// ------------------- INVERSE MATRIX ---------------------------
// TO DO: make without minor matrix copy, but with minor matrix itself. Economy of the memory.
/// <summary>
/// Calculates the inverse matrix using the LUP-factorization for calculating matrix determinants
/// and algebraic supplements of its elements.
/// It takes O(n^5) operations to run.
///
/// Works only for square, non-singular matrices.
/// If these requirements are not met, either a MatrixSizeException
/// or a MatrixSingularityException shall be thrown.
/// </summary>
public Matrix <T, C> inverseMatrix_LUP_Factorization()
{
this.checkSquare();
Numeric <T, C> determinant = this.Determinant_LUP_Factorization();
if (determinant == Numeric <T, C> .Zero)
{
throw new MatrixSingularityException("cannot calculate the inverse matrix.");
}
// to do: make matrix type
Matrix <T, C> inverse = new Matrix_SDA <T, C>(this.rows, this.columns);
for (int i = 0; i < rows; i++)
{
for (int j = 0; j < columns; j++)
{
Numeric <T, C> minor = this.getMinorMatrix(j, i).Determinant_LUP_Factorization();
if (((i + j) & 1) == 1)
{
minor = -minor;
}
inverse.setItemAt(i, j, minor / determinant);
}
}
return(inverse);
}
/// <summary>
/// Preforms the LUP-factorization of a matrix (let it be A)
/// in the form of:
///
/// P*A = L*U.
///
/// The P is an identity matrix with a plenty of row inversions.
/// For the economy of space it is provided as a single-dimensional array of integers:
///
/// (0, 1, 2, 3, ..., n).
///
/// Element indices of this array stand for matrix rows, and elements value
/// mean the position of '1' in a row.
///
/// Requirements: works for any square and nonsingular matrix (having a non-zero determinant).
/// If these requirements aren't met, either MatrixSingularException of MatrixSizeException
/// would be thrown.
/// </summary>
/// <param name="L">The lower-triangular matrix with '1' on the main diagonal.</param>
/// <param name="U">The upper-triangular matrix.</param>
/// <param name="P">The identity matrix with a plenty of row inversions in the form of array.</param>
public void LUP_Factorization(out int[] P, out Matrix_SDA <T, C> L, out Matrix_SDA <T, C> U)
{
Matrix_SDA <T, C> C;
this.LUP_Factorization(out P, out C);
int n = this.rows;
L = new Matrix_SDA <T, C>(n, n);
U = new Matrix_SDA <T, C>(n, n);
Numeric <T, C> one = (Numeric <T, C>) 1;
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
if (i < j)
{
U.setItemAt(i, j, C.getItemAt(i, j));
}
else if (i > j)
{
L.setItemAt(i, j, C.getItemAt(i, j));
}
else
{
L.setItemAt(i, j, one); // здесь единицы на диагонали
U.setItemAt(i, j, C.getItemAt(i, j)); // E возместить не надо.
}
}
}
return;
}
protected override Matrix <T, C> sum(Matrix <T, C> another)
{
if (this.rows != another.RowCount || this.columns != another.ColumnCount)
{
throw new MatrixSizeException("Matrices must be of the same size in order to sum.");
}
// If the matrix is SDA, we can do it quick'n'lucky.
if (this.GetType().IsInstanceOfType(another))
{
Matrix_SDA <T, C> temp = (Matrix_SDA <T, C>)another;
Matrix_SDA <T, C> newMatrix = new Matrix_SDA <T, C>(this.rows, this.columns);
for (int i = 0; i < this.ElementCount; i++)
{
newMatrix.matrixArray[i] = this.matrixArray[i] + temp.matrixArray[i];
}
return(newMatrix);
}
// Here comes the bad case
else
{
Matrix_SDA <T, C> newMatrix = new Matrix_SDA <T, C>(this.rows, this.columns);
for (int i = 0; i < this.ElementCount; i++)
{
newMatrix.matrixArray[i] = this.matrixArray[i] + another.getItemAt(i / columns, i % columns);
}
return(newMatrix);
}
}
/// <summary>
/// Provides a deep clone of the current matrix.
/// </summary>
/// <returns>The cloned matrix.</returns>
public override object Clone()
{
Matrix_SDA <T, C> temp = new Matrix_SDA <T, C>();
temp.matrixArray = (Numeric <T, C>[]) this.matrixArray.Clone();
temp.rows = this.rows;
temp.columns = this.columns;
return(temp);
}
/// <summary>
/// Provides a deep clone of the current matrix.
