diff --git a/MPI-Codes/Collectives/Makefile b/MPI-Codes/Collectives/Makefile
new file mode 100755
index 0000000..24f3302
--- /dev/null
+++ b/MPI-Codes/Collectives/Makefile
@@ -0,0 +1,21 @@
+MPICC = mpicc			#The wrapper for the compiler
+CC = gcc
+CFLAGS += -g			#Useful for debug symbols
+LDLIBS += -lm
+
+all: sequentialderiv parallelderiv
+
+sequentialderiv: sequentialderiv.c
+	$(CC) $(CFLAGS) $(LDFLAGS) $? $(LDLIBS) -o $@
+
+parallelderiv: parallelderiv.c
+	$(MPICC) $(CFLAGS) $(LDFLAGS) $? $(LDLIBS) -o $@
+
+midpointintegral: midpointintegral.c
+		$(MPICC) $(CFLAGS) $(LDFLAGS) $? $(LDLIBS) -o $@
+
+montecarlostyle: montecarlostyle.c
+		$(MPICC) $(CFLAGS) $(LDFLAGS) $? $(LDLIBS) -o $@
+
+clean:
+	rm -f easysendrecv nonblockingsendrecv sequentialderiv
diff --git a/MPI-Codes/Collectives/Solution/Makefile b/MPI-Codes/Collectives/Solution/Makefile
new file mode 100755
index 0000000..f5118d3
--- /dev/null
+++ b/MPI-Codes/Collectives/Solution/Makefile
@@ -0,0 +1,18 @@
+MPICC = mpicc			#The wrapper for the compiler
+CC = gcc
+CFLAGS += -g			#Useful for debug symbols
+LDLIBS += -lm
+
+all: parallelderiv midpointintegral montecarlostyle
+
+parallelderiv: parallelderiv.c
+	$(MPICC) $(CFLAGS) $(LDFLAGS) $? $(LDLIBS) -o $@
+
+midpointintegral: midpointintegral.c
+		$(MPICC) $(CFLAGS) $(LDFLAGS) $? $(LDLIBS) -o $@
+
+montecarlostyle: montecarlostyle.c
+		$(MPICC) $(CFLAGS) $(LDFLAGS) $? $(LDLIBS) -o $@
+
+clean:
+	rm -f parallelderiv midpointintegral montecarlostyle
diff --git a/MPI-Codes/Collectives/Solution/midpointintegral.c b/MPI-Codes/Collectives/Solution/midpointintegral.c
new file mode 100755
index 0000000..d0e8e82
--- /dev/null
+++ b/MPI-Codes/Collectives/Solution/midpointintegral.c
@@ -0,0 +1,43 @@
+#include <stdio.h>
+#include <stdlib.h>
+#include <math.h>
+#include <mpi.h>
+
+double f( double x);
+
+int main( int argc, char *argv[]){
+int n, mynode, totalnodes, i;
+double PI25DT = 3.141592653589793238462643;
+double mypi, pi, h, sum, x;
+
+MPI_Init(&argc,&argv);
+MPI_Comm_size(MPI_COMM_WORLD,&totalnodes);
+MPI_Comm_rank(MPI_COMM_WORLD,&mynode);
+
+if(mynode == 0){
+  if(argc != 2){
+   n = 20;
+  }else{
+   n = atoi(argv[1]);
+  }
+}
+MPI_Bcast(&n, 1, MPI_INT, 0, MPI_COMM_WORLD);
+h   = 1.0 / ((double) n*totalnodes);
+sum = 0.0;
+for (i = 1+mynode*n; i <= n*(mynode+1); i++){
+ x = h * ((double)i - 0.5);
+ sum += f(x);
+}
+mypi = h*sum;
+MPI_Reduce(&mypi, &pi, 1, MPI_DOUBLE, MPI_SUM, 0, MPI_COMM_WORLD);
+if (mynode == 0){
+ printf("pi is approximately %.16f, Error is %e\n",pi, fabs(pi - PI25DT));
+}
+MPI_Finalize();
+return 0;
+}
+
+double f( double x )
+{
+    return (4.0 / (1.0 + x*x));
+}
diff --git a/MPI-Codes/Collectives/Solution/montecarlostyle.c b/MPI-Codes/Collectives/Solution/montecarlostyle.c
new file mode 100755
index 0000000..64e5622
--- /dev/null
+++ b/MPI-Codes/Collectives/Solution/montecarlostyle.c
@@ -0,0 +1,58 @@
+#include <stdio.h>
+#include <stdlib.h>
+#include <math.h>
+#include <mpi.h>
+#include <time.h>
+
+int main( int argc, char *argv[]){
