/// <summary>
/// Solve a square Toeplitz system with a right-side matrix.
/// </summary>
/// <param name="Y">The right-side matrix</param>
/// <returns>The solution matrix.</returns>
/// <exception cref="ArgumentNullException">
/// Parameter <B>Y</B> is a null reference.
/// </exception>
/// <exception cref="RankException">
/// The number of rows in <B>Y</B> is not equal to the number of rows in the Toeplitz matrix.
/// </exception>
/// <exception cref="SingularMatrixException">
/// The Toeplitz matrix or one of the the leading sub-matrices is singular.
/// </exception>
/// <remarks>
/// This member solves the linear system <B>TX</B> = <B>Y</B>, where <B>T</B> is
/// a square Toeplitz matrix, <B>X</B> is the unknown solution matrix
/// and <B>Y</B> is a known matrix.
/// <para>
/// The class implicitly decomposes the inverse Toeplitz matrix into a <b>UDL</b> factorisation
/// using the Levinson algorithm, before calculating the solution vector.
/// </para>
/// </remarks>
public FloatMatrix Solve(IROFloatMatrix Y)
{
FloatMatrix X;
float Inner;
float[] a, b, x, y;
int i, j, l;
// check parameters
if (Y == null)
{
throw new System.ArgumentNullException("Y");
}
else if (m_Order != Y.Columns)
{
throw new RankException("The numer of rows in Y is not equal to the number of rows in the Toeplitz matrix.");
}
Compute();
if (m_IsSingular == true)
{
throw new SingularMatrixException("One of the leading sub-matrices is singular.");
}
// allocate memory for solution
X = new FloatMatrix(m_Order, Y.Rows);
x = new float[m_Order];
for (l = 0; l < Y.Rows; l++)
{
// get right-side column
y = FloatVector.GetColumnAsArray(Y,l);
// solve left-side column
for (i = 0; i < m_Order; i++)
{
a = m_LowerTriangle[i];
b = m_UpperTriangle[i];
Inner = y[i];
for (j = 0; j < i; j++)
{
Inner += a[j] * y[j];
}
Inner *= m_Diagonal[i];
x[i] = Inner;
for (j = 0; j < i; j++)
{
x[j] += Inner * b[j];
}
}
// insert left-side column into the matrix
X.SetColumn(l, x);
}
return X;
}
public void SetColumnArrayWrongRank()
{
FloatMatrix a = new FloatMatrix(2, 2);
float[] b = { 1, 2, 3 };
a.SetColumn(1, b);
}
/// <summary>
/// Solve a square Toeplitz system with a right-side matrix.
/// </summary>
/// <param name="col">The left-most column of the Toeplitz matrix.</param>
/// <param name="row">The top-most row of the Toeplitz matrix.</param>
/// <param name="Y">The right-side matrix of the system.</param>
/// <returns>The solution matrix.</returns>
/// <exception cref="ArgumentNullException">
/// <EM>col</EM> is a null reference,
/// <para>or</para>
/// <para><EM>row</EM> is a null reference,</para>
/// <para>or</para>
/// <para><EM>Y</EM> is a null reference.</para>
/// </exception>
/// <exception cref="RankException">
/// The length of <EM>col</EM> is 0,
/// <para>or</para>
/// <para>the lengths of <EM>col</EM> and <EM>row</EM> are not equal,</para>
/// <para>or</para>
/// <para>the number of rows in <EM>Y</EM> does not the length of <EM>col</EM> and <EM>row</EM>.</para>
/// </exception>
/// <exception cref="SingularMatrixException">
/// The Toeplitz matrix or one of the the leading sub-matrices is singular.
/// </exception>
/// <exception cref="ArithmeticException">
/// The values of the first element of <EM>col</EM> and <EM>row</EM> are not equal.
/// </exception>
/// <remarks>
/// This method solves the linear system <B>AX</B> = <B>Y</B>. Where
/// <B>T</B> is a square Toeplitz matrix, <B>X</B> is an unknown
/// matrix and <B>Y</B> is a known matrix.
/// <para>
/// The classic Levinson algorithm is used to solve the system. The algorithm
/// assumes that all the leading sub-matrices of the Toeplitz matrix are
/// non-singular. When a sub-matrix is near singular, accuracy will
/// be degraded. This member requires approximately <B>N</B> squared
/// FLOPS to calculate a solution, where <B>N</B> is the matrix order.
