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_serinv#

This backend implements the methods present in Serinv for the specific case of solving lesser, greater, and retarded Green's functions.

For more information see https://github.com/vincent-maillou/serinv

Classes:

  • ReducedSystem –

    Auxiliary class to handle the reduced system.

Functions:

ReducedSystem #

ReducedSystem(selected_solve: bool = False)

Auxiliary class to handle the reduced system.

This is basically a container for the diagonal and off-diagonal blocks of the reduced system with some extra functionality.

Parameters:

  • selected_solve (bool, default: False ) –

    Whether the ReducedSystem should also store blocks for solving the quadratic systems associated with the lesser and greater right-hand-sides, by default False.

Attributes:

  • comm (Comm) –

    The intranode MPI communicator.

  • num_diags (int) –

    The number of diagonal blocks in the reduced system.

  • diag_blocks (list[NDArray | None]) –

    The diagonal blocks of the reduced system.

  • upper_blocks (list[NDArray | None]) –

    The upper off-diagonal blocks of the reduced system.

  • lower_blocks (list[NDArray | None]) –

    The lower off-diagonal blocks of the reduced system

  • solve_lesser (bool) –

    Whether to solve the quadratic system associated with the lesser righ-hand-sideS.

  • diag_blocks_lesser (list[NDArray | None]) –

    The diagonal blocks of the reduced system associated with the lesser right-hand-side.

  • upper_blocks_lesser (list[NDArray | None]) –

    The upper off-diagonal blocks of the reduced system associated with the lesser right-hand-side.

  • lower_blocks_lesser (list[NDArray | None]) –

    The lower off-diagonal blocks of the reduced system associated with the lesser right-hand-side.

  • solve_greater (bool) –

    Whether to solve the quadratic system associated with the greater right-hand-side.

  • diag_blocks_greater (list[NDArray | None]) –

    The diagonal blocks of the reduced system associated with the greater right-hand-side.

  • upper_blocks_greater (list[NDArray | None]) –

    The upper off-diagonal blocks of the reduced system associated with the greater right-hand-side.

  • lower_blocks_greater (list[NDArray | None]) –

    The lower off-diagonal blocks of the reduced system associated with the greater right-hand-side.

Methods:

  • gather –

    Gathers the reduced system across all ranks.

  • gather_constant_block_size –

    Gathers the reduced system across all ranks.

  • solve –

    Solves the reduced system on all ranks.

  • scatter –

    Scatters the reduced system across all ranks.

gather #

gather(a: DSDBSparse | _DStackView, xr_diag_blocks: list[NDArray], xr_buffer_upper: list[NDArray], xr_buffer_lower: list[NDArray], sigma_lesser: DSDBSparse | _DStackView = None, xl_diag_blocks: list[NDArray] = None, xl_buffer_upper: list[NDArray] = None, xl_buffer_lower: list[NDArray] = None, sigma_greater: DSDBSparse | _DStackView = None, xg_diag_blocks: list[NDArray] = None, xg_buffer_upper: list[NDArray] = None, xg_buffer_lower: list[NDArray] = None)

Gathers the reduced system across all ranks.

Parameters:

  • a (DSDBSparse) –

    The system matrix.

  • xr_diag_blocks (list[NDArray]) –

    The diagonal blocks of the Schur factorization of the retarded system.

  • xr_buffer_upper (list[NDArray]) –

    The super-diagonal blocks of the Schur factorization of the retarded system.

  • xr_buffer_lower (list[NDArray]) –

    The sub-diagonal blocks of the Schur factorization of the retarded system.

  • sigma_lesser (DSDBSparse, default: None ) –

    The self-energy matrix for the lesser Green's function, by default None.

  • xl_diag_blocks (list[NDArray], default: None ) –

    The diagonal blocks of the Schur factorization of the lesser system, by default None.

  • xl_buffer_upper (list[NDArray], default: None ) –

    The super-diagonal blocks of the Schur factorization of the lesser system, by default None.

  • xl_buffer_lower (list[NDArray], default: None ) –

    The sub-diagonal blocks of the Schur factorization of the lesser system, by default None.

  • sigma_greater (DSDBSparse, default: None ) –

    The self-energy matrix for the greater Green's function, by default None.

  • xg_diag_blocks (list[NDArray], default: None ) –

    The diagonal blocks of the Schur factorization of the greater system, by default None.

