/** regularized kernel for even kernels with K_I even
* and K_B mirrored smooth to K(1/2) (used in dD, d>1)
*/
double _Complex regkern3(kernel k, double xx, int p, const double *param, double a, double b)
{
int r;
double _Complex sum=0.0;
xx=fabs(xx);
if (xx>=0.5) {
/*return kern(typ,c,0,0.5);*/
xx=0.5;
}
/* else */
if ((a<=xx) && (xx<=0.5-b)) {
return k(xx,0,param);
}
else if (xx<a) {
for (r=0; r<p; r++) {
sum+=pow(-a,(double)r)*k(a,r,param)
*(BasisPoly(p-1,r,xx/a)+BasisPoly(p-1,r,-xx/a));
}
/*sum=kern(typ,c,0,xx); */
return sum;
}
else if ((0.5-b<xx) && (xx<=0.5)) {
sum=k(0.5,0,param)*BasisPoly(p-1,0,-2.0*xx/b+(1.0-b)/b);
/* sum=regkern2(typ,c,p,a,b, 0.5)*BasisPoly(p-1,0,-2.0*xx/b+(1.0-b)/b); */
for (r=0; r<p; r++) {
sum+=pow(b/2.0,(double)r)
*k(0.5-b,r,param)
*BasisPoly(p-1,r,2.0*xx/b-(1.0-b)/b);
}
return sum;
}
return 0.0;
}
/** Use Lagrange basis polynomials for evaluating the interpolation polynomial P(x) that satisfies
* the non-symmetric Hermite interpolation conditions in two nodes up to the p-th derivative, i.e.,
* P^k(x0) = kernel^k(x0), for all k=0,...,p,
* P(x1) = c,
* P^k(x1) = 0, for all k=1,...,p
*/
static fcs_float interpolate_lagrange_ec(
const fcs_float *kernel, fcs_int p,
fcs_float x0, fcs_float x1, fcs_float x
)
{
fcs_float sum=0.0;
/* canonicalize x to y \in [-1,1] */
fcs_float r = 0.5*(x1-x0);
fcs_float m = 0.5*(x1+x0);
fcs_float y = (x-m)/r;
for (fcs_int i=0; i<p+1; i++)
sum += BasisPoly(p,i,y) * fcs_pow(r,(fcs_float)i) * kernel[i];
return sum + kernel[p+1] * BasisPoly(p,0,-y);
}
/** regularized kernel for even kernels with K_I even and K_B mirrored */
double _Complex regkern2(kernel k, double xx, int p, const double *param, double a, double b)
{
int r;
double _Complex sum=0.0;
xx=fabs(xx);
if (xx>0.5) {
for (r=0; r<p; r++) {
sum+=pow(b,(double)r)*k(0.5-b,r,param)
*(BasisPoly(p-1,r,0)+BasisPoly(p-1,r,0));
}
return sum;
}
else if ((a<=xx) && (xx<=0.5-b)) {
return k(xx,0,param);
}
else if (xx<a) {
for (r=0; r<p; r++) {
sum+=pow(-a,(double)r)*k(a,r,param)
*(BasisPoly(p-1,r,xx/a)+BasisPoly(p-1,r,-xx/a));
}
return sum;
}
else if ((0.5-b<xx) && (xx<=0.5)) {
for (r=0; r<p; r++) {
sum+=pow(b,(double)r)*k(0.5-b,r,param)
*(BasisPoly(p-1,r,(xx-0.5)/b)+BasisPoly(p-1,r,-(xx-0.5)/b));
}
return sum;
}
return 0.0;
}
/** regularized kernel with K_I arbitrary and K_B smooth to zero */
double _Complex regkern(kernel k, double xx, int p, const double *param, double a, double b)
{
int r;
double _Complex sum=0.0;
if (xx<-0.5)
xx=-0.5;
if (xx>0.5)
xx=0.5;
if ((xx>=-0.5+b && xx<=-a) || (xx>=a && xx<=0.5-b)) {
return k(xx,0,param);
}
else if (xx<-0.5+b) {
sum=(k(-0.5,0,param)+k(0.5,0,param))/2.0
*BasisPoly(p-1,0,2.0*xx/b+(1.0-b)/b);
for (r=0; r<p; r++) {
sum+=pow(-b/2.0,(double)r)
*k(-0.5+b,r,param)
*BasisPoly(p-1,r,-2.0*xx/b+(b-1.0)/b);
}
return sum;
}
else if ((xx>-a) && (xx<a)) {
for (r=0; r<p; r++) {
sum+=pow(a,(double)r)
*( k(-a,r,param)*BasisPoly(p-1,r,xx/a)
+k( a,r,param)*BasisPoly(p-1,r,-xx/a)*(r & 1 ? -1 : 1));
}
return sum;
}
else if (xx>0.5-b) {
sum=(k(-0.5,0,param)+k(0.5,0,param))/2.0
*BasisPoly(p-1,0,-2.0*xx/b+(1.0-b)/b);
for (r=0; r<p; r++) {
sum+=pow(b/2.0,(double)r)
*k(0.5-b,r,param)
*BasisPoly(p-1,r,2.0*xx/b-(1.0-b)/b);
}
return sum;
}
return k(xx,0,param);
}
/** regularized kernel with K_I arbitrary and K_B periodized
* (used in 1D)
*/
double _Complex regkern1(kernel k, double xx, int p, const double *param, double a, double b)
{
int r;
double _Complex sum=0.0;
if (xx<-0.5)
xx=-0.5;
if (xx>0.5)
xx=0.5;
if ((xx>=-0.5+b && xx<=-a) || (xx>=a && xx<=0.5-b))
{
return k(xx,0,param);
}
else if ((xx>-a) && (xx<a))
{
for (r=0; r<p; r++) {
sum+=pow(a,(double)r)
*( k(-a,r,param)*BasisPoly(p-1,r,xx/a)
+k( a,r,param)*BasisPoly(p-1,r,-xx/a)*(r & 1 ? -1 : 1));
}
return sum;
}
else if (xx<-0.5+b)
{
for (r=0; r<p; r++) {
sum+=pow(b,(double)r)
*( k(0.5-b,r,param)*BasisPoly(p-1,r,(xx+0.5)/b)
+k(-0.5+b,r,param)*BasisPoly(p-1,r,-(xx+0.5)/b)*(r & 1 ? -1 : 1));
}
return sum;
}
else if (xx>0.5-b)
{
for (r=0; r<p; r++) {
sum+=pow(b,(double)r)
*( k(0.5-b,r,param)*BasisPoly(p-1,r,(xx-0.5)/b)
+k(-0.5+b,r,param)*BasisPoly(p-1,r,-(xx-0.5)/b)*(r & 1 ? -1 : 1));
}
return sum;
}
return k(xx,0,param);
}
