def energy_prolongation_smoother(A, T, Atilde, B, Bf, Cpt_params,
krylov='cg', maxiter=4, tol=1e-8,
degree=1, weighting='local',
prefilter={}, postfilter={}):
"""Minimize the energy of the coarse basis functions (columns of T). Both
root-node and non-root-node style prolongation smoothing is available, see
Cpt_params description below.
Parameters
----------
A : {csr_matrix, bsr_matrix}
Sparse NxN matrix
T : {bsr_matrix}
Tentative prolongator, a NxM sparse matrix (M < N)
Atilde : {csr_matrix}
Strength of connection matrix
B : {array}
Near-nullspace modes for coarse grid. Has shape (M,k) where
k is the number of coarse candidate vectors.
Bf : {array}
Near-nullspace modes for fine grid. Has shape (N,k) where
k is the number of coarse candidate vectors.
Cpt_params : {tuple}
Tuple of the form (bool, dict). If the Cpt_params[0] = False, then the
standard SA prolongation smoothing is carried out. If True, then
root-node style prolongation smoothing is carried out. The dict must
be a dictionary of parameters containing, (1) P_I: P_I.T is the
injection matrix for the Cpts, (2) I_F: an identity matrix for only the
F-points (i.e. I, but with zero rows and columns for C-points) and I_C:
the C-point analogue to I_F. See Notes below for more information.
krylov : {string}
'cg' : for SPD systems. Solve A T = 0 in a constraint space with CG
'cgnr' : for nonsymmetric and/or indefinite systems.
Solve A T = 0 in a constraint space with CGNR
'gmres' : for nonsymmetric and/or indefinite systems.
Solve A T = 0 in a constraint space with GMRES
maxiter : integer
Number of energy minimization steps to apply to the prolongator
tol : {scalar}
Minimization tolerance
degree : {int}
Generate sparsity pattern for P based on (Atilde^degree T)
weighting : {string}
'block', 'diagonal' or 'local' construction of the
diagonal preconditioning
'local': Uses a local row-wise weight based on the Gershgorin estimate.
Avoids any potential under-damping due to inaccurate spectral
radius estimates.
'block': If A is a BSR matrix, use a block diagonal inverse of A
'diagonal': Use inverse of the diagonal of A
prefilter : {dictionary} : Default {}
Filters elements by row in sparsity pattern for P to reduce operator and
setup complexity. If None or empty dictionary, no dropping in P is done.
If postfilter has key 'k', then the largest 'k' entries are kept in each
row. If postfilter has key 'theta', all entries such that
P[i,j] < kwargs['theta']*max(abs(P[i,:]))
are dropped. If postfilter['k'] and postfiler['theta'] are present, then
they are used in conjunction, with the union of their patterns used.
postfilter : {dictionary} : Default {}
Filters elements by row in smoothed P to reduce operator complexity.
Only supported if using the rootnode_solver. If None or empty dictionary,
no dropping in P is done. If postfilter has key 'k', then the largest 'k'
entries are kept in each row. If postfilter has key 'theta', all entries
such that
P[i,j] < kwargs['theta']*max(abs(P[i,:]))
are dropped. If postfilter['k'] and postfiler['theta'] are present, then
they are used in conjunction, with the union of their patterns used.
Returns
-------
T : {bsr_matrix}
Smoothed prolongator
Notes
-----
Only 'diagonal' weighting is supported for the CGNR method, because
we are working with A^* A and not A.
When Cpt_params[0] == True, root-node style prolongation smoothing
is used to minimize the energy of columns of T. Essentially, an
identity block is maintained in T, corresponding to injection from
the coarse-grid to the fine-grid root-nodes. See [2] for more details,
and see util.utils.get_Cpt_params for the helper function to generate
Cpt_params.
If Cpt_params[0] == False, the energy of columns of T are still
minimized, but without maintaining the identity block.
Examples
--------
>>> from pyamg.aggregation import energy_prolongation_smoother
>>> from pyamg.gallery import poisson
>>> from scipy.sparse import coo_matrix
>>> import numpy as np
>>> data = np.ones((6,))
>>> row = np.arange(0,6)
>>> col = np.kron([0,1],np.ones((3,)))
>>> T = coo_matrix((data,(row,col)),shape=(6,2)).tocsr()
>>> print T.todense()
[[ 1. 0.]
