# scipy/scipy

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 #!/usr/bin/env python # # Created by: Pearu Peterson, March 2002 # """ Test functions for linalg.basic module """ """ Bugs: 1) solve.check_random_sym_complex fails if a is complex and transpose(a) = conjugate(a) (a is Hermitian). """ __usage__ = """ Build linalg: python setup_linalg.py build Run tests if scipy is installed: python -c 'import scipy;scipy.linalg.test()' Run tests if linalg is not installed: python tests/test_basic.py """ import numpy as np from numpy import arange, array, dot, zeros, identity, conjugate, transpose, \ float32 import numpy.linalg as linalg from numpy.testing import TestCase, rand, run_module_suite, assert_raises, \ assert_equal, assert_almost_equal, assert_array_almost_equal, assert_, \ assert_allclose from scipy.linalg import solve, inv, det, lstsq, pinv, pinv2, pinvh, norm,\ solve_banded, solveh_banded, solve_triangular from scipy.linalg._testutils import assert_no_overwrite def random(size): return rand(*size) class TestSolveBanded(TestCase): def test_real(self): a = array([[ 1.0, 20, 0, 0], [ -30, 4, 6, 0], [ 2, 1, 20, 2], [ 0, -1, 7, 14]]) ab = array([[ 0.0, 20, 6, 2], [ 1, 4, 20, 14], [ -30, 1, 7, 0], [ 2, -1, 0, 0]]) l,u = 2,1 b4 = array([10.0, 0.0, 2.0, 14.0]) b4by1 = b4.reshape(-1,1) b4by2 = array([[ 2, 1], [-30, 4], [ 2, 3], [ 1, 3]]) b4by4 = array([[1, 0, 0, 0], [0, 0, 0, 1], [0, 1, 0, 0], [0, 1, 0, 0]]) for b in [b4, b4by1, b4by2, b4by4]: x = solve_banded((l, u), ab, b) assert_array_almost_equal(dot(a, x), b) def test_complex(self): a = array([[ 1.0, 20, 0, 0], [ -30, 4, 6, 0], [ 2j, 1, 20, 2j], [ 0, -1, 7, 14]]) ab = array([[ 0.0, 20, 6, 2j], [ 1, 4, 20, 14], [ -30, 1, 7, 0], [ 2j, -1, 0, 0]]) l,u = 2,1 b4 = array([10.0, 0.0, 2.0, 14.0j]) b4by1 = b4.reshape(-1,1) b4by2 = array([[ 2, 1], [-30, 4], [ 2, 3], [ 1, 3]]) b4by4 = array([[1, 0, 0, 0], [0, 0, 0,1j], [0, 1, 0, 0], [0, 1, 0, 0]]) for b in [b4, b4by1, b4by2, b4by4]: x = solve_banded((l, u), ab, b) assert_array_almost_equal(dot(a, x), b) def test_bad_shape(self): ab = array([[ 0.0, 20, 6, 2], [ 1, 4, 20, 14], [ -30, 1, 7, 0], [ 2, -1, 0, 0]]) l,u = 2,1 bad = array([1.0, 2.0, 3.0, 4.0]).reshape(-1,4) assert_raises(ValueError, solve_banded, (l, u), ab, bad) assert_raises(ValueError, solve_banded, (l, u), ab, [1.0, 2.0]) # Values of (l,u) are not compatible with ab. assert_raises(ValueError, solve_banded, (1, 1), ab, [1.0, 2.0]) class TestSolveHBanded(TestCase): def test_01_upper(self): # Solve # [ 4 1 0] [1] # [ 1 4 1] X = [4] # [ 0 1 4] [1] # with the RHS as a 1D array. ab = array([[-99, 1.0, 1.0], [4.0, 4.0, 4.0]]) b = array([1.0, 4.0, 1.0]) x = solveh_banded(ab, b) assert_array_almost_equal(x, [0.0, 1.0, 0.0]) def test_02_upper(self): # Solve # [ 4 1 0] [1 4] # [ 1 4 1] X = [4 2] # [ 0 1 4] [1 4] # ab = array([[-99, 1.0, 1.0], [4.0, 4.0, 4.0]]) b = array([[1.0, 4.0], [4.0, 2.0], [1.0, 4.0]]) x = solveh_banded(ab, b) expected = array([[0.0, 1.0], [1.0, 0.0], [0.0, 1.0]]) assert_array_almost_equal(x, expected) def test_03_upper(self): # Solve # [ 4 1 0] [1] # [ 1 4 1] X = [4] # [ 0 1 