/// <summary>
/// Creates a new householder decomposition.
/// </summary>
/// <param name="A">The matrix to decompose.</param>
public HouseholderDecomposition(MatrixValue A)
: base(A)
{
QR = A.GetComplexMatrix();
Rdiag = new ScalarValue[n];
// Main loop.
for (int k = 0; k < n; k++)
{
var nrm = 0.0;
for (int i = k; i < m; i++)
{
nrm = Helpers.Hypot(nrm, QR[i][k].Re);
}
if (nrm != 0.0)
{
// Form k-th Householder vector.
if (QR[k][k].Re < 0)
{
nrm = -nrm;
}
for (int i = k; i < m; i++)
{
QR[i][k] /= nrm;
}
QR[k][k] += ScalarValue.One;
// Apply transformation to remaining columns.
for (int j = k + 1; j < n; j++)
{
var s = ScalarValue.Zero;
for (int i = k; i < m; i++)
{
s += QR[i][k] * QR[i][j];
}
s = (-s) / QR[k][k];
for (int i = k; i < m; i++)
{
QR[i][j] += s * QR[i][k];
}
}
}
else
{
FullRank = false;
}
Rdiag[k] = new ScalarValue(-nrm);
}
}
/// <summary>
/// Least squares solution of A * X = B
/// </summary>
/// <param name="b">A Matrix with as many rows as A and any number of columns.</param>
/// <returns>X that minimizes the two norm of Q*R*X-B.</returns>
public override MatrixValue Solve(MatrixValue b)
{
if (b.DimensionY != m)
{
throw new YAMPDifferentDimensionsException(m, 1, b.DimensionY, 1);
}
if (!this.FullRank)
{
throw new YAMPMatrixFormatException(SpecialMatrixFormat.NonSingular.ToString());
}
// Copy right hand side
var nx = b.DimensionX;
var X = b.GetComplexMatrix();
// Compute Y = transpose(Q)*B
for (var k = 0; k < n; k++)
{
for (var j = 0; j < nx; j++)
{
var s = ScalarValue.Zero;
for (var i = k; i < m; i++)
{
s += QR[i][k] * X[i][j];
}
s = (-s) / QR[k][k];
for (var i = k; i < m; i++)
{
X[i][j] += s * QR[i][k];
}
}
}
// Solve R * X = Y;
for (var k = n - 1; k >= 0; k--)
{
for (var j = 0; j < nx; j++)
{
X[k][j] /= Rdiag[k];
}
for (var i = 0; i < k; i++)
{
for (var j = 0; j < nx; j++)
{
X[i][j] -= X[k][j] * QR[i][k];
}
}
}
return(new MatrixValue(X, n, nx).GetSubMatrix(0, n, 0, nx));
}
/// <summary>Solve A*X = B</summary>
/// <param name="B"> A Matrix with as many rows as A and any number of columns.
/// </param>
/// <returns> X so that L*L'*X = B
/// </returns>
/// <exception cref="System.ArgumentException"> Matrix row dimensions must agree.
/// </exception>
/// <exception cref="System.SystemException"> Matrix is not symmetric positive definite.
/// </exception>
public override MatrixValue Solve(MatrixValue B)
{
if (B.DimensionY != n)
{
throw new YAMPDifferentDimensionsException(n, 1, B.DimensionY, 1);
}
if (!isspd)
{
throw new YAMPMatrixFormatException(SpecialMatrixFormat.SymmetricPositiveDefinite);
}
// Copy right hand side.
var X = B.GetComplexMatrix();
int nx = B.DimensionX;
// Solve L*Y = B;
for (int k = 0; k < n; k++)
{
for (int i = k + 1; i < n; i++)
{
for (int j = 0; j < nx; j++)
{
X[i][j] -= X[k][j] * L[i][k];
}
}
for (int j = 0; j < nx; j++)
{
X[k][j] /= L[k][k];
}
}
// Solve L'*X = Y;
for (int k = n - 1; k >= 0; k--)
{
for (int j = 0; j < nx; j++)
{
X[k][j] /= L[k][k];
}
for (int i = 0; i < k; i++)
{
for (int j = 0; j < nx; j++)
{
X[i][j] -= X[k][j] * L[k][i];
}
}
}
return(new MatrixValue(X, n, nx));
}
/// <summary>
/// Cholesky algorithm for symmetric and positive definite matrix.