/// </summary>
/// <returns>The cloned matrix.</returns>
public override object Clone()
{
Matrix_SDA temp = new Matrix_SDA();
temp.matrixArray = (double[])this.matrixArray.Clone();
temp.rows = this.rows;
temp.columns = this.columns;
return(temp);
}
protected override Matrix <double> multiply(Matrix <double> another)
{
if (this.columns != another.RowCount)
{
throw new ArgumentException("The column count of the first matrix and the row count of the second matrix must match.");
}
Matrix_SDA temp = new Matrix_SDA(this.rows, another.ColumnCount);
MatrixHelper.multiplySimple(this, another, temp);
return(temp);
}
// ------------------ OPERATORS
/// <summary>
/// Negates the matrix so that all the elements change their sign to the opposite.
/// </summary>
/// <returns></returns>
protected override Matrix <T, C> negate()
{
Matrix_SDA <T, C> temp = new Matrix_SDA <T, C>(rows, columns);
temp.matrixArray = (Numeric <T, C>[]) this.matrixArray.Clone();
for (int i = 0; i < this.ElementCount; i++)
{
temp.matrixArray[i] = -temp.matrixArray[i];
}
return(temp);
}
/// <summary>
/// Negates the matrix so that all the elements change their sign to the opposite.
/// </summary>
/// <returns></returns>
protected override Matrix <double> negate()
{
Matrix_SDA temp = new Matrix_SDA(rows, columns);
temp.matrixArray = (double[])this.matrixArray.Clone();
for (int i = 0; i < this.ElementCount; i++)
{
temp.matrixArray[i] *= -1;
}
return(temp);
}
/// <summary>
/// Preforms the LUP-factorization of a matrix (let it be A)
/// in the form of:
///
/// P*A = L*U.
///
/// The P is an identity matrix with a plenty of row inversions.
///
/// Requirements: works for any square and nonsingular matrix (having a non-zero determinant).
/// If these requirements aren't met, either MatrixSingularException of MatrixSizeException
/// would be thrown.
/// </summary>
/// <param name="L">The lower-triangular matrix with '1' on the main diagonal.</param>
/// <param name="U">The upper-triangular matrix.</param>
/// <param name="P">The identity matrix with a plenty of row inversions.</param>
public void LUP_Factorization(out Matrix_SDA <T, C> P, out Matrix_SDA <T, C> L, out Matrix_SDA <T, C> U)
{
int[] arr;
int n = this.rows;
Numeric <T, C> one = (Numeric <T, C>) 1;
LUP_Factorization(out arr, out L, out U);
P = new Matrix_SDA <T, C>(n, n);
for (int i = 0; i < n; i++)
{
P.setItemAt(i, arr[i], one);
}
return;
}
// ---------------------------------------------
// MATRIX ARITHMETIC
// ---------------------------------------------
/// <summary>
/// Performs the matrix multiplication using the quick Strassen algorithm.
/// Consumes more memory than simple iterational method, but much quicker
/// on bigger dimensions (128+).
///
/// If the dimension of method parameters passed is N, then maximum additional object memory consumption is
/// <value>9N + [if N>64: (log2(N)-6)*f] + o(N)</value>
/// Where f == (nearly) 4.5*(new Matrix of 2N*2N memory consumption).
/// </summary>
/// <param name="A">The matrix to multiply.</param>
/// <param name="B">The matrix to multiply by.</param>
/// <param name="result">The resulting matrix of [A.Rows x B.Columns] size.</param>
public static void multiplyStrassen(Matrix <T, C> A, Matrix <T, C> B, Matrix <T, C> result)
{
int exp = 1;
do
{
exp *= 2;
} while (A.ColumnCount > exp || A.RowCount > exp || B.ColumnCount > exp || B.RowCount > exp);
Matrix_SDA <T, C> Anew = new Matrix_SDA <T, C>(exp, exp);
Matrix_SDA <T, C> Bnew = new Matrix_SDA <T, C>(exp, exp);
Matrix_SDA <T, C> ResNew = new Matrix_SDA <T, C>(exp, exp);
Anew.layMatrixAt(A, 0, 0);
Bnew.layMatrixAt(B, 0, 0);
strassenSkeleton(Anew, Bnew, ResNew, exp);
result.layMatrixAt(ResNew.getSubMatrixCopyAt(0, 0, result.RowCount, result.ColumnCount), 0, 0);
}
/// <summary>
/// Overloaded. Gets the submatrix at the specified point and with specified size.
