/// <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 ComplexDoubleMatrix Solve(IROComplexDoubleMatrix Y)
{
ComplexDoubleMatrix X;
Complex Inner;
Complex[] 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 ComplexDoubleMatrix(m_Order, Y.Rows);
x = new Complex[m_Order];
for (l = 0; l < Y.Rows; l++)
{
// get right-side column
y = ComplexDoubleVector.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);
}
///<summary>Constructor for QR decomposition class. The constructor performs the factorization and the upper and
///lower matrices are accessible by the <c>Q</c> and <c>R</c> properties.</summary>
///<param name="matrix">The matrix to factor.</param>
///<exception cref="ArgumentNullException">matrix is null.</exception>
public ComplexDoubleQRDecomp(IROComplexDoubleMatrix matrix)
{
if (matrix == null)
{
throw new System.ArgumentNullException("matrix cannot be null.");
}
this.matrix = matrix;
}
///<summary>Solves a system on linear equations, AX=B, where A is the factored matrixed.</summary>
///<param name="B">RHS side of the system.</param>
///<returns>the solution matrix, X.</returns>
///<exception cref="ArgumentNullException">B is null.</exception>
///<exception cref="SingularMatrixException">Ais singular.</exception>
///<exception cref="ArgumentException">The number of rows of A and B must be the same.</exception>
public ComplexDoubleMatrix Solve(IROComplexDoubleMatrix B)
{
if (B == null)
{
throw new System.ArgumentNullException("B cannot be null.");
}
Compute();
if (singular)
{
throw new SingularMatrixException();
}
else
{
if (B.Rows != order)
{
throw new System.ArgumentException("Matrix row dimensions must agree.");
}
#if MANAGED
// Copy right hand side with pivoting
int nx = B.Columns;
ComplexDoubleMatrix X = Pivot(B);
// Solve L*Y = B(piv,:)
for (int k = 0; k < order; k++)
{
for (int i = k + 1; i < order; i++)
{
for (int j = 0; j < nx; j++)
{
X.data[i][j] -= X.data[k][j] * factor[i][k];
}
}
}
// Solve U*X = Y;
for (int k = order - 1; k >= 0; k--)
{
for (int j = 0; j < nx; j++)
{
X.data[k][j] /= factor[k][k];
}
for (int i = 0; i < k; i++)
{
for (int j = 0; j < nx; j++)
{
X.data[i][j] -= X.data[k][j] * factor[i][k];
}
}
}
return(X);
#else
Complex[] rhs = ComplexDoubleMatrix.ToLinearComplexArray(B);
Lapack.Getrs.Compute(Lapack.Transpose.NoTrans, order, B.Columns, factor, order, pivots, rhs, B.Rows);
ComplexDoubleMatrix ret = new ComplexDoubleMatrix(order, B.Columns);
ret.data = rhs;
return(ret);
#endif
}
}
///<summary>Constructor for SVD decomposition class.</summary>
///<param name="matrix">The matrix to decompose.</param>
///<param name="computeVectors">Whether to compute the singular vectors or not.</param>
///<exception cref="ArgumentNullException">matrix is null.</exception>
public ComplexDoubleSVDDecomp(IROComplexDoubleMatrix matrix, bool computeVectors)
{
if (matrix == null)
{
throw new System.ArgumentNullException("matrix cannot be null.");
}
this.matrix = new ComplexDoubleMatrix(matrix);
this.computeVectors = computeVectors;
}
///<summary>Solves a system on linear equations, AX=B, where A is the factored matrixed.</summary>
///<param name="B">RHS side of the system.</param>
///<returns>the solution matrix, X.</returns>
///<exception cref="ArgumentNullException">B is null.</exception>
///<exception cref="NotPositiveDefiniteException">A is not positive definite.</exception>
///<exception cref="ArgumentException">The number of rows of A and B must be the same.</exception>
public ComplexDoubleMatrix Solve(IROComplexDoubleMatrix B)
{
if (B == null)
{
throw new System.ArgumentNullException("B cannot be null.");
}
Compute();
if (!ispd)
{
throw new NotPositiveDefiniteException();
}
else
{
if (B.Rows != order)
{
throw new System.ArgumentException("Matrix row dimensions must agree.");
}
#if MANAGED
// Copy right hand side.