/// </summary>
/// <param name="Arg">Square, symmetric matrix.</param>
/// <returns>Structure to access L and isspd flag.</returns>
public CholeskyDecomposition(MatrixValue Arg)
{
// Initialize.
var A = Arg.GetComplexMatrix();
n = Arg.DimensionY;
L = new ScalarValue[n][];
for (int i = 0; i < n; i++)
{
L[i] = new ScalarValue[n];
}
isspd = Arg.DimensionX == n;
// Main loop.
for (int i = 0; i < n; i++)
{
var Lrowi = L[i];
var d = ScalarValue.Zero;
for (int j = 0; j < i; j++)
{
var Lrowj = L[j];
var s = new ScalarValue();
for (int k = 0; k < j; k++)
{
s += Lrowi[k] * Lrowj[k].Conjugate();
}
s = (A[i][j] - s) / L[j][j];
Lrowi[j] = s;
d += s * s.Conjugate();
isspd = isspd && (A[j][i] == A[i][j]);
}
d = A[i][i] - d;
isspd = isspd & (d.Abs() > 0.0);
L[i][i] = d.Sqrt();
for (int k = i + 1; k < n; k++)
{
L[i][k] = ScalarValue.Zero;
}
}
}
/// <summary>
/// Creates a new householder decomposition.
/// </summary>
/// <param name="A">The matrix to decompose.</param>
public HouseholderDecomposition(MatrixValue A)
: base(A)
{
QR = A.GetComplexMatrix();
Rdiag = new ScalarValue[n];
// Main loop.
for (int k = 0; k < n; k++)
{
var nrm = 0.0;
for (int i = k; i < m; i++)
nrm = Helpers.Hypot(nrm, QR[i][k].Re);
if (nrm != 0.0)
{
// Form k-th Householder vector.
if (QR[k][k].Re < 0)
nrm = -nrm;
for (int i = k; i < m; i++)
QR[i][k] /= nrm;
QR[k][k] += ScalarValue.One;
// Apply transformation to remaining columns.
for (int j = k + 1; j < n; j++)
{
var s = ScalarValue.Zero;
for (int i = k; i < m; i++)
s += QR[i][k] * QR[i][j];
s = (-s) / QR[k][k];
for (int i = k; i < m; i++)
QR[i][j] += s * QR[i][k];
}
}
else
FullRank = false;
Rdiag[k] = new ScalarValue(-nrm);
}
}
/// <summary>
/// LU Decomposition
/// </summary>
/// <param name="A">Rectangular matrix</param>
/// <returns>Structure to access L, U and piv.</returns>
public LUDecomposition(MatrixValue A)
{
// Use a "left-looking", dot-product, Crout / Doolittle algorithm.
LU = A.GetComplexMatrix();
m = A.DimensionY;
n = A.DimensionX;
piv = new int[m];
for (int i = 0; i < m; i++)
{
piv[i] = i;
}
pivsign = 1;
var LUrowi = new ScalarValue[0];
var LUcolj = new ScalarValue[m];
// Outer loop.
for (int j = 0; j < n; j++)
{
// Make a copy of the j-th column to localize references.
for (int i = 0; i < m; i++)
{
LUcolj[i] = LU[i][j];
}
// Apply previous transformations.
for (int i = 0; i < m; i++)
{
LUrowi = LU[i];
// Most of the time is spent in the following dot product.
var kmax = Math.Min(i, j);
var s = ScalarValue.Zero;
for (int k = 0; k < kmax; k++)
{
s += LUrowi[k] * LUcolj[k];
}
LUrowi[j] = LUcolj[i] -= s;
}
// Find pivot and exchange if necessary.
