/// <summary>
/// Creates the eigenvalue decomposition of the given matrix.
/// </summary>
/// <param name="matrixA">The square matrix A.</param>
/// <exception cref="ArgumentNullException">
/// <paramref name="matrixA"/> is <see langword="null"/>.
/// </exception>
/// <exception cref="ArgumentException">
/// <paramref name="matrixA"/> is non-square (rectangular).
/// </exception>
public EigenvalueDecompositionF(MatrixF matrixA)
{
if (matrixA == null)
{
throw new ArgumentNullException("matrixA");
}
if (matrixA.IsSquare == false)
{
throw new ArgumentException("The matrix A must be square.", "matrixA");
}
_n = matrixA.NumberOfColumns;
_d = new VectorF(_n);
_e = new VectorF(_n);
_isSymmetric = matrixA.IsSymmetric;
if (_isSymmetric)
{
_v = matrixA.Clone();
// Tridiagonalize.
ReduceToTridiagonal();
// Diagonalize.
TridiagonalToQL();
}
else
{
_v = new MatrixF(_n, _n);
// Abort if A contains NaN values.
// If we continue with NaN values, we run into an infinite loop.
for (int i = 0; i < _n; i++)
{
for (int j = 0; j < _n; j++)
{
if (Numeric.IsNaN(matrixA[i, j]))
{
_e.Set(float.NaN);
_v.Set(float.NaN);
_d.Set(float.NaN);
return;
}
}
}
// Storage of nonsymmetric Hessenberg form.
MatrixF matrixH = matrixA.Clone();
// Working storage for nonsymmetric algorithm.
float[] ort = new float[_n];
// Reduce to Hessenberg form.
ReduceToHessenberg(matrixH, ort);
// Reduce Hessenberg to real Schur form.
HessenbergToRealSchur(matrixH);
}
}
/// <summary>
/// Creates the QR decomposition of the given matrix.
/// </summary>
/// <param name="matrixA">
/// The matrix A. (Can be rectangular. NumberOfRows must be ≥ NumberOfColumns.)
/// </param>
/// <remarks>
/// The QR decomposition is computed by Householder reflections.
/// </remarks>
/// <exception cref="ArgumentNullException">
/// <paramref name="matrixA"/> is <see langword="null"/>.
/// </exception>
/// <exception cref="ArgumentException">
/// The number of rows must be greater than or equal to the number of columns.
/// </exception>
public QRDecompositionF(MatrixF matrixA)
{
if (matrixA == null)
{
throw new ArgumentNullException("matrixA");
}
if (matrixA.NumberOfRows < matrixA.NumberOfColumns)
{
throw new ArgumentException("The number of rows must be greater than or equal to the number of columns.", "matrixA");
}
// Initialize.
_qr = matrixA.Clone();
_m = matrixA.NumberOfRows;
_n = matrixA.NumberOfColumns;
_rDiagonal = new float[_n];
// Main loop.
for (int k = 0; k < _n; k++)
{
// Compute 2-norm of k-th column without under/overflow.
float norm = 0;
for (int i = k; i < _m; i++)
{
norm = MathHelper.Hypotenuse(norm, _qr[i, k]);
}
if (norm != 0) // TODO: Maybe a comparison with an epsilon tolerance is required here!?
{
// Form k-th Householder vector.
if (_qr[k, k] < 0)
{
norm = -norm;
}
for (int i = k; i < _m; i++)
{
_qr[i, k] /= norm;
}
_qr[k, k] += 1;
// Apply transformation to remaining columns.
for (int j = k + 1; j < _n; j++)
{
float s = 0;
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];
}
}
}
_rDiagonal[k] = -norm;
}
}
//--------------------------------------------------------------
/// <summary>
/// Creates the eigenvalue decomposition of the given matrix.
/// </summary>
/// <param name="matrixA">The square matrix A.</param>
/// <exception cref="ArgumentNullException">
/// <paramref name="matrixA"/> is <see langword="null"/>.
/// </exception>
/// <exception cref="ArgumentException">
/// <paramref name="matrixA"/> is non-square (rectangular).
/// </exception>
public EigenvalueDecompositionF(MatrixF matrixA)
{
if (matrixA == null)
throw new ArgumentNullException("matrixA");
if (matrixA.IsSquare == false)
throw new ArgumentException("The matrix A must be square.", "matrixA");
_n = matrixA.NumberOfColumns;
_d = new VectorF(_n);
_e = new VectorF(_n);
_isSymmetric = matrixA.IsSymmetric;
if (_isSymmetric)
{
_v = matrixA.Clone();
// Tridiagonalize.
ReduceToTridiagonal();
// Diagonalize.
TridiagonalToQL();
}
else
{
_v = new MatrixF(_n, _n);
// Abort if A contains NaN values.
// If we continue with NaN values, we run into an infinite loop.
for (int i = 0; i < _n; i++)
{
for (int j = 0; j < _n; j++)
{
if (Numeric.IsNaN(matrixA[i, j]))
{
_e.Set(float.NaN);
_v.Set(float.NaN);
_d.Set(float.NaN);
return;
}
}
}
// Storage of nonsymmetric Hessenberg form.
MatrixF matrixH = matrixA.Clone();
// Working storage for nonsymmetric algorithm.
float[] ort = new float[_n];
// Reduce to Hessenberg form.
ReduceToHessenberg(matrixH, ort);
// Reduce Hessenberg to real Schur form.
HessenbergToRealSchur(matrixH);
}
}
/// <summary>
/// Returns the least squares solution for the equation <c>A * X = B</c>.
/// </summary>
/// <param name="matrixB">The matrix B with as many rows as A and any number of columns.</param>
/// <returns>X with the least squares solution.</returns>
/// <exception cref="ArgumentNullException">
/// <paramref name="matrixB"/> is <see langword="null"/>.
/// </exception>
/// <exception cref="ArgumentException">
/// The number of rows does not match.
/// </exception>
/// <exception cref="MathematicsException">
/// The matrix A does not have full rank.
