/**
* Finds a path which traces the 0 contour of f, traversing from A to B as a single d2<sbasis>.
* degmax specifies the degree (degree = 2*degmax-1, so a degmax of 2 generates a cubic fit).
* The algorithm is based on dividing out derivatives at each end point and does not use the curvature for fitting.
* It is less accurate than sb2d_cubic_solve, although this may be fixed in the future.
*/
D2<SBasis>
sb2dsolve(SBasis2d const &f, Geom::Point const &A, Geom::Point const &B, unsigned degmax){
//g_warning("check f(A)= %f = f(B) = %f =0!", f.apply(A[X],A[Y]), f.apply(B[X],B[Y]));
SBasis2d dfdu = partial_derivative(f, 0);
SBasis2d dfdv = partial_derivative(f, 1);
Geom::Point dfA(dfdu.apply(A[X],A[Y]),dfdv.apply(A[X],A[Y]));
Geom::Point dfB(dfdu.apply(B[X],B[Y]),dfdv.apply(B[X],B[Y]));
Geom::Point nA = dfA/(dfA[X]*dfA[X]+dfA[Y]*dfA[Y]);
Geom::Point nB = dfB/(dfB[X]*dfB[X]+dfB[Y]*dfB[Y]);
D2<SBasis>result(SBasis(degmax, Linear()), SBasis(degmax, Linear()));
double fact_k=1;
double sign = 1.;
for(int dim = 0; dim < 2; dim++)
result[dim][0] = Linear(A[dim],B[dim]);
for(unsigned k=1; k<degmax; k++){
// these two lines make the solutions worse!
//fact_k *= k;
//sign = -sign;
SBasis f_on_curve = compose(f,result);
Linear reste = f_on_curve[k];
double ax = -reste[0]/fact_k*nA[X];
double ay = -reste[0]/fact_k*nA[Y];
double bx = -sign*reste[1]/fact_k*nB[X];
double by = -sign*reste[1]/fact_k*nB[Y];
result[X][k] = Linear(ax,bx);
result[Y][k] = Linear(ay,by);
//sign *= 3;
}
return result;
}
SBasis2d partial_derivative(SBasis2d const &f, int dim) {
SBasis2d result;
for(unsigned i = 0; i < f.size(); i++) {
result.push_back(Linear2d(0,0,0,0));
}
result.us = f.us;
result.vs = f.vs;
for(unsigned i = 0; i < f.us; i++) {
for(unsigned j = 0; j < f.vs; j++) {
Linear2d lin = f.index(i,j);
Linear2d dlin(lin[1+dim]-lin[0], lin[1+2*dim]-lin[dim], lin[3-dim]-lin[2*(1-dim)], lin[3]-lin[2-dim]);
result[i+j*result.us] += dlin;
unsigned di = dim?j:i;
if (di>=1){
float motpi = dim?-1:1;
Linear2d ds_lin_low( lin[0], -motpi*lin[1], motpi*lin[2], -lin[3] );
result[(i+dim-1)+(j-dim)*result.us] += di*ds_lin_low;
Linear2d ds_lin_hi( lin[1+dim]-lin[0], lin[1+2*dim]-lin[dim], lin[3]-lin[2-dim], lin[3-dim]-lin[2-dim] );
result[i+j*result.us] += di*ds_lin_hi;
}
}
}
return result;
}
void
v_coef(SBasis2d f, unsigned deg, SBasis &a, SBasis &b) {
a = SBasis(f.us, Linear());
b = SBasis(f.us, Linear());
for (unsigned u=0; u<f.us; u++){
a[u] = Linear(f.index(deg,u)[0], f.index(deg,u)[1]);
b[u] = Linear(f.index(deg,u)[2], f.index(deg,u)[3]);
}
}
//see a sb2d as an sb of u with coef in sbasis of v.
void
u_coef(SBasis2d f, unsigned deg, SBasis &a, SBasis &b) {
a = SBasis(f.vs, Linear());
b = SBasis(f.vs, Linear());
for (unsigned v=0; v<f.vs; v++){
a[v] = Linear(f.index(deg,v)[0], f.index(deg,v)[2]);
b[v] = Linear(f.index(deg,v)[1], f.index(deg,v)[3]);
}
}
void
v_coef(SBasis2d f, unsigned deg, SBasis &a, SBasis &b) {
a = SBasis();
b = SBasis();
for (unsigned u=0; u<f.us; u++){
a.push_back(Linear(f.index(deg,u)[0], f.index(deg,u)[1]));
b.push_back(Linear(f.index(deg,u)[2], f.index(deg,u)[3]));
}
}
//see a sb2d as an sb of u with coef in sbasis of v.
