//*----------------------------------------------------------------------* //* This program is free software: you can redistribute it and/or modify * //* it under the terms of the GNU General Public License as published by * //* the Free Software Foundation, either version 3 of the License, or * //* (at your option) any later version. * //* * //* This program is distributed in the hope that it will be useful, * //* but WITHOUT ANY WARRANTY; without even the implied warranty of * //* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * //* GNU General Public License for more details. * //* * //* You should have received a copy of the GNU General Public License * //* along with this program. If not, see . * //*----------------------------------------------------------------------* //*----------------------------------------------------------------------* //* "The purpose of computing is insight, not numbers." - R.W. Hamming * //* Hermite polynomials, Hermite functions * //* and their respective arbitrary derivatives * //* Copyright 2011-2014 Konrad Griessinger * //* (konradg(at)gmx.net) * //*----------------------------------------------------------------------* // TODO: // - array functions for derivatives of Hermite functions // - asymptotic approximation for derivatives of Hermite functions // - refine existing asymptotic approximations, especially around x=sqrt(2*n+1) or x=sqrt(2*n+1)*sqrt(2), respectively #include #include #include #include // #include #include #include #include "error.h" #include "eval.h" static double pow2(int c) // Small function to calculate integer powers of 2 quickly by bit-shifting when in the standard integer range. Otherwise repeated squaring via gsl_sf_pow_int is used. { if(c<0 && c>-31){ return 1/((double)(1 << -c)); } else if(c>=0 && c<31){ return (double)(1 << c); } else{ return gsl_sf_pow_int(2,c); } } static int gsl_sf_hermite_prob_iter_e(const int n, const double x, gsl_sf_result * result) // Evaluates the probabilists' Hermite polynomial of order n at position x using upward recurrence. { // return gsl_sf_hyperg_U(-n/2.,1./2.,x*x/2.)*gsl_sf_pow_int(2,n/2)*(GSL_IS_ODD(n)?M_SQRT2:1); result->val = 0.; result->err = 0.; if(n < 0) { DOMAIN_ERROR(result); // GSL_ERROR ("domain error", GSL_EDOM); } else if(n == 0) { result->val = 1.; result->err = 0.; return GSL_SUCCESS; // return 1.0; } else if(n == 1) { result->val = x; result->err = 0.; return GSL_SUCCESS; // return x; } else if(x == 0.){ if(GSL_IS_ODD(n)){ result->val = 0.; result->err = 0.; return GSL_SUCCESS; // return 0.; } else{ if(n < 301){ /* double f; int j; f = (GSL_IS_ODD(n/2)?-1.:1.); for(j=1; j < n; j+=2) { f*=j; } result->val = f; result->err = 0.; */ if(n < 297){ gsl_sf_doublefact_e(n-1, result); (GSL_IS_ODD(n/2)?result->val = -result->val:1.); } else if (n == 298){ result->val = (GSL_IS_ODD(n/2)?-1.:1.)*1.25527562259930633890922678431e304; result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); } else{ result->val = (GSL_IS_ODD(n/2)?-1.:1.)*3.7532741115719259533385880851e306; result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); } } else{ result->val = (GSL_IS_ODD(n/2)?GSL_NEGINF:GSL_POSINF); result->err = GSL_POSINF; } return GSL_SUCCESS; // return f; } } /* else if(x*x < 4.0*n && n > 100000) { // asymptotic formula double f = 1.0; int j; if(GSL_IS_ODD(n)) { f=gsl_sf_fact((n-1)/2)*gsl_sf_pow_int(2,n/2)*M_SQRT2/M_SQRTPI; } else { for(j=1; j < n; j+=2) { f*=j; } } return f*exp(x*x/4)*cos(x*sqrt(n)-(n%4)*M_PI_2)/sqrt(sqrt(1-x*x/4.0/n)); // return f*exp(x*x/4)*cos(x*sqrt(n)-n*M_PI_2)/sqrt(sqrt(1-x*x/4.0/n)); } */ else{ // printf("recurrence, n= %d\n",n); // upward recurrence: He_{n+1} = x He_n - n He_{n-1} double p_n0 = 1.0; // He_0(x) double p_n1 = x; // He_1(x) double p_n = p_n1; double e_n0 = GSL_DBL_EPSILON; double e_n1 = fabs(x)*GSL_DBL_EPSILON; double e_n = e_n1; int j=0, c=0; for(j=1; j <= n-1; j++){ if (gsl_isnan(p_n) == 1){ // printf("break at j= %d\n",j); break; } p_n = x*p_n1-j*p_n0; p_n0 = p_n1; p_n1 = p_n; e_n = (fabs(x)*e_n1+j*e_n0); e_n0 = e_n1; e_n1 = e_n; while(( fmin(fabs(p_n0),fabs(p_n1)) > 2.0e-100 ) && ( fmax(fabs(p_n0),fabs(p_n1)) > 1.0e100 )){ p_n0 = p_n0/2; p_n1 = p_n1/2; p_n = p_n1; e_n0 = e_n0/2; e_n1 = e_n1/2; e_n = e_n1; c++; } while(( ( fabs(p_n0) < 1.0e-100 ) && ( p_n0 != 0) && ( fabs(p_n1) < 1.0e-100 ) && ( p_n1 != 0) ) && ( fmax(fabs(p_n0),fabs(p_n1)) < 2.0e100 )){ p_n0 = p_n0*2; p_n1 = p_n1*2; p_n = p_n1; e_n0 = e_n0*2; e_n1 = e_n1*2; e_n = e_n1; c--; } } /* // check to see that the correct values are computed, even when overflow strikes in the end; works, thus very large results are accessible by determining mantissa and exponent separately double lg2 = 0.30102999566398119521467838; double ln10 = 2.3025850929940456840179914546843642076011014886; printf("res= %g\n", p_n*pow(10.,((lg2*c)-((long)(lg2*c)))) ); printf("res= %g * 10^(%ld)\n", p_n*pow(10.,((lg2*c)-((long)(lg2*c))))/pow(10.,((long)(log(fabs(p_n*pow(10.,((lg2*c)-((long)(lg2*c))))))/ln10))), ((long)(log(fabs(p_n*pow(10.,((lg2*c)-((long)(lg2*c))))))/ln10))+((long)(lg2*c)) ); */ result->val = pow2(c)*p_n; // result->err = n*GSL_DBL_EPSILON*fabs(result->val); result->err = pow2(c)*e_n + fabs(result->val)*GSL_DBL_EPSILON; /* result->err = e_n + n*fabs(p_n)*GSL_DBL_EPSILON; no idea, where the factor n came from => removed */ if (gsl_isnan(result->val) != 1){ return GSL_SUCCESS; // return p_n; } else{ return GSL_ERANGE; } } } static int gsl_sf_hermite_prob_appr_e(const int n, const double x, gsl_sf_result * result) // Approximatively evaluates the probabilists' Hermite polynomial of order n at position x. // An approximation depending on the x-range (see Szego, Gabor (1939, 1958, 1967), Orthogonal Polynomials, American Mathematical Society) is used. { // Plancherel-Rotach approximation (note: Szego defines the Airy function differently!) // printf("approx, n= %d\n",n); const double aizero1 = -2.3381074104597670384891972524467; // first zero of the Airy function Ai //const double aizero1 = -2.3381074104597670384891972524467354406385401456723878524838544372; // first zero of the Airy function Ai double z = fabs(x)*M_SQRT1_2; double f = 1.; int j; for(j=1; j <= n; j++) { f*=sqrt(j); } if (z < sqrt(2*n+1.)+aizero1/M_SQRT2/pow(n,1/6.)){ // printf("trig\n"); double phi = acos(z/sqrt(2*n+1.)); result->val = f*(GSL_IS_ODD(n)&&(x<0.)?-1.:1.)*pow(2./n,0.25)/sqrt(M_SQRTPI*sin(phi))*sin(M_PI*0.75+(n/2.