From b6ca15ed2e71ffdd35fdddcfc1df61802e76e322 Mon Sep 17 00:00:00 2001
From: chenyixuan0119 <100676477+chenyixuan0119@users.noreply.github.com>
Date: Fri, 4 Mar 2022 23:55:07 +0800
Subject: [PATCH 1/9] Update 5_practice.c
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

直接計算版本
---
 5_practice.c | 34 +++++++++++++++++++++-------------
 1 file changed, 21 insertions(+), 13 deletions(-)

diff --git a/5_practice.c b/5_practice.c
index 70d6651..b5eaf72 100644
--- a/5_practice.c
+++ b/5_practice.c
@@ -9,31 +9,39 @@ int main() {
 	time_t t;
 
 	// Input the number N
-	
+	printf("Input the number N :");
+	scanf("%d",&N);
 	
 	// Locate the memory for x, yr, yi;
-
-	// Initial setting for x, for example, x[k] = k
-	
-	
+	x = (double *) malloc(N*sizeof(double));
+	yr = (double *) malloc(N*sizeof(double));
+	yi = (double *) malloc(N*sizeof(double));
 	
+	// Initial setting for x, for example, x[k] = k
+	for(k=0;k<N;k++){
+		x[k] = k;
+	}
 	
 	t = clock();	
 	// yr[n]+i*yi[n] = sum(exp(-i 2 Pi k n / N)*x[k], k=0..N-1), n=0..N-1 
-
-
-
-
-
-
+	for(n=0;n<N;n++){
+		yr[n] = 0.0;
+		yi[n] = 0.0;
+		for(k=0;k<N;k++){
+			yr[n] += cos(2.0*M_PI*k*n/N)*x[k];
+			yi[n] -= sin(2.0*M_PI*k*n/N)*x[k];
+		}
+		
+	}
 
 	// output the results
 	t = clock() - t;
 	printf("%d ms for discrete Fourier Transform of %d elements\n", t, N);
 
 	// free the memory located by x, yr, yi
-	
-	
+	free(x);
+	free(yr);
+	free(yi);
 	
 	return 100;
 }

From f4a3e64883ec60a32dcee13d0bc80dc0d0ca5683 Mon Sep 17 00:00:00 2001
From: chenyixuan0119 <100676477+chenyixuan0119@users.noreply.github.com>
Date: Sat, 5 Mar 2022 00:32:46 +0800
Subject: [PATCH 2/9] Revert "Update 5_practice.c"

This reverts commit b6ca15ed2e71ffdd35fdddcfc1df61802e76e322.
---
 5_practice.c | 34 +++++++++++++---------------------
 1 file changed, 13 insertions(+), 21 deletions(-)

diff --git a/5_practice.c b/5_practice.c
index b5eaf72..70d6651 100644
--- a/5_practice.c
+++ b/5_practice.c
@@ -9,39 +9,31 @@ int main() {
 	time_t t;
 
 	// Input the number N
-	printf("Input the number N :");
-	scanf("%d",&N);
 	
-	// Locate the memory for x, yr, yi;
-	x = (double *) malloc(N*sizeof(double));
-	yr = (double *) malloc(N*sizeof(double));
-	yi = (double *) malloc(N*sizeof(double));
 	
+	// Locate the memory for x, yr, yi;
+
 	// Initial setting for x, for example, x[k] = k
-	for(k=0;k<N;k++){
-		x[k] = k;
-	}
+	
+	
+	
 	
 	t = clock();	
 	// yr[n]+i*yi[n] = sum(exp(-i 2 Pi k n / N)*x[k], k=0..N-1), n=0..N-1 
-	for(n=0;n<N;n++){
-		yr[n] = 0.0;
-		yi[n] = 0.0;
-		for(k=0;k<N;k++){
-			yr[n] += cos(2.0*M_PI*k*n/N)*x[k];
-			yi[n] -= sin(2.0*M_PI*k*n/N)*x[k];
-		}
-		
-	}
+
+
+
+
+
+
 
 	// output the results
 	t = clock() - t;
 	printf("%d ms for discrete Fourier Transform of %d elements\n", t, N);
 
