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minimatrix-polyroots v3.0.3

Implementation of Kahan's polynomial root finders for polynomials up to degree 4.


Implementation of Kahan's polynomial root finders for polynomials up to degree 4.


According to Kahan, standard cubic root finders based on classical methods are prone to numerical inaccuracies. This is an implementation of a numerically stable method illustrated in Kahan's notes titled "To Solve a Real Cubic Equation".

The notes can be found here:

Kahan's cubic root finder is based on a quadratic root finder. In like fashion, the quartic root finder is based on Kahan's cubic root finder. The linear root finder is easy to calculate and is added for completeness.


The linear equation solved is Ax + B = 0.

The quadratic equation solved is Ax2 + Bx + C = 0.

The cubic equation solved is Ax3 + Bx2 + Cx + D = 0.

The quartic equation solved is Ax4 + Bx3 + Cx2 + Dx + E = 0.

  • getLinearRoot(A, B)
  • getQuadraticRoots(A, B, C)
  • getCubicRoots(A, B, C, D)
  • getQuarticRoots(A, B, C, D, E)

The roots returned are in an unsorted array. Since roots can be complex, they are given in a complex form:

    real: <number>,
    imag: <number>


Algorithm and pseudocode by William Kahan. Implementation in Javascript by Tim Bright.


(c) Tim Bright, 2017. MIT License.

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