+0.25)*(sin(2*phi)-2*phi))*exp(z*z/2.); result->err = 2. * GSL_DBL_EPSILON * fabs(result->val); return GSL_SUCCESS; // return f*(GSL_IS_ODD(n)&&(x<0.)?-1.:1.)*pow(2./n,0.25)/sqrt(M_SQRTPI*sin(phi))*sin(M_PI*0.75+(n/2.+0.25)*(sin(2*phi)-2*phi))*exp(z*z/2.); } else if (z > sqrt(2*n+1.)-aizero1/M_SQRT2/pow(n,1/6.)){ // printf("hyp\n"); // double phi = gsl_acosh(z/sqrt(2*n+1.)); double phi = acosh(z/sqrt(2*n+1.)); result->val = f*(GSL_IS_ODD(n)&&(x<0.)?-1.:1.)*pow(n,-0.25)/M_SQRT2/sqrt(M_SQRT2*M_SQRTPI*sinh(phi))*exp((n/2.+0.25)*(2*phi-sinh(2*phi)))*exp(z*z/2.); result->err = 2. * GSL_DBL_EPSILON * fabs(result->val); return GSL_SUCCESS; // return f*(GSL_IS_ODD(n)&&(x<0.)?-1.:1.)*pow(0.125/n,0.25)/sqrt(M_SQRTPI*sinh(phi))*exp((n/2.+0.25)*(2*phi-sinh(2*phi)))*exp(z*z/2.); } else{ // printf("Airy\n"); gsl_sf_result Ai; // int tmp_Ai = gsl_sf_airy_Ai_e((z-sqrt(2*n+1.))*pow(8.*n,1/6.),0,&Ai); gsl_sf_airy_Ai_e((z-sqrt(2*n+1.))*M_SQRT2*pow(n,1/6.),0,&Ai); result->val = f*(GSL_IS_ODD(n)&&(x<0.)?-1.:1.)*sqrt(M_SQRTPI*M_SQRT2)*pow(n,-1/12.)*Ai.val*exp(z*z/2.); result->err = f*sqrt(M_SQRTPI*M_SQRT2)*pow(n,-1/12.)*Ai.err*exp(z*z/2.) + GSL_DBL_EPSILON*fabs(result->val); return GSL_SUCCESS; // return f*(GSL_IS_ODD(n)&&(x<0.)?-1.:1.)*sqrt(M_SQRTPI)*pow(2.,0.25)*pow(n,-1/12.)*gsl_sf_airy_Ai((z-sqrt(2*n+1.))*pow(8.*n,1/6.),0)*exp(z*z/2.); } } int gsl_sf_hermite_prob_e(const int n, const double x, gsl_sf_result * result) // Evaluates the probabilists' Hermite polynomial of order n at position x. // For small n upward recurrence is employed, while for large n and NaNs from the iteration an approximation depending on the x-range (see Szego, Gabor (1939, 1958, 1967), Orthogonal Polynomials, American Mathematical Society) is used. { // return gsl_sf_hyperg_U(-n/2.,1./2.,x*x/2.)*gsl_sf_pow_int(2,n/2)*(GSL_IS_ODD(n)?M_SQRT2:1); if( (x==0. || n<=100000) && (gsl_sf_hermite_prob_iter_e(n,x,result)==GSL_SUCCESS) ){ return GSL_SUCCESS; } else{ return gsl_sf_hermite_prob_appr_e(n,x,result); } } double gsl_sf_hermite_prob(const int n, const double x) { EVAL_RESULT(gsl_sf_hermite_prob_e(n, x, &result)); } int gsl_sf_hermite_prob_der_e(const int m, const int n, const double x, gsl_sf_result * result) // Evaluates the m-th derivative of the probabilists' Hermite polynomial of order n at position x. // The direct formula He^{(m)}_n = n!/(n-m)!*He_{n-m}(x) (where He_j(x) is the j-th probabilists' Hermite polynomial and He^{(m)}_j(x) its m-th derivative) is employed. { if(n < 0 || m < 0) { // GSL_ERROR ("domain error", GSL_EDOM); DOMAIN_ERROR(result); } else if(n < m) { result->val = 0.; result->err = 0.; return GSL_SUCCESS; // return 0.; } else{ double f = gsl_sf_choose(n,m)*gsl_sf_fact(m); gsl_sf_result He; gsl_sf_hermite_prob_e(n-m,x,&He); result->val = He.val*f; result->err = He.err*f + GSL_DBL_EPSILON*fabs(result->val); return GSL_SUCCESS; // return gsl_sf_hermite_prob(n-m,x)*gsl_sf_choose(n,m)*gsl_sf_fact(m); } } double gsl_sf_hermite_prob_der(const int m, const int n, const double x) { EVAL_RESULT(gsl_sf_hermite_prob_der_e(m, n, x, &result)); } int gsl_sf_hermite_phys_e(const int n, const double x, gsl_sf_result * result) // Evaluates the physicists' Hermite polynomial of order n at position x. // For small n upward recurrence is employed, while for large n and NaNs from the iteration an approximation depending on the x-range (see Szego, Gabor (1939, 1958, 1967), Orthogonal Polynomials, American Mathematical Society) is used. { // return gsl_sf_hyperg_U(-n/2.,1./2.,x*x)*gsl_sf_pow_int(2,n); result->val = 0.; result->err = 0.; if(n < 0) { DOMAIN_ERROR(result); // GSL_ERROR ("domain error", GSL_EDOM); } else if(n == 0) { result->val = 1.; result->err = 0.; return GSL_SUCCESS; // return 1.0; } else if(n == 1) { result->val = 2.0*x; result->err = 0.; return GSL_SUCCESS; // return 2.0*x; } else if(x == 0.){ if(GSL_IS_ODD(n)){ result->val = 0.; result->err = 0.; return GSL_SUCCESS; // return 0.; } else{ if(n < 269){ // double f = gsl_sf_pow_int(2,n/2); double f = pow2(n/2); gsl_sf_doublefact_e(n-1, result); result->val *= f; result->err *= f; (GSL_IS_ODD(n/2)?result->val = -result->val:1.); /* double f; int j; f = (GSL_IS_ODD(n/2)?-1.:1.); for(j=1; j < n; j+=2) { f*=2*j; } result->val = f; result->err = 0.; */ } else{ result->val = (GSL_IS_ODD(n/2)?GSL_NEGINF:GSL_POSINF); result->err = GSL_POSINF; } return GSL_SUCCESS; // return gsl_sf_pow_int(2,n/2)*f; } } /* else if(x*x < 2.0*n && n > 100000) { // asymptotic formula double f = 1.0; int j; if(GSL_IS_ODD(n)) { f=gsl_sf_fact((n-1)/2)*gsl_sf_pow_int(2,n)/M_SQRTPI; } else { for(j=1; j < n; j+=2) { f*=j; } f*=gsl_sf_pow_int(2,n/2); } return f*exp(x*x/2)*cos(x*sqrt(2.0*n)-(n%4)*M_PI_2)/sqrt(sqrt(1-x*x/2.0/n)); // return f*exp(x*x/2)*cos(x*sqrt(2.0*n)-n*M_PI_2)/sqrt(sqrt(1-x*x/2.0/n)); } */ else if (n <= 100000){ // upward recurrence: H_{n+1} = 2x H_n - 2j H_{n-1} double p_n0 = 1.0; // H_0(x) double p_n1 = 2.0*x; // H_1(x) double p_n = p_n1; double e_n0 = GSL_DBL_EPSILON; double e_n1 = 2.*fabs(x)*GSL_DBL_EPSILON; double e_n = e_n1; int j=0, c=0; for(j=1; j <= n-1; j++){ if (gsl_isnan(p_n) == 1){ break; } p_n = 2.0*x*p_n1-2.0*j*p_n0; p_n0 = p_n1; p_n1 = p_n; e_n = 2.*(fabs(x)*e_n1+j*e_n0); e_n0 = e_n1; e_n1 = e_n; while(( fmin(fabs(p_n0),fabs(p_n1)) > 2.0e-100 ) && ( fmax(fabs(p_n0),fabs(p_n1)) > 1.0e100 )){ p_n0 = p_n0/2; p_n1 = p_n1/2; p_n = p_n1; e_n0 = e_n0/2; e_n1 = e_n1/2; e_n = e_n1; c++; } while(( ( fabs(p_n0) < 1.0e-100 ) && ( p_n0 != 0) && ( fabs(p_n1) < 1.0e-100 ) && ( p_n1 != 0) ) && ( fmax(fabs(p_n0),fabs(p_n1)) < 2.0e100 )){ p_n0 = p_n0*2; p_n1 = p_n1*2; p_n = p_n1; e_n0 = e_n0*2; e_n1 = e_n1*2; e_n = e_n1; c--; } } result->val = pow2(c)*p_n; // result->err = n*fabs(result->val)*GSL_DBL_EPSILON; result->err = pow2(c)*e_n + fabs(result->val)*GSL_DBL_EPSILON; /* result->err = e_n + n*fabs(p_n)*GSL_DBL_EPSILON; no idea, where the factor n came from => removed */ if (gsl_isnan(result->val) != 1){ return GSL_SUCCESS; // return p_n; } } // the following condition is implied by the logic above // if (n > 10000 || gsl_isnan(result->val) == 1){ // Plancherel-Rotach approximation (note: Szego defines the Airy function differently!) const double aizero1 = -2.3381074104597670384891972524467; // first zero of the Airy function Ai //const double aizero1 = -2.3381074104597670384891972524467354406385401456723878524838544372; // first zero of the Airy function Ai double z = fabs(x); double f = 1.; int j; for(j=1; j <= n; j++) { f*=sqrt(j); } if (z < sqrt(2*n+1.)