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//Last modified on February 8th, 2010 -- F. Gelis
// This code is based on the paper "An elegant and fast method to
// solve QCD evolution equations" by Laurent Schoeffel. The resolution
// of the ODE relies on routines of the GNU Scientific Library.
#include <stdio.h>
#include <math.h>
#include <stdlib.h>
#include <string.h>
#include <signal.h>
#include <unistd.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_odeiv.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_sf.h>
#include <gsl/gsl_complex.h>
#include <gsl/gsl_complex_math.h>
#include <gsl/gsl_sf_dilog.h>
#include <gsl/gsl_integration.h>
// Nc=3 is fixed once for all, Nf remains free
#define Nc 3.0
#define Cf 1.333333333333333333333
#define Cg 3.0
#define Tr 0.5
// this counter is used by the function sigalarm_handler() to display
// something about the progress of the calculation
static int counter;
// this global variable is a flag to tell other routines whether we do
// LO or NLO calculations. It should be set by the function set_LO()
static double NLO=1.0; // by default, we do NLO calculations
// this global variable is used to force a "pure glue" evolution of
// xG(x,Q^2) in order to test the hypothetical situation where the
// quarks would be infinitely massive
static double QUARKS=1.0; // by default, we keep the quarks
// this global variable is used to force a constant alpha_s
static double RUNNING=1.0; // by default, alpha_s is running
// global variable to compute the x-derivative of the gluon distribution
static int GLUON_DERIVATIVE=0; // by default, we do not compute the derivative
// Internal data structure used to pass extra arguments to some
// functions
typedef struct {
double nf; // number of flavors
int nlag; // Laguerre order
} data_pair;
// Internal structure used to pass arguments to the routine that
// calculates Laguerre coefficients
typedef struct {
int nlag; // Laguerre order
double (*function)(double); // a function of a real variable
} lag_pair;
// Structure that holds the Laguerre coefficients of the splitting
// functions. Defined in the file laguerre.c
typedef struct {
double nf; // number of flavors
double nmax; // maximum Laguerre order
double *pns0; // The splitting functions....
double *pns1; // 0=LO, 1=NLO
double *psqq0;
double *psqq1;
double *psqg0;
double *psqg1;
double *psgq0;
double *psgq1;
double *psgg0;
double *psgg1;
} tab_Pij;
// A structure that holds the Laguerre transform of some PDFs
typedef struct {
int nmax; // maximum Laguerre order
double q2; // the value of Q^2
double *qns; // quark non singlet xQns(x)
double *qs; // quark singlet xQs(x)
double *g; // gluon xG(x)
} tab_dist;
// A structure that holds the values of the PDFs at some x as a
// function of Q^2
typedef struct {
int nq2; // number of values of Q^2
double x; // value of x
double nf; // number of flavors in the evolution
double *q2; // table of values of Q^2 q2[0..nq2]
double *qns; // corresponding values of xQns(x)
double *qs; // ....................... xQs(x)
double *g; // ....................... xG(x)
double *dg; // ....................... d[xG(x)]/dx
} tab_dist_x;
// A structure that holds the values of the PDFs on a grid in x and
// Q^2.
typedef struct {
int nq2; // number of values of Q^2
int nx; // number of values of x
double nf; // number of flavors in the evolution
double xmin; // minimum value of x (the max value is x=1.0)
double *q2; // table of values of Q^2 q2[0..nq2]
double *x; // table of values of x x[0..nx]
// Then, we have the values of the PDFs at the corresponding points.
// They are 2d tables table[0..nx][0..nq2] such that table[i][j]
// is the value of the PDF at x=x[i] and Q^2=q2[j]. BEWARE that since
// they are declared as pointers to a double and not as a 2d table, the
// compiler does not know the length of a row in order to make any sense
// of the notation table[i][j]. Instead, one should use table[i*(1+nq2)+j]
double *qns; // values of xQns(x,Q^2) qns[0..nx][0..nq2]
double *qs; // values of xQs(x,Q^2) qs[0..nx][0..nq2]
double *g; // values of xG(x,Q^2) g[0..nx][0..nq2]
double *dg; // values of d[xG(x,Q^2)]/dx dg[0..nx][0..nq2]
} grid_dist;
// A structure that holds the values of the PDFs at a single point (x,Q^2)
typedef struct {
double x; // value of x
double q2; // value of Q^2
double qns; // value of xQns(x,Q^2)
double qs; // value of xQs(x,Q^2)
double g; // value of xG(x,Q^2)
double dg; // value of d[xG(x,Q^2)]/dx
} value_dist;
// A structure that holds the solution of the differential equations
// for the Laguerre coefficients of the PDFs
typedef struct {
int nmax; // maximum Laguerre order
int nq2; // number of values of Q^2
double nf; // value of Nf used in the evolution
double *q2; // table of values of Q^2 q2[0..nq2]
// The corresponding values of the Laguerre coefficients of the PDFs.
// table[i][j] is the j-th Laguerre coefficient of the PDF at Q^2=q2[i].
