def test_full_rank(self): eps = 1.0e-12 # fixed precision A = np.random.rand(16, 8) k, idx, proj = pymatrixid.interp_decomp(A, eps) assert_(k == A.shape[1]) P = pymatrixid.reconstruct_interp_matrix(idx, proj) B = pymatrixid.reconstruct_skel_matrix(A, k, idx) assert_allclose(A, B.dot(P)) # fixed rank idx, proj = pymatrixid.interp_decomp(A, k) P = pymatrixid.reconstruct_interp_matrix(idx, proj) B = pymatrixid.reconstruct_skel_matrix(A, k, idx) assert_allclose(A, B.dot(P))
def test_full_rank(self): eps = 1.0e-12 # fixed precision A = np.random.rand(16, 8) k, idx, proj = pymatrixid.interp_decomp(A, eps) assert_equal(k, A.shape[1]) P = pymatrixid.reconstruct_interp_matrix(idx, proj) B = pymatrixid.reconstruct_skel_matrix(A, k, idx) assert_allclose(A, B @ P) # fixed rank idx, proj = pymatrixid.interp_decomp(A, k) P = pymatrixid.reconstruct_interp_matrix(idx, proj) B = pymatrixid.reconstruct_skel_matrix(A, k, idx) assert_allclose(A, B @ P)
def test_real_id_skel_and_interp_matrices(self, A, L, eps, rank, rand, lin_op): k = rank A_or_L = A if not lin_op else L idx, proj = pymatrixid.interp_decomp(A_or_L, k, rand=rand) P = pymatrixid.reconstruct_interp_matrix(idx, proj) B = pymatrixid.reconstruct_skel_matrix(A, k, idx) assert_allclose(B, A[:, idx[:k]], rtol=eps, atol=1e-08) assert_allclose(B @ P, A, rtol=eps, atol=1e-08)
def __row_extraction(self): A = self.A Q = self.Q m, n = A.shape m, k = Q.shape idx, proj = sli.interp_decomp(Q, k, rand=False) Qj = sli.reconstruct_skel_matrix(Q, k, idx) X = sli.reconstruct_interp_matrix(idx, proj) Qj = Qj.T X = X.T V, R = linalg.qr(X) Ajj = A[np.ix_(idx, idx)] Z = np.dot(R, np.dot(Ajj, R)) w, W = linalg.eig(Z) U = np.dot(V, W) return w, U
def __row_extraction(self): A = self.A Q = self.Q m, n = A.shape m, k = Q.shape idx, proj = sli.interp_decomp(Q, k, rand=False) Qj = sli.reconstruct_skel_matrix(Q, k, idx) X = sli.reconstruct_interp_matrix(idx, proj) Qj = Qj.T X = X.T Aj = A[idx, :] W, R = linalg.qr(Aj.T) Z = np.dot(X, R.T) U, sigma, Vt = linalg.svd(Z) V = np.dot(W, Vt.T) Vt = V.T return U, sigma, Vt
def check_id(self, dtype): # Test ID routines on a Hilbert matrix. # set parameters n = 300 eps = 1e-12 # construct Hilbert matrix A = hilbert(n).astype(dtype) if np.issubdtype(dtype, np.complexfloating): A = A * (1 + 1j) L = aslinearoperator(A) # find rank S = np.linalg.svd(A, compute_uv=False) try: rank = np.nonzero(S < eps)[0][0] except: rank = n # print input summary _debug_print("Hilbert matrix dimension: %8i" % n) _debug_print("Working precision: %8.2e" % eps) _debug_print("Rank to working precision: %8i" % rank) # set print format fmt = "%8.2e (s) / %5s" # test real ID routines _debug_print("-----------------------------------------") _debug_print("Real ID routines") _debug_print("-----------------------------------------") # fixed precision _debug_print("Calling iddp_id / idzp_id ...", ) t0 = time.clock() k, idx, proj = pymatrixid.interp_decomp(A, eps, rand=False) t = time.clock() - t0 B = pymatrixid.reconstruct_matrix_from_id(A[:, idx[:k]], idx, proj) _debug_print(fmt % (t, np.allclose(A, B, eps))) assert_(np.allclose(A, B, eps)) _debug_print("Calling iddp_aid / idzp_aid ...", ) t0 = time.clock() k, idx, proj = pymatrixid.interp_decomp(A, eps) t = time.clock() - t0 