solvespace/src/system.cpp at wip-intersection · ruevs/solvespace · GitHub

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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.

//-----------------------------------------------------------------------------

#include "solvespace.h"

// This tolerance is used to determine whether two (linearized) constraints

// are linearly dependent. If this is too small, then we will attempt to

// solve truly inconsistent systems and fail. But if it's too large, then

// we will give up on legitimate systems like a skinny right angle triangle by

// its hypotenuse and long side.

const double System::RANK_MAG_TOLERANCE = 1e-4;

// 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));

bool System::WriteJacobian(int tag) {

int j = 0;

for(auto &p : param) {

if(j >= MAX_UNKNOWNS)

return false;

if(p.tag != tag)

continue;

mat.param[j] = p.h;

j++;

}

mat.n = j;

int i = 0;

for(auto &e : eq) {

if(i >= MAX_UNKNOWNS) return false;

if(e.tag != tag)

continue;

mat.eq[i] = e.h;

Expr *f = e.e->DeepCopyWithParamsAsPointers(&param, &(SK.param));

f = f->FoldConstants();

// Hash table (61 bits) to accelerate generation of zero partials.

uint64_t scoreboard = f->ParamsUsed();

for(j = 0; j < mat.n; j++) {

Expr *pd;

if(scoreboard & ((uint64_t)1 << (mat.param[j].v % 61)) &&

f->DependsOn(mat.param[j]))

{

pd = f->PartialWrt(mat.param[j]);

pd = pd->FoldConstants();

pd = pd->DeepCopyWithParamsAsPointers(&param, &(SK.param));

} else {

pd = Expr::From(0.0);

}

mat.A.sym[i][j] = pd;

}

mat.B.sym[i] = f;

i++;

}

mat.m = i;

return true;

}

void System::EvalJacobian() {

int i, j;

for(i = 0; i < mat.m; i++) {

for(j = 0; j < mat.n; j++) {

mat.A.num[i][j] = (mat.A.sym[i][j])->Eval();

}

bool System::IsDragged(hParam p) {

hParam *pp;

for(pp = dragged.First(); pp; pp = dragged.NextAfter(pp)) {

if(p == *pp) return true;

}

return false;

}

void System::SolveBySubstitution() {

for(auto &teq : eq) {

Expr *tex = teq.e;

if(tex->op == Expr::Op::MINUS &&

tex->a->op == Expr::Op::PARAM &&

tex->b->op == Expr::Op::PARAM)

{

hParam a = tex->a->parh;

hParam b = tex->b->parh;

if(!(param.FindByIdNoOops(a) && param.FindByIdNoOops(b))) {

// 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(IsDragged(a)) {

// A is being dragged, so A should stay, and B should go

std::swap(a, b);

}

for(auto &req : eq) {

req.e->Substitute(a, b); // A becomes B, B unchanged

}

for(auto &rp : param) {

if(rp.substd == a) {

rp.substd = b;

}

Param *ptr = param.FindById(a);

ptr->tag = VAR_SUBSTITUTED;

ptr->substd = b;

teq.tag = EQ_SUBSTITUTED;

}

//-----------------------------------------------------------------------------

// Calculate the rank of the Jacobian matrix, by Gram-Schimdt orthogonalization

// in place. A row (~equation) is considered to be all zeros if its magnitude

// is less than the tolerance RANK_MAG_TOLERANCE.

//-----------------------------------------------------------------------------

int System::CalculateRank() {

// Actually work with magnitudes squared, not the magnitudes

double rowMag[MAX_UNKNOWNS] = {};

double tol = RANK_MAG_TOLERANCE*RANK_MAG_TOLERANCE;

int i, iprev, j;

int rank = 0;

for(i = 0; i < mat.m; i++) {

// Subtract off this row's component in the direction of any

// previous rows

for(iprev = 0; iprev < i; iprev++) {

if(rowMag[iprev] <= tol) continue; // ignore zero rows

double dot = 0;

for(j = 0; j < mat.n; j++) {

dot += (mat.A.num[iprev][j]) * (mat.A.num[i][j]);

}

for(j = 0; j < mat.n; j++) {

mat.A.num[i][j] -= (dot/rowMag[iprev])*mat.A.num[iprev][j];

}

// Our row is now normal to all previous rows; calculate the

// magnitude of what's left

double mag = 0;

for(j = 0; j < mat.n; j++) {

mag += (mat.A.num[i][j]) * (mat.A.num[i][j]);

}

if(mag > tol) {

rank++;

}

rowMag[i] = mag;

}

return rank;

}

bool System::TestRank(int *rank) {

EvalJacobian();

int jacobianRank = CalculateRank();

if(rank) *rank = jacobianRank;

return jacobianRank == mat.m;

}

bool System::SolveLinearSystem(double X[], double A[][MAX_UNKNOWNS],

double B[], int n)

{

// Gaussian elimination, with partial pivoting. It's an error if the

// matrix is singular, because that means two constraints are

// equivalent.

int i, j, ip, jp, imax = 0;

double max, temp;

for(i = 0; i < n; i++) {

// We are trying eliminate the term in column i, for rows i+1 and

// greater. First, find a pivot (between rows i and N-1).