// Because they are declared as pointers to a double, these tables
// cannot be accessed as table[i][j], but instead as table[i*(1+nmax)+j]
double *qns; // table qns[0..nq2][0..nmax]
double *qs; // table qs[0..nq2][0..nmax]
double *g; // table g[0..nq2][0..nmax]
} tab_evol;
// A structure that holds the running coupling constant as a function
// of Q^2
typedef struct {
int nq2; // number of values of Q^2
double *q2; // table of values of Q^2: q2[0..nq2]
double *alpha; // table of values of alpha: alpha[0..nq2]
} tab_alpha;
// Internal data structure for the ODEs
typedef struct {
tab_Pij *pij;
tab_alpha *alpha;
} ode_pair;
// A structure that holds a pointers to a variable of type tab_evol
// and one to a variable of type tab_alpha
typedef struct {
tab_evol *pdf;
tab_alpha *alpha;
} evol_pair;
// This function is defined in the file laguerre.c. Therefore, any
// program that uses the functions in the present file must also be
// linked with laguerre.o
extern double Lag(int,double);
// A function which is called every 10 seconds and prints something
// about the state of the calculation, for the routines that are
// particularly slow
static void sigalarm_handler(int sig){
(void) sig; // silence clang warnings
fprintf(stderr,"%3d%% done\n",counter);
alarm(10); /* let's request another alarm 5 seconds from now */
#ifdef not_linux // On SOLARIS at least, one needs to reinstall the
// signal handler after every use. Just give the extra option
// "-Dnot_linux" to the compiler when compiling this file
signal( SIGALRM, sigalarm_handler);
#endif
}
// Set up for LO calculations
void set_LO(void){
NLO=0.0;
}
// Set things up so that quarks are neglected in the evolution of gluons
void set_NO_QUARKS(void){
QUARKS=0.0;
}
// Set things up so that alpha_s is taken to be constant
void set_NO_RUNNING(void){
RUNNING=0.0;
}
// Set things up to compute the x-derivative of xG(x,Q^2)
void compute_GLUON_DERIVATIVE(void){
GLUON_DERIVATIVE=1;
}
// An intermediate function used in the computation of the Laguerre
// coefficients of a given function. It is not meant to be called
// directly.
double Lag_integrand(double z,void *params){
double x=exp(-z);
lag_pair *data=(lag_pair *)params;
int n=data->nlag;
double (*f)(double)=data->function;
return x*f(x)*Lag(n,z);
}
// Calculate the n-th Laguerre coefficient for the function f(x)
double Lag_coeff(int n,double (*f)(double)){
double epsilon=1.0e-12;
gsl_integration_workspace *wksp=gsl_integration_workspace_alloc(200);
double result,abserr;
lag_pair params;
gsl_function F;
params.nlag=n;
params.function=f;
F.function=Lag_integrand;
F.params=&params;
gsl_integration_qag(&F,0.0,100.0,epsilon,epsilon,200,6,wksp,&result,&abserr);
gsl_integration_workspace_free(wksp);
return result;
}
// Reconstruct a function at point x from its Laguerre coefficients
double Lag_inv(double x,int N,double *coeff){
// x: point at which we want the value of the function
// N: maximum Laguerre order
// coeff: the table[0..N] of the Laguerre coefficients
double t=-log(x);
double lag0[N+1];
double sum=0.0;
int i;
double a1,a2;
lag0[0]=1.0;
lag0[1]=1.0-t;
for (i=2;i<=N;i++){
a1=((double)(2*i-1)-t)/((double)(i));
a2=((double)(i-1))/((double)(i));
lag0[i]=a1*lag0[i-1]-a2*lag0[i-2];
}
for (i=N;i>=0;i--) sum+=coeff[i]*lag0[i];
return sum;
}
// Reconstruct the derivative of a function at point x from its Laguerre coefficients
double diff_Lag_inv(double x,int N,double *coeff){
// x: point at which we want the derivative of the function
// N: maximum Laguerre order
// coeff: the table[0..N] of the Laguerre coefficients
double t=-log(x);
double lag0[N+1];
double dlag0[N+1];
double sum=0.0;
int i;
double a1,a2,da1;
lag0[0] = 1.0;
dlag0[0] = 0.0;
lag0[1] = 1.0-t;
dlag0[1] = -1.0;
for (i=2;i<=N;i++){
a1 = ((double)(2*i-1)-t)/((double)(i));
da1 = -1.0/((double)(i));
a2 = ((double)(i-1))/((double)(i));
lag0[i] = a1*lag0[i-1] - a2*lag0[i-2];
dlag0[i] = da1*lag0[i-1] + a1*dlag0[i-1] - a2*dlag0[i-2];
}
for (i=N;i>=0;i--) sum+=coeff[i]*dlag0[i];
return -sum/x;
}
// Calculate the Laguerre coefficients of some PDFs passed as arguments
tab_dist *Lag_dist(int N,double (*qns)(double),double (*qs)(double),double (*g)(double)){
// N: maximum Laguerre order
// qns: pointer to the non singlet quark PDF
// qs: pointer to the singlet quark PDF
// g: pointer to the gluon PDF
int i;
tab_dist *tt=(tab_dist *)malloc(sizeof(tab_dist));
fprintf(stderr,"Calculating Laguerre coefficients of the initial PDFs...");
tt->qns=(double *)malloc((1+N)*sizeof(double));
tt->qs=(double *)malloc((1+N)*sizeof(double));
tt->g=(double *)malloc((1+N)*sizeof(double));
tt->nmax=N;
tt->q2=1.0;
for (i=0;i<=N;i++){
(tt->qns)[i]=Lag_coeff(i,qns);
(tt->qs)[i]=Lag_coeff(i,qs);
(tt->g)[i]=Lag_coeff(i,g);
}
fprintf(stderr,"done\n");
return tt;
}
// Calculate the PDFs at some x from a table of Laguerre coefficients
value_dist Lag_dist_inv(double x,tab_dist *tt){
value_dist out;
out.x=x;
out.q2=tt->q2;
out.qns=Lag_inv(x,tt->nmax,tt->qns);
out.qs=Lag_inv(x,tt->nmax,tt->qs);
out.g=Lag_inv(x,tt->nmax,tt->g);
if (GLUON_DERIVATIVE) {
out.dg=diff_Lag_inv(x,tt->nmax,tt->g);
} else {
out.dg=0.0;
}
return out;
}
// The jacobian function, empty here since we don't need it (we use
// only explicit ODE solvers).
