_-Ph"dZddlmZddlZddlmZddlmZ ddl m Z m Z m Z ddlmZdd lmZdd lmZd d Zd!dZ d"dZd!dZd#dZd$dZ d%dZd&dZd!dZd#dZ d'dZ d'dZ d(dZ d#dZ! d)dZ" d)dZ# d*dZ$ d*dZ%dS)+z&Relaxation methods for linear systems.)warnN)sparse)lapack) type_prep get_diagonalget_block_diag)set_tol)norm)amg_corec|n|dgkrmtj|rntj|r|}nzt dtjtj|}nKtj|r |j|vrn-tj| |d}t|tj stdt|tj std|j\}}||krtd|j|f|dffvrtd |j|f|dffvrtd |j|jks|j|jkrt!d |jjstd tj|}tj|}|||fS) aWReturn A,x,b suitable for relaxation or raise an exception. Parameters ---------- A : sparse-matrix n x n system x : array n-vector, initial guess b : array n-vector, right-hand side formats: {'csr', 'csc', 'bsr', 'lil', 'dok',...} desired sparse matrix format default is no change to A's format Returns ------- (A,x,b), where A is in the desired sparse-matrix format and x and b are "raveled", i.e. (n,) vectors. Notes ----- Does some rudimentary error checking on the system, such as checking for compatible dimensions and checking for compatible type, i.e. float or complex. Examples -------- >>> from pyamg.relaxation.relaxation import make_system >>> from pyamg.gallery import poisson >>> import numpy as np >>> A = poisson((10,10), format='csr') >>> x = np.zeros((A.shape[0],1)) >>> b = np.ones((A.shape[0],1)) >>> (A,x,b) = make_system(A,x,b,formats=['csc']) >>> print(x.shape) (100,) >>> print(b.shape) (100,) >>> print(A.format) csc Ncsrzimplicit conversion to CSRrz#expected numpy array for argument xz#expected numpy array for argument bzexpected square matrixzx has invalid dimensionszb has invalid dimensionsz.arguments A, x, and b must have the same dtypezx must be contiguous in memory)risspmatrix_csrisspmatrix_bsrtocsrrSparseEfficiencyWarning csr_matrix isspmatrixformatasformat isinstancenpndarray ValueErrorshapedtype TypeErrorflagscarrayravel)AxbformatsMNs [/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/pyamg/relaxation/relaxation.py make_systemr)sV UG     # # %   "1 % % % AA -v/M N N N!!$$AA  Q   :AH$7$7 !!$$--gaj99A a $ $@>??? a $ $@>??? 7DAqAvv1222wtaVn$$3444wtaVn$$3444w!'QW//HIII 7>;9:::  A  A a7Nrforwardct|||ddg\}}}tj|}t|D]/}||dd<t |||d|||z}|d|z z}||z }0dS)aPerform SOR iteration on the linear system Ax=b. Parameters ---------- A : csr_matrix, bsr_matrix Sparse NxN matrix x : ndarray Approximate solution (length N) b : ndarray Right-hand side (length N) omega : scalar Damping parameter iterations : int Number of iterations to perform sweep : {'forward','backward','symmetric'} Direction of sweep Returns ------- Nothing, x will be modified in place. Notes ----- When omega=1.0, SOR is equivalent to Gauss-Seidel. Examples -------- >>> # Use SOR as stand-along solver >>> from pyamg.relaxation.relaxation import sor >>> from pyamg.gallery import poisson >>> from pyamg.util.linalg import norm >>> import numpy as np >>> A = poisson((10,10), format='csr') >>> x0 = np.zeros((A.shape[0],1)) >>> b = np.ones((A.shape[0],1)) >>> sor(A, x0, b, 1.33, iterations=10) >>> print(f'{norm(b-A*x0):2.4}') 3.039 >>> # >>> # Use SOR as the multigrid smoother >>> from pyamg import smoothed_aggregation_solver >>> sa = smoothed_aggregation_solver(A, B=np.ones((A.shape[0],1)), ... coarse_solver='pinv', max_coarse=50, ... presmoother=('sor', {'sweep':'symmetric', 'omega' : 1.33}), ... postsmoother=('sor', {'sweep':'symmetric', 'omega' : 1.33})) >>> x0 = np.zeros((A.shape[0],1)) >>> residuals=[] >>> x = sa.solve(b, x0=x0, tol=1e-8, residuals=residuals) rbsrr%Nr iterationssweep)r)r empty_likerange gauss_seidel)r"r#r$omegar0r1x_old_is r(sorr8esf!QE5>:::GAq! M!  