int cols = B.Columns;
var X = new ComplexDoubleMatrix(B);
for (int c = 0; c < cols; c++)
{
// Solve L*Y = B;
for (int i = 0; i < order; i++)
{
Complex sum = B[i, c];
for (int k = i - 1; k >= 0; k--)
{
sum -= l.data[i][k] * X.data[k][c];
}
X.data[i][c] = sum / l.data[i][i];
}
// Solve L'*X = Y;
for (int i = order - 1; i >= 0; i--)
{
Complex sum = X.data[i][c];
for (int k = i + 1; k < order; k++)
{
sum -= ComplexMath.Conjugate(l.data[k][i]) * X.data[k][c];
}
X.data[i][c] = sum / ComplexMath.Conjugate(l.data[i][i]);
}
}
return(X);
#else
Complex[] rhs = ComplexDoubleMatrix.ToLinearComplexArray(B);
Lapack.Potrs.Compute(Lapack.UpLo.Lower, order, B.Columns, l.data, order, rhs, B.Rows);
ComplexDoubleMatrix ret = new ComplexDoubleMatrix(order, B.Columns);
ret.data = rhs;
return(ret);
#endif
}
}
/// <summary>
/// Returns the column of a <see cref="IROComplexDoubleMatrix" /> as a new <c>Complex[]</c> array.
/// </summary>
/// <param name="mat">The matrix to copy the column from.</param>
/// <param name="col">Number of column to copy from the matrix.</param>
/// <returns>A new array of <c>Complex</c> with the same elements as the column of the given matrix.</returns>
public static Complex[] GetColumnAsArray(IROComplexDoubleMatrix mat, int col)
{
Complex[] result = new Complex[mat.Rows];
for (int i = 0; i < result.Length; ++i)
{
result[i] = mat[i, col];
}
return(result);
}
/// <summary>
/// Returns the row of a <see cref="IROComplexDoubleMatrix" /> as a new <c>ComplexDoubleVector.</c>
/// </summary>
/// <param name="mat">The matrix to copy the column from.</param>
/// <param name="row">Number of row to copy from the matrix.</param>
/// <returns>A new <c>ComplexDoubleVector</c> with the same elements as the row of the given matrix.</returns>
public static ComplexDoubleVector GetRow(IROComplexDoubleMatrix mat, int row)
{
ComplexDoubleVector result = new ComplexDoubleVector(mat.Columns);
for (int i = 0; i < result.data.Length; ++i)
{
result.data[i] = mat[row, i];
}
return(result);
}
/// <summary>
/// Returns the column of a <see cref="IROComplexDoubleMatrix" /> as a new <c>ComplexDoubleVector.</c>
/// </summary>
/// <param name="mat">The matrix to copy the column from.</param>
/// <param name="col">Number of column to copy from the matrix.</param>
/// <returns>A new <c>ComplexDoubleVector</c> with the same elements as the column of the given matrix.</returns>
public static ComplexDoubleVector GetColumn(IROComplexDoubleMatrix mat, int col)
{
ComplexDoubleVector result = new ComplexDoubleVector(mat.Rows);
for (int i = 0; i < result.data.Length; ++i)
{
result.data[i] = mat[i, col];
}
return(result);
}
///<summary>Constructor for Cholesky decomposition class. The constructor performs the factorization of a Hermitian positive
///definite matrax and the Cholesky factored matrix is accessible by the <c>Factor</c> property. The factor is the lower
///triangular factor.</summary>
///<param name="matrix">The matrix to factor.</param>
///<exception cref="ArgumentNullException">matrix is null.</exception>
///<exception cref="NotSquareMatrixException">matrix is not square.</exception>
///<remarks>This class only uses the lower triangle of the input matrix. It ignores the
///upper triangle.</remarks>
public ComplexDoubleCholeskyDecomp(IROComplexDoubleMatrix matrix)
{
if (matrix == null)
{
throw new System.ArgumentNullException("matrix cannot be null.");
}
if (matrix.Rows != matrix.Columns)
{
throw new NotSquareMatrixException("Matrix must be square.");
}
order = matrix.Columns;