+aizero1/M_SQRT2/pow(n,1/6.)){ double phi = acos(z/sqrt(2*n+1.)); result->val = f*(GSL_IS_ODD(n)&&(x<0.)?-1.:1.)*(GSL_IS_ODD(n)?M_SQRT2:1.)*pow2(n/2)*pow(2./n,0.25)/sqrt(M_SQRTPI*sin(phi))*sin(M_PI*0.75+(n/2.+0.25)*(sin(2*phi)-2*phi))*exp(z*z/2.); result->err = 2. * GSL_DBL_EPSILON * fabs(result->val); return GSL_SUCCESS; // return f*(GSL_IS_ODD(n)&&(x<0.)?-1.:1.)*(GSL_IS_ODD(n)?M_SQRT2:1.)*gsl_sf_pow_int(2,n/2)*pow(2./n,0.25)/sqrt(M_SQRTPI*sin(phi))*sin(M_PI*0.75+(n/2.+0.25)*(sin(2*phi)-2*phi))*exp(z*z/2.); } else if (z > sqrt(2*n+1.)-aizero1/M_SQRT2/pow(n,1/6.)){ // double phi = gsl_acosh(z/sqrt(2*n+1.)); double phi = acosh(z/sqrt(2*n+1.)); result->val = f*(GSL_IS_ODD(n)&&(x<0.)?-1.:1.)*(GSL_IS_ODD(n)?1.:M_SQRT1_2)*pow2(n/2)*pow(n,-0.25)/sqrt(M_SQRT2*M_SQRTPI*sinh(phi))*exp((n/2.+0.25)*(2*phi-sinh(2*phi)))*exp(z*z/2.); result->err = 2. * GSL_DBL_EPSILON * fabs(result->val); return GSL_SUCCESS; // return f*(GSL_IS_ODD(n)&&(x<0.)?-1.:1.)*(GSL_IS_ODD(n)?M_SQRT2:1.)*gsl_sf_pow_int(2,n/2)*pow(0.125/n,0.25)/sqrt(M_SQRTPI*sinh(phi))*exp((n/2.+0.25)*(2*phi-sinh(2*phi)))*exp(z*z/2.); } else{ gsl_sf_result Ai; gsl_sf_airy_Ai_e((z-sqrt(2*n+1.))*M_SQRT2*pow(n,1/6.),0,&Ai); result->val = f*(GSL_IS_ODD(n)&&(x<0.)?-1.:1.)*(GSL_IS_ODD(n)?M_SQRT2:1.)*sqrt(M_SQRTPI*M_SQRT2)*pow2(n/2)*pow(n,-1/12.)*Ai.val*exp(z*z/2.); result->err = f*(GSL_IS_ODD(n)?M_SQRT2:1.)*pow2(n/2)*sqrt(M_SQRTPI*M_SQRT2)*pow(n,-1/12.)*Ai.err*exp(z*z/2.) + GSL_DBL_EPSILON*fabs(result->val); return GSL_SUCCESS; // return f*(GSL_IS_ODD(n)&&(x<0.)?-1.:1.)*(GSL_IS_ODD(n)?M_SQRT2:1.)*sqrt(M_SQRTPI)*gsl_sf_pow_int(2,n/2)*pow(2.,0.25)*pow(n,-1/12.)*gsl_sf_airy_Ai((z-sqrt(2*n+1.))*pow(8.*n,1/6.),0)*exp(z*z/2.); } // } } double gsl_sf_hermite_phys(const int n, const double x) { EVAL_RESULT(gsl_sf_hermite_phys_e(n, x, &result)); } int gsl_sf_hermite_phys_der_e(const int m, const int n, const double x, gsl_sf_result * result) // Evaluates the m-th derivative of the physicists' Hermite polynomial of order n at position x. // The direct formula H^{(m)}_n = 2**m*n!/(n-m)!*H_{n-m}(x) (where H_j(x) is the j-th physicists' Hermite polynomial and H^{(m)}_j(x) its m-th derivative) is employed. { if(n < 0 || m < 0) { DOMAIN_ERROR(result); // GSL_ERROR ("domain error", GSL_EDOM); } else if(n < m) { result->val = 0.; result->err = 0.; return GSL_SUCCESS; // return 0.; } else{ // double f = gsl_sf_choose(n,m)*gsl_sf_fact(m)*gsl_sf_pow_int(2,m); double f = gsl_sf_choose(n,m)*gsl_sf_fact(m)*pow2(m); gsl_sf_result H; gsl_sf_hermite_phys_e(n-m,x,&H); result->val = H.val*f; result->err = H.err*f + GSL_DBL_EPSILON*fabs(result->val); return GSL_SUCCESS; // return gsl_sf_hermite_phys(n-m,x)*gsl_sf_choose(n,m)*gsl_sf_fact(m)*gsl_sf_pow_int(2,m); } } double gsl_sf_hermite_phys_der(const int m, const int n, const double x) { EVAL_RESULT(gsl_sf_hermite_phys_der_e(m, n, x, &result)); } int gsl_sf_hermite_func_e(const int n, const double x, gsl_sf_result * result) // Evaluates the Hermite function of order n at position x. // For large n an approximation depending on the x-range (see Szego, Gabor (1939, 1958, 1967), Orthogonal Polynomials, American Mathematical Society) is used, while for small n the direct formula via the probabilists' Hermite polynomial is applied. { /* if (x*x < 2.0*n && n > 100000){ // asymptotic formula double f = 1.0; int j; // return f*exp(x*x/4)*cos(x*sqrt(n)-n*M_PI_2)/sqrt(sqrt(1-x*x/4.0/n)); return cos(x*sqrt(2.0*n)-(n%4)*M_PI_2)/sqrt(sqrt(n/M_PI/2.0*(1-x*x/2.0/n)))/M_PI; } */ if (n < 0){ DOMAIN_ERROR(result); // GSL_ERROR ("domain error", GSL_EDOM); } else if(n == 0 && x != 0.) { result->val = exp(-x*x/2.)/sqrt(M_SQRTPI); result->err = GSL_DBL_EPSILON*fabs(result->val); return GSL_SUCCESS; // return 1.0; } else if(n == 1 && x != 0.) { result->val = M_SQRT2*x*exp(-x*x/2.)/sqrt(M_SQRTPI); result->err = GSL_DBL_EPSILON*fabs(result->val); return GSL_SUCCESS; // return 2.0*x; } else if (x == 0.){ if (GSL_IS_ODD(n)){ result->val = 0.; result->err = 0.; return GSL_SUCCESS; // return 0.; } else{ double f; int j; f = (GSL_IS_ODD(n/2)?-1.:1.); for(j=1; j < n; j+=2) { f*=sqrt(j/(j+1.)); } result->val = f/sqrt(M_SQRTPI); result->err = GSL_DBL_EPSILON*fabs(result->val); return GSL_SUCCESS; // return f/sqrt(M_SQRTPI); } } else if (n <= 100000){ double f = exp(-x*x/2)/sqrt(M_SQRTPI*gsl_sf_fact(n)); gsl_sf_result He; gsl_sf_hermite_prob_iter_e(n,M_SQRT2*x,&He); result->val = He.val*f; result->err = He.err*f + GSL_DBL_EPSILON*fabs(result->val); if (gsl_isnan(result->val) != 1 && f > 1.0e-300 && gsl_finite(He.val) == 1){ return GSL_SUCCESS; } // return gsl_sf_hermite_prob(n,M_SQRT2*x)*exp(-x*x/2)/sqrt(M_SQRTPI*gsl_sf_fact(n)); } // the following condition is implied by the logic above // else if (n > 100000 || gsl_isnan(result->val) == 1){ // upward recurrence: Psi_{n+1} = sqrt(2/(n+1))*x Psi_n - sqrt(n/(n+1)) Psi_{n-1} double tw = exp(-x*x/2./n); // "twiddle factor" (in the spirit of FFT) double p_n0 = tw/sqrt(M_SQRTPI); // Psi_0(x) // double tw = 1.; // double p_n0 = exp(-x*x/2.)/sqrt(M_SQRTPI); // Psi_0(x) double p_n1 = p_n0*M_SQRT2*x; // Psi_1(x) double p_n = p_n1; double e_n0 = p_n0*GSL_DBL_EPSILON; double e_n1 = p_n1*GSL_DBL_EPSILON; double e_n = e_n1; int j; int c = 0; for (j=1;j 2.0e-100 ) && ( fmax(fabs(p_n0),fabs(p_n1)) > 1.0e100 )){ p_n0 = p_n0/2; p_n1 = p_n1/2; p_n = p_n1; e_n0 = e_n0/2; e_n1 = e_n1/2; e_n = e_n1; c++; } while(( ( fabs(p_n0) < 1.0e-100 ) && ( p_n0 != 0) && ( fabs(p_n1) < 1.0e-100 ) && ( p_n1 != 0) ) && ( fmax(fabs(p_n0),fabs(p_n1)) < 2.0e100 )){ p_n0 = p_n0*2; p_n1 = p_n1*2; p_n = p_n1; e_n0 = e_n0*2; e_n1 = e_n1*2; e_n = e_n1; c--; } } result->val = p_n*pow2(c); // result->err = e_n*pow2(c) + fabs(result->val)*GSL_DBL_EPSILON; result->err = n*fabs(result->val)*GSL_DBL_EPSILON; if (gsl_isnan(result->val) != 1){ return GSL_SUCCESS; // return p_n; } // Plancherel-Rotach approximation (note: Szego defines the Airy function differently!) const double aizero1 = -2.3381074104597670384891972524467; // first zero of the Airy function Ai //const double aizero1 = -2.3381074104597670384891972524467354406385401456723878524838544372; // first zero of the Airy function Ai double z = fabs(x); if (z < sqrt(2*n+1.)+aizero1/M_SQRT2/pow(n,1/6.)){ // printf("hermite func trig approx\n"); double phi = acos(z/sqrt(2*n+1.)); result->val = (GSL_IS_ODD(n)&&(x<0.)?-1.:1.)