Error when passing 2D array to function in c - arrays

I am writing a program to calculate matrix multiplication but it does not work. When I debug and check each value of the array a and b in function printMatrixMultiplication (which are entered by user), GDB prints out "cannot perform pointer math on incomplete type try casting". (I have searched for it but I still don't get it.) The function only works when the input is predefined in main.
This is my code
#include <stdio.h>
void input(int m, int n, double a[m][n]);
void output(int m, int n, double a[m][n]);
void printMatrixMultiplication(int row_a, int col_a, double a[row_a][col_a], int row_b, int col_b, double b[row_b][col_b]);
int main()
{
int row_a, col_a, row_b, col_b;
// get value of matrix a
printf("row_a = ");
scanf("%d", &row_a);
printf("col_a = ");
scanf("%d", &col_a);
double a[row_a][col_a];
input(row_a, col_a, a);
// output(row_a, col_a, a);
// get value of matrix b
printf("row_b = ");
scanf("%d", &row_b);
printf("col_b = ");
scanf("%d", &col_b);
double b[row_b][col_b];
input(row_b, col_b, a);
// output(row_b, col_b, a);
printMatrixMultiplication(row_a, col_a, a, row_b, col_b, b);
//test
// double a[2][2]={1,2,3,4};
// double b[2][3]={1,2,3,4,5,6};
// printMatrixMultiplication(2,2,a,2,3,b);
return 0;
}
void input(int m, int n, double a[m][n])
{
int i, j;
for (i = 0; i < m; i++)
{
for (j = 0; j < n; j++)
{
scanf("%lf", &a[i][j]);
}
}
}
void output(int m, int n, double a[m][n])
{
int i, j;
for (i = 0; i < m; i++)
{
for (j = 0; j < n; j++)
{
printf("%.2f ", a[i][j]);
}
printf("\n");
}
}
void printMatrixMultiplication(int row_a, int col_a, double a[row_a][col_a], int row_b, int col_b, double b[row_b][col_b])
{
if (col_a != row_b)
{
return;
}
double res[row_a][col_b]; //this matrix store results
for (int i = 0; i < row_a; i++) //the values be stored line by line, this
{ //operation is controled by i and j loops.
for (int j = 0; j < col_b; j++) //the k loop helps calculate dot_product.
{
double dot_product = 0;
for (int k = 0; k < col_a; k++)
{
dot_product += a[i][k] * b[k][j]; //ERROR HERE
}
res[i][j] = dot_product;
}
}
output(row_a, col_b, res);
}
So, where does the error come from and how to fix it?
Irrelevant, but the function is not well implemented so if possible, I would really appreciate if anyone gives me a hint to improve it.
I am using GCC version 6.3.0.

It's typo in your code when reading matrix b.
Just replace:
input(row_b, col_b, a);
with
input(row_b, col_b, b);