EJaaaQ1%8888 U  !E' U r*c t|||dg\}}}|||tdt|||||\}}}}|dkrd|jddz d} } } nt|dkr|jdd z d d } } } nY|d krDt |D]2} t |||d||||d t |||d||||d 3dStd t |D]K} tj|j |j |j |||||||jddz |jd| | | LdS)a Perform Overlapping multiplicative Schwarz on the linear system Ax=b. Parameters ---------- A : csr_matrix, bsr_matrix Sparse NxN matrix x : ndarray Approximate solution (length N) b : ndarray Right-hand side (length N) iterations : int Number of iterations to perform subdomain : int array Linear array containing each subdomain's elements subdomain_ptr : int array Pointer in subdomain, such that subdomain[subdomain_ptr[i]:subdomain_ptr[i+1]]] contains the _sorted_ indices in subdomain i inv_subblock : int_array Linear array containing each subdomain's inverted diagonal block of A inv_subblock_ptr : int array Pointer in inv_subblock, such that inv_subblock[inv_subblock_ptr[i]:inv_subblock_ptr[i+1]]] contains the inverted diagonal block of A for the i-th subdomain in _row_ major order sweep : {'forward','backward','symmetric'} Direction of sweep Returns ------- Nothing, x will be modified in place. Notes ----- If subdomains is None, then a point-wise iteration takes place, with the overlapping region defined by each degree-of-freedom's neighbors in the matrix graph. If subdomains is not None, but subblocks is, then the subblocks are formed internally. Currently only supports CSR matrices Examples -------- >>> # Use Overlapping Schwarz as a Stand-Alone Solver >>> from pyamg.relaxation.relaxation import schwarz >>> from pyamg.gallery import poisson >>> from pyamg.util.linalg import norm >>> import numpy as np >>> A = poisson((10,10), format='csr') >>> x0 = np.zeros((A.shape[0],1)) >>> b = np.ones((A.shape[0],1)) >>> schwarz(A, x0, b, iterations=10) >>> print(f'{norm(b-A*x0):2.4}') 0.1263 >>> # >>> # Schwarz as the Multigrid Smoother >>> from pyamg import smoothed_aggregation_solver >>> sa = smoothed_aggregation_solver(A, B=np.ones((A.shape[0],1)), ... coarse_solver='pinv', max_coarse=50, ... presmoother='schwarz', ... postsmoother='schwarz') >>> x0=np.zeros((A.shape[0],1)) >>> residuals=[] >>> x = sa.solve(b, x0=x0, tol=1e-8, residuals=residuals) rr.Nz.inv_subblock must be None if subdomain is Noner+rrbackwardr symmetric)r0 subdomain subdomain_ptr inv_subblockinv_subblock_ptrr1z>valid sweep directions: 'forward', 'backward', and 'symmetric') r) sort_indicesrschwarz_parametersrr3schwarzr overlapping_schwarz_csrindptrindicesdata) r"r#r$r0r=r>r?r@r1 row_startrow_stoprow_step_iters r(rCrCsN!QE7333GAq!NN\5IJJJ 1i')9 ; ;?Y |-= ()=+>q+A!+CQX8 *  (5(;A(>q(@"bX8 +  :&& I IE Aq!Y"/l%5Y H H H H Aq!Y"/l%5Z I I I I I YZZZz""HH(19af)*A|=M)2M)6)t|D],} tj|j |j |j |||| | -d St|D]?} tj |j |j tj|j |||| | | @d S)aPerform Gauss-Seidel iteration on the linear system Ax=b. Parameters ---------- A : csr_matrix, bsr_matrix Sparse NxN matrix x : ndarray Approximate solution (length N) b : ndarray Right-hand side (length N) iterations : int Number of iterations to perform sweep : {'forward','backward','symmetric'} Direction of sweep Returns ------- Nothing, x will be modified in place. Examples -------- >>> # Use Gauss-Seidel as a Stand-Alone Solver >>> from pyamg.relaxation.relaxation import gauss_seidel >>> from pyamg.gallery import poisson >>> from pyamg.util.linalg import norm >>> import numpy as np >>> A = poisson((10,10), format='csr') >>> x0 = np.zeros((A.shape[0],1)) >>> b = np.ones((A.shape[0],1)) >>> gauss_seidel(A, x0, b, iterations=10) >>> print(f'{norm(b-A*x0):2.4}') 4.007 >>> # >>> # Use Gauss-Seidel as the Multigrid Smoother >>> from pyamg import smoothed_aggregation_solver >>> sa = smoothed_aggregation_solver(A, B=np.ones((A.shape[0],1)), ... coarse_solver='pinv', max_coarse=50, ... presmoother=('gauss_seidel', {'sweep':'symmetric'}), ... postsmoother=('gauss_seidel', {'sweep':'symmetric'})) >>> x0=np.zeros((A.shape[0],1)) >>> residuals=[] >>> x = sa.solve(b, x0=x0, tol=1e-8, residuals=residuals) rr-r.rBSR blocks must be squarer+rr:r;r<r/N>valid sweep directions: "forward", "backward", and "symmetric")r)rr blocksizerintlenr3r4r rErFrGbsr_gauss_seidelrr!) r"r#r$r0r1rORCrHrIrJrKs r(r4r4s Z!QE5>:::GAq! Q {1 66899 9  ()3s1vvi/?