*pow(2./n,0.25)/M_SQRTPI/sqrt(sin(phi))*sin(M_PI*0.75+(n/2.+0.25)*(sin(2*phi)-2*phi)); result->err = 2. * GSL_DBL_EPSILON * fabs(result->val); return GSL_SUCCESS; // return (GSL_IS_ODD(n)&&(x<0.)?-1.:1.)*pow(2./n,0.25)/M_SQRTPI/sqrt(sin(phi))*sin(M_PI*0.75+(n/2.+0.25)*(sin(2*phi)-2*phi)); } else if (z > sqrt(2*n+1.)-aizero1/M_SQRT2/pow(n,1/6.)){ // printf("hermite func hyp approx\n"); // double phi = gsl_acosh(z/sqrt(2*n+1.)); double phi = acosh(z/sqrt(2*n+1.)); result->val = (GSL_IS_ODD(n)&&(x<0.)?-1.:1.)*pow(n,-0.25)/ 2/M_SQRTPI/sqrt(sinh(phi)/M_SQRT2)*exp((n/2.+0.25)*(2*phi-sinh(2*phi))); result->err = 2. * GSL_DBL_EPSILON * fabs(result->val); return GSL_SUCCESS; // return (GSL_IS_ODD(n)&&(x<0.)?-1.:1.)*pow(0.125/n,0.25)/M_SQRTPI/sqrt(sinh(phi))*exp((n/2.+0.25)*(2*phi-sinh(2*phi))); } else{ gsl_sf_result Ai; // printf("hermite func Airy approx\n"); // int tmp_Ai = gsl_sf_airy_Ai_e((z-sqrt(2*n+1.))*pow(8.*n,1/6.),0,&Ai); gsl_sf_airy_Ai_e((z-sqrt(2*n+1.))*M_SQRT2*pow(n,1/6.),0,&Ai); result->val = (GSL_IS_ODD(n)&&(x<0.)?-1.:1.)*sqrt(M_SQRT2)*pow(n,-1/12.)*Ai.val; result->err = sqrt(M_SQRT2)*pow(n,-1/12.)*Ai.err + GSL_DBL_EPSILON*fabs(result->val); return GSL_SUCCESS; // return (GSL_IS_ODD(n)&&(x<0.)?-1.:1.)*pow(2.,0.25)*pow(n,-1/12.)*gsl_sf_airy_Ai((z-sqrt(2*n+1.))*pow(8.*n,1/6.),0); } // } } double gsl_sf_hermite_func(const int n, const double x) { EVAL_RESULT(gsl_sf_hermite_func_e(n, x, &result)); } int gsl_sf_hermite_prob_array(const int nmax, const double x, double * result_array) // Evaluates all probabilists' Hermite polynomials up to order nmax at position x. The results are stored in result_array. // Since all polynomial orders are needed, upward recurrence is employed. { // CHECK_POINTER(result_array) if(nmax < 0) { GSL_ERROR ("domain error", GSL_EDOM); } else if(nmax == 0) { result_array[0] = 1.0; return GSL_SUCCESS; } else if(nmax == 1) { result_array[0] = 1.0; result_array[1] = x; return GSL_SUCCESS; } else { // upward recurrence: He_{n+1} = x He_n - n He_{n-1} double p_n0 = 1.0; // He_0(x) double p_n1 = x; // He_1(x) double p_n = p_n1; int j=0, c=0; result_array[0] = 1.0; result_array[1] = x; for(j=1; j <= nmax-1; j++){ p_n = x*p_n1-j*p_n0; p_n0 = p_n1; p_n1 = p_n; while(( fmin(fabs(p_n0),fabs(p_n1)) > 2.0e-100 ) && ( fmax(fabs(p_n0),fabs(p_n1)) > 1.0e100 )){ p_n0 = p_n0/2; p_n1 = p_n1/2; p_n = p_n1; c++; } while(( ( fabs(p_n0) < 1.0e-100 ) && ( p_n0 != 0) && ( fabs(p_n1) < 1.0e-100 ) && ( p_n1 != 0) ) && ( fmax(fabs(p_n0),fabs(p_n1)) < 2.0e100 )){ p_n0 = p_n0*2; p_n1 = p_n1*2; p_n = p_n1; c--; } result_array[j+1] = pow2(c)*p_n; } return GSL_SUCCESS; } } int gsl_sf_hermite_prob_array_der(const int m, const int nmax, const double x, double * result_array) // Evaluates the m-th derivative of all probabilists' Hermite polynomials up to order nmax at position x. The results are stored in result_array. // Since all polynomial orders are needed, upward recurrence is employed. { // CHECK_POINTER(result_array) if(nmax < 0 || m < 0) { GSL_ERROR ("domain error", GSL_EDOM); } else if(m == 0) { gsl_sf_hermite_prob_array(nmax, x, result_array); return GSL_SUCCESS; } else if(nmax < m) { int j; for(j=0; j <= nmax; j++){ result_array[j] = 0.0; } return GSL_SUCCESS; } else if(nmax == m) { int j; for(j=0; j < m; j++){ result_array[j] = 0.0; } result_array[nmax] = gsl_sf_fact(m); return GSL_SUCCESS; } else if(nmax == m+1) { int j; for(j=0; j < m; j++){ result_array[j] = 0.0; } result_array[nmax-1] = gsl_sf_fact(m); result_array[nmax] = result_array[nmax-1]*(m+1)*x; return GSL_SUCCESS; } else { // upward recurrence: He^{(m)}_{n+1} = (n+1)/(n-m+1)*(x He^{(m)}_n - n He^{(m)}_{n-1}) double p_n0 = gsl_sf_fact(m); // He^{(m)}_{m}(x) double p_n1 = p_n0*(m+1)*x; // He^{(m)}_{m+1}(x) double p_n = p_n1; int j=0, c=0; for(j=0; j < m; j++){ result_array[j] = 0.0; } result_array[m] = p_n0; result_array[m+1] = p_n1; for(j=m+1; j <= nmax-1; j++){ p_n = (x*p_n1-j*p_n0)*(j+1)/(j-m+1); p_n0 = p_n1; p_n1 = p_n; while(( fmin(fabs(p_n0),fabs(p_n1)) > 2.0e-100 ) && ( fmax(fabs(p_n0),fabs(p_n1)) > 1.0e100 )){ p_n0 = p_n0/2; p_n1 = p_n1/2; p_n = p_n1; c++; } while(( ( fabs(p_n0) < 1.0e-100 ) && ( p_n0 != 0) && ( fabs(p_n1) < 1.0e-100 ) && ( p_n1 != 0) ) && ( fmax(fabs(p_n0),fabs(p_n1)) < 2.0e100 )){ p_n0 = p_n0*2; p_n1 = p_n1*2; p_n = p_n1; c--; } result_array[j+1] = pow2(c)*p_n; } return GSL_SUCCESS; } } int gsl_sf_hermite_prob_der_array(const int mmax, const int n, const double x, double * result_array) // Evaluates all derivatives (starting from 0) up to the mmax-th derivative of the probabilists' Hermite polynomial of order n at position x. The results are stored in result_array. // Since all polynomial orders are needed, upward recurrence is employed. { // CHECK_POINTER(result_array) if(n < 0 || mmax < 0) { GSL_ERROR ("domain error", GSL_EDOM); } else if(n == 0) { result_array[0] = 1.0; int j; for(j=1; j <= mmax; j++){ result_array[j] = 0.0; } return GSL_SUCCESS; } else if(n == 1 && mmax > 0) { result_array[0] = x; result_array[1] = 1.0; int j; for(j=2; j <= mmax; j++){ result_array[j] = 0.0; } return GSL_SUCCESS; } else if( mmax == 0) { result_array[0] = gsl_sf_hermite_prob(n,x); return GSL_SUCCESS; } else if( mmax == 1) { result_array[0] = gsl_sf_hermite_prob(n,x); result_array[1] = n*gsl_sf_hermite_prob(n-1,x); return GSL_SUCCESS; } else { // upward recurrence int k = GSL_MAX_INT(0,n-mmax); // Getting a bit lazy here... double p_n0 = gsl_sf_hermite_prob(k,x); // He_k(x) double p_n1 = gsl_sf_hermite_prob(k+1,x); // He_{k+1}(x) double p_n = p_n1; int j=0, c=0; for(j=n+1; j <= mmax; j++){ result_array[j] = 0.0; } result_array[GSL_MIN_INT(n,mmax)] = p_n0; result_array[GSL_MIN_INT(n,mmax)-1] = p_n1; for(j=GSL_MIN_INT(mmax,n)-1; j > 0; j--){ k++; p_n = x*p_n1-k*p_n0; p_n0 = p_n1; p_n1 = p_n; while(( fmin(fabs(p_n0),fabs(p_n1)) > 2.0e-100 ) && ( fmax(fabs(p_n0),fabs(p_n1)) > 1.0e100 )){ p_n0 = p_n0/2; p_n1 = p_n1/2; p_n = p_n1; c++; } while(( ( fabs(p_n0) < 1.0e-100 ) && ( p_n0 != 0) && ( fabs(p_n1) < 1.0e-100 ) && ( p_n1 != 0) ) && ( fmax(fabs(p_n0),fabs(p_n1)) < 2.0e100 )){ p_n0 = p_n0*2; p_n1 = p_n1*2; p_n = p_n1; c--; } result_array[j-1] = pow2(c)*p_n; } p_n = 1.0; for(j=1; j <= GSL_MIN_INT(n,mmax); j++){ p_n = p_n*(n-j+1); result_array[j] = p_n*result_array[j]; } return GSL_SUCCESS; } } int gsl_sf_hermite_prob_series_e(const int n, const double x, const double * a, gsl_sf_result * result) // Evaluates the series sum_{j=0}^n a_j*He_j(x) with He_j being the j-th probabilists' Hermite polynomial. // For improved numerical stability the Clenshaw algorithm (Clenshaw, C. W. (July 1955). "A note on the summation of Chebyshev series". Mathematical Tables and other Aids to Computation 9 (51): 118–110.) adapted to probabilists' Hermite polynomials is used. { // CHECK_POINTER(a) if(n < 0) { DOMAIN_ERROR(result); // GSL_ERROR ("domain error", GSL_EDOM); } else if(n == 0) { result->val = a[0]; result->err = 0.; return GSL_SUCCESS; // return a[0]; } else if(n == 1) { result->val = a[0]+a[1]*x; result->err = 2.