➗Approximation Theory Unit 5 – Padé approximation

What's Padé Approximation?

• Rational function approximation technique that approximates a function by a ratio of two polynomials
• Approximates a function $f(x)$ by a rational function $R(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials
• Polynomials $P(x)$ and $Q(x)$ are constructed so that the Taylor series expansion of $R(x)$ matches the Taylor series of $f(x)$ up to a certain order
• Provides a local approximation of a function near a specific point (usually around $x=0$)
• Often gives better approximations than Taylor series for functions with poles or singularities
• Can be used to approximate functions with known Taylor series expansions or data points
• Particularly useful for approximating functions with asymptotic behavior or those that are difficult to evaluate directly

Why It's Useful

• Provides a compact and efficient representation of a function using rational functions
• Often requires fewer terms than Taylor series to achieve a desired level of accuracy
• Can accurately capture the behavior of functions with poles, singularities, or other complex features
• Useful for approximating functions with known Taylor series expansions or data points
• Enables the development of efficient numerical algorithms for function evaluation and integration
• Finds applications in various fields, including numerical analysis, physics, engineering, and signal processing
• Can be used to solve differential equations, integral equations, and other mathematical problems

The Math Behind It

• Constructs a rational function $R(x) = \frac{P(x)}{Q(x)}$ that approximates a given function $f(x)$
• $P(x)$ and $Q(x)$ are polynomials of degree $m$ and $n$, respectively: $P(x) = \sum_{i=0}^{m} a_i x^i$ and $Q(x) = \sum_{j=0}^{n} b_j x^j$
• Coefficients $a_i$ and $b_j$ are determined by matching the Taylor series expansion of $R(x)$ with that of $f(x)$ up to order $m+n$
• Leads to a system of linear equations that can be solved for the coefficients $a_i$ and $b_j$
  • The system of equations is obtained by equating the coefficients of like powers of $x$ in the Taylor series expansions
• Normalizes the approximant by setting $Q(0) = 1$ to ensure uniqueness
• The choice of $m$ and $n$ determines the accuracy and complexity of the approximation
  • Higher values of $m$ and $n$ generally lead to better approximations but also increase the computational cost

How to Construct a Padé Approximant

• Start with a function $f(x)$ that you want to approximate
• Choose the degrees $m$ and $n$ of the polynomials $P(x)$ and $Q(x)$, respectively
• Expand $f(x)$ in a Taylor series around a chosen point (usually $x=0$): $f(x) = \sum_{k=0}^{\infty} c_k x^k$
• Equate the coefficients of the Taylor series expansion of $R(x) = \frac{P(x)}{Q(x)}$ with those of $f(x)$ up to order $m+n$
• Solve the resulting system of linear equations for the coefficients $a_i$ and $b_j$
  • Can be done using various methods, such as Gaussian elimination or LU decomposition
• Normalize the approximant by setting $Q(0) = 1$ to ensure uniqueness
• The resulting rational function $R(x) = \frac{P(x)}{Q(x)}$ is the Padé approximant of $f(x)$ of order $[m/n]$

Common Applications

• Approximating special functions, such as the exponential, logarithm, and trigonometric functions
• Modeling physical systems and phenomena in engineering and physics (electrical circuits, fluid dynamics)
• Developing numerical methods for solving differential equations and integral equations
• Approximating transfer functions in control theory and signal processing
• Extrapolating and interpolating data points in numerical analysis and data fitting
• Accelerating the convergence of series expansions and improving their accuracy
• Approximating functions with branch cuts, poles, or other singularities that are difficult to handle with other methods

Pros and Cons

• Pros:
  • Often provides better approximations than Taylor series, especially for functions with poles or singularities
  • Requires fewer terms than Taylor series to achieve a desired level of accuracy
  • Can accurately capture the behavior of functions with complex features
  • Useful for approximating functions with known Taylor series expansions or data points
  • Enables the development of efficient numerical algorithms
• Cons:
  • The choice of the degrees $m$ and $n$ can significantly affect the accuracy and complexity of the approximation
  • Finding the optimal values of $m$ and $n$ can be challenging and may require trial and error
  • The approximant may have poles that are not present in the original function, leading to spurious singularities
  • The approximation is local and may not be accurate far from the expansion point
  • The construction of Padé approximants can be computationally expensive for high-degree polynomials

Comparing to Other Approximations

• Taylor series:
  • Padé approximants often provide better approximations than Taylor series, especially for functions with poles or singularities
  • Padé approximants typically require fewer terms than Taylor series to achieve a desired level of accuracy
• Chebyshev approximation:
  • Chebyshev approximation minimizes the maximum error over an interval, while Padé approximation focuses on local accuracy near a specific point
  • Chebyshev approximation is particularly useful for approximating functions over a wide range of values
• Least-squares approximation:
  • Least-squares approximation minimizes the sum of squared errors between the approximation and the original function
  • Padé approximation focuses on matching the derivatives of the function at a specific point
• Spline interpolation:
  • Spline interpolation uses piecewise polynomial functions to interpolate data points
  • Padé approximation uses a single rational function to approximate the function locally

Advanced Topics and Extensions

• Multipoint Padé approximation: Constructs a rational function that matches the function and its derivatives at multiple points
• Padé-Hermite approximation: Incorporates derivative information at the expansion point to improve accuracy
• Padé-Chebyshev approximation: Combines the ideas of Padé approximation and Chebyshev approximation to minimize the maximum error over an interval
• Rational Chebyshev approximation: Finds the best rational function approximation that minimizes the maximum error over an interval
• Padé-Laplace approximation: Applies Padé approximation to the Laplace transform of a function to solve differential equations
• Padé-Borel summation: Uses Padé approximation to improve the convergence of divergent series and to sum asymptotic expansions
• Padé-based model reduction: Reduces the order of high-dimensional systems using Padé approximation to preserve essential properties