 	// free the memory located by x, yr, yi
-	free(x);
-	free(yr);
-	free(yi);
+	
+	
 	
 	return 100;
 }

From feb5bc497b4e80432ed3d45da52b73d18992fc0b Mon Sep 17 00:00:00 2001
From: chenyixuan0119 <100676477+chenyixuan0119@users.noreply.github.com>
Date: Sat, 5 Mar 2022 03:41:38 +0800
Subject: [PATCH 3/9] l16091019
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

利用尤拉公式把e^(-i*2*pi*k*n/N)轉換成cos跟sin直接計算
---
 5_practice.c | 40 +++++++++++++++++++++++++---------------
 1 file changed, 25 insertions(+), 15 deletions(-)

diff --git a/5_practice.c b/5_practice.c
index 70d6651..4aa2d34 100644
--- a/5_practice.c
+++ b/5_practice.c
@@ -2,38 +2,48 @@
 #include <stdlib.h>	// for memory allocation
 #include <time.h>	// for time calculation
 #include <math.h>	// for sine and cosine functions
+
 int main() {
 	// Declare all the variables
 	int k, n, N;
 	double *x, *yr, *yi;
-	time_t t;
+	time_t t, t1;
+	int j;
 
 	// Input the number N
-	
+	printf("Input the number N :");
+	scanf("%d",&N);
 	
 	// Locate the memory for x, yr, yi;
-
-	// Initial setting for x, for example, x[k] = k
-	
-	
+	x = (double *) malloc(N*sizeof(double));
+	yr = (double *) malloc(N*sizeof(double));
+	yi = (double *) malloc(N*sizeof(double));
 	
+	// Initial setting for x, for example, x[k] = k
+	for(k=0;k<N;k++){
+		x[k] = k;
+	}
 	
 	t = clock();	
 	// yr[n]+i*yi[n] = sum(exp(-i 2 Pi k n / N)*x[k], k=0..N-1), n=0..N-1 
-
-
-
-
-
-
-
+	for(n=0;n<N;n++){
+		yr[n] = 0.0;
+		yi[n] = 0.0;
+		for(k=0;k<N;k++){
+			yr[n] += cos(2.0*M_PI*k*n/N)*x[k];
+			yi[n] -= sin(2.0*M_PI*k*n/N)*x[k];
+		} 
+		
+	}
 	// output the results
 	t = clock() - t;
 	printf("%d ms for discrete Fourier Transform of %d elements\n", t, N);
-
-	// free the memory located by x, yr, yi
 	
 	
+	// free the memory located by x, yr, yi
+	free(x);
+	free(yr);
+	free(yi);
 	
 	return 100;
 }

From 62c621737b3188be5a5353135aeab24fce94e376 Mon Sep 17 00:00:00 2001
From: chenyixuan0119 <100676477+chenyixuan0119@users.noreply.github.com>
Date: Sat, 5 Mar 2022 03:45:18 +0800
Subject: [PATCH 4/9] Update 5_practice.c

---
 5_practice.c | 1 -
 1 file changed, 1 deletion(-)

diff --git a/5_practice.c b/5_practice.c
index 4aa2d34..299ae30 100644
--- a/5_practice.c
+++ b/5_practice.c
@@ -8,7 +8,6 @@ int main() {
 	int k, n, N;
 	double *x, *yr, *yi;
 	time_t t, t1;
-	int j;
 
 	// Input the number N
 	printf("Input the number N :");

From 579dfbd2759a65619671ca65eccebecb67ee4e2b Mon Sep 17 00:00:00 2001
From: chenyixuan0119 <100676477+chenyixuan0119@users.noreply.github.com>
Date: Sat, 5 Mar 2022 03:45:52 +0800
Subject: [PATCH 5/9] l16091019

---
 5_practice.c | 1 -
 1 file changed, 1 deletion(-)

diff --git a/5_practice.c b/5_practice.c
index 299ae30..caa4256 100644
--- a/5_practice.c
+++ b/5_practice.c
@@ -2,7 +2,6 @@
 #include <stdlib.h>	// for memory allocation
 #include <time.h>	// for time calculation
 #include <math.h>	// for sine and cosine functions
-
 int main() {
 	// Declare all the variables
 	int k, n, N;

