def test_svd_test_case():
# a test case from Golub and Reinsch
# (see wilkinson/reinsch: handbook for auto. comp., vol ii-linear algebra, 134-151(1971).)
eps = mp.exp(0.8 * mp.log(mp.eps))
a = [[22, 10, 2, 3, 7],
[14, 7, 10, 0, 8],
[-1, 13, -1, -11, 3],
[-3, -2, 13, -2, 4],
[ 9, 8, 1, -2, 4],
[ 9, 1, -7, 5, -1],
[ 2, -6, 6, 5, 1],
[ 4, 5, 0, -2, 2]]
a = mp.matrix(a)
b = mp.matrix([mp.sqrt(1248), 20, mp.sqrt(384), 0, 0])
S = mp.svd_r(a, compute_uv = False)
S -= b
assert mp.mnorm(S) < eps
S = mp.svd_c(a, compute_uv = False)
S -= b
assert mp.mnorm(S) < eps
def run_gauss(qtype, a, b):
eps = 1e-5
d, e = mp.gauss_quadrature(len(a), qtype)
d -= mp.matrix(a)
e -= mp.matrix(b)
assert mp.mnorm(d) < eps
assert mp.mnorm(e) < eps
def run_gauss(qtype, a, b):
eps = 1e-5
d, e = mp.gauss_quadrature(len(a), qtype)
d -= mp.matrix(a)
e -= mp.matrix(b)
assert mp.mnorm(d) < eps
assert mp.mnorm(e) < eps
def test_eighe_fixed_matrix():
A = mp.matrix([[2, 3], [3, 5]])
run_eigsy(A)
run_eighe(A)
A = mp.matrix([[7, -11], [-11, 13]])
run_eigsy(A)
run_eighe(A)
A = mp.matrix([[2, 11, 7], [11, 3, 13], [7, 13, 5]])
run_eigsy(A)
run_eighe(A)
A = mp.matrix([[2, 0, 7], [0, 3, 1], [7, 1, 5]])
run_eigsy(A)
run_eighe(A)
#
A = mp.matrix([[2, 3 + 7j], [3 - 7j, 5]])
run_eighe(A)
A = mp.matrix([[2, -11j, 0], [+11j, 3, 29j], [0, -29j, 5]])
run_eighe(A)
A = mp.matrix([[2, 11 + 17j, 7 + 19j], [11 - 17j, 3, -13 + 23j],
[7 - 19j, -13 - 23j, 5]])
run_eighe(A)
def test_eighe_fixed_matrix():
A = mp.matrix([[2, 3], [3, 5]])
run_eigsy(A)
run_eighe(A)
A = mp.matrix([[7, -11], [-11, 13]])
run_eigsy(A)
run_eighe(A)
A = mp.matrix([[2, 11, 7], [11, 3, 13], [7, 13, 5]])
run_eigsy(A)
run_eighe(A)
A = mp.matrix([[2, 0, 7], [0, 3, 1], [7, 1, 5]])
run_eigsy(A)
run_eighe(A)
#
A = mp.matrix([[2, 3+7j], [3-7j, 5]])
run_eighe(A)
A = mp.matrix([[2, -11j, 0], [+11j, 3, 29j], [0, -29j, 5]])
run_eighe(A)
A = mp.matrix([[2, 11 + 17j, 7 + 19j], [11 - 17j, 3, -13 + 23j], [7 - 19j, -13 - 23j, 5]])
run_eighe(A)
def irandmatrix(n, range=10):
"""
random matrix with integer entries
"""
A = mp.matrix(n, n)
for i in xrange(n):
for j in xrange(n):
A[i, j] = int((2 * mp.rand() - 1) * range)
return A
def irandmatrix(n, range = 10):
"""
random matrix with integer entries
"""
A = mp.matrix(n, n)
for i in xrange(n):
for j in xrange(n):
A[i,j]=int( (2 * mp.rand() - 1) * range)
return A
def test_svd_test_case():