this.matrix = new ComplexDoubleMatrix(matrix);
}
///<summary>Constructor for Cholesky decomposition class. The constructor performs the factorization of a Hermitian positive
///definite matrax and the Cholesky factored matrix is accessible by the <c>Factor</c> property. The factor is the lower
///triangular factor.</summary>
///<param name="matrix">The matrix to factor.</param>
///<exception cref="ArgumentNullException">matrix is null.</exception>
///<exception cref="NotSquareMatrixException">matrix is not square.</exception>
///<remarks>This class only uses the lower triangle of the input matrix. It ignores the
///upper triangle.</remarks>
public ComplexDoubleCholeskyDecomp(IROComplexDoubleMatrix matrix)
{
if ( matrix == null )
{
throw new System.ArgumentNullException("matrix cannot be null.");
}
if ( matrix.Rows != matrix.Columns )
{
throw new NotSquareMatrixException("Matrix must be square.");
}
order = matrix.Columns;
this.matrix = new ComplexDoubleMatrix(matrix);
}
private ComplexDoubleMatrix Pivot(IROComplexDoubleMatrix B)
{
int m = B.Rows;
int n = B.Columns;
ComplexDoubleMatrix ret = new ComplexDoubleMatrix(m, n);
for (int i = 0; i < pivots.Length; i++)
{
for (int j = 0; j < n; j++)
{
ret.data[i][j] = B[pivots[i], j];
}
}
return(ret);
}
///<summary>Solves a system on linear equations, AX=B, where A is the factored matrixed.</summary>
///<param name="B">RHS side of the system.</param>
///<returns>the solution matrix, X.</returns>
///<exception cref="ArgumentNullException">B is null.</exception>
///<exception cref="NotPositiveDefiniteException">A is not positive definite.</exception>
///<exception cref="ArgumentException">The number of rows of A and B must be the same.</exception>
public ComplexDoubleMatrix Solve (IROComplexDoubleMatrix B)
{
if ( B == null )
{
throw new System.ArgumentNullException("B cannot be null.");
}
Compute();
if ( !ispd )
{
throw new NotPositiveDefiniteException();
}
else
{
if ( B.Rows != order )
{
throw new System.ArgumentException("Matrix row dimensions must agree." );
}
#if MANAGED
// Copy right hand side.
int cols = B.Columns;
ComplexDoubleMatrix X = new ComplexDoubleMatrix(B);
for (int c = 0; c < cols; c++ )
{
// Solve L*Y = B;
for (int i = 0; i < order; i++)
{
Complex sum = B[i,c];
for (int k = i-1; k >= 0; k--)
{
sum -= l.data[i][k] * X.data[k][c];
}
X.data[i][c] = sum / l.data[i][i];
}
// Solve L'*X = Y;
for (int i =order-1; i >= 0; i--)
{
Complex sum = X.data[i][c];
for (int k = i+1; k < order; k++)
{
sum -= ComplexMath.Conjugate(l.data[k][i]) * X.data[k][c];
}
X.data[i][c] = sum / ComplexMath.Conjugate(l.data[i][i]);
}
}
return X;
#else
Complex[] rhs = ComplexDoubleMatrix.ToLinearComplexArray(B);
Lapack.Potrs.Compute(Lapack.UpLo.Lower,order,B.Columns,l.data,order,rhs,B.Rows);
ComplexDoubleMatrix ret = new ComplexDoubleMatrix(order,B.Columns);
ret.data = rhs;
return ret;
#endif
}
}
public static string MatrixToString(string name, IROComplexDoubleMatrix a)
{
if (null == name)
name = "";
if (a.Rows == 0 || a.Columns == 0)
return string.Format("EmptyMatrix {0}({1},{2})", name, a.Rows, a.Columns);
System.Text.StringBuilder s = new System.Text.StringBuilder();
s.Append("Matrix " + name + ":");
for (int i = 0; i < a.Rows; i++)
{
s.Append("\n(");
for (int j = 0; j < a.Columns; j++)
{
s.Append(a[i, j].ToString());
if (j + 1 < a.Columns)
s.Append(";");
else
s.Append(")");
}
}
return s.ToString();
}
/// <summary>
/// Returns the column of a <see cref="IROComplexDoubleMatrix" /> as a new <c>Complex[]</c> array.