[ 1. 0.]
[ 1. 0.]
[ 0. 1.]
[ 0. 1.]
[ 0. 1.]]
>>> A = poisson((6,),format='csr')
>>> B = np.ones((2,1),dtype=float)
>>> P = energy_prolongation_smoother(A,T,A,B, None, (False,{}))
>>> print P.todense()
[[ 1. 0. ]
[ 1. 0. ]
[ 0.66666667 0.33333333]
[ 0.33333333 0.66666667]
[ 0. 1. ]
[ 0. 1. ]]
References
----------
.. [1] Jan Mandel, Marian Brezina, and Petr Vanek
"Energy Optimization of Algebraic Multigrid Bases"
Computing 62, 205-228, 1999
http://dx.doi.org/10.1007/s006070050022
.. [2] Olson, L. and Schroder, J. and Tuminaro, R.,
"A general interpolation strategy for algebraic
multigrid using energy minimization", SIAM Journal
on Scientific Computing (SISC), vol. 33, pp.
966--991, 2011.
"""
# Test Inputs
if maxiter < 0:
raise ValueError('maxiter must be > 0')
if tol > 1:
raise ValueError('tol must be <= 1')
if sparse.isspmatrix_csr(A):
A = A.tobsr(blocksize=(1, 1), copy=False)
elif sparse.isspmatrix_bsr(A):
pass
else:
raise TypeError("A must be csr_matrix or bsr_matrix")
if sparse.isspmatrix_csr(T):
T = T.tobsr(blocksize=(1, 1), copy=False)
elif sparse.isspmatrix_bsr(T):
pass
else:
raise TypeError("T must be csr_matrix or bsr_matrix")
if T.blocksize[0] != A.blocksize[0]:
raise ValueError("T row-blocksize should be the same as A blocksize")
if B.shape[0] != T.shape[1]:
raise ValueError("B is the candidates for the coarse grid. \
num_rows(b) = num_cols(T)")
if min(T.nnz, A.nnz) == 0:
return T
if not sparse.isspmatrix_csr(Atilde):
raise TypeError("Atilde must be csr_matrix")
if ('theta' in prefilter) and (prefilter['theta'] == 0):
prefilter.pop('theta', None)
if ('theta' in postfilter) and (postfilter['theta'] == 0):
postfilter.pop('theta', None)
# Prepocess Atilde, the strength matrix
if Atilde is None:
Atilde = sparse.csr_matrix((np.ones(len(A.indices)),
A.indices.copy(), A.indptr.copy()),
shape=(A.shape[0]/A.blocksize[0],
A.shape[1]/A.blocksize[1]))
# If Atilde has no nonzeros, then return T
if min(T.nnz, A.nnz) == 0:
return T
# Expand allowed sparsity pattern for P through multiplication by Atilde
if degree > 0:
# Construct Sparsity_Pattern by multiplying with Atilde
T.sort_indices()
shape = (int(T.shape[0]/T.blocksize[0]), int(T.shape[1]/T.blocksize[1]))
Sparsity_Pattern = sparse.csr_matrix((np.ones(T.indices.shape),
T.indices, T.indptr),
shape=shape)
AtildeCopy = Atilde.copy()
AtildeCopy.data[:] = 1.0
for i in range(degree):
Sparsity_Pattern = AtildeCopy*Sparsity_Pattern
# Optional filtering of sparsity pattern before smoothing
if 'theta' in prefilter and 'k' in prefilter:
Sparsity_theta = filter_matrix_rows(Sparsity_Pattern, prefilter['theta'])
Sparsity_Pattern = truncate_rows(Sparsity_Pattern, prefilter['k'])
# Union two sparsity patterns
Sparsity_Pattern += Sparsity_theta
elif 'k' in prefilter:
Sparsity_Pattern = truncate_rows(Sparsity_Pattern, prefilter['k'])
elif 'theta' in prefilter:
Sparsity_Pattern = filter_matrix_rows(Sparsity_Pattern, prefilter['theta'])
elif len(prefilter) > 0:
raise ValueError("Unrecognized prefilter option")
# UnAmal returns a BSR matrix with 1's in the nonzero locations
Sparsity_Pattern = UnAmal(Sparsity_Pattern,
T.blocksize[0], T.blocksize[1])
Sparsity_Pattern.sort_indices()
else:
# If degree is 0, just copy T for the sparsity pattern
Sparsity_Pattern = T.copy()
if 'theta' in prefilter and 'k' in prefilter:
Sparsity_theta = filter_matrix_rows(Sparsity_Pattern, prefilter['theta'])
Sparsity_Pattern = truncate_rows(Sparsity_Pattern, prefilter['k'])
# Union two sparsity patterns
Sparsity_Pattern += Sparsity_theta
elif 'k' in prefilter:
Sparsity_Pattern = truncate_rows(Sparsity_Pattern, prefilter['k'])
elif 'theta' in prefilter:
Sparsity_Pattern = filter_matrix_rows(Sparsity_Pattern, prefilter['theta'])
elif len(prefilter) > 0:
raise ValueError("Unrecognized prefilter option")
Sparsity_Pattern.data[:] = 1.0
Sparsity_Pattern.sort_indices()
# If using root nodes, enforce identity at C-points
if Cpt_params[0]:
Sparsity_Pattern = Cpt_params[1]['I_F'] * Sparsity_Pattern
Sparsity_Pattern = Cpt_params[1]['P_I'] + Sparsity_Pattern
# Construct array of inv(Bi'Bi), where Bi is B restricted to row i's
# sparsity pattern in Sparsity Pattern. This array is used multiple times
# in Satisfy_Constraints(...).
BtBinv = compute_BtBinv(B, Sparsity_Pattern)
# If using root nodes and B has more columns that A's blocksize, then
# T must be updated so that T*B = Bfine. Note, if this is a 'secondpass'
# after dropping entries in P, then we must re-enforce the constraints
if (Cpt_params[0] and (B.shape[1] > A.blocksize[0])) or ('secondpass' in postfilter):
T = filter_operator(T, Sparsity_Pattern, B, Bf, BtBinv)
# Ensure identity at C-pts
if Cpt_params[0]:
T = Cpt_params[1]['I_F']*T + Cpt_params[1]['P_I']
# Iteratively minimize the energy of T subject to the constraints of
# Sparsity_Pattern and maintaining T's effect on B, i.e. T*B =
# (T+Update)*B, i.e. Update*B = 0
if krylov == 'cg':
T = cg_prolongation_smoothing(A, T, B, BtBinv, Sparsity_Pattern,
maxiter, tol, weighting, Cpt_params)
elif krylov == 'cgnr':
T = cgnr_prolongation_smoothing(A, T, B, BtBinv, Sparsity_Pattern,
maxiter, tol, weighting, Cpt_params)
elif krylov == 'gmres':
T = gmres_prolongation_smoothing(A, T, B, BtBinv, Sparsity_Pattern,
maxiter, tol, weighting, Cpt_params)
T.eliminate_zeros()
# Filter entries in P, only in the rootnode case, i.e., Cpt_params[0] == True
if (len(postfilter) == 0) or ('secondpass' in postfilter) or (Cpt_params[0] is False):
return T
else:
if 'theta' in postfilter and 'k' in postfilter:
T_theta = filter_matrix_rows(T, postfilter['theta'])
T_k = truncate_rows(T, postfilter['k'])
# Union two sparsity patterns
T_theta.data[:] = 1.0
T_k.data[:] = 1.0
T_filter = T_theta + T_k
T_filter.data[:] = 1.0
T_filter = T.multiply(T_filter)
elif 'k' in postfilter:
T_filter = truncate_rows(T, postfilter['k'])
elif 'theta' in postfilter:
T_filter = filter_matrix_rows(T, postfilter['theta'])
else:
raise ValueError("Unrecognized postfilter option")
# Re-smooth T_filter and re-fit the modes B into the span.
# Note, we set 'secondpass', because this is the second
# filtering pass
T = energy_prolongation_smoother(A, T_filter,
Atilde, B, Bf, Cpt_params,
krylov=krylov, maxiter=1,
tol=1e-8, degree=0,
weighting=weighting,
prefilter={},
postfilter={'secondpass' : True} )
return T
def energy_prolongation_smoother(A, T, Atilde, B, Bf, Cpt_params, krylov='cg', maxiter=4,
tol=1e-8, degree=1, weighting='local'):
"""Minimize the energy of the coarse basis functions (columns of T). Both
root-node and non-root-node style prolongation smoothing is available, see
Cpt_params description below.
