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OCCT/src/FoundationClasses/TKMath/MathSys/MathSys_Newton4D.hxx
Pasukhin Dmitry d515004353 Foundation Classes - Add Laguerre polynomial solver and refactor specialized Newton APIs (#1086) 
Add a general Laguerre polynomial solver to MathPoly and migrate specialized
MathSys Newton solvers (2D/3D/4D) to a unified fixed-size API aligned with
MathUtils statuses and configuration.

MathPoly:
- Add MathPoly_Laguerre.hxx with Laguerre + deflation for higher-degree polynomials.
- Add GeneralPolyResult with real and complex root outputs.
- Add Quintic/Sextic/Octic helpers built on the general solver.
- Add MathPoly_Laguerre gtests and update FILES.cmake/README.

MathSys:
- Add MathSys_NewtonTypes.hxx with NewtonResultN, NewtonBoundsN, NewtonOptions.
- Add/replace specialized APIs:
  - Solve2D(), Solve2DSymmetric()
  - Solve3D(), SolveCurveSurfaceExtrema3D()
  - Solve4D(), SolveSurfaceSurfaceExtrema4D()
- Add dedicated 2D/3D/4D Newton gtests and update FILES.cmake.
- Restore detailed solver comments/docs and update MathSys README.

MathUtils alignment:
- Add Status::NonDescentDirection.
- Make NewtonOptions derive from MathUtils::Config and use FTolerance/
  XTolerance/MaxIterations consistently.
- Add shared Newton-related constants in MathUtils_Config.hxx.
- Add Domain1D::IsEqual().
- Update MathUtils README accordingly.

Correctness updates in specialized Newton:
- Convergence is strictly residual-based.
- Armijo directional derivative in symmetric 2D uses J^T*F.
- Singular Jacobian handling keeps robust fallback directions.
- NewtonResultN iteration counter renamed to NbIterations for consistency.

Also updated docs and tests to remove old specialized Newton status/options
references and use MathUtils::Status + FTolerance/XTolerance.
2026-02-14 13:05:07 +00:00

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// Copyright (c) 2025 OPEN CASCADE SAS
//
// This file is part of Open CASCADE Technology software library.
//
// This library is free software; you can redistribute it and/or modify it under
// the terms of the GNU Lesser General Public License version 2.1 as published
// by the Free Software Foundation, with special exception defined in the file
// OCCT_LGPL_EXCEPTION.txt. Consult the file LICENSE_LGPL_21.txt included in OCCT
// distribution for complete text of the license and disclaimer of any warranty.
//
// Alternatively, this file may be used under the terms of Open CASCADE
// commercial license or contractual agreement.
#ifndef _MathSys_Newton4D_HeaderFile
#define _MathSys_Newton4D_HeaderFile
#include <MathSys_NewtonTypes.hxx>
#include <gp_Pnt.hxx>
#include <gp_Vec.hxx>
#include <algorithm>
#include <array>
#include <cmath>
//! @file MathSys_Newton4D.hxx
//! @brief Optimized 4D Newton-Raphson solver for systems of 4 equations in 4 unknowns.
//!
//! This solver is specifically optimized for 4D problems like:
//! - Surface-surface extrema (find u1,v1,u2,v2 where distance is extremal)
//! - Curve-surface intersection with tangent constraints
//! - Two-curve closest point with additional constraints
//!
//! Optimizations compared to general Newton:
//! - No math_Vector/math_Matrix allocation overhead
//! - Gaussian elimination with partial pivoting for 4x4 system
//! - Squared norm comparisons (avoid sqrt in convergence check)
//! - Step limiting to prevent wild oscillations
//! - Gradient descent fallback for singular Jacobian
namespace MathSys
{
namespace detail
{
//! Check that NewtonBoundsN<4> has valid (min <= max) ranges.
inline bool IsBoundsValid4D(const NewtonBoundsN<4>& theBounds)
{
if (!theBounds.HasBounds)
{
return true;
}
return theBounds.Min[0] <= theBounds.Max[0] && theBounds.Min[1] <= theBounds.Max[1]
&& theBounds.Min[2] <= theBounds.Max[2] && theBounds.Min[3] <= theBounds.Max[3];
}
//! Check that NewtonOptions fields are positive and valid.