*GSL_DBL_EPSILON * (fabs(a[0]) + fabs(a[1]*x)) ; return GSL_SUCCESS; // return a[0]+a[1]*x; } else { // downward recurrence: b_n = a_n + x b_{n+1} - (n+1) b_{n+2} double b0 = 0.; double b1 = 0.; double btmp = 0.; double e0 = 0.; double e1 = 0.; double etmp = e1; int j; for(j=n; j >= 0; j--){ btmp = b0; b0 = a[j]+x*b0-(j+1)*b1; b1 = btmp; etmp = e0; e0 = (GSL_DBL_EPSILON*fabs(a[j])+fabs(x)*e0+(j+1)*e1); e1 = etmp; } result->val = b0; result->err = e0 + fabs(b0)*GSL_DBL_EPSILON; return GSL_SUCCESS; // return b0; } } double gsl_sf_hermite_prob_series(const int n, const double x, const double * a) { EVAL_RESULT(gsl_sf_hermite_prob_series_e(n, x, a, &result)); } int gsl_sf_hermite_phys_array(const int nmax, const double x, double * result_array) // Evaluates all physicists' Hermite polynomials up to order nmax at position x. The results are stored in result_array. // Since all polynomial orders are needed, upward recurrence is employed. { // CHECK_POINTER(result_array) if(nmax < 0) { GSL_ERROR ("domain error", GSL_EDOM); } else if(nmax == 0) { result_array[0] = 1.0; return GSL_SUCCESS; } else if(nmax == 1) { result_array[0] = 1.0; result_array[1] = 2.0*x; return GSL_SUCCESS; } else { // upward recurrence: H_{n+1} = 2x H_n - 2n H_{n-1} double p_n0 = 1.0; // H_0(x) double p_n1 = 2.0*x; // H_1(x) double p_n = p_n1; int j=0, c=0; result_array[0] = 1.0; result_array[1] = 2.0*x; for(j=1; j <= nmax-1; j++){ p_n = 2.0*x*p_n1-2.0*j*p_n0; p_n0 = p_n1; p_n1 = p_n; while(( fmin(fabs(p_n0),fabs(p_n1)) > 2.0e-100 ) && ( fmax(fabs(p_n0),fabs(p_n1)) > 1.0e100 )){ p_n0 = p_n0/2; p_n1 = p_n1/2; p_n = p_n1; c++; } while(( ( fabs(p_n0) < 1.0e-100 ) && ( p_n0 != 0) && ( fabs(p_n1) < 1.0e-100 ) && ( p_n1 != 0) ) && ( fmax(fabs(p_n0),fabs(p_n1)) < 2.0e100 )){ p_n0 = p_n0*2; p_n1 = p_n1*2; p_n = p_n1; c--; } result_array[j+1] = pow2(c)*p_n; } return GSL_SUCCESS; } } int gsl_sf_hermite_phys_array_der(const int m, const int nmax, const double x, double * result_array) // Evaluates the m-th derivative of all physicists' Hermite polynomials up to order nmax at position x. The results are stored in result_array. // Since all polynomial orders are needed, upward recurrence is employed. { // CHECK_POINTER(result_array) if(nmax < 0 || m < 0) { GSL_ERROR ("domain error", GSL_EDOM); } else if(m == 0) { gsl_sf_hermite_phys_array(nmax, x, result_array); return GSL_SUCCESS; } else if(nmax < m) { int j; for(j=0; j <= nmax; j++){ result_array[j] = 0.0; } return GSL_SUCCESS; } else if(nmax == m) { int j; for(j=0; j < m; j++){ result_array[j] = 0.0; } // result_array[nmax] = gsl_sf_pow_int(2,m)*gsl_sf_fact(m); result_array[nmax] = pow2(m)*gsl_sf_fact(m); return GSL_SUCCESS; } else if(nmax == m+1) { int j; for(j=0; j < m; j++){ result_array[j] = 0.0; } // result_array[nmax-1] = gsl_sf_pow_int(2,m)*gsl_sf_fact(m); result_array[nmax-1] = pow2(m)*gsl_sf_fact(m); result_array[nmax] = result_array[nmax-1]*2*(m+1)*x; return GSL_SUCCESS; } else { // upward recurrence: H^{(m)}_{n+1} = 2(n+1)/(n-m+1)*(x H^{(m)}_n - n H^{(m)}_{n-1}) // double p_n0 = gsl_sf_pow_int(2,m)*gsl_sf_fact(m); // H^{(m)}_{m}(x) double p_n0 = pow2(m)*gsl_sf_fact(m); // H^{(m)}_{m}(x) double p_n1 = p_n0*2*(m+1)*x; // H^{(m)}_{m+1}(x) double p_n = p_n1; int j=0, c=0; for(j=0; j < m; j++){ result_array[j] = 0.0; } result_array[m] = p_n0; result_array[m+1] = p_n1; for(j=m+1; j <= nmax-1; j++){ p_n = (x*p_n1-j*p_n0)*2*(j+1)/(j-m+1); p_n0 = p_n1; p_n1 = p_n; while(( fmin(fabs(p_n0),fabs(p_n1)) > 2.0e-100 ) && ( fmax(fabs(p_n0),fabs(p_n1)) > 1.0e100 )){ p_n0 = p_n0/2; p_n1 = p_n1/2; p_n = p_n1; c++; } while(( ( fabs(p_n0) < 1.0e-100 ) && ( p_n0 != 0) && ( fabs(p_n1) < 1.0e-100 ) && ( p_n1 != 0) ) && ( fmax(fabs(p_n0),fabs(p_n1)) < 2.0e100 )){ p_n0 = p_n0*2; p_n1 = p_n1*2; p_n = p_n1; c--; } result_array[j+1] = pow2(c)*p_n; } return GSL_SUCCESS; } } int gsl_sf_hermite_phys_der_array(const int mmax, const int n, const double x, double * result_array) // Evaluates all derivatives (starting from 0) up to the mmax-th derivative of the physicists' Hermite polynomial of order n at position x. The results are stored in result_array. // Since all polynomial orders are needed, upward recurrence is employed. { // CHECK_POINTER(result_array) if(n < 0 || mmax < 0) { GSL_ERROR ("domain error", GSL_EDOM); } else if(n == 0) { result_array[0] = 1.0; int j; for(j=1; j <= mmax; j++){ result_array[j] = 0.0; } return GSL_SUCCESS; } else if(n == 1 && mmax > 0) { result_array[0] = 2*x; result_array[1] = 1.0; int j; for(j=2; j <= mmax; j++){ result_array[j] = 0.0; } return GSL_SUCCESS; } else if( mmax == 0) { result_array[0] = gsl_sf_hermite_phys(n,x); return GSL_SUCCESS; } else if( mmax == 1) { result_array[0] = gsl_sf_hermite_phys(n,x); result_array[1] = 2*n*gsl_sf_hermite_phys(n-1,x); return GSL_SUCCESS; } else { // upward recurrence int k = GSL_MAX_INT(0,n-mmax); // Getting a bit lazy here... double p_n0 = gsl_sf_hermite_phys(k,x); // H_k(x) double p_n1 = gsl_sf_hermite_phys(k+1,x); // H_{k+1}(x) double p_n = p_n1; int j=0, c=0; for(j=n+1; j <= mmax; j++){ result_array[j] = 0.0; } result_array[GSL_MIN_INT(n,mmax)] = p_n0; result_array[GSL_MIN_INT(n,mmax)-1] = p_n1; for(j=GSL_MIN_INT(mmax,n)-1; j > 0; j--){ k++; p_n = 2*x*p_n1-2*k*p_n0; p_n0 = p_n1; p_n1 = p_n; while(( fmin(fabs(p_n0),fabs(p_n1)) > 2.0e-100 ) && ( fmax(fabs(p_n0),fabs(p_n1)) > 1.0e100 )){ p_n0 = p_n0/2; p_n1 = p_n1/2; p_n = p_n1; c++; } while(( ( fabs(p_n0) < 1.0e-100 ) && ( p_n0 != 0) && ( fabs(p_n1) < 1.0e-100 ) && ( p_n1 != 0) ) && ( fmax(fabs(p_n0),fabs(p_n1)) < 2.0e100 )){ p_n0 = p_n0*2; p_n1 = p_n1*2; p_n = p_n1; c--; } result_array[j-1] = pow2(c)*p_n; } p_n = 1.0; for(j=1; j <= GSL_MIN_INT(n,mmax); j++){ p_n = p_n*(n-j+1)*2; result_array[j] = p_n*result_array[j]; } return GSL_SUCCESS; } } int gsl_sf_hermite_phys_series_e(const int n, const double x, const double * a, gsl_sf_result * result) // Evaluates the series sum_{j=0}^n a_j*H_j(x) with H_j being the j-th physicists' Hermite polynomial. // For improved numerical stability the Clenshaw algorithm (Clenshaw, C. W. (July 1955). "A note on the summation of Chebyshev series". Mathematical Tables and other Aids to Computation 9 (51): 118–110.) adapted to physicists' Hermite polynomials is used. { // CHECK_POINTER(a) if(n < 0) { DOMAIN_ERROR(result); // GSL_ERROR ("domain error", GSL_EDOM); } else if(n == 0) { result->val = a[0]; result->err = 0.; return GSL_SUCCESS; // return a[0]; } else if(n == 1) { result->val = a[0]+a[1]*2.