# a test case from Golub and Reinsch
# (see wilkinson/reinsch: handbook for auto. comp., vol ii-linear algebra, 134-151(1971).)
eps = mp.exp(0.8 * mp.log(mp.eps))
a = [[22, 10, 2, 3, 7], [14, 7, 10, 0, 8], [-1, 13, -1, -11, 3],
[-3, -2, 13, -2, 4], [9, 8, 1, -2, 4], [9, 1, -7, 5, -1],
[2, -6, 6, 5, 1], [4, 5, 0, -2, 2]]
a = mp.matrix(a)
b = mp.matrix([mp.sqrt(1248), 20, mp.sqrt(384), 0, 0])
S = mp.svd_r(a, compute_uv=False)
S -= b
assert mp.mnorm(S) < eps
S = mp.svd_c(a, compute_uv=False)
S -= b
assert mp.mnorm(S) < eps
def test_eig():
v = 0
AS = []
A = mp.matrix([
[2, 1, 0], # jordan block of size 3
[0, 2, 1],
[0, 0, 2]
])
AS.append(A)
AS.append(A.transpose())
A = mp.matrix([
[2, 0, 0], # jordan block of size 2
[0, 2, 1],
[0, 0, 2]
])
AS.append(A)
AS.append(A.transpose())
A = mp.matrix([
[2, 0, 1], # jordan block of size 2
[0, 2, 0],
[0, 0, 2]
])
AS.append(A)
AS.append(A.transpose())
A = mp.matrix([
[0, 0, 1], # cyclic
[1, 0, 0],
[0, 1, 0]
])
AS.append(A)
AS.append(A.transpose())
for A in AS:
run_hessenberg(A, verbose=v)
run_schur(A, verbose=v)
run_eig(A, verbose=v)
def _nodal_basis(self, pts, ptsord):
# Obtain an orthonormal basis
ob = self._orthonormal_basis(ptsord)
p, q = self._dims
# Evaluate each basis function at each point
V = self._eval_lbasis_at(lambdify_mpf(self._dims, ob), pts)
# Invert this matrix to obtain the expansion coefficients
Vinv = mp.matrix(V.T)**-1
# Each nodal basis function is a linear combination of
# orthonormal basis functions
nb = [sum(c*b for c, b in zip(Vinv[i,:], ob))
for i in xrange(len(pts))]
return np.array(nb, dtype=np.object)
subroutine = '%s_quadrature_' % q
with open('%s.hpp' % q, 'w') as f:
f.write('#ifndef %s_HPP\n' % q.upper())
f.write('#define %s_HPP\n\n' % q.upper())
f.write('// this file was generated by "mksmat.py"\n')
f.write('#include<cassert>\n\n')
f.write('namespace detail{\n\n')
f.write(' template<int sdcnodes>\n')
f.write(' inline long double %smatrix(unsigned i, unsigned j);\n\n' % subroutine)
f.write(' template<int sdcnodes>\n')
f.write(' inline long double %snode(unsigned i);\n\n' % subroutine)
f.write(' template<int sdcnodes>\n')
f.write(' inline long double %sdt(unsigned i);\n\n' % subroutine)
for n in quads[q]:
dt = mp.matrix(1, n-1)
for r in quads[q][n]:
f.write(' template<>\n')
f.writelines([' inline long double %smatrix<%d>(unsigned i, unsigned j)\n {\n' % (subroutine, n),
' static long double integration_matrix[%d][%d] = \n {\n' % (n-1, n)
])
(nodes, smat) = quads[q][n][r]
rows, cols = smat.shape
for row in range(rows):
dt[row] = sum(smat[row])
f.write(' {')
for col in range(cols-1):
f.write('%sL,' % str(smat[row][col]))
f.write('%sL}' % str(smat[row][cols-1]))
if row == rows-1:
f.write('\n };\n')