+ int n, mynode, totalnodes, i;
+ double PI25DT = 3.141592653589793238462643;
+ double mypi, pi, h;
+ double x, y, x1, x2, y1, y2;
+ int SqPoints, CiPoints, my_SqPoints, my_CiPoints;
+ unsigned int seed = (unsigned) time(NULL);
+
+ MPI_Init(&argc,&argv);
+ MPI_Comm_size(MPI_COMM_WORLD,&totalnodes);
+ MPI_Comm_rank(MPI_COMM_WORLD,&mynode);
+
+ if(mynode == 0){
+   if(argc != 2){
+    n = 20;
+   }else{
+    n = atoi(argv[1]);
+   }
+ }
+ MPI_Bcast(&n, 1, MPI_INT, 0, MPI_COMM_WORLD);
+
+ h  = 2.0 / (double) totalnodes ;
+ x1 = -1.0 + mynode * h;
+ x2 = x1 + h;
+ y1 = -1.0;
+ y2 = 1.0;
+ my_SqPoints  = 0;
+ my_CiPoints  = 0;
+
+ for (i = 1; i <= n; i++)
+ {
+    srand(seed + mynode);
+    x = rand_r(&seed); x = x / RAND_MAX; x = x1 + x * (x2 - x1);
+    y = rand_r(&seed); y = y / RAND_MAX; y = y1 + y * (y2 - y1);
+    my_SqPoints++;
+    if ( ( x*x + y*y ) <= 1.0 ) my_CiPoints++;
+}
+SqPoints = 0;
+CiPoints = 0;
+MPI_Reduce(&my_SqPoints, &SqPoints, 1, MPI_INT, MPI_SUM, 0, MPI_COMM_WORLD);
+MPI_Reduce(&my_CiPoints, &CiPoints, 1, MPI_INT, MPI_SUM, 0, MPI_COMM_WORLD);
+
+if (mynode == 0)
+{
+   pi = 4.0 * (double)CiPoints / (double)SqPoints;
+   printf("Pi is approximately %.16f = 4.0 %d/%d, Error is %e (h = %e)\n",pi,CiPoints,SqPoints,fabs(pi - PI25DT),1.0/sqrt( (double) SqPoints) );
+}
+
+MPI_Finalize();
+
+return 0;
+}
diff --git a/MPI-Codes/Collectives/Solution/parallelderiv.c b/MPI-Codes/Collectives/Solution/parallelderiv.c
new file mode 100755
index 0000000..a62bc30
--- /dev/null
+++ b/MPI-Codes/Collectives/Solution/parallelderiv.c
@@ -0,0 +1,98 @@
+#include <stdio.h>
+#include <stdlib.h>
+#include <math.h>
+#include "mpi.h"
+
+void firstderiv1Dp_vec(int n, double dx, double *f,double *fx, int mynode, int totalnodes);
+
+double fun(double x);
+double funprime( double x);
+
+int main(int argc, char **argv){
+  double globala,globalb,a,b,dx,*f,*fx, error;
+  int n, mynode, totalnodes;
+  MPI_Status status;
+
+  MPI_Init( &argc, &argv );
+  MPI_Comm_rank( MPI_COMM_WORLD, &mynode );
+  MPI_Comm_size( MPI_COMM_WORLD, &totalnodes );
+
+  n = 20;
+
+  globala = 0;
+  globalb = 1;
+
+  a = globala + ((double) mynode)*(globalb - globala)/( (double) totalnodes);
+  b = globala + ((double) mynode+1)*(globalb - globala)/( (double) totalnodes);
+
+  f  = (double *) malloc(sizeof(double)*(n));
+  fx = (double *) malloc(sizeof(double)*(n));
+
+  dx = (b-a)/((double) n);
+  for( int i = 0; i < n; i++){
+    f[i] = fun(a+((double) i)*dx);
+  }
+  firstderiv1Dp_vec( n, dx, f, fx, mynode, totalnodes);
+
+  error = 0.0;
+  for(int i = 0; i < n; i++){
+    error += pow( fx[i]-funprime(a+((b-a)*((double) i))/((double) n)),2.0);
+  }
+  error = sqrt(dx*error);
+  printf("Node %d ||f' - fx||_2 = %e\n",mynode,error);
+
+
+  free(f);
+  free(fx);
+  MPI_Finalize();
+
+  return 0;
+}
+
+
+void firstderiv1Dp_vec(int n, double dx, double *f,double *fx, int mynode, int totalnodes){
+double scale = 1.0/(2.0*dx);
+double mpitemp;
+MPI_Status status;
+
+if(mynode == 0){
+ fx[0] = (-3.0*f[0] + 4.0*f[1] - f[2])*scale;
