Glossary

1.5 Hermite interpolation

7 min read•august 20, 2024

is a powerful method that uses both function values and derivatives to create a unique polynomial. It's especially useful when we know or can easily calculate derivatives, offering more accurate results than simpler interpolation techniques.

This method fits into the broader context of polynomial interpolation, which constructs functions passing through given data points. Hermite interpolation stands out by matching both function values and derivatives, providing a more precise representation of the original function.

Definition of Hermite interpolation

  • Hermite interpolation is a method of polynomial interpolation that uses both function values and derivatives at the interpolation points
  • It is named after , a French mathematician who developed the method in the late 19th century
  • Hermite interpolation is particularly useful when the derivatives of the function are known or can be easily computed

Interpolation vs approximation

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  • Interpolation constructs a function that passes exactly through a set of given data points
  • Approximation finds a function that is close to the given data points but may not pass through them exactly
  • Hermite interpolation is a form of interpolation that matches both function values and derivatives at the interpolation points

Polynomial interpolation

  • Polynomial interpolation constructs a polynomial function that passes through a set of given data points
  • The interpolating polynomial is unique for a given set of distinct interpolation points
  • Polynomial interpolation can be performed using various methods such as , Newton's divided differences, and Hermite interpolation

Hermite interpolation conditions

  • Hermite interpolation requires the interpolating polynomial to match the function values at the interpolation points
  • It also requires the interpolating polynomial to match the derivatives of the function up to a certain order at the interpolation points
  • The number of conditions imposed by Hermite interpolation is twice the number of interpolation points (one for the function value and one for the derivative at each point)

Hermite interpolating polynomial

  • The Hermite interpolating polynomial is a polynomial function that satisfies the Hermite interpolation conditions
  • It is unique for a given set of distinct interpolation points and the corresponding function values and derivatives
  • The Hermite interpolating polynomial can be constructed using various methods such as the Hermite basis functions or the Newton-Hermite formula

Existence and uniqueness

  • The Hermite interpolating polynomial exists and is unique for a given set of distinct interpolation points and the corresponding function values and derivatives
  • The existence and uniqueness of the Hermite interpolating polynomial can be proved using the Hermite interpolation conditions and the properties of polynomials
  • The uniqueness of the Hermite interpolating polynomial implies that it is independent of the method used to construct it

Degree of Hermite polynomial

  • The degree of the Hermite interpolating polynomial depends on the number of interpolation points and the order of derivatives matched at each point
  • If there are nn interpolation points and derivatives up to order mm are matched at each point, the degree of the Hermite interpolating polynomial is 2mn−12mn-1
  • The high degree of the Hermite interpolating polynomial can lead to oscillations and numerical instability, especially when the number of interpolation points is large

Construction of Hermite polynomial

  • The Hermite interpolating polynomial can be constructed using the Hermite basis functions, which are polynomials that satisfy the Hermite interpolation conditions
  • Another method to construct the Hermite interpolating polynomial is the Newton-Hermite formula, which uses divided differences to compute the coefficients of the polynomial
  • The choice of the method to construct the Hermite interpolating polynomial depends on the specific problem and the available data

Divided differences in Hermite interpolation

  • Divided differences are a recursive method to compute the coefficients of the interpolating polynomial
  • They are particularly useful in Hermite interpolation because they can handle repeated interpolation points and derivatives
  • Divided differences can also be used to estimate the error in Hermite interpolation

Definition of divided differences

  • The divided difference of a function ff with respect to a set of points x0,x1,…,xnx_0, x_1, \ldots, x_n is defined recursively as:
    • f[xi]=f(xi)f[x_i] = f(x_i) for i=0,1,…,ni=0,1,\ldots,n
    • f[xi,xi+1,…,xi+k]=f[xi+1,…,xi+k]−f[xi,…,xi+k−1]xi+k−xif[x_i, x_{i+1}, \ldots, x_{i+k}] = \frac{f[x_{i+1}, \ldots, x_{i+k}] - f[x_i, \ldots, x_{i+k-1}]}{x_{i+k} - x_i} for k=1,2,…,nk=1,2,\ldots,n
  • Divided differences can be arranged in a tabular form called the divided difference table

Recursive formula for divided differences

  • The recursive formula for divided differences allows to compute higher-order divided differences from lower-order ones
  • It is particularly useful when the interpolation points are not evenly spaced or when there are repeated points
  • The recursive formula for divided differences can be derived from the definition of divided differences and the properties of polynomials

Properties of divided differences

  • Divided differences are symmetric with respect to the order of the interpolation points
  • They satisfy a Leibniz-like rule for the product of two functions
  • Divided differences can be used to estimate the derivatives of a function at the interpolation points
  • They are related to the coefficients of the Newton form of the interpolating polynomial

Error in Hermite interpolation

  • The error in Hermite interpolation is the difference between the interpolating polynomial and the actual function
  • It can be estimated using the remainder term of the Hermite interpolation formula
  • The error in Hermite interpolation depends on the of the function and the spacing of the interpolation points