Related

Segmentation Fault During Matrix Multiplication

I am new to C coding, and am trying to implement standard matrix multiplication. My code works fine for square matrices, but refuses to accept a column vector. Here is my attempt at the code. Any help would be much appreciated.
//---------------------------------------IMPORTING NECESSARY C PACKAGES AND DEFINING EXECUTION CONSTANTS-------------------------------------------//
#include <stdio.h> // Standard input output library
#include <math.h> // Mathematical function library
#include <stdlib.h> // General purpose standard library
#define true 1
#define false 0
typedef long double numeric; // Using the long double datatype to avoid overflows during computations
//-------------------------------------------------------------------------------------------------------------------------------------------------//
//----------------------------------------------------------------FUNCTION DECLERATION-------------------------------------------------------------//
numeric **create_matrix(int x, int y); // To dynamically allocate memory and create a matrix
void input_matrix(numeric **matrix, int m, int n); // To accept a matrix
void print_matrix(numeric **l, int x, int y); // To print a matrix
numeric **standard_matrix_multiplication(int m, int n, int l); // To multiply two matrices
//-------------------------------------------------------------------------------------------------------------------------------------------------//
//------------------------------------------------------------------DRIVER CODE--------------------------------------------------------------------//
int main(int argc, char *argv[]) {
int m, n, l; int choice;
printf("Enter the matrix operation to be performed using the corresponding index number.\n");
printf("\n");
printf("1.\tMatrix Multiplication");
printf("\n");
scanf("%d", &choice);
switch(choice) {
case 1 :
printf("Enter the number of rows in the first matrix\n");
scanf("%d", &m);
printf("Enter the number of columns in the first matrix\n");
scanf("%d", &n);
printf("Enter the number of columns in the second matrix\n");
scanf("%d", &l);
printf("Enter both matrices.\n");
numeric **matrix_x;
matrix_x = create_matrix(m, l);
matrix_x = standard_matrix_multiplication(m, n, l);
print_matrix(matrix_x, m, l);
break;
}
}
//-------------------------------------------------------------------------------------------------------------------------------------------------//
//----------------------------------------------------------MATRIX MULTIPLICATION IMPLEMENTATIONS--------------------------------------------------//
numeric **standard_matrix_multiplication(int m, int n, int l) {
numeric **matrix_a; numeric **matrix_b; numeric **matrix_k;
matrix_a = create_matrix(m, n);
matrix_b = create_matrix(n, l);
matrix_k = create_matrix(m, l);
input_matrix(matrix_a, m, n);
print_matrix(matrix_a, m, n);
input_matrix(matrix_b, n, l);
for(int i = 0; i < m; i++) {
for (int j = 0; j < n; j ++) {
for (int k = 0; k < l; k++) {
matrix_k[i][j] += matrix_a[i][k] * matrix_b[k][j];
}
}
}
return matrix_k;
}
//-------------------------------------------------------------------------------------------------------------------------------------------------//
//---------------------------------------------------------------HELPER FUNCTIONS------------------------------------------------------------------//
numeric **create_matrix(int x, int y) {
numeric **matrix = (numeric**)malloc(x * sizeof(numeric*)); // Dynamically creating an array of pointers
for (int i = 0; i < y; i++) {
matrix[i] = (numeric*)malloc(y * sizeof(numeric)); // Dynamically allocating memory for each columns of the matrix
}
return matrix;
}
void input_matrix(numeric **matrix, int m, int n) {
printf("Enter the elements of the matrix, row wise.\n"); // Instructing the user on matrix entry
printf("For example, to enter the matrix\n");
printf("\t\t1\t2\n");
printf("\t\t3\t4\n");
printf("enter 1, 2, 3, and 4 in that order.\n");
for (int i = 0; i < m; i++) { // Iterating through the rows and columns of the matrix
for (int j = 0; j < n; j++) {
scanf("%Lf", &matrix[i][j]); // Accepting each element
}
}
}
void print_matrix(numeric **l, int x, int y) { // To print a matrix
for (int i = 0; i < x; i++) {
for (int j = 0; j < y; j++) {
printf("%0.10Lf\t", l[i][j]); // Printing numeric type values
}
printf("\n");
}
printf("\n");
}
As of now, I have only written one switch case, and that is for matrix multiplication. So I chose 1. I gave 2, 1, 2 as my inputs for the number of rows in the first matrix, number of columns in the first matrix, and number of columns in the second matrix respectively. I have given a print statement in line 52, and it isn't executing it for the above input, giving a segmentation fault instead. Could someone please help me out?
Yes there are some issues with your code that gives segfault error during runtime.
The Matrix multiplication logic part of the code needs to be corrected as follows
for(int i = 0; i < m; i++) {
for (int j = 0; j < l; j ++) {
