/// <summary>
/// Get the 1D Laplacian matrix (with Dirichlet boundary conditions).
/// </summary>
/// <param name="nx">Grid size.</param>
/// <param name="eigenvalues">Vector to store eigenvalues (optional).</param>
/// <returns>Laplacian sparse matrix.</returns>
public static SparseMatrix Laplacian(int nx, DenseVector eigenvalues = null)
{
if (nx == 1)
{
// Handle special case n = 1.
var A = new SparseMatrix(nx, nx);
A.At(0, 0, 2.0);
return(A);
}
var C = new CoordinateStorage <Complex>(nx, nx);
for (int i = 0; i < nx; i++)
{
C.At(i, i, 2.0);
if (i == 0)
{
C.At(i, i + 1, -1.0);
}
else if (i == (nx - 1))
{
C.At(i, i - 1, -1.0);
}
else
{
C.At(i, i - 1, -1.0);
C.At(i, i + 1, -1.0);
}
}
if (eigenvalues != null)
{
// Compute eigenvalues.
int count = Math.Min(nx, eigenvalues.Count);
var eigs = new double[nx];
for (int i = 0; i < count; i++)
{
eigs[i] = 4 * Math.Pow(Math.Sin((i + 1) * Math.PI / (2 * (nx + 1))), 2);
}
Array.Sort(eigs);
for (int i = 0; i < count; ++i)
{
eigenvalues.At(i, eigs[i]);
}
}
return((SparseMatrix)C.ToSparseMatrix());
}
/// <summary>
/// Get the 2D Laplacian matrix (with Dirichlet boundary conditions).
/// </summary>
/// <param name="nx">Number of elements in x direction.</param>
/// <param name="ny">Number of elements in y direction.</param>
/// <param name="eigenvalues">Vector to store eigenvalues (optional).</param>
/// <returns>Laplacian sparse matrix.</returns>
public static SparseMatrix Laplacian(int nx, int ny, DenseVector eigenvalues = null)
{
var Ix = Eye(nx);
var Iy = Eye(ny);
var Dx = Laplacian(nx);
var Dy = Laplacian(ny);
if (eigenvalues != null)
{
// Compute eigenvalues.
int count = Math.Min(nx * ny, eigenvalues.Count);
int index = 0;
var eigs = new double[nx * ny];
double ax, ay;
for (int i = 0; i < nx; i++)
{
ax = 4 * Math.Pow(Math.Sin((i + 1) * Math.PI / (2 * (nx + 1))), 2);
for (int j = 0; j < ny; j++)
{
ay = 4 * Math.Pow(Math.Sin((j + 1) * Math.PI / (2 * (ny + 1))), 2);
eigs[index++] = ax + ay;
}
}
Array.Sort(eigs);
for (int i = 0; i < count; ++i)
{
eigenvalues.At(i, eigs[i]);
}
}
return((SparseMatrix)(Kron(Iy, Dx).FastAdd(Kron(Dy, Ix))));
}