def create_add_functions(header, coeffs, pairwise):
''' Generate all of the custom add functions.
header is the file to which we are writing
coeffs is the set of all coefficients
'''
def all_adds(coeffs, name, bfs_par_avail):
for i, coeff_set in enumerate(coeffs):
if len(coeff_set) > 0:
if pairwise:
write_pairwise_add_func(header, coeff_set, i + 1, name)
else:
write_add_func(header, coeff_set, i + 1, name,
bfs_par_avail)
write_break(header)
# S matrices formed from A subblocks
all_adds(subexpr_elim.transpose(coeffs[0]), 'S', False)
# T matrices formed from B subblocks
all_adds(subexpr_elim.transpose(coeffs[1]), 'T', False)
# Output C formed from multiplications
all_adds(coeffs[2], 'M', True)
# If there was CSE, create more add functions for the temporary matrices.
if len(coeffs) > 3:
all_adds(coeffs[3], 'SX', True)
if len(coeffs) > 4:
all_adds(coeffs[4], 'TX', True)
if len(coeffs) > 5:
all_adds(coeffs[5], 'MX', True)
def compute_phi(coeffs):
''' Compute phi (in Bini et al. notation), which is defined as
\phi = max { z | u_i^{(r)}v_j^{(r)}w_k^{(r)} = O(\epsilon^{-z}) },
where u_i^{(r)}, v_j^{(r)}, and w_k^{(r)} are entries of the rth column of U, V, and W.
'''
U = transpose(coeffs[0])
V = transpose(coeffs[1])
W = transpose(coeffs[2])
def smallest_exponent(u, v, w):
def min_exp(num):
# num.val is a dictionary of (exponent, coefficient)
all_viable = filter(lambda keyval: keyval[1] != 0, num.val.items())
if len(all_viable) == 0:
return 0
return min([key for key, val in all_viable])
all_nums = [Number(x1) * Number(x2) * Number(x3) \
for x1 in u for x2 in v for x3 in w]
return min([min_exp(num) for num in all_nums])
return -min([smallest_exponent(u, v, w) for u, v, w in zip(U, V, W)])
def compute_phi(coeffs):
''' Compute phi (in Bini et al. notation), which is defined as
\phi = max { z | u_i^{(r)}v_j^{(r)}w_k^{(r)} = O(\epsilon^{-z}) },
where u_i^{(r)}, v_j^{(r)}, and w_k^{(r)} are entries of the rth column of U, V, and W.
'''
U = transpose(coeffs[0])
V = transpose(coeffs[1])
W = transpose(coeffs[2])
def smallest_exponent(u, v, w):
def min_exp(num):
# num.val is a dictionary of (exponent, coefficient)
all_viable = filter(lambda keyval: keyval[1] != 0, num.val.items())
if len(all_viable) == 0:
return 0
return min([key for key, val in all_viable])
all_nums = [Number(x1) * Number(x2) * Number(x3) \
for x1 in u for x2 in v for x3 in w]
return min([min_exp(num) for num in all_nums])
return -min([smallest_exponent(u, v, w) for u, v, w in zip(U, V, W)])
def main():
try:
coeff_file = sys.argv[1]
dims = tuple([int(d) for d in sys.argv[2].split(',')])
except:
raise Exception(
'USAGE: python relative_quantities.py coeff_file m,k,n')
full_stab_mat = 0
if len(sys.argv) > 3:
full_stab_mat = sys.argv[3]
coeffs = read_coeffs(coeff_file)
# Using the notation from our paper
a_vec = np.array(
[sum(np.abs([float(x) for x in row])) for row in transpose(coeffs[0])])
b_vec = np.array(
[sum(np.abs([float(x) for x in row])) for row in transpose(coeffs[1])])
e_vec = [
np.dot(np.abs([float(x) for x in row]), a_vec * b_vec)
for row in coeffs[2]
]
R = len(coeffs[0][0])
print R, dims[0], dims[1], dims[2], int(max(e_vec))
def main():
try:
coeff_file = sys.argv[1]
dims = tuple([int(d) for d in sys.argv[2].split(',')])
except:
raise Exception('USAGE: python stability_vector.py coeff_file m,k,n')
coeffs = read_coeffs(coeff_file)
# Using the notation from our paper
a_vec = np.array([
sum(np.abs([zero_one(x) for x in row])) for row in transpose(coeffs[0])
])
b_vec = np.array([
sum(np.abs([zero_one(x) for x in row])) for row in transpose(coeffs[1])
])
e_vec = [
np.dot(np.abs([float(x) for x in row]), a_vec * b_vec)
for row in coeffs[2]
]
mn = max_norm(coeffs[0]) * max_norm(coeffs[1]) * max_norm(coeffs[2])