*x; result->err = 2.*GSL_DBL_EPSILON * (fabs(a[0]) + fabs(a[1]*2.*x)) ; return GSL_SUCCESS; // return a[0]+a[1]*2.*x; } else { // downward recurrence: b_n = a_n + 2x b_{n+1} - 2(n+1) b_{n+2} double b0 = 0.; double b1 = 0.; double btmp = 0.; double e0 = 0.; double e1 = 0.; double etmp = e1; int j; for(j=n; j >= 0; j--){ btmp = b0; b0 = a[j]+2.*x*b0-2.*(j+1)*b1; b1 = btmp; etmp = e0; e0 = (GSL_DBL_EPSILON*fabs(a[j])+fabs(2.*x)*e0+2.*(j+1)*e1); e1 = etmp; } result->val = b0; result->err = e0 + fabs(b0)*GSL_DBL_EPSILON; return GSL_SUCCESS; // return b0; } } double gsl_sf_hermite_phys_series(const int n, const double x, const double * a) { EVAL_RESULT(gsl_sf_hermite_phys_series_e(n, x, a, &result)); } int gsl_sf_hermite_func_array(const int nmax, const double x, double * result_array) // Evaluates all Hermite functions up to order nmax at position x. The results are stored in result_array. // Since all polynomial orders are needed, upward recurrence is employed. { // CHECK_POINTER(result_array) if(nmax < 0) { GSL_ERROR ("domain error", GSL_EDOM); } else if(nmax == 0) { result_array[0] = exp(-x*x/2.)/sqrt(M_SQRTPI); return GSL_SUCCESS; } else if(nmax == 1) { result_array[0] = exp(-x*x/2.)/sqrt(M_SQRTPI); result_array[1] = result_array[0]*M_SQRT2*x; return GSL_SUCCESS; } else { // upward recurrence: Psi_{n+1} = sqrt(2/(n+1))*x Psi_n - sqrt(n/(n+1)) Psi_{n-1} double p_n0 = exp(-x*x/2.)/sqrt(M_SQRTPI); // Psi_0(x) double p_n1 = p_n0*M_SQRT2*x; // Psi_1(x) double p_n = p_n1; int j=0, c=0; result_array[0] = p_n0; result_array[1] = p_n1; for (j=1;j<=nmax-1;j++) { p_n=(M_SQRT2*x*p_n1-sqrt(j)*p_n0)/sqrt(j+1.); p_n0=p_n1; p_n1=p_n; while(( fmin(fabs(p_n0),fabs(p_n1)) > 2.0e-100 ) && ( fmax(fabs(p_n0),fabs(p_n1)) > 1.0e100 )){ p_n0 = p_n0/2; p_n1 = p_n1/2; p_n = p_n1; c++; } while(( ( fabs(p_n0) < 1.0e-100 ) && ( p_n0 != 0) && ( fabs(p_n1) < 1.0e-100 ) && ( p_n1 != 0) ) && ( fmax(fabs(p_n0),fabs(p_n1)) < 2.0e100 )){ p_n0 = p_n0*2; p_n1 = p_n1*2; p_n = p_n1; c--; } result_array[j+1] = pow2(c)*p_n; } return GSL_SUCCESS; } } int gsl_sf_hermite_func_series_e(const int n, const double x, const double * a, gsl_sf_result * result) // Evaluates the series sum_{j=0}^n a_j*Psi_j(x) with Psi_j being the j-th Hermite function. // For improved numerical stability the Clenshaw algorithm (Clenshaw, C. W. (July 1955). "A note on the summation of Chebyshev series". Mathematical Tables and other Aids to Computation 9 (51): 118–110.) adapted to Hermite functions is used. { // CHECK_POINTER(a) if(n < 0) { DOMAIN_ERROR(result); // GSL_ERROR ("domain error", GSL_EDOM); } else if(n == 0) { result->val = a[0]*exp(-x*x/2.)/sqrt(M_SQRTPI); result->err = GSL_DBL_EPSILON*fabs(result->val); return GSL_SUCCESS; // return a[0]*exp(-x*x/2.)/sqrt(M_SQRTPI); } else if(n == 1) { result->val = (a[0]+a[1]*M_SQRT2*x)*exp(-x*x/2.)/sqrt(M_SQRTPI); result->err = 2.*GSL_DBL_EPSILON*(fabs(a[0])+fabs(a[1]*M_SQRT2*x))*exp(-x*x/2.)/sqrt(M_SQRTPI); return GSL_SUCCESS; // return (a[0]+a[1]*M_SQRT2*x)*exp(-x*x/2.)/sqrt(M_SQRTPI); } else { // downward recurrence: b_n = a_n + sqrt(2/(n+1))*x b_{n+1} - sqrt((n+1)/(n+2)) b_{n+2} double b0 = 0.; double b1 = 0.; double btmp = 0.; double e0 = 0.; double e1 = 0.; double etmp = e1; int j; for(j=n; j >= 0; j--){ btmp = b0; b0 = a[j]+sqrt(2./(j+1))*x*b0-sqrt((j+1.)/(j+2.))*b1; b1 = btmp; etmp = e0; e0 = (GSL_DBL_EPSILON*fabs(a[j])+sqrt(2./(j+1))*fabs(x)*e0+sqrt((j+1.)/(j+2.))*e1); e1 = etmp; } result->val = b0*exp(-x*x/2.)/sqrt(M_SQRTPI); result->err = e0 + fabs(result->val)*GSL_DBL_EPSILON; return GSL_SUCCESS; // return b0*exp(-x*x/2.)/sqrt(M_SQRTPI); } } double gsl_sf_hermite_func_series(const int n, const double x, const double * a) { EVAL_RESULT(gsl_sf_hermite_func_series_e(n, x, a, &result)); } int gsl_sf_hermite_func_der_e(const int m, const int n, const double x, gsl_sf_result * result) // Evaluates the m-th derivative of the Hermite function of order n at position x. // A summation including upward recurrences is used. { // FIXME: asymptotic formula! if(m < 0 || n < 0) { DOMAIN_ERROR(result); // GSL_ERROR ("domain error", GSL_EDOM); } else if(m == 0){ return gsl_sf_hermite_func_e(n,x,result); } else{ int j=0, c=0; double r,er,b; double h0 = 1.; double h1 = x; double eh0 = GSL_DBL_EPSILON; double eh1 = GSL_DBL_EPSILON; double p0 = 1.; double p1 = M_SQRT2*x; double ep0 = GSL_DBL_EPSILON; double ep1 = M_SQRT2*GSL_DBL_EPSILON; double f = 1.; for (j=GSL_MAX_INT(1,n-m+1);j<=n;j++) { //f*=2.*j; f *= sqrt(2.*j); } //f*=gsl_sf_pow_int(2,GSL_MIN_INT(n,m)/2)*(GSL_IS_ODD(GSL_MIN_INT(n,m))?M_SQRT2:1.); //f*=pow2(GSL_MIN_INT(n,m)/2)*(GSL_IS_ODD(GSL_MIN_INT(n,m))?M_SQRT2:1.); //f=sqrt(f); if (m>n) { f = (GSL_IS_ODD(m-n)?-f:f); for (j=0;j 2.0e-100 ) && ( fmax(fabs(h0),fabs(h1)) > 1.0e100 )){ h0 = h0/2; h1 = h1/2; eh0 = eh0/2; eh1 = eh1/2; c++; } while(( (fabs(h0) < 1.0e-100) && (h0 != 0) && (fabs(h1) < 1.0e-100) && (h1 != 0) ) && ( fmax(fabs(h0),fabs(h1)) < 2.0e100 )){ h0 = h0*2; h1 = h1*2; eh0 = eh0*2; eh1 = eh1*2; c--; } } h0 *= pow2(c); h1 *= pow2(c); eh0 *= pow2(c); eh1 *= pow2(c); b = 0.; c = 0; for (j=1;j<=n-m;j++) { b = (M_SQRT2*x*p1-sqrt(j)*p0)/sqrt(j+1.); p0 = p1; p1 = b; b = (M_SQRT2*fabs(x)*ep1+sqrt(j)*ep0)/sqrt(j+1.); ep0 = ep1; ep1 = b; while(( fmin(fabs(p0),fabs(p1)) > 2.0e-100 ) && ( fmax(fabs(p0),fabs(p1)) > 1.0e100 )){ p0 = p0/2; p1 = p1/2; ep0 = ep0/2; ep1 = ep1/2; c++; } while(( (fabs(p0) < 1.0e-100) && (p0 != 0) && (fabs(p1) < 1.0e-100) && (p1 != 0) ) && ( fmax(fabs(p0),fabs(p1)) < 2.0e100 )){ p0 = p0*2; p1 = p1*2; ep0 = ep0*2; ep1 = ep1*2; c--; } } p0 *= pow2(c); p1 *= pow2(c); ep0 *= pow2(c); ep1 *= pow2(c); c = 0; b = 0.; r = 0.; er = 0.; for (j=GSL_MAX_INT(0,m-n);j<=m;j++) { r += f*h0*p0; er += eh0*fabs(f*p0)+ep0*fabs(f*h0)+GSL_DBL_EPSILON*fabs(f*h0*p0); b = x*h1-(j+1.)