/// </summary>
/// <param name="i">Row index of the upper-left corner element</param>
/// <param name="j">Columnt index of the upper-left corner element</param>
/// <param name="rows">Row count of the submatrix</param>
/// <param name="columns">Column count of the submatrix</param>
/// <returns>The submatrix of specified size</returns>
public override Matrix <double> getSubMatrixCopyAt(int i, int j, int rows, int columns)
{
checkPositive(rows, columns);
checkBounds(i + rows, j + columns);
double[] ma = new double[rows * columns];
int z = 0;
for (int k = i; k < i + rows; k++)
{
Array.Copy(this.matrixArray, this.columns * k + j, ma, z++ *columns, columns);
}
Matrix_SDA temp = new Matrix_SDA();
temp.matrixArray = ma;
temp.rows = rows;
temp.columns = columns;
return(temp);
}
/// <summary>
/// Overloaded. Gets the stand-alone submatrix copy at the specified point and with specified size.
/// </summary>
/// <param name="i">Row index of the upper-left corner element.</param>
/// <param name="j">Columnt index of the upper-left corner element.</param>
/// <param name="rows">Row count of the submatrix.</param>
/// <param name="columns">Column count of the submatrix.</param>
/// <returns>The submatrix of specified size.</returns>
public virtual Matrix <T, C> getSubMatrixCopyAt(int i, int j, int rows, int columns)
{
checkPositive(rows, columns);
checkBounds(i + rows, j + columns);
Numeric <T, C>[] ma = new Numeric <T, C> [rows * columns];
for (int k = i; k < i + rows; k++)
{
for (int m = j; m < j + rows; m++)
{
ma[k * rows + m] = calc.getCopy(this.getItemAt(k, m));
}
}
Matrix_SDA <T, C> temp = new Matrix_SDA <T, C>();
temp.matrixArray = ma;
temp.rows = rows;
temp.columns = columns;
return(temp);
}
// ------------------- FACTORIZATION ----------------------------
/// <summary>
/// Preforms the LUP-factorization of a matrix (let it be A)
/// in the form of:
///
/// P*A = L*U.
///
/// The P is an identity matrix with a plenty of row inversions.
/// For the economy of space it is provided as a single-dimensional array of integers:
///
/// (0, 1, 2, 3, ..., n).
///
/// Element indices of this array stand for matrix rows, and elements value
/// mean the position of '1' in a row.
///
/// Requirements: works for any square and nonsingular matrix (having a non-zero determinant).
/// If these requirements aren't met, either a MatrixSingularException or a MatrixSizeException
/// would be thrown.
/// </summary>
/// <param name="C">The matrix C containing L + U - E. It is clear that both L and U can be easily extracted from this matrix.</param>
/// <param name="P">The identity matrix with a plenty of row inversions in the form of array.</param>
public void LUP_Factorization(out int[] P, out Matrix_SDA <T, C> C)
{
if (this.rows != this.columns)
{
throw new MatrixSizeException("The matrix is not square an thus cannot be factorized.");
}
int n = this.rows; // размер матрицы
C = new Matrix_SDA <T, C>(n, n);
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
C.setItemAt(i, j, this.getItemAt(i, j));
}
}
P = new int[n];
P.FillByAssign(delegate(int i) { return(i); });
// ----- пошел ----------
Numeric <T, C> pivot;
Numeric <T, C> abs;
int pivotIndex;
for (int i = 0; i < n; i++)
{
pivot = Numeric <T, C> .Zero;
pivotIndex = -1;
for (int row = i; row < n; row++)
{
abs = WhiteMath <T, C> .Abs(this.getItemAt(row, i));
if (abs > pivot)
{
pivot = abs;
pivotIndex = row;
}
}
if (pivot == Numeric <T, C> .Zero)
{
throw new MatrixSingularityException("The matrix is singular. It cannot be factorized.");
}
if (pivotIndex != i)
{
P.Swap(pivotIndex, i);
C.swapRows(pivotIndex, i);
}
try
{
for (int j = i + 1; j < n; j++)
{
C.setItemAt(j, i, C.getItemAt(j, i) / C.getItemAt(i, i));
if (Numeric <T, C> .isInfinity(C.getItemAt(j, i)) || Numeric <T, C> .isNaN(C.getItemAt(j, i)))
{
throw new DivideByZeroException();
}
for (int k = i + 1; k < n; k++)
{
C.setItemAt(j, k, C.getItemAt(j, k) - C.getItemAt(j, i) * C.getItemAt(i, k));
}
}
}
catch (DivideByZeroException)
{
throw new MatrixSingularityException("The matrix is singular. It cannot be factorized.");
}
}
return;
}
/// <summary>
/// Overloaded. Gets the submatrix at the specified point and with specified size.