/// </summary>
/// <param name="mat">The matrix to copy the column from.</param>
/// <param name="col">Number of column to copy from the matrix.</param>
/// <returns>A new array of <c>Complex</c> with the same elements as the column of the given matrix.</returns>
public static Complex[] GetColumnAsArray(IROComplexDoubleMatrix mat, int col)
{
Complex[] result = new Complex[mat.Rows];
for(int i=0;i<result.Length;++i)
result[i] = mat[i,col];
return result;
}
///<summary>Constructor for SVD decomposition class.</summary>
///<param name="matrix">The matrix to decompose.</param>
///<exception cref="ArgumentNullException">matrix is null.</exception>
public ComplexDoubleSVDDecomp(IROComplexDoubleMatrix matrix)
{
if ( matrix == null )
{
throw new System.ArgumentNullException("matrix cannot be null.");
}
this.matrix = new ComplexDoubleMatrix(matrix);
}
/// <summary>
/// Returns the row of a <see cref="IROComplexDoubleMatrix" /> as a new <c>ComplexDoubleVector.</c>
/// </summary>
/// <param name="mat">The matrix to copy the column from.</param>
/// <param name="row">Number of row to copy from the matrix.</param>
/// <returns>A new <c>ComplexDoubleVector</c> with the same elements as the row of the given matrix.</returns>
public static ComplexDoubleVector GetRow(IROComplexDoubleMatrix mat, int row)
{
ComplexDoubleVector result = new ComplexDoubleVector(mat.Columns);
for(int i=0;i<result.data.Length;++i)
result.data[i] = mat[row,i];
return result;
}
/// <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 ComplexDoubleMatrix Solve(IROComplexDoubleMatrix Y)
{
ComplexDoubleMatrix X;
Complex Inner;
Complex[] 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 ComplexDoubleMatrix(m_Order, Y.Rows);
x = new Complex[m_Order];
for (l = 0; l < Y.Rows; l++)
{
// get right-side column
y = ComplexDoubleVector.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;
}
/// <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 ComplexDoubleMatrix Solve(IROComplexDoubleVector col, IROComplexDoubleVector row, IROComplexDoubleMatrix 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] == Complex.Zero)
{
throw new SingularMatrixException("One of the leading sub-matrices is singular.");
}
// decompose matrix
int order = col.Length;
Complex[] A = new Complex[order];
Complex[] B = new Complex[order];
Complex[] Z = new Complex[order];
ComplexDoubleMatrix X = new ComplexDoubleMatrix(order);
Complex Q, S, Ke, Kr, e;
Complex Inner;
int i, j, l;
// setup the zero order solution
A[0] = Complex.One;
B[0] = Complex.One;
e = Complex.One / col[0];
X.SetRow(0, e * ComplexDoubleVector.GetRow(Y, 0));
for (i = 1; i < order; i++)
{
// calculate inner products
Q = Complex.Zero;
for (j = 0, l = 1; j < i; j++, l++)
{
Q += col[l] * A[j];
}
S = Complex.Zero;
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] = Complex.Zero;
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] = Complex.Zero;
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 == Complex.One)
{
throw new SingularMatrixException("One of the leading sub-matrices is singular.");
}
// update diagonal
e = e / (Complex.One - Ke * Kr);
for (l = 0; l < Y.Rows; l++)
{
ComplexDoubleVector W = X.GetColumn(l);
ComplexDoubleVector M = ComplexDoubleVector.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;
}
/// <summary>
/// Solve a symmetric square Toeplitz system with a right-side matrix.
/// </summary>
/// <param name="T">The left-most column of the Toeplitz matrix.</param>
/// <param name="Y">The right-side matrix of the system.</param>
/// <returns>The solution matrix.</returns>
/// <exception cref="ArgumentNullException">
/// <B>T</B> and/or <B>Y</B> are null references
/// </exception>
/// <exception cref="RankException">
/// The length of <B>T</B> does not match the number of rows in <B>Y</B>.
/// </exception>
/// <exception cref="SingularMatrixException">
/// The Toeplitz matrix or one of the the leading sub-matrices is singular.
/// </exception>
/// <remarks>
/// This method solves the linear system <B>AX</B> = <B>Y</B>. Where
/// <B>T</B> is a symmetric square Toeplitz matrix, <B>X</B> is an unknown
/// matrix and <B>Y</B> is a known matrix.
/// <para>
/// This static member combines the <b>UDL</b> decomposition and the calculation of the solution into a
/// single algorithm. When compared to the non-static member it requires minimal data storage
/// and suffers from no speed penalty.