/// </para>
/// <para>
/// This static method has minimal storage requirements as it combines
/// the <b>UDL</b> decomposition with the calculation of the solution vector
/// in a single algorithm.
/// </para>
/// </remarks>
public static FloatMatrix Solve(IROFloatVector col, IROFloatVector row, IROFloatMatrix Y)
{
// check parameters
if (col == null)
{
throw new System.ArgumentNullException("col");
}
else if (col.Length == 0)
{
throw new RankException("The length of col is zero.");
}
else if (row == null)
{
throw new System.ArgumentNullException("row");
}
else if (col.Length != row.Length)
{
throw new RankException("The lengths of col and row are not equal.");
}
else if (col[0] != row[0])
{
throw new ArithmeticException("The values of the first element of col and row are not equal.");
}
else if (Y == null)
{
throw new System.ArgumentNullException("Y");
}
else if (col.Length != Y.Columns)
{
throw new RankException("The numer of rows in Y does not match the length of col and row.");
}
// check if leading diagonal is zero
if (col[0] == 0.0f)
{
throw new SingularMatrixException("One of the leading sub-matrices is singular.");
}
// decompose matrix
int order = col.Length;
float[] A = new float[order];
float[] B = new float[order];
float[] Z = new float[order];
FloatMatrix X = new FloatMatrix(order);
float Q, S, Ke, Kr, e;
float Inner;
int i, j, l;
// setup the zero order solution
A[0] = 1.0f;
B[0] = 1.0f;
e = 1.0f / col[0];
X.SetRow(0, e * FloatVector.GetRow(Y,0));
for (i = 1; i < order; i++)
{
// calculate inner products
Q = 0.0f;
for ( j = 0, l = 1; j < i; j++, l++)
{
Q += col[l] * A[j];
}
S = 0.0f;
for ( j = 0, l = 1; j < i; j++, l++)
{
S += row[l] * B[j];
}
// reflection coefficients
Kr = -S * e;
Ke = -Q * e;
// update lower triangle (in temporary storage)
Z[0] = 0.0f;
Array.Copy(A, 0, Z, 1, i);
for (j = 0, l = i - 1; j < i; j++, l--)
{
Z[j] += Ke * B[l];
}
// update upper triangle
for (j = i; j > 0; j--)
{
B[j] = B[j-1];
}
B[0] = 0.0f;
for (j = 0, l = i - 1; j < i; j++, l--)
{
B[j] += Kr * A[l];
}
// copy from temporary storage to lower triangle
Array.Copy(Z, 0, A, 0, i + 1);
// check for singular sub-matrix)
if (Ke * Kr == 1.0f)
{
throw new SingularMatrixException("One of the leading sub-matrices is singular.");
}
// update diagonal
e = e / (1.0f - Ke * Kr);
for (l = 0; l < Y.Rows; l++)
{
FloatVector W = X.GetColumn(l);
FloatVector M = FloatVector.GetColumn(Y,l);
Inner = M[i];
for (j = 0; j < i; j++)
{
Inner += A[j] * M[j];
}
Inner *= e;
W[i] = Inner;
for (j = 0; j < i; j++)
{
W[j] += Inner * B[j];
}
X.SetColumn(l, W);
}
}
return X;
}
public void SetColumnArrayOutOfRange()
{
FloatMatrix a = new FloatMatrix(2, 2);
float[] b = { 1, 2 };
a.SetColumn(2, b);
}
public void SetColumnArray()
{
FloatMatrix a = new FloatMatrix(2, 2);
float[] b = { 1, 2 };
a.SetColumn(0, b);
Assert.AreEqual(b[0], a[0, 0]);
Assert.AreEqual(b[1], a[1, 0]);
}
public void SetColumnWrongRank()
{
FloatMatrix a = new FloatMatrix(2, 2);
FloatVector b = new FloatVector(3);
a.SetColumn(1, b);
}
public void SetColumnOutOfRange()
{
FloatMatrix a = new FloatMatrix(2, 2);
FloatVector b = new FloatVector(2);
a.SetColumn(2, b);
}
public void SetColumn()
{
FloatMatrix a = new FloatMatrix(2, 2);
FloatVector b = new FloatVector(2);
b[0] = 1;
b[1] = 2;
a.SetColumn(0, b);
Assert.AreEqual(b[0], a[0, 0]);
Assert.AreEqual(b[1], a[1, 0]);
}
