public static MMatrix EchelonForm(MMatrix matrix)
{
try
{
MMatrix EchelonMatrix = new MMatrix(matrix);
for (int i = 0; i < matrix.row; ++i)
{
/*Overcome round-off error to approximate values
for (int j = 0; j < EchelonMatrix.row; ++j)
for (int k = 0 ; k < EchelonMatrix.col; ++k)
if (Math.Abs(EchelonMatrix[j, k]) <= ZERO)
EchelonMatrix[j, k] = 0;
*/
if (i < EchelonMatrix.col)
{
if (EchelonMatrix[i, i] == 0) // if diagonal entry is zero,
for (int j = i + 1; j < EchelonMatrix.row; ++j)
if (EchelonMatrix[j, i] != 0) //check if some below entry is non-zero
EchelonMatrix.InterchangeRow(i, j); // then interchange the two rows
if (EchelonMatrix[i, i] == 0) // if not found any non-zero diagonal entry
continue; // increment i;
if (EchelonMatrix[i, i] != 1) // if diagonal entry is not 1 ,
for (int j = i + 1; j < EchelonMatrix.row; ++j)
if (EchelonMatrix[j, i] == 1) //check if some below entry is 1
EchelonMatrix.InterchangeRow(i, j); // then interchange the two rows
EchelonMatrix.MultiplyRow(i, 1 / (EchelonMatrix[i, i]));
for (int j = i + 1; j < EchelonMatrix.row; ++j)
EchelonMatrix.AddRow(j, i, -EchelonMatrix[j, i]);
}
}
return EchelonMatrix;
}
catch (Exception)
{
throw new MMatrixException("MMatrix can not be reduced to Echelon form");
}
}
public static MMatrix ReducedEchelonForm(MMatrix matrix)
{
try
{
MMatrix ReducedEchelonMatrix = new MMatrix(matrix);
for (int i = 0; i < matrix.row; ++i)
{
if (i < ReducedEchelonMatrix.col)
{
if (ReducedEchelonMatrix[i, i] == 0) // if diagonal entry is zero,
for (int j = i + 1; j < ReducedEchelonMatrix.row; ++j)
if (ReducedEchelonMatrix[j, i] != 0) //check if some below entry is non-zero
ReducedEchelonMatrix.InterchangeRow(i, j); // then interchange the two rows
if (ReducedEchelonMatrix[i, i] == 0) // if not found any non-zero diagonal entry
continue; // increment i;
if (ReducedEchelonMatrix[i, i] != 1) // if diagonal entry is not 1 ,
for (int j = i + 1; j < ReducedEchelonMatrix.row; ++j)
if (ReducedEchelonMatrix[j, i] == 1) //check if some below entry is 1
ReducedEchelonMatrix.InterchangeRow(i, j); // then interchange the two rows
ReducedEchelonMatrix.MultiplyRow(i, 1 / (ReducedEchelonMatrix[i, i]));
for (int j = i + 1; j < ReducedEchelonMatrix.row; ++j)
ReducedEchelonMatrix.AddRow(j, i, -ReducedEchelonMatrix[j, i]);
for (int j = i - 1; j >= 0; j--)
ReducedEchelonMatrix.AddRow(j, i, -ReducedEchelonMatrix[j, i]);
}
}
return ReducedEchelonMatrix;
}
catch (Exception)
{
throw new MMatrixException("MMatrix can not be reduced to Echelon form");
}
}
//Solve A*x = b
//After solving the system, Argument Matrix will contain the reduced system
public static Vector SolveSystemOfEquations(MMatrix CoefficientMatrix, Vector b,ref MMatrix ArgumentMatrix, ref bool InfiniteSolution)
{
if (CoefficientMatrix.row != b.Elements.Length)
throw new MMatrixException("The coefficient matrix A and vector b are different size.");
Vector x = new Vector(CoefficientMatrix.col);
Vector xSetFlag = new Vector(CoefficientMatrix.col);