/// </exception>
public MatrixF SolveLinearEquations(MatrixF matrixB)
{
if (matrixB == null)
{
throw new ArgumentNullException("matrixB");
}
if (matrixB.NumberOfRows != _m)
{
throw new ArgumentException("The number of rows does not match.", "matrixB");
}
if (HasNumericallyFullRank == false)
{
throw new MathematicsException("The matrix does not have full rank.");
}
// Copy right hand side
MatrixF x = matrixB.Clone();
// Compute Y = transpose(Q)*B
for (int k = 0; k < _n; k++)
{
for (int j = 0; j < matrixB.NumberOfColumns; j++)
{
float s = 0;
for (int i = k; i < _m; i++)
{
s += _qr[i, k] * x[i, j];
}
s = -s / _qr[k, k];
for (int i = k; i < _m; i++)
{
x[i, j] += s * _qr[i, k];
}
}
}
// Solve R*X = Y.
for (int k = _n - 1; k >= 0; k--)
{
for (int j = 0; j < matrixB.NumberOfColumns; j++)
{
x[k, j] /= _rDiagonal[k];
}
for (int i = 0; i < k; i++)
{
for (int j = 0; j < matrixB.NumberOfColumns; j++)
{
x[i, j] -= x[k, j] * _qr[i, k];
}
}
}
return(x.GetSubmatrix(0, _n - 1, 0, matrixB.NumberOfColumns - 1));
}
/// <summary>
/// Solves the equation <c>A * X = B</c>.
/// </summary>
/// <param name="matrixB">The matrix B with as many rows as A and any number of columns.</param>
/// <returns>X, so that <c>A * X = B</c>.</returns>
/// <exception cref="ArgumentNullException">
/// <paramref name="matrixB"/> is <see langword="null"/>.
/// </exception>
/// <exception cref="ArgumentException">
/// The number of rows does not match.
/// </exception>
/// <exception cref="MathematicsException">
/// The matrix A is not symmetric and positive definite.
/// </exception>
public MatrixF SolveLinearEquations(MatrixF matrixB)
{
if (matrixB == null)
{
throw new ArgumentNullException("matrixB");
}
if (matrixB.NumberOfRows != L.NumberOfRows)
{
throw new ArgumentException("The number of rows does not match.", "matrixB");
}
if (IsSymmetricPositiveDefinite == false)
{
throw new MathematicsException("The original matrix A is not symmetric and positive definite.");
}
// Initialize x as a copy of B.
MatrixF x = matrixB.Clone();
// Solve L*Y = B.
for (int k = 0; k < L.NumberOfRows; k++)
{
for (int j = 0; j < matrixB.NumberOfColumns; j++)
{
for (int i = 0; i < k; i++)
{
x[k, j] -= x[i, j] * L[k, i];
}
x[k, j] /= L[k, k];
}
}
// Solve transpose(L) * X = Y.
for (int k = L.NumberOfRows - 1; k >= 0; k--)
{
for (int j = 0; j < matrixB.NumberOfColumns; j++)
{
for (int i = k + 1; i < L.NumberOfRows; i++)
{
x[k, j] -= x[i, j] * L[i, k];
}
x[k, j] /= L[k, k];
}
}
return(x);
}
//--------------------------------------------------------------
/// <summary>
/// Creates the QR decomposition of the given matrix.
/// </summary>
/// <param name="matrixA">
/// The matrix A. (Can be rectangular. NumberOfRows must be ≥ NumberOfColumns.)
/// </param>
/// <remarks>
/// The QR decomposition is computed by Householder reflections.
/// </remarks>
/// <exception cref="ArgumentNullException">
/// <paramref name="matrixA"/> is <see langword="null"/>.
/// </exception>
/// <exception cref="ArgumentException">
/// The number of rows must be greater than or equal to the number of columns.
/// </exception>
public QRDecompositionF(MatrixF matrixA)
{
if (matrixA == null)
throw new ArgumentNullException("matrixA");
if (matrixA.NumberOfRows < matrixA.NumberOfColumns)
throw new ArgumentException("The number of rows must be greater than or equal to the number of columns.", "matrixA");
// Initialize.
_qr = matrixA.Clone();
_m = matrixA.NumberOfRows;
_n = matrixA.NumberOfColumns;
_rDiagonal = new float[_n];
// Main loop.
for (int k = 0; k < _n; k++)
{
// Compute 2-norm of k-th column without under/overflow.
float norm = 0;
for (int i = k; i < _m; i++)
norm = MathHelper.Hypotenuse(norm, _qr[i, k]);
if (norm != 0) // TODO: Maybe a comparison with an epsilon tolerance is required here!?
{
// Form k-th Householder vector.
if (_qr[k, k] < 0)
norm = -norm;
for (int i = k; i < _m; i++)
_qr[i, k] /= norm;
_qr[k, k] += 1;
// Apply transformation to remaining columns.
for (int j = k + 1; j < _n; j++)
{
float s = 0;
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];
}
}
_rDiagonal[k] = -norm;
}
}
public void Absolute()
{
float[] values = new float[] { -1.0f, -2.0f, -3.0f,
-4.0f, -5.0f, -6.0f,
-7.0f, -8.0f, -9.0f };
MatrixF m = new MatrixF(3, 3, values, MatrixOrder.RowMajor);
MatrixF absolute = m.Clone();
absolute.Absolute();
for (int i = 0; i < absolute.NumberOfRows; i++)
for (int j = 0; j < absolute.NumberOfColumns; j++)
Assert.AreEqual(i * absolute.NumberOfColumns + j + 1, absolute[i, j]);
absolute = MatrixF.Absolute(m);
for (int i = 0; i < absolute.NumberOfRows; i++)
for (int j = 0; j < absolute.NumberOfColumns; j++)
Assert.AreEqual(i * absolute.NumberOfColumns + j + 1, absolute[i, j]);
values = new float[] { 1.0f, 2.0f, 3.0f,
4.0f, 5.0f, 6.0f,
7.0f, 8.0f, 9.0f };
m = new MatrixF(3, 3, values, MatrixOrder.RowMajor);
absolute = m.Clone();
absolute.Absolute();
for (int i = 0; i < absolute.NumberOfRows; i++)
for (int j = 0; j < absolute.NumberOfColumns; j++)
Assert.AreEqual(i * absolute.NumberOfColumns + j + 1, absolute[i, j]);
absolute = MatrixF.Absolute(m);
for (int i = 0; i < absolute.NumberOfRows; i++)
for (int j = 0; j < absolute.NumberOfColumns; j++)
Assert.AreEqual(i * absolute.NumberOfColumns + j + 1, absolute[i, j]);
Assert.IsNull(MatrixF.Absolute(null));
}
//--------------------------------------------------------------
/// <summary>
/// Returns the least squares solution for the equation <c>A * X = B</c>.