+}
+if(mynode == (totalnodes-1)){
+ fx[n-1] = (3.0*f[n-1] - 4.0*f[n-2] + f[n-3])*scale;
+}
+for(int i=1;i<n-1;i++){
+ fx[i] = (f[i+1]-f[i-1])*scale;
+}
+if(mynode == 0){
+   mpitemp = f[n-1];
+   MPI_Send(&mpitemp,1,MPI_DOUBLE,1,1,MPI_COMM_WORLD);
+   MPI_Recv(&mpitemp,1,MPI_DOUBLE,1,1,MPI_COMM_WORLD,&status);
+   fx[n-1] = (mpitemp - f[n-2])*scale;
+}
+else if(mynode == (totalnodes-1)){
+  MPI_Recv(&mpitemp,1,MPI_DOUBLE,mynode-1,1,MPI_COMM_WORLD, &status);
+  fx[0] = (f[1]-mpitemp)*scale;
+  mpitemp = f[0];
+  MPI_Send(&mpitemp,1,MPI_DOUBLE,mynode-1,1,MPI_COMM_WORLD);
+}
+else{
+ MPI_Recv(&mpitemp,1,MPI_DOUBLE,mynode-1,1,MPI_COMM_WORLD, &status);
+ fx[0] = (f[1]-mpitemp)*scale;
+ mpitemp = f[0];
+ MPI_Send(&mpitemp,1,MPI_DOUBLE,mynode-1,1,MPI_COMM_WORLD);
+ mpitemp = f[n-1];
+ MPI_Send(&mpitemp,1,MPI_DOUBLE,mynode+1,1,MPI_COMM_WORLD);
+ MPI_Recv(&mpitemp,1,MPI_DOUBLE,mynode+1,1,MPI_COMM_WORLD, &status);
+ fx[n-1] = (mpitemp-f[n-2])*scale;
+}
+return;
+}
+
+double fun(double x){
+  return(sin(x));
+}
+
+double funprime( double x){
+  return(cos(x));
+}
diff --git a/MPI-Codes/Collectives/midpointintegral.c b/MPI-Codes/Collectives/midpointintegral.c
new file mode 100755
index 0000000..0ec3aa9
--- /dev/null
+++ b/MPI-Codes/Collectives/midpointintegral.c
@@ -0,0 +1,45 @@
+#include <stdio.h>
+#include <stdlib.h>
+#include <math.h>
+#include <mpi.h>
+
+double f( double x);
+
+int main( int argc, char *argv[]){
+int n, mynode, totalnodes, i;
+double PI25DT = 3.141592653589793238462643;
+double mypi, pi, h, sum, x;
+
+MPI_Init(&argc,&argv);
+MPI_Comm_size(MPI_COMM_WORLD,&totalnodes);
+MPI_Comm_rank(MPI_COMM_WORLD,&mynode);
+
+if(mynode == 0){
+  if(argc != 2){
+   n = 20;
+  }else{
+   n = atoi(argv[1]);
+  }
+}
+// Broadcast the value to every process
+
+h   = 1.0 / ((double) n*totalnodes);
+sum = 0.0;
+for (i = 1+mynode*n; i <= n*(mynode+1); i++){
+   // compute the local quadrature rule
+}
+mypi = h * sum;
+
+// sum all the local sums via reduction
+
+if (mynode == 0){
+ printf("pi is approximately %.16f, Error is %e\n",pi, fabs(pi - PI25DT));
+}
+MPI_Finalize();
+return 0;
+}
+
+double f( double x )
+{
+    return (4.0 / (1.0 + x*x));
+}
diff --git a/MPI-Codes/Collectives/montecarlostyle.c b/MPI-Codes/Collectives/montecarlostyle.c
new file mode 100755
index 0000000..7975c20
--- /dev/null
+++ b/MPI-Codes/Collectives/montecarlostyle.c
@@ -0,0 +1,58 @@
+#include <stdio.h>
+#include <stdlib.h>
+#include <math.h>
+#include <mpi.h>
+#include <time.h>
+
+int main( int argc, char *argv[]){
+ int n, mynode, totalnodes, i;
+ double PI25DT = 3.141592653589793238462643;
+ double mypi, pi, h;
+ double x, y, x1, x2, y1, y2;
+ int SqPoints, CiPoints, my_SqPoints, my_CiPoints;
+ unsigned int seed = (unsigned) time(NULL);
+
+ MPI_Init(&argc,&argv);
+ MPI_Comm_size(MPI_COMM_WORLD,&totalnodes);
+ MPI_Comm_rank(MPI_COMM_WORLD,&mynode);
+
+ if(mynode == 0){
+   if(argc != 2){
+    n = 20;
+   }else{
+    n = atoi(argv[1]);
+   }
+ }
+ // Broadcast the number of intervals everywhere
+
