Shared types for specialized small-dimension Newton solvers. More...

Namespaces
namespace	detail

Data Structures
struct	LMConfig
	Configuration for Levenberg-Marquardt algorithm. Extends base Config with damping parameter settings. More...

struct	NewtonBoundsN
	Box bounds for N-dimensional solver. More...

struct	NewtonOptions
	Solver options for small-dimension Newton methods. More...

struct	NewtonResultN
	Result of N-dimensional Newton solver. More...

Functions
template<typename FuncSetType >
VectorResult	Newton (FuncSetType &theFunc, const math_Vector &theStart, const math_Vector &theTolX, double theTolF, size_t theMaxIter=100)
	Newton-Raphson method for solving systems of nonlinear equations.

template<typename FuncSetType >
VectorResult	NewtonBounded (FuncSetType &theFunc, const math_Vector &theStart, const math_Vector &theInfBound, const math_Vector &theSupBound, const math_Vector &theTolX, double theTolF, size_t theMaxIter=100)
	Newton-Raphson method with bounds constraints.

template<typename FuncSetType >
VectorResult	Newton (FuncSetType &theFunc, const math_Vector &theStart, double theTolX, double theTolF, size_t theMaxIter=100)
	Simplified Newton method with uniform tolerances.

template<typename Function >
NewtonResultN< 2 >	Solve2D (const Function &theFunc, const std::array< double, 2 > &theX0, const NewtonBoundsN< 2 > &theBounds, const NewtonOptions &theOptions=NewtonOptions())
	Solve a general 2x2 nonlinear system by Newton iteration. Function contract: bool operator()(double u, double v, double f[2], double j[2][2]) const;.

template<typename Function >
NewtonResultN< 2 >	Solve2DSymmetric (const Function &theFunc, const std::array< double, 2 > &theX0, const NewtonBoundsN< 2 > &theBounds, const NewtonOptions &theOptions=NewtonOptions())
	Solve a 2x2 system with symmetric Jacobian by robust Newton iteration. Function contract: bool ValueAndJacobian(double u, double v, double& f1, double& f2, double& j11, double& j12, double& j22) const; bool Value(double u, double v, double& f1, double& f2) const; // required if line search is enabled.

template<typename Function >
NewtonResultN< 3 >	Solve3D (const Function &theFunc, const std::array< double, 3 > &theX0, const NewtonBoundsN< 3 > &theBounds, const NewtonOptions &theOptions=NewtonOptions())
	Solve a 3x3 nonlinear system by Newton iteration with bounds.

template<typename CurveEvaluator , typename SurfaceEvaluator >
NewtonResultN< 3 >	SolveCurveSurfaceExtrema3D (const CurveEvaluator &theCurve, const SurfaceEvaluator &theSurface, const std::array< double, 3 > &theX0, const NewtonBoundsN< 3 > &theBounds, const NewtonOptions &theOptions=NewtonOptions())
	Optimized 3D Newton solver for curve-surface extrema.

template<typename Function >
NewtonResultN< 4 >	Solve4D (const Function &theFunc, const std::array< double, 4 > &theX0, const NewtonBoundsN< 4 > &theBounds, const NewtonOptions &theOptions=NewtonOptions())
	Solve a 4x4 nonlinear system by Newton iteration with bounds.

template<typename SurfaceEvaluator1 , typename SurfaceEvaluator2 >
NewtonResultN< 4 >	SolveSurfaceSurfaceExtrema4D (const SurfaceEvaluator1 &theSurf1, const SurfaceEvaluator2 &theSurf2, const std::array< double, 4 > &theX0, const NewtonBoundsN< 4 > &theBounds, const NewtonOptions &theOptions=NewtonOptions())
	Optimized 4D Newton solver for surface-surface extrema.

template<typename FuncSetType >
VectorResult	LevenbergMarquardt (FuncSetType &theFunc, const math_Vector &theStart, const LMConfig &theConfig=LMConfig())
	Levenberg-Marquardt algorithm for nonlinear least squares.

template<typename FuncSetType >
VectorResult	LevenbergMarquardtBounded (FuncSetType &theFunc, const math_Vector &theStart, const math_Vector &theInfBound, const math_Vector &theSupBound, const LMConfig &theConfig=LMConfig())
	Levenberg-Marquardt with bounds constraints.

Detailed Description

Shared types for specialized small-dimension Newton solvers.

Function Documentation

◆ LevenbergMarquardt()

template<typename FuncSetType >

VectorResult MathSys::LevenbergMarquardt	(	FuncSetType &	theFunc,
		const math_Vector &	theStart,
		const LMConfig &	theConfig = LMConfig() )

Levenberg-Marquardt algorithm for nonlinear least squares.

Minimizes ||F(X)||^2 where F is a vector function of vector X. Combines Gauss-Newton method (fast near minimum) with gradient descent (robust far from minimum) using adaptive damping.