Remainder term for Hermite interpolation

  • The remainder term for Hermite interpolation is a formula that expresses the error in terms of the (2mn)(2mn)-th derivative of the function, where mm is the order of the highest derivative matched at each interpolation point
  • It is similar to the remainder term for Taylor series expansion, but it involves the divided differences of the function
  • The remainder term can be used to derive bounds on the interpolation error and to study the of Hermite interpolation

Bound on interpolation error

  • A bound on the interpolation error can be obtained from the remainder term by estimating the (2mn)(2mn)-th derivative of the function
  • The bound depends on the spacing of the interpolation points and the maximum value of the (2mn)(2mn)-th derivative of the function in the interpolation interval
  • The bound can be used to choose the interpolation points and the order of derivatives matched at each point to achieve a desired level of accuracy

Convergence of Hermite interpolation

  • Hermite interpolation converges to the actual function as the number of interpolation points increases and the spacing between them decreases
  • The rate of convergence depends on the smoothness of the function and the order of derivatives matched at each point
  • In general, matching higher-order derivatives at each point leads to faster convergence, but it also increases the computational cost and the risk of numerical instability

Applications of Hermite interpolation

  • Hermite interpolation has various applications in and scientific computing
  • It can be used to approximate functions, compute derivatives and integrals, and solve differential equations
  • Hermite interpolation is particularly useful when the function is known only at a few points and its derivatives are available or can be estimated

Numerical differentiation

  • Hermite interpolation can be used to approximate the derivatives of a function from its values and derivatives at a few points
  • The approximation is based on the derivative of the Hermite interpolating polynomial
  • Numerical differentiation using Hermite interpolation is more accurate than finite difference methods, especially when the function is not smooth or the data points are not evenly spaced

Numerical integration

  • Hermite interpolation can be used to approximate the integral of a function from its values and derivatives at a few points
  • The approximation is based on the integral of the Hermite interpolating polynomial
  • Numerical integration using Hermite interpolation is more accurate than Newton-Cotes formulas, especially when the function is not smooth or the data points are not evenly spaced

Solving initial value problems

  • Hermite interpolation can be used to solve initial value problems for ordinary differential equations
  • The idea is to interpolate the solution and its derivative at a few points and use the interpolating polynomial to approximate the solution at other points
  • Hermite interpolation can be combined with numerical integration methods (Runge-Kutta) to obtain a high-order method for solving initial value problems

Comparison of interpolation methods

  • Hermite interpolation can be compared with other interpolation methods such as Lagrange interpolation and cubic spline interpolation
  • The choice of the interpolation method depends on the specific problem, the available data, and the desired properties of the interpolating function
  • In general, Hermite interpolation is more accurate than Lagrange interpolation but less smooth than cubic spline interpolation

Lagrange vs Hermite interpolation

  • Lagrange interpolation uses only the function values at the interpolation points, while Hermite interpolation uses both the function values and the derivatives
  • Lagrange interpolation has a lower degree than Hermite interpolation for the same number of interpolation points
  • Hermite interpolation is more accurate than Lagrange interpolation, especially when the function is not smooth or the data points are not evenly spaced

Cubic spline vs Hermite interpolation

  • Cubic spline interpolation constructs a piecewise cubic polynomial that is continuous and has continuous first and second derivatives at the interpolation points
  • Hermite interpolation constructs a single polynomial that matches the function values and derivatives at the interpolation points
  • Cubic spline interpolation produces a smoother interpolating function than Hermite interpolation, but it requires solving a system of linear equations

Piecewise Hermite interpolation

  • Piecewise Hermite interpolation combines the ideas of Hermite interpolation and piecewise polynomial interpolation
  • The idea is to divide the interpolation interval into subintervals and construct a Hermite interpolating polynomial on each subinterval
  • Piecewise Hermite interpolation can achieve a higher accuracy than global Hermite interpolation while maintaining the smoothness of the interpolating function
  • It is particularly useful when the function has different behavior in different regions of the interpolation interval

Key Terms to Review (18)

