Task: Reduced row echelon form
C: #include <stdio.h>
#define TALLOC(n,typ) malloc(n*sizeof(typ))

#define EL_Type int

typedef struct sMtx {
    int     dim_x, dim_y;
    EL_Type *m_stor;
    EL_Type **mtx;
} *Matrix, sMatrix;

typedef struct sRvec {
    int     dim_x;
    EL_Type *m_stor;
} *RowVec, sRowVec;

Matrix NewMatrix( int x_dim, int y_dim )
{
    int n;
    Matrix m;
    m = TALLOC( 1, sMatrix);
    n = x_dim * y_dim;
    m->dim_x = x_dim;
    m->dim_y = y_dim;
    m->m_stor = TALLOC(n, EL_Type);
    m->mtx = TALLOC(m->dim_y, EL_Type *);
    for(n=0; n<y_dim; n++) {
        m->mtx[n] = m->m_stor+n*x_dim;
    }
    return m;
}

void MtxSetRow(Matrix m, int irow, EL_Type *v)
{
    int ix;
    EL_Type *mr;
    mr = m->mtx[irow];
    for(ix=0; ix<m->dim_x; ix++)
        mr[ix] = v[ix];
}

Matrix InitMatrix( int x_dim, int y_dim, EL_Type **v)
{
    Matrix m;
    int iy;
    m = NewMatrix(x_dim, y_dim);
    for (iy=0; iy<y_dim; iy++) 
        MtxSetRow(m, iy, v[iy]);
    return m;
}

void MtxDisplay( Matrix m )
{
    int iy, ix;
    const char *sc;
    for (iy=0; iy<m->dim_y; iy++) {
        printf("   ");
        sc = " ";
        for (ix=0; ix<m->dim_x; ix++) {
            printf("%s %3d", sc, m->mtx[iy][ix]);
            sc = ",";
        }
        printf("\n");
    }
    printf("\n");
}

void MtxMulAndAddRows(Matrix m, int ixrdest, int ixrsrc, EL_Type mplr)
{
    int ix;
    EL_Type *drow, *srow;
    drow = m->mtx[ixrdest];
    srow = m->mtx[ixrsrc];
    for (ix=0; ix<m->dim_x; ix++) 
        drow[ix] += mplr * srow[ix];


}

void MtxSwapRows( Matrix m, int rix1, int rix2)
{
    EL_Type *r1, *r2, temp;
    int ix;
    if (rix1 == rix2) return;
    r1 = m->mtx[rix1];
    r2 = m->mtx[rix2];
    for (ix=0; ix<m->dim_x; ix++)
        temp = r1[ix]; r1[ix]=r2[ix]; r2[ix]=temp;


}

void MtxNormalizeRow( Matrix m, int rix, int lead)
{
    int ix;
    EL_Type *drow;
    EL_Type lv;
    drow = m->mtx[rix];
    lv = drow[lead];
    for (ix=0; ix<m->dim_x; ix++)
        drow[ix] /= lv;


}

#define MtxGet( m, rix, cix ) m->mtx[rix][cix]

void MtxToReducedREForm(Matrix m)
{
    int lead;
    int rix, iix;
    EL_Type lv;
    int rowCount = m->dim_y;

    lead = 0;
    for (rix=0; rix<rowCount; rix++) {
        if (lead >= m->dim_x)
            return;
        iix = rix;
        while (0 == MtxGet(m, iix,lead)) {
            iix++;
            if (iix == rowCount) {
                iix = rix;
                lead++;
                if (lead == m->dim_x)
                    return;
            }
        }
        MtxSwapRows(m, iix, rix );
        MtxNormalizeRow(m, rix, lead );
        for (iix=0; iix<rowCount; iix++) {
            if ( iix != rix ) {
                lv = MtxGet(m, iix, lead );
                MtxMulAndAddRows(m,iix, rix, -lv) ;
            }
        }
        lead++;
    }
}

int main()
{
    Matrix m1;
    static EL_Type r1[] = {1,2,-1,-4};
    static EL_Type r2[] = {2,3,-1,-11};
    static EL_Type r3[] = {-2,0,-3,22};
    static EL_Type *im[] = { r1, r2, r3 };

    m1 = InitMatrix( 4,3, im );
    printf("Initial\n");
    MtxDisplay(m1);
    MtxToReducedREForm(m1);
    printf("Reduced R-E form\n");
    MtxDisplay(m1);
    return 0;
}

Java: import java.util.*;
import java.lang.Math;
import org.apache.commons.math.fraction.Fraction;
import org.apache.commons.math.fraction.FractionConversionException;


class Matrix {
	LinkedList<LinkedList<Fraction>> matrix;
	int numRows;
	int numCols;	
	
	static class Coordinate {
		int row;
		int col;

		Coordinate(int r, int c) {
			row = r;
			col = c;
		}

		public String toString() {
			return "(" + row + ", " + col + ")";
		}
	}

	Matrix(double [][] m) {
		numRows = m.length;	
		numCols = m[0].length;

		matrix = new LinkedList<LinkedList<Fraction>>();

		for (int i = 0; i < numRows; i++) {
			matrix.add(new LinkedList<Fraction>());
			for (int j = 0; j < numCols; j++) {
				try {
					matrix.get(i).add(new Fraction(m[i][j]));
				} catch (FractionConversionException e) {
					System.err.println("Fraction could not be converted from double by apache commons . . .");
				}
			}
		}
	}

	public void Interchange(Coordinate a, Coordinate b) {
		LinkedList<Fraction> temp = matrix.get(a.row);
		matrix.set(a.row, matrix.get(b.row));		
		matrix.set(b.row, temp);

