public void SolveMatrix()
{
DoubleMatrix b = new DoubleMatrix(3);
b[0, 0] = 2;
b[0, 1] = 2;
b[0, 2] = 2;
b[1, 0] = 13;
b[1, 1] = 13;
b[1, 2] = 13;
b[2, 0] = 25;
b[2, 1] = 25;
b[2, 2] = 25;
DoubleMatrix x = lu.Solve(b);
Assert.AreEqual(x[0, 0], 2.965, TOLERENCE);
Assert.AreEqual(x[0, 1], 2.965, TOLERENCE);
Assert.AreEqual(x[0, 2], 2.965, TOLERENCE);
Assert.AreEqual(x[1, 0], -0.479, TOLERENCE);
Assert.AreEqual(x[1, 1], -0.479, TOLERENCE);
Assert.AreEqual(x[1, 2], -0.479, TOLERENCE);
Assert.AreEqual(x[2, 0], 1.227, TOLERENCE);
Assert.AreEqual(x[2, 1], 1.227, TOLERENCE);
Assert.AreEqual(x[2, 2], 1.227, TOLERENCE);
b = new DoubleMatrix(3, 2);
b[0, 0] = 2;
b[0, 1] = 2;
b[1, 0] = 13;
b[1, 1] = 13;
b[2, 0] = 25;
b[2, 1] = 25;
x = lu.Solve(b);
Assert.AreEqual(x[0, 0], 2.965, TOLERENCE);
Assert.AreEqual(x[0, 1], 2.965, TOLERENCE);
Assert.AreEqual(x[1, 0], -0.479, TOLERENCE);
Assert.AreEqual(x[1, 1], -0.479, TOLERENCE);
Assert.AreEqual(x[2, 0], 1.227, TOLERENCE);
Assert.AreEqual(x[2, 1], 1.227, TOLERENCE);
b = new DoubleMatrix(3, 4);
b[0, 0] = 2;
b[0, 1] = 2;
b[0, 2] = 2;
b[0, 3] = 2;
b[1, 0] = 13;
b[1, 1] = 13;
b[1, 2] = 13;
b[1, 3] = 13;
b[2, 0] = 25;
b[2, 1] = 25;
b[2, 2] = 25;
b[2, 3] = 25;
x = lu.Solve(b);
Assert.AreEqual(x[0, 0], 2.965, TOLERENCE);
Assert.AreEqual(x[0, 1], 2.965, TOLERENCE);
Assert.AreEqual(x[0, 2], 2.965, TOLERENCE);
Assert.AreEqual(x[0, 3], 2.965, TOLERENCE);
Assert.AreEqual(x[1, 0], -0.479, TOLERENCE);
Assert.AreEqual(x[1, 1], -0.479, TOLERENCE);
Assert.AreEqual(x[1, 2], -0.479, TOLERENCE);
Assert.AreEqual(x[1, 3], -0.479, TOLERENCE);
Assert.AreEqual(x[2, 0], 1.227, TOLERENCE);
Assert.AreEqual(x[2, 1], 1.227, TOLERENCE);
Assert.AreEqual(x[2, 2], 1.227, TOLERENCE);
Assert.AreEqual(x[2, 3], 1.227, TOLERENCE);
}
public static void Test01e_2()
{
var X = new DoubleMatrix(new double[,] { { 73, 746, 743, 106 }, { 584, 531, 420, 579 }, { 255, 562, 234, 693 }, { 360, 474, 381, 484 }, { 301, 78, 68, 313 } });
var y = new DoubleMatrix(new double[,] { { 803 }, { 292 }, { 230 }, { 469 }, { 655 } });
var XtX = X.GetTranspose() * X;
var Xty = X.GetTranspose() * y;
var solver = new DoubleLUDecomp(XtX);
var expected = solver.Solve(Xty);
IMatrix x, w;
FastNonnegativeLeastSquares.Execution(XtX, Xty, (i) => false, null, out x, out w);
Assert.AreEqual(expected[0, 0], x[0, 0], 1e-4);
Assert.AreEqual(expected[1, 0], x[1, 0], 1e-4);
Assert.AreEqual(expected[2, 0], x[2, 0], 1e-4);
Assert.AreEqual(expected[3, 0], x[3, 0], 1e-4);
Assert.AreEqual(0, w[0, 0], 1e-8);
Assert.AreEqual(0, w[1, 0], 1e-8);
Assert.AreEqual(0, w[2, 0], 1e-8);
Assert.AreEqual(0, w[3, 0], 1e-8);
}
/// <summary>
/// Calculate the matrix for the polyharmonic spline and solves the linear equation to calculate the coefficients.