Charles Hermite: Charles Hermite was a French mathematician known for his significant contributions to various branches of mathematics, particularly in approximation theory and interpolation methods. His work laid the foundation for Hermite interpolation, which involves constructing polynomials that match both function values and derivatives at given data points. This technique is crucial for ensuring that approximated functions maintain the same behavior as the original function around those points.
Chebyshev nodes: Chebyshev nodes are specific points used in polynomial interpolation, chosen to minimize the error in approximation. They are the roots of Chebyshev polynomials, which are optimal for reducing oscillations and improving the accuracy of polynomial interpolations compared to evenly spaced points. This makes them particularly useful for various types of interpolation techniques.
Computer Graphics: Computer graphics refers to the creation, manipulation, and representation of visual images using computers. It plays a vital role in various fields like video games, simulations, and digital art, helping to visualize complex data and create immersive experiences. Techniques in computer graphics rely heavily on mathematical principles, making it essential for the representation and rendering of curves and surfaces.
Continuity: Continuity refers to the property of a function where small changes in the input lead to small changes in the output. In approximation theory, continuity is crucial because it ensures that approximating functions behave predictably and smoothly, making them suitable for tasks such as interpolation and geometric modeling.
Convergence: Convergence refers to the process of a sequence or function approaching a limit or a desired value as the number of iterations or data points increases. This concept is critical across various approximation methods, as it indicates how closely an approximation represents the true function or value being estimated, thereby establishing the reliability and effectiveness of the approximation techniques used.
Cubic Hermite Spline: A cubic Hermite spline is a piecewise-defined curve that uses cubic polynomials to interpolate a set of points while also ensuring control over the tangents at those points. This type of spline is especially useful for creating smooth curves in graphics and animation, as it allows for precise manipulation of the slope at each interpolation point, resulting in a visually appealing transition between segments.
Degree of Approximation: The degree of approximation refers to a measure of how well a function can be approximated by a simpler function, often using polynomials or other types of interpolants. It quantifies the accuracy of the approximation and indicates the closeness of the approximation to the original function over a specified interval. This concept is critical in understanding how well different interpolation methods, such as polynomial interpolation and Bernstein polynomials, can represent functions in numerical analysis.
Derivative matching: Derivative matching is a technique used in interpolation methods, especially in Hermite interpolation, where not only the function values at given points are matched, but also the derivatives at those points. This method allows for a smoother and more accurate representation of functions by ensuring that both the function and its derivative values align at specified nodes, resulting in better approximation properties compared to simpler interpolation techniques.
Error Analysis: Error analysis is the process of quantifying the difference between an approximate solution and the exact solution in mathematical computations. It helps identify the sources of errors, allowing for improvements in approximation methods and techniques. This analysis is crucial for understanding the reliability and accuracy of various approximation strategies used across different mathematical applications.
F^(k)(x): The notation f^(k)(x) represents the k-th derivative of a function f at the point x. This means it captures how the function changes at that specific point through its derivatives, which are key to understanding the behavior of functions, particularly in interpolation contexts where derivatives provide crucial information for constructing approximating polynomials.
H_n(x): In the context of Hermite interpolation, $h_n(x)$ represents the Hermite interpolating polynomial of degree at most $n$. This polynomial not only approximates a function at given points but also ensures that both the function values and some of their derivatives match at those points. The construction of $h_n(x)$ is crucial for achieving a higher accuracy in approximation compared to simple polynomial interpolation, particularly when dealing with functions that have known derivative values at interpolation nodes.
Hermite Interpolation: Hermite interpolation is a method of constructing a polynomial that not only passes through given data points but also matches specified derivative values at those points. This technique allows for more accurate approximations than standard polynomial interpolation by ensuring that both the function's value and its slope are taken into account, which is particularly useful in applications requiring smoothness and continuity.
Interpolation Polynomial: An interpolation polynomial is a polynomial function that passes through a given set of points, providing an estimate or approximation of the function's values at those points. It is widely used in numerical analysis to construct functions that closely approximate other functions based on discrete data, making it essential for various applications like curve fitting and numerical integration.
Joseph-Louis Lagrange: Joseph-Louis Lagrange was an influential mathematician and astronomer in the 18th century, known for his contributions to various fields, including calculus, number theory, and approximation theory. He developed important concepts such as Lagrange interpolation, which forms the foundation of Hermite interpolation, and made significant advancements in the use of continued fractions for approximating functions and solving equations. His work laid the groundwork for future developments in these mathematical areas.
Lagrange Interpolation: Lagrange interpolation is a polynomial interpolation method that allows the construction of a polynomial that passes through a given set of data points. This technique is particularly useful for estimating values and constructing functions based on discrete data, connecting closely to concepts like Hermite interpolation, which enhances the polynomial fit by incorporating derivative information, and Padé approximation, which provides rational function approximations for better convergence properties.
Newton's Divided Difference: Newton's divided difference is a method used to compute the coefficients of a polynomial interpolation based on a set of given data points. This technique allows for the construction of the Newton interpolation polynomial, which can efficiently approximate functions and handle new data points with relative ease. The divided difference is especially useful in Hermite interpolation, where it aids in maintaining the values of both the function and its derivatives at specified points.
Numerical Analysis: Numerical analysis is the study of algorithms that use numerical approximation to solve mathematical problems. It focuses on finding approximate solutions to complex equations that cannot be solved analytically, providing methods to analyze the accuracy and efficiency of these approximations. This field is essential in applied mathematics and connects deeply with concepts like polynomial approximation and interpolation methods, helping to address practical problems in engineering, physics, and other scientific domains.
Smoothness: Smoothness refers to the property of a function or curve that indicates how continuously it can be represented, particularly in terms of derivatives. In approximation theory, smoothness is crucial as it impacts the quality of interpolation and approximation, affecting how well a function can be modeled by simpler functions or curves. A smoother function will generally lead to better approximations and more visually appealing curves, which is especially important in applications like computer graphics and geometric modeling.
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