/// <summary>
/// Generates tri-diagonal square matrix with overloaded vectors
/// as main and secondary diagonals. The dimension of the output
/// matrix is determined by the length of d.
/// </summary>
/// <param name="l">Lower secondary diagonal vector.</param>
/// <param name="d">Main diagonal vector.</param>
/// <param name="u">Upper secondary diagonal vector.</param>
/// <returns></returns>
public static Matrix TriDiag(Matrix l, Matrix d, Matrix u)
{
int sizeL = l.VectorLength();
int sizeD = d.VectorLength();
int sizeU = u.VectorLength();
if (sizeL * sizeD * sizeU == 0)
throw new ArgumentException("At least one of the paramter matrices is not a vector.");
if (sizeL != sizeU)
throw new ArgumentException("Lower and upper secondary diagonal must have the same length.");
if (sizeL + 1 != sizeD)
throw new ArgumentException("Main diagonal must have exactly one element more than the secondary diagonals.");
return Diag(l, -1) + Diag(d) + Diag(u, 1);
}
/// <summary>
/// Implements the dot product of two vectors.
/// </summary>
/// <param name="v">Row or column vector.</param>
/// <param name="w">Row or column vector.</param>
/// <returns>Dot product.</returns>
public static Complex Dot(Matrix v, Matrix w)
{
int m = v.VectorLength();
int n = w.VectorLength();
if (m == 0 || n == 0)
throw new ArgumentException("Arguments need to be vectors.");
else if (m != n)
throw new ArgumentException("Vectors must be of the same length.");
Complex buf = Complex.Zero;
for (int i = 1; i <= m; i++)
{
buf += v[i] * w[i];
}
return buf;
}
/// <summary>
/// Generates diagonal matrix
/// </summary>
/// <param name="diag_vector">column vector containing the diag elements</param>
/// <returns></returns>
public static Matrix Diag(Matrix diag_vector)
{
int dim = diag_vector.VectorLength();
if(dim == 0)
throw new ArgumentException("diag_vector must be 1xN or Nx1");
Matrix M = new Matrix(dim, dim);
for (int i = 1; i <= dim; i++)
{
M[i, i] = diag_vector[i];
}
return M;
}
/// <summary>
/// Generates diagonal matrix
/// </summary>
/// <param name="diag_vector">column vector containing the diag elements</param>
/// <returns></returns>
public static Matrix Diag(Matrix diag_vector, int offset)
{
int dim = diag_vector.VectorLength();
if(dim == 0)
throw new ArgumentException("diag_vector must be 1xN or Nx1.");
//if (Math.Abs(offset) >= dim)
// throw new ArgumentException("Absolute value of offset must be less than length of diag_vector.");
Matrix M = new Matrix(dim + Math.Abs(offset), dim + Math.Abs(offset));
dim = M.RowCount;
if (offset >= 0)
{
for (int i = 1; i <= dim - offset; i++)
{
M[i, i + offset] = diag_vector[i];
}
}
else
{
for (int i = 1; i <= dim + offset; i++)
{
M[i - offset, i] = diag_vector[i];
}
}
return M;
}
/// <summary>
/// Performs backward insertion for regular upper triangular matrix
/// and right side b, such that the solution is saved right within b.
/// The matrix is not changed.
/// </summary>
/// <param name="b">Vector of height n, if matrix is n by n.</param>
public void BackwardInsertion(Matrix b)
{
if (!this.IsUpperTriangular())
throw new InvalidOperationException("Cannot perform backward insertion for matrix not being upper triangular.");
if (/*this.Determinant*/this.DiagProd() == 0)
throw new InvalidOperationException("Warning: Matrix is nearly singular.");
int n = rowCount;
if (b.VectorLength() != n)
throw new ArgumentException("Parameter must vector of the same height as matrix.");
for (int j = n; j >= 2; j--)
{
b[j] /= this[j, j];
for (int i = 1; i <= j - 1; i++)
b[i] -= b[j] * this[i, j];
}
b[1] /= this[1, 1];
}
/// <summary>
/// Performs forward insertion for regular lower triangular matrix
/// and right side b, such that the solution is saved right within b.