inline bool IsOptionsValid4D(const NewtonOptions& theOptions)
{
return theOptions.FTolerance > 0.0 && theOptions.XTolerance > 0.0 && theOptions.MaxIterations > 0
&& theOptions.MaxStepRatio > 0.0 && theOptions.SoftBoundsExtension >= 0.0;
}
//! Return the largest domain extent across all 4 dimensions (min 1.0).
inline double MaxDomainSize4D(const NewtonBoundsN<4>& theBounds)
{
if (!theBounds.HasBounds)
{
return 1.0;
}
const double aD0 = theBounds.Max[0] - theBounds.Min[0];
const double aD1 = theBounds.Max[1] - theBounds.Min[1];
const double aD2 = theBounds.Max[2] - theBounds.Min[2];
const double aD3 = theBounds.Max[3] - theBounds.Min[3];
return std::max(1.0, std::max(aD0, std::max(aD1, std::max(aD2, aD3))));
}
//! Clamp solution array to bounds, optionally extending by soft-bounds ratio.
inline void Clamp4D(std::array<double, 4>& theX,
const NewtonBoundsN<4>& theBounds,
bool theUseSoftBounds,
double theSoftExtRatio)
{
if (!theBounds.HasBounds)
{
return;
}
for (int i = 0; i < 4; ++i)
{
const double aExt =
theUseSoftBounds ? (theBounds.Max[i] - theBounds.Min[i]) * theSoftExtRatio : 0.0;
theX[i] = std::clamp(theX[i], theBounds.Min[i] - aExt, theBounds.Max[i] + aExt);
}
}
//! Solve 4x4 linear system using Gaussian elimination with partial pivoting.
//! @param[in] theJ 4x4 Jacobian matrix
//! @param[in] theF 4-element right-hand side
//! @param[out] theDelta 4-element solution vector
//! @return true if system was solved successfully
inline bool Solve4x4(const double theJ[4][4], const double theF[4], double theDelta[4])
{
// Augmented matrix [J | -F]
double A[4][5];
for (int i = 0; i < 4; ++i)
{
for (int j = 0; j < 4; ++j)
{
A[i][j] = theJ[i][j];
}
A[i][4] = -theF[i];
}
// Forward elimination with partial pivoting
for (int k = 0; k < 4; ++k)
{
// Find pivot
int aMaxRow = k;
double aMaxVal = std::abs(A[k][k]);
for (int i = k + 1; i < 4; ++i)
{
const double aVal = std::abs(A[i][k]);
if (aVal > aMaxVal)
{
aMaxVal = aVal;
aMaxRow = i;
}
}
// Check for singularity
if (aMaxVal < 1.0e-30)
{
return false;
}
// Swap rows if needed
if (aMaxRow != k)
{
for (int j = k; j <= 4; ++j)
{
std::swap(A[k][j], A[aMaxRow][j]);
}
}
// Eliminate column
const double aInvPivot = 1.0 / A[k][k];
for (int i = k + 1; i < 4; ++i)
{
const double aFactor = A[i][k] * aInvPivot;
for (int j = k + 1; j <= 4; ++j)
{
A[i][j] -= aFactor * A[k][j];
}
A[i][k] = 0.0;
}
}
// Back substitution
for (int i = 3; i >= 0; --i)
{
double aSum = A[i][4];
for (int j = i + 1; j < 4; ++j)
{
aSum -= A[i][j] * theDelta[j];
}
theDelta[i] = aSum / A[i][i];
}
return true;
}
} // namespace detail
//! 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:
//! @code
//! bool operator()(double theX1, double theX2, double theX3, double theX4,
//! double theF[4], double theJ[4][4]) const;
//! @endcode
//! where theF is the function values and theJ is the 4x4 Jacobian matrix.
//!
//! @tparam Function callable type (functor, lambda, or function pointer)
//! @param[in] theFunc function to solve (provides F and Jacobian)
//! @param[in] theX0 initial guess {x1, x2, x3, x4}
//! @param[in] theBounds box bounds for each variable
//! @param[in] theOptions solver options (tolerances, max iterations, etc.)