emax = int(np.max(e_vec))
print emax, mn, emax * mn
def main():
try:
coeff_file = sys.argv[1]
dims = tuple([int(d) for d in sys.argv[2].split(',')])
except:
raise Exception('USAGE: python stability_vector.py coeff_file m,k,n')
full_stab_mat = 0
if len(sys.argv) > 3:
full_stab_mat = sys.argv[3]
coeffs = read_coeffs(coeff_file)
# Using the notation from our paper
a_vec = np.array(
[sum(np.abs([float(x) for x in row])) for row in transpose(coeffs[0])])
b_vec = np.array(
[sum(np.abs([float(x) for x in row])) for row in transpose(coeffs[1])])
e_vec = [
np.dot(np.abs([float(x) for x in row]), a_vec * b_vec)
for row in coeffs[2]
]
alpha_vec = np.array([num_nonzero(row) for row in transpose(coeffs[0])])
beta_vec = np.array([num_nonzero(row) for row in transpose(coeffs[1])])
ab_vec = alpha_vec + beta_vec
gamma_vec = [num_nonzero(row) for row in coeffs[2]]
W_ind = np.array([[float(val) for val in row] for row in coeffs[2]])
W_ind[np.nonzero(W_ind)] = 1
q_vec = [gamma_vec[k] + np.max(ab_vec * W_ind[k,:]) \
for k in range(W_ind.shape[0])]
# Number of additions
nnz = sum([val for val in ab_vec]) + sum([val for val in gamma_vec])
# print as q_vec emax
rank = len(coeffs[0][0])
mkn = dims[0] * dims[1] * dims[2]
print mkn, rank, nnz, int(np.max(q_vec)), int(np.max(e_vec))
if full_stab_mat:
# Print in the same style as D'Alberto presents in his 2014 paper.
print 'e vector:'
for i in range(dims[0]):
out_format = '\t' + ' '.join(['%4d' for _ in range(dims[2])])
vals = e_vec[(i * dims[2]):(i * dims[2] + dims[2])]
print out_format % tuple(vals)
print 'q vector:'
for i in range(dims[0]):
out_format = '\t' + ' '.join(['%4d' for _ in range(dims[2])])
vals = q_vec[(i * dims[2]):(i * dims[2] + dims[2])]
print out_format % tuple(vals)
def main():
try:
coeff_file = sys.argv[1]
dims = tuple([int(d) for d in sys.argv[2].split(',')])
except:
raise Exception('USAGE: python stability_vector.py coeff_file m,k,n')
full_stab_mat = 0
if len(sys.argv) > 3:
full_stab_mat = sys.argv[3]
coeffs = read_coeffs(coeff_file)
# Using the notation from our paper
a_vec = np.array([sum(np.abs([float(x) for x in row])) for row in transpose(coeffs[0])])
b_vec = np.array([sum(np.abs([float(x) for x in row])) for row in transpose(coeffs[1])])
e_vec = [np.dot(np.abs([float(x) for x in row]), a_vec * b_vec) for row in coeffs[2]]
alpha_vec = np.array([num_nonzero(row) for row in transpose(coeffs[0])])
beta_vec = np.array([num_nonzero(row) for row in transpose(coeffs[1])])
ab_vec = alpha_vec + beta_vec
gamma_vec = [num_nonzero(row) for row in coeffs[2]]
W_ind = np.array([[float(val) for val in row] for row in coeffs[2]])
W_ind[np.nonzero(W_ind)] = 1
q_vec = [gamma_vec[k] + np.max(ab_vec * W_ind[k,:]) \
for k in range(W_ind.shape[0])]
# Number of additions
nnz = sum([val for val in ab_vec]) + sum([val for val in gamma_vec])
# print as q_vec emax
rank = len(coeffs[0][0])
mkn = dims[0] * dims[1] * dims[2]
print mkn, rank, nnz, int(np.max(q_vec)), int(np.max(e_vec))
if full_stab_mat:
# Print in the same style as D'Alberto presents in his 2014 paper.
print 'e vector:'
for i in range(dims[0]):
out_format = '\t' + ' '.join(['%4d' for _ in range(dims[2])])
vals = e_vec[(i * dims[2]):(i * dims[2] + dims[2])]
print out_format % tuple(vals)
print 'q vector:'
for i in range(dims[0]):
out_format = '\t' + ' '.join(['%4d' for _ in range(dims[2])])
vals = q_vec[(i * dims[2]):(i * dims[2] + dims[2])]
print out_format % tuple(vals)
def need_streaming_cse_tmp(ind, coeffs, mat_dims):
''' Given an index, determine whether or not we need an extra temporary
matrix for streaming additions. This occurs when we have an expression
that is a eliminated through common subexpression elimination and the
expression is part of a length-1 addition string in a multiplication.