for (int k = 0; k < n; k++) {
matrix_k[i][j] += matrix_a[i][k] * matrix_b[k][j];
because k should iterate till the no of columns in 1st matrix which is n but not till l.
And there is a slight correction needed in create_matrix function.
You use y (i.e. no of columns in the matrix) in your for instead of x (no of rows in matrix).
If x < y, you end up outside of the memory you allocated.
If x > y, you end up with uninitialized pointers.
So change it as follows
for (int i = 0; i < x; i++) {
matrix[i] = (numeric*)malloc(y * sizeof(numeric));
After these corrections try executing the code, you should get the expected results without any segfault errors
Here is the complete working code
#include <stdio.h> // Standard input output library
#include <math.h> // Mathematical function library
#include <stdlib.h> // General purpose standard library
#define true 1
#define false 0
typedef long double numeric; // Using the long double datatype to avoid overflows during computations
//-------------------------------------------------------------------------------------------------------------------------------------------------//
//----------------------------------------------------------------FUNCTION DECLERATION-------------------------------------------------------------//
numeric **create_matrix(int x, int y); // To dynamically allocate memory and create a matrix
void input_matrix(numeric **matrix, int m, int n); // To accept a matrix
void print_matrix(numeric **l, int x, int y); // To print a matrix
numeric **standard_matrix_multiplication(int m, int n, int l); // To multiply two matrices
//-------------------------------------------------------------------------------------------------------------------------------------------------//
//------------------------------------------------------------------DRIVER CODE--------------------------------------------------------------------//
int main(int argc, char *argv[]) {
int m, n, l; int choice;
printf("Enter the matrix operation to be performed using the corresponding index number.\n");
printf("\n");
printf("1.\tMatrix Multiplication");
printf("\n");
scanf("%d", &choice);
switch(choice) {
case 1 :
printf("Enter the number of rows in the first matrix\n");
scanf("%d", &m);
printf("Enter the number of columns in the first matrix\n");
scanf("%d", &n);
printf("Enter the number of columns in the second matrix\n");
scanf("%d", &l);
printf("Enter both matrices.\n");
numeric **matrix_x;
matrix_x = create_matrix(m, l);
matrix_x = standard_matrix_multiplication(m, n, l);
print_matrix(matrix_x, m, l);
break;
}
}
//-------------------------------------------------------------------------------------------------------------------------------------------------//
//----------------------------------------------------------MATRIX MULTIPLICATION IMPLEMENTATIONS--------------------------------------------------//
numeric **standard_matrix_multiplication(int m, int n, int l) {
numeric **matrix_a; numeric **matrix_b; numeric **matrix_k;
matrix_a = create_matrix(m, n);
matrix_b = create_matrix(n, l);
matrix_k = create_matrix(m, l);
input_matrix(matrix_a, m, n);
print_matrix(matrix_a, m, n);
input_matrix(matrix_b, n, l);
//print_matrix(matrix_b, n, l);
for(int i = 0; i < m; i++) {
for (int j = 0; j < l; j ++) {
for (int k = 0; k < n; k++) {
matrix_k[i][j] += matrix_a[i][k] * matrix_b[k][j];
}
}
}
return matrix_k;
}
//-------------------------------------------------------------------------------------------------------------------------------------------------//
//---------------------------------------------------------------HELPER FUNCTIONS------------------------------------------------------------------//
numeric **create_matrix(int x, int y) {
numeric **matrix = (numeric**)malloc(x * sizeof(numeric*)); // Dynamically creating an array of pointers
for (int i = 0; i < x; i++) {
matrix[i] = (numeric*)malloc(y * sizeof(numeric)); // Dynamically allocating memory for each columns of the matrix
}
return matrix;
}
void input_matrix(numeric **matrix, int m, int n) {
printf("Enter the elements of the matrix, row wise.\n"); // Instructing the user on matrix entry
printf("For example, to enter the matrix\n");
printf("\t\t1\t2\n");
printf("\t\t3\t4\n");
printf("enter 1, 2, 3, and 4 in that order.\n");
for (int i = 0; i < m; i++) { // Iterating through the rows and columns of the matrix
for (int j = 0; j < n; j++) {
scanf("%Lf", &matrix[i][j]); // Accepting each element
}
}
}
void print_matrix(numeric **l, int x, int y) { // To print a matrix
for (int i = 0; i < x; i++) {
for (int j = 0; j < y; j++) {
printf("%0.10Lf\t", l[i][j]); // Printing numeric type values
}
printf("\n");
}
printf("\n");
}
You are not allocating enough memory for matrix_k. You are allocating memory for m rows and l columns, but you are trying to access m rows and n columns. You need to allocate memory for m rows and n columns. You can do this by changing the line
matrix_k = create_matrix(m, l);
to
matrix_k = create_matrix(m, n);
This is the same problem you had in your other question.

