1.5 Hermite interpolation

Hermite interpolation 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 Charles Hermite, 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 Lagrange interpolation, 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 $n$ interpolation points and derivatives up to order $m$ are matched at each point, the degree of the Hermite interpolating polynomial is $2mn-1$
• 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 $f$ with respect to a set of points $x_0, x_1, \ldots, x_n$ is defined recursively as:
  • $f[x_i] = f(x_i)$ for $i=0,1,\ldots,n$
  • $f[x_i, x_{i+1}, \ldots, x_{i+k}] = \frac{f[x_{i+1}, \ldots, x_{i+k}] - f[x_i, \ldots, x_{i+k-1}]}{x_{i+k} - x_i}$ for $k=1,2,\ldots,n$
• 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 smoothness 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 $(2mn)$-th derivative of the function, where $m$ 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 convergence of Hermite interpolation

Bound on interpolation error

• A bound on the interpolation error can be obtained from the remainder term by estimating the $(2mn)$-th derivative of the function
• The bound depends on the spacing of the interpolation points and the maximum value of the $(2mn)$-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 numerical analysis 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