MathPoly Namespace Reference

Polynomial root finding algorithms. More...

Namespaces
namespace	detail

Data Structures
struct	GeneralPolyResult
	Result for general polynomial solver. More...

Functions
MathUtils::PolyResult	Linear (double theA, double theB)
	Solve linear equation: a*x + b = 0 Handles degenerate cases (a = 0).
MathUtils::PolyResult	Quadratic (double theA, double theB, double theC)
	Solve quadratic equation: a*x^2 + b*x + c = 0 Uses numerically stable formulas to avoid catastrophic cancellation.
MathUtils::PolyResult	Cubic (double theA, double theB, double theC, double theD)
	Solve cubic equation: a*x^3 + b*x^2 + c*x + d = 0 Uses Cardano's method with Vieta substitution and trigonometric solution.
MathUtils::PolyResult	Quartic (double theA, double theB, double theC, double theD, double theE)
	Solve quartic equation: a*x^4 + b*x^3 + c*x^2 + d*x + e = 0 Uses Ferrari's method with modern numerical enhancements.
GeneralPolyResult	Laguerre (const double *theCoeffs, int theDegree, double theTol=1.0e-12)
	Solve polynomial equation using Laguerre's method with deflation. Works for any degree up to THE_MAX_POLY_DEGREE.
GeneralPolyResult	LaguerreN (const double *theCoeffs, size_t theSize, double theTol=1.0e-12)
	Convenience function: solve polynomial given as array with specified size.
MathUtils::PolyResult	Sextic (double theA, double theB, double theC, double theD, double theE, double theF, double theG)
	Solve sextic (degree 6) polynomial: a*x^6 + b*x^5 + c*x^4 + d*x^3 + e*x^2 + f*x + g = 0.
MathUtils::PolyResult	Quintic (double theA, double theB, double theC, double theD, double theE, double theF)
	Solve quintic (degree 5) polynomial: a*x^5 + b*x^4 + c*x^3 + d*x^2 + e*x + f = 0.
GeneralPolyResult	Octic (const double theCoeffs[9])
	Solve octic (degree 8) polynomial using Laguerre's method. Useful for Circle-Ellipse extrema after Weierstrass substitution.

Variables
constexpr int	THE_MAX_POLY_DEGREE = 20
	Maximum polynomial degree supported by Laguerre solver.

Detailed Description

Polynomial root finding algorithms.

Polynomial root finding algorithms using Laguerre's method.

Function Documentation

Cubic()

MathUtils::PolyResult MathPoly::Cubic ( double theA, double theB, double theC, double theD )

inline

Solve cubic equation: a*x^3 + b*x^2 + c*x + d = 0 Uses Cardano's method with Vieta substitution and trigonometric solution.

Algorithm:

1. Handle lower degree cases (a = 0 -> quadratic)
2. Transform to depressed cubic: t^3 + pt + q = 0 via x = t - b/(3a)
3. Compute discriminant Delta = q^2/4 + p^3/27
4. For Delta > 0: one real root (Cardano's formula)
5. For Delta < 0: three real roots (trigonometric method)
6. For Delta = 0: multiple roots
7. Apply Newton-Raphson refinement

Parameters

theA	coefficient of x^3
theB	coefficient of x^2
theC	coefficient of x
theD	constant term

Returns

result containing 1, 2, or 3 real roots (sorted in ascending order)

Laguerre()

GeneralPolyResult MathPoly::Laguerre ( const double * theCoeffs, int theDegree, double theTol = 1.0e-12 )

inline

Solve polynomial equation using Laguerre's method with deflation. Works for any degree up to THE_MAX_POLY_DEGREE.

Algorithm:

1. Use Laguerre iteration to find one root (complex)
2. If root is nearly real, treat it as real and deflate
3. If root is complex, deflate by quadratic factor (conjugate pair)
4. Repeat until all roots found
5. Refine real roots using Newton on original polynomial

Parameters

theCoeffs	coefficients array [a0, a1, a2, ..., an] for a0 + a1*x + ... + an*x^n
theDegree	polynomial degree
theTol	tolerance for convergence and real/complex discrimination

Returns

GeneralPolyResult containing real and complex roots

LaguerreN()

GeneralPolyResult MathPoly::LaguerreN ( const double * theCoeffs, size_t theSize, double theTol = 1.0e-12 )

inline

Convenience function: solve polynomial given as array with specified size.

