int main(int argc, char **argv)
{
ap::real_1d_array y;
int n;
int i;
double t;
barycentricinterpolant p;
double v;
double dv;
double d2v;
double err;
double maxerr;
//
// Demonstration
//
printf("POLYNOMIAL INTERPOLATION\n\n");
printf("F(x)=sin(x), [0, pi]\n");
printf("Second degree polynomial is used\n\n");
//
// Create polynomial interpolant
//
n = 3;
y.setlength(n);
for(i = 0; i <= n-1; i++)
{
y(i) = sin(0.5*ap::pi()*(1.0+cos(ap::pi()*i/(n-1))));
}
polynomialbuildcheb2(double(0), ap::pi(), y, n, p);
//
// Output results
//
barycentricdiff2(p, double(0), v, dv, d2v);
printf(" P(x) F(x) \n");
printf("function %6.3lf %6.3lf \n",
double(barycentriccalc(p, double(0))),
double(0));
printf("d/dx(0) %6.3lf %6.3lf \n",
double(dv),
double(1));
printf("d2/dx2(0) %6.3lf %6.3lf \n",
double(d2v),
double(0));
printf("\n\n");
return 0;
}
/*************************************************************************
Weighted fitting by Chebyshev polynomial in barycentric form, with
constraints on function values or first derivatives.
Small regularizing term is used when solving constrained tasks (to improve
stability).
Task is linear, so linear least squares solver is used. Complexity of this
computational scheme is O(N*M^2), mostly dominated by least squares solver
SEE ALSO:
PolynomialFit()
INPUT PARAMETERS:
X - points, array[0..N-1].
Y - function values, array[0..N-1].
W - weights, array[0..N-1]
Each summand in square sum of approximation deviations from
given values is multiplied by the square of corresponding
weight. Fill it by 1's if you don't want to solve weighted
task.
N - number of points, N>0.
XC - points where polynomial values/derivatives are constrained,
array[0..K-1].
YC - values of constraints, array[0..K-1]
DC - array[0..K-1], types of constraints:
* DC[i]=0 means that P(XC[i])=YC[i]
* DC[i]=1 means that P'(XC[i])=YC[i]
SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
K - number of constraints, 0<=K<M.
K=0 means no constraints (XC/YC/DC are not used in such cases)
M - number of basis functions (= polynomial_degree + 1), M>=1
OUTPUT PARAMETERS:
Info- same format as in LSFitLinearW() subroutine:
* Info>0 task is solved
* Info<=0 an error occured:
-4 means inconvergence of internal SVD
-3 means inconsistent constraints
-1 means another errors in parameters passed
(N<=0, for example)
P - interpolant in barycentric form.
Rep - report, same format as in LSFitLinearW() subroutine.
Following fields are set:
* RMSError rms error on the (X,Y).
* AvgError average error on the (X,Y).
* AvgRelError average relative error on the non-zero Y
* MaxError maximum error
NON-WEIGHTED ERRORS ARE CALCULATED
IMPORTANT:
this subroitine doesn't calculate task's condition number for K<>0.
SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:
Setting constraints can lead to undesired results, like ill-conditioned
behavior, or inconsistency being detected. From the other side, it allows
us to improve quality of the fit. Here we summarize our experience with
constrained regression splines:
* even simple constraints can be inconsistent, see Wikipedia article on
this subject: http://en.wikipedia.org/wiki/Birkhoff_interpolation
* the greater is M (given fixed constraints), the more chances that
constraints will be consistent
* in the general case, consistency of constraints is NOT GUARANTEED.
* in the one special cases, however, we can guarantee consistency. This
case is: M>1 and constraints on the function values (NOT DERIVATIVES)
Our final recommendation is to use constraints WHEN AND ONLY when you
can't solve your task without them. Anything beyond special cases given
above is not guaranteed and may result in inconsistency.
-- ALGLIB PROJECT --
Copyright 10.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialfitwc(ap::real_1d_array x,
ap::real_1d_array y,
const ap::real_1d_array& w,
int n,
ap::real_1d_array xc,
ap::real_1d_array yc,
const ap::integer_1d_array& dc,
int k,
int m,
int& info,
barycentricinterpolant& p,
polynomialfitreport& rep)
{
double xa;
double xb;
double sa;
double sb;
ap::real_1d_array xoriginal;
ap::real_1d_array yoriginal;
ap::real_1d_array y2;
ap::real_1d_array w2;
ap::real_1d_array tmp;
ap::real_1d_array tmp2;
ap::real_1d_array tmpdiff;
ap::real_1d_array bx;
ap::real_1d_array by;