+@+@!X8 *  (+CFF9,<(=(=a(?RX8 +  :&& B BE AqQi @ @ @ @ AqQj A A A A AYZZZ QN:&& A AE  !!(AIqvq!"+Xx A A A A A A:&& N NE  %ah 28AF;K;K&'Ix1 N N N N N Nr*?ct|||ddg\}}}td}||jd\}}}||z |zdkrdSt j|} t |j|g\}tj |r@t|D].} tj |j |j|j||| |||| /dS|j\} } | | krt!dt#|| z }t#|| z }t|D]A} tj|j |jt j|j||| |||| | BdS)abPerform Jacobi iteration on the linear system Ax=b. Parameters ---------- A : csr_matrix Sparse NxN matrix x : ndarray Approximate solution (length N) b : ndarray Right-hand side (length N) iterations : int Number of iterations to perform omega : scalar Damping parameter Returns ------- Nothing, x will be modified in place. Examples -------- >>> # Use Jacobi as a Stand-Alone Solver >>> from pyamg.relaxation.relaxation import jacobi >>> from pyamg.gallery import poisson >>> from pyamg.util.linalg import norm >>> import numpy as np >>> A = poisson((10,10), format='csr') >>> x0 = np.zeros((A.shape[0],1)) >>> b = np.ones((A.shape[0],1)) >>> jacobi(A, x0, b, iterations=10, omega=1.0) >>> print(f'{norm(b-A*x0):2.4}') 5.835 >>> # >>> # Use Jacobi as the Multigrid Smoother >>> from pyamg import smoothed_aggregation_solver >>> sa = smoothed_aggregation_solver(A, B=np.ones((A.shape[0],1)), ... coarse_solver='pinv', max_coarse=50, ... presmoother=('jacobi', {'omega': 4.0/3.0, 'iterations' : 2}), ... postsmoother=('jacobi', {'omega': 4.0/3.0, 'iterations' : 2})) >>> x0=np.zeros((A.shape[0],1)) >>> residuals=[] >>> x = sa.solve(b, x0=x0, tol=1e-8, residuals=residuals) rr-r.NrrM)r)slicerFrrr2rrrrr3r jacobirErGrOrrP bsr_jacobir!) r"r#r$r0r5r1rHrIrJtemprKrSrTs r(rXrX_sZ!QE5>:::GAq! $KKE&+mmAGAJ&?&?#Y(9(A-- =  D%))GU Q 4:&& B BE OAHaiAt%x5 B B B B B B{1 66899 9 A && x!|$$:&& 4 4E  !)RXaf5E5E !1dIx (!U 4 4 4 4 4 4r*c,t|||ddg\}}}|||f}|t||d}nl|jdt |jd|z krt d |jd |ks|jd |krt d t d}|t |jd|z \}} } | |z | zdkrdStj |} t|j |g\}t|D]T} tj|j|jtj|j||tj|| || | || UdS) aVPerform block Jacobi iteration on the linear system Ax=b. Parameters ---------- A : csr_matrix or bsr_matrix Sparse NxN matrix x : ndarray Approximate solution (length N) b : ndarray Right-hand side (length N) Dinv : array Array holding block diagonal inverses of A size (N/blocksize, blocksize, blocksize) blocksize : int Desired dimension of blocks iterations : int Number of iterations to perform omega : scalar Damping parameter Returns ------- Nothing, x will be modified in place. Examples -------- >>> # Use block Jacobi as a Stand-Alone Solver >>> from pyamg.relaxation.relaxation import block_jacobi >>> from pyamg.gallery import poisson >>> from pyamg.util.linalg import norm >>> import numpy as np >>> A = poisson((10,10), format='csr') >>> x0 = np.zeros((A.shape[0],1)) >>> b = np.ones((A.shape[0],1)) >>> block_jacobi(A, x0, b, blocksize=4, iterations=10, omega=1.0) >>> print(f'{norm(b-A*x0):2.4}') 4.665 >>> # >>> # Use block Jacobi as the Multigrid Smoother >>> from pyamg import smoothed_aggregation_solver >>> opts = {'omega': 4.0/3.0, 'iterations' : 2, 'blocksize' : 4} >>> sa = smoothed_aggregation_solver(A, B=np.ones((A.shape[0],1)), ... coarse_solver='pinv', max_coarse=50, ... presmoother=('block_jacobi', opts), ... postsmoother=('block_jacobi', opts)) >>> x0=np.zeros((A.shape[0],1)) >>> residuals=[] >>> x = sa.solve(b, x0=x0, tol=1e-8, residuals=residuals) rr-r.rONTrOinv_flagr'Dinv and A have incompatible dimensionsrr#Dinv and blocksize are incompatible)r)tobsrr rrPrrWrFrr2rrr3r block_jacobirEr!rG) r"r#r$DinvrOr0r5r1rHrIrJrZrKs r(rbrbsf!QE5>:::GAq! 9i011A |a9tDDD A#agaj233 3 3BCCC *Q-9 $ $$*Q-9*D*D>??? $KKE&+mmC 98L4M4M&N&N#Y(9(A-- =  D%))GUz""00ah 28AF3C3CBHTNND'8#Y 0 0 0 000r*c t|||ddg\}}}|||f}|t||d}nl|jdt |jd|z krt d |jd |ks|jd |krt d |d kr$dt t ||z d } }}n|dkr't t ||z d z dd} }}nU|dkr@t|D].