var p = j;
for (int i = j + 1; i < m; i++)
{
if (LUcolj[i].Abs() > LUcolj[p].Abs())
{
p = i;
}
}
if (p != j)
{
for (int k = 0; k < n; k++)
{
var t = LU[p][k];
LU[p][k] = LU[j][k];
LU[j][k] = t;
}
var k2 = piv[p];
piv[p] = piv[j];
piv[j] = k2;
pivsign = -pivsign;
}
// Compute multipliers.
if (j < m & LU[j][j] != 0.0)
{
for (int i = j + 1; i < m; i++)
{
LU[i][j] = LU[i][j] / LU[j][j];
}
}
}
}
/// <summary>
/// LU Decomposition
/// </summary>
/// <param name="A">Rectangular matrix</param>
/// <returns>Structure to access L, U and piv.</returns>
public LUDecomposition(MatrixValue A)
{
// Use a "left-looking", dot-product, Crout / Doolittle algorithm.
LU = A.GetComplexMatrix();
m = A.DimensionY;
n = A.DimensionX;
piv = new int[m];
for (int i = 0; i < m; i++)
piv[i] = i;
pivsign = 1;
var LUrowi = new ScalarValue[0];
var LUcolj = new ScalarValue[m];
// Outer loop.
for (int j = 0; j < n; j++)
{
// Make a copy of the j-th column to localize references.
for (int i = 0; i < m; i++)
LUcolj[i] = LU[i][j];
// Apply previous transformations.
for (int i = 0; i < m; i++)
{
LUrowi = LU[i];
// Most of the time is spent in the following dot product.
var kmax = Math.Min(i, j);
var s = ScalarValue.Zero;
for (int k = 0; k < kmax; k++)
s += LUrowi[k] * LUcolj[k];
LUrowi[j] = LUcolj[i] -= s;
}
// Find pivot and exchange if necessary.
var p = j;
for (int i = j + 1; i < m; i++)
{
if (LUcolj[i].Abs() > LUcolj[p].Abs())
p = i;
}
if (p != j)
{
for (int k = 0; k < n; k++)
{
var t = LU[p][k];
LU[p][k] = LU[j][k];
LU[j][k] = t;
}
var k2 = piv[p];
piv[p] = piv[j];
piv[j] = k2;
pivsign = -pivsign;
}
// Compute multipliers.
if (j < m & LU[j][j] != 0.0)
{
for (int i = j + 1; i < m; i++)
LU[i][j] = LU[i][j] / LU[j][j];
}
}
}
/// <summary>
/// Least squares solution of A * X = B
/// </summary>
/// <param name="b">A Matrix with as many rows as A and any number of columns.</param>
/// <returns>X that minimizes the two norm of Q*R*X-B.</returns>
public override MatrixValue Solve(MatrixValue b)
{
if (b.DimensionY != m)
throw new YAMPDifferentDimensionsException(m, 1, b.DimensionY, 1);
if (!this.FullRank)
throw new YAMPMatrixFormatException(SpecialMatrixFormat.NonSingular.ToString());
// Copy right hand side
var nx = b.DimensionX;
var X = b.GetComplexMatrix();
// Compute Y = transpose(Q)*B
for (var k = 0; k < n; k++)
{
for (var j = 0; j < nx; j++)
{
var s = ScalarValue.Zero;
for (var i = k; i < m; i++)
{
s += QR[i][k] * X[i][j];
}
s = (-s) / QR[k][k];
for (var i = k; i < m; i++)
{
X[i][j] += s * QR[i][k];
}
}
}
// Solve R * X = Y;
for (var k = n - 1; k >= 0; k--)
{
for (var j = 0; j < nx; j++)
{
X[k][j] /= Rdiag[k];
}
for (var i = 0; i < k; i++)
{
for (var j = 0; j < nx; j++)
{
X[i][j] -= X[k][j] * QR[i][k];
}
}
}
return new MatrixValue(X, n, nx).GetSubMatrix(0, n, 0, nx);
}
