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#include <stdio.h>
#include <stdlib.h>
#include <math.h>
/*given function f on domain [0,D-1], compute derivatives f^{(0)} until f^{(K)} and store them in df*/
void differentiate(double* f, double** df, int D, int K){
df[0] = f;
int i;
for(int k=1; k<=K; k++){
/*compute kth derivative of f from (k-1)th*/
/*only compute derivatives at arguments i < D-k, because for larger i we would need data from outside the domain of f*/
for(i=0; i<D-k; i++){
df[k][i] = df[k-1][i+1] - df[k-1][i];
}
}
}
/*given function f on domain [0,D-1] compute pth root of integral of |f|^p*/
double integrate(double* f, double p, int D){
double sum = 0.0;
for(int i=0;i<D;i++){
sum += pow(fabs(f[i]),p);
}
return pow(sum,1/p);
}
/*given integer valued function f on domain D, compute ||Mf^{(K)}||_p/||f^{(K)}||_p*/
/*All the other arguments are pointers to storage for intermediate variables. The purpose is that we do not have to allocate new storage with each invocation. Probably there's a more user friendly way but I don't know much about garbage collection.*/
double compute_derivatives(int* f, int** Sf, double** Af, double* Mf, double** df, double** dMf, double p, double* intdf, double* intdMf, int D, int K){
/*Convert integer valued f to float valued df[0]*/
for(int i=0;i<D;i++) df[0][i] = (double) f[i];
/*Sf[i][j] will be the integral of f on [min(i,j),max(i,j)]*/
/*Af[i][j] will be the average of f on [min(i,j),max(i,j)]*/
for(int i=0;i<D;i++) {
Sf[i][i] = f[i];
Af[i][i] = 1.0*Sf[i][i];
}
for(int n=1;n<D;n++){
/*Recursively compute all integrals and averages over intervals of increasing length*/
for(int i=0;i+n<D;i++){
Sf[i][i+n] = Sf[i][i+n-1]+f[i+n];
Sf[i+n][i] = Sf[i][i+n];
Af[i][i+n] = 1.0*Sf[i][i+n]/(n+1);
Af[i+n][i] = Af[i][i+n];
}
}
/*Compute maximal function by picking the largest average*/
for(int i=0;i<D;i++){
Mf[i] = Af[i][i];
for(int j=0;j<D;j++){
if(Af[i][j] > Mf[i]) Mf[i] = Af[i][j];
}
}
/*Print computed functions and averages*/
//printf("Mf ");
//for(int i=0;i<D;i++) printf("%+0.1f ",Mf[i]);
//printf("\n");
//for(int i=0;i<D;i++){
//for(int j=0;j<D;j++) printf("%0.1f ",Af[i][j]);
//printf("\n");
//}
/*Compute 0th until Kth derivative of f and Mf*/
differentiate(df[0],df,D,K);
differentiate(dMf[0],dMf,D,K);
/*Print derivatives*/
//for(int k=0;k<=K;k++){
//printf("f %d: ",k);
//for(int i=0;i<D;i++) printf("%+0.1f ",df[k][i]);
//printf("\n");
//printf("Mf %d: ",k);
//for(int i=0;i<D;i++) printf("%+0.1f ",dMf[k][i]);
//printf("\n");
//}
//for(int k=0;k<=K;k++){
/*Compute L^p norm of derivatives*/
int k=K;
intdf[k] = integrate(df[k],p,D);
intdMf[k] = integrate(dMf[k],p,D);
//printf("%d: %f / %f = %f\n",k,intdMf[k],intdf[k],intdMf[k]/intdf[k]);
//}
/*Return ratio of L^p norms*/
return intdMf[k]/intdf[k];
}
int main() {
/*length of the support of f*/
int N=16;
/*length of the domain*/
int D=5*N;
/*order of the derivative to consider. Should not be larger than (D-N)/2 because then the support of f^{(K)} reaches outside of our domain.*/
int K=3;
/*exponent p of the L^p norm to consider*/
double p = 1.0;
/*allocate memory for f*/
int* f = malloc(D*sizeof(int));
/*allocate memory for temporary variables, such as averages*/
int** Sf = malloc(D*sizeof(int*));
double** Af = malloc(D*sizeof(double*));
double* Mf = malloc(D*sizeof(double));
for(int i=0;i<D;i++){
Sf[i] = malloc(D*sizeof(int));
Af[i] = malloc(D*sizeof(double));
}
/*Allocate memory for derivatives*/
double** df = malloc(D*sizeof(double*));
double** dMf = malloc(D*sizeof(double*));
for(int k=0;k<=K;k++){
df[k] = malloc(D*sizeof(double));
dMf[k] = malloc(D*sizeof(double));
for(int i=0;i<D;i++){
df[k][i] = 0;
dMf[k][i] = 0;
}
}
dMf[0] = Mf;
/*Allocate memory for integrals*/
double* intdf = malloc(K*sizeof(double));
double* intdMf = malloc(K*sizeof(double));
/*Allocate memory for ||Mf^{(k)}||_p/||f^{(k)}||_p.*/
double r = 0.0;
/*Iterate over all strings of 0s and 1s with length N. Those will represent f.*/
for(int t=1; t<=(1 << N)-1; t++){
/*Initiate f to be zero everywhere.*/
for(int i=0;i<D;i++) f[i]=0;
//for(int i=0;i<N;i++) f[N+i] = rand() %2;
/*In the middle of the domain set f to the values encoded in bit string t*/
for(int i=0;i<N;i++){
/*Since we care about the Kth derivative, which in i depends on f on [i,i+K], shift f to the right by K/2 so that the most interesting part of f^{(K)} and Mf^{(K)} will be around the center of the domain*/
f[2*N+i+K/2] = (t >> i) & 1;
//if(i%3==0) f[2*N+i+K/2] = 1;
}
/*Compute ||Mf^{(k)}||_p/||f^{(k)}||_p.*/
r = compute_derivatives(f, Sf, Af, Mf, df, dMf, p, intdf, intdMf, D, K);
//printf("%.3d: %.3f \n",t,r);
/*Print f and ||Mf^{(k)}||_p/||f^{(k)}||_p if the latter is close to 1/2.*/
if(r>.4997){
printf("f: ");
for(int i=0;i<D;i++) printf("%.1d ",f[i]);
printf("\n");
printf("%.4f\n",r);
}
}
return 0;
}
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