From ac64c7b70dab450c07a65d484227f662d5bb782f Mon Sep 17 00:00:00 2001
From: chenyixuan0119 <100676477+chenyixuan0119@users.noreply.github.com>
Date: Tue, 8 Mar 2022 21:05:11 +0800
Subject: [PATCH 6/9] L16091019_update2_5_practice.c
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

method1跟上個版本一樣直接計算，method2為上課所說的利用遞迴方式計算
---
 5_practice.c | 57 ++++++++++++++++++++++++++++++++++++++++++++--------
 1 file changed, 49 insertions(+), 8 deletions(-)

diff --git a/5_practice.c b/5_practice.c
index caa4256..1c4b351 100644
--- a/5_practice.c
+++ b/5_practice.c
@@ -2,20 +2,26 @@
 #include <stdlib.h>	// for memory allocation
 #include <time.h>	// for time calculation
 #include <math.h>	// for sine and cosine functions
+
 int main() {
 	// Declare all the variables
-	int k, n, N;
+	int k, n, N, i;
 	double *x, *yr, *yi;
-	time_t t, t1;
-
+	double *yr1,*yi1;
+	double an,bn,a0,b0,a,b;
+	double temp,dr,di;
+	time_t t,t1;
+	
 	// Input the number N
 	printf("Input the number N :");
 	scanf("%d",&N);
-	
+
 	// Locate the memory for x, yr, yi;
 	x = (double *) malloc(N*sizeof(double));
 	yr = (double *) malloc(N*sizeof(double));
 	yi = (double *) malloc(N*sizeof(double));
+	yr1 = (double *) malloc(N*sizeof(double));
+	yi1 = (double *) malloc(N*sizeof(double));
 	
 	// Initial setting for x, for example, x[k] = k
 	for(k=0;k<N;k++){
@@ -24,6 +30,7 @@ int main() {
 	
 	t = clock();	
 	// yr[n]+i*yi[n] = sum(exp(-i 2 Pi k n / N)*x[k], k=0..N-1), n=0..N-1 
+	// Method 1 : �����p�� 
 	for(n=0;n<N;n++){
 		yr[n] = 0.0;
 		yi[n] = 0.0;
@@ -35,13 +42,47 @@ int main() {
 	}
 	// output the results
 	t = clock() - t;
-	printf("%d ms for discrete Fourier Transform of %d elements\n", t, N);
-	
 	
-	// free the memory located by x, yr, yi
-	free(x);
+	// Method 2
+	a = cos(2.0*M_PI*k/N);
+	b = -sin(2.0*M_PI*k/N);
+	an = 1.0;
+	bn = 0.0;
+	t1 = clock();
+	for(n=0;n<N;n++){
+		yr1[n] = 0.0;
+		yi1[n] = 0.0;
+		a0 = 1.0;
+		b0 = 0.0;
+		for(k=0;k<N;k++){
+			yr1[n] += a0*x[k];
+			yi1[n] -= b0*x[k];
+			temp = an*a0-bn*b0;
+			b0 = an*b0+bn*a0;
+			a0 = temp;
+		}
+		temp = an*a - bn*b;
+		bn = an*b + bn*a;  
+		an = temp;
+	}
+	// output the results
+	t1 = clock() - t1;
+	printf("Method 1 : %e ms for discrete Fourier Transform of %d elements\n", t, N);
+	printf("Method 2 : %e ms for discrete Fourier Transform of %d elements\n", t1, N);
+	dr = 0.0;
+	di = 0.0;
+	for(i=0;i<N;i++){
+		dr += pow(yr1[i]-yr[i],2);
+		di += pow(yi1[i]-yi[i],2);
+	}
+	printf("the difference of real number for two methods: %e\n",sqrt(dr));
+	printf("the difference of imaginary numberfor two methods: %e\n",sqrt(di));
+	// free the memory located by x, yr, yi	
 	free(yr);
 	free(yi);
+	free(yr1);
+	free(yi1);
+	free(x);
 	
 	return 100;
 }

From b4bbdd71a724d4bd81ec9a256bee16785dc38bfe Mon Sep 17 00:00:00 2001
From: chenyixuan0119 <100676477+chenyixuan0119@users.noreply.github.com>
Date: Tue, 15 Mar 2022 22:59:38 +0800
Subject: [PATCH 7/9] Add files via upload

