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Lab 04

Root Geometry Lab

How does the position of the roots control a rational function's derivatives?

Analyze a quadratic by its discriminant and root form, see the zeros of the derivatives of 1/(ax² + bx + c) arise from equally spaced angles at the complex root, and compute certified numerical roots of real polynomials up to degree 6.

  • Computational Demonstration
  • Published Book Content
  • Classical Foundation
p + iqxₖ = p + q cot(kπ/(n+1))

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Scope of this instrument

Supported

  • Quadratics ax2+bx+cax^2+bx+c with real coefficients, a≠0a \neq 0: discriminant, exact roots, the trigonometric form re±iθre^{\pm i\theta} when Δ<0\Delta<0, and the hyperbolic forms ±re±η\pm re^{\pm\eta} or reη,−re−ηre^{\eta}, -re^{-\eta} when Δ>0\Delta>0.
  • For Δ<0\Delta<0 and 1≤n≤101 \le n \le 10: f(n)f^{(n)} of f=1/(ax2+bx+c)f=1/(ax^2+bx+c) in angular form, and the zeros xk=p+qcot⁡(kπ/(n+1))x_k=p+q\cot(k\pi/(n+1)), k=1,…,nk=1,\dots,n, each confirmed by a sign change of an independent numerical derivative.
  • Real polynomials of degree 1 to 6: Durand–Kerner roots with multiplicity detection, conjugate-symmetry check, and a certificate that the roots rebuild the coefficients.

Not supported

  • A leading coefficient of 0, non-finite coefficients, or iterations that fail to certify (the lab says so and reports no roots).
  • Complex coefficients and degrees above 6.
  • Distinct roots closer than about 10−510^{-5} (relative), which may be reported as one repeated root.
  • The zero law for Δ≥0\Delta \ge 0, where ff has real poles.

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