Algorithm:

Compute F(X) and Jacobian J at current point
Solve (J^T*J + lambda*I) * dX = -J^T*F for correction dX
If ||F(X+dX)||^2 < ||F(X)||^2: accept step, decrease lambda
Otherwise: reject step, increase lambda
Repeat until convergence

Template Parameters

FuncSetType type with NbVariables(), NbEquations(), Value(const math_Vector& X, math_Vector& F) and Derivatives(const math_Vector& X, math_Matrix& J) or Values(const math_Vector& X, math_Vector& F, math_Matrix& J)

Parameters

theFunc	function set providing residuals and Jacobian
theStart	initial guess vector
theConfig	Levenberg-Marquardt configuration

Returns: result containing solution vector if converged

◆ LevenbergMarquardtBounded()

template<typename FuncSetType >

VectorResult MathSys::LevenbergMarquardtBounded	(	FuncSetType &	theFunc,
		const math_Vector &	theStart,
		const math_Vector &	theInfBound,
		const math_Vector &	theSupBound,
		const LMConfig &	theConfig = LMConfig() )

Levenberg-Marquardt with bounds constraints.

Minimizes ||F(X)||^2 subject to theInfBound <= X <= theSupBound. Solution is clamped to bounds after each step.

Parameters

theFunc	function set providing residuals and Jacobian
theStart	initial guess vector
theInfBound	lower bounds for solution
theSupBound	upper bounds for solution
theConfig	Levenberg-Marquardt configuration

Returns: result containing solution vector if converged

◆ Newton() [1/2]

template<typename FuncSetType >

VectorResult MathSys::Newton	(	FuncSetType &	theFunc,
		const math_Vector &	theStart,
		const math_Vector &	theTolX,
		double	theTolF,
		size_t	theMaxIter = 100 )

Newton-Raphson method for solving systems of nonlinear equations.

Solves F(X) = 0 where F is a vector function of vector X. The method iteratively improves an initial guess by solving the linear system J(X)*dX = -F(X) where J is the Jacobian matrix.

Parameters

theFunc	function set with derivatives (Jacobian)
theStart	initial guess vector
theTolX	tolerance for solution change \|\|X(n+1) - X(n)\|\| < tolX
theTolF	tolerance for function values \|\|F(X)\|\| < tolF
theMaxIter	maximum number of iterations

Returns: result containing solution vector if converged

◆ Newton() [2/2]

template<typename FuncSetType >

VectorResult MathSys::Newton	(	FuncSetType &	theFunc,
		const math_Vector &	theStart,
		double	theTolX,
		double	theTolF,
		size_t	theMaxIter = 100 )

Simplified Newton method with uniform tolerances.

Parameters

theFunc	function set with derivatives
theStart	initial guess vector
theTolX	uniform tolerance for all variables
theTolF	tolerance for function values
theMaxIter	maximum number of iterations

Returns: result containing solution vector if converged

◆ NewtonBounded()

template<typename FuncSetType >

VectorResult MathSys::NewtonBounded	(	FuncSetType &	theFunc,
		const math_Vector &	theStart,
		const math_Vector &	theInfBound,
		const math_Vector &	theSupBound,
		const math_Vector &	theTolX,
		double	theTolF,
		size_t	theMaxIter = 100 )

Newton-Raphson method with bounds constraints.

Solves F(X) = 0 subject to InfBound <= X <= SupBound. If the Newton step would take X outside bounds, the solution is clamped to the boundary.

Parameters

theFunc	function set with derivatives (Jacobian)
theStart	initial guess vector
theInfBound	lower bounds for solution
theSupBound	upper bounds for solution
theTolX	tolerance for solution change
theTolF	tolerance for function values
theMaxIter	maximum number of iterations

Returns: result containing solution vector if converged

◆ Solve2D()

template<typename Function >

NewtonResultN< 2 > MathSys::Solve2D	(	const Function &	theFunc,
		const std::array< double, 2 > &	theX0,
		const NewtonBoundsN< 2 > &	theBounds,
		const NewtonOptions &	theOptions = NewtonOptions() )

Solve a general 2x2 nonlinear system by Newton iteration. Function contract: bool operator()(double u, double v, double f[2], double j[2][2]) const;.

◆ Solve2DSymmetric()

template<typename Function >

NewtonResultN< 2 > MathSys::Solve2DSymmetric	(	const Function &	theFunc,
		const std::array< double, 2 > &	theX0,
		const NewtonBoundsN< 2 > &	theBounds,
		const NewtonOptions &	theOptions = NewtonOptions() )

Solve a 2x2 system with symmetric Jacobian by robust Newton iteration. Function contract: bool ValueAndJacobian(double u, double v, double& f1, double& f2, double& j11, double& j12, double& j22) const; bool Value(double u, double v, double& f1, double& f2) const; // required if line search is enabled.

◆ Solve3D()

template<typename Function >

NewtonResultN< 3 > MathSys::Solve3D	(	const Function &	theFunc,
		const std::array< double, 3 > &	theX0,
		const NewtonBoundsN< 3 > &	theBounds,
		const NewtonOptions &	theOptions = NewtonOptions() )

Solve a 3x3 nonlinear system by Newton iteration with bounds.

Solves the system [F1, F2, F3] = [0, 0, 0] using Newton-Raphson iteration with Cramer's rule for the 3x3 linear system at each step.