int jac(double t,const double y[],double *dfdy,double dfdt[],void *params){
// silence clang warnings
(void) t;
(void) y;
(void) dfdy;
(void) dfdt;
(void) params;
return GSL_SUCCESS;
}
// The r.h.s. for the ODE that controls the evolution of
// alpha_s. NOTE: I do not take into account the heavy quark masses in
// the evolution, nor the fact that the number of active flavors
// change with Q^2.
int rhs_alpha(double q2,const double y[],double f[],void *params){
double Nf=*((double *)params);
double beta0=11.0-2.0*Nf/3.0;
double beta1=102.0-38.0*Nf/3.0;
double four_PI=4.0*M_PI;
double alpha2=y[0]*y[0]/four_PI;
double alpha3=alpha2*y[0]/four_PI;
// If the global variable NLO is 0.0, we include only the LO beta
// function
f[0]=-RUNNING*(beta0*alpha2+NLO*beta1*alpha3)/q2;
return GSL_SUCCESS;
}
// The routine that actually solves the ODE in order to get the
// running of alpha_s
tab_alpha *alpha_s_evol(double q20,double alpha0,double q2min,double q2max,int NQ2,double Nf){
// q20: the value of Q^2 at which alpha_s is given
// alpha0: the corresponding value of alpha_s
// q2min: the min value of Q^2 in the produced table
// q2max: the maximum value of Q^2..................
// NQ2: the number of value of Q^2
// Nf: the number of flavors
double params[1]={Nf};
const gsl_odeiv_step_type *T= gsl_odeiv_step_rk8pd;
gsl_odeiv_step *s=gsl_odeiv_step_alloc(T,1);
double ode_eps=1.0e-6;
gsl_odeiv_control *c=gsl_odeiv_control_y_new(ode_eps,ode_eps);
gsl_odeiv_evolve *e=gsl_odeiv_evolve_alloc(1);
gsl_odeiv_system sys={rhs_alpha,jac,1,params};
tab_alpha *tt=(tab_alpha *)malloc(sizeof(tab_alpha));
int i,j;
double h;
double y[1];
double q2;
double dlq2=exp(log(q2max/q2min)/((double)NQ2));
fprintf(stderr,"Calculating the evolution of alpha_s...");
// Allocate some memory
tt->q2=(double *)malloc((1+NQ2)*sizeof(double));
tt->alpha=(double *)malloc((1+NQ2)*sizeof(double));
tt->nq2=NQ2;
// initialize the table of the Q^2's
q2=q2min;
j=-1;
for (i=0;i<=NQ2;i++) {
tt->q2[i]=q2;
if (q2<q20) j=i; // this determines the index in the table just
// before q2[i]
q2*=dlq2;
}
// Forward evolution of alpha_s
q2=q20;
y[0]=alpha0;
h=1.0e-10;
for (i=j+1;i<=NQ2;i++) {//if Q0^2>Qmax^2, then j=NQ2 and this loop is skept
while (q2<tt->q2[i]) gsl_odeiv_evolve_apply(e,c,s,&sys,&q2,tt->q2[i],&h,y);
tt->alpha[i]=y[0];
}
// Backward evolution of alpha_s
q2=q20;
y[0]=alpha0;
h=-1.0e-10;
for (i=j;i>=0;i--) {//if Q0^2<Qmin^2, then j=-1 and this loop is skept
while (q2>tt->q2[i]) gsl_odeiv_evolve_apply(e,c,s,&sys,&q2,tt->q2[i],&h,y);
tt->alpha[i]=y[0];
}
// Free the allocated workspace
gsl_odeiv_evolve_free(e);
gsl_odeiv_step_free(s);
gsl_odeiv_control_free(c);
fprintf(stderr,"done\n");
return tt;
}
// Running alpha_s obtained by interpolation from a table of
// precomputed values
double alpha_s(double Q2,tab_alpha *tt){
int NQ2=tt->nq2;