+ h  = 2.0 / (double) totalnodes;
+ x1 = -1.0 + mynode * h;
+ x2 = x1 + h;
+ y1 = -1.0;
+ y2 = 1.0;
+ my_SqPoints  = 0;
+ my_CiPoints  = 0;
+
+ for (i = 1; i <= n; i++)
+ {
+    srand(seed + mynode);
+    x = rand(&seed); x = x / RAND_MAX; x = x1 + x * (x2 - x1);
+    y = rand(&seed); y = y / RAND_MAX; y = y1 + y * (y2 - y1);
+    // do the check on the local Montecarlo Integration : my_SqPoints, my_CiPoints should store
+    // the number of points into the square, and into the circle (respectively)
+}
+SqPoints = 0;
+CiPoints = 0;
+
+// Perform reduce operation to put into SqPoints and CiPoints the values
+
+if (mynode == 0)
+{
+   pi = 4.0 * (double)CiPoints / (double)SqPoints;
+   printf("Pi is approximately %.16f = 4.0 %d/%d, Error is %e (h = %e)\n",pi,CiPoints,SqPoints,fabs(pi - PI25DT),1.0/sqrt( (double) SqPoints) );
+}
+
+MPI_Finalize();
+
+return 0;
+}
diff --git a/MPI-Codes/Collectives/parallelderiv.c b/MPI-Codes/Collectives/parallelderiv.c
new file mode 100755
index 0000000..85c76ba
--- /dev/null
+++ b/MPI-Codes/Collectives/parallelderiv.c
@@ -0,0 +1,98 @@
+#include <stdio.h>
+#include <stdlib.h>
+#include <math.h>
+#include "mpi.h"
+
+void firstderiv1Dp_vec(int n, double dx, double *f,double *fx, int mynode, int totalnodes);
+
+double fun(double x);
+double funprime( double x);
+
+int main(int argc, char **argv){
+  double globala,globalb,a,b,dx,*f,*fx, error;
+  int n, mynode, totalnodes;
+  MPI_Status status;
+
+  MPI_Init( &argc, &argv );
+  MPI_Comm_rank( MPI_COMM_WORLD, &mynode );
+  MPI_Comm_size( MPI_COMM_WORLD, &totalnodes );
+
+  n = 20;
+
+  globala = 0;
+  globalb = 1;
+
+  a = globala + ((double) mynode)*(globalb - globala)/( (double) totalnodes);
+  b = globala + ((double) mynode+1)*(globalb - globala)/( (double) totalnodes);
+
+  f  = (double *) malloc(sizeof(double)*(n));
+  fx = (double *) malloc(sizeof(double)*(n));
+
+  dx = (b-a)/((double) n);
+  for( int i = 0; i < n; i++){
+    f[i] = fun(a+((double) i)*dx);
+  }
+  firstderiv1Dp_vec( n, dx, f, fx, mynode, totalnodes);
+
+  error = 0.0;
+  for(int i = 0; i < n; i++){
+    error += pow( fx[i]-funprime(a+((b-a)*((double) i))/((double) n)),2.0);
+  }
+  error = sqrt(dx*error);
+  printf("Node %d ||f' - fx||_2 = %e\n",mynode,error);
+
+
+  free(f);
+  free(fx);
+  MPI_Finalize();
+
+  return 0;
+}
+
+
+void firstderiv1Dp_vec(int n, double dx, double *f,double *fx, int mynode, int totalnodes){
+double scale = 1.0/(2.0*dx);
+double mpitemp;
+MPI_Status status;
+
+if(mynode == 0){
+ fx[0] = (-3.0*f[0] + 4.0*f[1] - f[2])*scale;
+}
+if(mynode == (totalnodes-1)){
+ fx[n-1] = (3.0*f[n-1] - 4.0*f[n-2] + f[n-3])*scale;
+}
+for(int i=1;i<n-1;i++){
+ fx[i] = (f[i+1]-f[i-1])*scale;
+}
+if(mynode == 0){
+   mpitemp = f[n-1];
+   MPI_Send();
+   MPI_Recv();
+   fx[n-1] = (mpitemp - f[n-2])*scale;
+}
+else if(mynode == (totalnodes-1)){
+  MPI_Recv();
+  fx[0] = (f[1]-mpitemp)*scale;
+  mpitemp = f[0];
+  MPI_Send();
+}
+else{
+ MPI_Recv();
+ fx[0] = (f[1]-mpitemp)*scale;
+ mpitemp = f[0];
+ MPI_Send();
