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system.cpp
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614 lines (531 loc) · 20.6 KB
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//-----------------------------------------------------------------------------
// Once we've written our constraint equations in the symbolic algebra system,
// these routines linearize them, and solve by a modified Newton's method.
// This also contains the routines to detect non-convergence or inconsistency,
// and report diagnostics to the user.
//
// Copyright 2008-2013 Jonathan Westhues.
//-----------------------------------------------------------------------------
#include "solvespace.h"
#include <Eigen/Core>
#include <Eigen/SparseQR>
namespace SolveSpace {
// The solver will converge all unknowns to within this tolerance. This must
// always be much less than LENGTH_EPS, and in practice should be much less.
const double System::CONVERGE_TOLERANCE = (LENGTH_EPS/(1e2));
constexpr size_t LikelyPartialCountPerEq = 10;
bool System::WriteJacobian(int tag) {
// Clear all
mat.param.clear();
mat.eq.clear();
mat.A.sym.setZero();
mat.B.sym.clear();
for(Equation &e : eq) {
if(e.tag != tag) continue;
mat.eq.push_back(&e);
}
if(mat.eq.size() >= MAX_UNKNOWNS) {
return false;
}
mat.m = mat.eq.size();
std::unordered_map<uint32_t, int> paramToIndex;
for(Param &p : param) {
if(p.tag != tag) continue;
// Fill the param id to index map
paramToIndex[p.h.v] = mat.param.size();
mat.param.push_back(p.h);
}
mat.n = mat.param.size();
// In some experimenting, this is almost always the right size.
// Value is usually between 0 and 20, comes from number of constraints?
mat.A.sym.resize(mat.m, mat.n);
mat.A.sym.reserve(Eigen::VectorXi::Constant(mat.n, LikelyPartialCountPerEq));
mat.B.sym.reserve(mat.eq.size());
for(size_t i = 0; i < mat.eq.size(); i++) {
Equation *e = mat.eq[i];
// Deep-copy and simplify (fold) the current equation.
Expr *f = e->e->DeepCopyWithParamsAsPointers(¶m, &(SK.param), /*foldConstants=*/true);
ParamSet paramsUsed;
f->ParamsUsedList(¶msUsed);
for(hParam p : paramsUsed) {
// Find the index of this parameter
auto it = paramToIndex.find(p.v);
if(it == paramToIndex.end()) continue;
// this is the parameter index
const int j = it->second;
// compute partial derivative of f
Expr *pd = f->PartialWrt(p);
pd = pd->FoldConstants(/*allocCopy=*/false);
if(pd->IsZeroConst())
continue;
mat.A.sym.insert(i, j) = pd;
}
mat.B.sym.push_back(f);
}
return true;
}
void System::EvalJacobian() {
using namespace Eigen;
mat.A.num.setZero();
mat.A.num.resize(mat.m, mat.n);
const int size = mat.A.sym.outerSize();
for(int k = 0; k < size; k++) {
for(SparseMatrix <Expr *>::InnerIterator it(mat.A.sym, k); it; ++it) {
double value = it.value()->Eval();
if(EXACT(value == 0.0)) continue;
mat.A.num.insert(it.row(), it.col()) = value;
}
}
mat.A.num.makeCompressed();
}
bool System::IsDragged(hParam p) {
return dragged.find(p) != dragged.end();
}
SubstitutionMap System::SolveBySubstitution() {
// Contains pointers to last substitutions in a substitution chain
std::vector<Param *> subVec;
// Maps a parameter to the index of its last substitution in `subVec`
std::unordered_map<hParam, size_t, HandleHasher<hParam>> leaves;
// Tracks how many slots in `subVec` contain a specific last substitution
// (this can happen as slots that once contained a specific substitution
// are updated to point to another over the run of the substitution algorithm)
std::unordered_map<hParam, std::vector<size_t>, HandleHasher<hParam>> slotTrack;
for(auto &teq : eq) {
Expr *tex = teq.e;
// If we have `(a - b) = 0` where both a and b are parameters, then `a = b` and we can substitute
if(tex->op == Expr::Op::MINUS &&
tex->a->op == Expr::Op::PARAM &&
tex->b->op == Expr::Op::PARAM)
{
Param *sub = param.FindByIdNoOops(tex->a->parh);
Param *by = param.FindByIdNoOops(tex->b->parh);
if(!sub || !by) {
// Don't substitute unless they're both solver params;
// otherwise it's an equation that can be solved immediately,
// or an error to flag later.