*h0; h0 = h1; h1 = b; b = 0.5*(fabs(x)*eh1+(j+1.)*eh0); eh0 = eh1; eh1 = b; b = (M_SQRT2*x*p1-sqrt(n-m+j+1.)*p0)/sqrt(n-m+j+2.); p0 = p1; p1 = b; b = 0.5*(M_SQRT2*fabs(x)*ep1+sqrt(n-m+j+1.)*ep0)/sqrt(n-m+j+2.); ep0 = ep1; ep1 = b; f *= -(m-j)/(j+1.)/sqrt(n-m+j+1.)*M_SQRT1_2; while(( (fabs(h0) > 2.0e-100) && (fabs(h1) > 2.0e-100) && (fabs(p0) > 2.0e-100) && (fabs(p1) > 2.0e-100) && (fabs(r) > 2.0e-100) ) && ( (fabs(h0) > 1.0e100) || (fabs(h1) > 1.0e100) || (fabs(p0) > 1.0e100) || (fabs(p1) > 1.0e100) || (fabs(r) > 1.0e100) )){ h0 = h0/2; h1 = h1/2; eh0 = eh0/2; eh1 = eh1/2; p0 = p0/2; p1 = p1/2; ep0 = ep0/2; ep1 = ep1/2; r = r/4; er = er/4; c++; } while(( ( (fabs(h0) < 1.0e-100) && (h0 != 0) )|| ( (fabs(h1) < 1.0e-100) && (h1 != 0) ) || ( (fabs(p0) < 1.0e-100) && (p0 != 0) ) || ( (fabs(p1) < 1.0e-100) && (p1 != 0) ) || ( (fabs(r) < 1.0e-100) && (r != 0) ) ) && ( (fabs(h0) < 2.0e100) && (fabs(h1) < 2.0e100) || (fabs(p0) < 2.0e100) || (fabs(p1) < 2.0e100) || (fabs(r) < 2.0e100) )){ p0 = p0*2; p1 = p1*2; ep0 = ep0*2; ep1 = ep1*2; h0 = h0*2; h1 = h1*2; eh0 = eh0*2; eh1 = eh1*2; r = r*4; er = er*4; c--; } } r *= pow2(2*c); er *= pow2(2*c); result->val = r*exp(-x*x/2.)/sqrt(M_SQRTPI); result->err = er*fabs(exp(-x*x/2.)/sqrt(M_SQRTPI)) + GSL_DBL_EPSILON*fabs(result->val); return GSL_SUCCESS; // return r*exp(-x*x/2.)/sqrt(M_SQRTPI); } } double gsl_sf_hermite_func_der(const int m, const int n, const double x) { EVAL_RESULT(gsl_sf_hermite_func_der_e(m, n, x, &result)); } static double H_zero_init(const int n, const int k) { double p = 1., x = 1., y = 1.; if (k == 1 && n > 50) { x = (GSL_IS_ODD(n)?1./sqrt((n-1)/6.):1./sqrt(n/2.)); } else { // p = -pow(2.,-1/3.)*gsl_sf_airy_zero_Ai(n/2-k+1); p = -0.7937005259840997373758528196*gsl_sf_airy_zero_Ai(n/2-k+1); x = sqrt(2*n+1.); y = pow(2*n+1.,1/6.); x = x - p/y - p*p/x/y/y/10. + (9/280. - p*p*p*11/350.)/x/x/x + (p*277/12600. - gsl_sf_pow_int(p,4)*823/63000.)/gsl_sf_pow_int(x,4)/y; } p = acos(x/sqrt(2*n+1.)); y = M_PI*(-2*(n/2-k)-1.5)/(n+0.5); if(gsl_fcmp(y,sin(2.*p)-2*p,GSL_SQRT_DBL_EPSILON)==0) return x; // initial approx sufficiently accurate if (y > -GSL_DBL_EPSILON) return sqrt(2*n+1.); // p = fmax(0.,p); // if (p < GSL_DBL_EPSILON) return sqrt(2*n+1.); if (p < GSL_DBL_EPSILON) p = GSL_DBL_EPSILON; // p = fmin(M_PI_2,p); if (p > M_PI_2) p = M_PI_2; if (sin(2.*p)-2*p > y){ x = fmax((sin(2.*p)-2*p-y)/4.,GSL_SQRT_DBL_EPSILON); // printf("< dp= %g\n",p); do{ x *= 2.; p += x; } while (sin(2.*p)-2*p > y); } do { x = p; p -= (sin(2.*p)-2.*p-y)/(2.*cos(2.*p)-2.); if (p<0.||p>M_PI_2) p = M_PI_2; } while (gsl_fcmp(x,p,100*GSL_DBL_EPSILON)!=0); // printf("p= %g\n",p); return sqrt(2*n+1.)*cos(p); } // lookup table for the positive zeros of the probabilists' Hermite polynomials of order 3 through 20 static double He_zero_tab[99] = { 1.73205080756887729352744634151, 0.741963784302725857648513596726, 2.33441421833897723931751226721, 1.35562617997426586583052129087, 2.85697001387280565416230426401, 0.616706590192594152193686099399, 1.88917587775371067550566789858, 3.32425743355211895236183546247, 1.154405394739968127239597758838, 2.36675941073454128861885646856, 3.75043971772574225630392202571, 0.539079811351375108072461918694, 1.63651904243510799922544657297, 2.80248586128754169911301080618, 4.14454718612589433206019783917, 1.023255663789132524828148225810, 2.07684797867783010652215614374, 3.20542900285646994336567590292, 4.51274586339978266756667884317, 0.484935707515497653046233483105, 1.46598909439115818325066466416, 2.48432584163895458087625118368, 3.58182348355192692277623675546, 4.85946282833231215015516494660, 0.928868997381063940144111999584, 1.87603502015484584534137013967, 2.86512316064364499771968407254, 3.93616660712997692868589612142, 5.18800122437487094818666404539, 0.444403001944138945299732445510, 1.34037519715161672153112945211, 2.25946445100079912386492979448, 3.22370982877009747166319001956, 4.27182584793228172295999293076, 5.50090170446774760081221630899, 0.856679493519450033897376121795, 1.72541837958823916151095838741, 2.62068997343221478063807762201, 3.56344438028163409162493844661, 4.59139844893652062705231872720, 5.80016725238650030586450565322, 0.412590457954601838167454145167, 1.24268895548546417895063983219, 2.08834474570194417097139675101, 2.96303657983866750254927123447, 3.88692457505976938384755016476, 4.89693639734556468372449782879, 6.08740954690129132226890147034, 0.799129068324547999424888414207, 1.60671006902872973652322479373, 2.43243682700975804116311571682, 3.28908242439876638890856229770, 4.19620771126901565957404160583, 5.19009359130478119946445431715, 6.36394788882983831771116094427, 0.386760604500557347721047189801, 1.16382910055496477419336819907, 1.95198034571633346449212362880, 2.76024504763070161684598142269, 3.60087362417154828824902745506, 4.49295530252001124266582263095, 5.47222570594934308841242925805, 6.63087819839312848022981922233, 0.751842600703896170737870774614, 1.50988330779674075905491513417, 2.28101944025298889535537879396, 3.07379717532819355851658337833, 3.90006571719800990903311840097, 4.77853158962998382710540812497, 5.74446007865940618125547815768, 6.88912243989533223256205432938, 0.365245755507697595916901619097, 1.09839551809150122773848360538, 1.83977992150864548966395498992, 2.59583368891124032910545091458, 3.37473653577809099529779309480, 4.18802023162940370448450911428, 5.05407268544273984538327527397, 6.00774591135959752029303858752, 7.13946484914647887560975631213, 0.712085044042379940413609979021, 1.42887667607837287134157901452, 2.15550276131693514033871248449, 2.89805127651575312007902775275, 3.66441654745063847665304033851, 4.46587262683103133615452574019, 5.32053637733603803162823765939, 6.26289115651325170419416064557, 7.38257902403043186766326977122, 0.346964157081355927973322447164, 1.04294534880275103146136681143, 1.74524732081412671493067861704, 2.45866361117236775131735057433, 3.18901481655338941485371744116, 3.94396735065731626033176813604, 4.73458133404605534390170946748, 5.57873880589320115268040332802, 6.51059015701365448636289263918, 7.61904854167975829138128156060 }; int gsl_sf_hermite_prob_zero_e(const int n, const int s, gsl_sf_result * result) /* Computes the s-th zero the probabilists' Hermite polynomial of order n. A Newton iteration using a continued fraction representation adapted from [E.T. Whittaker (1914), On the continued fractions which represent the