-1.0);
- std::vector<double> v(n);
- for(size_t i = 0; i < n; ++i) {
- double t = t0 + i*h;
- v[i] = bump(t);
- }
- auto ws = whittaker_shannon(std::move(v), t0, h);
- double y = ws(0.3);
- The derivative of the interpolant can also be evaluated, but the accuracy is not as high:
- double yp = ws.prime(0.3);
- [heading Complexity and Performance]
- The call to the constructor requires [bigo](1) operations, simply moving data into the class.
- Each call to the interpolant is [bigo](/n/), where /n/ is the number of points to interpolate.
- [endsect] [/section:whittaker_shannon]
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