int i=(int)floor(((double)NQ2)*log(Q2/tt->q2[0])/log(tt->q2[NQ2]/tt->q2[0]));
double f1,f2,x1,x2;
x1=tt->q2[i];
x2=tt->q2[i+1];
f1=tt->alpha[i];
f2=tt->alpha[i+1];
return f1+(f2-f1)*(Q2-x1)/(x2-x1);
}
// LO running alpha_s -- direct calculation
double alpha_s_LO(double Q2,double Nf){
double Mz=91.18; // Z0 mass in GeV
double inv_alpha_s_MZ=1.0/0.113;
double beta0=(11.0-2.0*Nf/3.0)/(4.0*M_PI);
double dlog=log(Mz*Mz/Q2);
return 1.0/(inv_alpha_s_MZ-beta0*dlog);
}
// The r.h.s. of the ODE in the non singlet case
// y[0]..y[N]=qns
int rhs_NS(double q2,const double y[],double f[],void *params){
int i,j;
tab_Pij *tt=((ode_pair *)params)->pij;
tab_alpha *tta=((ode_pair *)params)->alpha;
double alpha_s_2pi0=alpha_s(q2,tta)/(2.0*M_PI);
double alpha_s_2pi=NLO*alpha_s_2pi0;
double sum;
int N=tt->nmax;
f[0]=alpha_s_2pi0*y[0]*((tt->pns0)[0]+alpha_s_2pi*(tt->pns1)[0])/q2;
for (i=1;i<=N;i++){
sum=0.0;
for (j=0;j<i;j++) {
sum+=y[j]*((tt->pns0)[i-j]+alpha_s_2pi*(tt->pns1)[i-j]\
-(tt->pns0)[i-j-1]-alpha_s_2pi*(tt->pns1)[i-j-1]);
}
sum+=y[i]*((tt->pns0)[0]+alpha_s_2pi*(tt->pns1)[0]);
f[i]=sum*alpha_s_2pi0/q2;
}
return GSL_SUCCESS;
}
// The r.h.s. of the ODE in the singlet case
// y[0]..y[N]=qs
// y[N+1]..y[2N+1]=g
int rhs_S(double q2,const double y[],double f[],void *params){
int i,j;
tab_Pij *tt=((ode_pair *)params)->pij;
tab_alpha *tta=((ode_pair *)params)->alpha;
double alpha_s_2pi0=alpha_s(q2,tta)/(2.0*M_PI);
double alpha_s_2pi=NLO*alpha_s_2pi0;
double sum_q;
double sum_g;
//double Nf=tt->nf;
int N=tt->nmax;
f[0]=alpha_s_2pi0*(y[0]*((tt->psqq0)[0]+alpha_s_2pi*(tt->psqq1)[0])\
+y[N+1]*((tt->psqg0)[0]+alpha_s_2pi*(tt->psqg1)[0]))/q2;
f[N+1]=alpha_s_2pi0*(QUARKS*y[0]*((tt->psgq0)[0]+alpha_s_2pi*(tt->psgq1)[0])\
+y[N+1]*((tt->psgg0)[0]+alpha_s_2pi*(tt->psgg1)[0]))/q2;
for (i=1;i<=N;i++){
sum_q=0.0;
sum_g=0.0;
for (j=0;j<i;j++) {
sum_q+=y[j]*((tt->psqq0)[i-j]+alpha_s_2pi*(tt->psqq1)[i-j]\
-(tt->psqq0)[i-j-1]-alpha_s_2pi*(tt->psqq1)[i-j-1])\
+y[j+N+1]*((tt->psqg0)[i-j]+alpha_s_2pi*(tt->psqg1)[i-j]\
-(tt->psqg0)[i-j-1]-alpha_s_2pi*(tt->psqg1)[i-j-1]);
sum_g+=QUARKS*y[j]*((tt->psgq0)[i-j]+alpha_s_2pi*(tt->psgq1)[i-j]\
-(tt->psgq0)[i-j-1]-alpha_s_2pi*(tt->psgq1)[i-j-1])\
+y[j+N+1]*((tt->psgg0)[i-j]+alpha_s_2pi*(tt->psgg1)[i-j]\
-(tt->psgg0)[i-j-1]-alpha_s_2pi*(tt->psgg1)[i-j-1]);
}
sum_q+=y[i]*((tt->psqq0)[0]+alpha_s_2pi*(tt->psqq1)[0])\
+y[i+N+1]*((tt->psqg0)[0]+alpha_s_2pi*(tt->psqg1)[0]);
sum_g+=QUARKS*y[i]*((tt->psgq0)[0]+alpha_s_2pi*(tt->psgq1)[0])\
+y[i+N+1]*((tt->psgg0)[0]+alpha_s_2pi*(tt->psgg1)[0]);
f[i]=sum_q*alpha_s_2pi0/q2;
f[i+N+1]=sum_g*alpha_s_2pi0/q2;
}
return GSL_SUCCESS;
}
// DGLAP evolution of the PDFs. This routine actually solves the ODEs
// for the Laguerre coefficients of the PDFs. It returns a structure
// of type evol_pair which contains the evolution with Q^2 of the
// Laguerre coefficients of the PDFs, and the evolution with Q^2 of
// alpha_s.