+ mpitemp = f[n-1];
+ MPI_Send();
+ MPI_Recv();
+ fx[n-1] = (mpitemp-f[n-2])*scale;
+}
+return;
+}
+
+double fun(double x){
+  return(sin(x));
+}
+
+double funprime( double x){
+  return(cos(x));
+}
diff --git a/MPI-Codes/Collectives/sequentialderiv.c b/MPI-Codes/Collectives/sequentialderiv.c
new file mode 100755
index 0000000..39d43bc
--- /dev/null
+++ b/MPI-Codes/Collectives/sequentialderiv.c
@@ -0,0 +1,65 @@
+#include <stdio.h>
+#include <stdlib.h>
+#include <math.h>
+
+void firstderiv1D(int n, double a, double b, double *fx);
+void firstderiv1D_vec(int n, double dx, double *f, double *fx);
+double fun(double x);
+double funprime( double x);
+
+int main(int argc, char **argv){
+  double a,b,dx,*f,*fx;
+  int n;
+  if(argc != 2){
+    n = 20;
+  }else{
+    n = atoi(argv[1]);
+  }
+  a = 0.0;
+  b = 1.0;
+  f  = (double *) malloc(sizeof(double)*n);
+  fx = (double *) malloc(sizeof(double)*n);
+  // firstderiv1D(n, a, b, fx);
+  dx = (b-a)/((double) n);
+  for( int i = 0; i <= n; i++){
+    f[i] = fun(a+((double) i)*dx);
+  }
+  firstderiv1D_vec(n, dx, f, fx);
+  for(int i = 0; i < n; i++){
+    printf("fx[%d] = %1.2f f'[%d] = %1.2f |fx - f'| = %1.1e\n",
+      i,fx[i],i,funprime(a+((b-a)*i)/n),abs(fx[i]-
+        funprime(a+((b-a)*((double) i))/((double) (n-1)))));
+  }
+
+  return 1;
+}
+
+void firstderiv1D(int n, double a, double b, double *fx){
+  double dx = (b-a)/((double) n-1);
+  double scale = 1.0/(2.0*dx);
+  for (int i = 1; i < n-1; i++){
+    fx[i] = (fun(a + (i+1)*dx) - fun(a + (i-1)*dx))*scale;
+  }
+  fx[0] = (-3.0*fun(a) + 4.0*fun(a + dx) - fun(a + 2.0*dx))*scale;
+  fx[n-1] = (3.0*fun(a+ (n-1)*dx) - 4.0*fun(a + (n-2)*dx) + fun(a + (n-3)*dx))*scale;
+  return;
+}
+
+void firstderiv1D_vec(int n, double dx, double *f, double *fx){
+  double scale;
+  scale = 1.0/(2.0*dx);
+  for (int i = 1; i < n-1; i++){
+    fx[i] = (f[i+1] - f[i-1])*scale;
+  }
+  fx[0] = (-3.0*f[0] + 4.0*f[1] - f[2])*scale;
+  fx[n-1] = (3.0*f[n-1] - 4.0*f[n-2] + f[n-3])*scale;
+  return;
+}
+
+double fun(double x){
+  return(x*x + 1);
+}
+
+double funprime( double x){
+  return(2*x);
+}
diff --git a/README.md b/README.md
index 979a9c2..c683f44 100644
--- a/README.md
+++ b/README.md
@@ -6,7 +6,9 @@ Progetto speciale per la didattica dell'Università di Pisa, a.a. 2022/23.
 - Giovedì 30 Marzo 2023, ore 16.00: Introduzione e uso di GIT
 - Giovedì 20 Aprile 2023, ore 11.00: Introduzione ad MPI
 - Venerdì 28 Aprile 2023, ore 11.00: Assemblaggio e prime configurazioni
-- Giovedì 4 Maggio 2023, ore 16.00: Configurazione cluster (Slurm&GlusterFS: prime run)
+- Giovedì 4 Maggio 2023, ore 16.00: Comunicazioni collettive
+- Giovedì 18 Maggio 2023, ore 16.00: Test con il cluster Steffé e uso di Slurm
+- Giovedì 25 Maggio 2023, ore 16.00: *Guest lecture*: Prof. S. Filippone
 
 Il materiale si trova nella cartella `Slides` del `repository`.
 
diff --git a/Slides/Incontro-4.pdf b/Slides/Incontro-4.pdf
new file mode 100644
index 0000000..f769e29
Binary files /dev/null and b/Slides/Incontro-4.pdf differ