/// <summary>
/// Generates diagonal matrix
/// </summary>
/// <param name="diag_vector">column vector containing the diag elements</param>
/// <returns></returns>
public static ArrayMatrix Diag(ArrayMatrix diag_vector, int offset)
{
var 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.");
var M = new ArrayMatrix(dim + Math.Abs(offset), dim + Math.Abs(offset));
dim = M.RowCount;
if (offset >= 0) for (var i = 1; i <= dim - offset; i++) M[i, i + offset] = diag_vector[i];
else for (var i = 1; i <= dim + offset; i++) M[i - offset, i] = diag_vector[i];
return M;
}
/// <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(ArrayMatrix v, ArrayMatrix w)
{
var m = v.VectorLength();
var 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.");
var buf = Complex.Zero;
for (var i = 1; i <= m; i++) buf += v[i]*w[i];
return buf;
}
/// <summary>
/// Computes the Householder vector.
/// </summary>
/// <param name="x"></param>
/// <returns></returns>
static ArrayMatrix[] HouseholderVector(ArrayMatrix x)
{
//throw new NotImplementedException("Supposingly buggy!");
//if (!x.IsReal())
// throw new ArgumentException("Cannot compute housholder vector of non-real vector.");
var n = x.VectorLength();
if (n == 0) throw new InvalidOperationException("Expected vector as argument.");
var y = x/x.Norm();
var buf = y.Extract(2, n, 1, 1);
var s = Dot(buf, buf);
var v = Zeros(n, 1);
v[1] = Complex.One;
v.Insert(2, 1, buf);
double beta = 0;
if (s != 0)
{
var mu = Complex.Sqrt(y[1]*y[1] + s);
if (y[1].Real <= 0) v[1] = y[1] - mu;
else v[1] = -s/(y[1] + mu);
beta = 2*v[1].Real*v[1].Real/(s.Real + v[1].Real*v[1].Real);
v = v/v[1];
}
return new ArrayMatrix[] {v, new ArrayMatrix(beta)};
}
/// <summary>
/// Generates diagonal matrix
/// </summary>
/// <param name="diag_vector">column vector containing the diag elements</param>
/// <returns></returns>
public static ArrayMatrix Diag(ArrayMatrix diag_vector)
{
var dim = diag_vector.VectorLength();
if (dim == 0) throw new ArgumentException("diag_vector must be 1xN or Nx1");
var M = new ArrayMatrix(dim, dim);
for (var i = 1; i <= dim; i++) M[i, i] = diag_vector[i];
return M;
}
/// <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(ArrayMatrix row, int i)
{
var 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 (var k = 1; k <= size; k++) ((ArrayList) Values[i]).Add(row[k]);
// fill w/ zeros if vector row is too short
for (var k = size; k < ColumnCount; k++) ((ArrayList) Values[i]).Add(Complex.Zero);
}
/// <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(ArrayMatrix col, int j)
{
var 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 (var k = 0; k < size; k++) ((ArrayList) Values[k]).Insert(j, col[k + 1]);
// fill w/ zeros if vector col too short
for (var k = size; k < RowCount; k++) ((ArrayList) Values[k]).Insert(j, 0);
}
/// <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(ArrayMatrix b)
{
if (!IsLowerTriangular()) throw new InvalidOperationException("Cannot perform forward insertion for matrix not being lower triangular.");
if ( /*this.Determinant*/DiagProd() == 0) throw new InvalidOperationException("Warning: Matrix is nearly singular.");
var n = RowCount;
if (b.VectorLength() != n) throw new ArgumentException("Parameter must vector of the same height as matrix.");
for (var j = 1; j <= n - 1; j++)
{
b[j] /= this[j, j];
for (var i = 1; i <= n - j; i++) b[j + i] -= b[j]*this[j + i, j];
}
b[n] /= this[n, n];
}
/// <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(ArrayMatrix b)
{
if (!IsUpperTriangular())
throw new InvalidOperationException(
"Cannot perform backward insertion for matrix not being upper triangular.");
if ( /*this.Determinant*/DiagProd() == 0) throw new InvalidOperationException("Warning: Matrix is nearly singular.");
var n = RowCount;
if (b.VectorLength() != n) throw new ArgumentException("Parameter must vector of the same height as matrix.");
for (var j = n; j >= 2; j--)
{
b[j] /= this[j, j];
for (var i = 1; i <= j - 1; i++) b[i] -= b[j]*this[i, j];
}
b[1] /= this[1, 1];
}
/// <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 ArrayMatrix TriDiag(ArrayMatrix l, ArrayMatrix d, ArrayMatrix u)
{
var sizeL = l.VectorLength();
var sizeD = d.VectorLength();
var 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);
}