Function inputElements() taking input only once

I am trying to create a program to find the transpose of a matrix my dynamic memory allocation. However, while entering the elements of the matrix I can't input more than one element, its only taking the a[0][0] as the input and the program is ending after that.
#include <stdio.h>
#include <stdlib.h>
void createMatrix(int **, int, int);
void inputElements(int **, int, int);
void transpose(int **, int **, int, int);
void display(int **, int, int);
void main()
{
int **matrix, **trans, rows, cols;
printf("\nEnter number of rows in the matrix: ");
scanf("%d", &rows);
printf("\nEnter number of columns in the matrix: ");
scanf("%d", &cols);
createMatrix(matrix, rows, cols);
createMatrix(trans, cols, rows);
inputElements(matrix, rows, cols);
transpose(matrix, trans, rows, cols);
printf("\nMATRIX:\n");
display(matrix, rows, cols);
printf("\nTRANSPOSE OF THE MATRIX:\n");
display(trans, rows, cols);
}
void createMatrix(int **a, int r, int c) //for allocating memory for the matrix
{
int i, j;
a = (int **)malloc(sizeof(int *) * r);
for(i = 0; i < r; i++)
a[i] = (int *)malloc(sizeof(int) * c);
}
void inputElements(int **a, int r, int c) //for entering matrix elements
{
int i, j, t;
for(i = 0; i < r; i++)
{
for(j = 0; j < c; j++)
{
printf("\nEnter matrix element[%d][%d]: ", i + 1, j + 1);
fflush(stdin);
getchar();
scanf("%d", &(a[i][j]));
}
}
}
void transpose(int **a, int **t, int r, int c) //for finding out the transpose of the matrix
{
int i, j;
for (i = 0; i < c; i++)
{
for (j = 0; j < r; j++)
t[i][j] = a[j][i];
}
}
void display(int **a, int r, int c) //for displaying the matrix
{
int i, j;
for (i = 0; i < r; i++)
{
printf("\n");
for (j = 0; j < c; j++)
printf("\t%d", a[i][j]);
}
}
There are many posts about scanf in loops I've seen, but I wasn't able to connect my issue with any of them.

i have problem with acessing 2d array via function

i am learning pointers and i write these code to add two martix via function, it compile well and give correct answer for 2x2 matrix but for more than two row and col it fails when i try it with 3x3 matrix [0][0] element for the first matrix it automaticaly become 6 and [1][0] it became 9 every time, i can't figure out why this happen
#include <stdio.h>
//-----------------Function------------------
void add(int (*p1)[10], int (*p2)[10], int (*p3)[10], int r, int c);
void get(int (*p1)[10], int r, int c);
void print(int (*p1)[10], int r, int c);
//---------------Main Program----------------
int main()
{
//declartion
int i, j, r, c;
printf("\n\tenter the row and column of matrix\n\t");
scanf("%d %d", &r, &c);
int m1[r][c], m2[r][c], m3[r][c];
printf("\n\tenter the element of first matrix\n\t");
get(m1, r, c);
print(m1, r, c);
printf("\n\tenter the element of second matrix\n\t");
get(m2, r, c);
print(m2, r, c);
add(m1, m2, m3, r, c);
printf("\n");
print(m3, r, c);
return(0);
}
//-------------Define Function---------------
//get()
void get(int (*p1)[10], int r, int c)
{
int i, j;
for(i = 0; i < r; i++)
{
printf("\n\t");
for(j = 0; j < c; j++)
{
scanf("%d", (*(p1+i)+j));
}
}
}
//--------------------X----------------------
//add()
void add(int (*p1)[10], int (*p2)[10], int (*p3)[10], int r, int c)
{
int i, j;
for(i = 0; i < r; i++)
{
for(j = 0; j < c; j++)
{
//printf("\n%d %d = %d & %d", i, j, *(*(p1+i)+j), *(*(p2+i)+j));
*(*(p3+i)+j) = *(*(p1+i)+j) + *(*(p2+i)+j);
}
}
}
//--------------------X----------------------
//print()
void print(int (*p1)[10], int r, int c)
{
int i, j;
for(i = 0; i < r; i++)
{
printf("\n");
for (j = 0; j < c; j++)
{
printf("\t%d", *(*(p1+i)+j));
}
}
}
//--------------------X----------------------
Drop the array pointer notation and the fixed size. Use variable-length arrays based on the parameters.
Remove useless comments like // get ... void get...
Don't use unreadable de-referncing with *(arr+i) when you could be writing readable arr[i].
Print new line after each row, not before each row.
Here's a cleaned up program which compiles. I haven't tested it.
#include <stdio.h>
void add (int r, int c, int p1[r][c], int p2[r][c], int p3[r][c]);
void get (int r, int c, int p1[r][c]);
void print (int r, int c, int p1[r][c]);
int main (void)
{
int i, j, r, c;
printf("\n\tenter the row and column of matrix\n\t");
scanf("%d %d", &r, &c);
int m1[r][c], m2[r][c], m3[r][c];
printf("\n\tenter the element of first matrix\n\t");
get(r, c, m1);
print(r, c, m1);
printf("\n\tenter the element of second matrix\n\t");
get(r, c, m2);
print(r, c, m2);
add(r, c, m1, m2, m3);
printf("\n");
print(r, c, m3);
return(0);
}
void get (int r, int c, int p1[r][c])
{
for(int i = 0; i < r; i++)
{
for(int j = 0; j < c; j++)
{
scanf("%d", &p1[i][j]);
}
}
}
void add (int r, int c, int p1[r][c], int p2[r][c], int p3[r][c])
{
for(int i = 0; i < r; i++)
{
for(int j = 0; j < c; j++)
{
p3[i][j] = p1[i][j] + p2[i][j];
}
}
}
void print (int r, int c, int p1[r][c])
{
for(int i = 0; i < r; i++)
{
for (int j = 0; j < c; j++)
{
printf("\t%d", p1[i][j]);
}
printf("\n");
}
}