/// </summary>
/// <param name="matrixB">The matrix B with as many rows as A and any number of columns.</param>
/// <returns>X with the least squares solution.</returns>
/// <exception cref="ArgumentNullException">
/// <paramref name="matrixB"/> is <see langword="null"/>.
/// </exception>
/// <exception cref="ArgumentException">
/// The number of rows does not match.
/// </exception>
/// <exception cref="MathematicsException">
/// The matrix A does not have full rank.
/// </exception>
public MatrixF SolveLinearEquations(MatrixF matrixB)
{
if (matrixB == null)
throw new ArgumentNullException("matrixB");
if (matrixB.NumberOfRows != _m)
throw new ArgumentException("The number of rows does not match.", "matrixB");
if (HasNumericallyFullRank == false)
throw new MathematicsException("The matrix does not have full rank.");
// Copy right hand side
MatrixF x = matrixB.Clone();
// Compute Y = transpose(Q)*B
for (int k = 0; k < _n; k++)
{
for (int j = 0; j < matrixB.NumberOfColumns; j++)
{
float s = 0;
for (int i = k; i < _m; i++)
s += _qr[i, k] * x[i, j];
s = -s / _qr[k, k];
for (int i = k; i < _m; i++)
x[i, j] += s * _qr[i, k];
}
}
// Solve R*X = Y.
for (int k = _n - 1; k >= 0; k--)
{
for (int j = 0; j < matrixB.NumberOfColumns; j++)
x[k, j] /= _rDiagonal[k];
for (int i = 0; i < k; i++)
for (int j = 0; j < matrixB.NumberOfColumns; j++)
x[i, j] -= x[k, j] * _qr[i, k];
}
return x.GetSubmatrix(0, _n - 1, 0, matrixB.NumberOfColumns - 1);
}
//--------------------------------------------------------------
/// <summary>
/// Solves the equation <c>A * X = B</c>.
/// </summary>
/// <param name="matrixB">The matrix B with as many rows as A and any number of columns.</param>
/// <returns>X, so that <c>A * X = B</c>.</returns>
/// <exception cref="ArgumentNullException">
/// <paramref name="matrixB"/> is <see langword="null"/>.
/// </exception>
/// <exception cref="ArgumentException">
/// The number of rows does not match.
/// </exception>
/// <exception cref="MathematicsException">
/// The matrix A is not symmetric and positive definite.
/// </exception>
public MatrixF SolveLinearEquations(MatrixF matrixB)
{
if (matrixB == null)
throw new ArgumentNullException("matrixB");
if (matrixB.NumberOfRows != L.NumberOfRows)
throw new ArgumentException("The number of rows does not match.", "matrixB");
if (IsSymmetricPositiveDefinite == false)
throw new MathematicsException("The original matrix A is not symmetric and positive definite.");
// Initialize x as a copy of B.
MatrixF x = matrixB.Clone();
// Solve L*Y = B.
for (int k = 0; k < L.NumberOfRows; k++)
{
for (int j = 0; j < matrixB.NumberOfColumns; j++)
{
for (int i = 0; i < k; i++)
x[k, j] -= x[i, j] * L[k, i];
x[k, j] /= L[k, k];
}
}
// Solve transpose(L) * X = Y.
for (int k = L.NumberOfRows - 1; k >= 0; k--)
{
for (int j = 0; j < matrixB.NumberOfColumns; j++)
{
for (int i = k + 1; i < L.NumberOfRows; i++)
x[k, j] -= x[i, j] * L[i, k];
x[k, j] /= L[k, k];
}
}
return x;
}
public void Invert()
{
Assert.AreEqual(MatrixF.CreateIdentity(3, 3), MatrixF.CreateIdentity(3, 3).Inverse);
MatrixF m = new MatrixF(new float[,] {{1, 2, 3, 4},
{2, 5, 8, 3},
{7, 6, -1, 1},
{4, 9, 7, 7}});
MatrixF inverse = m.Clone();
m.Invert();
VectorF v = new VectorF(4, 1);
VectorF w = m * v;
Assert.IsTrue(VectorF.AreNumericallyEqual(v, inverse * w));
Assert.IsTrue(MatrixF.AreNumericallyEqual(MatrixF.CreateIdentity(4, 4), m * inverse));
m = new MatrixF(new float[,] {{1, 2, 3},
{2, 5, 8},
{7, 6, -1},
{4, 9, 7}});
// To check the pseudo-inverse we use the definition: A*A.Transposed*A = A
// see http://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse
inverse = m.Clone();
inverse.Invert();
Assert.IsTrue(MatrixF.AreNumericallyEqual(m, m * inverse * m));
}
public void Clone()
{
MatrixF m = new MatrixF(3, 4, rowMajor, MatrixOrder.RowMajor);
var o = m.Clone();
Assert.AreEqual(m, o);
}
public void TryInvert()
{
// Regular, square
MatrixF m = new MatrixF(new float[,] {{1, 2, 3, 4},
{2, 5, 8, 3},
{7, 6, -1, 1},
{4, 9, 7, 7}});
MatrixF inverse = m.Clone();
Assert.AreEqual(true, m.TryInvert());
Assert.IsTrue(MatrixF.AreNumericallyEqual(MatrixF.CreateIdentity(4, 4), m * inverse));
// Full column rank, rectangular
m = new MatrixF(new float[,] {{1, 2, 3},
{2, 5, 8},
{7, 6, -1},
{4, 9, 7}});
inverse = m.Clone();
Assert.AreEqual(true, m.TryInvert());
Assert.IsTrue(MatrixF.AreNumericallyEqual(m, m * inverse * m));
// singular
m = new MatrixF(new float[,] {{1, 2, 3},
{2, 5, 8},
{3, 7, 11}});
inverse = m.Clone();
Assert.AreEqual(false, m.TryInvert());
}
public SingularValueDecompositionF(MatrixF matrixA)
{
if (matrixA == null)
throw new ArgumentNullException("matrixA");
// Derived from LINPACK code.