---
 6_sorting.c | 90 +++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 90 insertions(+)
 create mode 100644 6_sorting.c

diff --git a/6_sorting.c b/6_sorting.c
new file mode 100644
index 0000000..d50b77d
--- /dev/null
+++ b/6_sorting.c
@@ -0,0 +1,90 @@
+#include <stdio.h>	// for printf function
+#include <stdlib.h>	// for memory allocation
+#include <time.h>	// for time calculation
+#include <math.h>	// for sine and cosine functions
+int quick_sort(double *x, int L, int R);
+int main() {
+	// Declare all the variables
+	int k, m, n, N;
+	double *x, *y, z, p;
+	time_t t;
+
+	// Input the number N
+	printf("Input N: ");
+	scanf("%d",&N);
+	
+	// Locate the memory for x and y;
+	x = (double *) malloc(N*sizeof(double));
+	y = (double *) malloc(N*sizeof(double));
+
+	// Initial setting for x, for example, x[k] = 1.0*rand()/RAND_MAX
+	srand( time(NULL) );
+	for(k=0;k<N;k++){
+		x[k] = y[k] = 1.0*rand()/RAND_MAX;
+	}
+	
+	t = clock();	
+	// sorting x;  
+	for(n=0;n<N;n++) {
+		for(k=n+1;k<N;k++) {
+			if (x[n] > x[k]) {
+				z = x[n];
+				x[n] = x[k];
+				x[k] = z;
+			}
+		}
+	}
+	t = clock() - t;
+	// print y, x, and time
+	printf("Sorting %d elements: %f s\n", N, 1.0*t/CLOCKS_PER_SEC);
+	if(N<10) {
+		printf("y \t\t x\n");
+		for(k=0;k<N;++k) {
+			printf("%f\t%f\n",y[k],x[k]);
+		}
+	}
+
+	for(k=0;k<N;++k){
+		x[k] = y[k];
+	}
+		
+	t = clock();
+	
+	
+	// x[0] is at the correct location!
+	if(N<10) {
+		printf("y \t\t x\n");
+		for(k=0;k<N;++k) {
+			printf("%f\t%f\n",y[k],x[k]);
+		}
+	}
+	// sorting
+
+
+	t = clock() - t;	
+	
+
+
+	// free the memory located by x, y
+	free(x);
+	free(y);	
+	
+	return 100;
+}
+
+int quick_sort(double *x, int L, int R){
+	p = x[N-1]; n = 0;
+	for(k=0;k<N-1;k++) {
+		// put all elements smaller than p from the beginning
+		if (x[k] < p) {
+			z = x[n];
+			x[n] = x[k];
+			x[k] = z;
+			n++;
+		}
+		
+	}
+	z = x[n];
+	x[n] = x[N-1];
+	x[N-1] = z;
+}

From 5093d67d278924cc24a11b53711853cb0b6cd8a1 Mon Sep 17 00:00:00 2001
From: chenyixuan0119 <100676477+chenyixuan0119@users.noreply.github.com>
Date: Wed, 16 Mar 2022 02:30:35 +0800
Subject: [PATCH 8/9] update1_6_sorting.c

---
 6_sorting.c | 103 ++++++++++++++++++++++++++++++++++++++++++----------
 1 file changed, 83 insertions(+), 20 deletions(-)

diff --git a/6_sorting.c b/6_sorting.c
index d50b77d..ec832e3 100644
--- a/6_sorting.c
+++ b/6_sorting.c
@@ -3,10 +3,11 @@
 #include <time.h>	// for time calculation
 #include <math.h>	// for sine and cosine functions
 int quick_sort(double *x, int L, int R);
+double quick_sort_median(double *x1, int L, int R, int M1,int bl);
 int main() {
 	// Declare all the variables
 	int k, m, n, N;
-	double *x, *y, z, p;
+	double *x, *y,*x1, z, p , med, med1;
 	time_t t;
 