/// The matrix is not changed.
/// </summary>
/// <param name="b">Vector of height n, if matrix is n by n.</param>
public void ForwardInsertion(Matrix b)
{
if (!this.IsLowerTriangular())
throw new InvalidOperationException("Cannot perform forward insertion for matrix not being lower triangular.");
if (/*this.Determinant*/this.DiagProd() == 0)
throw new InvalidOperationException("Warning: Matrix is nearly singular.");
int n = rowCount;
if (b.VectorLength() != n)
throw new ArgumentException("Parameter must vector of the same height as matrix.");
for (int j = 1; j <= n - 1; j++)
{
b[j] /= this[j, j];
for (int i = 1; i <= n - j; i++)
b[j + i] -= b[j] * this[j + i, j];
}
b[n] /= this[n, n];
}
/// <summary>
/// Inserts column at specified index.
/// </summary>
/// <param name="col">Vector to insert</param>
/// <param name="j">One-based index at which to insert</param>
public void InsertColumn(Matrix col, int j)
{
int size = col.VectorLength();
if (size == 0)
throw new InvalidOperationException("Row must be a vector of length > 0.");
if (j <= 0)
throw new ArgumentException("Row index must be positive.");
if (j > columnCount)
{
this[size, j] = Complex.Zero;
}
else
columnCount++;
if (size > rowCount)
{
this[size, j] = Complex.Zero;
}
j--;
for (int k = 0; k < size; k++)
{
((ArrayList)Values[k]).Insert(j, col[k + 1]);
}
// fill w/ zeros if vector col too short
for (int k = size; k < rowCount; k++)
{
((ArrayList)Values[k]).Insert(j, 0);
}
}
/// <summary>
/// Inserts row at specified index.
/// </summary>
/// <param name="row">Vector to insert</param>
/// <param name="i">One-based index at which to insert</param>
public void InsertRow(Matrix row, int i)
{
int size = row.VectorLength();
if (size == 0)
throw new InvalidOperationException("Row must be a vector of length > 0.");
if (i <= 0)
throw new ArgumentException("Row index must be positive.");
if (i > rowCount)
this[i, size] = Complex.Zero;
else if (size > columnCount)
{
this[i, size] = Complex.Zero;
rowCount++;
}
else
rowCount++;
Values.Insert(--i, new ArrayList(size));
//Debug.WriteLine(Values.Count.ToString());
for (int k = 1; k <= size; k++)
{
((ArrayList)Values[i]).Add(row[k]);
}
// fill w/ zeros if vector row is too short
for (int k = size; k < columnCount; k++)
{
((ArrayList)Values[i]).Add(Complex.Zero);
}
}
/// <summary>
/// Computes the Householder vector.
/// </summary>
/// <param name="x"></param>
/// <returns></returns>
private static Matrix[] HouseholderVector(Matrix x)
{
//throw new NotImplementedException("Supposingly buggy!");
//if (!x.IsReal())
// throw new ArgumentException("Cannot compute housholder vector of non-real vector.");
int n = x.VectorLength();
if (n == 0)
throw new InvalidOperationException("Expected vector as argument.");
Matrix y = x / x.Norm();
Matrix buf = y.Extract(2, n, 1, 1);
Complex s = Dot(buf, buf);
Matrix v = Zeros(n, 1);
v[1] = Complex.One;
v.Insert(2, 1, buf);
double beta = 0;
if (s != 0)
{
Complex mu = Complex.Sqrt(y[1] * y[1] + s);
if (y[1].Re <= 0)
v[1] = y[1] - mu;
else
v[1] = -s / (y[1] + mu);
beta = 2 * v[1].Re * v[1].Re / (s.Re + v[1].Re * v[1].Re);
v = v / v[1];
}
return new Matrix[] { v, new Matrix(beta) };
}