/// </para>
/// </remarks>
public static ComplexDoubleMatrix Solve(IROComplexDoubleVector T, IROComplexDoubleMatrix Y)
{
ComplexDoubleMatrix X;
// check parameters
if (T == null)
{
throw new System.ArgumentNullException("T");
}
else if (Y == null)
{
throw new System.ArgumentNullException("Y");
}
else if (T.Length != Y.Columns)
{
throw new RankException("The length of T and Y are not equal.");
}
else
{
// allocate memory
int N = T.Length;
int M = Y.Rows;
X = new ComplexDoubleMatrix(N, M); // solution matrix
ComplexDoubleVector Z = new ComplexDoubleVector(N); // temporary storage vector
Complex e; // prediction error
int i, j, l, m;
// setup zero order solution
e = T[0];
if (e == Complex.Zero)
{
throw new SingularMatrixException("The Toeplitz matrix or one of the the leading sub-matrices is singular.");
}
for (m = 0; m < M; m++)
{
X[0, m] = Y[0, m] / T[0];
}
if (N > 1)
{
ComplexDoubleVector a = new ComplexDoubleVector(N - 1); // prediction coefficients
Complex p; // reflection coefficient
Complex inner; // inner product
Complex k;
// calculate solution for successive orders
for (i = 1; i < N; i++)
{
// calculate first inner product
inner = T[i];
for (j = 0, l = i - 1; j < i - 1; j++, l--)
{
inner += a[j] * T[l];
}
// update predictor coefficients
p = -(inner / e);
for (j = 0, l = i - 2; j < i - 1; j++, l--)
{
Z[j] = a[j] + p * a[l];
}
// copy vector
for (j = 0; j < i - 1; j++)
{
a[j] = Z[j];
}
a[i - 1] = p;
e *= (Complex.One - p * p);
if (e == Complex.Zero)
{
throw new SingularMatrixException("The Toeplitz matrix or one of the the leading sub-matrices is singular.");
}
// update the solution matrix
for (m = 0; m < M; m++)
{
// retrieve a copy of solution column
for (j = 0; j < i; j++)
{
Z[j] = X[j, m];
}
// calculate second inner product
inner = Y[i, m];
for (j = 0, l = i; j < i; j++, l--)
{
inner -= Z[j] * T[l];
}
// update solution vector
k = inner / e;
for (j = 0, l = i - 1; j < i; j++, l--)
{
Z[j] = Z[j] + k * a[l];
}
Z[j] = k;
// store solution column in matrix
for (j = 0; j <= i; j++)
{
X[j, m] = Z[j];
}
}
}
}
}
return(X);
}
/// <summary>
/// Returns the column of a <see cref="IROComplexDoubleMatrix" /> as a new <c>ComplexDoubleVector.</c>
/// </summary>
/// <param name="mat">The matrix to copy the column from.</param>
/// <param name="col">Number of column to copy from the matrix.</param>
/// <returns>A new <c>ComplexDoubleVector</c> with the same elements as the column of the given matrix.</returns>
public static ComplexDoubleVector GetColumn(IROComplexDoubleMatrix mat, int col)
{
ComplexDoubleVector result = new ComplexDoubleVector(mat.Rows);
for(int i=0;i<result.data.Length;++i)
result.data[i] = mat[i,col];
return result;
}
/// <summary>
/// Solve a symmetric square Toeplitz system with a right-side matrix.
/// </summary>
/// <param name="T">The left-most column of the Toeplitz matrix.</param>
/// <param name="Y">The right-side matrix of the system.</param>
/// <returns>The solution matrix.</returns>
/// <exception cref="ArgumentNullException">
/// <B>T</B> and/or <B>Y</B> are null references
/// </exception>
/// <exception cref="RankException">
/// The length of <B>T</B> does not match the number of rows in <B>Y</B>.
/// </exception>
/// <exception cref="SingularMatrixException">
/// The Toeplitz matrix or one of the the leading sub-matrices is singular.
/// </exception>
/// <remarks>
/// This method solves the linear system <B>AX</B> = <B>Y</B>. Where
/// <B>T</B> is a symmetric square Toeplitz matrix, <B>X</B> is an unknown
/// matrix and <B>Y</B> is a known matrix.
/// <para>
/// This static member combines the <b>UDL</b> decomposition and the calculation of the solution into a
/// single algorithm. When compared to the non-static member it requires minimal data storage
/// and suffers from no speed penalty.