// Initialize.
_m = matrixA.NumberOfRows;
_n = matrixA.NumberOfColumns;
MatrixF matrixAClone = matrixA.Clone();
if (_m < _n)
throw new ArgumentException("The number of rows must be greater than or equal to the number of columns.", "matrixA");
int nu = Math.Min(_m, _n);
_s = new VectorF(Math.Min(_m + 1, _n));
_u = new MatrixF(_m, nu); //Jama getU() returns new Matrix(U,_m,Math.min(_m+1,_n)) ?!
_v = new MatrixF(_n, _n);
float[] e = new float[_n];
float[] work = new float[_m];
// Abort if A contains NaN values.
// If we continue with NaN values, we run into an infinite loop.
for (int i = 0; i < _m; i++)
{
for (int j = 0; j < _n; j++)
{
if (Numeric.IsNaN(matrixA[i, j]))
{
_u.Set(float.NaN);
_v.Set(float.NaN);
_s.Set(float.NaN);
return;
}
}
}
// By default, we calculate U and V. To calculate only U or V we can set one of the following
// two constants to false. (This optimization is not yet tested.)
const bool wantu = true;
const bool wantv = true;
// Reduce A to bidiagonal form, storing the diagonal elements
// in s and the super-diagonal elements in e.
int nct = Math.Min(_m - 1, _n);
int nrt = Math.Max(0, Math.Min(_n - 2, _m));
for (int k = 0; k < Math.Max(nct, nrt); k++)
{
if (k < nct)
{
// Compute the transformation for the k-th column and
// place the k-th diagonal in s[k].
// Compute 2-norm of k-th column without under/overflow.
_s[k] = 0;
for (int i = k; i < _m; i++)
_s[k] = MathHelper.Hypotenuse(_s[k], matrixAClone[i, k]);
if (_s[k] != 0)
{
if (matrixAClone[k, k] < 0)
_s[k] = -_s[k];
for (int i = k; i < _m; i++)
matrixAClone[i, k] /= _s[k];
matrixAClone[k, k] += 1;
}
_s[k] = -_s[k];
}
for (int j = k + 1; j < _n; j++)
{
if ((k < nct) && (_s[k] != 0))
{
// Apply the transformation.
float t = 0;
for (int i = k; i < _m; i++)
t += matrixAClone[i, k] * matrixAClone[i, j];
t = -t / matrixAClone[k, k];
for (int i = k; i < _m; i++)
matrixAClone[i, j] += t * matrixAClone[i, k];
}
// Place the k-th row of A into e for the
// subsequent calculation of the row transformation.
e[j] = matrixAClone[k, j];
}
if (wantu & (k < nct))
{
// Place the transformation in U for subsequent back
// multiplication.
for (int i = k; i < _m; i++)
_u[i, k] = matrixAClone[i, k];
}
if (k < nrt)
{
// Compute the k-th row transformation and place the
// k-th super-diagonal in e[k].
// Compute 2-norm without under/overflow.
e[k] = 0;
for (int i = k + 1; i < _n; i++)
e[k] = MathHelper.Hypotenuse(e[k], e[i]);
if (e[k] != 0)
{
if (e[k + 1] < 0)
e[k] = -e[k];
for (int i = k + 1; i < _n; i++)
e[i] /= e[k];
e[k + 1] += 1;
}
e[k] = -e[k];
if ((k + 1 < _m) && (e[k] != 0))
{
// Apply the transformation.
for (int i = k + 1; i < _m; i++)
work[i] = 0;
for (int j = k + 1; j < _n; j++)
for (int i = k + 1; i < _m; i++)
work[i] += e[j] * matrixAClone[i, j];
for (int j = k + 1; j < _n; j++)
{
float t = -e[j] / e[k + 1];
for (int i = k + 1; i < _m; i++)
matrixAClone[i, j] += t * work[i];
}
}
if (wantv)
{
// Place the transformation in V for subsequent
// back multiplication.
for (int i = k + 1; i < _n; i++)
_v[i, k] = e[i];
}
}
}
// Set up the final bidiagonal matrix or order p.
int p = Math.Min(_n, _m + 1);
if (nct < _n)
_s[nct] = matrixAClone[nct, nct];
if (_m < p)
_s[p - 1] = 0;
if (nrt + 1 < p)
e[nrt] = matrixAClone[nrt, p - 1];
e[p - 1] = 0;
// If required, generate U.
if (wantu)
{
for (int j = nct; j < nu; j++)
{
for (int i = 0; i < _m; i++)
_u[i, j] = 0;
_u[j, j] = 1;
}
for (int k = nct - 1; k >= 0; k--)
{
if (_s[k] != 0)
{
for (int j = k + 1; j < nu; j++)
{
float t = 0;
for (int i = k; i < _m; i++)
t += _u[i, k] * _u[i, j];
t = -t / _u[k, k];
for (int i = k; i < _m; i++)
_u[i, j] += t * _u[i, k];
}
for (int i = k; i < _m; i++)
_u[i, k] = -_u[i, k];
_u[k, k] = 1 + _u[k, k];
for (int i = 0; i < k - 1; i++)
_u[i, k] = 0;
}
else
{
for (int i = 0; i < _m; i++)
_u[i, k] = 0;
_u[k, k] = 1;
}
}
}
// If required, generate V.
if (wantv)
{
for (int k = _n - 1; k >= 0; k--)
{
if ((k < nrt) & (e[k] != 0.0))
{
for (int j = k + 1; j < nu; j++)
{
float t = 0;
for (int i = k + 1; i < _n; i++)
t += _v[i, k] * _v[i, j];
t = -t / _v[k + 1, k];
for (int i = k + 1; i < _n; i++)
_v[i, j] += t * _v[i, k];
}
}
for (int i = 0; i < _n; i++)
_v[i, k] = 0;
_v[k, k] = 1;
}
}
// Main iteration loop for the singular values.
int pp = p - 1;
int iter = 0;
double eps = (float)Math.Pow(2, -23); // 2^-52 for double
double tiny = (float)Math.Pow(2, -100); // 2^-100 for float is an estimation by HelmutG
// Original: 2^-966 for double
while (p > 0)
{
int k, kase;
// Here is where a test for too many iterations would go.