ap::real_1d_array bw;
ap::real_2d_array fmatrix;
ap::real_2d_array cmatrix;
int i;
int j;
double mx;
double decay;
double u;
double v;
double s;
int relcnt;
lsfitreport lrep;
if( m<1||n<1||k<0||k>=m )
{
info = -1;
return;
}
for(i = 0; i <= k-1; i++)
{
info = 0;
if( dc(i)<0 )
{
info = -1;
}
if( dc(i)>1 )
{
info = -1;
}
if( info<0 )
{
return;
}
}
//
// weight decay for correct handling of task which becomes
// degenerate after constraints are applied
//
decay = 10000*ap::machineepsilon;
//
// Scale X, Y, XC, YC
//
lsfitscalexy(x, y, n, xc, yc, dc, k, xa, xb, sa, sb, xoriginal, yoriginal);
//
// allocate space, initialize/fill:
// * FMatrix- values of basis functions at X[]
// * CMatrix- values (derivatives) of basis functions at XC[]
// * fill constraints matrix
// * fill first N rows of design matrix with values
// * fill next M rows of design matrix with regularizing term
// * append M zeros to Y
// * append M elements, mean(abs(W)) each, to W
//
y2.setlength(n+m);
w2.setlength(n+m);
tmp.setlength(m);
tmpdiff.setlength(m);
fmatrix.setlength(n+m, m);
if( k>0 )
{
cmatrix.setlength(k, m+1);
}
//
// Fill design matrix, Y2, W2:
// * first N rows with basis functions for original points
// * next M rows with decay terms
//
for(i = 0; i <= n-1; i++)
{
//
// prepare Ith row
// use Tmp for calculations to avoid multidimensional arrays overhead
//
for(j = 0; j <= m-1; j++)
{
if( j==0 )
{
tmp(j) = 1;
}
else
{
if( j==1 )
{
tmp(j) = x(i);
}
else
{
tmp(j) = 2*x(i)*tmp(j-1)-tmp(j-2);
}
}
}
ap::vmove(&fmatrix(i, 0), &tmp(0), ap::vlen(0,m-1));
}
for(i = 0; i <= m-1; i++)
{
for(j = 0; j <= m-1; j++)
{
if( i==j )
{
fmatrix(n+i,j) = decay;
}
else
{
fmatrix(n+i,j) = 0;
}
}
}
ap::vmove(&y2(0), &y(0), ap::vlen(0,n-1));
ap::vmove(&w2(0), &w(0), ap::vlen(0,n-1));
mx = 0;
for(i = 0; i <= n-1; i++)
{
mx = mx+fabs(w(i));
}
mx = mx/n;
for(i = 0; i <= m-1; i++)
{
y2(n+i) = 0;
w2(n+i) = mx;
}
//
// fill constraints matrix
//
for(i = 0; i <= k-1; i++)
{
//
// prepare Ith row
// use Tmp for basis function values,
// TmpDiff for basos function derivatives
//
for(j = 0; j <= m-1; j++)
{
if( j==0 )
{
tmp(j) = 1;
tmpdiff(j) = 0;
}
else
{
if( j==1 )
{
tmp(j) = xc(i);
tmpdiff(j) = 1;
}
else
{
tmp(j) = 2*xc(i)*tmp(j-1)-tmp(j-2);
tmpdiff(j) = 2*(tmp(j-1)+xc(i)*tmpdiff(j-1))-tmpdiff(j-2);
}
}
}
if( dc(i)==0 )
{
ap::vmove(&cmatrix(i, 0), &tmp(0), ap::vlen(0,m-1));
}
if( dc(i)==1 )
{
ap::vmove(&cmatrix(i, 0), &tmpdiff(0), ap::vlen(0,m-1));
}
cmatrix(i,m) = yc(i);
}
//
// Solve constrained task
//
if( k>0 )
{
//
// solve using regularization
//
lsfitlinearwc(y2, w2, fmatrix, cmatrix, n+m, m, k, info, tmp, lrep);
}
else
{
//
// no constraints, no regularization needed
//
lsfitlinearwc(y, w, fmatrix, cmatrix, n, m, 0, info, tmp, lrep);
}
if( info<0 )
{
return;
}
//
// Generate barycentric model and scale it
// * BX, BY store barycentric model nodes
// * FMatrix is reused (remember - it is at least MxM, what we need)
//
// Model intialization is done in O(M^2). In principle, it can be
// done in O(M*log(M)), but before it we solved task with O(N*M^2)
// complexity, so it is only a small amount of total time spent.
//
bx.setlength(m);
by.setlength(m);
bw.setlength(m);
tmp2.setlength(m);
s = 1;
for(i = 0; i <= m-1; i++)
{
if( m!=1 )
{
u = cos(ap::pi()*i/(m-1));
}
else
{
u = 0;
}
v = 0;
for(j = 0; j <= m-1; j++)
{
if( j==0 )
{
tmp2(j) = 1;
}
else
{
if( j==1 )
{
tmp2(j) = u;
}
else
{
tmp2(j) = 2*u*tmp2(j-1)-tmp2(j-2);
}
}
v = v+tmp(j)*tmp2(j);
}
bx(i) = u;
by(i) = v;
bw(i) = s;
if( i==0||i==m-1 )
{
bw(i) = 0.5*bw(i);
}
s = -s;
}
barycentricbuildxyw(bx, by, bw, m, p);
barycentriclintransx(p, 2/(xb-xa), -(xa+xb)/(xb-xa));
barycentriclintransy(p, sb-sa, sa);
//
// Scale absolute errors obtained from LSFitLinearW.
// Relative error should be calculated separately
// (because of shifting/scaling of the task)
//
rep.taskrcond = lrep.taskrcond;
rep.rmserror = lrep.rmserror*(sb-sa);
rep.avgerror = lrep.avgerror*(sb-sa);
rep.maxerror = lrep.maxerror*(sb-sa);
rep.avgrelerror = 0;
relcnt = 0;
for(i = 0; i <= n-1; i++)
{
if( ap::fp_neq(yoriginal(i),0) )
{
rep.avgrelerror = rep.avgrelerror+fabs(barycentriccalc(p, xoriginal(i))-yoriginal(i))/fabs(yoriginal(i));
relcnt = relcnt+1;
}
}
if( relcnt!=0 )
{
rep.avgrelerror = rep.avgrelerror/relcnt;
}
}
