def main():
print(header.lstrip())
print('all_bfs = {')
for dim, _dbfs in all_bfs.items():
print(' {} : {{'.format(dim))
for order, _bfs in _dbfs.items():
print(' {} : ['.format(order))
vs = [x, y, z][:dim]
bfs = [
sm.horner(
sm.simplify(
bf.subs({
x: -1 + 2 * x,
y: -1 + 2 * y,
z: -1 + 2 * z
}))) for bf in _bfs
]
bfgs = [[sm.horner(sm.simplify(bf.diff(v))) for v in vs]
for bf in bfs]
print(' [')
for bf in bfs:
print((' ' * 16), str(bf).replace('1.0*', ''), ',')
print(' ],')
print(' [')
for bfg in bfgs:
print((' ' * 16), str(bfg).replace('1.0*', ''), ',')
print(' ],')
print(' ],')
print(' },')
print('}')
def sympy_get_monomials(dimension, degree, vector=False):
xs = smp.symbols(['x[{:d}]'.format(ii) for ii in range(dimension)],
real=True)
monos = smp.polys.monomials.itermonomials(xs, degree)
for xx in xs:
monos = [smp.horner(mm, wrt=xx) for mm in monos]
monos = sorted(monos,
key=smp.polys.orderings.monomial_key('grlex', xs[::-1]))[1:]
if vector:
expressions = [
Constant([0 for ll in range(kk)] + [1] +
[0 for ll in range(kk + 1, dimension)])
for kk in range(dimension)
]
expressions += [
Expression(['0' for ll in range(kk)] + [smp.ccode(mm)] +
['0' for ll in range(kk + 1, dimension)],
degree=int(smp.degree(mm))) for mm in monos
for kk in range(dimension)
]
else:
expressions = [Constant(1)]
expressions += [
Expression(smp.ccode(mm), degree=int(smp.degree(mm)))
for mm in monos
]
return expressions
def compute_fast_beta(self):
self.fast_beta = []
for i in range(len(self.beta)):
self.fast_beta.append([[sp.utilities.lambdify((self.xi),
sp.horner(sp.Poly((self.beta[i][j][k].coef[::-1]), self.xi)), np)
for k in range(len(self.beta[i][j]))]
for j in range(self.m + 1)])
return None
def poly_factorize(poly):
"""Factorize multivariate polynomials into a sum of products of monomials.
This function can be used to decompose polynomials into a form which
minimizes the number of additions and multiplications, and which thus
can be evaluated efficently."""
max_deg = {}
if 'horner' in dir(sympy):
return sympy.horner(poly)
if not isinstance(poly, Poly):
poly = Poly(sympy.expand(poly), *poly.atoms(Symbol))
denom, poly = poly.as_integer()
# Determine the order of factorization. We proceed through the
# symbols, starting with the one present in the highest order
# in the polynomial.
for i, sym in enumerate(poly.symbols):
max_deg[i] = 0
for monom in poly.monoms:
for i, symvar in enumerate(monom):
max_deg[i] = max(max_deg[i], symvar)
ret_poly = 0
monoms = list(poly.monoms)
for isym, maxdeg in sorted(max_deg.items(),
key=itemgetter(1),
reverse=True):
drop_idx = []
new_poly = []
for i, monom in enumerate(monoms):
if monom[isym] > 0:
drop_idx.append(i)
new_poly.append((poly.coeff(*monom), monom))
if not new_poly:
continue
ret_poly += sympy.factor(Poly(new_poly, *poly.symbols))
for idx in reversed(drop_idx):
del monoms[idx]
# Add any remaining O(1) terms.
new_poly = []
for i, monom in enumerate(monoms):
new_poly.append((poly.coeff(*monom), monom))
if new_poly:
ret_poly += Poly(new_poly, *poly.symbols)
return ret_poly / denom
def poly_factorize(poly):
"""Factorize multivariate polynomials into a sum of products of monomials.
This function can be used to decompose polynomials into a form which
minimizes the number of additions and multiplications, and which thus
can be evaluated efficently."""
max_deg = {}
if 'horner' in dir(sympy):
return sympy.horner(poly)
if not isinstance(poly, Poly):
poly = Poly(sympy.expand(poly), *poly.atoms(Symbol))
denom, poly = poly.as_integer()