} t|||d d ||t|||d d||/dSt dt|D]R} tj|j |j tj |j||tj |||| | SdS)aPerform block Gauss-Seidel iteration on the linear system Ax=b. Parameters ---------- A : csr_matrix, bsr_matrix Sparse NxN matrix x : ndarray Approximate solution (length N) b : ndarray Right-hand side (length N) iterations : int Number of iterations to perform sweep : {'forward','backward','symmetric'} Direction of sweep Dinv : array Array holding block diagonal inverses of A size (N/blocksize, blocksize, blocksize) blocksize : int Desired dimension of blocks Returns ------- Nothing, x will be modified in place. Examples -------- >>> # Use Gauss-Seidel as a Stand-Alone Solver >>> from pyamg.relaxation.relaxation import block_gauss_seidel >>> from pyamg.gallery import poisson >>> from pyamg.util.linalg import norm >>> import numpy as np >>> A = poisson((10,10), format='csr') >>> x0 = np.zeros((A.shape[0],1)) >>> b = np.ones((A.shape[0],1)) >>> block_gauss_seidel(A, x0, b, iterations=10, blocksize=4, sweep='symmetric') >>> print(f'{norm(b-A*x0):2.4}') 0.9583 >>> # >>> # Use Gauss-Seidel as the Multigrid Smoother >>> from pyamg import smoothed_aggregation_solver >>> opts = {'sweep':'symmetric', 'blocksize' : 4} >>> sa = smoothed_aggregation_solver(A, B=np.ones((A.shape[0],1)), ... coarse_solver='pinv', max_coarse=50, ... presmoother=('block_gauss_seidel', opts), ... postsmoother=('block_gauss_seidel', opts)) >>> x0=np.zeros((A.shape[0],1)) >>> residuals=[] >>> x = sa.solve(b, x0=x0, tol=1e-8, residuals=residuals) rr-r.r\NTr]rr_rrr`r+r:r;r<)r0r1rOrcrN)r)rar rrPrrQr3block_gauss_seidelr rErFrr!rG) r"r#r$r0r1rOrcrHrIrJrKs r(reresj!QE5>:::GAq! 9i011A |a9tDDD A#agaj233 3 3BCCC *Q-9 $ $$*Q-9*D*D>??? ()3s1vvi/?+@+@!X8 *  (+CFF9,<(=(=a(?RX8 +  :&& ? ?E q!Q1I)2 ? ? ? ? q!Q1J)2 ? ? ? ? ?YZZZz""NN#AHai!&9I9I$%q"(4..$-x9 N N N NNNr*ct|||d\}}}t|D]H}t|dkr|}n|||zz }|d|z}|ddD] }||z||zz}||z }IdS)aApply a polynomial smoother to the system Ax=b. Parameters ---------- A : sparse matrix Sparse NxN matrix x : ndarray Approximate solution (length N) b : ndarray Right-hand side (length N) coefficients : array_like Coefficients of the polynomial. See Notes section for details. iterations : int Number of iterations to perform Returns ------- Nothing, x will be modified in place. Notes ----- The smoother has the form x[:] = x + p(A) (b - A*x) where p(A) is a polynomial in A whose scalar coefficients are specified (in descending order) by argument 'coefficients'. - Richardson iteration p(A) = c_0: polynomial_smoother(A, x, b, [c_0]) - Linear smoother p(A) = c_1*A + c_0: polynomial_smoother(A, x, b, [c_1, c_0]) - Quadratic smoother p(A) = c_2*A^2 + c_1*A + c_0: polynomial_smoother(A, x, b, [c_2, c_1, c_0]) Here, Horner's Rule is applied to avoid computing A^k directly. For efficience, the method detects the case x = 0 one matrix-vector product is avoided (since (b - A*x) is b). Examples -------- >>> # The polynomial smoother is not currently used directly >>> # in PyAMG. It is only used by the chebyshev smoothing option, >>> # which automatically calculates the correct coefficients. >>> from pyamg.gallery import poisson >>> from pyamg.util.linalg import norm >>> import numpy as np >>> from pyamg.aggregation import smoothed_aggregation_solver >>> A = poisson((10,10), format='csr') >>> b = np.ones((A.shape[0],1)) >>> sa = smoothed_aggregation_solver(A, B=np.ones((A.shape[0],1)), ... coarse_solver='pinv', max_coarse=50, ... presmoother=('chebyshev', {'degree':3, 'iterations':1}), ... postsmoother=('chebyshev', {'degree':3, 'iterations':1})) >>> x0=np.zeros((A.shape[0],1)) >>> residuals=[] >>> x = sa.solve(b, x0=x0, tol=1e-8, residuals=residuals) Nr.rr)r)r3r ) r"r#r$ coefficientsr0r7residualhcs r( polynomialrkKsx!Q4000GAq!J   77a<<HH1Q3wH OH $abb! ! !A( QqS AA Q  r*c t|||dg\}}}tj|d}|dkrdt|d}}}np|dkrt|dz d d }}}nS|d kr>t |D],} t ||||dd t ||||dd -d St d t |D]-} tj|j|j |j |||||| .d S)aPerform indexed Gauss-Seidel iteration on the linear system Ax=b. In indexed Gauss-Seidel, the sequence in which unknowns are relaxed is specified explicitly. In contrast, the standard Gauss-Seidel method always performs complete sweeps of all variables in increasing or decreasing order. The indexed method may be used to implement specialized smoothers, like F-smoothing in Classical AMG. Parameters ---------- A : csr_matrix Sparse NxN matrix x : ndarray Approximate solution (length N) b : ndarray Right-hand side (length N) indices : ndarray Row indices to relax. iterations : int Number of iterations to perform sweep : {'forward','backward','symmetric'} Direction of sweep Returns ------- Nothing, x will be modified in place. Examples -------- >>> from pyamg.gallery import poisson >>> from pyamg.relaxation.relaxation import gauss_seidel_indexed >>> import numpy as np >>> A = poisson((4,), format='csr') >>> x = np.array([0.0, 0.0, 0.0, 0.0]) >>> b = np.array([0.0, 1.0, 2.0, 3.0]) >>> gauss_seidel_indexed(A, x, b, [0,1,2,3]) # relax all rows in order >>> gauss_seidel_indexed(A, x, b, [0,1]) # relax first two rows >>> gauss_seidel_indexed(A, x, b, [2,0]) # relax row 2, then row 0 >>> gauss_seidel_indexed(A, x, b, [2,3], sweep='backward') # 3, then 2 >>> gauss_seidel_indexed(A, x, b, [2,0,2]) # relax row 2, 0, 2 rr.intcrr+rrr:r;r<r/NrN) r)rasarrayrQr3gauss_seidel_indexedrr rErFrG) r"r#r$rFr0r1rHrIrJrKs r(rprps]V!QE7333GAq!j///G ()3w<<X8 *  (+G QBX8 +  :&& 3 3E Aq'a'0 2 2 2 2 Aq'a'1 3 3 3 3 3YZZZz""EE%ah 16&'G&/8 E E E EEEr*c$t|||dg\}}}td}||jd\}}}t j|} t |dd} t|j|g\}t|D]v} t j |||zz t j | z |j} tj |j|j|j||| | |||| wdS)aPerform Jacobi iterations on the linear system A A.H x = A.H b. Also known as Cimmino relaxation Parameters ---------- A : csr_matrix Sparse NxN matrix x : ndarray Approximate solution (length N) b : ndarray Right-hand side (length N) iterations : int Number of iterations to perform omega : scalar Damping parameter Returns ------- Nothing, x will be modified in place. References ---------- .. [1] Brandt, Ta'asan. "Multigrid Method For Nearly Singular And Slightly Indefinite Problems." 1985. NASA Technical Report Numbers: ICASE-85-57; NAS 1.26:178026; NASA-CR-178026; .. [2] Kaczmarz. Angenaeherte Aufloesung von Systemen Linearer Gleichungen. Bull. Acad. Polon. Sci. Lett. A 35, 355-57. 1937 .. [3] Cimmino. La ricerca scientifica ser. II 1. Pubbliz. dell'Inst. pre le Appl. del Calculo 34, 326-333, 1938. Examples -------- >>> # Use NE Jacobi as a Stand-Alone Solver >>> from pyamg.relaxation.relaxation import jacobi_ne >>> from pyamg.gallery import poisson >>> from pyamg.util.linalg import norm >>> import numpy as np >>> A = poisson((50,50), format='csr') >>> x0 = np.zeros((A.shape[0],1)) >>> b = np.ones((A.shape[0],1)) >>> jacobi_ne(A, x0, b, iterations=10, omega=2.0/3.0) >>> print(f'{norm(b-A*x0):2.4}') 49.39 >>> # >>> # Use NE Jacobi as the Multigrid Smoother >>> from pyamg import smoothed_aggregation_solver >>> opts = {'iterations' : 2, 'omega' : 4.0/3.0} >>> sa = smoothed_aggregation_solver(A, B=np.ones((A.shape[0],1)), ... coarse_solver='pinv', max_coarse=50, ... presmoother=('jacobi_ne', opts), ... postsmoother=('jacobi_ne', opts)) >>> x0=np.zeros((A.shape[0],1)) >>> residuals=[] >>> x = sa.solve(b, x0=x0, tol=1e-8, residuals=residuals) rr.NrrTnorm_eqinv)r)rWrFrr zeros_likerrrr3r!astyper jacobi_nerErG) r"r#r$r0r5r1rHrIrJrZrcr7deltas r(rwrwsz!QE7333GAq! $KKE&+mmAGAJ&?