/// </summary>
private void InternalCompute()
{
var N = _numberOfControlPoints;
// Allocate the matrix and vector
var mtx_l = new DoubleMatrix(N + 1 + _coordDim, N + 1 + _coordDim);
// there is no need for this matrix if we don't need to calculate the bending energy
_mtx_orig_k = new DoubleMatrix(N, N);
// Fill K (p x p, upper left of L)
// K is symmetrical so we really have to calculate only about half of the coefficients.
double a = 0.0;
for (int i = 0; i < N; ++i)
{
for (int j = i + 1; j < N; ++j)
{
double distance = DistanceBetweenControlPoints(i, j);
mtx_l[i, j] = mtx_l[j, i] = _mtx_orig_k[i, j] = _mtx_orig_k[j, i] = _cachedTpsFunc(distance);
a += distance * 2; // same for upper & lower triangle
}
}
a /= ((double)N) * N;
// Fill the rest of L with the values to do regularization and linear interpolation
for (int i = 0; i < N; ++i)
{
// diagonal: regularization parameter (lambda * a^2)
mtx_l[i, i] = _mtx_orig_k[i, i] = _regularizationParameter * (a * a);
// P (N x (nCoordDim+1), upper right)
mtx_l[i, N + 0] = 1; // for the intercept
for (int d = 0; d < _coordDim; d++)
mtx_l[i, N + 1 + d] = _coordinates[d][i];
// P transposed ((nCoordDim+1) x N, bottom left)
mtx_l[N + 0, i] = 1; // for the intercept
for (int d = 0; d < _coordDim; ++d)
mtx_l[N + 1 + d, i] = _coordinates[d][i];
}
// Zero ((nCoordDim+1) x (nCoordDim+1), lower right)
for (int i = N; i < N + _coordDim + 1; ++i)
for (int j = N; j < N + _coordDim + 1; ++j)
mtx_l[i, j] = 0;
// Solve the linear system
var solver = new DoubleLUDecomp(mtx_l);
if (solver.IsSingular)
throw new ArgumentException("The provided points lead to a singular matrix");
// Fill the right hand vector V with the values to spline; the last nCoordDim+1 elements are zero
_mtx_v = new DoubleVector(N + 1 + _coordDim);
for (int i = 0; i < N; ++i)
_mtx_v[i] = _values[i];
for (int d = 0; d <= _coordDim; ++d) // comparison '<=' is ok here because of additional intercept
_mtx_v[N + d] = 0;
_mtx_v = solver.Solve(_mtx_v);
}
/// <summary>
/// Calculate the matrix for the polyharmonic spline and solves the linear equation to calculate the coefficients.
/// </summary>
private void InternalCompute()
{
var N = _numberOfControlPoints;
// Allocate the matrix and vector
var mtx_l = new DoubleMatrix(N + 1 + _coordDim, N + 1 + _coordDim);
// there is no need for this matrix if we don't need to calculate the bending energy
_mtx_orig_k = new DoubleMatrix(N, N);
// Fill K (p x p, upper left of L)
// K is symmetrical so we really have to calculate only about half of the coefficients.