# Determine the order of factorization. We proceed through the
# symbols, starting with the one present in the highest order
# in the polynomial.
for i, sym in enumerate(poly.symbols):
max_deg[i] = 0
for monom in poly.monoms:
for i, symvar in enumerate(monom):
max_deg[i] = max(max_deg[i], symvar)
ret_poly = 0
monoms = list(poly.monoms)
for isym, maxdeg in sorted(max_deg.items(), key=itemgetter(1), reverse=True):
drop_idx = []
new_poly = []
for i, monom in enumerate(monoms):
if monom[isym] > 0:
drop_idx.append(i)
new_poly.append((poly.coeff(*monom), monom))
if not new_poly:
continue
ret_poly += sympy.factor(Poly(new_poly, *poly.symbols))
for idx in reversed(drop_idx):
del monoms[idx]
# Add any remaining O(1) terms.
new_poly = []
for i, monom in enumerate(monoms):
new_poly.append((poly.coeff(*monom), monom))
if new_poly:
ret_poly += Poly(new_poly, *poly.symbols)
return ret_poly / denom
def compute_fast_beta(self):
self.fast_beta = []
for i in range(len(self.beta)):
self.fast_beta.append([[
sp.utilities.lambdify(
(self.xi),
sp.horner(sp.Poly((self.beta[i][j][k].coef[::-1]),
self.xi)), np)
for k in range(len(self.beta[i][j]))
] for j in range(self.m + 1)])
return None
def _parse(cls, raw):
raw = list(raw)
x = sympy.Symbol(cls._spce.x)
exprs = list(map(sympy.parse_expr, raw))
# attemps polynom extraction
try:
# polys = [sympy.poly(f, x, domain="RR") for f in exprs]
polys = sympy.parallel_poly_from_expr(exprs, x, domain="RR")[0]
except sympy.PolynomialError:
pass
else:
ncoeff = max(len(p.all_coeffs()) for p in polys)
coeffs = np.zeros((len(polys), ncoeff), dtype=cls.dtype)
coeffs[:, 0] = 1
for i, p in enumerate(polys):
c = p.all_coeffs()[::-1]
coeffs[i, : len(c)] = c
expr = sympy.horner(sympy.Poly.from_list(cls._spce.symbols(ncoeff)[::-1], x))
return expr, coeffs
# extraction
for formulas in list(map(sympy.simplify, exprs)), exprs:
expr = set()
coeffs = []
for e, c in map(cls._spce.extract, formulas):
expr.add(e)
if len(expr) > 1:
break
coeffs.append(c)
else:
(expr,) = list(expr)
coeff0 = np.array(coeffs[0], dtype=object)
coeffs = np.array(coeffs, dtype=cls.dtype)
const = np.all(coeffs[:1] == coeffs, axis=0)
symbols = np.array(cls._spce.symbols(len(const)))
# put back constants
expr = expr.xreplace(dict(zip(symbols[const], coeff0[const])))
# shift symbols
for old, new in zip(symbols[~const], symbols):
if old != new:
assert new not in expr.free_symbols
expr.replace(old, new)
# shift coeffs
coeffs = coeffs[:, ~const]
# horner-ify
expr = expr.replace(cls._horner_test, sympy.horner)
return expr, coeffs
return RuntimeError("\t\n".join(["can't unify expression:"] + raw))
def run_transmute_test(data,
degree,
prec,
time,
*,
expr=None,
plot=True,
_print=False,
run_all=True,
use_cache=True,
thetas=None,
alphas=None,
alpha0=None):
"""
Run transmute test on the data
If run_all=True, runs the test on all forms of the exponential. Otherwise,
only use part_frac_complex (the fastest).
"""
nucs, matrix = load_sparse_csr(data)
expr = get_CRAM_from_cache(degree,
prec,
expr=expr,
plot=plot,
use_cache=use_cache)
num, den = fraction(expr)
if not (thetas and alphas and alpha0):
thetas, alphas, alpha0 = thetas_alphas(expr, prec)
part_frac = thetas_alphas_to_expr_real(thetas, alphas, alpha0)
part_frac_complex = thetas_alphas_to_expr_complex(thetas, alphas, alpha0)
# part_frac_complex2 = thetas_alphas_to_expr_complex2(thetas, alphas, alpha0)
e = {}
from ..py_solve import py_solve
py_solve_expm = getattr(py_solve, 'expmI%d' % degree)
e['transmutagen generated C solver'] = lambda m: py_solve_expm(m).T
e['part_frac_complex UMFPACK'] = lambda m: (use_solver(
useUmfpack=True) or lambdify_expr(part_frac_complex)(m))
e['part_frac_complex SuperLU'] = lambda m: (use_solver(
useUmfpack=False) or lambdify_expr(part_frac_complex)(m))
e['scipy.sparse.linalg.expm'] = lambda m: expm(-m)
if run_all:
e['rat_func'] = lambdify_expr(expr)
e['rat_func_horner'] = lambdify_expr(horner(num) / horner(den))
e['part_frac'] = lambdify_expr(part_frac)
# e['part_frac_complex2'] = lambdify_expr(part_frac_complex2)
res = {}
for func in sorted(e):
if _print:
print(func)
arg = -matrix * time
res[func] = time_and_run(e[func], arg, _print=_print)
return res
def gen_lobatto(max_order):
assert max_order > 2
x = sm.symbols('x')
lobs = [0, 1]
lobs[0] = (1 - x) / 2
lobs[1] = (1 + x) / 2
dlobs = [lob.diff('x') for lob in lobs]
legs = [sm.legendre(0, 'y')]
clegs = [sm.ccode(legs[0])]
dlegs = [sm.legendre(0, 'y').diff('y')]
cdlegs = [sm.ccode(dlegs[0])]
clobs = [sm.ccode(lob) for lob in lobs]
cdlobs = [sm.ccode(dlob) for dlob in dlobs]
denoms = [] # for lobs.