// This section of the program inspects for
// negligible elements in the s and e arrays. On
// completion the variables kase and k are set as follows.
// kase = 1 if s(p) and e[k-1] are negligible and k<p
// kase = 2 if s(k) is negligible and k<p
// kase = 3 if e[k-1] is negligible, k<p, and
// s(k), ..., s(p) are not negligible (qr step).
// kase = 4 if e(p-1) is negligible (convergence).
for (k = p - 2; k >= -1; k--)
{
if (k == -1)
break;
if (Math.Abs(e[k]) <= tiny + eps * (Math.Abs(_s[k]) + Math.Abs(_s[k + 1])))
{
e[k] = 0;
break;
}
}
if (k == p - 2)
{
kase = 4;
}
else
{
int ks;
for (ks = p - 1; ks >= k; ks--)
{
if (ks == k)
break;
float t = (ks != p ? Math.Abs(e[ks]) : 0) + (ks != k + 1 ? Math.Abs(e[ks - 1]) : 0);
if (Math.Abs(_s[ks]) <= tiny + eps * t)
{
_s[ks] = 0;
break;
}
}
if (ks == k)
{
kase = 3;
}
else if (ks == p - 1)
{
kase = 1;
}
else
{
kase = 2;
k = ks;
}
}
k++;
// Perform the task indicated by kase.
switch (kase)
{
// Deflate negligible s(p).
case 1:
{
float f = e[p - 2];
e[p - 2] = 0;
for (int j = p - 2; j >= k; j--)
{
float t = MathHelper.Hypotenuse(_s[j], f);
float cs = _s[j] / t;
float sn = f / t;
_s[j] = t;
if (j != k)
{
f = -sn * e[j - 1];
e[j - 1] = cs * e[j - 1];
}
if (wantv)
{
for (int i = 0; i < _n; i++)
{
t = cs * _v[i, j] + sn * _v[i, p - 1];
_v[i, p - 1] = -sn * _v[i, j] + cs * _v[i, p - 1];
_v[i, j] = t;
}
}
}
}
break;
// Split at negligible s(k).
case 2:
{
float f = e[k - 1];
e[k - 1] = 0;
for (int j = k; j < p; j++)
{
float t = MathHelper.Hypotenuse(_s[j], f);
float cs = _s[j] / t;
float sn = f / t;
_s[j] = t;
f = -sn * e[j];
e[j] = cs * e[j];
if (wantu)
{
for (int i = 0; i < _m; i++)
{
t = cs * _u[i, j] + sn * _u[i, k - 1];
_u[i, k - 1] = -sn * _u[i, j] + cs * _u[i, k - 1];
_u[i, j] = t;
}
}
}
}
break;
// Perform one qr step.
case 3:
{
// Calculate the shift.
float scale = Math.Max(Math.Max(Math.Max(Math.Max(
Math.Abs(_s[p - 1]), Math.Abs(_s[p - 2])), Math.Abs(e[p - 2])),
Math.Abs(_s[k])), Math.Abs(e[k]));
float sp = _s[p - 1] / scale;
float spm1 = _s[p - 2] / scale;
float epm1 = e[p - 2] / scale;
float sk = _s[k] / scale;
float ek = e[k] / scale;
float b = ((spm1 + sp) * (spm1 - sp) + epm1 * epm1) / 2;
float c = (sp * epm1) * (sp * epm1);
float shift = 0;
if ((b != 0.0) | (c != 0.0))
{
shift = (float)Math.Sqrt(b * b + c);
if (b < 0.0)
shift = -shift;
shift = c / (b + shift);
}
float f = (sk + sp) * (sk - sp) + shift;
float g = sk * ek;
// Chase zeros.
for (int j = k; j < p - 1; j++)
{
float t = MathHelper.Hypotenuse(f, g);
float cs = f / t;
float sn = g / t;
if (j != k)
e[j - 1] = t;
f = cs * _s[j] + sn * e[j];
e[j] = cs * e[j] - sn * _s[j];
g = sn * _s[j + 1];
_s[j + 1] = cs * _s[j + 1];
if (wantv)
{
for (int i = 0; i < _n; i++)
{
t = cs * _v[i, j] + sn * _v[i, j + 1];
_v[i, j + 1] = -sn * _v[i, j] + cs * _v[i, j + 1];
_v[i, j] = t;
}
}
t = MathHelper.Hypotenuse(f, g);
cs = f / t;
sn = g / t;
_s[j] = t;
f = cs * e[j] + sn * _s[j + 1];
_s[j + 1] = -sn * e[j] + cs * _s[j + 1];
g = sn * e[j + 1];
e[j + 1] = cs * e[j + 1];
if (wantu && (j < _m - 1))
{
for (int i = 0; i < _m; i++)
{
t = cs * _u[i, j] + sn * _u[i, j + 1];
_u[i, j + 1] = -sn * _u[i, j] + cs * _u[i, j + 1];
_u[i, j] = t;
}
}
}
e[p - 2] = f;
iter = iter + 1;
}
break;
// Convergence.
case 4:
{
// Make the singular values positive.
if (_s[k] <= 0.0)
{
_s[k] = (_s[k] < 0.0 ? -_s[k] : 0);
if (wantv)
{
for (int i = 0; i <= pp; i++)
_v[i, k] = -_v[i, k];
}
}
// Order the singular values.
while (k < pp)
{
if (_s[k] >= _s[k + 1])
break;
float t = _s[k];
_s[k] = _s[k + 1];
_s[k + 1] = t;
if (wantv && (k < _n - 1))
{
for (int i = 0; i < _n; i++)
{
t = _v[i, k + 1];
_v[i, k + 1] = _v[i, k];
_v[i, k] = t;
}
}
if (wantu && (k < _m - 1))
{
for (int i = 0; i < _m; i++)
{
t = _u[i, k + 1];
_u[i, k + 1] = _u[i, k];
_u[i, k] = t;
}
}
k++;
}
iter = 0;
p--;
}
break;
}
}
}
//--------------------------------------------------------------
#region Creation and Cleanup
//--------------------------------------------------------------
/// <summary>
/// Creates the LU decomposition of the given matrix.