For example,
M1 = (A11 + A12 + A13) * (B11 + B12)
M2 = (A11 + A12) * (B21 + B22)
We would eliminate A_X = A11 + A12 and have:
M1 = (A_X + A13) * (B11 + B12)
M2 = (A_X) * (B21 + B22)
A_X gets used as part of a length-1 addition string for M2.
ind is the index (zero-indexed linearly) of the of the matrix in the
coefficient file
coeffs is the set of coefficients for the U or V matrix
mat_dims is 2-tuple of the matrix dimensions (A or B corresponding to U or V)
'''
if ind < mat_dims[0] * mat_dims[1]:
return False
for col in subexpr_elim.transpose(coeffs):
data = [(i, val) for i, val in enumerate(col) if is_nonzero(val)]
if len(data) == 1 and data[0][0] == ind:
return True
return False
def main():
try:
coeff_file = sys.argv[1]
dims = tuple([int(d) for d in sys.argv[2].split(",")])
except:
raise Exception("USAGE: python counts.py coeff_file m,k,n")
coeffs = convert.read_coeffs(coeff_file)
coeffs[2] = subexpr_elim.transpose(coeffs[2])
counts = [base_counts(coeffs[i]) for i in range(3)]
counts_X = [(0, 0, 0) for i in range(3, 6)]
if len(coeffs) > 3:
counts_X = [elim_counts(coeffs[i]) for i in range(3, 6)]
xtotal = tuple([sum(cnts) for cnts in zip(*(counts_X))])
total = tuple([sum(cnts) for cnts in zip(*(counts + counts_X))])
print " + r w"
print "A %3d %3d %3d" % counts[0]
print "B %3d %3d %3d" % counts[1]
print "C %3d %3d %3d" % counts[2]
print "AX %3d %3d %3d" % counts_X[0]
print "BX %3d %3d %3d" % counts_X[1]
print "CX %3d %3d %3d" % counts_X[2]
print "xtot %3d %3d %3d" % xtotal
print "tot %3d %3d %3d" % total
def main():
try:
coeff_file = sys.argv[1]
dims = tuple([int(d) for d in sys.argv[2].split(',')])
except:
raise Exception('USAGE: python stability_vector.py coeff_file m,k,n')
coeffs = read_coeffs(coeff_file)
# Using the notation from our paper
a_vec = np.array([sum(np.abs([zero_one(x) for x in row])) for row in transpose(coeffs[0])])
b_vec = np.array([sum(np.abs([zero_one(x) for x in row])) for row in transpose(coeffs[1])])
e_vec = [np.dot(np.abs([float(x) for x in row]), a_vec * b_vec) for row in coeffs[2]]
mn = max_norm(coeffs[0]) * max_norm(coeffs[1]) * max_norm(coeffs[2])
emax = int(np.max(e_vec))
print emax, mn, emax * mn
def base_counts(coeff_set):
""" Count the number of additions, reads, and writes for this coefficient set (A, B, or C).
For A and B, these are the adds used to form the 'S' and 'T' matrices.
If an S or T matrix is just a length-1 addition chain, then no additions are counted.
"""
cols = subexpr_elim.transpose(coeff_set)
num_adds = sum([gen.num_nonzero(col) - 1 for col in cols])
num_reads = sum([gen.num_nonzero(col) for col in cols])
num_writes = sum([1 for col in cols if gen.num_nonzero(col) > 1])
return (num_adds, num_reads, num_writes)
def main():
try:
coeff_file = sys.argv[1]
dims = tuple([int(d) for d in sys.argv[2].split(',')])
except:
raise Exception('USAGE: python relative_quantities.py coeff_file m,k,n')
full_stab_mat = 0
if len(sys.argv) > 3:
full_stab_mat = sys.argv[3]
coeffs = read_coeffs(coeff_file)
# Using the notation from our paper
a_vec = np.array([sum(np.abs([float(x) for x in row])) for row in transpose(coeffs[0])])
b_vec = np.array([sum(np.abs([float(x) for x in row])) for row in transpose(coeffs[1])])
e_vec = [np.dot(np.abs([float(x) for x in row]), a_vec * b_vec) for row in coeffs[2]]
R = len(coeffs[0][0])
print R, dims[0], dims[1], dims[2], int(max(e_vec))
def streaming_additions(header,
coeff_set,
mat_name,
tmp_name,
mat_dims,
is_output,
num_multiplies,
sub_coeffs=None):
'''
Write the streaming additions for the matrices to be used in the multiplications.
