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| # TODO: don't use round | |
| from __future__ import division | |
| import pytest | |
| from mpmath import * | |
| xrange = libmp.backend.xrange | |
| # XXX: these shouldn't be visible(?) | |
| LU_decomp = mp.LU_decomp | |
| L_solve = mp.L_solve | |
| U_solve = mp.U_solve | |
| householder = mp.householder | |
| improve_solution = mp.improve_solution | |
| A1 = matrix([[3, 1, 6], | |
| [2, 1, 3], | |
| [1, 1, 1]]) | |
| b1 = [2, 7, 4] | |
| A2 = matrix([[ 2, -1, -1, 2], | |
| [ 6, -2, 3, -1], | |
| [-4, 2, 3, -2], | |
| [ 2, 0, 4, -3]]) | |
| b2 = [3, -3, -2, -1] | |
| A3 = matrix([[ 1, 0, -1, -1, 0], | |
| [ 0, 1, 1, 0, -1], | |
| [ 4, -5, 2, 0, 0], | |
| [ 0, 0, -2, 9,-12], | |
| [ 0, 5, 0, 0, 12]]) | |
| b3 = [0, 0, 0, 0, 50] | |
| A4 = matrix([[10.235, -4.56, 0., -0.035, 5.67], | |
| [-2.463, 1.27, 3.97, -8.63, 1.08], | |
| [-6.58, 0.86, -0.257, 9.32, -43.6 ], | |
| [ 9.83, 7.39, -17.25, 0.036, 24.86], | |
| [-9.31, 34.9, 78.56, 1.07, 65.8 ]]) | |
| b4 = [8.95, 20.54, 7.42, 5.60, 58.43] | |
| A5 = matrix([[ 1, 2, -4], | |
| [-2, -3, 5], | |
| [ 3, 5, -8]]) | |
| A6 = matrix([[ 1.377360, 2.481400, 5.359190], | |
| [ 2.679280, -1.229560, 25.560210], | |
| [-1.225280+1.e6, 9.910180, -35.049900-1.e6]]) | |
| b6 = [23.500000, -15.760000, 2.340000] | |
| A7 = matrix([[1, -0.5], | |
| [2, 1], | |
| [-2, 6]]) | |
| b7 = [3, 2, -4] | |
| A8 = matrix([[1, 2, 3], | |
| [-1, 0, 1], | |
| [-1, -2, -1], | |
| [1, 0, -1]]) | |
| b8 = [1, 2, 3, 4] | |
| A9 = matrix([[ 4, 2, -2], | |
| [ 2, 5, -4], | |
| [-2, -4, 5.5]]) | |
| b9 = [10, 16, -15.5] | |
| A10 = matrix([[1.0 + 1.0j, 2.0, 2.0], | |
| [4.0, 5.0, 6.0], | |
| [7.0, 8.0, 9.0]]) | |
| b10 = [1.0, 1.0 + 1.0j, 1.0] | |
| def test_LU_decomp(): | |
| A = A3.copy() | |
| b = b3 | |
| A, p = LU_decomp(A) | |
| y = L_solve(A, b, p) | |
| x = U_solve(A, y) | |
| assert p == [2, 1, 2, 3] | |
| assert [round(i, 14) for i in x] == [3.78953107960742, 2.9989094874591098, | |
| -0.081788440567070006, 3.8713195201744801, 2.9171210468920399] | |
| A = A4.copy() | |
| b = b4 | |
| A, p = LU_decomp(A) | |
| y = L_solve(A, b, p) | |
| x = U_solve(A, y) | |
| assert p == [0, 3, 4, 3] | |
| assert [round(i, 14) for i in x] == [2.6383625899619201, 2.6643834462368399, | |
| 0.79208015947958998, -2.5088376454101899, -1.0567657691375001] | |
| A = randmatrix(3) | |
| bak = A.copy() | |
| LU_decomp(A, overwrite=1) | |
| assert A != bak | |
| def test_inverse(): | |
| for A in [A1, A2, A5]: | |
| inv = inverse(A) | |
| assert mnorm(A*inv - eye(A.rows), 1) < 1.e-14 | |
| def test_householder(): | |
| mp.dps = 15 | |
| A, b = A8, b8 | |
| H, p, x, r = householder(extend(A, b)) | |
| assert H == matrix( | |
| [[mpf('3.0'), mpf('-2.0'), mpf('-1.0'), 0], | |
| [-1.0,mpf('3.333333333333333'),mpf('-2.9999999999999991'),mpf('2.0')], | |
| [-1.0, mpf('-0.66666666666666674'),mpf('2.8142135623730948'), | |
| mpf('-2.8284271247461898')], | |
| [1.0, mpf('-1.3333333333333333'),mpf('-0.20000000000000018'), | |
| mpf('4.2426406871192857')]]) | |
| assert p == [-2, -2, mpf('-1.4142135623730949')] | |
| assert round(norm(r, 2), 10) == 4.2426406870999998 | |
| y = [102.102, 58.344, 36.463, 24.310, 17.017, 12.376, 9.282, 7.140, 5.610, | |
| 4.488, 3.6465, 3.003] | |
| def coeff(n): | |
| # similiar to Hilbert matrix | |
| A = [] | |
| for i in range(1, 13): | |