4] [1] # with the RHS as a 2D array with shape (3,1). ab = array([[-99, 1.0, 1.0], [4.0, 4.0, 4.0]]) b = array([1.0, 4.0, 1.0]).reshape(-1,1) x = solveh_banded(ab, b) assert_array_almost_equal(x, array([0.0, 1.0, 0.0]).reshape(-1,1)) def test_01_lower(self): # Solve # [ 4 1 0] [1] # [ 1 4 1] X = [4] # [ 0 1 4] [1] # ab = array([[4.0, 4.0, 4.0], [1.0, 1.0, -99]]) b = array([1.0, 4.0, 1.0]) x = solveh_banded(ab, b, lower=True) assert_array_almost_equal(x, [0.0, 1.0, 0.0]) def test_02_lower(self): # Solve # [ 4 1 0] [1 4] # [ 1 4 1] X = [4 2] # [ 0 1 4] [1 4] # ab = array([[4.0, 4.0, 4.0], [1.0, 1.0, -99]]) b = array([[1.0, 4.0], [4.0, 2.0], [1.0, 4.0]]) x = solveh_banded(ab, b, lower=True) expected = array([[0.0, 1.0], [1.0, 0.0], [0.0, 1.0]]) assert_array_almost_equal(x, expected) def test_01_float32(self): # Solve # [ 4 1 0] [1] # [ 1 4 1] X = [4] # [ 0 1 4] [1] # ab = array([[-99, 1.0, 1.0], [4.0, 4.0, 4.0]], dtype=float32) b = array([1.0, 4.0, 1.0], dtype=float32) x = solveh_banded(ab, b) assert_array_almost_equal(x, [0.0, 1.0, 0.0]) def test_02_float32(self): # Solve # [ 4 1 0] [1 4] # [ 1 4 1] X = [4 2] # [ 0 1 4] [1 4] # ab = array([[-99, 1.0, 1.0], [4.0, 4.0, 4.0]], dtype=float32) b = array([[1.0, 4.0], [4.0, 2.0], [1.0, 4.0]], dtype=float32) x = solveh_banded(ab, b) expected = array([[0.0, 1.0], [1.0, 0.0], [0.0, 1.0]]) assert_array_almost_equal(x, expected) def test_01_complex(self): # Solve # [ 4 -j 0] [ -j] # [ j 4 -j] X = [4-j] # [ 0 j 4] [4+j] # ab = array([[-99, -1.0j, -1.0j], [4.0, 4.0, 4.0]]) b = array([-1.0j, 4.0-1j, 4+1j]) x = solveh_banded(ab, b) assert_array_almost_equal(x, [0.0, 1.0, 1.0]) def test_02_complex(self): # Solve # [ 4 -j 0] [ -j 4j] # [ j 4 -j] X = [4-j -1-j] # [ 0 j 4] [4+j 4 ] # ab = array([[-99, -1.0j, -1.0j], [4.0, 4.0, 4.0]]) b = array([[ -1j, 4.0j], [4.0-1j, -1.0-1j], [4.0+1j, 4.0]]) x = solveh_banded(ab, b) expected = array([[0.0, 1.0j], [1.0, 0.0], [1.0, 1.0]]) assert_array_almost_equal(x, expected) def test_bad_shapes(self): ab = array([[-99, 1.0, 1.0], [4.0, 4.0, 4.0]]) b = array([[1.0, 4.0], [4.0, 2.0]]) assert_raises(ValueError, solveh_banded, ab, b) assert_raises(ValueError, solveh_banded, ab, [1.0, 2.0]) assert_raises(ValueError, solveh_banded, ab, [1.0]) class TestSolve(TestCase): def test_20Feb04_bug(self): a = [[1,1],[1.0,0]] # ok x0 = solve(a,[1,0j]) assert_array_almost_equal(dot(a,x0),[1,0]) a = [[1,1],[1.2,0]] # gives failure with clapack.zgesv(..,rowmajor=0) b = [1,0j] x0 = solve(a,b) assert_array_almost_equal(dot(a,x0),[1,0]) def test_simple(self): a = [[1,20],[-30,4]] for b in ([[1,0],[0,1]],[1,0], [[2,1],[-30,4]]): x = solve(a,b) assert_array_almost_equal(dot(a,x),b) def test_simple_sym(self): a = [[2,3],[3,5]] for lower in [0,1]: for b in ([[1,0],[0,1]],[1,0]): x = solve(a,b,sym_pos=1,lower=lower) assert_array_almost_equal(dot(a,x),b) def test_simple_sym_complex(self): a = [[5,2],[2,4]] for b in [[1j,0], [[1j,1j], [0,2]], ]: x = solve(a,b,sym_pos=1) assert_array_almost_equal(dot(a,x),b) def test_simple_complex(self): a = array([[5,2],[2j,4]],'D') for b in [[1j,0], [[1j,1j], [0,2]], [1,0j], array([1,0],'D'), ]: x = solve(a,b) assert_array_almost_equal(dot(a,x),b) def test_nils_20Feb04(self): n = 2 A = random([n,n])+random([n,n])*1j X = zeros((n,n),'D') Ainv = inv(A) R = identity(n)+identity(n)*0j for i in arange(0,n): r = R[:,i] X[:,i] = solve(A,r) assert_array_almost_equal(X,Ainv) def test_random(self): n = 20 a = random([n,n]) for i in range(n): a[i,i] = 20*(.1+a[i,i]) for i in range(4): b = random([n,3]) x = solve(a,b) assert_array_almost_equal(dot(a,x),b) def test_random_complex(self): n = 20 a = random([n,n]) + 1j * random([n,n]) for i in range(n): a[i,i] = 20*(.1+a[i,i]) for i in range(2): b = random([n,3]) x = solve(a,b) assert_array_almost_equal(dot(a,x),b) def test_random_sym(self): n = 20 a = random([n,n]) for i in range(n): a[i,i] = abs(20*(.1+a[i,i])) for j in range(i): a[i,j] = a[j,i] for i in range(4): b = random([n]) x = solve(a,b,sym_pos=1) assert_array_almost_equal(dot(a,x),b) def test_random_sym_complex(self): n = 20 a = random([n,n]) #a = a + 1j*random([n,n]) # XXX: with this the accuracy will be very low for i in range(n): a[i,i] = abs(20*(.1+a[i,i])) for j in range(i): a[i,j] = conjugate(a[j,i]) b = random([n])+2j*random([n]) for i in range(2): x = solve(a,b,sym_pos=1) assert_array_almost_equal(dot(a,x),b) class TestSolveTriangular(TestCase): def test_simple(self): """ solve_triangular on a simple 2x2 matrix. """ A = array([[1,0], [1,2]]) b = [1, 1] sol = solve_triangular(A, b, lower=True) assert_array_almost_equal(sol, [1, 0]) # check that it works also for non-contiguous matrices sol = solve_triangular(A.T, b, lower=False) assert_array_almost_equal(sol, [.5, .5]) # and that it gives the same result as trans=1 sol = solve_triangular(A, b, lower=True, trans=1) assert_array_almost_equal(sol, [.5, .5]) b = identity(2) sol = solve_triangular(A, b, lower=True, trans=1) assert_array_almost_equal(sol, [[1., -.5], [0, 0.5]]) def test_simple_complex(self): """ solve_triangular on a simple 2x2 complex matrix """ A = array([[1+1j, 0], [1j, 2]]) b = identity(2) sol = solve_triangular(A, b, lower=True, trans=1) assert_array_almost_equal(sol, [[.5-.5j, -.25-.25j], [0, 0.5]]) class TestInv(TestCase): def test_simple(self): a = [[1,2],[3,4]] a_inv = inv(a) assert_array_almost_equal(dot(a,a_inv), [[1,0],[0,1]]) a = [[1,2,3],[4,5,6],[7,8,10]] a_inv = inv(a) assert_array_almost_equal(dot(a,a_inv), [[1,0,0],[0,1,0],[0,0,1]]) def test_random(self): n = 20 for i in range(4): a = random([n,n]) for i in range(n): a[i,i] = 20*(.1+a[i,i]) a_inv = inv(a) assert_array_almost_equal(dot(a,a_inv), identity(n)) def test_simple_complex(self): a = [[1,2],[3,4j]] a_inv = inv(a) assert_array_almost_equal(dot(a,a_inv), [[1,0],[0,1]]) def test_random_complex(self): n = 20 for i in range(4): a = random([n,n])+2j*random([n,n]) for i in range(n): a[i,i] = 20*(.1+a[i,i]) a_inv = inv(a) assert_array_almost_equal(dot(a,a_inv), identity(n)) class TestDet(TestCase): def test_simple(self): a = [[1,2],[3,4]] a_det = det(a) assert_almost_equal(a_det,-2.0) def test_simple_complex(self): a = [[1,2],[3,4j]] a_det = det(a) assert_almost_equal(a_det,-6+4j) def