  • xg_buffer_upper (list[NDArray], default: None ) –

    The super-diagonal blocks of the Schur factorization of the greater system, by default None.

  • xg_buffer_lower (list[NDArray], default: None ) –

    The sub-diagonal blocks of the Schur factorization of the greater system, by default None.

gather_constant_block_size #

gather_constant_block_size(a: DSDBSparse | _DStackView, xr_diag_blocks: list[NDArray], xr_buffer_upper: list[NDArray], xr_buffer_lower: list[NDArray], sigma_lesser: DSDBSparse | _DStackView = None, xl_diag_blocks: list[NDArray] = None, xl_buffer_upper: list[NDArray] = None, xl_buffer_lower: list[NDArray] = None, sigma_greater: DSDBSparse | _DStackView = None, xg_diag_blocks: list[NDArray] = None, xg_buffer_upper: list[NDArray] = None, xg_buffer_lower: list[NDArray] = None)

Gathers the reduced system across all ranks.

Parameters:

  • a (DSDBSparse) –

    The system matrix.

  • xr_diag_blocks (list[NDArray]) –

    The diagonal blocks of the Schur factorization of the retarded system.

  • xr_buffer_upper (list[NDArray]) –

    The super-diagonal blocks of the Schur factorization of the retarded system.

  • xr_buffer_lower (list[NDArray]) –

    The sub-diagonal blocks of the Schur factorization of the retarded system.

  • sigma_lesser (DSDBSparse, default: None ) –

    The self-energy matrix for the lesser Green's function, by default None.

  • xl_diag_blocks (list[NDArray], default: None ) –

    The diagonal blocks of the Schur factorization of the lesser system, by default None.

  • xl_buffer_upper (list[NDArray], default: None ) –

    The super-diagonal blocks of the Schur factorization of the lesser system, by default None.

  • xl_buffer_lower (list[NDArray], default: None ) –

    The sub-diagonal blocks of the Schur factorization of the lesser system, by default None.

  • sigma_greater (DSDBSparse, default: None ) –

    The self-energy matrix for the greater Green's function, by default None.

  • xg_diag_blocks (list[NDArray], default: None ) –

    The diagonal blocks of the Schur factorization of the greater system, by default None.

  • xg_buffer_upper (list[NDArray], default: None ) –

    The super-diagonal blocks of the Schur factorization of the greater system, by default None.

  • xg_buffer_lower (list[NDArray], default: None ) –

    The sub-diagonal blocks of the Schur factorization of the greater system, by default None.

solve #

solve()

Solves the reduced system on all ranks.

scatter #

scatter(xr_diag_blocks: list[NDArray], xr_buffer_upper: list[NDArray], xr_buffer_lower: list[NDArray], xr_out: DSDBSparse | _DStackView, return_retarded: bool = True, xl_diag_blocks: list[NDArray] = None, xl_buffer_lower: list[NDArray] = None, xl_buffer_upper: list[NDArray] = None, xl_out: DSDBSparse | _DStackView = None, xg_diag_blocks: list[NDArray] = None, xg_buffer_lower: list[NDArray] = None, xg_buffer_upper: list[NDArray] = None, xg_out: DSDBSparse | _DStackView = None)

Scatters the reduced system across all ranks.

Parameters:

  • xr_diag_blocks (list[NDArray]) –

    The diagonal blocks of the Schur factorization of the retarded system.

  • xr_buffer_upper (list[NDArray]) –

    The super-diagonal blocks of the Schur factorization of the retarded system.

  • xr_buffer_lower (list[NDArray]) –

    The sub-diagonal blocks of the Schur factorization of the retarded system.

  • xr_out (DSDBSparse | _DStackView) –

    The output matrix for the retarded system.

  • return_retarded (bool, default: True ) –

    Whether to write the retarded Green's function to the output matrix, by default True.

  • xl_diag_blocks (list[NDArray], default: None ) –

    The diagonal blocks of the Schur factorization of the lesser system, by default None.

  • xl_buffer_lower (list[NDArray], default: None ) –

    The sub-diagonal blocks of the Schur factorization of the lesser system, by default None.

  • xl_buffer_upper (list[NDArray], default: None ) –

    The super-diagonal blocks of the Schur factorization of the lesser system, by default None.

  • xl_out (DSDBSparse | _DStackView, default: None ) –

    The output matrix for the lesser system, by default None.

