How to use the petsc4py.PETSc.Mat.Stencil function in petsc4py

north_diag = vs.hur[2:-2, 3:-1] / vs.dyu[np.newaxis, 2:-2] / \
            vs.dyt[np.newaxis, 3:-1] * vs.cost[np.newaxis, 3:-1] / vs.cosu[np.newaxis, 2:-2]
        south_diag = vs.hur[2:-2, 2:-2] / vs.dyu[np.newaxis, 2:-2] / \
            vs.dyt[np.newaxis, 2:-2] * vs.cost[np.newaxis, 2:-2] / vs.cosu[np.newaxis, 2:-2]

        # construct sparse matrix
        cf = tuple(diag for diag in (
            boundary_mask * main_diag + (1 - boundary_mask),
            boundary_mask * east_diag,
            boundary_mask * west_diag,
            boundary_mask * north_diag,
            boundary_mask * south_diag
        ))

        row = PETSc.Mat.Stencil()
        col = PETSc.Mat.Stencil()

        ij_offsets = [(0, 0), (1, 0), (-1, 0), (0, 1), (0, -1)]

        (i0, i1), (j0, j1) = self._da.getRanges()
        for j in range(j0, j1):
            for i in range(i0, i1):
                iloc, jloc = i % (vs.nx // rs.num_proc[0]), j % (vs.ny // rs.num_proc[1])
                row.index = (i, j)

                for diag, offset in zip(cf, ij_offsets):
                    io, jo = (i + offset[0], j + offset[1])
                    col.index = (io, jo)
                    matrix.setValueStencil(row, col, diag[iloc, jloc])

        matrix.assemble()

vs.dxt[2:-2, np.newaxis] / vs.cosu[np.newaxis, 2:-2]**2
        north_diag = vs.hur[2:-2, 3:-1] / vs.dyu[np.newaxis, 2:-2] / \
            vs.dyt[np.newaxis, 3:-1] * vs.cost[np.newaxis, 3:-1] / vs.cosu[np.newaxis, 2:-2]
        south_diag = vs.hur[2:-2, 2:-2] / vs.dyu[np.newaxis, 2:-2] / \
            vs.dyt[np.newaxis, 2:-2] * vs.cost[np.newaxis, 2:-2] / vs.cosu[np.newaxis, 2:-2]

        # construct sparse matrix
        cf = tuple(diag for diag in (
            boundary_mask * main_diag + (1 - boundary_mask),
            boundary_mask * east_diag,
            boundary_mask * west_diag,
            boundary_mask * north_diag,
            boundary_mask * south_diag
        ))

        row = PETSc.Mat.Stencil()
        col = PETSc.Mat.Stencil()

        ij_offsets = [(0, 0), (1, 0), (-1, 0), (0, 1), (0, -1)]

        (i0, i1), (j0, j1) = self._da.getRanges()
        for j in range(j0, j1):
            for i in range(i0, i1):
                iloc, jloc = i % (vs.nx // rs.num_proc[0]), j % (vs.ny // rs.num_proc[1])
                row.index = (i, j)

                for diag, offset in zip(cf, ij_offsets):
                    io, jo = (i + offset[0], j + offset[1])
                    col.index = (io, jo)
                    matrix.setValueStencil(row, col, diag[iloc, jloc])

        matrix.assemble()

col.field = 1
                    A.setValueStencil(row, col, value)

    A.assemble()
    A.view()
    exit()

    Id = da.createMatrix()
    Id.setType('aij')  # sparse
    Id.setFromOptions()
    Id.setPreallocationNNZ((5, 5))
    Id.setUp()

    Id.zeroEntries()
    row = PETSc.Mat.Stencil()
    col = PETSc.Mat.Stencil()
    mx, my = da.getSizes()
    (xs, xe), (ys, ye) = da.getRanges()
    for j in range(ys, ye):
        for i in range(xs, xe):
            row.index = (i, j)
            row.field = 0
            col.index = (i, j)
            col.field = 0
            Id.setValueStencil(row, row, 1.0)
            row.field = 1
            col.field = 1
            Id.setValueStencil(row, col, 1.0)
    Id.assemble()

    # (xs, xe), (ys, ye) = da.getRanges()
    # print(A.getValues(range(n*n), range(n*n)))

Helper function to assemble PETSc matrix A

        Returns:
            PETSc matrix object
        """
        # create matrix and set basic options
        A = self.init.createMatrix()
        A.setType('aij')  # sparse
        A.setFromOptions()
        A.setPreallocationNNZ((5, 5))
        A.setUp()

        # fill matrix
        A.zeroEntries()
        row = PETSc.Mat.Stencil()
        col = PETSc.Mat.Stencil()
        mx, my = self.init.getSizes()
        (xs, xe), (ys, ye) = self.init.getRanges()
        for j in range(ys, ye):
            for i in range(xs, xe):
                row.index = (i, j)
                row.field = 0
                if i == 0 or j == 0 or i == mx - 1 or j == my - 1:
                    A.setValueStencil(row, row, 1.0)
                else:
                    diag = self.params.nu * (-2.0 / self.dx ** 2 - 2.0 / self.dy ** 2)
                    for index, value in [
                        ((i, j - 1), self.params.nu / self.dy ** 2),
                        ((i - 1, j), self.params.nu / self.dx ** 2),
                        ((i, j), diag),
                        ((i + 1, j), self.params.nu / self.dx ** 2),
                        ((i, j + 1), self.params.nu / self.dy ** 2),

"""
        Function to return the Jacobian matrix

        Args:
            snes: nonlinear solver object
            X: input vector
            J: Jacobian matrix (current?)
            P: Jacobian matrix (new)

