C Program to Perform LU Decomposition of Any Matrix

This is a C Program to perform LU decomposition of a given matrix. Every square matrix A can be decomposed into a product of a lower triangular matrix L and a upper triangular matrix U, as described in LU decomposition.
A = LU

Here is source code of the C Program to Perform LU Decomposition of any Matrix. The C program is successfully compiled and run on a Linux system. The program output is also shown below.

```
#include <stdio.h>
```
```
#include <stdlib.h>
```
```
#include <math.h>
```
```
 
```

#define foreach(a, b, c) for (int a = b; a < c; a++)

```
#define for_i foreach(i, 0, n)
```
```
#define for_j foreach(j, 0, n)
```
```
#define for_k foreach(k, 0, n)
```
```
#define for_ij for_i for_j
```
```
#define for_ijk for_ij for_k
```
```
#define _dim int n
```

#define _swap(x, y) { typeof(x) tmp = x; x = y; y = tmp; }

#define _sum_k(a, b, c, s) { s = 0; foreach(k, a, b) s+= c; }

```
 
```
```
typedef double **mat;
```
```
 
```
```
#define _zero(a) mat_zero(a, n)
```
```
void mat_zero(mat x, int n) {
```
```
    for_ij
```
```
            x[i][j] = 0;
```
```
}
```
```
 
```
```
#define _new(a) a = mat_new(n)
```
```
mat mat_new(_dim) {
```

    mat x = malloc(sizeof(double*) * n);

    x[0] = malloc(sizeof(double) * n * n);

```
 
```
```
    for_i
```
```
        x[i] = x[0] + n * i;
```
```
    _zero(x);
```
```
 
```
```
    return x;
```
```
}
```
```
 
```
```
#define _copy(a) mat_copy(a, n)
```
```
mat mat_copy(void *s, _dim) {
```
```
    mat x = mat_new(n);
```
```
    for_ij
```

            x[i][j] = ((double(*)[n]) s)[i][j];

```
    return x;
```
```
}
```
```
 
```
```
#define _del(x) mat_del(x)
```
```
void mat_del(mat x) {
```
```
    free(x[0]);
```
```
    free(x);
```
```
}
```
```
 
```
```
#define _QUOT(x) #x
```
```
#define QUOTE(x) _QUOT(x)
```

#define _show(a) printf(QUOTE(a)" =");mat_show(a, 0, n)

```
void mat_show(mat x, char *fmt, _dim) {
```
```
    if (!fmt)
```
```
        fmt = "%8.4g";
```
```
    for_i {
```
```
        printf(i ? "      " : " [ ");
```
```
        for_j {
```
```
            printf(fmt, x[i][j]);
```

            printf(j < n - 1 ? "  " : i == n - 1 ? " ]\n" : "\n");

```
        }
```
```
    }
```
```
}
```
```
 
```
```
#define _mul(a, b) mat_mul(a, b, n)
```
```
mat mat_mul(mat a, mat b, _dim) {
```
```
    mat c = _new(c);
```
```
    for_ijk
```

                c[i][j] += a[i][k] * b[k][j];

```
    return c;
```
```
}
```
```
 
```
```
#define _pivot(a, b) mat_pivot(a, b, n)
```
```
void mat_pivot(mat a, mat p, _dim) {
```
```
    for_ij
```
```
        {
```
```
            p[i][j] = (i == j);
```
```
        }
```
```
    for_i {
```
```
        int max_j = i;
```
```
        foreach(j, i, n)
```

            if (fabs(a[j][i]) > fabs(a[max_j][i]))

```
                max_j = j;
```
```
 
```
```
        if (max_j != i)
```
```
            for_k {
```

                _swap(p[i][k], p[max_j][k]);

```
            }
```
```
    }
```
```
}
```
```
 
```

#define _LU(a, l, u, p) mat_LU(a, l, u, p, n)

void mat_LU(mat A, mat L, mat U, mat P, _dim) {

```
    _zero(L);
```
```
    _zero(U);
```
```
    _pivot(A, P);
```
```
 
```
```
    mat Aprime = _mul(P, A);
```
```
 
```
```
    for_i {
```
```
        L[i][i] = 1;
```
```
    }
```
```
    for_ij
```
```
        {
```
```
            double s;
```
```
            if (j <= i) {
```

                _sum_k(0, j, L[j][k] * U[k][i], s)

                U[j][i] = Aprime[j][i] - s;

```
            }
```
```
            if (j >= i) {
```

                _sum_k(0, i, L[j][k] * U[k][i], s);

                L[j][i] = (Aprime[j][i] - s) / U[i][i];

```
            }
```
```
        }
```
```
 
```
```
    _del(Aprime);
```
```
}
```
```
 
```

double A3[][3] = { { 1, 3, 5 }, { 2, 4, 7 }, { 1, 1, 0 } };

double A4[][4] = { { 11, 9, 24, 2 }, { 1, 5, 2, 6 }, { 3, 17, 18, 1 }, { 2, 5,

```
        7, 1 } };
```
```
 
```
```
int main() {
```
```
    int n = 3;
```
```
    mat A, L, P, U;
```
```
 
```
```
    _new(L);
```
```
    _new(P);
```
```
    _new(U);
```
```
    A = _copy(A3);
```
```
    _LU(A, L, U, P);
```
```
    _show(A);
```
```
    _show(L);
```
```
    _show(U);
```
```
    _show(P);
```
```
    _del(A);
```
```
    _del(L);
```
```
    _del(U);
```
```
    _del(P);
```
```
 
```
```
    printf("\n");
```
```
 
```
```
    n = 4;
```
```
 
```
```
    _new(L);
```
```
    _new(P);
```
```
    _new(U);
```
```
    A = _copy(A4);
```
```
    _LU(A, L, U, P);
```
```
    _show(A);
```
```
    _show(L);
```
```
    _show(U);
```
```
    _show(P);
```
```
    _del(A);
```
```
    _del(L);
```
```
    _del(U);
```
```
    _del(P);
```
```
 
```
```
    return 0;
```
```
}
```

Output:

$ gcc LUDecomposition.cpp
$ ./a.out
 
A
 
1   3   5
2   4   7
1   1   0
 
L
 
1.00000   0.00000   0.00000
0.50000   1.00000   0.00000
0.50000  -1.00000   1.00000
 
U
 
2.00000   4.00000   7.00000
0.00000   1.00000   1.50000
0.00000   0.00000  -2.00000
 
P
 
0   1   0
1   0   0
0   0   1

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