Polynomial Laguerre(unsigned i)
{
if(i==0)
return Polynomial({1.0f});
else if(i==1)
return Polynomial({1.0f, -1.0f});
else
return (Laguerre(i-1)*Polynomial({2.0f*(i-1)+1.0f, -1.0f})-Laguerre(i-2)*float(i-1))/float(i);
}
/* find all roots */
void CLPCAnal::AllRoots(COMPLEX *ap,int m,COMPLEX *roots)
{
int k,j,i;
COMPLEX x,bp,c;
COMPLEX ad[MAXPOLY];
for (j=0;j<=m;j++) ad[j] = ap[j];
for (j=m;j>=1;j--) {
/* find root */
x = CMake(0.0,0.0);
Laguerre(ad,j,&x);
if (fabs(x.im) <= (IM_RANGE*fabs(x.re)))
x.im = 0.0;
roots[j] = x;
/* deflation */
bp = ad[j];
for (k=j-1;k>=0;k--) {
c = ad[k];
ad[k] = bp;
bp = CAdd(CMult(x,bp),c);
}
}
/* sort into increasing root.real */
for (j=2;j<=m;j++) {
x = roots[j];
i = j;
while ((i > 1) && (x.re < roots[i-1].re)) {
roots[i] = roots[i-1];
i = i - 1;
}
roots[i] = x;
}
}
double motional_rabi_factor(int n1, int n2, double z0, double k)
{
//see Experimental Issues, Eq. 18, section 2.3.1
double eta = z0*k;
return exp(-0.5*eta*eta) * sqrt(factorial(n1)/ factorial(n2)) * pow(eta,(int)(n2-n1)) * fabs(Laguerre(n1, n2-n1, eta*eta));
}
main()
{
short m;
double abscis[N];
double wt[N];
double left, right;
printf("Table 25.8, k= %1d, n=%1d\n\n", k, N);
Jacobi(N, FLOATk, ZERO, abscis, wt);
for (m=0; m<N; m++) wt[m] /= (k+1);
for (m=0; m<N; m++) printf("%2d %20.18g %20.18g\n", m, abscis[m], wt[m]);
getchar();
printf("\n\n\nTable 25.4, n=9\n\n");
Radau_Jacobi(4, -0.5, ZERO, abscis, wt, &left);
left *= 2;
for (m=0; m<4; m++) abscis[m]= sqrt(abscis[m]);
printf("%2d %20.18g %20.18g\n", 0, ZERO, left);
for (m=0; m<4; m++) printf("%2d %20.18g %20.18g\n", m, abscis[m], wt[m]);
getchar();
printf("\n\n\nTable 25.9, n= %1d\n\n", N);
Laguerre(N, ZERO, abscis, wt);
for (m=0; m<N; m++) printf("%2d %20.18g %20.18g\n", m, abscis[m], wt[m]);
getchar();
printf("\n\n\nTable 25.10, n= %1d\n\n", N);
Hermite(N, ZERO, abscis, wt);
for (m=0; m<N; m++) wt[m] *= SqrtPI;
for (m=0; m<N; m++) printf("%2d %20.18g %20.18g\n", m, abscis[m], wt[m]);
getchar();
printf("\n\n\nSame, n= %1d\n\n", 2*N);
Even_Hermite(2*N, ZERO, abscis, wt);
for (m=0; m<N; m++) wt[m] *= SqrtPI;
for (m=0; m<N; m++) printf("%2d %20.18g %20.18g\n", m, abscis[m], wt[m]);
getchar();
printf("\n\n\nSame, n= %1d\n\n", 9);
Odd_Hermite(9, ZERO, abscis, wt, &left);
left *= SqrtPI;
for (m=0; m<4; m++) wt[m] *= SqrtPI;
printf("%2d %20.18g %20.18g\n", 0, ZERO, left);
for (m=0; m<4; m++) printf("%2d %20.18g %20.18g\n", m, abscis[m], wt[m]);
getchar();
printf("\n\n\nTable 25.6, n= %1d\n\n", N+1);
Lobatto_Jacobi(N-1, ZERO, ZERO, abscis, wt, &left, &right);
for (m=0; m<(N-1); m++) wt[m] *= 2;
left *= 2; right *= 2;
for (m=0; m<(N-1); m++) abscis[m] = (2*abscis[m] - 1);
printf(" %20.18g %20.18g\n", -1.0, left);
for (m=0; m<(N-1); m++)
printf("%2d %20.18g %20.18g\n", m, abscis[m], wt[m]);
printf(" %20.18g %20.18g\n", 1.0, right);
}