double sum;
InfiniteSolution = false;
ArgumentMatrix = CalculateArgumentMatrix(CoefficientMatrix,b);
try
{
//Gaussian elimination
for (int i = 0; i < CoefficientMatrix.row; ++i)
{
if (i < ArgumentMatrix.col - 1)
{
if (ArgumentMatrix[i, i] == 0) // if diagonal entry is zero,
for (int j = i + 1; j < ArgumentMatrix.row; ++j)
if (ArgumentMatrix[j, i] != 0) //check if some below entry is non-zero
ArgumentMatrix.InterchangeRow(i, j); // then interchange the two rows
if (ArgumentMatrix[i, i] == 0) // if not found any non-zero diagonal entry
continue; // increment i;
if (ArgumentMatrix[i, i] != 1) // if diagonal entry is not 1 ,
for (int j = i + 1; j < ArgumentMatrix.row; ++j)
if (ArgumentMatrix[j, i] == 1) //check if some below entry is 1
ArgumentMatrix.InterchangeRow(i, j); // then interchange the two rows
ArgumentMatrix.MultiplyRow(i, 1 / (ArgumentMatrix[i, i]));
for (int j = i + 1; j < ArgumentMatrix.row; ++j)
ArgumentMatrix.AddRow(j, i, -ArgumentMatrix[j, i]);
}
}
//Finding solution based on the reduced system == Argument matrix
//More equations than unknown varriables x[i]
if (CoefficientMatrix.row > CoefficientMatrix.col)
for (int i = CoefficientMatrix.col; i < CoefficientMatrix.row; ++i)
{
if (ArgumentMatrix[i, CoefficientMatrix.col - 1] == 0)
if (ArgumentMatrix[i, ArgumentMatrix.col - 1] != 0) //b[i] == 0
return null;
}
for (int i = Math.Min(CoefficientMatrix.row, CoefficientMatrix.col) - 1; i >= 0; i--)
{
//The system has more than one solutions. Get one.
if (ArgumentMatrix[i, i] == 0)
{
//More unknowns than equations
int j;
for (j = i+1; j < CoefficientMatrix.col; ++j)
{
if (ArgumentMatrix[i, j] != 0)
{
sum = 0;
for (int k = j + 1; k < CoefficientMatrix.col; ++k)
{
sum += ArgumentMatrix[i, k] * x[k];
}
if (xSetFlag[j] == 0)
{
//this is the 1st time setting the solution
x[j] = (ArgumentMatrix[i, ArgumentMatrix.col - 1] - sum) / ArgumentMatrix[i, j];
xSetFlag[j] = 1;
}
//this is not the 1st time setting the solution
else
if (x[j] != (ArgumentMatrix[i, ArgumentMatrix.col - 1] - sum) / CoefficientMatrix[i, j])
{
return null;
}
else
InfiniteSolution = true;
break;
}
}
//All coefficient == 0 & b[i] == 0 -> The system has infinitely many solutions
if ((j == CoefficientMatrix.col) && (ArgumentMatrix[i, ArgumentMatrix.col - 1] == 0))
{
x[i] = 0;
//xSetFlag[i] = 1;-> since we may use other equations to update this x[i] later, so don't set flag for case 0 = 0.
InfiniteSolution = true;
}
else if ((j == CoefficientMatrix.col) && (ArgumentMatrix[i, ArgumentMatrix.col - 1] != 0))
return null; //All coefficient == 0 & b[i] != 0 -> The system has no solutions
}
else
{
sum = 0;
for (int j = i + 1; j < CoefficientMatrix.col; ++j)
{
sum += ArgumentMatrix[i, j] * x[j];
}
x[i] = (ArgumentMatrix[i, ArgumentMatrix.col - 1] - sum) / ArgumentMatrix[i, i];
xSetFlag[i] = 1;
}
}
return x;
}
catch (Exception exp)
{
throw new MMatrixException("The system of equation cannot be solved:" + exp.Message);
}
}