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Keywords
- Bernstein coefficient (1)
- Bernstein coefficients (1)
- Bernstein polynomial (4)
- Bernstein polynomials (1)
- Cauchon algorithm (2)
- Checkerboard ordering (2)
- Checkerboard partial ordering (1)
- Complex interval (2)
- Complex polynomial (2)
- Convex optimization (1)
- Cyclic sign variation (1)
- Enclosure of the range (1)
- Hurwitz matrix (1)
- Interval matrix (2)
- Interval property (1)
- Linear independence (1)
- Matrix interval (2)
- Multivariate complex polynomial (1)
- Multivariate polynomial (1)
- Quasi-stable polynomial (1)
- Range bounding (1)
- Range enclosure (2)
- Rank (1)
- Rational function (1)
- Sign regular matrix (1)
- Sign variation (1)
- Sign-regular matrix (1)
- Stable polynomial (1)
- Subdivision (1)
- Totally nonnegative matrix (3)
- Totally positive matrix (1)
Institute
The expansion of a given multivariate polynomial into Bernstein polynomials is considered. Matrix methods for the calculation of the Bernstein expansion of the product of two polynomials and of the Bernstein expansion of a polynomial from the expansion of one of its partial derivatives are provided which allow also a symbolic computation.
A method is investigated by which tight bounds on the range of a multivariate rational function over a box can be computed. The approach relies on the expansion of the numerator and denominator polynomials in Bernstein polynomials. Convergence of the bounds to the range with respect to degree elevation of the Bernstein expansion, to the width of the box and to subdivision are proven and the inclusion isotonicity of the related enclosure function is shown.