 	// Input the number N
@@ -16,11 +17,11 @@ int main() {
 	// Locate the memory for x and y;
 	x = (double *) malloc(N*sizeof(double));
 	y = (double *) malloc(N*sizeof(double));
-
+	x1 = (double *) malloc(N*sizeof(double));
 	// Initial setting for x, for example, x[k] = 1.0*rand()/RAND_MAX
 	srand( time(NULL) );
 	for(k=0;k<N;k++){
-		x[k] = y[k] = 1.0*rand()/RAND_MAX;
+		x[k] = y[k] = x1[k] = 1.0*rand()/RAND_MAX;
 	}
 	
 	t = clock();	
@@ -36,8 +37,8 @@ int main() {
 	}
 	t = clock() - t;
 	// print y, x, and time
-	printf("Sorting %d elements: %f s\n", N, 1.0*t/CLOCKS_PER_SEC);
-	if(N<10) {
+	printf("Sorting %d elements: %e s\n", N, 1.0*t/CLOCKS_PER_SEC);
+	if(N<=10) {
 		printf("y \t\t x\n");
 		for(k=0;k<N;++k) {
 			printf("%f\t%f\n",y[k],x[k]);
@@ -47,34 +48,54 @@ int main() {
 	for(k=0;k<N;++k){
 		x[k] = y[k];
 	}
-		
-	t = clock();
-	
 	
+	// quick sort x 
+	t = clock();
+	quick_sort(x,0,N);
+	t = clock() - t;
 	// x[0] is at the correct location!
-	if(N<10) {
+	printf("Quick Sorting %d elements: %e s\n", N, 1.0*t/CLOCKS_PER_SEC);
+	if(N<=10) {
 		printf("y \t\t x\n");
 		for(k=0;k<N;++k) {
 			printf("%f\t%f\n",y[k],x[k]);
 		}
 	}
-	// sorting
-
-
-	t = clock() - t;	
 	
-
+	t = clock();
+	if( N % 2 != 0){
+		med1 = quick_sort_median(x1,0,N,(N-1)/2,1);
+	}
+	else{
+		med1 = quick_sort_median(x1,0,N,(N-1)/2,0);
+	}
+	t = clock() - t;
+	if( N % 2 != 0){
+		med = x[N/2];
+	}
+	else{
+		med = (x[(N/2)-1]+x[N/2])/2.0;
+	}
+	printf("quick_sort_median spends : %e s\n", N, 1.0*t/CLOCKS_PER_SEC);
+	printf("The error is %f \n",med-med1);
 
 	// free the memory located by x, y
 	free(x);
 	free(y);	
-	
+	free(x1);
 	return 100;
 }
 
 int quick_sort(double *x, int L, int R){
-	p = x[N-1]; n = 0;
-	for(k=0;k<N-1;k++) {
+	int k, n;
+	double z, p;
+	if(L >= R){
+		return 0;
+	}
+	p = x[R-1];
+	n = L;
+	
+	for(k=L;k<R-1;k++) {
 		// put all elements smaller than p from the beginning
 		if (x[k] < p) {
 			z = x[n];
@@ -82,9 +103,51 @@ int quick_sort(double *x, int L, int R){
 			x[k] = z;
 			n++;
 		}
-		
 	}
 	z = x[n];
-	x[n] = x[N-1];
-	x[N-1] = z;
+	x[n] = x[R-1];
+	x[R-1] = z;
+
+	quick_sort(x,L,n);
+	quick_sort(x,n+1,R);
+	return 0;
+}
+
+double quick_sort_median(double *x1, int L, int R, int M1,int bl){
+	int k, n,m;
+	double z, p;
+	if(L >= R){
+		return 0;
+	}
+
+	p = x1[R-1];
+	n = L;
+	for(k=L;k<R-1;k++) {
+		// put all elements smaller than p from the beginning
+		if (x1[k] < p) {
+			z = x1[n];
+			x1[n] = x1[k];
+			x1[k] = z;
+			n++;
+		}
+	}
+	z = x1[n];
+	x1[n] = x1[R-1];
+	x1[R-1] = z;
+
+	if(n > M1){
+		quick_sort_median(x1,L,n,M1,bl);
+	}
+	else if(n < M1){
+		quick_sort_median(x1,n+1,R,M1,bl);
+	}
+	else{
+		if(bl){
+			return x1[n];
+		}
+		else{
+			quick_sort_median(x1,n+1,R,M1+1,bl);
+			return (x1[n]+x1[n+1])/2.0;
+		}
+	}
 }

From 5153c81b572b1bbac30648967660329b3a8d4b10 Mon Sep 17 00:00:00 2001
From: chenyixuan0119 <100676477+chenyixuan0119@users.noreply.github.com>
Date: Wed, 16 Mar 2022 04:44:53 +0800
Subject: [PATCH 9/9] update2_6_sorting.c
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