void
u_coef(SBasis2d f, unsigned deg, SBasis &a, SBasis &b) {
a = SBasis();
b = SBasis();
for (unsigned v=0; v<f.vs; v++){
a.push_back(Linear(f.index(deg,v)[0], f.index(deg,v)[2]));
b.push_back(Linear(f.index(deg,v)[1], f.index(deg,v)[3]));
}
}
SBasis extract_u(SBasis2d const &a, double u) {
SBasis sb(a.vs, Linear());
double s = u*(1-u);
for(unsigned vi = 0; vi < a.vs; vi++) {
double sk = 1;
Linear bo(0,0);
for(unsigned ui = 0; ui < a.us; ui++) {
bo += (extract_u(a.index(ui, vi), u))*sk;
sk *= s;
}
sb[vi] = bo;
}
return sb;
}
SBasis extract_v(SBasis2d const &a, double v) {
SBasis sb(a.us, Linear());
double s = v*(1-v);
for(unsigned ui = 0; ui < a.us; ui++) {
double sk = 1;
Linear bo(0,0);
for(unsigned vi = 0; vi < a.vs; vi++) {
bo += (extract_v(a.index(ui, vi), v))*sk;
sk *= s;
}
sb[ui] = bo;
}
return sb;
}
virtual void draw(cairo_t *cr, std::ostringstream *notify, int width, int height, bool save, std::ostringstream *timer_stream) {
double slider_top = width/4.;
double slider_bot = width*3./4.;
double slider_margin = width/8.;
if(hand.pts.empty()) {
hand.pts.push_back(Geom::Point(width*3./16., 3*width/4.));
hand.pts.push_back(hand.pts[0] + Geom::Point(width/2., 0));
hand.pts.push_back(hand.pts[0] + Geom::Point(width/8., -width/12.));
hand.pts.push_back(hand.pts[0] + Geom::Point(0,-width/4.));
hand.pts.push_back(Geom::Point(slider_margin,slider_bot));
hand.pts.push_back(Geom::Point(width-slider_margin,slider_top));
}
hand.pts[4][X] = slider_margin;
if (hand.pts[4][Y]<slider_top) hand.pts[4][Y] = slider_top;
if (hand.pts[4][Y]>slider_bot) hand.pts[4][Y] = slider_bot;
hand.pts[5][X] = width-slider_margin;
if (hand.pts[5][Y]<slider_top) hand.pts[5][Y] = slider_top;
if (hand.pts[5][Y]>slider_bot) hand.pts[5][Y] = slider_bot;
double tA = (slider_bot-hand.pts[4][Y])/(slider_bot-slider_top);
double tB = (slider_bot-hand.pts[5][Y])/(slider_bot-slider_top);
cairo_move_to(cr,Geom::Point(slider_margin,slider_bot));
cairo_line_to(cr,Geom::Point(slider_margin,slider_top));
cairo_move_to(cr,Geom::Point(width-slider_margin,slider_bot));
cairo_line_to(cr,Geom::Point(width-slider_margin,slider_top));
cairo_set_line_width(cr,.5);
cairo_set_source_rgba (cr, 0., 0.3, 0., 1.);
cairo_stroke(cr);
Frame frame;
frame.O = hand.pts[0];//
frame.x = hand.pts[1]-hand.pts[0];//
frame.y = hand.pts[2]-hand.pts[0];//
frame.z = hand.pts[3]-hand.pts[0];//
/*
SBasis2d f = y_x2();
D2<SBasis> true_solution(Linear(0,1),Linear(0,1));
true_solution[Y].push_back(Linear(-1,-1));
SBasis zero = SBasis(Linear(0.));
Geom::Point A = true_solution(tA);
Geom::Point B = true_solution(tB);
*/
SBasis2d f = x2_plus_y2_1();
D2<Piecewise<SBasis> > true_solution;
true_solution[X] = cos(SBasis(Linear(0,3.141592/2)));
true_solution[Y] = sin(SBasis(Linear(0,3.141592/2)));
Piecewise<SBasis> zero = Piecewise<SBasis>(SBasis(Linear(0.)));
//Geom::Point A(cos(tA*M_PI/2), sin(tA*M_PI/2));// = true_solution(tA);
//Geom::Point B(cos(tB*M_PI/2), sin(tB*M_PI/2));// = true_solution(tB);
Geom::Point A = true_solution(tA);
Geom::Point B = true_solution(tB);
Point dA(-sin(tA*M_PI/2), cos(tA*M_PI/2));
Geom::Point dB(-sin(tB*M_PI/2), cos(tB*M_PI/2));
SBasis2d dfdu = partial_derivative(f, 0);
SBasis2d dfdv = partial_derivative(f, 1);
Geom::Point dfA(dfdu.apply(A[X],A[Y]),dfdv.apply(A[X],A[Y]));
Geom::Point dfB(dfdu.apply(B[X],B[Y]),dfdv.apply(B[X],B[Y]));
dA = rot90(dfA);
dB = rot90(dfB);
plot3d(cr,Linear(0,1),Linear(0,0),Linear(0,0),frame);