test_random(self): basic_det = linalg.det n = 20 for i in range(4): a = random([n,n]) d1 = det(a) d2 = basic_det(a) assert_almost_equal(d1,d2) def test_random_complex(self): basic_det = linalg.det n = 20 for i in range(4): a = random([n,n]) + 2j*random([n,n]) d1 = det(a) d2 = basic_det(a) assert_almost_equal(d1,d2) def direct_lstsq(a,b,cmplx=0): at = transpose(a) if cmplx: at = conjugate(at) a1 = dot(at, a) b1 = dot(at, b) return solve(a1, b1) class TestLstsq(TestCase): def test_random_overdet_large(self): #bug report: Nils Wagner n = 200 a = random([n,2]) for i in range(2): a[i,i] = 20*(.1+a[i,i]) b = random([n,3]) x = lstsq(a,b)[0] assert_array_almost_equal(x,direct_lstsq(a,b)) def test_simple_exact(self): a = [[1,20],[-30,4]] for b in ([[1,0],[0,1]],[1,0], [[2,1],[-30,4]]): x = lstsq(a,b)[0] assert_array_almost_equal(dot(a,x),b) def test_simple_overdet(self): a = [[1,2],[4,5],[3,4]] b = [1,2,3] x,res,r,s = lstsq(a,b) assert_array_almost_equal(x,direct_lstsq(a,b)) assert_almost_equal((abs(dot(a,x) - b)**2).sum(axis=0), res) def test_simple_overdet_complex(self): a = [[1+2j,2],[4,5],[3,4]] b = [1,2+4j,3] x,res,r,s = lstsq(a,b) assert_array_almost_equal(x,direct_lstsq(a,b,cmplx=1)) assert_almost_equal(res, (abs(dot(a,x) - b)**2).sum(axis=0)) def test_simple_underdet(self): a = [[1,2,3],[4,5,6]] b = [1,2] x,res,r,s = lstsq(a,b) #XXX: need independent check assert_array_almost_equal(x,[-0.05555556, 0.11111111, 0.27777778]) def test_random_exact(self): n = 20 a = random([n,n]) for i in range(n): a[i,i] = 20*(.1+a[i,i]) for i in range(4): b = random([n,3]) x = lstsq(a,b)[0] assert_array_almost_equal(dot(a,x),b) def test_random_complex_exact(self): n = 20 a = random([n,n]) + 1j * random([n,n]) for i in range(n): a[i,i] = 20*(.1+a[i,i]) for i in range(2): b = random([n,3]) x = lstsq(a,b)[0] assert_array_almost_equal(dot(a,x),b) def test_random_overdet(self): n = 20 m = 15 a = random([n,m]) for i in range(m): a[i,i] = 20*(.1+a[i,i]) for i in range(4): b = random([n,3]) x,res,r,s = lstsq(a,b) assert_(r == m, 'unexpected efficient rank') #XXX: check definition of res assert_array_almost_equal(x,direct_lstsq(a,b)) def test_random_complex_overdet(self): n = 20 m = 15 a = random([n,m]) + 1j * random([n,m]) for i in range(m): a[i,i] = 20*(.1+a[i,i]) for i in range(2): b = random([n,3]) x,res,r,s = lstsq(a,b) assert_(r == m, 'unexpected efficient rank') #XXX: check definition of res assert_array_almost_equal(x,direct_lstsq(a,b,1)) class TestPinv(TestCase): def test_simple_real(self): a = array([[1, 2, 3], [4, 5, 6], [7, 8, 10]], dtype=float) a_pinv = pinv(a) assert_array_almost_equal(dot(a,a_pinv), np.eye(3)) a_pinv = pinv2(a) assert_array_almost_equal(dot(a,a_pinv), np.eye(3)) def test_simple_complex(self): a = (array([[1, 2, 3], [4, 5, 6], [7, 8, 10]], dtype=float) + 1j * array([[10, 8, 7], [6, 5, 4], [3, 2, 1]], dtype=float)) a_pinv = pinv(a) assert_array_almost_equal(dot(a, a_pinv), np.eye(3)) a_pinv = pinv2(a) assert_array_almost_equal(dot(a, a_pinv), np.eye(3)) def test_simple_singular(self): a = array([[1, 2, 3], [4, 5, 6], [7, 8, 9]], dtype=float) a_pinv = pinv(a) a_pinv2 = pinv2(a) assert_array_almost_equal(a_pinv,a_pinv2) def test_simple_cols(self): a = array([[1, 2, 3], [4, 5, 6]], dtype=float) a_pinv = pinv(a) a_pinv2 = pinv2(a) assert_array_almost_equal(a_pinv,a_pinv2) def test_simple_rows(self): a = array([[1, 2], [3, 4], [5, 6]], dtype=float) a_pinv = pinv(a) a_pinv2 = pinv2(a) assert_array_almost_equal(a_pinv,a_pinv2) class TestPinvSymmetric(TestCase): def test_simple_real(self): a = array([[1, 2, 3], [4, 5, 6], [7, 8, 10]], dtype=float) a = np.dot(a, a.T) a_pinv = pinvh(a) assert_array_almost_equal(np.dot(a, a_pinv), np.eye(3)) def test_nonpositive(self): a = array([[1, 2, 3], [4, 5, 6], [7, 8, 9]], dtype=float) a = np.dot(a, a.T) u, s, vt = np.linalg.svd(a) s[0] *= -1 a = np.dot(u * s, vt) # a is now symmetric non-positive and singular a_pinv = pinv2(a) a_pinvh = pinvh(a) assert_array_almost_equal(a_pinv, a_pinvh) def test_simple_complex(self): a = (array([[1, 2, 3], [4, 5, 6], [7, 8, 10]], dtype=float) + 1j * array([[10, 8, 7], [6, 5, 4], [3, 2, 1]], dtype=float)) a = np.dot(a, a.conj().T) a_pinv = pinvh(a) assert_array_almost_equal(np.dot(a, a_pinv), np.eye(3)) class TestNorm(object): def test_types(self): for dtype in np.typecodes['AllFloat']: x = np.array([1,2,3], dtype=dtype) tol = max(1e-15, np.finfo(dtype).eps.real * 20) assert_allclose(norm(x), np.sqrt(14), rtol=tol) assert_allclose(norm(x, 2), np.sqrt(14), rtol=tol) for dtype in np.typecodes['Complex']: x = np.array([1j,2j,3j], dtype=dtype) tol = max(1e-15, np.finfo(dtype).eps.real * 20) assert_allclose(norm(x), np.sqrt(14), rtol=tol) assert_allclose(norm(x, 2), np.sqrt(14), rtol=tol) def test_overflow(self): # unlike numpy's norm, this one is # safer on overflow a = array([1e20], dtype=float32) assert_almost_equal(norm(a), a) def test_stable(self): # more stable than numpy's norm a = array([1e4] + [1]*10000, dtype=float32) try: # snrm in double precision; we obtain the same as for float64 assert_almost_equal(norm(a) - 1e4, 0.5) except AssertionError: # snrm implemented in single precision, == np.linalg.norm result msg = ": Result should equal either 0.0 or 0.5 (depending on " \ "implementation of snrm2)." assert_almost_equal(norm(a) - 1e4, 0.0, err_msg=msg) def test_zero_norm(self): assert_equal(norm([1,0,3], 0), 2) assert_equal(norm([1,2,3], 0), 3) class TestOverwrite(object): def test_solve(self): assert_no_overwrite(solve, [(3,3), (3,)]) def test_solve_triangular(self): assert_no_overwrite(solve_triangular, [(3,3), (3,)]) def test_solve_banded(self): assert_no_overwrite(lambda ab, b: solve_banded((2,1), ab, b), [(4,6), (6,)]) def test_solveh_banded(self): assert_no_overwrite(solveh_banded, [(2,6), (6,)]) def test_inv(self): assert_no_overwrite(inv, [(3,3)]) def test_det(self): assert_no_overwrite(det, [(3,3)]) def test_lstsq(self): assert_no_overwrite(lstsq, [(3,2), (3,)]) def test_pinv(self): assert_no_overwrite(pinv, [(3,3)]) def test_pinv2(self): assert_no_overwrite(pinv2, [(3,3)]) def test_pinvh(self): assert_no_overwrite(pinvh, [(3,3)]) if __name__ == "__main__": run_module_suite()
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