gauss_seidel_y¶

emg3d.core.gauss_seidel_y(ex, ey, ez, sx, sy, sz, eta_x, eta_y, eta_z, zeta, hx, hy, hz, nu)[source]¶

Gauss-Seidel method with line relaxation in y-direction.

This is the equivalent to gauss_seidel(), but with line relaxation in the y-direction. See gauss_seidel() for more details.

The resulting system A x = b to solve consists of n unknowns (x-vector), and the corresponding matrix A is a banded matrix with the main diagonal and five upper and lower diagonals:

.-0
|X|\   0
0-.-0       left:  middle:  right:
 \|X|\                      (not used)
  0-.-0      0-     .-      0
   \|X|\      \     |X      |\
    0-.-0
 0   \|X|
      0-.

. 1*1, - 4*1, | 1*4, X 4*4, \ 4*4 upper or lower

The matrix A is complex and symmetric (A = A^T), and therefore only the main diagonal and the lower five off-diagonals are required.

The right-hand-side b has length 5*nCy-4 (nCy even).
The matrix A has length of b and 1+2*5 diagonals; we use for it an array of length 6*len(b).

The values are computed in rows of 5 lines, with the indicated middle and left matrices as indicated in the above scheme. These blocks are filled into the main matrix A and vector b, and subsequently solved with a non-standard Cholesky factorisation, solve().

Note: The smoothing with linerelaxation in y-direction is carried out in reversed lexicographical order, in order to improve speed (memory access). All other smoothers (gauss_seidel(), gauss_seidel_x(), and gauss_seidel_z()) use lexicographical order.

Tangential components at the boundaries are assumed to be 0 (PEC boundaries).

The result is stored in the provided electric fields ex, ey, and ez.

Parameters:

ex, ey, ez : ndarray: Electric fields in x-, y-, and z-directions, as obtained from emg3d.fields.Field.
sx, sy, sz :: Source fields in x-, y-, and z-directions, as obtained from emg3d.fields.Field.
eta_x, eta_y, eta_z, zeta :: VolumeModel parameters (multiplied by volumes) as obtained from emg3d.models.VolumeModel().
hx, hy, hz : ndarray: Cell widths in x-, y-, and z-directions.
nu : int: Number of Gauss-Seidel iterations.