//! @return NewtonResultN<4> containing solution, status, and diagnostics
template <typename Function>
NewtonResultN<4> Solve4D(const Function& theFunc,
const std::array<double, 4>& theX0,
const NewtonBoundsN<4>& theBounds,
const NewtonOptions& theOptions = NewtonOptions())
{
NewtonResultN<4> aRes;
aRes.X = theX0;
if (!detail::IsOptionsValid4D(theOptions) || !detail::IsBoundsValid4D(theBounds))
{
aRes.Status = MathUtils::Status::InvalidInput;
return aRes;
}
detail::Clamp4D(aRes.X, theBounds, theOptions.AllowSoftBounds, theOptions.SoftBoundsExtension);
const double aTolSq = theOptions.FTolerance * theOptions.FTolerance;
const double aMaxStep = theOptions.MaxStepRatio * detail::MaxDomainSize4D(theBounds);
for (int anIter = 0; anIter < theOptions.MaxIterations; ++anIter)
{
aRes.NbIterations = static_cast<size_t>(anIter + 1);
double aF[4];
double aJ[4][4];
if (!theFunc(aRes.X[0], aRes.X[1], aRes.X[2], aRes.X[3], aF, aJ))
{
aRes.Status = MathUtils::Status::NumericalError;
return aRes;
}
// Check convergence using squared norm (avoid sqrt)
const double aFNormSq = aF[0] * aF[0] + aF[1] * aF[1] + aF[2] * aF[2] + aF[3] * aF[3];
aRes.ResidualNorm = std::sqrt(aFNormSq);
if (aFNormSq <= aTolSq)
{
aRes.Status = MathUtils::Status::OK;
return aRes;
}
// Solve 4x4 linear system: J * delta = -F
double aDelta[4];
if (!detail::Solve4x4(aJ, aF, aDelta))
{
// Singular Jacobian - try steepest descent direction: -J^T * F
double aGrad[4] = {0.0, 0.0, 0.0, 0.0};
for (int c = 0; c < 4; ++c)
{
for (int r = 0; r < 4; ++r)
{
aGrad[c] += aJ[r][c] * aF[r];
}
}
const double aGradSq =
aGrad[0] * aGrad[0] + aGrad[1] * aGrad[1] + aGrad[2] * aGrad[2] + aGrad[3] * aGrad[3];
if (aGradSq < 1.0e-60)
{
aRes.Status = MathUtils::Status::Singular;
return aRes;
}
const double aAlpha = std::min(1.0, std::sqrt(aFNormSq / aGradSq) * 0.1);
for (int i = 0; i < 4; ++i)
{
aDelta[i] = -aAlpha * aGrad[i];
}
}
// Limit step size to prevent wild oscillations
const double aStepNormSq =
aDelta[0] * aDelta[0] + aDelta[1] * aDelta[1] + aDelta[2] * aDelta[2] + aDelta[3] * aDelta[3];
const double aStepNorm = std::sqrt(aStepNormSq);
if (aStepNorm > aMaxStep)
{
const double aScale = aMaxStep / aStepNorm;
for (int i = 0; i < 4; ++i)
{
aDelta[i] *= aScale;
}
}
// Update and clamp to bounds
std::array<double, 4> aNewX = {aRes.X[0] + aDelta[0],
aRes.X[1] + aDelta[1],
aRes.X[2] + aDelta[2],
aRes.X[3] + aDelta[3]};
detail::Clamp4D(aNewX, theBounds, theOptions.AllowSoftBounds, theOptions.SoftBoundsExtension);
aRes.StepNorm = std::sqrt((aNewX[0] - aRes.X[0]) * (aNewX[0] - aRes.X[0])
+ (aNewX[1] - aRes.X[1]) * (aNewX[1] - aRes.X[1])
+ (aNewX[2] - aRes.X[2]) * (aNewX[2] - aRes.X[2])
+ (aNewX[3] - aRes.X[3]) * (aNewX[3] - aRes.X[3]));
aRes.X = aNewX;
const double aScaleRef = std::max(
1.0,
std::max(std::abs(aRes.X[0]),
std::max(std::abs(aRes.X[1]), std::max(std::abs(aRes.X[2]), std::abs(aRes.X[3])))));
if (aRes.StepNorm <= theOptions.XTolerance * aScaleRef)
{
aRes.Status = MathUtils::Status::MaxIterations;
return aRes;
}
}
// Final convergence check after max iterations
double aF[4];
double aJ[4][4];
if (!theFunc(aRes.X[0], aRes.X[1], aRes.X[2], aRes.X[3], aF, aJ))
{
aRes.Status = MathUtils::Status::NumericalError;
return aRes;
}
aRes.ResidualNorm = std::sqrt(aF[0] * aF[0] + aF[1] * aF[1] + aF[2] * aF[2] + aF[3] * aF[3]);
aRes.Status = (aRes.ResidualNorm <= theOptions.FTolerance) ? MathUtils::Status::OK
: MathUtils::Status::MaxIterations;
return aRes;
}
//! 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.