double a = 0.0;
for (int i = 0; i < N; ++i)
{
for (int j = i + 1; j < N; ++j)
{
double distance = DistanceBetweenControlPoints(i, j);
mtx_l[i, j] = mtx_l[j, i] = _mtx_orig_k[i, j] = _mtx_orig_k[j, i] = _cachedTpsFunc(distance);
a += distance * 2; // same for upper & lower triangle
}
}
a /= ((double)N) * N;
// Fill the rest of L with the values to do regularization and linear interpolation
for (int i = 0; i < N; ++i)
{
// diagonal: regularization parameter (lambda * a^2)
mtx_l[i, i] = _mtx_orig_k[i, i] = _regularizationParameter * (a * a);
// P (N x (nCoordDim+1), upper right)
mtx_l[i, N + 0] = 1; // for the intercept
for (int d = 0; d < _coordDim; d++)
{
mtx_l[i, N + 1 + d] = _coordinates[d][i];
}
// P transposed ((nCoordDim+1) x N, bottom left)
mtx_l[N + 0, i] = 1; // for the intercept
for (int d = 0; d < _coordDim; ++d)
{
mtx_l[N + 1 + d, i] = _coordinates[d][i];
}
}
// Zero ((nCoordDim+1) x (nCoordDim+1), lower right)
for (int i = N; i < N + _coordDim + 1; ++i)
{
for (int j = N; j < N + _coordDim + 1; ++j)
{
mtx_l[i, j] = 0;
}
}
// Solve the linear system
var solver = new DoubleLUDecomp(mtx_l);
if (solver.IsSingular)
{
throw new ArgumentException("The provided points lead to a singular matrix");
}
// Fill the right hand vector V with the values to spline; the last nCoordDim+1 elements are zero
_mtx_v = new DoubleVector(N + 1 + _coordDim);
for (int i = 0; i < N; ++i)
{
_mtx_v[i] = _values[i];
}
for (int d = 0; d <= _coordDim; ++d) // comparison '<=' is ok here because of additional intercept
{
_mtx_v[N + d] = 0;
}
_mtx_v = solver.Solve(_mtx_v);
}
/// <summary>
/// Execution of the fast nonnegative least squares algorithm. The algorithm finds a vector x with all elements xi>=0 which minimizes |X*x-y|.
/// </summary>
/// <param name="XtX">X transposed multiplied by X, thus a square matrix.</param>
/// <param name="Xty">X transposed multiplied by Y, thus a matrix with one column and same number of rows as X.</param>
/// <param name="isRestrictedToPositiveValues">Function that takes the parameter index as argument and returns true if the parameter at this index is restricted to positive values; otherwise the return value must be false.</param>
/// <param name="tolerance">Used to decide if a solution element is less than or equal to zero. If this is null, a default tolerance of tolerance = MAX(SIZE(XtX)) * NORM(XtX,1) * EPS is used.</param>
/// <param name="x">Output: solution vector (matrix with one column and number of rows according to dimension of X.</param>
/// <param name="w">Output: Lagrange vector. Elements which take place in the fit are set to 0. Elements fixed to zero contain a negative number.</param>
/// <remarks>
/// <para>
/// Literature: Rasmus Bro and Sijmen De Jong, 'A fast non-negativity-constrained least squares algorithm', Journal of Chemometrics, Vol. 11, 393-401 (1997)
/// </para>
/// <para>
/// Algorithm modified by Dirk Lellinger 2015 to allow a mixture of restricted and unrestricted parameters.
/// </para>
/// </remarks>
public static void Execution(IROMatrix XtX, IROMatrix Xty, Func<int, bool> isRestrictedToPositiveValues, double? tolerance, out IMatrix x, out IMatrix w)
{
if (null == XtX)
throw new ArgumentNullException(nameof(XtX));
if (null == Xty)
throw new ArgumentNullException(nameof(Xty));
if (null == isRestrictedToPositiveValues)
throw new ArgumentNullException(nameof(isRestrictedToPositiveValues));
if (XtX.Rows != XtX.Columns)
throw new ArgumentException("Matrix should be a square matrix", nameof(XtX));
if (Xty.Columns != 1)