continue;
}
if(sub->h == by->h) {
teq.tag = EQ_SUBSTITUTED;
continue;
}
// Take the last substitution of parameter a
size_t subIdx = 0;
auto it = leaves.find(sub->h);
if(it != leaves.end()) {
subIdx = it->second;
sub = subVec.at(it->second - 1);
}
// Take the last substitution of parameter b
size_t byIdx = 0;
it = leaves.find(by->h);
if(it != leaves.end()) {
byIdx = it->second;
by = subVec.at(it->second - 1);
}
// If the last substituton of `a` is a dragged param, keep it
// and substitute the other param
if(IsDragged(sub->h)) {
std::swap(sub, by);
std::swap(subIdx, byIdx);
}
if(subIdx == 0) {
if(byIdx == 0) {
// Neither `sub` nor `by` are in the map, so add them and
// set the target index
subVec.push_back(by);
leaves[by->h] = leaves[sub->h] = subVec.size();
} else {
// `sub` isn't in the map, but `by` is, so just add `sub` to
// the map with `by` as the target
leaves[sub->h] = byIdx;
}
} else {
// `sub` already exists in the map, so just update any slots
// that point to it as the last substitution to point to `by`
// instead
auto it = slotTrack.find(sub->h);
if(it == slotTrack.end()) {
// There's only this one slot, so just replace it with `by`
subVec[subIdx - 1] = by;
// If `by` was already in the map, that means we now have
// an additional slot where it resides, so add it to the
// slot tracker
if(byIdx != 0) {
// If `by` is already in the slot tracker, we'll get
// back a vector with at least two elements; otherwise
// this access will add a new item to the slot tracker
// with an empty vector
auto &bySlots = slotTrack[by->h];
if(bySlots.empty()) {
bySlots.push_back(byIdx);
}
bySlots.push_back(subIdx);
}
} else {
// We have more than one slot pointing to `sub`, so update
// all of the slots to point to `by`
for(size_t i : it->second) {
subVec[i - 1] = by;
}
// No more slots are pointing to `sub`, so extract the slot list
// and erase `sub` from the tracker
auto subSlots = std::move(it->second);
slotTrack.erase(it);
// Same as above: this access either gives us an existing vector
// with at least two elements, or creates an empty vector
auto &bySlots = slotTrack[by->h];
if(bySlots.empty()) {
bySlots = std::move(subSlots);
if(byIdx != 0) {
bySlots.push_back(byIdx);
}
} else {
bySlots.insert(bySlots.end(), subSlots.begin(), subSlots.end());
}
}
leaves[by->h] = subIdx;
}
sub->tag = VAR_SUBSTITUTED;
teq.tag = EQ_SUBSTITUTED;
}
}
SubstitutionMap subMap;
for(auto &sub : leaves) {
Param *by = subVec[sub.second - 1];
if(sub.first != by->h) {
subMap[sub.first] = by;
}
}
// Substitute all the equations
for(auto &req : eq) {
req.e->Substitute(subMap);
}
return subMap;
}
//-----------------------------------------------------------------------------
// Calculate the rank of the Jacobian matrix
//-----------------------------------------------------------------------------
int System::CalculateRank() {
using namespace Eigen;
if(mat.n == 0 || mat.m == 0) return 0;
SparseQR <SparseMatrix<double>, COLAMDOrdering<int>> solver;
solver.compute(mat.A.num);
int result = solver.rank();
return result;
}
bool System::TestRank(int *dof, int *rank) {
EvalJacobian();
int jacobianRank = CalculateRank();
// We are calculating dof based on real rank, not mat.m.
// Using this approach we can calculate real dof even when redundant is allowed.
if(dof != NULL) *dof = mat.n - jacobianRank;
if(rank) {
*rank = jacobianRank;
}
return jacobianRank == mat.m;
}
bool System::SolveLinearSystem(const Eigen::SparseMatrix <double> &A,
const Eigen::VectorXd &B, Eigen::VectorXd *X)
{
if(A.outerSize() == 0) return true;
using namespace Eigen;
SparseQR<SparseMatrix<double>, COLAMDOrdering<int>> solver;
//SimplicialLDLT<SparseMatrix<double>> solver;
solver.compute(A);
*X = solver.solve(B);
return (solver.info() == Success);
}
bool System::SolveLeastSquares() {
using namespace Eigen;
// Scale the columns; this scale weights the parameters for the least
// squares solve, so that we can encourage the solver to make bigger
// changes in some parameters, and smaller in others.
VectorXd scale = VectorXd::Ones(mat.n);
for(int c = 0; c < mat.n; c++) {
if(IsDragged(mat.param[c])) {
// It's least squares, so this parameter doesn't need to be all
// that big to get a large effect.
scale[c] = 1 / 20.0;
}
}
const int size = mat.A.num.outerSize();
for(int k = 0; k < size; k++) {
for(SparseMatrix<double>::InnerIterator it(mat.A.num, k); it; ++it) {
it.valueRef() *= scale[it.col()];
}
}
SparseMatrix<double> AAt = mat.A.num * mat.A.num.transpose();
AAt.makeCompressed();
VectorXd z(mat.n);
if(!SolveLinearSystem(AAt, mat.B.num, &z)) return false;
mat.X = mat.A.num.transpose() * z;
for(int c = 0; c < mat.n; c++) {
mat.X[c] *= scale[c];
}
return true;
}
bool System::NewtonSolve() {
int iter = 0;
bool converged = false;
int i;
// Evaluate the functions at our operating point.
mat.B.num = Eigen::VectorXd(mat.m);
for(i = 0; i < mat.m; i++) {
mat.B.num[i] = (mat.B.sym[i])->Eval();
}
do {
// And evaluate the Jacobian at our initial operating point.