Parameters
----------
A : {csr_matrix, bsr_matrix}
Sparse NxN matrix
T : {bsr_matrix}
Tentative prolongator, a NxM sparse matrix (M < N)
Atilde : {csr_matrix}
Strength of connection matrix
B : {array}
Near-nullspace modes for coarse grid. Has shape (M,k) where
k is the number of coarse candidate vectors.
Bf : {array}
Near-nullspace modes for fine grid. Has shape (N,k) where
k is the number of coarse candidate vectors.
Cpt_params : {tuple}
Tuple of the form (bool, dict). If the Cpt_params[0] = False, then the
standard SA prolongation smoothing is carried out. If True, then
root-node style prolongation smoothing is carried out. The dict must
be a dictionary of parameters containing, (1) P_I: P_I.T is the
injection matrix for the Cpts, (2) I_F: an identity matrix for only the
F-points (i.e. I, but with zero rows and columns for C-points) and I_C:
the C-point analogue to I_F. See Notes below for more information.
krylov : {string}
'cg' : for SPD systems. Solve A T = 0 in a constraint space with CG
'cgnr' : for nonsymmetric and/or indefinite systems.
Solve A T = 0 in a constraint space with CGNR
'gmres' : for nonsymmetric and/or indefinite systems.
Solve A T = 0 in a constraint space with GMRES
maxiter : integer
Number of energy minimization steps to apply to the prolongator
tol : {scalar}
Minimization tolerance
degree : {int}
Generate sparsity pattern for P based on (Atilde^degree T)
weighting : {string}
'block', 'diagonal' or 'local' construction of the diagonal preconditioning
'local': Uses a local row-wise weight based on the Gershgorin estimate.
Avoids any potential under-damping due to inaccurate spectral radius
estimates.
'block': If A is a BSR matrix, use a block diagonal inverse of A
'diagonal': Use inverse of the diagonal of A
Returns
-------
T : {bsr_matrix}
Smoothed prolongator
Notes
-----
Only 'diagonal' weighting is supported for the CGNR method, because
we are working with A^* A and not A.
When Cpt_params[0] == True, root-node style prolongation smoothing
is used to minimize the energy of columns of T. Essentially, an
identity block is maintained in T, corresponding to injection from
the coarse-grid to the fine-grid root-nodes. See [2] for more details,
and see util.utils.get_Cpt_params for the helper function to generate
Cpt_params.
If Cpt_params[0] == False, the energy of columns of T are still
minimized, but without maintaining the identity block.
Examples
--------
>>> from pyamg.aggregation import energy_prolongation_smoother
>>> from pyamg.gallery import poisson
>>> from scipy.sparse import coo_matrix
>>> import numpy
>>> data = numpy.ones((6,))
>>> row = numpy.arange(0,6)
>>> col = numpy.kron([0,1],numpy.ones((3,)))
>>> T = coo_matrix((data,(row,col)),shape=(6,2)).tocsr()
>>> print T.todense()
[[ 1. 0.]
[ 1. 0.]
[ 1. 0.]
[ 0. 1.]
[ 0. 1.]
[ 0. 1.]]
>>> A = poisson((6,),format='csr')
>>> P = energy_prolongation_smoother(A,T,A,numpy.ones((2,1),dtype=float), None, (False,{}) )
>>> print P.todense()
[[ 1. 0. ]
[ 1. 0. ]
[ 0.66666667 0.33333333]
[ 0.33333333 0.66666667]
[ 0. 1. ]
[ 0. 1. ]]
References
----------
.. [1] Jan Mandel, Marian Brezina, and Petr Vanek
"Energy Optimization of Algebraic Multigrid Bases"
Computing 62, 205-228, 1999
http://dx.doi.org/10.1007/s006070050022
.. [2] Olson, L. and Schroder, J. and Tuminaro, R.,
"A general interpolation strategy for algebraic
multigrid using energy minimization", SIAM Journal
on Scientific Computing (SISC), vol. 33, pp.
966--991, 2011.