&?#Y( =  D 1$ / / /D%))GUJ66!ac'""28D>>199!'BB18QYai#Xu 6 6 6 666r*c t|||dg\}}}|$tjt|dd}|dkrdt |d } }}nr|d krt |d z d d } }}nU|d kr@t |D].} t |||d d|| t |||d d || /dStdt |D].} tj|j |j |j ||||| || /dS)a!Perform Gauss-Seidel iterations on the linear system A A.H y = b, where x = A.H y. Also known as Kaczmarz relaxation Parameters ---------- A : csr_matrix Sparse NxN matrix x : ndarray Approximate solution (length N) b : ndarray Right-hand side (length N) iterations : int Number of iterations to perform sweep : {'forward','backward','symmetric'} Direction of sweep omega : float Relaxation parameter typically in (0, 2) if omega != 1.0, then algorithm becomes SOR on A A.H Dinv : ndarray Inverse of diag(A A.H), (length N) Returns ------- Nothing, x will be modified in place. References ---------- .. [1] Brandt, Ta'asan. "Multigrid Method For Nearly Singular And Slightly Indefinite Problems." 1985. NASA Technical Report Numbers: ICASE-85-57; NAS 1.26:178026; NASA-CR-178026; .. [2] Kaczmarz. Angenaeherte Aufloesung von Systemen Linearer Gleichungen. Bull. Acad. Polon. Sci. Lett. A 35, 355-57. 1937 Examples -------- >>> # Use NE Gauss-Seidel as a Stand-Alone Solver >>> from pyamg.relaxation.relaxation import gauss_seidel_ne >>> from pyamg.gallery import poisson >>> from pyamg.util.linalg import norm >>> import numpy as np >>> A = poisson((10,10), format='csr') >>> x0 = np.zeros((A.shape[0],1)) >>> b = np.ones((A.shape[0],1)) >>> gauss_seidel_ne(A, x0, b, iterations=10, sweep='symmetric') >>> print(f'{norm(b-A*x0):2.4}') 8.476 >>> # >>> # Use NE Gauss-Seidel as the Multigrid Smoother >>> from pyamg import smoothed_aggregation_solver >>> sa = smoothed_aggregation_solver(A, B=np.ones((A.shape[0],1)), ... coarse_solver='pinv', max_coarse=50, ... presmoother=('gauss_seidel_ne', {'sweep' : 'symmetric'}), ... postsmoother=('gauss_seidel_ne', {'sweep' : 'symmetric'})) >>> x0=np.zeros((A.shape[0],1)) >>> residuals=[] >>> x = sa.solve(b, x0=x0, tol=1e-8, residuals=residuals) rr.NrTrrr+rrr:r;r<r0r1r5rcrN) r)rr!rrQr3gauss_seidel_nerr rErFrG) r"r#r$r0r1r5rcrHrIrJrKr7s r(r{r{1sr~!QE7333GAq! |x Qt<<<== ()3q661X8 *  (+Aq"bX8 +  :&& 4 4E Aq!"'d 4 4 4 4 Aq!"'d 4 4 4 4 4YZZZJBB 19af!"Ay!)8T5 B B B BBBr*c &t|||dg\}}}|$tjt|dd}|dkrdt |d} }}nr|d krt |dz d d } }}nU|d kr@t |D].} t |||dd|| t |||dd || /dStd |||zz } t |D].} tj|j |j |j || ||| || /dS)a2Perform Gauss-Seidel iterations on the linear system A.H A x = A.H b. Parameters ---------- A : csr_matrix Sparse NxN matrix x : ndarray Approximate solution (length N) b : ndarray Right-hand side (length N) iterations : int Number of iterations to perform sweep : {'forward','backward','symmetric'} Direction of sweep omega : float Relaxation parameter typically in (0, 2) if omega != 1.0, then algorithm becomes SOR on A.H A Dinv : ndarray Inverse of diag(A.H A), (length N) Returns ------- Nothing, x will be modified in place. References ---------- .. [1] Yousef Saad, "Iterative Methods for Sparse Linear Systems, Second Edition", SIAM, pp. 247-9, 2003 http://www-users.cs.umn.edu/~saad/books.html Examples -------- >>> # Use NR Gauss-Seidel as a Stand-Alone Solver >>> from pyamg.relaxation.relaxation import gauss_seidel_nr >>> from pyamg.gallery import poisson >>> from pyamg.util.linalg import norm >>> import numpy as np >>> A = poisson((10,10), format='csr') >>> x0 = np.zeros((A.shape[0],1)) >>> b = np.ones((A.shape[0],1)) >>> gauss_seidel_nr(A, x0, b, iterations=10, sweep='symmetric') >>> print(f'{norm(b-A*x0):2.4}') 8.45 >>> # >>> # Use NR Gauss-Seidel as the Multigrid Smoother >>> from pyamg import smoothed_aggregation_solver >>> sa = smoothed_aggregation_solver(A, B=np.ones((A.shape[0],1)), ... coarse_solver='pinv', max_coarse=50, ... presmoother=('gauss_seidel_nr', {'sweep' : 'symmetric'}), ... postsmoother=('gauss_seidel_nr', {'sweep' : 'symmetric'})) >>> x0=np.zeros((A.shape[0],1)) >>> residuals=[] >>> x = sa.solve(b, x0=x0, tol=1e-8, residuals=residuals) cscr.NrTrrr+rr:r;r<rzrN) r)rr!rrQr3gauss_seidel_nrrr rErFrG) r"r#r$r0r1r5rc col_startcol_stopcol_steprKrr7s r(r~r~st!QE7333GAq! |x Qt<<<== ()3q661X8 *  (+Aq"bX8 +  :&& 4 4E Aq!"'d 4 4 4 4 Aq!"'d 4 4 4 4 4YZZZ AaCAJBB 19af!"Ay!)8T5 B B B BBBr*c t|dr}|t|rtj|jd|krr?r@rOrmy_pinvimrhsj0j1 gelssoutputs r(rBrBs,q&''(  ]%>x,Q/9<==AACC ,x,Q/=@AAEEGG ,++' 'M1  INN$$ /78M$7*+)/;;;!!""% crc(:: !y9)<==x!1"!5 7qwGGG "18QY #3Y #&}':1'=a'?#@#@!'!* N N Nqw&y(*AG(D(D(DGG}*1-a/00 ; ;A! A&AQW---C!!$B!!A#&B!',r"u"5"="=a"C"C"%Dd.2444K#%(;q>":":LB  %}l,.A r*c t|||ddg\}}}tj|d}t|j|g\}t j|r=t|D]+}tj |j |j |j ||||,d S|j \}}||krtd||jd|z dz krtd t|D]>}tj|j |j |j |||||?d S) apPerform indexed Jacobi iteration on the linear system Ax=b. The indexed method may be used to implement specialized smoothers, like F-smoothing in classical AMG. Parameters ---------- A : csr_matrix Sparse NxN matrix x : ndarray Approximate solution (length N) b : ndarray Right-hand side (length N) indices : ndarray Row indices to relax. iterations : int Number of iterations to perform omega : scalar Damping parameter Returns ------- Nothing, x will be modified in place. Examples -------- >>> from pyamg.gallery import poisson >>> from pyamg.relaxation.relaxation import jacobi_indexed >>> import numpy as np >>> A = poisson((4,), format='csr') >>> x = np.array([0.0, 0.0, 0.0, 0.0]) >>> b = np.array([0.0, 1.0, 2.0, 3.0]) >>> jacobi_indexed(A, x, b, [0,1,2,3]) # relax all rows in order >>> jacobi_indexed(A, x, b, [0,1]) # relax first two rows >>> jacobi_indexed(A, x, b, [2,0]) # relax row 2, then row 0 >>> jacobi_indexed(A, x, b, [2,3]) # relax 2 and 3 rr-r.rmrnrMrrz6Indices must range from 0, ..., numrows/blocksize - 1)N)r)rrorrrrr3r jacobi_indexedrErFrGrOrmaxrbsr_jacobi_indexedr!) r"r#r$rFr0r5rKrSrTs r(rr2s`N!QE5>:::GAq!j///G%))GU Q ;:&& W WE  #AHaiAwPU V V V V W W{1 66899 9 ;;==171:a} tj|j|j tj|j |||| |?t|D]>} tj|j|j tj|j |||| |?dS)aPerform CF Jacobi iteration on the linear system Ax=b. CF Jacobi executes xc = (1-omega)xc + omega*Dff^{-1}(bc - Acf*xf - Acc*xc) xf = (1-omega)xf + omega*Dff^{-1}(bf - Aff*xf - Afc*xc) where xf is x restricted to F-points, and likewise for c subscripts. Parameters ---------- A : csr_matrix Sparse NxN matrix x : ndarray Approximate solution (length N) b : ndarray Right-hand side (length N) Cpts : array ints List of C-points Fpts : array ints List of F-points iterations : int Number of iterations to perform of total CF-cycle f_iterations : int Number of sweeps of F-relaxation to perform c_iterations : int Number of sweeps of C-relaxation to perform omega : scalar Damping parameter Returns ------- Nothing, x will be modified in place. rr-r.rMNr)rvrErrrrr3r rrFrGrOrrrr!)r"r#r$CptsFptsr0 f_iterations c_iterationsr5rK_citer_fiterrSrTs r( cf_jacobirn%H!QE5>:::GAq! ;;qx~ & &D ;;qx~ & &D%))GU QB:&& X XE -- X X'!)QVQ4QVWWWW -- X X'!)QVQ4QVWWWW X X X {1 66899 9:&& B BE -- B B+AHai!&AQAQ,-q$5BBBB -- B B+AHai!