EvalJacobian();
if(!SolveLeastSquares()) break;
// Take the Newton step;
// J(x_n) (x_{n+1} - x_n) = 0 - F(x_n)
for(i = 0; i < mat.n; i++) {
Param *p = param.FindById(mat.param[i]);
p->val -= mat.X[i];
if(IsReasonable(p->val)) {
// Very bad, and clearly not convergent
return false;
}
}
// Re-evalute the functions, since the params have just changed.
for(i = 0; i < mat.m; i++) {
mat.B.num[i] = (mat.B.sym[i])->Eval();
if(IsReasonable(mat.B.num[i])) {
// Very bad, and clearly not convergent
return false;
}
}
// Check for convergence
converged = true;
for(i = 0; i < mat.m; i++) {
if(fabs(mat.B.num[i]) > CONVERGE_TOLERANCE) {
converged = false;
break;
}
}
} while(iter++ < 50 && !converged);
return converged;
}
void System::WriteEquationsExceptFor(hConstraint hc, Group *g) {
// Generate all the equations from constraints in this group
for(auto &con : SK.constraint) {
ConstraintBase *c = &con;
if(c->group != g->h) continue;
if(c->h == hc) continue;
if(c->HasLabel() && c->type != Constraint::Type::COMMENT &&
g->allDimsReference)
{
// When all dimensions are reference, we adjust them to display
// the correct value, and then don't generate any equations.
c->ModifyToSatisfy();
continue;
}
if(g->relaxConstraints && c->type != Constraint::Type::POINTS_COINCIDENT) {
// When the constraints are relaxed, we keep only the point-
// coincident constraints, and the constraints generated by
// the entities and groups.
continue;
}
c->GenerateEquations(&eq);
}
// And the equations from entities
for(auto &ent : SK.entity) {
EntityBase *e = &ent;
if(e->group != g->h) continue;
e->GenerateEquations(&eq);
}
// And from the groups themselves
g->GenerateEquations(&eq);
}
void System::FindWhichToRemoveToFixJacobian(Group *g, List<hConstraint> *bad, bool forceDofCheck) {
auto time = GetMilliseconds();
g->solved.timeout = false;
int a;
for(a = 0; a < 2; a++) {
for(auto &con : SK.constraint) {
if((GetMilliseconds() - time) > g->solved.findToFixTimeout) {
g->solved.timeout = true;
return;
}
ConstraintBase *c = &con;
if(c->group != g->h) continue;
if((c->type == Constraint::Type::POINTS_COINCIDENT && a == 0) ||
(c->type != Constraint::Type::POINTS_COINCIDENT && a == 1))
{
// Do the constraints in two passes: first everything but
// the point-coincident constraints, then only those
// constraints (so they appear last in the list).
continue;
}
param.ClearTags();
eq.Clear();
WriteEquationsExceptFor(c->h, g);
eq.ClearTags();
// It's a major speedup to solve the easy ones by substitution here,
// and that doesn't break anything.
if(!forceDofCheck) {
SolveBySubstitution();
}
WriteJacobian(0);
EvalJacobian();
int rank = CalculateRank();
if(rank == mat.m) {
// We fixed it by removing this constraint
bad->Add(&(c->h));
}
}
}
}
SolveResult System::Solve(Group *g, int *dof, List<hConstraint> *bad,
bool andFindBad, bool andFindFree, bool forceDofCheck)
{
WriteEquationsExceptFor(Constraint::NO_CONSTRAINT, g);
bool rankOk;
// int x;
// printf("%d equations", eq.n);
// for(x = 0; x < eq.n; x++) {
// printf(" %.3f = %s = 0", eq[x].e->Eval(), eq[x].e->Print().c_str());
// }
// printf("%d parameters", param.n);
// for(x = 0; x < param.n; x++) {
// printf(" param %08x at %.3f", param[x].h.v, param[x].val);
// }
// All params and equations are assigned to group zero.
param.ClearTags();
eq.ClearTags();
SubstitutionMap subMap;
// Since we are suppressing dof calculation or allowing redundant, we
// can't / don't want to catch result of dof checking without substitution
if(g->suppressDofCalculation || g->allowRedundant || !forceDofCheck) {
subMap = SolveBySubstitution();
}
// Before solving the big system, see if we can find any equations that
// are soluble alone. This can be a huge speedup. We don't know whether
// the system is consistent yet, but if it isn't then we'll catch that
// later.