| A.append([1. / (i + j - 1) for j in range(1, n + 1)]) | |
| return matrix(A) | |
| residuals = [] | |
| refres = [] | |
| for n in range(2, 7): | |
| A = coeff(n) | |
| H, p, x, r = householder(extend(A, y)) | |
| x = matrix(x) | |
| y = matrix(y) | |
| residuals.append(norm(r, 2)) | |
| refres.append(norm(residual(A, x, y), 2)) | |
| assert [round(res, 10) for res in residuals] == [15.1733888877, | |
| 0.82378073210000002, 0.302645887, 0.0260109244, | |
| 0.00058653999999999998] | |
| assert norm(matrix(residuals) - matrix(refres), inf) < 1.e-13 | |
| def hilbert_cmplx(n): | |
| # Complexified Hilbert matrix | |
| A = hilbert(2*n,n) | |
| v = randmatrix(2*n, 2, min=-1, max=1) | |
| v = v.apply(lambda x: exp(1J*pi()*x)) | |
| A = diag(v[:,0])*A*diag(v[:n,1]) | |
| return A | |
| residuals_cmplx = [] | |
| refres_cmplx = [] | |
| for n in range(2, 10): | |
| A = hilbert_cmplx(n) | |
| H, p, x, r = householder(A.copy()) | |
| residuals_cmplx.append(norm(r, 2)) | |
| refres_cmplx.append(norm(residual(A[:,:n-1], x, A[:,n-1]), 2)) | |
| assert norm(matrix(residuals_cmplx) - matrix(refres_cmplx), inf) < 1.e-13 | |
| def test_factorization(): | |
| A = randmatrix(5) | |
| P, L, U = lu(A) | |
| assert mnorm(P*A - L*U, 1) < 1.e-15 | |
| def test_solve(): | |
| assert norm(residual(A6, lu_solve(A6, b6), b6), inf) < 1.e-10 | |
| assert norm(residual(A7, lu_solve(A7, b7), b7), inf) < 1.5 | |
| assert norm(residual(A8, lu_solve(A8, b8), b8), inf) <= 3 + 1.e-10 | |
| assert norm(residual(A6, qr_solve(A6, b6)[0], b6), inf) < 1.e-10 | |
| assert norm(residual(A7, qr_solve(A7, b7)[0], b7), inf) < 1.5 | |
| assert norm(residual(A8, qr_solve(A8, b8)[0], b8), 2) <= 4.3 | |
| assert norm(residual(A10, lu_solve(A10, b10), b10), 2) < 1.e-10 | |
| assert norm(residual(A10, qr_solve(A10, b10)[0], b10), 2) < 1.e-10 | |
| def test_solve_overdet_complex(): | |
| A = matrix([[1, 2j], [3, 4j], [5, 6]]) | |
| b = matrix([1 + j, 2, -j]) | |
| assert norm(residual(A, lu_solve(A, b), b)) < 1.0208 | |
| def test_singular(): | |
| mp.dps = 15 | |
| A = [[5.6, 1.2], [7./15, .1]] | |
| B = repr(zeros(2)) | |
| b = [1, 2] | |
| for i in ['lu_solve(%s, %s)' % (A, b), 'lu_solve(%s, %s)' % (B, b), | |
| 'qr_solve(%s, %s)' % (A, b), 'qr_solve(%s, %s)' % (B, b)]: | |
| pytest.raises((ZeroDivisionError, ValueError), lambda: eval(i)) | |
| def test_cholesky(): | |
| assert fp.cholesky(fp.matrix(A9)) == fp.matrix([[2, 0, 0], [1, 2, 0], [-1, -3/2, 3/2]]) | |
| x = fp.cholesky_solve(A9, b9) | |
| assert fp.norm(fp.residual(A9, x, b9), fp.inf) == 0 | |
| def test_det(): | |
| assert det(A1) == 1 | |
| assert round(det(A2), 14) == 8 | |
| assert round(det(A3)) == 1834 | |
| assert round(det(A4)) == 4443376 | |
| assert det(A5) == 1 | |
| assert round(det(A6)) == 78356463 | |
| assert det(zeros(3)) == 0 | |
| def test_cond(): | |
| mp.dps = 15 | |
| A = matrix([[1.2969, 0.8648], [0.2161, 0.1441]]) | |
| assert cond(A, lambda x: mnorm(x,1)) == mpf('327065209.73817754') | |
| assert cond(A, lambda x: mnorm(x,inf)) == mpf('327065209.73817754') | |
| assert cond(A, lambda x: mnorm(x,'F')) == mpf('249729266.80008656') | |
| def test_precision(): | |
| A = randmatrix(10, 10) | |
| assert mnorm(inverse(inverse(A)) - A, 1) < 1.e-45 | |
| def test_interval_matrix(): | |
| mp.dps = 15 | |
| iv.dps = 15 | |
| a = iv.matrix([['0.1','0.3','1.0'],['7.1','5.5','4.8'],['3.2','4.4','5.6']]) | |
| b = iv.matrix(['4','0.6','0.5']) | |
