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Answer :
Final answer:
Piecewise cubic Hermite interpolation is a method used to approximate a smooth curve or function using a series of cubic polynomials. It ensures that the interpolated curve passes through the given data points and has continuous first derivatives at the points of division.
Explanation:
Piecewise Cubic Hermite Interpolation and its Relationship to MATLAB Routines Spline and Pchip
Piecewise cubic Hermite interpolation is a method used to approximate a smooth curve or function using a series of cubic polynomials. It is commonly used in computer graphics and numerical analysis. The interpolation is performed by dividing the domain into smaller intervals and constructing a cubic polynomial for each interval. The polynomials are chosen such that they pass through the given data points and have continuous first derivatives at the points of division. This ensures a smooth and visually pleasing curve.
In MATLAB, the routines spline and pchip are used to implement piecewise polynomial interpolation. The spline function uses a cubic spline interpolation method, which is a type of piecewise cubic Hermite interpolation. It constructs a smooth curve by fitting cubic polynomials to the given data points. The pchip function, on the other hand, uses a shape-preserving piecewise cubic Hermite interpolation method. It ensures that the interpolated curve preserves the shape of the data, avoiding overshoots or undershoots.
Both spline and pchip provide efficient and accurate methods for interpolating data using piecewise cubic polynomials. They are widely used in various fields, including engineering, physics, and computer science.
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