		int t = a.row;
		a.row = b.row;
		b.row = t;
	} 

	public void Scale(Coordinate x, Fraction d) {
		LinkedList<Fraction> row = matrix.get(x.row);
		for (int i = 0; i < numCols; i++) {
			row.set(i, row.get(i).multiply(d));
		}
	}

	public void MultiplyAndAdd(Coordinate to, Coordinate from, Fraction scalar) {
		LinkedList<Fraction> row = matrix.get(to.row);
		LinkedList<Fraction> rowMultiplied = matrix.get(from.row);

		for (int i = 0; i < numCols; i++) {
			row.set(i, row.get(i).add((rowMultiplied.get(i).multiply(scalar))));
		}
	}

	public void RREF() {
		Coordinate pivot = new Coordinate(0,0);

		int submatrix = 0;
		for (int x = 0; x < numCols; x++) {
			pivot = new Coordinate(pivot.row, x);
			
				
				for (int i = x; i < numCols; i++) {
					if (isColumnZeroes(pivot) == false) {
						break;	
					} else {
						pivot.col = i;
					}
				}
			
				
				pivot = findPivot(pivot);
			
				if (getCoordinate(pivot).doubleValue() == 0.0) {
					pivot.row++;
					continue;
				}

				
				
				if (pivot.row != submatrix) {
					Interchange(new Coordinate(submatrix, pivot.col), pivot);
				}
		
				
				if (getCoordinate(pivot).doubleValue() != 1) {
					
					Fraction scalar = getCoordinate(pivot).reciprocal();
					Scale(pivot, scalar);
				}
			
				
				
				for (int i = pivot.row; i < numRows; i++) {
					if (i == pivot.row) {
						continue;
					}
					Coordinate belowPivot = new Coordinate(i, pivot.col);
					Fraction complement = (getCoordinate(belowPivot).negate().divide(getCoordinate(pivot)));
					MultiplyAndAdd(belowPivot, pivot, complement);
				}
			
				
				
					
				for (int i = pivot.row; i >= 0; i--) {
					if (i == pivot.row) {
						if (getCoordinate(pivot).doubleValue() != 1.0) {
							Scale(pivot, getCoordinate(pivot).reciprocal());	
						}
						continue;
					}
					if (i == pivot.row) {
						continue;
					}
				
					Coordinate abovePivot = new Coordinate(i, pivot.col);
					Fraction complement = (getCoordinate(abovePivot).negate().divide(getCoordinate(pivot)));
					MultiplyAndAdd(abovePivot, pivot, complement);
				}
			
				
				
				if ((pivot.row + 1) >= numRows || isRowZeroes(new Coordinate(pivot.row+1, pivot.col))) {
					break;
				}

				submatrix++;
				pivot.row++;
		}
	}
	
	public boolean isColumnZeroes(Coordinate a) {
		for (int i = 0; i < numRows; i++) {
			if (matrix.get(i).get(a.col).doubleValue() != 0.0) {
				return false;
			}
		}

		return true;
	}

	public boolean isRowZeroes(Coordinate a) {
		for (int i = 0; i < numCols; i++) {
			if (matrix.get(a.row).get(i).doubleValue() != 0.0) {
				return false;
			}
		}

		return true;
	}

	public Coordinate findPivot(Coordinate a) {
		int first_row = a.row;
		Coordinate pivot = new Coordinate(a.row, a.col);
		Coordinate current = new Coordinate(a.row, a.col);	

		for (int i = a.row; i < (numRows - first_row); i++) {
			current.row = i;
			if (getCoordinate(current).doubleValue() == 1.0) {
				Interchange(current, a);
			}
		}

		current.row = a.row;
		for (int i = current.row; i < (numRows - first_row); i++) {
			current.row = i;
			if (getCoordinate(current).doubleValue() != 0) {
				pivot.row = i;
				break;
			}
		}
	
		
		return pivot;	
	}	

	public Fraction getCoordinate(Coordinate a) {
		return matrix.get(a.row).get(a.col);
	}

	public String toString() {
		return matrix.toString().replace("], ", "]\n");
	}

	public static void main (String[] args) {
        	double[][] matrix_1 = {
			{1, 2, -1, -4},
			{2, 3, -1, -11},
			{-2, 0, -3, 22}
		};

		Matrix x = new Matrix(matrix_1);
		System.out.println("before\n" + x.toString() + "\n");
		x.RREF();
		System.out.println("after\n" + x.toString() + "\n");

		double matrix_2 [][] = {
			{2, 0, -1, 0, 0},
			{1, 0, 0, -1, 0},
			{3, 0, 0, -2, -1},
			{0, 1, 0, 0, -2},
			{0, 1, -1, 0, 0}
		};
	
		Matrix y = new Matrix(matrix_2);
		System.out.println("before\n" + y.toString() + "\n");
		y.RREF();
		System.out.println("after\n" + y.toString() + "\n");

		double matrix_3 [][] = {
			{1, 2, 3, 4, 3, 1},
			{2, 4, 6, 2, 6, 2},
			{3, 6, 18, 9, 9, -6},
			{4, 8, 12, 10, 12, 4},
			{5, 10, 24, 11, 15, -4}
		};

		Matrix z = new Matrix(matrix_3);
		System.out.println("before\n" + z.toString() + "\n");
		z.RREF();
		System.out.println("after\n" + z.toString() + "\n");

		double matrix_4 [][] = {
			{0, 1},
			{1, 2},
			{0,5}
		};

		Matrix a = new Matrix(matrix_4);
		System.out.println("before\n" + a.toString() + "\n");
		a.RREF();
		System.out.println("after\n" + a.toString() + "\n");
	}	
}