/// </para>
/// </remarks>
public static ComplexDoubleMatrix Solve(IROComplexDoubleVector T, IROComplexDoubleMatrix Y)
{
ComplexDoubleMatrix X;
// check parameters
if (T == null)
{
throw new System.ArgumentNullException("T");
}
else if (Y == null)
{
throw new System.ArgumentNullException("Y");
}
else if (T.Length != Y.Columns)
{
throw new RankException("The length of T and Y are not equal.");
}
else
{
// allocate memory
int N = T.Length;
int M = Y.Rows;
X = new ComplexDoubleMatrix(N, M); // solution matrix
ComplexDoubleVector Z = new ComplexDoubleVector(N); // temporary storage vector
Complex e; // prediction error
int i, j, l, m;
// setup zero order solution
e = T[0];
if (e == Complex.Zero)
{
throw new SingularMatrixException("The Toeplitz matrix or one of the the leading sub-matrices is singular.");
}
for (m = 0; m < M; m++)
{
X[0, m] = Y[0,m] / T[0];
}
if (N > 1)
{
ComplexDoubleVector a = new ComplexDoubleVector(N - 1); // prediction coefficients
Complex p; // reflection coefficient
Complex inner; // inner product
Complex k;
// calculate solution for successive orders
for (i = 1; i < N; i++)
{
// calculate first inner product
inner = T[i];
for (j = 0, l = i - 1; j < i - 1; j++, l--)
{
inner += a[j] * T[l];
}
// update predictor coefficients
p = -(inner / e);
for (j = 0, l = i - 2; j < i - 1; j++, l--)
{
Z[j] = a[j] + p * a[l];
}
// copy vector
for (j = 0; j < i - 1; j++)
{
a[j] = Z[j];
}
a[i - 1] = p;
e *= (Complex.One - p * p);
if (e == Complex.Zero)
{
throw new SingularMatrixException("The Toeplitz matrix or one of the the leading sub-matrices is singular.");
}
// update the solution matrix
for (m = 0; m < M; m++)
{
// retrieve a copy of solution column
for (j = 0; j < i; j++)
{
Z[j] = X[j, m];
}
// calculate second inner product
inner = Y[i, m];
for (j = 0, l = i; j < i; j++, l--)
{
inner -= Z[j] * T[l];
}
// update solution vector
k = inner / e;
for (j = 0, l = i - 1; j < i; j++, l--)
{
Z[j] = Z[j] + k * a[l];
}
Z[j] = k;
// store solution column in matrix
for (j = 0; j <= i; j++)
{
X[j, m] = Z[j];
}
}
}
}
}
return X;
}
/// <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 ComplexDoubleMatrix Solve(IROComplexDoubleVector col, IROComplexDoubleVector row, IROComplexDoubleMatrix 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] == Complex.Zero)
{
throw new SingularMatrixException("One of the leading sub-matrices is singular.");
}
// decompose matrix
int order = col.Length;
Complex[] A = new Complex[order];
Complex[] B = new Complex[order];
Complex[] Z = new Complex[order];
ComplexDoubleMatrix X = new ComplexDoubleMatrix(order);
Complex Q, S, Ke, Kr, e;
Complex Inner;
int i, j, l;
// setup the zero order solution
A[0] = Complex.One;
B[0] = Complex.One;
e = Complex.One / col[0];
X.SetRow(0, e * ComplexDoubleVector.GetRow(Y, 0));
for (i = 1; i < order; i++)
{
// calculate inner products
Q = Complex.Zero;
for (j = 0, l = 1; j < i; j++, l++)
{
Q += col[l] * A[j];
}
S = Complex.Zero;
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] = Complex.Zero;
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] = Complex.Zero;
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 == Complex.One)
{
throw new SingularMatrixException("One of the leading sub-matrices is singular.");
}
// update diagonal
e = e / (Complex.One - Ke * Kr);
for (l = 0; l < Y.Rows; l++)
{
ComplexDoubleVector W = X.GetColumn(l);
ComplexDoubleVector M = ComplexDoubleVector.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);
}
/// <summary>
/// Solve a symmetric square Toeplitz system with a right-side matrix.
/// </summary>
/// <param name="Y">The right-hand side of the system.</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 symmetric 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, and then calculates the solution matrix.
