Examples
Examples
Try the starting example
- Polynomial coefficients, highest degree first
- 1,-6,11,-6
- Relative residual acceptance target
- 1e-08
- Complex roots as real, imaginary pairs
- 1, 0; 2, 0; 3, 0
- Root count including multiplicity
- 3
- Maximum scaled polynomial residual
- 0
- Residual check
- Residual target met
Try another scenario
- Polynomial coefficients, highest degree first
- 1,0,1
- Relative residual acceptance target
- 1e-08
- Complex roots as real, imaginary pairs
- 0, -1; 0, 1
- Root count including multiplicity
- 2
- Maximum scaled polynomial residual
- 0
- Residual check
- Residual target met
Calculation method
Calculation method
Complex roots as real, imaginary pairs
Build the monic companion matrix and obtain its real/complex eigenvalues. Degree is limited to 1–12.
Root count including multiplicity
Return one eigenvalue per polynomial degree, retaining multiplicity.
Maximum scaled polynomial residual
For each root divide |P(root)| by the coefficient-magnitude Horner scale at |root|; report the maximum.
Residual check
Compare the computed scaled residual with the supplied target; a residual is not a forward-error bound.
Symbols and units
- a
- Polynomial coefficients, highest degree first
- b
- Relative residual acceptance target
Use the units shown. A list keeps the order in which you enter its values. The symbol ^ means “raised to the power of”.
Assumptions & limits
Assumptions & limits
- Real coefficient polynomials of degree 1–12, with a nonzero leading coefficient. Multiple or badly conditioned roots can be inaccurate despite small residuals. Double-precision results are approximate. Numeric tables use 12 significant digits and display components at roundoff scale (32 machine epsilons relative to their row or relevant calculation scale) as zero; scalar diagnostics retain full precision.
Are these inputs private?
These calculations run in your browser. A shared link contains the values you choose to share. Only questions submitted to the separate AI finder are sent to OpenAI.
Further reading
ml-matrix: numerical decompositions ↗
Background reading; this does not imply endorsement or independent review of SumFizz.