for ii in range(2, max_order + 1):
coef = sm.sympify('sqrt(2 * (2 * %s - 1)) / 2' % ii)
leg = sm.legendre(ii - 1, 'y')
pleg = leg.as_poly()
coefs = pleg.all_coeffs()
denom = max(sm.denom(val) for val in coefs)
cleg = sm.ccode(sm.horner(leg * denom) / denom)
dleg = leg.diff('y')
cdleg = sm.ccode(sm.horner(dleg * denom) / denom)
lob = sm.simplify(coef * sm.integrate(leg, ('y', -1, x)))
lobnc = sm.simplify(sm.integrate(leg, ('y', -1, x)))
plobnc = lobnc.as_poly()
coefs = plobnc.all_coeffs()
denom = sm.denom(coef) * max(sm.denom(val) for val in coefs)
clob = sm.ccode(sm.horner(lob * denom) / denom)
dlob = lob.diff('x')
cdlob = sm.ccode(sm.horner(dlob * denom) / denom)
legs.append(leg)
clegs.append(cleg)
dlegs.append(dleg)
cdlegs.append(cdleg)
lobs.append(lob)
clobs.append(clob)
dlobs.append(dlob)
cdlobs.append(cdlob)
denoms.append(denom)
coef = sm.sympify('sqrt(2 * (2 * %s - 1)) / 2' % (max_order + 1))
leg = sm.legendre(max_order, 'y')
pleg = leg.as_poly()
coefs = pleg.all_coeffs()
denom = max(sm.denom(val) for val in coefs)
cleg = sm.ccode(sm.horner(leg * denom) / denom)
dleg = leg.diff('y')
cdleg = sm.ccode(sm.horner(dleg * denom) / denom)
legs.append(leg)
clegs.append(cleg)
dlegs.append(dleg)
cdlegs.append(cdleg)
kerns = []
ckerns = []
dkerns = []
cdkerns = []
for ii, lob in enumerate(lobs[2:]):
kern = sm.simplify(lob / (lobs[0] * lobs[1]))
dkern = kern.diff('x')
denom = denoms[ii] / 4
ckern = sm.ccode(sm.horner(kern * denom) / denom)
cdkern = sm.ccode(sm.horner(dkern * denom) / denom)
kerns.append(kern)
ckerns.append(ckern)
dkerns.append(dkern)
cdkerns.append(cdkern)
return (legs, clegs, dlegs, cdlegs, lobs, clobs, dlobs, cdlobs, kerns,
ckerns, dkerns, cdkerns, denoms)
def gen_lobatto(max_order):
assert max_order > 2
x = sm.symbols('x')
lobs = [0, 1]
lobs[0] = (1 - x) / 2
lobs[1] = (1 + x) / 2
dlobs = [lob.diff('x') for lob in lobs]
legs = [sm.legendre(0, 'y')]
clegs = [sm.ccode(legs[0])]
dlegs = [sm.legendre(0, 'y').diff('y')]
cdlegs = [sm.ccode(dlegs[0])]
clobs = [sm.ccode(lob) for lob in lobs]
cdlobs = [sm.ccode(dlob) for dlob in dlobs]
denoms = [] # for lobs.