// NOTE: this routine relies heavily on the ODE solvers in the GNU
// scientific library. In principle, there is no problem to rewrite it
// with some "homemade" ODE solver, but that would make the code
// unecessarily messy...
evol_pair DGLAP_evol(tab_dist *dist0,double Q2max,int NQ2,tab_Pij *pij){
// dist0: Laguerre coefficients of the initial PDFs. They must be
// computed with the function Lag_dist. Note: the value of Q0^2 must
// be defined in the field dist0->q2.
// Q2max: value of Q^2 where we stop the evolution
// NQ2: number of steps in the evolution. The intermediate values of
// Q^2 will be equally spaced on a log scale.
// pij: the table of Laguerre coefficients of the splitting
// functions. This is calculated with the function
// create_Lag_Pij_table in laguerre.c
int N=dist0->nmax;
ode_pair params;
const gsl_odeiv_step_type *T= gsl_odeiv_step_rk8pd;
gsl_odeiv_step *s_NS=gsl_odeiv_step_alloc(T,N+1);
gsl_odeiv_step *s_S=gsl_odeiv_step_alloc(T,2*N+2);
double ode_eps=1.0e-6;
gsl_odeiv_control *c_NS=gsl_odeiv_control_y_new(ode_eps,ode_eps);
gsl_odeiv_control *c_S=gsl_odeiv_control_y_new(ode_eps,ode_eps);
gsl_odeiv_evolve *e_NS=gsl_odeiv_evolve_alloc(N+1);
gsl_odeiv_evolve *e_S=gsl_odeiv_evolve_alloc(2*N+2);
gsl_odeiv_system sys_NS={rhs_NS,jac,N+1,&params};
gsl_odeiv_system sys_S={rhs_S,jac,2*N+2,&params};
double y_NS[N+1];
double y_S[2*N+2];
double h_NS,h_S;
double q2_NS,q2_S;
int i,j;
double q2;
double q2i=dist0->q2;
double fact_q2=exp(log(Q2max/q2i)/((double)(NQ2)));
tab_evol *tt=(tab_evol *)malloc(sizeof(tab_evol));
evol_pair out;
double Mz=91.18; // The mass od the Z0
params.pij=pij;
// first, compute the running alpha_s as a function of Q^2. We set
// the value of alpha_s in the MSBAR scheme at the Z0 mass.
params.alpha=alpha_s_evol(Mz*Mz,0.113,q2i,Q2max,NQ2,pij->nf);
tt->nmax=N;
tt->nq2=NQ2;
tt->nf=pij->nf;
fprintf(stderr,"Solving the ODE...");
// Allocate memory for the result
tt->q2=(double *)malloc((NQ2+1)*sizeof(double));
tt->qns=(double *)malloc((N+1)*(NQ2+1)*sizeof(double));
tt->qs=(double *)malloc((N+1)*(NQ2+1)*sizeof(double));
tt->g=(double *)malloc((N+1)*(NQ2+1)*sizeof(double));
// Set the initial conditions
for (i=0;i<=N;i++) {
y_NS[i]=(dist0->qns)[i];
y_S[i]=(dist0->qs)[i];
y_S[N+1+i]=(dist0->g)[i];
}
q2_NS=q2i;
q2_S=q2i;
q2=q2i;
h_NS=1.0e-10;
h_S=1.0e-10;
// Store the initial condition
j=0;
(tt->q2)[0]=q2;
for (i=0;i<=N;i++){
(tt->qns)[i]=y_NS[i];
(tt->qs)[i]=y_S[i];
(tt->g)[i]=y_S[N+1+i];
}
// The main loop: NQ2 steps
for (j=1;j<=NQ2;j++){
q2=q2*fact_q2;
while (q2_NS<q2) gsl_odeiv_evolve_apply(e_NS,c_NS,s_NS,&sys_NS,&q2_NS,q2,&h_NS,y_NS);
while (q2_S<q2) gsl_odeiv_evolve_apply(e_S,c_S,s_S,&sys_S,&q2_S,q2,&h_S,y_S);
// Store the result of this step. See the function "get_step"
// below in order to understand why it is better to use
// [i+(1+N)*j] instead of [j+(1+NQ2)*i].
(tt->q2)[j]=q2;
for (i=0;i<=N;i++){
(tt->qns)[i+(1+N)*j]=y_NS[i];
(tt->qs)[i+(1+N)*j]= y_S[i];
(tt->g)[i+(1+N)*j]= y_S[N+1+i];
}
}
out.alpha=params.alpha;
out.pdf=tt;
// Free the allocated workspace
gsl_odeiv_evolve_free(e_NS);
gsl_odeiv_evolve_free(e_S);
gsl_odeiv_step_free(s_NS);
gsl_odeiv_step_free(s_S);
gsl_odeiv_control_free(c_NS);
gsl_odeiv_control_free(c_S);
fprintf(stderr,"done\n");
return out;
}
// extract step #j of the Q^2 evolution - useful if one needs one
// value of Q^2 that sits on the grid and many values of x
tab_dist *get_step(int j,tab_evol *tt_evol){
// j: the # of the step
// tt_evol: the structure returned by DGLAP_evol
int N=tt_evol->nmax;
tab_dist *tt=(tab_dist *)malloc(sizeof(tab_dist));
tt->nmax=N;
tt->q2=(tt_evol->q2)[j];
// We do not allocate any memory for the tables in tt. Instead, we
// make them point to the correct location in the structure
// tt_evol. This saves some memory and makes things faster, but
// assumes that the variable tt_evol is not altered in any way
// before one is done using the variable tt...