max = 0;

for(ip = i; ip < n; ip++) {

if(fabs(A[ip][i]) > max) {

imax = ip;

max = fabs(A[ip][i]);

}

// Don't give up on a singular matrix unless it's really bad; the

// assumption code is responsible for identifying that condition,

// so we're not responsible for reporting that error.

if(fabs(max) < 1e-20) continue;

// Swap row imax with row i

for(jp = 0; jp < n; jp++) {

swap(A[i][jp], A[imax][jp]);

}

swap(B[i], B[imax]);

// For rows i+1 and greater, eliminate the term in column i.

for(ip = i+1; ip < n; ip++) {

temp = A[ip][i]/A[i][i];

for(jp = i; jp < n; jp++) {

A[ip][jp] -= temp*(A[i][jp]);

}

B[ip] -= temp*B[i];

}

// We've put the matrix in upper triangular form, so at this point we

// can solve by back-substitution.

for(i = n - 1; i >= 0; i--) {

if(fabs(A[i][i]) < 1e-20) continue;

temp = B[i];

for(j = n - 1; j > i; j--) {

temp -= X[j]*A[i][j];

}

X[i] = temp / A[i][i];

}

return true;

}

bool System::SolveLeastSquares() {

int r, c, i;

// 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.

for(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.

mat.scale[c] = 1/20.0;

} else {

mat.scale[c] = 1;

}

for(r = 0; r < mat.m; r++) {

mat.A.num[r][c] *= mat.scale[c];

}

// Write A*A'

for(r = 0; r < mat.m; r++) {

for(c = 0; c < mat.m; c++) { // yes, AAt is square

double sum = 0;

for(i = 0; i < mat.n; i++) {

sum += mat.A.num[r][i]*mat.A.num[c][i];

}

mat.AAt[r][c] = sum;

}

if(!SolveLinearSystem(mat.Z, mat.AAt, mat.B.num, mat.m)) return false;

// And multiply that by A' to get our solution.

for(c = 0; c < mat.n; c++) {

double sum = 0;

for(i = 0; i < mat.m; i++) {

sum += mat.A.num[i][c]*mat.Z[i];

}

mat.X[c] = sum * mat.scale[c];

}

return true;

}

bool System::NewtonSolve(int tag) {

int iter = 0;

bool converged = false;

int i;

// Evaluate the functions at our operating point.

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(isnan(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();

}

// Check for convergence

converged = true;

for(i = 0; i < mat.m; i++) {

if(isnan(mat.B.num[i])) {

return false;

}

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) {

int a;

for(a = 0; a < 2; a++) {

for(auto &con : SK.constraint) {

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 *rank, int *dof, List<hConstraint> *bad,

bool andFindBad, bool andFindFree, bool forceDofCheck)

{

WriteEquationsExceptFor(Constraint::NO_CONSTRAINT, g);

int i;

bool rankOk;

dbp("%d equations", eq.n);

for(i = 0; i < eq.n; i++) {

dbp(" %.3f = %s = 0", eq[i].e->Eval(), eq[i].e->Print());

}

dbp("%d parameters", param.n);

for(i = 0; i < param.n; i++) {

dbp(" param %08x at %.3f", param[i].h.v, param[i].val);

} */

// All params and equations are assigned to group zero.

param.ClearTags();

eq.ClearTags();

// Solving by substitution eliminates duplicate e.g. H/V constraints, which can cause rank test

// to succeed even on overdefined systems, which will fail later.

if(!forceDofCheck) {

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(&param);

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(alone)) {

// 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;

}

rankOk = TestRank(rank);

// And do the leftovers as one big system

if(!NewtonSolve(0)) {

goto didnt_converge;

}

rankOk = TestRank(rank);

if(!rankOk) {

if(andFindBad) FindWhichToRemoveToFixJacobian(g, bad, forceDofCheck);

} else {

// This is not the full Jacobian, but any substitutions or single-eq

// solves removed one equation and one unknown, therefore no effect

// on the number of DOF.

if(dof) *dof = CalculateDof();

MarkParamsFree(andFindFree);

}

// System solved correctly, so write the new values back in to the

// main parameter table.

for(auto &p : param) {

double val;

if(p.tag == VAR_SUBSTITUTED) {

val = param.FindById(p.substd)->val;

} else {

val = p.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(i = 0; i < eq.n; i++) {

if(fabs(mat.B.num[i]) > CONVERGE_TOLERANCE || isnan(mat.B.num[i])) {

// This constraint is unsatisfied.

if(!mat.eq[i].isFromConstraint()) continue;

hConstraint hc = mat.eq[i].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(rank);

if(!rankOk) {

if(andFindBad) FindWhichToRemoveToFixJacobian(g, bad, /*forceDofCheck=*/true);

} else {

if(dof) *dof = CalculateDof();

MarkParamsFree(andFindFree);

}

return rankOk ? SolveResult::OKAY : SolveResult::REDUNDANT_OKAY;

}

void System::Clear() {

entity.Clear();

param.Clear();

eq.Clear();

dragged.Clear();

}

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;

}

int System::CalculateDof() {

return mat.n - mat.m;

}

solvespace/src/system.cpp at wip-intersection · ruevs/solvespace · GitHub

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