throw new ArgumentException(nameof(Xty) + " should be a column vector (number of columns should be equal to 1)", nameof(Xty));
if (Xty.Rows != XtX.Columns)
throw new ArgumentException("Number of rows in " + nameof(Xty) + " should match number of columns in " + nameof(XtX), nameof(Xty));
var matrixGenerator = new Func<int, int, DoubleMatrix>((rows, cols) => new DoubleMatrix(rows, cols));
// if nargin < 3
// tol = 10 * eps * norm(XtX, 1) * length(XtX);
// end
double tol = tolerance.HasValue ? tolerance.Value : 10 * DoubleConstants.DBL_EPSILON * MatrixMath.Norm(XtX, MatrixNorm.M1Norm) * Math.Max(XtX.Rows, XtX.Columns);
// [m, n] = size(XtX);
int n = XtX.Columns;
// P = zeros(1, n);
// Z = 1:n;
var P = new bool[n]; // POSITIVE SET: all indices which are currently not fixed are marked with TRUE (Negative set is simply this, but inverted)
bool initializationOfSolutionRequired = false;
for (int i = 0; i < n; ++i)
{
bool isNotRestricted = !isRestrictedToPositiveValues(i);
P[i] = isNotRestricted;
initializationOfSolutionRequired |= isNotRestricted;
}
// x = P';
x = matrixGenerator(n, 1);
// w = Xty-XtX*x;
w = matrixGenerator(n, 1);
MatrixMath.Copy(Xty, w);
var helper_n_1 = matrixGenerator(n, 1);
MatrixMath.Multiply(XtX, x, helper_n_1);
MatrixMath.Subtract(w, helper_n_1, w);
// set up iteration criterion
int iter = 0;
int itmax = 30 * n;
// outer loop to put variables into set to hold positive coefficients
// while any(Z) & any(w(ZZ) > tol)
while (initializationOfSolutionRequired || (P.Any(ele => false == ele) && w.Any((r, c, ele) => false == P[r] && ele > tol)))
{
if (initializationOfSolutionRequired)
{
initializationOfSolutionRequired = false;
}
else
{
// [wt, t] = max(w(ZZ));
// t = ZZ(t);
int t = -1; // INDEX
double wt = double.NegativeInfinity;
for (int i = 0; i < n; ++i)
{
if (!P[i])
{
if (w[i, 0] > wt)
{
wt = w[i, 0];
t = i;
}
}
}
// P(1, t) = t;
// Z(t) = 0;
P[t] = true;
}
// z(PP')=(Xty(PP)'/XtX(PP,PP)');
var subXty = Xty.SubMatrix(P, 0, matrixGenerator); // Xty(PP)'
var subXtX = XtX.SubMatrix(P, P, matrixGenerator);
var solver = new DoubleLUDecomp(subXtX);
var subSolution = solver.Solve(subXty);
var z = matrixGenerator(n, 1);
for (int i = 0, ii = 0; i < n; ++i)
z[i, 0] = P[i] ? subSolution[ii++, 0] : 0;
// C. Inner loop (to remove elements from the positive set which no longer belong to)
while (z.Any((r, c, ele) => true == P[r] && ele <= tol && isRestrictedToPositiveValues(r)) && iter < itmax)
{
++iter;
// QQ = find((z <= tol) & P');
//alpha = min(x(QQ)./ (x(QQ) - z(QQ)));
double alpha = double.PositiveInfinity;
for (int i = 0; i < n; ++i)
{
if ((z[i, 0] <= tol && true == P[i] && isRestrictedToPositiveValues(i)))
{
alpha = Math.Min(alpha, x[i, 0] / (x[i, 0] - z[i, 0]));
}
}
// x = x + alpha * (z - x);
for (int i = 0; i < n; ++i)
x[i, 0] += alpha * (z[i, 0] - x[i, 0]);
// ij = find(abs(x) < tol & P' ~= 0);
// Z(ij) = ij';
// P(ij) = zeros(1, length(ij));
for (int i = 0; i < n; ++i)
{
if (Math.Abs(x[i, 0]) < tol && P[i] == true && isRestrictedToPositiveValues(i))
{
P[i] = false;
}
}
//PP = find(P);
//ZZ = find(Z);
//nzz = size(ZZ);
//z(PP) = (Xty(PP)'/XtX(PP,PP)');
subXty = Xty.SubMatrix(P, 0, matrixGenerator);
subXtX = XtX.SubMatrix(P, P, matrixGenerator);
solver = new DoubleLUDecomp(subXtX);
subSolution = solver.Solve(subXty);
for (int i = 0, ii = 0; i < n; ++i)
z[i, 0] = P[i] ? subSolution[ii++, 0] : 0;
} // end inner loop
MatrixMath.Copy(z, x);
MatrixMath.Copy(Xty, w);
MatrixMath.Multiply(XtX, x, helper_n_1);
MatrixMath.Subtract(w, helper_n_1, w);
}
}