tt->qns=&((tt_evol->qns)[j*(1+N)]); // the data in the table tt_evol->qns
tt->qs = &((tt_evol->qs)[j*(1+N)]); // is laid out in such a way that the
tt->g = &((tt_evol->g)[j*(1+N)]); // Laguerre coefficients for the j-th
// value of Q^2 form a sub-vector of
// length 1+N that starts at the index
// j*(1+N). Hence that trick to get
// a pointer to this sub-vector.
return tt;
}
// Get the Laguerre coefficients of the PDFs at some intermediate Q^2
// by interpolation -- useful if all one needs is a specific value of
// Q^2 and many values of x
tab_dist *PDF_at_q2(double q2,tab_evol *tt_evol){
// q2: the value of Q^2
// tt_evol: the structure returned by DGLAP_evol
int N=tt_evol->nmax;
int NQ2=tt_evol->nq2;
tab_dist *tt=(tab_dist *)malloc(sizeof(tab_dist));
int j,i;
double dq2;
tt->nmax=N;
tt->q2=q2;
if ((q2<=(tt_evol->q2)[0])||(q2>=(tt_evol->q2)[NQ2])) {
fprintf(stderr,"ERROR: q2=%.5e out of range\n",q2);
free(tt);
return NULL;
}
tt->qns=(double *)malloc((1+N)*sizeof(double));
tt->qs =(double *)malloc((1+N)*sizeof(double));
tt->g =(double *)malloc((1+N)*sizeof(double));
j=floor(NQ2*log(q2/(tt_evol->q2)[0])/log((tt_evol->q2)[NQ2]/(tt_evol->q2)[0]));
dq2=(q2-(tt_evol->q2)[j])/((tt_evol->q2)[j+1]-(tt_evol->q2)[j]);
for (i=0;i<=N;i++){
(tt->qns)[i]=(1-dq2)*(tt_evol->qns)[j*(1+N)+i]\
+dq2*(tt_evol->qns)[(j+1)*(1+N)+i];
(tt->qs)[i]=(1-dq2)*(tt_evol->qs)[j*(1+N)+i]\
+dq2*(tt_evol->qs)[(j+1)*(1+N)+i];
(tt->g)[i]=(1-dq2)*(tt_evol->g)[j*(1+N)+i]\
+dq2*(tt_evol->g)[(j+1)*(1+N)+i];
}
return tt;
}
// Calculate PDFs at some value of x as a function of Q^2
tab_dist_x *PDF_at_x(double x,tab_evol *tt_evol){
// x: the value of x
// tt_evol: the structure returned by DGLAP_evol
tab_dist_x *tt=(tab_dist_x *)malloc(sizeof(tab_dist_x));
int NQ2=tt_evol->nq2;
int N=tt_evol->nmax;
int j;
// allocate memory
tt->qns=(double *)malloc((1+NQ2)*sizeof(double));
tt->qs =(double *)malloc((1+NQ2)*sizeof(double));
tt->g =(double *)malloc((1+NQ2)*sizeof(double));
if (GLUON_DERIVATIVE) {
tt->dg =(double *)malloc((1+NQ2)*sizeof(double));
} else {
tt->dg=(double *)NULL;
}
tt->nq2=NQ2;
tt->x=x;
tt->q2=tt_evol->q2;
tt->nf=tt_evol->nf;
for (j=0;j<=NQ2;j++){
(tt->qns)[j]=Lag_inv(x,N,&((tt_evol->qns)[(1+N)*j]));
(tt->qs)[j]= Lag_inv(x,N,&((tt_evol->qs)[(1+N)*j]));
(tt->g)[j]= Lag_inv(x,N,&((tt_evol->g)[(1+N)*j]));
if (GLUON_DERIVATIVE) (tt->dg)[j]= diff_Lag_inv(x,N,&((tt_evol->g)[(1+N)*j]));
}
return tt;
}
// Calculate the values of the PDFs on a rectangular grid in x and
// Q^2, from the table of Laguerre coefficients. In principle, one can
// compute values at any point x directly from the table of Laguerre
// coefficients. However, this is a rather slow process since one has
// to sum over the Laguerre polynomials. Therefore, it is better to
// precompute a table of values directly as a function of x. Then,
// this table can be stored in a file on disk, and re-read later.
