Project: s41467-024-45514-6 | Function: solve (python)

system: You are an expert Python programmer implementing algorithms from scientific papers.
Complete the masked function based on the algorithm description.
Follow Python best practices and maintain consistent code style.
Return **only** the complete Python function code.

user: Complete this Python function:

**DESCRIPTION:** Solve DGA with memory for projection and memory-correction coefficients using matrix inversion and linear algebra

**KEY TERMS:** {
  "a": "matrices for the homogeneous term",
  "b": "matrices for the nonhomogeneous term",
  "c0": "matrix of inner products of basis functions",
  "coef": "projection coefficients",
  "mem_coef": "memory-correction coefficients"
}

**CONTEXT CODE:**         Maximum lag time in units of frames.
    mem : int
        Number of memory terms to use. These are evaluated at equally
        spaced times between time 0 and time `lag`, so `mem+1` must
        evenly divide `lag`. For example, with a `lag=32`, `mem=3` and
        `mem=7` are fine since 7+1=8 and 3+1=4 evenly divide 32. Setting
        `mem=0` corresponds to not using memory.
    coef : NDArray[float], shape=(n_basis,)
        Projection coefficients.
    mem_coef : NDArray[float], shape=(mem, n_basis)
        Memory-correction coefficients.

    Returns
    -------
    Sequence[NDArray[float]], shape=(n_trajs, n_frames[traj])
        Estimate of the aftcast.

    """
    if not np.iterable(function):
        function = [np.broadcast_to(function, g.shape[0] - 1) for g in guess]

    assert lag % (mem + 1) == 0
    dlag = lag // (mem + 1)
    n_basis = basis[0].shape[1]

    out = []
    for y, d, f, g in zip(basis, in_domain, function, guess, strict=True):
        n_frames = y.shape[0]
        assert y.shape == (n_frames, n_basis)
        assert d.shape == (n_frames,)
        assert f.shape == (n_frames - 1,)
        assert g.shape == (n_frames,)

        if n_frames <= lag:
            out.append(np.full(n_frames, np.nan))
            continue

        stop = np.maximum(np.arange(len(d) - dlag), backward_stop(d)[dlag:])
        intf = np.insert(np.cumsum(f), 0, 0.0)
        r = intf[dlag:] - intf[stop]
        pad = np.full(dlag, np.nan)
        u = y @ coef + g
        for v in mem_coef:
            u = np.concatenate([pad, u[stop] + r])
            u -= y @ v
        u = np.concatenate([pad, u[stop] + r])
        out.append(u)
    return out


def solve(a, b, c0):
    """
    Solve DGA with memory for projection and memory-correction
    coefficients.

    Parameters
    ----------
    a : NDArray[float], shape=(mem+1, n_basis, n_basis)
        Matrices for the homogeneous term.
    b : NDArray[float], shape=(mem+1, n_basis)
        Matrices for the nonhomogeneous term.
    c0 : NDArray[float], shape=(n_basis, n_basis)
        Matrix of inner products of basis functions.

    Returns
    -------
    coef : NDArray[float], shape=(n_basis,)
        Projection coefficients.
    mem_coef : NDArray[float], shape=(mem, n_basis)
        Memory-correction coefficients.

    """
    # MASKED: Solve DGA with memory for projection and memory-correction coefficients using matrix inversion and linear algebra implementation
    raise NotImplementedError("This function is masked for testing purposes.")


Complete the function implementation replacing "raise NotImplementedError".