int alone = 1;
for(auto &e : eq) {
if(e.tag != 0)
continue;
hParam hp = e.e->ReferencedParams(¶m);
if(hp == Expr::NO_PARAMS) continue;
if(hp == Expr::MULTIPLE_PARAMS) continue;
Param *p = param.FindById(hp);
if(p->tag != 0) continue; // let rank test catch inconsistency
e.tag = alone;
p->tag = alone;
WriteJacobian(alone);
if(!NewtonSolve()) {
// We don't do the rank test, so let's arbitrarily return
// the DIDNT_CONVERGE result here.
rankOk = true;
// Failed to converge, bail out early
goto didnt_converge;
}
alone++;
}
// Now write the Jacobian for what's left, and do a rank test; that
// tells us if the system is inconsistently constrained.
if(!WriteJacobian(0)) {
return SolveResult::TOO_MANY_UNKNOWNS;
}
// Clear dof value in order to have indication when dof is actually not calculated
if(dof != NULL) *dof = -1;
// We are suppressing or allowing redundant, so we no need to catch unsolveable + redundant
rankOk = (!g->suppressDofCalculation && !g->allowRedundant) ? TestRank(dof) : true;
// And do the leftovers as one big system
if(!NewtonSolve()) {
goto didnt_converge;
}
// Here we are want to calculate dof even when redundant is allowed, so just handle suppressing
rankOk = (!g->suppressDofCalculation) ? TestRank(dof) : true;
if(!rankOk) {
if(andFindBad) FindWhichToRemoveToFixJacobian(g, bad, forceDofCheck);
} else {
MarkParamsFree(andFindFree);
}
// System solved correctly, so write the new values back in to the
// main parameter table.
for(auto &p : param) {
auto it = subMap.find(p.h);
double val = it == subMap.end() ? p.val : it->second->val;
Param *pp = SK.GetParam(p.h);
pp->val = val;
pp->known = true;
pp->free = p.free;
}
return rankOk ? SolveResult::OKAY : SolveResult::REDUNDANT_OKAY;
didnt_converge:
SK.constraint.ClearTags();
// Not using range-for here because index is used in additional ways
for(size_t i = 0; i < mat.eq.size(); i++) {
if(fabs(mat.B.num[i]) > CONVERGE_TOLERANCE || IsReasonable(mat.B.num[i])) {
// This constraint is unsatisfied.
if(!mat.eq[i]->h.isFromConstraint()) continue;
hConstraint hc = mat.eq[i]->h.constraint();
ConstraintBase *c = SK.constraint.FindByIdNoOops(hc);
if(!c) continue;
// Don't double-show constraints that generated multiple
// unsatisfied equations
if(!c->tag) {
bad->Add(&(c->h));
c->tag = 1;
}
}
}
return rankOk ? SolveResult::DIDNT_CONVERGE : SolveResult::REDUNDANT_DIDNT_CONVERGE;
}
SolveResult System::SolveRank(Group *g, int *rank, int *dof, List<hConstraint> *bad,
bool andFindBad, bool andFindFree)
{
WriteEquationsExceptFor(Constraint::NO_CONSTRAINT, g);
// All params and equations are assigned to group zero.
param.ClearTags();
eq.ClearTags();
// Now write the Jacobian, and do a rank test; that
// tells us if the system is inconsistently constrained.
if(!WriteJacobian(0)) {
return SolveResult::TOO_MANY_UNKNOWNS;
}
bool rankOk = TestRank(dof, rank);
if(!rankOk) {
// When we are testing with redundant allowed, we don't want to have additional info
// about redundants since this test is working only for single redundant constraint
if(!g->suppressDofCalculation && !g->allowRedundant) {
if(andFindBad) FindWhichToRemoveToFixJacobian(g, bad, true);
}
} else {
MarkParamsFree(andFindFree);
}
return rankOk ? SolveResult::OKAY : SolveResult::REDUNDANT_OKAY;
}
void System::Clear() {
entity.Clear();
param.Clear();
eq.Clear();
dragged.clear();
mat.A.num.setZero();
mat.A.sym.setZero();
}
void System::MarkParamsFree(bool find) {
// If requested, find all the free (unbound) variables. This might be
// more than the number of degrees of freedom. Don't always do this,
// because the display would get annoying and it's slow.
for(auto &p : param) {
p.free = false;
if(find) {
if(p.tag == 0) {
p.tag = VAR_DOF_TEST;
WriteJacobian(0);
EvalJacobian();
int rank = CalculateRank();
if(rank == mat.m) {
p.free = true;
}
p.tag = 0;
}
}
}
}
} // namespace SolveSpace