def test_Givens_rotation_definition():
'verify if Givens rotation satisfies its definition'
np.random.rand(3)
# general a and b
a = 10*np.random.rand()
b = 10*np.random.rand()
c, s = qra.Givens_rotation(a=a, b=b)
G = np.array([[ c, s],
[-s, c]])
v = np.array([a, b])
Gv = np.dot(G.T, v)
aae(Gv[1], 0, decimal=10)
# b = 0
a = 10*np.random.rand()
b = 0
c, s = qra.Givens_rotation(a=a, b=b)
G = np.array([[ c, s],
[-s, c]])
v = np.array([a, b])
Gv = np.dot(G.T, v)
aae(Gv[1], 0, decimal=10)
# |b| > |a|
a = 7*np.random.rand()
b = 10*np.random.rand()
c, s = qra.Givens_rotation(a=a, b=b)
G = np.array([[ c, s],
[-s, c]])
v = np.array([a, b])
Gv = np.dot(G.T, v)
aae(Gv[1], 0, decimal=10)
def test_Givens_matvec_matmat():
'verify matrix-matrix product with Givens rotations'
np.random.seed(3)
M = 5
N = 7
A = np.round(np.random.rand(M,N), decimals=3)
i = 2
k = 3
c, s = qra.Givens_rotation(a=A[i,3], b=A[k,3])
# verify product GTA
G = np.identity(M)
G[i,i] = c
G[i,k] = s
G[k,i] = -s
G[k,k] = c
A2 = A.copy()
qra.Givens_matvec(A=A2, c=c, s=s, i=i, k=k, order='GTA')
aae(A2, np.dot(G.T,A), decimal=10)
# verify AG
G = np.identity(N)
G[i,i] = c
G[i,k] = s
G[k,i] = -s
G[k,k] = c
A2 = A.copy()
qra.Givens_matvec(A=A2, c=c, s=s, i=i, k=k, order='AG')
aae(A2, np.dot(A,G), decimal=10)
def test_Givens_cs2rho_Givens_rho2cs():
'verify consistency'
np.random.seed(11)
a = 10*np.random.rand()
b = 10*np.random.rand()
c, s = qra.Givens_rotation(a=a, b=b)
rho = qra.Givens_cs2rho(c=c, s=s)
c2, s2 = qra.Givens_rho2cs(rho=rho)
aae(c, c2, decimal=10)
aae(s, s2, decimal=10)