// NOTE: the points x and Q^2 are linearly spaced on a log scale.
grid_dist *PDF_grid_calc(int Nx, double xmin,tab_evol *tt_evol){
// Nx: the number of points in the x direction
// xmin: the minimum value of x (xmax is always 1.0)
// tt_evol: the structure returned by tt_evol
grid_dist *tt=(grid_dist *)malloc(sizeof(grid_dist));
int NQ2=tt_evol->nq2;
int N=tt_evol->nmax;
int i,j;
double dlx=exp(log(xmin)/((double)Nx));
double x;
fprintf(stderr,"Calculating a grid of values...\n");
// Install a signal handler
signal( SIGALRM, sigalarm_handler );
alarm(10);
// Allocate memory
tt->q2=(double *)malloc((1+NQ2)*sizeof(double));
tt->x =(double *)malloc((1+Nx)*sizeof(double));
tt->qns=(double *)malloc((1+NQ2)*(1+Nx)*sizeof(double));
tt->qs =(double *)malloc((1+NQ2)*(1+Nx)*sizeof(double));
tt->g =(double *)malloc((1+NQ2)*(1+Nx)*sizeof(double));
if (GLUON_DERIVATIVE) {
tt->dg =(double *)malloc((1+NQ2)*(1+Nx)*sizeof(double));
} else {
tt->dg =(double *)NULL;
}
tt->nq2=NQ2;
tt->nx=Nx;
tt->xmin=xmin;
tt->nf=tt_evol->nf;
// Build the table
memcpy(tt->q2,tt_evol->q2,(1+NQ2)*sizeof(double));
x=1.0;
for (i=Nx;i>=0;i--){
(tt->x)[i]=x;
for (j=0;j<=NQ2;j++){
counter=(int)(100.0*((double)((Nx-i)*(1+NQ2)+j))/((double)((1+NQ2)*(1+Nx))));
(tt->qns)[i*(1+NQ2)+j]= Lag_inv(x,N,&((tt_evol->qns)[(1+N)*j]));
(tt->qs)[i*(1+NQ2)+j]= Lag_inv(x,N,&((tt_evol->qs)[(1+N)*j]));
(tt->g)[i*(1+NQ2)+j]= Lag_inv(x,N,&((tt_evol->g)[(1+N)*j]));
if (GLUON_DERIVATIVE) (tt->dg)[i*(1+NQ2)+j]= diff_Lag_inv(x,N,&((tt_evol->g)[(1+N)*j]));
}
x*=dlx;
}
// remove the signal handler
signal( SIGALRM, SIG_IGN);
return tt;
}
// Write the grid on disk. NOTE: the file generated by this routine is
// a binary file in order to simplify the parsing when it is read by
// PDF_grid_read. This means that it can be read only on computers
// where the binary representation of floating point numbers uses the
// same byte ordering... This file can then be read with the function
// PDF_grid_read, which returns a pointer to a structure of type
// "grid_dist".
void PDF_grid_write(grid_dist *tt,char *name){
// tt: the structure returned by PDF_grid_calc
// name: some identifier for the file. The filename will be
// "table_PDF_name.dat"
int Nx=tt->nx;
int NQ2=tt->nq2;
double xmin=tt->xmin;
double Nf=tt->nf;
char filename[50];
char *base_name="table_PDF_";
char *suffix=".dat";
FILE *fd;
strcpy(filename,base_name);
strcat(filename,name);
strcat(filename,suffix);
fd=fopen(filename,"w");
fwrite(&GLUON_DERIVATIVE,sizeof(int),1,fd);
fwrite(&Nx,sizeof(int),1,fd);
fwrite(&NQ2,sizeof(int),1,fd);
fwrite(&xmin,sizeof(double),1,fd);
fwrite(&Nf,sizeof(double),1,fd);
fwrite(tt->q2,sizeof(double),1+NQ2,fd);
fwrite(tt->x,sizeof(double),1+Nx,fd);
fwrite(tt->qns,sizeof(double),(1+NQ2)*(1+Nx),fd);
fwrite(tt->qs,sizeof(double),(1+NQ2)*(1+Nx),fd);
fwrite(tt->g,sizeof(double),(1+NQ2)*(1+Nx),fd);
if (GLUON_DERIVATIVE) fwrite(tt->dg,sizeof(double),(1+NQ2)*(1+Nx),fd);
fclose(fd);
}
// Read the grid from disk
grid_dist *PDF_grid_read(char *name){
// name: the name that has been used to identify the file in
// PDF_dist_write.