"""
#====================================================================
#Test Inputs
if maxiter < 0:
raise ValueError('maxiter must be > 0')
if tol > 1:
raise ValueError('tol must be <= 1')
if isspmatrix_csr(A):
A = A.tobsr(blocksize=(1,1), copy=False)
elif isspmatrix_bsr(A):
pass
else:
raise TypeError("A must be csr_matrix or bsr_matrix")
if isspmatrix_csr(T):
T = T.tobsr(blocksize=(1,1), copy=False)
elif isspmatrix_bsr(T):
pass
else:
raise TypeError("T must be csr_matrix or bsr_matrix")
if Atilde is None:
AtildeCopy = csr_matrix( (numpy.ones(len(A.indices)), A.indices.copy(), A.indptr.copy()),
shape=(A.shape[0]/A.blocksize[0], A.shape[1]/A.blocksize[1]))
else:
AtildeCopy = Atilde.copy()
if not isspmatrix_csr(AtildeCopy):
raise TypeError("Atilde must be csr_matrix")
if T.blocksize[0] != A.blocksize[0]:
raise ValueError("T's row-blocksize should be the same as A's blocksize")
if B.shape[0] != T.shape[1]:
raise ValueError("B is the candidates for the coarse grid. \
num_rows(b) = num_cols(T)")
if min(T.nnz, AtildeCopy.nnz, A.nnz) == 0:
return T
##
# Expand the allowed sparsity pattern for P through multiplication by Atilde
T.sort_indices()
Sparsity_Pattern = csr_matrix( (numpy.ones(T.indices.shape), T.indices, T.indptr),
shape=(T.shape[0]/T.blocksize[0],T.shape[1]/T.blocksize[1]) )
AtildeCopy.data[:] = 1.0
for i in range(degree):
Sparsity_Pattern = AtildeCopy*Sparsity_Pattern
##
#UnAmal returns a BSR matrix
Sparsity_Pattern = UnAmal(Sparsity_Pattern, T.blocksize[0], T.blocksize[1])
Sparsity_Pattern.sort_indices()
##
#If using root nodes, enforce identity at C-points
if Cpt_params[0]:
Sparsity_Pattern = Cpt_params[1]['I_F']*Sparsity_Pattern
Sparsity_Pattern = Cpt_params[1]['P_I'] + Sparsity_Pattern
##
# Construct array of inv(Bi'Bi), where Bi is B restricted to row i's sparsity pattern in
# Sparsity Pattern. This array is used multiple times in Satisfy_Constraints(...).
BtBinv = compute_BtBinv(B, Sparsity_Pattern)
##
# If using root nodes and B has more columns that A's blocksize, then
# T must be updated so that T*B = Bfine
if Cpt_params[0] and (B.shape[1] > A.blocksize[0]):
T = filter_operator(T, Sparsity_Pattern, B, Bf, BtBinv)
# Ensure identity at C-pts
if Cpt_params[0]:
T = Cpt_params[1]['I_F']*T + Cpt_params[1]['P_I']
##
# Iteratively minimize the energy of T subject to the constraints of Sparsity_Pattern
# and maintaining T's effect on B, i.e. T*B = (T+Update)*B, i.e. Update*B = 0
if krylov == 'cg':
T = cg_prolongation_smoothing(A, T, B, BtBinv, Sparsity_Pattern, maxiter, tol, weighting, Cpt_params)
elif krylov == 'cgnr':
T = cgnr_prolongation_smoothing(A, T, B, BtBinv, Sparsity_Pattern, maxiter, tol, weighting, Cpt_params)
elif krylov == 'gmres':
T = gmres_prolongation_smoothing(A, T, B, BtBinv, Sparsity_Pattern, maxiter, tol, weighting, Cpt_params)
T.eliminate_zeros()
return T
def jacobi_prolongation_smoother(S,
T,
C,
B,
omega=4.0 / 3.0,
degree=1,
filter=False,
weighting='diagonal'):
"""Jacobi prolongation smoother
Parameters
----------
S : {csr_matrix, bsr_matrix}
Sparse NxN matrix used for smoothing. Typically, A.
T : {csr_matrix, bsr_matrix}
Tentative prolongator
C : {csr_matrix, bsr_matrix}
Strength-of-connection matrix
B : {array}
Near nullspace modes for the coarse grid such that T*B
exactly reproduces the fine grid near nullspace modes
omega : {scalar}
Damping parameter
filter : {boolean}
If true, filter S before smoothing T. This option can greatly control
complexity.
weighting : {string}
'block', 'diagonal' or 'local' weighting for constructing the Jacobi D
'local': Uses a local row-wise weight based on the Gershgorin estimate.