B = pymatrixid.reconstruct_matrix_from_id(A[:, idx[:k]], idx, proj) _debug_print(fmt % (t, np.allclose(A, B, eps))) assert_(np.allclose(A, B, eps)) _debug_print("Calling iddp_rid / idzp_rid ...", ) t0 = time.clock() k, idx, proj = pymatrixid.interp_decomp(L, eps) t = time.clock() - t0 B = pymatrixid.reconstruct_matrix_from_id(A[:, idx[:k]], idx, proj) _debug_print(fmt % (t, np.allclose(A, B, eps))) assert_(np.allclose(A, B, eps)) # fixed rank k = rank _debug_print("Calling iddr_id / idzr_id ...", ) t0 = time.clock() idx, proj = pymatrixid.interp_decomp(A, k, rand=False) t = time.clock() - t0 B = pymatrixid.reconstruct_matrix_from_id(A[:, idx[:k]], idx, proj) _debug_print(fmt % (t, np.allclose(A, B, eps))) assert_(np.allclose(A, B, eps)) _debug_print("Calling iddr_aid / idzr_aid ...", ) t0 = time.clock() idx, proj = pymatrixid.interp_decomp(A, k) t = time.clock() - t0 B = pymatrixid.reconstruct_matrix_from_id(A[:, idx[:k]], idx, proj) _debug_print(fmt % (t, np.allclose(A, B, eps))) assert_(np.allclose(A, B, eps)) _debug_print("Calling iddr_rid / idzr_rid ...", ) t0 = time.clock() idx, proj = pymatrixid.interp_decomp(L, k) t = time.clock() - t0 B = pymatrixid.reconstruct_matrix_from_id(A[:, idx[:k]], idx, proj) _debug_print(fmt % (t, np.allclose(A, B, eps))) assert_(np.allclose(A, B, eps)) # check skeleton and interpolation matrices idx, proj = pymatrixid.interp_decomp(A, k, rand=False) P = pymatrixid.reconstruct_interp_matrix(idx, proj) B = pymatrixid.reconstruct_skel_matrix(A, k, idx) assert_(np.allclose(B, A[:, idx[:k]], eps)) assert_(np.allclose(B.dot(P), A, eps)) # test SVD routines _debug_print("-----------------------------------------") _debug_print("SVD routines") _debug_print("-----------------------------------------") # fixed precision _debug_print("Calling iddp_svd / idzp_svd ...", ) t0 = time.clock() U, S, V = pymatrixid.svd(A, eps, rand=False) t = time.clock() - t0 B = np.dot(U, np.dot(np.diag(S), V.T.conj())) _debug_print(fmt % (t, np.allclose(A, B, eps))) assert_(np.allclose(A, B, eps)) _debug_print("Calling iddp_asvd / idzp_asvd...", ) t0 = time.clock() U, S, V = pymatrixid.svd(A, eps) t = time.clock() - t0 B = np.dot(U, np.dot(np.diag(S), V.T.conj())) _debug_print(fmt % (t, np.allclose(A, B, eps))) assert_(np.allclose(A, B, eps)) _debug_print("Calling iddp_rsvd / idzp_rsvd...", ) t0 = time.clock() U, S, V = pymatrixid.svd(L, eps) t = time.clock() - t0 B = np.dot(U, np.dot(np.diag(S), V.T.conj())) _debug_print(fmt % (t, np.allclose(A, B, eps))) assert_(np.allclose(A, B, eps)) # fixed rank k = rank _debug_print("Calling iddr_svd / idzr_svd ...", ) t0 = time.clock() U, S, V = pymatrixid.svd(A, k, rand=False) t = time.clock() - t0 B = np.dot(U, np.dot(np.diag(S), V.T.conj())) _debug_print(fmt % (t, np.allclose(A, B, eps))) assert_(np.allclose(A, B, eps)) _debug_print("Calling iddr_asvd / idzr_asvd ...", ) t0 = time.clock() U, S, V = pymatrixid.svd(A, k) t = time.clock() - t0 B = np.dot(U, np.dot(np.diag(S), V.T.conj())) _debug_print(fmt % (t, np.allclose(A, B, eps))) assert_(np.allclose(A, B, eps)) _debug_print("Calling iddr_rsvd / idzr_rsvd ...", ) t0 = time.clock() U, S, V = pymatrixid.svd(L, k) t = time.clock() - t0 B = np.dot(U, np.dot(np.diag(S), V.T.conj())) _debug_print(fmt % (t, np.allclose(A, B, eps))) assert_(np.allclose(A, B, eps)) # ID to SVD idx, proj = pymatrixid.interp_decomp(A, k, rand=False) Up, Sp, Vp = pymatrixid.id_to_svd(A[:, idx[:k]], idx, proj) B = U.dot(np.diag(S).dot(V.T.conj())) assert_(np.allclose(A, B, eps)) # Norm estimates s = svdvals(A) norm_2_est = pymatrixid.estimate_spectral_norm(A) assert_(np.allclose(norm_2_est, s[0], 1e-6)) B = A.copy() B[:, 0] *= 1.2 s = svdvals(A - B) norm_2_est = pymatrixid.estimate_spectral_norm_diff(A, B) assert_(np.allclose(norm_2_est, s[0], 1e-6)) # Rank estimates B = np.array([[1, 1, 0], [0, 0, 1], [0, 0, 1]], dtype=dtype) for M in [A, B]: ML = aslinearoperator(M) rank_tol = 1e-9 rank_np = np.linalg.matrix_rank(M, norm(M, 2) * rank_tol) rank_est = pymatrixid.estimate_rank(M, rank_tol) rank_est_2 = pymatrixid.estimate_rank(ML, rank_tol) assert_(rank_est >= rank_np) assert_(rank_est <= rank_np + 10) assert_(rank_est_2 >= rank_np - 4) assert_(rank_est_2 <= rank_np + 4)
def check_id(self, dtype): # Test ID routines on a Hilbert matrix. # set parameters n = 300 eps = 1e-12 # construct Hilbert matrix A = hilbert(n).astype(dtype) if np.issubdtype(dtype, np.complexfloating): A = A * (1 + 1j) L = aslinearoperator(A) # find rank S = np.linalg.svd(A, compute_uv=False) try: rank = np.nonzero(S < eps)[0][0] except: rank = n # print input summary _debug_print("Hilbert matrix dimension: %8i" % n) _debug_print("Working precision: %8.2e" % eps) _debug_print("Rank to working precision: %8i" % rank) # set print format fmt = "%8.2e (s) / %5s" # test real ID routines _debug_print("-----------------------------------------") _debug_print("Real ID routines") _debug_print("-----------------------------------------") # fixed precision _debug_print("Calling iddp_id / idzp_id ...",) t0 = time.clock() k, idx, proj = pymatrixid.interp_decomp(A, eps, rand=False) t = time.clock() - t0 B = pymatrixid.reconstruct_matrix_from_id(A[:, idx[:k]], idx, proj) _debug_print(fmt % (t, np.allclose(A, B, eps))) assert_(np.allclose(A, B, eps)) _debug_print("Calling iddp_aid / idzp_aid ...",) t0 = time.clock() k, idx, proj = pymatrixid.interp_decomp(A, eps) t = time.clock() - t0 B = pymatrixid.reconstruct_matrix_from_id(A[:, idx[:k]], idx, proj) _debug_print(fmt % (t, np.allclose(A, B, eps))) assert_(np.allclose(A, B, eps)) _debug_print("Calling iddp_rid / idzp_rid ...",) t0 = time.clock() k, idx, proj = pymatrixid.interp_decomp(L, eps) t = time.clock() - t0 B = pymatrixid.reconstruct_matrix_from_id(A[:, idx[:k]], idx, proj) _debug_print(fmt % (t, np.allclose(A, B, eps))) assert_(np.allclose(A, B, eps)) # fixed rank k = rank _debug_print("Calling iddr_id / idzr_id ...",) t0 = time.clock() idx, proj = pymatrixid.interp_decomp(A, k, rand=False) t = time.clock() - t0 B = pymatrixid.reconstruct_matrix_from_id(A[:, idx[:k]], idx, proj) _debug_print(fmt % (t, np.allclose(A, B, eps))) assert_(np.allclose(A, B, eps)) _debug_print("Calling iddr_aid / idzr_aid ...",) t0 = time.clock() idx, proj = pymatrixid.interp_decomp(A, k) t = time.clock() - t0 B = pymatrixid.reconstruct_matrix_from_id(A[:, idx[:k]], idx, proj) _debug_print(fmt % (t, np.allclose(A, B, eps))) assert_(np.allclose(A, B, eps)) _debug_print("Calling iddr_rid / idzr_rid ...",) t0 = time.clock() idx, proj = pymatrixid.interp_decomp(L, k) t = time.clock() - t0 B = pymatrixid.reconstruct_matrix_from_id(A[:, idx[:k]], idx, proj) _debug_print(fmt % (t, np.allclose(A, B, eps))) assert_(np.allclose(A, B, eps)) # check skeleton and interpolation matrices idx, proj = pymatrixid.interp_decomp(A, k, rand=False) P = pymatrixid.reconstruct_interp_matrix(idx, proj) B = pymatrixid.reconstruct_skel_matrix(A, k, idx) assert_(np.allclose(B, A[:,idx[:k]], eps)) assert_(np.allclose(B.dot(P), A, eps)) # test SVD routines _debug_print("-----------------------------------------") _debug_print("SVD routines") _debug_print("-----------------------------------------") # fixed precision _debug_print("Calling iddp_svd / idzp_svd ...",) t0 = time.clock() U, S, V = pymatrixid.svd(A, eps, rand=False) t = time.clock() - t0 B = np.dot(U, np.dot(np.diag(S), V.T.conj())) _debug_print(fmt % (t, np.allclose(A, B, eps))) assert_(np.allclose(A, B, eps)) _debug_print("Calling iddp_asvd / idzp_asvd...",) t0 = time.clock() U, S, V = pymatrixid.svd(A, eps) t = time.clock() - t0 B = np.dot(U, np.dot(np.diag(S), V.T.conj())) _debug_print(fmt % (t, np.allclose(A, B, eps))) assert_(np.allclose(A, B, eps)) _debug_print("Calling iddp_rsvd / idzp_rsvd...",) t0 = time.clock() U, S, V = pymatrixid.svd(L, eps) t = time.clock() - t0 B = np.dot(U, np.dot(np.diag(S), V.T.conj())) _debug_print(fmt % (t, np.allclose(A, B, eps))) assert_(np.allclose(A, B, eps)) # fixed rank k = rank _debug_print("Calling iddr_svd / idzr_svd ...",) t0 = time.clock() U, S, V = pymatrixid.svd(A, k, rand=False) t = time.clock() - t0 B = np.dot(U, np.dot(np.diag(S), V.T.conj())) _debug_print(fmt % (t, np.allclose(A, B, eps))) assert_(np.allclose(A, B, eps)) _debug_print("Calling iddr_asvd / idzr_asvd ...",) t0 = time.clock() U, S, V = pymatrixid.svd(A, k) t = time.clock() - t0 B = np.dot(U, np.dot(np.diag(S), V.T.conj())) _debug_print(fmt % (t, np.allclose(A, B, eps))) assert_(np.allclose(A, B, eps)) _debug_print("Calling iddr_rsvd / idzr_rsvd ...",) t0 = time.clock() U, S, V = pymatrixid.svd(L, k) t = time.clock() - t0 B = np.dot(U, np.dot(np.diag(S), V.T.conj())) _debug_print(fmt % (t, np.allclose(A, B, eps))) assert_(np.allclose(A, B, eps)) # ID to SVD idx, proj = pymatrixid.interp_decomp(A, k, rand=False) Up, Sp, Vp = pymatrixid.id_to_svd(A[:, idx[:k]], idx, proj) B = U.dot(np.diag(S).dot(V.T.conj())) assert_(np.allclose(A, B, eps)) # Norm estimates s = svdvals(A) norm_2_est = pymatrixid.estimate_spectral_norm(A) assert_(np.allclose(norm_2_est, s[0], 1e-6)) B = A.copy() B[:,0] *= 1.2 s = svdvals(A - B) norm_2_est = pymatrixid.estimate_spectral_norm_diff(A, B) assert_(np.allclose(norm_2_est, s[0], 1e-6)) # Rank estimates B = np.array([[1, 1, 0], [0, 0, 1], [0, 0, 1]], dtype=dtype) for M in [A, B]: ML = aslinearoperator(M) rank_np = np.linalg.matrix_rank(M, 1e-9) rank_est = pymatrixid.estimate_rank(M, 1e-9) rank_est_2 = pymatrixid.estimate_rank(ML, 1e-9) assert_(rank_est >= rank_np) assert_(rank_est <= rank_np + 10) assert_(rank_est_2 >= rank_np) assert_(rank_est_2 <= rank_np + 10)
def IDLeft(matrix, rank, random=True): idx, proj = sli.interp_decomp(matrix, rank, rand=random) Askel = sli.reconstruct_skel_matrix(matrix, rank, idx) return Askel