Parameters

theCoeffs	coefficients [a0, a1, ..., an]
theSize	size of coefficient array (degree + 1)
theTol	tolerance

Returns

GeneralPolyResult

Linear()

MathUtils::PolyResult MathPoly::Linear ( double theA, double theB )

inline

Solve linear equation: a*x + b = 0 Handles degenerate cases (a = 0).

Parameters

theA	coefficient of x
theB	constant term

Returns

result containing 0 or 1 root, or infinite solutions flag

Octic()

GeneralPolyResult MathPoly::Octic ( const double theCoeffs[9] )

inline

Solve octic (degree 8) polynomial using Laguerre's method. Useful for Circle-Ellipse extrema after Weierstrass substitution.

Parameters

theCoeffs

coefficients [a0, a1, ..., a8] where polynomial is a0 + a1*x + ... + a8*x^8

Returns

GeneralPolyResult containing all real roots

Quadratic()

MathUtils::PolyResult MathPoly::Quadratic ( double theA, double theB, double theC )

inline

Solve quadratic equation: a*x^2 + b*x + c = 0 Uses numerically stable formulas to avoid catastrophic cancellation.

Algorithm:

1. Handle linear case when a = 0
2. Compute discriminant D = b^2 - 4ac
3. For D < 0: no real roots
4. For D = 0: one double root
5. For D > 0: two roots using stable formula

Parameters

theA	coefficient of x^2
theB	coefficient of x
theC	constant term

Returns

result containing 0, 1, or 2 real roots (sorted in ascending order)

Quartic()

MathUtils::PolyResult MathPoly::Quartic ( double theA, double theB, double theC, double theD, double theE )

inline

Solve quartic equation: a*x^4 + b*x^3 + c*x^2 + d*x + e = 0 Uses Ferrari's method with modern numerical enhancements.

Algorithm:

1. Handle lower degree cases (a = 0 -> cubic)
2. Transform to depressed quartic: t^4 + pt^2 + qt + r = 0 via x = t - b/(4a)
3. Solve Ferrari's resolvent cubic
4. Factor quartic into two quadratics
5. Solve both quadratics
6. Apply Newton-Raphson refinement

Parameters

theA	coefficient of x^4
theB	coefficient of x^3
theC	coefficient of x^2
theD	coefficient of x
theE	constant term

Returns

result containing 0, 1, 2, 3, or 4 real roots (sorted in ascending order)

Quintic()

MathUtils::PolyResult MathPoly::Quintic ( double theA, double theB, double theC, double theD, double theE, double theF )

inline

Solve quintic (degree 5) polynomial: a*x^5 + b*x^4 + c*x^3 + d*x^2 + e*x + f = 0.

Parameters

theA	coefficient of x^5
theB	coefficient of x^4
theC	coefficient of x^3
theD	coefficient of x^2
theE	coefficient of x
theF	constant term

Returns

PolyResult containing real roots only

Sextic()

MathUtils::PolyResult MathPoly::Sextic ( double theA, double theB, double theC, double theD, double theE, double theF, double theG )

inline

Solve sextic (degree 6) polynomial: a*x^6 + b*x^5 + c*x^4 + d*x^3 + e*x^2 + f*x + g = 0.

Parameters

theA	coefficient of x^6
theB	coefficient of x^5
theC	coefficient of x^4
theD	coefficient of x^3
theE	coefficient of x^2
theF	coefficient of x
theG	constant term

Returns

PolyResult containing real roots only

Variable Documentation

THE_MAX_POLY_DEGREE

constexpr int MathPoly::THE_MAX_POLY_DEGREE = 20

constexpr

Maximum polynomial degree supported by Laguerre solver.