&AQAQ,-q$5BBBB B  B Br*c t|||ddg\}}}||jj}||jj}t |j|g\}t j|rt|D]x} t|D]+} tj |j|j |j ||||,t|D]+} tj |j|j |j ||||,ydS|j \} } | | krtdt|D]} t|D]>} tj|j|j tj|j |||| |?t|D]>} tj|j|j tj|j |||| |?dS)aPerform FC Jacobi iteration on the linear system Ax=b. FC Jacobi executes xf = (1-omega)xf + omega*Dff^{-1}(bf - Aff*xf - Afc*xc) xc = (1-omega)xc + omega*Dff^{-1}(bc - Acf*xf - Acc*xc) where xf is x restricted to F-points, and likewise for c subscripts. Parameters ---------- A : csr_matrix Sparse NxN matrix x : ndarray Approximate solution (length N) b : ndarray Right-hand side (length N) Cpts : array ints List of C-points Fpts : array ints List of F-points iterations : int Number of iterations to perform of total FC-cycle f_iterations : int Number of sweeps of F-relaxation to perform c_iterations : int Number of sweeps of C-relaxation to perform omega : scalar Damping parameter Returns ------- Nothing, x will be modified in place. rr-r.rMNr)r"r#r$rrr0rrr5rKrrrSrTs r( fc_jacobirrr*c t|||ddg\}}}|||f}||jj}||jj}|t ||d}nl|jdt|jd|z krtd |jd |ks|jd |krtd t|j| g\} t|D]} t| D]Q} tj |j|j tj|j||tj||| | Rt|D]Q} tj |j|j tj|j||tj||| | RdS) aPerform CF block Jacobi iteration on the linear system Ax=b. CF block Jacobi executes xc = (1-omega)xc + omega*Dff^{-1}(bc - Acf*xf - Acc*xc) xf = (1-omega)xf + omega*Dff^{-1}(bf - Aff*xf - Afc*xc) where xf is x restricted to F-blocks, and Dff^{-1} the block inverse of the block diagonal Dff, and likewise for c subscripts. Parameters ---------- A : csr_matrix or bsr_matrix Sparse NxN matrix x : ndarray Approximate solution (length N) b : ndarray Right-hand side (length N) Cpts : array ints List of C-blocks in A Fpts : array ints List of F-blocks in A Dinv : array Array holding block diagonal inverses of A size (N/blocksize, blocksize, blocksize) blocksize : int Desired dimension of blocks iterations : int Number of iterations to perform of total CF-cycle f_iterations : int Number of sweeps of F-relaxation to perform c_iterations : int Number of sweeps of C-relaxation to perform omega : scalar Damping parameter Returns ------- Nothing, x will be modified in place. rr-r.r\NTr]rr_rrr`r)rarvrErr rrPrrr3r block_jacobi_indexedrFrr!rG)r"r#r$rrrcrOr0rrr5rKrrs r(cf_block_jacobirV!QE5>:::GAq! 9i011A ;;qx~ & &D ;;qx~ & &D |a9tDDD A#agaj233 3 3BCCC *Q-9 $ $$*Q-9*D*D>???%))GUz""55L)) 5 5F  )!(AIrx?O?O*+Qe*3 5 5 5 5L)) 5 5F  )!(AIrx?O?O*+Qe*3 5 5 5 5 5 55r*c t|||ddg\}}}|||f}||jj}||jj}|t ||d}nl|jdt|jd|z krtd |jd |ks|jd |krtd t|j| g\} t|D]} t|D]Q} tj |j|j tj|j||tj||| | Rt| D]Q} tj |j|j tj|j||tj||| | RdS) aPerform FC block Jacobi iteration on the linear system Ax=b. FC block Jacobi executes xf = (1-omega)xf + omega*Dff^{-1}(bf - Aff*xf - Afc*xc) xc = (1-omega)xc + omega*Dff^{-1}(bc - Acf*xf - Acc*xc) where xf is x restricted to F-blocks, and Dff^{-1} the block inverse of the block diagonal Dff, and likewise for c subscripts. Parameters ---------- A : csr_matrix or bsr_matrix Sparse NxN matrix x : ndarray Approximate solution (length N) b : ndarray Right-hand side (length N) Cpts : array ints List of C-blocks in A Fpts : array ints List of F-blocks in A Dinv : array Array holding block diagonal inverses of A size (N/blocksize, blocksize, blocksize) blocksize : int Desired dimension of blocks iterations : int Number of iterations to perform of total FC-cycle f_iterations : int Number of sweeps of F-relaxation to perform c_iterations : int Number of sweeps of C-relaxation to perform omega : scalar Damping parameter Returns ------- Nothing, x will be modified in place. rr-r.r\NTr]rr_rrr`r)r"r#r$rrrcrOr0rrr5rKrrs r(fc_block_jacobir5rr*)N)rr+)rNNNNr+)rrU)NrrrU)rr+rNr)rr+rUN)NNNN)rrrrU)NrrrrrU)&__doc__warningsrnumpyrscipyr scipy.linalgrr util.utilsrrr util.paramsr util.linalgr r r)r8rCr4rXrbrerkrprwr{r~rBrrrrrr*r(rs,,%%%%%%@@@@@@@@@@!!!!!!SSSSl>>>>BBF