/// </summary>
/// <param name="i">Row index of the upper-left corner element</param>
/// <param name="j">Columnt index of the upper-left corner element</param>
/// <param name="rows">Row count of the submatrix</param>
/// <param name="columns">Column count of the submatrix</param>
/// <returns>The submatrix of specified size</returns>
public override Matrix<double> getSubMatrixCopyAt(int i, int j, int rows, int columns)
{
checkPositive(rows, columns);
checkBounds(i + rows, j + columns);
double[] ma = new double[rows * columns];
int z=0;
for (int k = i; k < i + rows; k++)
{
Array.Copy(this.matrixArray, this.columns*k + j, ma, z++*columns, columns);
}
Matrix_SDA temp = new Matrix_SDA();
temp.matrixArray = ma;
temp.rows = rows;
temp.columns = columns;
return temp;
}
/// <summary>
/// Private recursive Strassen multiplying method skeleton called implicitly by the wrapper method.
/// </summary>
/// <param name="A">The matrix to multiply</param>
/// <param name="B">The matrix to multiply by</param>
/// <param name="result">The resulting matrix</param>
/// <param name="curDim">Current matrix dimension</param>
private static void strassenSkeleton(Matrix <T, C> A, Matrix <T, C> B, Matrix <T, C> result, int curDim)
{
if (curDim <= 64)
{
multiplySimple(A, B, result);
return;
}
int size = curDim / 2;
Matrix <T, C> A11 = A.getSubMatrixAt(0, 0, size, size);
Matrix <T, C> A12 = A.getSubMatrixAt(0, size, size, size);
Matrix <T, C> A21 = A.getSubMatrixAt(size, 0, size, size);
Matrix <T, C> A22 = A.getSubMatrixAt(size, size, size, size);
Matrix <T, C> B11 = B.getSubMatrixAt(0, 0, size, size);
Matrix <T, C> B12 = B.getSubMatrixAt(0, size, size, size);
Matrix <T, C> B21 = B.getSubMatrixAt(size, 0, size, size);
Matrix <T, C> B22 = B.getSubMatrixAt(size, size, size, size);
Matrix <T, C> P1 = new Matrix_SDA <T, C>(size, size);
Matrix <T, C> P2 = new Matrix_SDA <T, C>(size, size);
Matrix <T, C> P3 = new Matrix_SDA <T, C>(size, size);
Matrix <T, C> P4 = new Matrix_SDA <T, C>(size, size);
Matrix <T, C> P5 = new Matrix_SDA <T, C>(size, size);
Matrix <T, C> P6 = new Matrix_SDA <T, C>(size, size);
Matrix <T, C> P7 = new Matrix_SDA <T, C>(size, size);
Matrix <T, C> temp1 = new Matrix_SDA <T, C>(size, size);
Matrix <T, C> temp2 = new Matrix_SDA <T, C>(size, size);
sum(A11, A22, temp1);
sum(B11, B22, temp2);
strassenSkeleton(temp1, temp2, P1, size);
sum(A21, A22, temp1);
strassenSkeleton(temp1, B11, P2, size);
dif(B12, B22, temp1);
strassenSkeleton(A11, temp1, P3, size);
dif(B21, B11, temp1);
strassenSkeleton(A22, temp1, P4, size);
sum(A11, A12, temp1);
strassenSkeleton(temp1, B22, P5, size);
dif(A21, A11, temp1);
sum(B11, B12, temp2);
strassenSkeleton(temp1, temp2, P6, size);
dif(A12, A22, temp1);
sum(B21, B22, temp2);
strassenSkeleton(temp1, temp2, P7, size);
sum(P1, P4, temp1);
dif(temp1, P5, temp1);
sum(temp1, P7, temp1);
result.layMatrixAt(temp1, 0, 0);
sum(P3, P5, temp1);
result.layMatrixAt(temp1, 0, size);
sum(P2, P4, temp1);
result.layMatrixAt(temp1, size, 0);
dif(P1, P2, temp1);
sum(temp1, P3, temp1);
sum(temp1, P6, temp1);
result.layMatrixAt(temp1, size, size);
}
/// <summary>
/// Provides a deep clone of the current matrix.