/// </summary>
/// <param name="matrixA">
/// The matrix A. (Can be rectangular. Number of rows ≥ number of columns.)
/// </param>
/// <exception cref="ArgumentNullException">
/// <paramref name="matrixA"/> is <see langword="null"/>.
/// </exception>
/// <exception cref="ArgumentException">
/// The number of rows must be greater than or equal to the number of columns.
/// </exception>
public LUDecompositionF(MatrixF matrixA)
{
if (matrixA == null)
{
throw new ArgumentNullException("matrixA");
}
if (matrixA.NumberOfColumns > matrixA.NumberOfRows)
{
throw new ArgumentException("The number of rows must be greater than or equal to the number of columns.", "matrixA");
}
// Use a "left-looking", dot-product, Crout/Doolittle algorithm.
_lu = matrixA.Clone();
_m = matrixA.NumberOfRows;
_n = matrixA.NumberOfColumns;
_pivotVector = new int[_m];
// Initialize with the 0 to m-1.
for (int i = 0; i < _m; i++)
{
_pivotVector[i] = i;
}
_pivotSign = 1;
// Outer loop.
for (int j = 0; j < _n; j++)
{
// Make a copy of the j-th column to localize references.
float[] luColumnJ = new float[_m];
for (int i = 0; i < _m; i++)
{
luColumnJ[i] = _lu[i, j];
}
// Apply previous transformations.
for (int i = 0; i < _m; i++)
{
// Most of the time is spent in the following dot product.
int kmax = Math.Min(i, j);
float s = 0;
for (int k = 0; k < kmax; k++)
{
s += _lu[i, k] * luColumnJ[k];
}
luColumnJ[i] -= s;
_lu[i, j] = luColumnJ[i];
}
// Find pivot and exchange if necessary.
int p = j;
for (int i = j + 1; i < _m; i++)
{
if (Math.Abs(luColumnJ[i]) > Math.Abs(luColumnJ[p]))
{
p = i;
}
}
// Swap lines p and k.
if (p != j)
{
for (int k = 0; k < _n; k++)
{
float dummy = _lu[p, k];
_lu[p, k] = _lu[j, k];
_lu[j, k] = dummy;
}
MathHelper.Swap(ref _pivotVector[p], ref _pivotVector[j]);
_pivotSign = -_pivotSign;
}
// Compute multipliers.
if (j < _m && _lu[j, j] != 0)
{
for (int i = j + 1; i < _m; i++)
{
_lu[i, j] /= _lu[j, j];
}
}
}
}
//--------------------------------------------------------------
/// <summary>
/// Creates the LU decomposition of the given matrix.
/// </summary>
/// <param name="matrixA">
/// The matrix A. (Can be rectangular. Number of rows ≥ number of columns.)
/// </param>
/// <exception cref="ArgumentNullException">
/// <paramref name="matrixA"/> is <see langword="null"/>.
/// </exception>
/// <exception cref="ArgumentException">
/// The number of rows must be greater than or equal to the number of columns.
/// </exception>
public LUDecompositionF(MatrixF matrixA)
{
if (matrixA == null)
throw new ArgumentNullException("matrixA");
if (matrixA.NumberOfColumns > matrixA.NumberOfRows)
throw new ArgumentException("The number of rows must be greater than or equal to the number of columns.", "matrixA");
// Use a "left-looking", dot-product, Crout/Doolittle algorithm.
_lu = matrixA.Clone();
_m = matrixA.NumberOfRows;
_n = matrixA.NumberOfColumns;
_pivotVector = new int[_m];
// Initialize with the 0 to m-1.
for (int i = 0; i < _m; i++)
_pivotVector[i] = i;
_pivotSign = 1;
// Outer loop.
for (int j = 0; j < _n; j++)
{
// Make a copy of the j-th column to localize references.
float[] luColumnJ = new float[_m];
for (int i = 0; i < _m; i++)
luColumnJ[i] = _lu[i, j];
// Apply previous transformations.
for (int i = 0; i < _m; i++)
{
// Most of the time is spent in the following dot product.
int kmax = Math.Min(i, j);
float s = 0;
for (int k = 0; k < kmax; k++)
s += _lu[i, k] * luColumnJ[k];
luColumnJ[i] -= s;
_lu[i, j] = luColumnJ[i];
}
// Find pivot and exchange if necessary.
int p = j;
for (int i = j + 1; i < _m; i++)
if (Math.Abs(luColumnJ[i]) > Math.Abs(luColumnJ[p]))
p = i;
// Swap lines p and k.
if (p != j)
{
for (int k = 0; k < _n; k++)
{
float dummy = _lu[p, k];
_lu[p, k] = _lu[j, k];
_lu[j, k] = dummy;
}
MathHelper.Swap(ref _pivotVector[p], ref _pivotVector[j]);
_pivotSign = -_pivotSign;
}
// Compute multipliers.
if (j < _m && _lu[j, j] != 0)
for (int i = j + 1; i < _m; i++)
_lu[i, j] /= _lu[j, j];
}
}
public SingularValueDecompositionF(MatrixF matrixA)
{
if (matrixA == null)
{
throw new ArgumentNullException("matrixA");
}
// Derived from LINPACK code.
// Initialize.
_m = matrixA.NumberOfRows;
_n = matrixA.NumberOfColumns;
MatrixF matrixAClone = matrixA.Clone();
if (_m < _n)
{
throw new ArgumentException("The number of rows must be greater than or equal to the number of columns.", "matrixA");
}
int nu = Math.Min(_m, _n);
_s = new VectorF(Math.Min(_m + 1, _n));
_u = new MatrixF(_m, nu); //Jama getU() returns new Matrix(U,_m,Math.min(_m+1,_n)) ?!