plot3d(cr,Linear(0,1),Linear(1,1),Linear(0,0),frame);
plot3d(cr,Linear(0,0),Linear(0,1),Linear(0,0),frame);
plot3d(cr,Linear(1,1),Linear(0,1),Linear(0,0),frame);
cairo_set_line_width(cr,.2);
cairo_set_source_rgba (cr, 0., 0., 0., 1.);
cairo_stroke(cr);
plot3d_top(cr,f,frame);
cairo_set_line_width(cr,1);
cairo_set_source_rgba (cr, .5, 0.5, 0.5, 1.);
cairo_stroke(cr);
plot3d(cr,f,frame);
cairo_set_line_width(cr,.2);
cairo_set_source_rgba (cr, .5, 0.5, 0.5, 1.);
cairo_stroke(cr);
plot3d(cr, true_solution[X], true_solution[Y], zero, frame);
cairo_set_line_width(cr,.5);
cairo_set_source_rgba (cr, 0., 0., 0., 1.);
cairo_stroke(cr);
double error;
for(int degree = 2; degree < 2; degree++) {
D2<SBasis> zeroset = sb2dsolve(f,A,B,degree);
plot3d(cr, zeroset[X], zeroset[Y], SBasis(Linear(0.)),frame);
cairo_set_line_width(cr,1);
cairo_set_source_rgba (cr, 0.9, 0., 0., 1.);
cairo_stroke(cr);
SBasis comp = compose(f,zeroset);
plot3d(cr, zeroset[X], zeroset[Y], comp, frame);
cairo_set_source_rgba (cr, 0.7, 0., 0.7, 1.);
cairo_stroke(cr);
//Fix Me: bounds_exact does not work here?!?!
Interval bounds = *bounds_fast(comp);
error = (bounds.max()>-bounds.min() ? bounds.max() : -bounds.min() );
}
if (1) {
bits_n_bobs par = {&f, A, B, dA, dB};
bits_n_bobs* bnb = ∥
std::cout << f[0] << "= intial f \n";
const gsl_multimin_fminimizer_type *T =
gsl_multimin_fminimizer_nmsimplex;
gsl_multimin_fminimizer *s = NULL;
gsl_vector *ss, *x;
gsl_multimin_function minex_func;
size_t iter = 0;
int status;
double size;
/* Starting point */
x = gsl_vector_alloc (2);
gsl_vector_set (x, 0, 0.552); // magic number for quarter circle
gsl_vector_set (x, 1, 0.552);
/* Set initial step sizes to 1 */
ss = gsl_vector_alloc (2);
gsl_vector_set_all (ss, 0.1);
/* Initialize method and iterate */
minex_func.n = 2;
minex_func.f = &my_f;
minex_func.params = (void *)∥
s = gsl_multimin_fminimizer_alloc (T, 2);
gsl_multimin_fminimizer_set (s, &minex_func, x, ss);
do
{
iter++;
status = gsl_multimin_fminimizer_iterate(s);
if (status)
break;
size = gsl_multimin_fminimizer_size (s);
status = gsl_multimin_test_size (size, 1e-7);
if (status == GSL_SUCCESS)
{
printf ("converged to minimum at\n");
}
}
while (status == GSL_CONTINUE && iter < 100);
printf ("%5lu %g %gf f() = %g size = %g\n",
iter,
gsl_vector_get (s->x, 0),
gsl_vector_get (s->x, 1),
s->fval, size);
{
double x = gsl_vector_get(s->x, 0);
double y = gsl_vector_get(s->x, 1);
Bezier b0(bnb->B[0], bnb->B[0]+bnb->dB[0]*x, bnb->A[0]+bnb->dA[0]*y, bnb->A[0]);
Bezier b1(bnb->B[1], bnb->B[1]+bnb->dB[1]*x, bnb->A[1]+bnb->dA[1]*y, bnb->A[1]);
D2<SBasis> zeroset(b0.toSBasis(), b1.toSBasis());
plot3d(cr, zeroset[X], zeroset[Y], SBasis(Linear(0.)),frame);
cairo_set_line_width(cr,1);
cairo_set_source_rgba (cr, 0.9, 0., 0., 1.);
cairo_stroke(cr);
SBasis comp = compose(f,zeroset);
plot3d(cr, zeroset[X], zeroset[Y], comp, frame);
cairo_set_source_rgba (cr, 0.7, 0., 0.7, 1.);
cairo_stroke(cr);
//Fix Me: bounds_exact does not work here?!?!
Interval bounds = *bounds_fast(comp);
error = (bounds.max()>-bounds.min() ? bounds.max() : -bounds.min() );
}
gsl_vector_free(x);
gsl_vector_free(ss);
gsl_multimin_fminimizer_free (s);
}
*notify << "Gray: f-graph and true solution,\n";
*notify << "Red: solver solution,\n";
*notify << "Purple: value of f over solver solution.\n";
*notify << " error: "<< error <<".\n";
Toy::draw(cr, notify, width, height, save,timer_stream);
}