        Returns:
            matrix status
        """
        self.da.globalToLocal(X, self.localX)
        x = self.da.getVecArray(self.localX)
        P.zeroEntries()
        row = PETSc.Mat.Stencil()
        col = PETSc.Mat.Stencil()
        (xs, xe), (ys, ye) = self.da.getRanges()

        for j in range(ys, ye):
            for i in range(xs, xe):

                # diagnoal 2-by-2 block (for u and v)
                row.index = (i, j)
                col.index = (i, j)
                row.field = 0
                col.field = 0
                val = 1.0 - self.factor * (self.params.Du * (-2.0 / self.dx ** 2 - 2.0 / self.dy ** 2) -
                                           x[i, j, 1] ** 2 - self.params.A)
                P.setValueStencil(row, col, val)
                row.field = 0
                col.field = 1

boundary_type=('ghosted', 'ghosted')) 

# Create the rhs vector based on the DA. 
b = da.createGlobalVec()
b_val = da.getVecArray(b) # Obtain access to elements of b.
b_val[50, 50] = 1; # Set central value to 1.

# Create (a vector to store) the solution vector.
x = da.createGlobalVec()

# Create the matrix.
A = da.getMatrix('aij')

# Stencil objects make it easy to set the values of the matrix elements.
row = PETSc.Mat.Stencil()
col = PETSc.Mat.Stencil()

# Set matrix elements to correct values.
(i0, i1), (j0, j1) = da.getRanges()
for j in range(j0, j1):
    for i in range(i0, i1):
        row.index = (i, j)
        for index, value in [((i, j), -4 + w**2),
                             ((i-1, j), 1),
                             ((i+1, j), 1),
                             ((i, j-1), 1),
                             ((i, j+1), 1)]:
            col.index = index
            A.setValueStencil(row, col, value) # Sets a single matrix element.
                            
A.assemblyBegin() # Make matrices useable.
A.assemblyEnd()

Helper function to assemble PETSc identity matrix

        Returns:
            PETSc matrix object
        """

        # create matrix and set basic options
        Id = self.init.createMatrix()
        Id.setType('aij')  # sparse
        Id.setFromOptions()
        Id.setPreallocationNNZ((1, 1))
        Id.setUp()

        # fill matrix
        Id.zeroEntries()
        row = PETSc.Mat.Stencil()
        (xs, xe), (ys, ye) = self.init.getRanges()
        for j in range(ys, ye):
            for i in range(xs, xe):
                row.index = (i, j)
                row.field = 0
                Id.setValueStencil(row, row, 1.0)

        Id.assemble()

        return Id

Helper function to assemble PETSc matrix A

        Returns:
            PETSc matrix object
        """
        # create matrix and set basic options
        A = self.init.createMatrix()
        A.setType('aij')  # sparse
        A.setFromOptions()
        A.setPreallocationNNZ((5, 5))
        A.setUp()

        # fill matrix
        A.zeroEntries()
        row = PETSc.Mat.Stencil()
        col = PETSc.Mat.Stencil()
        mx, my = self.init.getSizes()
        (xs, xe), (ys, ye) = self.init.getRanges()
        for j in range(ys, ye):
            for i in range(xs, xe):
                row.index = (i, j)
                row.field = 0
                A.setValueStencil(row, row, self.params.Du * (-2.0 / self.dx ** 2 - 2.0 / self.dy ** 2))
                row.field = 1
                A.setValueStencil(row, row, self.params.Dv * (-2.0 / self.dx ** 2 - 2.0 / self.dy ** 2))
                col.index = (i, j - 1)
                col.field = 0
                row.field = 0
                A.setValueStencil(row, col, self.params.Du / self.dy ** 2)
                col.field = 1
                row.field = 1
                A.setValueStencil(row, col, self.params.Dv / self.dy ** 2)

Helper function to assemble PETSc matrix A

        Returns:
            PETSc matrix object
        """
        # create matrix and set basic options
        A = self.init.createMatrix()
        A.setType('aij')  # sparse
        A.setFromOptions()
        A.setPreallocationNNZ((3, 3))
        A.setUp()

        # fill matrix
        A.zeroEntries()
        row = PETSc.Mat.Stencil()
        col = PETSc.Mat.Stencil()
        mx = self.init.getSizes()[0]
        (xs, xe) = self.init.getRanges()[0]
        for i in range(xs, xe):
            row.i = i
            row.field = 0
            if i == 0 or i == mx - 1:
                A.setValueStencil(row, row, 1.0)
            else:
                diag = -2.0 / self.dx ** 2
                for index, value in [
                    (i - 1, 1.0 / self.dx ** 2),
                    (i, diag),
                    (i + 1, 1.0 / self.dx ** 2),
                ]:
                    col.i = index
                    col.field = 0

params: problem parameters
            factor: temporal factor (dt*Qd)
            dx: grid spacing in x direction
        """
        assert da.getDim() == 1
        self.da = da
        self.mx = self.da.getSizes()[0]
        (self.xs, self.xe) = self.da.getRanges()[0]
        self.dx = 100 / (self.mx - 1)
        print(self.mx, self.dx, self.xs, self.xe)

        self.lambda0 = 2.0
        self.nu = 1

        self.localX = self.da.createLocalVec()
        self.row = PETSc.Mat.Stencil()
        self.col = PETSc.Mat.Stencil()

        self.mat = self.da.createMatrix()
        self.mat.setType('aij')  # sparse
        self.mat.setFromOptions()
        self.mat.setPreallocationNNZ((3, 3))
        self.mat.setUp()

        self.gvec = self.da.createGlobalVec()
        self.rhs = self.da.createGlobalVec()