grid_dist *tt=(grid_dist *)malloc(sizeof(grid_dist));
int Nx;
int NQ2;
int der;
double xmin,Nf;
char filename[50];
char *base_name="table_PDF_";
char *suffix=".dat";
FILE *fd;
strcpy(filename,base_name);
strcat(filename,name);
strcat(filename,suffix);
fd=fopen(filename,"r");
fread(&der,sizeof(int),1,fd);
fread(&Nx,sizeof(int),1,fd);
fread(&NQ2,sizeof(int),1,fd);
fread(&xmin,sizeof(double),1,fd);
fread(&Nf,sizeof(double),1,fd);
if (der==1) GLUON_DERIVATIVE=1;
tt->nx=Nx;
tt->xmin=xmin;
tt->nq2=NQ2;
tt->nf=Nf;
// Now that we know the sizes, allocate memory
tt->q2=(double *)malloc((1+NQ2)*sizeof(double));
tt->x =(double *)malloc((1+Nx)*sizeof(double));
tt->qns=(double *)malloc((1+NQ2)*(1+Nx)*sizeof(double));
tt->qs =(double *)malloc((1+NQ2)*(1+Nx)*sizeof(double));
tt->g =(double *)malloc((1+NQ2)*(1+Nx)*sizeof(double));
if (GLUON_DERIVATIVE) {
tt->dg =(double *)malloc((1+NQ2)*(1+Nx)*sizeof(double));
} else {
tt->dg = NULL;
}
// Read the rest of the data
fread(tt->q2,sizeof(double),1+NQ2,fd);
fread(tt->x,sizeof(double),1+Nx,fd);
fread(tt->qns,sizeof(double),(1+NQ2)*(1+Nx),fd);
fread(tt->qs,sizeof(double),(1+NQ2)*(1+Nx),fd);
fread(tt->g,sizeof(double),(1+NQ2)*(1+Nx),fd);
if (GLUON_DERIVATIVE) fread(tt->dg,sizeof(double),(1+NQ2)*(1+Nx),fd);
fclose(fd);
return tt;
}
// Get the distributions at some intermediate point (x,Q^2) by
// interpolation.
// FIXME: written as it is, this routine needs to have the complete
// table, which can be huge, loaded in memory first. One should write a
// version of this function that determines where in the file is the
// piece of data we need, and then read only the corresponding (small)
// part of the file...
value_dist PDF_interpolated(double x,double q2,grid_dist *tt){
// x: the value of x
// q2: the value of Q^2
// tt: the structure returned by PDF_grid_calc or by PDF_grid_read
value_dist out;
int Nx=tt->nx;
int NQ2=tt->nq2;
double xmin=tt->xmin;
double q2min=(tt->q2)[0];
double q2max=(tt->q2)[NQ2];
int i,j;
double f1,f2,f3,f4,t,u;
out.x=0.0;
out.q2=0.0;
out.qns=0.0;
out.qs=0.0;
out.g=0.0;
out.dg=0.0;
if ((q2<q2min)||(q2>=q2max)) {
fprintf(stderr,"ERROR: q2=%.5e out of range\n",q2);
return out;
}
if ((x>=208)||(x<xmin)) {
fprintf(stderr,"ERROR: x=%.5e out of range\n",x);
return out;
}
out.x=x;
out.q2=q2;
// Find the right spot in the table
i=(int)floor(Nx*log(x/xmin)/log(1.0/xmin));
j=(int)floor(NQ2*log(q2/q2min)/log(q2max/q2min));
// Linear interpolation
t=(x-tt->x[i])/(tt->x[i+1]-tt->x[i]);
u=(q2-tt->q2[j])/(tt->q2[j+1]-tt->q2[j]);
f1=tt->qns[(1+NQ2)*i+j];
f2=tt->qns[(1+NQ2)*(i+1)+j];
f3=tt->qns[(1+NQ2)*(i+1)+j+1];
f4=tt->qns[(1+NQ2)*i+j+1];
out.qns=(1.0-t)*(1.0-u)*f1\
+t*(1.0-u)*f2\
+t*u*f3\
+(1.0-t)*u*f4;
f1=tt->qs[(1+NQ2)*i+j];
f2=tt->qs[(1+NQ2)*(i+1)+j];
f3=tt->qs[(1+NQ2)*(i+1)+j+1];
f4=tt->qs[(1+NQ2)*i+j+1];
out.qs=(1.0-t)*(1.0-u)*f1\
+t*(1.0-u)*f2\
+t*u*f3\
+(1.0-t)*u*f4;
f1=tt->g[(1+NQ2)*i+j];
f2=tt->g[(1+NQ2)*(i+1)+j];
f3=tt->g[(1+NQ2)*(i+1)+j+1];
f4=tt->g[(1+NQ2)*i+j+1];
out.g=(1.0-t)*(1.0-u)*f1\
+t*(1.0-u)*f2\
+t*u*f3\
+(1.0-t)*u*f4;
if (GLUON_DERIVATIVE) {
f1=tt->dg[(1+NQ2)*i+j];
f2=tt->dg[(1+NQ2)*(i+1)+j];
f3=tt->dg[(1+NQ2)*(i+1)+j+1];
f4=tt->dg[(1+NQ2)*i+j+1];
out.dg=(1.0-t)*(1.0-u)*f1\
+t*(1.0-u)*f2\
+t*u*f3\
+(1.0-t)*u*f4;
}
return out;
}
// Then, we provide some routines to free the memory allocated for the
// "large" structures
// Free a variable of type grid_dist
void free_grid_dist(grid_dist *tt){
free(tt->qns);
free(tt->qs);
free(tt->g);
free(tt->q2);
free(tt->x);
if (GLUON_DERIVATIVE) free(tt->dg);
free(tt);
}
// Free a variable of type tab_evol
void free_tab_evol(tab_evol *tt){
free(tt->q2);
free(tt->qns);
free(tt->qs);
free(tt->g);
free(tt);
}

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