for ii in range(2, max_order + 1):
coef = sm.sympify('sqrt(2 * (2 * %s - 1)) / 2' % ii)
leg = sm.legendre(ii - 1, 'y')
pleg = leg.as_poly()
coefs = pleg.all_coeffs()
denom = max(sm.denom(val) for val in coefs)
cleg = sm.ccode(sm.horner(leg*denom)/denom)
dleg = leg.diff('y')
cdleg = sm.ccode(sm.horner(dleg*denom)/denom)
lob = sm.simplify(coef * sm.integrate(leg, ('y', -1, x)))
lobnc = sm.simplify(sm.integrate(leg, ('y', -1, x)))
plobnc = lobnc.as_poly()
coefs = plobnc.all_coeffs()
denom = sm.denom(coef) * max(sm.denom(val) for val in coefs)
clob = sm.ccode(sm.horner(lob*denom)/denom)
dlob = lob.diff('x')
cdlob = sm.ccode(sm.horner(dlob*denom)/denom)
legs.append(leg)
clegs.append(cleg)
dlegs.append(dleg)
cdlegs.append(cdleg)
lobs.append(lob)
clobs.append(clob)
dlobs.append(dlob)
cdlobs.append(cdlob)
denoms.append(denom)
coef = sm.sympify('sqrt(2 * (2 * %s - 1)) / 2' % (max_order + 1))
leg = sm.legendre(max_order, 'y')
pleg = leg.as_poly()
coefs = pleg.all_coeffs()
denom = max(sm.denom(val) for val in coefs)
cleg = sm.ccode(sm.horner(leg*denom)/denom)
dleg = leg.diff('y')
cdleg = sm.ccode(sm.horner(dleg*denom)/denom)
legs.append(leg)
clegs.append(cleg)
dlegs.append(dleg)
cdlegs.append(cdleg)
kerns = []
ckerns = []
dkerns = []
cdkerns = []
for ii, lob in enumerate(lobs[2:]):
kern = sm.simplify(lob / (lobs[0] * lobs[1]))
dkern = kern.diff('x')
denom = denoms[ii] / 4
ckern = sm.ccode(sm.horner(kern*denom)/denom)
cdkern = sm.ccode(sm.horner(dkern*denom)/denom)
kerns.append(kern)
ckerns.append(ckern)
dkerns.append(dkern)
cdkerns.append(cdkern)
return (legs, clegs, dlegs, cdlegs,
lobs, clobs, dlobs, cdlobs,
kerns, ckerns, dkerns, cdkerns,
denoms)
def parse(f, fac=1, vec=False, toOpenCL=True,
CLBuiltins=False, keep=False):
"""
Parsing function.
@param f : functions to parse as string
@param fac : numeric factor for all formulas
@param vec : is OpenCL output is generated to use vector builtin types
@param toOpenCL : is OpenCL output
@param CLBuiltins : is OpenCL output uses fma builtin function
@param keep : low parsing
"""
msg = 'Vector works only in OpenCL parsing'
assert not (vec and not toOpenCL), msg
assert not (CLBuiltins and not toOpenCL),\
"CLBuiltins only in OpenCL parsing"
t = "float__N__" if vec else "float"
cteEnd = ".0" if toOpenCL else "."
res = ""
# Split each line
fl = f.split('\n')
# sympy formulas
y = sp.symbols('y')
print (y)
sw = [None] * f.count(';')
i = 0
for wl in fl:
if len(wl) > 2:
# replace pow
power = re.search('pow\(y, ([0-9]+)\)', wl)
if power is not None:
np = "y" + "*y" * (int(power.group(1)) - 1)
wl = wl.replace(power.group(0), np)
sw[i] = '('
sw[i] += str(sp.horner(eval(wl.split(';')[0].split('=')[1]) * fac))
sw[i] += ')/' + str(fac)
i += 1
for i, s in enumerate(sw):
if not keep:
if toOpenCL:
res += "inline " + t + " "
res += weights_names[i] + "(" + t + " y){\n"
res += ' return '
else:
res += 'lambda y, s: s * '
res += '('
# replace y**n
power = re.findall('y\*\*[0-9]+', s)
if power is not None:
for pw in power:
n = int(pw.split('**')[1])
npower = 'y' + "*y" * (n - 1)
s = s.replace(pw, npower)
s = s.replace(' ', '')
if CLBuiltins:
s = createFMA(s)
# From integers to floats
s = re.sub(r"(?P<id>[0-9]+)", r"\g<id>" + cteEnd, s)
s = s.replace('*', ' * ')
s = s.replace('/', ' / ')
s = s.replace('+', ' + ')
s = s.replace('-', ' - ')
s = s.replace('( - ', '(-')
s = s.replace(' ', ' ')
s = s.replace(", - ", ", -")
res += s + ')'
if toOpenCL:
res += ";}"
if i < len(sw) - 1:
res += "\n"
else:
res += "w[{0}] = ".format(i)
# replace y**n
power = re.findall('y\*\*[0-9]+', s)
if power is not None:
for pw in power:
n = int(pw.split('**')[1])
npower = 'y' + "*y" * (n - 1)
s = s.replace(pw, npower)
# From integers to floats
s = re.sub(r"(?P<id>[0-9]+)", r"\g<id>.", s)
s = s.replace('*', ' * ')
s = s.replace('/', ' / ')
s = s.replace('+', ' + ')
s = s.replace('-', ' - ')
s = s.replace('( - ', '(-')
s = s.replace(' ', ' ')
s = s.replace(", - ", ", -")
res += s + "\n"
return res