| c = iv.lu_solve(a, b) | |
| assert c[0].delta < 1e-13 | |
| assert c[1].delta < 1e-13 | |
| assert c[2].delta < 1e-13 | |
| assert 5.25823271130625686059275 in c[0] | |
| assert -13.155049396267837541163 in c[1] | |
| assert 7.42069154774972557628979 in c[2] | |
| def test_LU_cache(): | |
| A = randmatrix(3) | |
| LU = LU_decomp(A) | |
| assert A._LU == LU_decomp(A) | |
| A[0,0] = -1000 | |
| assert A._LU is None | |
| def test_improve_solution(): | |
| A = randmatrix(5, min=1e-20, max=1e20) | |
| b = randmatrix(5, 1, min=-1000, max=1000) | |
| x1 = lu_solve(A, b) + randmatrix(5, 1, min=-1e-5, max=1.e-5) | |
| x2 = improve_solution(A, x1, b) | |
| assert norm(residual(A, x2, b), 2) < norm(residual(A, x1, b), 2) | |
| def test_exp_pade(): | |
| for i in range(3): | |
| dps = 15 | |
| extra = 15 | |
| mp.dps = dps + extra | |
| dm = 0 | |
| N = 3 | |
| dg = range(1,N+1) | |
| a = diag(dg) | |
| expa = diag([exp(x) for x in dg]) | |
| # choose a random matrix not close to be singular | |
| # to avoid adding too much extra precision in computing | |
| # m**-1 * M * m | |
| while abs(dm) < 0.01: | |
| m = randmatrix(N) | |
| dm = det(m) | |
| m = m/dm | |
| a1 = m**-1 * a * m | |
| e2 = m**-1 * expa * m | |
| mp.dps = dps | |
| e1 = expm(a1, method='pade') | |
| mp.dps = dps + extra | |
| d = e2 - e1 | |
| #print d | |
| mp.dps = dps | |
| assert norm(d, inf).ae(0) | |
| mp.dps = 15 | |
| def test_qr(): | |
| mp.dps = 15 # used default value for dps | |
| lowlimit = -9 # lower limit of matrix element value | |
| uplimit = 9 # uppter limit of matrix element value | |
| maxm = 4 # max matrix size | |
| flg = False # toggle to create real vs complex matrix | |
| zero = mpf('0.0') | |
| for k in xrange(0,10): | |
| exdps = 0 | |
| mode = 'full' | |
| flg = bool(k % 2) | |
| # generate arbitrary matrix size (2 to maxm) | |
| num1 = nint(maxm*rand()) | |
| num2 = nint(maxm*rand()) | |
| m = int(max(num1, num2)) | |
| n = int(min(num1, num2)) | |
| # create matrix | |
| A = mp.matrix(m,n) | |
| # populate matrix values with arbitrary integers | |
| if flg: | |
| flg = False | |
| dtype = 'complex' | |
| for j in xrange(0,n): | |
| for i in xrange(0,m): | |
| val = nint(lowlimit + (uplimit-lowlimit)*rand()) | |
| val2 = nint(lowlimit + (uplimit-lowlimit)*rand()) | |
| A[i,j] = mpc(val, val2) | |
| else: | |
| flg = True | |
| dtype = 'real' | |
| for j in xrange(0,n): | |
| for i in xrange(0,m): | |
| val = nint(lowlimit + (uplimit-lowlimit)*rand()) | |
| A[i,j] = mpf(val) | |
| # perform A -> QR decomposition | |
| Q, R = qr(A, mode, edps = exdps) | |
| #print('\n\n A = \n', nstr(A, 4)) | |
| #print('\n Q = \n', nstr(Q, 4)) | |
| #print('\n R = \n', nstr(R, 4)) | |
| #print('\n Q*R = \n', nstr(Q*R, 4)) | |
| maxnorm = mpf('1.0E-11') | |
| n1 = norm(A - Q * R) | |
| #print '\n Norm of A - Q * R = ', n1 | |
| assert n1 <= maxnorm | |
| if dtype == 'real': | |
| n1 = norm(eye(m) - Q.T * Q) | |
| #print ' Norm of I - Q.T * Q = ', n1 | |
| assert n1 <= maxnorm | |
| n1 = norm(eye(m) - Q * Q.T) | |
| #print ' Norm of I - Q * Q.T = ', n1 | |
| assert n1 <= maxnorm | |
| if dtype == 'complex': | |
| n1 = norm(eye(m) - Q.T * Q.conjugate()) | |
| #print ' Norm of I - Q.T * Q.conjugate() = ', n1 | |
| assert n1 <= maxnorm | |
| n1 = norm(eye(m) - Q.conjugate() * Q.T) | |
| #print ' Norm of I - Q.conjugate() * Q.T = ', n1 | |
| assert n1 <= maxnorm | |