Rational Least Squares Approximation

The Rational Approximation Problem

Given \(f \in C[a,b]\), we seek a rational function \(R(x) = P(x)/Q(x)\) of type \([m/n]\) (numerator degree \(m\), denominator degree \(n\)) that minimises the discrete \(L^2\) error over sample points \(x_i \in [a,b]\):

Linearised SVD Solver

The problem is inherently nonlinear in the denominator coefficients. We linearise by multiplying through by \(Q(x_i)\) and fixing \(q_0=1\), yielding an overdetermined linear system:

This system is solved via full SVD with pseudoinverse — no normal equations are formed, preserving numerical stability. The condition number and effective rank are reported.

Pole Detection

Zeros of the denominator \(Q(x)\) inside \([a,b]\) are poles of \(R(x)\). The tool locates them via sign-change scanning and bisection refinement. If internal poles are found, plotting is disabled to avoid singularities.

Options

• Chebyshev sampling: Uses Chebyshev nodes instead of uniform points. Recommended for higher degrees to mitigate Runge-like effects.
• Tikhonov regularisation: Adds \(\lambda\|\mathbf{q}\|^2\) penalty on denominator coefficients to improve conditioning in near-degenerate cases.

Why Rational over Polynomial?

Rational approximations often converge much faster than polynomials for functions with poles, branch points, or steep gradients near \([a,b]\). They are the backbone of Padé approximation and are widely used for special functions (exponential, trigonometric, Bessel, etc.).

Export

After fitting, \(R(x)\) can be exported in LaTeX, Python, or MATLAB format. The plot (SVG) and data (CSV) are also downloadable.