/// </para>
/// </remarks>
public ComplexDoubleMatrix Solve(IROComplexDoubleMatrix Y)
{
ComplexDoubleMatrix X;
// 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("The Toeplitz matrix or one of the the leading sub-matrices is singular.");
}
int M = Y.Rows;
int i, j, l, m; // index/loop variables
Complex[] Inner; // inner product
Complex[] G; // scaling constant
Complex[] A; // reference to current order coefficients
Complex scalar;
// allocate memory for solution
X = new ComplexDoubleMatrix(m_Order, M);
Inner = new Complex[M];
G = new Complex[M];
// setup zero order solution
scalar = Complex.One / m_LeftColumn[0];
for (m = 0; m < M; m++)
{
#if MANAGED
X.data[0][m] = scalar * Y[0, m];
#else
X.data[m * m_Order] = scalar * Y[0, m];
#endif
}
// solve systems of increasing order
for (i = 1; i < m_Order; i++)
{
// calculate inner product
for (m = 0; m < M; m++)
{
#if MANAGED
Inner[m] = Y[i, m];
#else
Inner[m] = Y[i, m];
#endif
}
for (j = 0, l = i; j < i; j++, l--)
{
scalar = m_LeftColumn[l];
for (m = 0; m < M; m++)
{
#if MANAGED
Inner[m] -= scalar * X.data[j][m];
#else
Inner[m] -= scalar * X.data[m * m_Order + j];
#endif
}
}
// get the current predictor coefficients row
A = m_LowerTriangle[i];
// update the solution matrix
for (m = 0; m < M; m++)
{
G[m] = m_Diagonal[i] * Inner[m];
}
for (j = 0; j <= i; j++)
{
scalar = A[j];
for (m = 0; m < M; m++)
{
#if MANAGED
X.data[j][m] += scalar * G[m];
#else
X.data[m * m_Order + j] += scalar * G[m];
#endif
}
}
}
return(X);
}
/// <summary>
/// Solve a symmetric square Toeplitz system with a right-side matrix.
/// </summary>
/// <param name="Y">The right-hand side of the system.</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 symmetric 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, and then calculates the solution matrix.
/// </para>
/// </remarks>
public ComplexDoubleMatrix Solve(IROComplexDoubleMatrix Y)
{
ComplexDoubleMatrix X;
// 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("The Toeplitz matrix or one of the the leading sub-matrices is singular.");
}
int M = Y.Rows;
int i, j, l, m; // index/loop variables
Complex[] Inner; // inner product
Complex[] G; // scaling constant
Complex[] A; // reference to current order coefficients
Complex scalar;
// allocate memory for solution
X = new ComplexDoubleMatrix(m_Order, M);
Inner = new Complex[M];
G = new Complex[M];
// setup zero order solution
scalar = Complex.One / m_LeftColumn[0];
for (m = 0; m < M; m++)
{
#if MANAGED
X.data[0][m] = scalar * Y[0,m];
#else
X.data[m*m_Order] = scalar * Y[0,m];
#endif
}
// solve systems of increasing order
for (i = 1; i < m_Order; i++)
{
// calculate inner product
for (m = 0; m < M; m++)
{
#if MANAGED
Inner[m] = Y[i,m];
#else
Inner[m] = Y[i,m];
#endif
}
for (j = 0, l = i; j < i; j++, l--)
{
scalar = m_LeftColumn[l];
for (m = 0; m < M; m++)
{
#if MANAGED
Inner[m] -= scalar * X.data[j][m];
#else
Inner[m] -= scalar * X.data[m*m_Order+j];
#endif
}
}
// get the current predictor coefficients row
A = m_LowerTriangle[i];
// update the solution matrix
for (m = 0; m < M; m++)
{
G[m] = m_Diagonal[i] * Inner[m];
}
for (j = 0; j <= i; j++)
{
scalar = A[j];
for (m = 0; m < M; m++)
{
#if MANAGED
X.data[j][m] += scalar * G[m];
#else
X.data[m*m_Order+j] += scalar * G[m];
#endif
}
}
}
return X;
}
///<summary>Constructor for QR decomposition class. The constructor performs the factorization and the upper and
///lower matrices are accessible by the <c>Q</c> and <c>R</c> properties.</summary>
///<param name="matrix">The matrix to factor.</param>
///<exception cref="ArgumentNullException">matrix is null.</exception>
public ComplexDoubleQRDecomp(IROComplexDoubleMatrix matrix)
{
if (matrix == null)
throw new System.ArgumentNullException("matrix cannot be null.");
this.matrix = matrix;
}