The function type must be callable with signature:

bool operator()(double theX1, double theX2, double theX3,

double theF[3], double theJ[3][3]) const;

NCollection_ForwardRangeIterator

STL input iterator that wraps an OCCT More()/Next() iterator.

Definition NCollection_ForwardRange.hxx:142

where theF is the function values and theJ is the 3x3 Jacobian matrix.

Template Parameters

Function callable type (functor, lambda, or function pointer)

Parameters

[in]	theFunc	function to solve (provides F and Jacobian)
[in]	theX0	initial guess {x1, x2, x3}
[in]	theBounds	box bounds for each variable
[in]	theOptions	solver options (tolerances, max iterations, etc.)

Returns: NewtonResultN<3> containing solution, status, and diagnostics

◆ Solve4D()

template<typename Function >

NewtonResultN< 4 > MathSys::Solve4D	(	const Function &	theFunc,
		const std::array< double, 4 > &	theX0,
		const NewtonBoundsN< 4 > &	theBounds,
		const NewtonOptions &	theOptions = NewtonOptions() )

Solve a 4x4 nonlinear system by Newton iteration with bounds.

Solves the system [F1, F2, F3, F4] = [0, 0, 0, 0] using Newton-Raphson iteration with Gaussian elimination for the 4x4 linear system at each step.

The function type must be callable with signature:

bool operator()(double theX1, double theX2, double theX3, double theX4,

double theF[4], double theJ[4][4]) const;

where theF is the function values and theJ is the 4x4 Jacobian matrix.

Template Parameters

Function callable type (functor, lambda, or function pointer)

Parameters

[in]	theFunc	function to solve (provides F and Jacobian)
[in]	theX0	initial guess {x1, x2, x3, x4}
[in]	theBounds	box bounds for each variable
[in]	theOptions	solver options (tolerances, max iterations, etc.)

Returns: NewtonResultN<4> containing solution, status, and diagnostics

◆ SolveCurveSurfaceExtrema3D()

template<typename CurveEvaluator , typename SurfaceEvaluator >

NewtonResultN< 3 > MathSys::SolveCurveSurfaceExtrema3D	(	const CurveEvaluator &	theCurve,
		const SurfaceEvaluator &	theSurface,
		const std::array< double, 3 > &	theX0,
		const NewtonBoundsN< 3 > &	theBounds,
		const NewtonOptions &	theOptions = NewtonOptions() )

Optimized 3D Newton solver for curve-surface extrema.

Specialized version for finding extrema between a curve C(t) and surface S(u,v). The function values are the gradient components of the squared distance:

F1 = (C-S) . dC/dt
F2 = (S-C) . dS/du
F3 = (S-C) . dS/dv

Template Parameters

CurveEvaluator	type providing D2(t, P, D1, D2) evaluation
SurfaceEvaluator	type providing D2(u, v, P, D1U, D1V, D2UU, D2VV, D2UV) evaluation

Parameters

[in]	theCurve	curve evaluator
[in]	theSurface	surface evaluator
[in]	theX0	initial guess {t, u, v}
[in]	theBounds	box bounds for {t, u, v}
[in]	theOptions	solver options

Returns: NewtonResultN<3> with t, u, v stored in X[0], X[1], X[2]

◆ SolveSurfaceSurfaceExtrema4D()

template<typename SurfaceEvaluator1 , typename SurfaceEvaluator2 >

NewtonResultN< 4 > MathSys::SolveSurfaceSurfaceExtrema4D	(	const SurfaceEvaluator1 &	theSurf1,
		const SurfaceEvaluator2 &	theSurf2,
		const std::array< double, 4 > &	theX0,
		const NewtonBoundsN< 4 > &	theBounds,
		const NewtonOptions &	theOptions = NewtonOptions() )

Optimized 4D Newton solver for surface-surface extrema.

Specialized version for finding extrema between two surfaces S1(u1,v1) and S2(u2,v2). The function values are the gradient components of the squared distance:

F1 = (S1-S2) . dS1/dU1
F2 = (S1-S2) . dS1/dV1
F3 = (S2-S1) . dS2/dU2
F4 = (S2-S1) . dS2/dV2

The Jacobian has a special block structure with 2x2 blocks.

Template Parameters

SurfaceEvaluator1	type providing D2 evaluation for first surface
SurfaceEvaluator2	type providing D2 evaluation for second surface

Parameters

[in]	theSurf1	first surface evaluator
[in]	theSurf2	second surface evaluator
[in]	theX0	initial guess {u1, v1, u2, v2}
[in]	theBounds	box bounds for {u1, v1, u2, v2}
[in]	theOptions	solver options

Returns: NewtonResultN<4> with u1, v1, u2, v2 stored in X[0..3]

Namespaces

Data Structures

Functions

Detailed Description

Function Documentation

◆ LevenbergMarquardt()

◆ LevenbergMarquardtBounded()

◆ Newton() [1/2]

◆ Newton() [2/2]

◆ NewtonBounded()

◆ Solve2D()

◆ Solve2DSymmetric()

◆ Solve3D()

◆ Solve4D()

◆ SolveCurveSurfaceExtrema3D()

◆ SolveSurfaceSurfaceExtrema4D()