Avoids any potential under-damping due to inaccurate spectral radius
estimates.
'block': If A is a BSR matrix, use a block diagonal inverse of A
'diagonal': Classic Jacobi D = diagonal(A)
Returns
-------
P : {csr_matrix, bsr_matrix}
Smoothed (final) prolongator defined by P = (I - omega/rho(K) K) * T
where K = diag(S)^-1 * S and rho(K) is an approximation to the
spectral radius of K.
Notes
-----
If weighting is not 'local', then results using Jacobi prolongation
smoother are not precisely reproducible due to a random initial guess used
for the spectral radius approximation. For precise reproducibility,
set numpy.random.seed(..) to the same value before each test.
Examples
--------
>>> from pyamg.aggregation import jacobi_prolongation_smoother
>>> from pyamg.gallery import poisson
>>> from scipy.sparse import coo_matrix
>>> import numpy as np
>>> data = np.ones((6,))
>>> row = np.arange(0,6)
>>> col = np.kron([0,1],np.ones((3,)))
>>> T = coo_matrix((data,(row,col)),shape=(6,2)).tocsr()
>>> T.todense()
matrix([[ 1., 0.],
[ 1., 0.],
[ 1., 0.],
[ 0., 1.],
[ 0., 1.],
[ 0., 1.]])
>>> A = poisson((6,),format='csr')
>>> P = jacobi_prolongation_smoother(A,T,A,np.ones((2,1)))
>>> P.todense()
matrix([[ 0.64930164, 0. ],
[ 1. , 0. ],
[ 0.64930164, 0.35069836],
[ 0.35069836, 0.64930164],
[ 0. , 1. ],
[ 0. , 0.64930164]])
"""
# preprocess weighting
if weighting == 'block':
if sparse.isspmatrix_csr(S):
weighting = 'diagonal'
elif sparse.isspmatrix_bsr(S):
if S.blocksize[0] == 1:
weighting = 'diagonal'
if filter:
# Implement filtered prolongation smoothing for the general case by
# utilizing satisfy constraints
if sparse.isspmatrix_bsr(S):
numPDEs = S.blocksize[0]
else:
numPDEs = 1
# Create a filtered S with entries dropped that aren't in C
C = UnAmal(C, numPDEs, numPDEs)
S = S.multiply(C)
S.eliminate_zeros()
if weighting == 'diagonal':
# Use diagonal of S
D_inv = get_diagonal(S, inv=True)
D_inv_S = scale_rows(S, D_inv, copy=True)
D_inv_S = (omega / approximate_spectral_radius(D_inv_S)) * D_inv_S
elif weighting == 'block':
# Use block diagonal of S
D_inv = get_block_diag(S, blocksize=S.blocksize[0], inv_flag=True)
D_inv = sparse.bsr_matrix(
(D_inv, np.arange(D_inv.shape[0]), np.arange(D_inv.shape[0] + 1)),
shape=S.shape)
D_inv_S = D_inv * S
D_inv_S = (omega / approximate_spectral_radius(D_inv_S)) * D_inv_S
elif weighting == 'local':
# Use the Gershgorin estimate as each row's weight, instead of a global
# spectral radius estimate
D = np.abs(S) * np.ones((S.shape[0], 1), dtype=S.dtype)
D_inv = np.zeros_like(D)
D_inv[D != 0] = 1.0 / np.abs(D[D != 0])
D_inv_S = scale_rows(S, D_inv, copy=True)
D_inv_S = omega * D_inv_S
else:
raise ValueError('Incorrect weighting option')
if filter:
# Carry out Jacobi, but after calculating the prolongator update, U,
# apply satisfy constraints so that U*B = 0
P = T
for i in range(degree):
U = (D_inv_S * P).tobsr(blocksize=P.blocksize)
# Enforce U*B = 0 (1) Construct array of inv(Bi'Bi), where Bi is B
# restricted to row i's sparsity pattern in Sparsity Pattern. This
# array is used multiple times in Satisfy_Constraints(...).
BtBinv = compute_BtBinv(B, U)
# (2) Apply satisfy constraints
Satisfy_Constraints(U, B, BtBinv)
# Update P
P = P - U
else:
# Carry out Jacobi as normal
P = T
for i in range(degree):
P = P - (D_inv_S * P)
return P