//!
//! @tparam SurfaceEvaluator1 type providing D2 evaluation for first surface
//! @tparam SurfaceEvaluator2 type providing D2 evaluation for second surface
//! @param[in] theSurf1 first surface evaluator
//! @param[in] theSurf2 second surface evaluator
//! @param[in] theX0 initial guess {u1, v1, u2, v2}
//! @param[in] theBounds box bounds for {u1, v1, u2, v2}
//! @param[in] theOptions solver options
//! @return NewtonResultN<4> with u1, v1, u2, v2 stored in X[0..3]
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())
{
auto aFunc = [&theSurf1, &theSurf2](double theU1,
double theV1,
double theU2,
double theV2,
double theF[4],
double theJ[4][4]) -> bool {
gp_Pnt aPt1, aPt2;
gp_Vec aD1U1, aD1V1, aD2UU1, aD2VV1, aD2UV1;
gp_Vec aD1U2, aD1V2, aD2UU2, aD2VV2, aD2UV2;
theSurf1.D2(theU1, theV1, aPt1, aD1U1, aD1V1, aD2UU1, aD2VV1, aD2UV1);
theSurf2.D2(theU2, theV2, aPt2, aD1U2, aD1V2, aD2UU2, aD2VV2, aD2UV2);
// D = P1 - P2
const gp_Vec aD(aPt2, aPt1);
// Function values: gradient of ||S1 - S2||^2
theF[0] = aD.Dot(aD1U1);
theF[1] = aD.Dot(aD1V1);
theF[2] = -aD.Dot(aD1U2);
theF[3] = -aD.Dot(aD1V2);
// Jacobian: Hessian of ||S1 - S2||^2
// Block [0:2, 0:2]: derivatives w.r.t. (u1, v1)
theJ[0][0] = aD1U1.Dot(aD1U1) + aD.Dot(aD2UU1);
theJ[0][1] = aD1V1.Dot(aD1U1) + aD.Dot(aD2UV1);
theJ[1][0] = theJ[0][1]; // Symmetric
theJ[1][1] = aD1V1.Dot(aD1V1) + aD.Dot(aD2VV1);
// Block [0:2, 2:4]: cross derivatives
theJ[0][2] = -aD1U2.Dot(aD1U1);
theJ[0][3] = -aD1V2.Dot(aD1U1);
theJ[1][2] = -aD1U2.Dot(aD1V1);
theJ[1][3] = -aD1V2.Dot(aD1V1);
// Block [2:4, 0:2]: cross derivatives (symmetric to above)
theJ[2][0] = theJ[0][2];
theJ[2][1] = theJ[1][2];
theJ[3][0] = theJ[0][3];
theJ[3][1] = theJ[1][3];
// Block [2:4, 2:4]: derivatives w.r.t. (u2, v2)
theJ[2][2] = aD1U2.Dot(aD1U2) - aD.Dot(aD2UU2);
theJ[2][3] = aD1V2.Dot(aD1U2) - aD.Dot(aD2UV2);
theJ[3][2] = theJ[2][3]; // Symmetric
theJ[3][3] = aD1V2.Dot(aD1V2) - aD.Dot(aD2VV2);
return true;
};
return Solve4D(aFunc, theX0, theBounds, theOptions);
}
} // namespace MathSys
#endif // _MathSys_Newton4D_HeaderFile
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