Using scanf() with a pointer to a double pointer

I feel like I've attempted every combination I know of to get this to work and can't figure it out. How can I scanf() into an int** passed as a pointer to a function? I tried searching but couldn't find this, if it's a duplicate please let me know and I'll delete. It begins to run and after entering a few values it segfaults.
Here's my code, I think it's messing up on the scanf() line of the setMatrix() function:
#include <stdio.h>
#include <stdlib.h>
// create zero initialized matrix
int** callocMatrix(int rmax, int colmax) {
int **mat = calloc(rmax, sizeof(int*));
for(int i = 0; i < rmax; i++) mat[i] = calloc(colmax, sizeof(int));
return mat;
}
// fill matrix
void setMatrix(int ***mat, int r, int c){
printf("Insert the elements of your matrix:\n");
for (int i = 0; i < r; i++) {
for (int j = 0; j < c; j++) {
printf("Insert element [%d][%d]: ", i, j);
scanf("%d", mat[i][j]); // problem here??
printf("matrix[%d][%d]: %d\n", i, j, (*mat)[i][j]);
}
}
return;
}
// print matrix
void printMatrix(int ***mat, int r, int c){
for (int i=0; i<r;i++){
for (int j=0; j<c;j++) {
printf("%d ", (*mat)[i][j]);
}
printf("\n");
}
}
int main(int argc, char *argv[]) {
int r = 3, c = 3;
int **mat = callocMatrix(r, c);
setMatrix(&mat, r, c);
printMatrix(&mat, r, c);
}
There is no need to use triple pointer ***. Passing two-dimensional array will work as is. Here is the code:
#include <stdio.h>
#include <stdlib.h>
// create zero initialized matrix
int** callocMatrix(int rmax, int colmax) {
int **mat = calloc(rmax, sizeof(int*));
for(int i = 0; i < rmax; i++) mat[i] = calloc(colmax, sizeof(int));
return mat;
}
// fill matrix
void setMatrix(int **mat, int r, int c){
printf("Insert the elements of your matrix:\n");
for (int i = 0; i < r; i++) {
for (int j = 0; j < c; j++) {
printf("Insert element [%d][%d]: ", i, j);
scanf("%d", &mat[i][j]); // no problem here
printf("matrix[%d][%d]: %d\n", i, j, mat[i][j]);
}
}
}
// print matrix
void printMatrix(int **mat, int r, int c){
for (int i=0; i<r;i++){
for (int j=0; j<c;j++) {
printf("%d ", mat[i][j]);
}
printf("\n");
}
}
int main(int argc, char *argv[]) {
int r = 3, c = 3;
int **mat = callocMatrix(r, c);
setMatrix(mat, r, c);
printMatrix(mat, r, c);
}
Should be:
scanf("%d", &(*mat)[i][j]);
You're passing a pointer to you matrix object, so you need to dereference it (with *) just as you do with printf. scanf then needs the address of the element to write into, so you need the &