M1是利用quick_sort_median求中位數，而M2是利用quick_sort_median1求中位數，差別在求n為偶數時，M1是在 function 裡從大於第n/2大的數中找第n/2+1大的數字，而M2則是直接在main裡面全部從x[0]到x[N-1]在重新跑過一遍找第n/2+1大的數字，但後來發現兩個會一樣快，是因為M2在前n/2項不會做更動，所以其實就跟M1在找第n/2+1大的數做的事情是一樣的，四分位數就直接利用 M1 去搜尋
---
 6_sorting.c | 146 +++++++++++++++++++++++++++++++++++-----------------
 1 file changed, 99 insertions(+), 47 deletions(-)

diff --git a/6_sorting.c b/6_sorting.c
index ec832e3..8c815d1 100644
--- a/6_sorting.c
+++ b/6_sorting.c
@@ -3,12 +3,13 @@
 #include <time.h>	// for time calculation
 #include <math.h>	// for sine and cosine functions
 int quick_sort(double *x, int L, int R);
-double quick_sort_median(double *x1, int L, int R, int M1,int bl);
+double quick_sort_median(double *x, int L, int R, int M1,int bl);
+double quick_sort_median1(double *x, int L, int R, int M1);
 int main() {
 	// Declare all the variables
 	int k, m, n, N;
-	double *x, *y,*x1, z, p , med, med1;
-	time_t t;
+	double *x, *y, z, p , med, med1, med2,q1,q3;
+	time_t t,t1,t2;
 
 	// Input the number N
 	printf("Input N: ");
@@ -17,14 +18,12 @@ int main() {
 	// Locate the memory for x and y;
 	x = (double *) malloc(N*sizeof(double));
 	y = (double *) malloc(N*sizeof(double));
-	x1 = (double *) malloc(N*sizeof(double));
 	// Initial setting for x, for example, x[k] = 1.0*rand()/RAND_MAX
+
 	srand( time(NULL) );
 	for(k=0;k<N;k++){
-		x[k] = y[k] = x1[k] = 1.0*rand()/RAND_MAX;
-	}
-	
-	t = clock();	
+		x[k] = y[k] = 1.0*rand()/RAND_MAX;
+	}	
 	// sorting x;  
 	for(n=0;n<N;n++) {
 		for(k=n+1;k<N;k++) {
@@ -35,54 +34,73 @@ int main() {
 			}
 		}
 	}
-	t = clock() - t;
-	// print y, x, and time
-	printf("Sorting %d elements: %e s\n", N, 1.0*t/CLOCKS_PER_SEC);
-	if(N<=10) {
-		printf("y \t\t x\n");
-		for(k=0;k<N;++k) {
-			printf("%f\t%f\n",y[k],x[k]);
-		}
-	}
-
 	for(k=0;k<N;++k){
 		x[k] = y[k];
 	}
 	
-	// quick sort x 
-	t = clock();
 	quick_sort(x,0,N);
-	t = clock() - t;
-	// x[0] is at the correct location!
-	printf("Quick Sorting %d elements: %e s\n", N, 1.0*t/CLOCKS_PER_SEC);
-	if(N<=10) {
+	
+	if(N<=20) {
 		printf("y \t\t x\n");
 		for(k=0;k<N;++k) {
-			printf("%f\t%f\n",y[k],x[k]);
+			printf("%d %f\t%f\n",k,y[k],x[k]);
 		}
 	}
 	
