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#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <pthread.h>
#ifndef EXACT
#define EXACT false
#endif
#if EXACT
#include "ratio.h"
#define VALUETYPE rational
#define EXPTYPE unsigned int
#else
#include "double.h"
#define VALUETYPE double
#define EXPTYPE double
#endif
/*given function df[0] on domain [0,D-1], compute derivatives f' until f^{(K)} and store them in df[1] to df[K]*/
void differentiate(VALUETYPE* f, VALUETYPE* df, int D, int K){
/*Set zeroth derivative to be f.*/
for(int i=0; i<D; i++){
df[i] = f[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(int i=0; i<D-k; i++){
df[i] = difference(df[i+1], df[i]);
}
/*Make them zero for arguments where their dependence reaches outside the domain.*/
for(int i=D-k; i<D; i++){
df[i] = convert_int(0);
}
}
}
/*given function f on domain [0,D-1] compute pth root of integral of |f|^p*/
VALUETYPE integratep(VALUETYPE* f, EXPTYPE p, int D){
VALUETYPE integralp = convert_int(0);
for(int i=0;i<D;i++){
integralp = sum(integralp,power(absolute(f[i]),p));
}
return integralp;
}
void compute_maximalfunction(VALUETYPE* f, VALUETYPE* Mf, int D){
/*Sf[i][j] will be the integral of f on [min(i,j),max(i,j)]*/
VALUETYPE** Sf = malloc(D*sizeof(VALUETYPE*));
/*Af[i][j] will be the average of f on [min(i,j),max(i,j)]*/
VALUETYPE** Af = malloc(D*sizeof(VALUETYPE*));
for(int i=0;i<D;i++){
Sf[i] = malloc(D*sizeof(VALUETYPE));
Af[i] = malloc(D*sizeof(VALUETYPE));
}
for(int i=0;i<D;i++) {
Sf[i][i] = f[i];
Af[i][i] = f[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] = sum(Sf[i][i+n-1], f[i+n]);
Sf[i+n][i] = Sf[i][i+n];
Af[i][i+n] = ratio(Sf[i][i+n],convert_int(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(is_greater(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 ",to_double(Mf[i]));
printf("\n");
for(int i=0;i<D;i++){
for(int j=0;j<D;j++) printf("%0.1f ",to_double(Af[i][j]));
printf("\n");
}
*/
for(int i=0;i<D;i++){
free(Sf[i]);
free(Af[i]);
}
free(Sf);
free(Af);
}
void compute(int N, int K, int D, EXPTYPE p, int t){
/*allocate memory for f*/
VALUETYPE* f = malloc(D*sizeof(VALUETYPE));
/*Initiate f to be zero everywhere.*/
for(int i=0;i<D;i++) f[i] = convert_int(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[(D-N)/2+i+K/2] = convert_int((t >> i) & 1);
//if(i%3==0) f[2*N+i+K/2] = 1;
}
VALUETYPE* Mf = malloc(D*sizeof(VALUETYPE));
compute_maximalfunction(f, Mf, D);
/*Allocate memory for derivatives.*/
VALUETYPE* df = malloc(D*sizeof(VALUETYPE));
VALUETYPE* dMf = malloc(D*sizeof(VALUETYPE));
/*Compute Kth derivative of f and Mf*/
differentiate(f,df,D,K);
differentiate(Mf,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");
}
*/
/*Compute L^p norm of derivatives*/
VALUETYPE intdfp = integratep(df,p,D);
VALUETYPE intdMfp = integratep(dMf,p,D);
//printf("%d: %f / %f = %f\n",k,intdMfp[k],intdfp[k],intdMfp[k]/intdfp[k]);
/*Compute ||Mf^{(k)}||_p/||f^{(k)}||_p.*/
VALUETYPE r = ratio(intdMfp, intdfp);
free(df);
free(dMf);
//printf("%.3d: %.3f \n",t,r);
/*Print f and ||Mf^{(k)}||_p/||f^{(k)}||_p if the latter is close to 1/2.*/
if(to_double(r)>.4997){
//if(to_double(r)>.7)
printf("f: ");
for(int i=0;i<D;i++) printf("%1.0f ",to_double(f[i]));
printf("\n");
printf("%.4f\n",to_double(r));
}
free(f);
free(Mf);
}
int main() {
/*length of the support of f*/
int N=12;
/*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;
/*length of the domain*/
int D=N+K+1;
/*exponent p of the L^p norm to consider*/
EXPTYPE p = 1;
/*Iterate over all strings of 0s and 1s with length N. Those will represent f.*/
for(int t=1; t<=(1 << N)-1; t++) compute(N,K,D,p,t);
return 0;
}
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