matrix multipication in c

i know there are similiar questions but i tried the suggestions there and it still does not work, here my complete code;
#include <stdio.h>
#include <math.h>
void ludec(int N, double lu[N][N]){
double temp;
int k,i,j;
for(k=0;k<N;k++){
for(i=k+1;i<N;i++){
temp=lu[i][k]/=lu[k][k];
for(j=k+1;j<N;j++){
lu[i][j]-=temp*lu[k][j];
}
}
}
}
void matmul(int u,int m,int n,double p[u][m],double bb[m][n]){
double mul[u][n];
int ii,jj,kk;
for(ii=0;ii<u;ii++){
for(jj=0;jj<n;jj++){
mul[ii][jj]=0.0;
}
}
for(ii=0;ii<u;ii++){
for(jj=0;jj<n;jj++){
for(kk=0;kk<m;kk++){
mul[ii][jj]=mul[ii][jj]+p[ii][kk]*bb[kk][jj];
printf("la %f ii %d jj%d kk%d\n",mul[ii][jj],ii,jj,kk);
}
}
}
}
int main(void){
double lu[3][3]={{4.0,3.0,-2.0},{-1.0,-1.0,3.0},{2.0,-1.0,5.0}};
double a[3][3]={{4.0,3.0,-2.0},{-1.0,-1.0,3.0},{2.0,-1.0,5.0}};
double y[3][1];
double b[3][1]={{9.0},{-4.0},{6.0}};
double x[3][1];
int i,j,N=3;
double f=0;
ludec(N, lu);
y[0][0]=b[0][0];
for(i=1;i<3;i++){
for(j=0;j<i;j++){
f+=lu[i][j]*y[j][0];
}
y[i][0]=(b[i][0]-f);
f=0;
}
f=0;
x[2][0]=y[2][0]/lu[2][2];
for(i=1;i>=0;i--){
for(j=1+i;j<3;j++){
f+=lu[i][j]*x[j][0];
}
x[i][0]=(y[i][0]-f)/lu[i][i];
f=0;
}
printf("%f\n %f\n %f\n",x[0][0],x[1][0],x[2][0]);
for(i=0;i<3;i++){
for(j=0;j<3;j++){
printf("%f %d %d\n",lu[i][j],i,j);
}
}
double mul[3][1];
matmul(3,3,1,a,x);
for(i=0;i<3;i++){
for(j=0;j<2;j++){
printf("%f %d %d\n",mul[i][j],i,j);
}
}
return 0.0;
}
here is the part responsible by multiplication;
the problem is
i have to get 3*1 matrix as result in matrix multiplication, but it gives bigger matrix with alot of garbage elements i monitor the matrix multiplication by printf in the matmul function and it seems to work but when i do same out side the matmul function it gives garbage only some values are correct with a lot of useless wrong elements.
void matmul(int u,int m,int n,double p[u][m],double bb[m][n]){
double mul[u][n];
int ii,jj,kk;
for(ii=0;ii<u;ii++){
for(jj=0;jj<n;jj++){
mul[ii][jj]=0.0;
}
}
for(ii=0;ii<u;ii++){
for(jj=0;jj<n;jj++){
for(kk=0;kk<m;kk++){
mul[ii][jj]=mul[ii][jj]+p[ii][kk]*bb[kk][jj];
printf("la %f ii %d jj%d kk%d\n",mul[ii][jj],ii,jj,kk);
}
}
}
}
the problem is matmul function and i dont know why it does not work i already initialized the matrix with zeros. am i using void function property wrong?
i am very new to c hence there may be something i am doing wrong. the overall code is for lu decompositation, and i want to see wheter the code works or not by multiplying result with initial matrix.
this is the out put
mul is a local variable in matmul, thus the result is not visible to the calling function. You should modify the matmul function to take the result matrix as an argument:
void matmul(int u, int m, int n,
double mul[u][n], const double p[u][m], const double bb[m][n])
{
for (int i = 0; i < u; i++) {
for (int j = 0; j < n; j++) {
mul[i][j] = 0.0;
for (int k = 0; k < m; k++) {
mul[i][j] += p[i][k] * bb[k][j];
}
}
}
}
Invoke from main as
double mul[3][1];
matmul(3, 3, 1, mul, a, x);
And print the output vector:
for (i = 0; i < 3; i++) {
for (j = 0; j < 1; j++) {
printf("%f %d %d\n", mul[i][j], i, j);
}
}

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