+	if( N % 2 != 0){
+		med = x[N/2];
+	}
+	else{
+		med = (x[(N/2)-1]+x[N/2])/2.0;
+	}
+
+	for(k=0;k<N;++k){
+		x[k] = y[k];
+	}
 	t = clock();
 	if( N % 2 != 0){
-		med1 = quick_sort_median(x1,0,N,(N-1)/2,1);
+		med1 = quick_sort_median(x,0,N,(N-1)/2,1);
 	}
 	else{
-		med1 = quick_sort_median(x1,0,N,(N-1)/2,0);
+		med1 = quick_sort_median(x,0,N,(N-1)/2,0);
 	}
 	t = clock() - t;
-	if( N % 2 != 0){
-		med = x[N/2];
+	
+	t2 = clock();
+	if( (((N-1)/2)%2 != 0 && N%2 != 0) || (N % 2 ==0 && (N/2)%2 != 0)){
+		q1 = quick_sort_median(x,0,N/2,((N/2)-1)/2,1); 
+		q3 = quick_sort_median(x,N/2+1,N,3*N/4,1); 
 	}
 	else{
-		med = (x[(N/2)-1]+x[N/2])/2.0;
+		q1 = quick_sort_median(x,0,N/2,((N/2)-1)/2,0);
+		q3 = quick_sort_median(x,(N-1)/2+1,N,(3*N-1)/4,0);
 	}
-	printf("quick_sort_median spends : %e s\n", N, 1.0*t/CLOCKS_PER_SEC);
-	printf("The error is %f \n",med-med1);
+	t2 = t2-clock();
+	
+	for(k=0;k<N;++k){
+		x[k] = y[k];
+	}
+	
+	t1 = clock();
+	if( N % 2 != 0){
+	med2 = quick_sort_median1(x,0,N,(N-1)/2);
+	  }
+	else{
+	  	med2 = quick_sort_median1(x,0,N,(N-1)/2);
+	   	med2 = med2 + quick_sort_median1(x,0,N,N/2);
+	   	med2 = med2/2.0;
+	}
+	t1 = t1-clock();
 
+	printf("quick_sort_median spends : %e s\n", N, 1.0*t/CLOCKS_PER_SEC);
+	printf("quick_sort_median1 spends : %e s\n", N, 1.0*t1/CLOCKS_PER_SEC);
+	printf("The error with quick_sort_median is %f \n",med-med1);
+	printf("The error with quick_sort_median1 is %f \n",med-med2);
+	printf("q1 = %f q3 = %f time : %e \n", q1,q3,1.0*t2/CLOCKS_PER_SEC);
 	// free the memory located by x, y
 	free(x);
 	free(y);	
-	free(x1);
+
 	return 100;
 }
 
@@ -113,41 +131,75 @@ int quick_sort(double *x, int L, int R){
 	return 0;
 }
 
-double quick_sort_median(double *x1, int L, int R, int M1,int bl){
+double quick_sort_median(double *x, int L, int R, int M1,int bl){
 	int k, n,m;
 	double z, p;
 	if(L >= R){
 		return 0;
 	}
 
-	p = x1[R-1];
+	p = x[R-1];
 	n = L;
 	for(k=L;k<R-1;k++) {
 		// put all elements smaller than p from the beginning
-		if (x1[k] < p) {
-			z = x1[n];
-			x1[n] = x1[k];
-			x1[k] = z;
+		if (x[k] < p) {
+			z = x[n];
+			x[n] = x[k];
+			x[k] = z;
 			n++;
 		}
 	}
-	z = x1[n];
-	x1[n] = x1[R-1];
-	x1[R-1] = z;
+	z = x[n];
+	x[n] = x[R-1];
+	x[R-1] = z;
 
 	if(n > M1){
-		quick_sort_median(x1,L,n,M1,bl);
+		quick_sort_median(x,L,n,M1,bl);
 	}
 	else if(n < M1){
-		quick_sort_median(x1,n+1,R,M1,bl);
+		quick_sort_median(x,n+1,R,M1,bl);
 	}
 	else{
 		if(bl){
-			return x1[n];
+			return x[n];
 		}
 		else{
-			quick_sort_median(x1,n+1,R,M1+1,bl);
-			return (x1[n]+x1[n+1])/2.0;
+			quick_sort_median(x,n+1,R,M1+1,bl);
+			return (x[n]+x[n+1])/2.0;
 		}
 	}
 }
+
+double quick_sort_median1(double *x, int L, int R, int M1){
+	int k, n,m;
+	double z, p;
+	if(L >= R){
+		return 0;
+	}
+
+	p = x[R-1];
+	n = L;
+	for(k=L;k<R-1;k++) {
+		// put all elements smaller than p from the beginning
+		if (x[k] < p) {
+			z = x[n];
+			x[n] = x[k];
+			x[k] = z;
+			n++;
+		}
+	}
+	z = x[n];
+	x[n] = x[R-1];
+	x[R-1] = z;
+
+	if(n > M1){
+		quick_sort_median1(x,L,n,M1);
+	}
+	else if(n < M1){
+		quick_sort_median1(x,n+1,R,M1);
+	}
+	else{
+		return x[n];
+	}
+	
+}