_v = new MatrixF(_n, _n);
float[] e = new float[_n];
float[] work = new float[_m];
// Abort if A contains NaN values.
// If we continue with NaN values, we run into an infinite loop.
for (int i = 0; i < _m; i++)
{
for (int j = 0; j < _n; j++)
{
if (Numeric.IsNaN(matrixA[i, j]))
{
_u.Set(float.NaN);
_v.Set(float.NaN);
_s.Set(float.NaN);
return;
}
}
}
// By default, we calculate U and V. To calculate only U or V we can set one of the following
// two constants to false. (This optimization is not yet tested.)
const bool wantu = true;
const bool wantv = true;
// Reduce A to bidiagonal form, storing the diagonal elements
// in s and the super-diagonal elements in e.
int nct = Math.Min(_m - 1, _n);
int nrt = Math.Max(0, Math.Min(_n - 2, _m));
for (int k = 0; k < Math.Max(nct, nrt); k++)
{
if (k < nct)
{
// Compute the transformation for the k-th column and
// place the k-th diagonal in s[k].
// Compute 2-norm of k-th column without under/overflow.
_s[k] = 0;
for (int i = k; i < _m; i++)
{
_s[k] = MathHelper.Hypotenuse(_s[k], matrixAClone[i, k]);
}
if (_s[k] != 0)
{
if (matrixAClone[k, k] < 0)
{
_s[k] = -_s[k];
}
for (int i = k; i < _m; i++)
{
matrixAClone[i, k] /= _s[k];
}
matrixAClone[k, k] += 1;
}
_s[k] = -_s[k];
}
for (int j = k + 1; j < _n; j++)
{
if ((k < nct) && (_s[k] != 0))
{
// Apply the transformation.
float t = 0;
for (int i = k; i < _m; i++)
{
t += matrixAClone[i, k] * matrixAClone[i, j];
}
t = -t / matrixAClone[k, k];
for (int i = k; i < _m; i++)
{
matrixAClone[i, j] += t * matrixAClone[i, k];
}
}
// Place the k-th row of A into e for the
// subsequent calculation of the row transformation.
e[j] = matrixAClone[k, j];
}
if (wantu & (k < nct))
{
// Place the transformation in U for subsequent back
// multiplication.
for (int i = k; i < _m; i++)
{
_u[i, k] = matrixAClone[i, k];
}
}
if (k < nrt)
{
// Compute the k-th row transformation and place the
// k-th super-diagonal in e[k].
// Compute 2-norm without under/overflow.
e[k] = 0;
for (int i = k + 1; i < _n; i++)
{
e[k] = MathHelper.Hypotenuse(e[k], e[i]);
}
if (e[k] != 0)
{
if (e[k + 1] < 0)
{
e[k] = -e[k];
}
for (int i = k + 1; i < _n; i++)
{
e[i] /= e[k];
}
e[k + 1] += 1;
}
e[k] = -e[k];
if ((k + 1 < _m) && (e[k] != 0))
{
// Apply the transformation.
for (int i = k + 1; i < _m; i++)
{
work[i] = 0;
}
for (int j = k + 1; j < _n; j++)
{
for (int i = k + 1; i < _m; i++)
{
work[i] += e[j] * matrixAClone[i, j];
}
}
for (int j = k + 1; j < _n; j++)
{
float t = -e[j] / e[k + 1];
for (int i = k + 1; i < _m; i++)
{
matrixAClone[i, j] += t * work[i];
}
}
}
if (wantv)
{
// Place the transformation in V for subsequent
// back multiplication.
for (int i = k + 1; i < _n; i++)
{
_v[i, k] = e[i];
}
}
}
}
// Set up the final bidiagonal matrix or order p.
int p = Math.Min(_n, _m + 1);
if (nct < _n)
{
_s[nct] = matrixAClone[nct, nct];
}
if (_m < p)
{
_s[p - 1] = 0;
}
if (nrt + 1 < p)
{
e[nrt] = matrixAClone[nrt, p - 1];
}
e[p - 1] = 0;
// If required, generate U.
if (wantu)
{
for (int j = nct; j < nu; j++)
{
for (int i = 0; i < _m; i++)
{
_u[i, j] = 0;
}
_u[j, j] = 1;
}
for (int k = nct - 1; k >= 0; k--)
{
if (_s[k] != 0)
{
for (int j = k + 1; j < nu; j++)
{
float t = 0;
for (int i = k; i < _m; i++)
{
t += _u[i, k] * _u[i, j];
}
t = -t / _u[k, k];
for (int i = k; i < _m; i++)
{
_u[i, j] += t * _u[i, k];
}
}
for (int i = k; i < _m; i++)
{
_u[i, k] = -_u[i, k];
}
_u[k, k] = 1 + _u[k, k];
for (int i = 0; i < k - 1; i++)
{
_u[i, k] = 0;
}
}
else
{
for (int i = 0; i < _m; i++)
{
_u[i, k] = 0;
}
_u[k, k] = 1;
}
}
}
// If required, generate V.
if (wantv)
{
for (int k = _n - 1; k >= 0; k--)
{
if ((k < nrt) & (e[k] != 0.0))
{
for (int j = k + 1; j < nu; j++)
{
float t = 0;
for (int i = k + 1; i < _n; i++)
{
t += _v[i, k] * _v[i, j];
}
t = -t / _v[k + 1, k];
for (int i = k + 1; i < _n; i++)
{
_v[i, j] += t * _v[i, k];
}
}
}
for (int i = 0; i < _n; i++)
{
_v[i, k] = 0;
}
_v[k, k] = 1;
}
}
// Main iteration loop for the singular values.
int pp = p - 1;
int iter = 0;
double eps = (float)Math.Pow(2, -23); // 2^-52 for double
double tiny = (float)Math.Pow(2, -100); // 2^-100 for float is an estimation by HelmutG
// Original: 2^-966 for double
while (p > 0)
{
int k, kase;
// Here is where a test for too many iterations would go.