  • xg_diag_blocks (list[NDArray], default: None ) –

    The diagonal blocks of the Schur factorization of the greater system, by default None.

  • xg_buffer_lower (list[NDArray], default: None ) –

    The sub-diagonal blocks of the Schur factorization of the greater system, by default None.

  • xg_buffer_upper (list[NDArray], default: None ) –

    The super-diagonal blocks of the Schur factorization of the greater system, by default None.

  • xg_out (DSDBSparse | _DStackView, default: None ) –

    The output matrix for the greater system, by default None.

downward_schur #

downward_schur(a: DSDBSparse | _DStackView, xr_diag_blocks: list[NDArray], obc_blocks: OBCBlocks, sigma_lesser: DSDBSparse | _DStackView = None, xl_diag_blocks: list[NDArray] = None, sigma_greater: DSDBSparse | _DStackView = None, xg_diag_blocks: list[NDArray] = None, stack_slice: slice = Ellipsis, invert_last_block: bool = True, selected_solve: bool = False)

Performs the downward Schur complement decomposition.

upward_schur #

upward_schur(a: DSDBSparse | _DStackView, xr_diag_blocks: list[NDArray], obc_blocks: OBCBlocks, sigma_lesser: DSDBSparse | _DStackView = None, xl_diag_blocks: list[NDArray] = None, sigma_greater: DSDBSparse | _DStackView = None, xg_diag_blocks: list[NDArray] = None, stack_slice: slice = Ellipsis, invert_last_block: bool = True, selected_solve: bool = False)

Performs the upward Schur complement decomposition.

permuted_schur #

permuted_schur(a: DSDBSparse | _DStackView, xr_diag_blocks: list[NDArray], xr_buffer_lower: list[NDArray], xr_buffer_upper: list[NDArray], obc_blocks: OBCBlocks, sigma_lesser: DSDBSparse | _DStackView = None, xl_diag_blocks: list[NDArray] = None, xl_buffer_lower: list[NDArray] = None, xl_buffer_upper: list[NDArray] = None, sigma_greater: DSDBSparse | _DStackView = None, xg_diag_blocks: list[NDArray] = None, xg_buffer_lower: list[NDArray] = None, xg_buffer_upper: list[NDArray] = None, stack_slice: slice = Ellipsis, selected_solve: bool = False)

Performs the permuted Schur complement decomposition.

downward_selinv #

downward_selinv(a: DSDBSparse | _DStackView, xr_diag_blocks: list[NDArray], xr_out: DSDBSparse | _DStackView, sigma_lesser: DSDBSparse | _DStackView = None, xl_diag_blocks: list[NDArray] = None, xl_out: DSDBSparse | _DStackView = None, sigma_greater: DSDBSparse | _DStackView = None, xg_diag_blocks: list[NDArray] = None, xg_out: DSDBSparse | _DStackView = None, selected_solve: bool = False, return_retarded: bool = True)

Performs the downward selected inversion.

upward_selinv #

upward_selinv(a: DSDBSparse, xr_diag_blocks: list[NDArray], xr_out: DSDBSparse, sigma_lesser: DSDBSparse = None, xl_diag_blocks: list[NDArray] = None, xl_out: DSDBSparse = None, sigma_greater: DSDBSparse = None, xg_diag_blocks: list[NDArray] = None, xg_out: DSDBSparse = None, selected_solve: bool = False, return_retarded: bool = True)

Performs the upward selected inversion.

permuted_selinv #

permuted_selinv(a: DSDBSparse | _DStackView, xr_diag_blocks: list[NDArray], xr_buffer_lower: list[NDArray], xr_buffer_upper: list[NDArray], xr_out: DSDBSparse | _DStackView, sigma_lesser: DSDBSparse | _DStackView = None, xl_diag_blocks: list[NDArray] = None, xl_buffer_lower: list[NDArray] = None, xl_buffer_upper: list[NDArray] = None, xl_out: DSDBSparse | _DStackView = None, sigma_greater: DSDBSparse | _DStackView = None, xg_diag_blocks: list[NDArray] = None, xg_buffer_lower: list[NDArray] = None, xg_buffer_upper: list[NDArray] = None, xg_out: DSDBSparse | _DStackView = None, selected_solve: bool = False, return_retarded: bool = True)

Performs the permuted selected inversion.