/// </summary>
/// <returns>The cloned matrix.</returns>
public override object Clone()
{
Matrix_SDA temp = new Matrix_SDA();
temp.matrixArray = (double[])this.matrixArray.Clone();
temp.rows = this.rows;
temp.columns = this.columns;
return temp;
}
protected override Matrix<double> sum(Matrix<double> another)
{
if (this.rows != another.RowCount || this.columns != another.ColumnCount)
throw new ArgumentException("Matrices must be of the same size in order to sum.");
// If the matrix is SDA, we can do it quick'n'lucky.
if (this.GetType().IsInstanceOfType(another))
{
Matrix_SDA temp = (Matrix_SDA)another;
Matrix_SDA newMatrix = new Matrix_SDA(this.rows, this.columns);
for (int i = 0; i < this.ElementCount; i++)
newMatrix.matrixArray[i] = this.matrixArray[i] + temp.matrixArray[i];
return newMatrix;
}
// Here comes the bad case
else
{
Matrix_SDA newMatrix = new Matrix_SDA(this.rows, this.columns);
for (int i = 0; i < this.ElementCount; i++)
newMatrix.matrixArray[i] = this.matrixArray[i] + another.getItemAt(i / columns, i % columns);
return newMatrix;
}
}
protected override Matrix<double> multiply(Matrix<double> another)
{
if (this.columns != another.RowCount)
throw new ArgumentException("The column count of the first matrix and the row count of the second matrix must match.");
Matrix_SDA temp = new Matrix_SDA(this.rows, another.ColumnCount);
MatrixHelper.multiplySimple(this, another, temp);
return temp;
}
/// <summary>
/// Negates the matrix so that all the elements change their sign to the opposite.
/// </summary>
/// <returns></returns>
protected override Matrix<double> negate()
{
Matrix_SDA temp = new Matrix_SDA(rows, columns);
temp.matrixArray = (double[])this.matrixArray.Clone();
for (int i = 0; i < this.ElementCount; i++)
temp.matrixArray[i] *= -1;
return temp;
}
/// <summary>
/// This algorithm uses the LU-Factorization of coefficient matrix in order
/// to calculate the solution of the equation system.
///
/// As LU-Factorization is not guaranteed to exist for every
/// non-singular matrix, this method may sometimes fail.
/// </summary>
/// <param name="coefficients">A square matrix of unknown terms' coefficients.</param>
/// <param name="freeTerm">A vector of free terms.</param>
/// <param name="x">The vector containing the solution of the equation system.</param>
public static void LU_FactorizationSolving <T, C>(Matrix <T, C> coefficients, Vector <T, C> freeTerm, out Vector <T, C> x) where C : ICalc <T>, new()
{
if (coefficients.RowCount != coefficients.ColumnCount)
{
throw new ArgumentException("Only square matrices are supported.");
}
int dim = coefficients.RowCount;
Matrix_SDA <T, C> K = new Matrix_SDA <T, C>(dim, dim);
Matrix_SDA <T, C> M = new Matrix_SDA <T, C>(dim, dim);
// Заполняем единичную диагональ матрицы K.
for (int i = 0; i < dim; i++)
{
for (int j = 0; j < dim; j++)
{
K[i, j] = M[i, j] = Numeric <T, C> .Zero;
}
M[i, i] = (Numeric <T, C>) 1;
K[i, 0] = coefficients[i, 0];
M[0, i] = coefficients[0, i] / K[0, 0];
}
// вектор промежуточных решений
Vector <T, C> y = new Vector <T, C>(dim);
y[0] = freeTerm[0] / K[0, 0];
Numeric <T, C> s; // вспомогательная
for (int i = 1; i < dim; i++)
{
int j;
for (j = 0; j < dim; j++)
{
s = (Numeric <T, C>) 0;
for (int z = 0; z <= j - 1; z++)
{
s += K[i, z] * M[z, j];
}
K[i, j] = coefficients[i, j] - s;
}
for (j = i; j < dim; j++)
{
Numeric <T, C> s1 = (Numeric <T, C>) 0;
Numeric <T, C> s2 = (Numeric <T, C>) 0;
for (int z = 0; z <= i - 1; z++)
{
s1 += K[i, z] * M[z, j];
s2 += K[i, z] * y[z];
}
M[i, j] = (coefficients[i, j] - s1) / K[i, i];
y[i] = (freeTerm[i] - s2) / K[i, i];
}
}
x = new Vector <T, C>(dim);
x[dim - 1] = y[dim - 1];
for (int i = dim - 2; i >= 0; i--)
{
s = (Numeric <T, C>) 0;
for (int z = i + 1; z < dim; z++)
{
s += M[i, z] * x[z];
}
x[i] = y[i] - s;
}
return;
}