// This section of the program inspects for
// negligible elements in the s and e arrays. On
// completion the variables kase and k are set as follows.
// kase = 1 if s(p) and e[k-1] are negligible and k<p
// kase = 2 if s(k) is negligible and k<p
// kase = 3 if e[k-1] is negligible, k<p, and
// s(k), ..., s(p) are not negligible (qr step).
// kase = 4 if e(p-1) is negligible (convergence).
for (k = p - 2; k >= -1; k--)
{
if (k == -1)
{
break;
}
if (Math.Abs(e[k]) <= tiny + eps * (Math.Abs(_s[k]) + Math.Abs(_s[k + 1])))
{
e[k] = 0;
break;
}
}
if (k == p - 2)
{
kase = 4;
}
else
{
int ks;
for (ks = p - 1; ks >= k; ks--)
{
if (ks == k)
{
break;
}
float t = (ks != p ? Math.Abs(e[ks]) : 0) + (ks != k + 1 ? Math.Abs(e[ks - 1]) : 0);
if (Math.Abs(_s[ks]) <= tiny + eps * t)
{
_s[ks] = 0;
break;
}
}
if (ks == k)
{
kase = 3;
}
else if (ks == p - 1)
{
kase = 1;
}
else
{
kase = 2;
k = ks;
}
}
k++;
// Perform the task indicated by kase.
switch (kase)
{
// Deflate negligible s(p).
case 1:
{
float f = e[p - 2];
e[p - 2] = 0;
for (int j = p - 2; j >= k; j--)
{
float t = MathHelper.Hypotenuse(_s[j], f);
float cs = _s[j] / t;
float sn = f / t;
_s[j] = t;
if (j != k)
{
f = -sn * e[j - 1];
e[j - 1] = cs * e[j - 1];
}
if (wantv)
{
for (int i = 0; i < _n; i++)
{
t = cs * _v[i, j] + sn * _v[i, p - 1];
_v[i, p - 1] = -sn * _v[i, j] + cs * _v[i, p - 1];
_v[i, j] = t;
}
}
}
}
break;
// Split at negligible s(k).
case 2:
{
float f = e[k - 1];
e[k - 1] = 0;
for (int j = k; j < p; j++)
{
float t = MathHelper.Hypotenuse(_s[j], f);
float cs = _s[j] / t;
float sn = f / t;
_s[j] = t;
f = -sn * e[j];
e[j] = cs * e[j];
if (wantu)
{
for (int i = 0; i < _m; i++)
{
t = cs * _u[i, j] + sn * _u[i, k - 1];
_u[i, k - 1] = -sn * _u[i, j] + cs * _u[i, k - 1];
_u[i, j] = t;
}
}
}
}
break;
// Perform one qr step.
case 3:
{
// Calculate the shift.
float scale = Math.Max(Math.Max(Math.Max(Math.Max(
Math.Abs(_s[p - 1]), Math.Abs(_s[p - 2])), Math.Abs(e[p - 2])),
Math.Abs(_s[k])), Math.Abs(e[k]));
float sp = _s[p - 1] / scale;
float spm1 = _s[p - 2] / scale;
float epm1 = e[p - 2] / scale;
float sk = _s[k] / scale;
float ek = e[k] / scale;
float b = ((spm1 + sp) * (spm1 - sp) + epm1 * epm1) / 2;
float c = (sp * epm1) * (sp * epm1);
float shift = 0;
if ((b != 0.0) | (c != 0.0))
{
shift = (float)Math.Sqrt(b * b + c);
if (b < 0.0)
{
shift = -shift;
}
shift = c / (b + shift);
}
float f = (sk + sp) * (sk - sp) + shift;
float g = sk * ek;
// Chase zeros.
for (int j = k; j < p - 1; j++)
{
float t = MathHelper.Hypotenuse(f, g);
float cs = f / t;
float sn = g / t;
if (j != k)
{
e[j - 1] = t;
}
f = cs * _s[j] + sn * e[j];
e[j] = cs * e[j] - sn * _s[j];
g = sn * _s[j + 1];
_s[j + 1] = cs * _s[j + 1];
if (wantv)
{
for (int i = 0; i < _n; i++)
{
t = cs * _v[i, j] + sn * _v[i, j + 1];
_v[i, j + 1] = -sn * _v[i, j] + cs * _v[i, j + 1];
_v[i, j] = t;
}
}
t = MathHelper.Hypotenuse(f, g);
cs = f / t;
sn = g / t;
_s[j] = t;
f = cs * e[j] + sn * _s[j + 1];
_s[j + 1] = -sn * e[j] + cs * _s[j + 1];
g = sn * e[j + 1];
e[j + 1] = cs * e[j + 1];
if (wantu && (j < _m - 1))
{
for (int i = 0; i < _m; i++)
{
t = cs * _u[i, j] + sn * _u[i, j + 1];
_u[i, j + 1] = -sn * _u[i, j] + cs * _u[i, j + 1];
_u[i, j] = t;
}
}
}
e[p - 2] = f;
iter = iter + 1;
}
break;
// Convergence.
case 4:
{
// Make the singular values positive.
if (_s[k] <= 0.0)
{
_s[k] = (_s[k] < 0.0 ? -_s[k] : 0);
if (wantv)
{
for (int i = 0; i <= pp; i++)
{
_v[i, k] = -_v[i, k];
}
}
}
// Order the singular values.
while (k < pp)
{
if (_s[k] >= _s[k + 1])
{
break;
}
float t = _s[k];
_s[k] = _s[k + 1];
_s[k + 1] = t;
if (wantv && (k < _n - 1))
{
for (int i = 0; i < _n; i++)
{
t = _v[i, k + 1];
_v[i, k + 1] = _v[i, k];
_v[i, k] = t;
}
}
if (wantu && (k < _m - 1))
{
for (int i = 0; i < _m; i++)
{
t = _u[i, k + 1];
_u[i, k + 1] = _u[i, k];
_u[i, k] = t;
}
}
k++;
}
iter = 0;
p--;
}
break;
}
}
}