functions of Hermite and other functions defined by differential equations, Proceedings of the Edinburgh Mathematical Society, 32, 65-74] is performed with the initial approximation from [Arpad Elbert and Martin E. Muldoon, Approximations for zeros of Hermite functions, pp. 117-126 in D. Dominici and R. S. Maier, eds, "Special Functions and Orthogonal Polynomials", Contemporary Mathematics, vol 471 (2008)] refined via the bisection method. */ { if(n <= 0 || s < 0 || s > n/2) { DOMAIN_ERROR(result); } else if(s == 0) { if (GSL_IS_ODD(n) == 1) { result->val = 0.; result->err = 0.; return GSL_SUCCESS; } else { DOMAIN_ERROR(result); } } else if(n == 2) { result->val = 1.; result->err = 0.; return GSL_SUCCESS; } else if(n < 21) { result->val = He_zero_tab[(GSL_IS_ODD(n)?n/2:0)+((n/2)*(n/2-1))+s-2]; result->err = GSL_DBL_EPSILON*(result->val); return GSL_SUCCESS; } else { double d = 1., x = 1., x0 = 1.; int j; x = H_zero_init(n,s) * M_SQRT2; do { x0 = x; d = 0.; for (j=1; jval = x; result->err = 2*GSL_DBL_EPSILON*x + fabs(x-x0); return GSL_SUCCESS; } } double gsl_sf_hermite_prob_zero(const int n, const int s) { EVAL_RESULT(gsl_sf_hermite_prob_zero_e(n, s, &result)); } // lookup table for the positive zeros of the physicists' Hermite polynomials of order 3 through 20 static double H_zero_tab[99] = { 1.22474487139158904909864203735, 0.524647623275290317884060253835, 1.65068012388578455588334111112, 0.958572464613818507112770593893, 2.02018287045608563292872408814, 0.436077411927616508679215948251, 1.335849074013696949714895282970, 2.35060497367449222283392198706, 0.816287882858964663038710959027, 1.67355162876747144503180139830, 2.65196135683523349244708200652, 0.381186990207322116854718885584, 1.157193712446780194720765779063, 1.98165675669584292585463063977, 2.93063742025724401922350270524, 0.723551018752837573322639864579, 1.46855328921666793166701573925, 2.26658058453184311180209693284, 3.19099320178152760723004779538, 0.342901327223704608789165025557, 1.03661082978951365417749191676, 1.75668364929988177345140122011, 2.53273167423278979640896079775, 3.43615911883773760332672549432, 0.656809566882099765024611575383, 1.32655708449493285594973473558, 2.02594801582575533516591283121, 2.78329009978165177083671870152, 3.66847084655958251845837146485, 0.314240376254359111276611634095, 0.947788391240163743704578131060, 1.59768263515260479670966277090, 2.27950708050105990018772856942, 3.02063702512088977171067937518, 3.88972489786978191927164274724, 0.605763879171060113080537108602, 1.22005503659074842622205526637, 1.85310765160151214200350644316, 2.51973568567823788343040913628, 3.24660897837240998812205115236, 4.10133759617863964117891508007, 0.291745510672562078446113075799, 0.878713787329399416114679311861, 1.47668273114114087058350654421, 2.09518325850771681573497272630, 2.74847072498540256862499852415, 3.46265693360227055020891736115, 4.30444857047363181262129810037, 0.565069583255575748526020337198, 1.13611558521092066631913490556, 1.71999257518648893241583152515, 2.32573248617385774545404479449, 2.96716692790560324848896036355, 3.66995037340445253472922383312, 4.49999070730939155366438053053, 0.273481046138152452158280401965, 0.822951449144655892582454496734, 1.38025853919888079637208966969, 1.95178799091625397743465541496, 2.54620215784748136215932870545, 3.17699916197995602681399455926, 3.86944790486012269871942409801, 4.68873893930581836468849864875, 0.531633001342654731349086553718, 1.06764872574345055363045773799, 1.61292431422123133311288254454, 2.17350282666662081927537907149, 2.75776291570388873092640349574, 3.37893209114149408338327069289, 4.06194667587547430689245559698, 4.87134519367440308834927655662, 0.258267750519096759258116098711, 0.776682919267411661316659462284, 1.30092085838961736566626555439, 1.83553160426162889225383944409, 2.38629908916668600026459301424, 2.96137750553160684477863254906, 3.57376906848626607950067599377, 4.24811787356812646302342016090, 5.04836400887446676837203757885, 0.503520163423888209373811765050, 1.01036838713431135136859873726, 1.52417061939353303183354859367, 2.04923170985061937575050838669, 2.59113378979454256492128084112, 3.15784881834760228184318034120, 3.76218735196402009751489394104, 4.42853280660377943723498532226, 5.22027169053748216460967142500, 0.245340708300901249903836530634, 0.737473728545394358705605144252, 1.23407621539532300788581834696, 1.73853771211658620678086566214, 2.25497400208927552308233334473, 2.78880605842813048052503375640, 3.34785456738321632691492452300, 3.94476404011562521037562880052, 4.60368244955074427307767524898, 5.38748089001123286201690041068 }; int gsl_sf_hermite_phys_zero_e(const int n, const int s, gsl_sf_result * result) /* Computes the s-th zero the physicists' Hermite polynomial of order n, thus also the s-th zero of the Hermite function of order n. A Newton iteration using a continued fraction representation adapted from [E.T. Whittaker (1914), On the continued fractions which represent the functions of Hermite and other functions defined by differential equations, Proceedings of the Edinburgh Mathematical Society, 32, 65-74] is performed with the initial approximation from [Arpad Elbert and Martin E. Muldoon, Approximations for zeros of Hermite functions, pp. 117-126 in D. Dominici and R. S. Maier, eds, "Special Functions and Orthogonal Polynomials", Contemporary Mathematics, vol 471 (2008)] refined via the bisection method. */ { if(n <= 0 || s < 0 || s > n/2) { DOMAIN_ERROR(result); } else if(s == 0) { if (GSL_IS_ODD(n) == 1) { result->val = 0.; result->err = 0.; return GSL_SUCCESS; } else { DOMAIN_ERROR(result); } } else if(n == 2) { result->val = M_SQRT1_2; result->err = 0.; return GSL_SUCCESS; } else if(n < 21) { result->val = H_zero_tab[(GSL_IS_ODD(n)?n/2:0)+((n/2)*(n/2-1))+s-2]; result->err = GSL_DBL_EPSILON*(result->val); return GSL_SUCCESS; } else { double d = 1., x = 1., x0 = 1.; int j; x = H_zero_init(n,s); do { x0 = x; d = 0.; for (j=1; jval = x; result->err = 2*GSL_DBL_EPSILON*x + fabs(x-x0); return GSL_SUCCESS; } } double gsl_sf_hermite_phys_zero(const int n, const int s) { EVAL_RESULT(gsl_sf_hermite_phys_zero_e(n, s, &result)); } int gsl_sf_hermite_func_zero_e(const int n, const int s, gsl_sf_result * result) // stupid wrapper { gsl_sf_hermite_phys_zero_e(n, s, result); } double gsl_sf_hermite_func_zero(const int n, const int s) { EVAL_RESULT(gsl_sf_hermite_func_zero_e(n, s, &result)); }