header is the file where the code is being generated
coeff_set is the set of coefficients (corresponding to U, V, or W)
mat_name is the name of the matrix we are working on (A, B, or C, that matches
U, V, or W where coeff_set comes from)
tmp_name is the base name to use for temporary variables (e.g., 'S' or 'T')
mat_dims is the dimension of the base case matrix
is_output indicates whether or not we are working on the matrix C
num_multiplies is the number of multiplications, i.e., the rank of the
algorithm sub_coeffs are the substitution coefficients used
for common subexpression elimination (CSE). This is an
optional argument, and should only be used if we are using CSE.
'''
def tmp_mat_name(i):
return tmp_name + str(i + 1)
def subblock_name(i):
return mat_name + get_suffix(i, mat_dims[0], mat_dims[1])
# Find indices of additional temporaries needed from subexpression elimination
if is_output:
additional_tmps = []
else:
additional_tmps = [i for i in range(len(coeff_set)) if \
need_streaming_cse_tmp(i, coeff_set, mat_dims)]
# All of the strides for the matrix subblocks
for i in range(len(coeff_set)):
subblock = subblock_name(i)
if i in additional_tmps:
write_line(header, 1, instantiate(subblock, mat_name))
if i < mat_dims[0] * mat_dims[1] or i in additional_tmps:
write_line(header, 1, stride_call(subblock))
write_line(header, 1, data_call(subblock))
# Data for the temporary matrices
if is_output:
for i in xrange(num_multiplies):
tmp_mat = tmp_mat_name(i)
write_line(header, 1, stride_call(tmp_mat))
write_line(header, 1, data_call(tmp_mat))
else:
for i, col in enumerate(subexpr_elim.transpose(coeff_set)):
if need_tmp_mat(col):
tmp_mat = tmp_mat_name(i)
write_line(header, 1, instantiate(tmp_mat, mat_name))
write_line(header, 1, stride_call(tmp_mat))
write_line(header, 1, data_call(tmp_mat))
if not is_output:
coeff_set = subexpr_elim.transpose(coeff_set)
def inner_loop(handle_beta=False):
write_line(header, 0, '#ifdef _PARALLEL_')
write_line(header, 0, '# pragma omp parallel for')
write_line(header, 0, '#endif')
write_line(header, 1,
'for (int j = 0; j < %s11.n(); ++j) {' % mat_name)
write_line(header, 2,
'for (int i = 0; i < %s11.m(); ++i) {' % mat_name)
# Deal with substitutions from CSE
if sub_coeffs != None:
for i, col in enumerate(sub_coeffs):
if i + mat_dims[0] * mat_dims[1] in additional_tmps:
add = data_access('%s_X%d' % (mat_name, i + 1)) + ' ='
else:
if is_output:
curr_name = tmp_name
else:
curr_name = mat_name
add = 'Scalar %s_X%d = ' % (curr_name, i + 1)
data = [(k, coeff) for k, coeff in enumerate(col)
if is_nonzero(coeff)]
for j, (ind, coeff) in enumerate(data):
if is_output:
data_name = tmp_mat_name(ind)
else:
data_name = subblock_name(ind)
add += arith_expression(coeff, data_access(data_name), j)
add += ';'
write_line(header, 3, add)
for i, col in enumerate(coeff_set):
if need_tmp_mat(col):
if is_output:
add = data_access(subblock_name(i))
else:
add = data_access(tmp_mat_name(i))
add += ' = '
data = [(k, coeff) for k, coeff in enumerate(col)
if is_nonzero(coeff)]
for j, (ind, coeff) in enumerate(data):
if is_output:
if ind >= num_multiplies:
data_name = 'M_X' + str(ind + 1 - num_multiplies)
else:
data_name = tmp_mat_name(ind)
else:
data_name = subblock_name(ind)
# Deal with subexpression elimination
if (ind >= mat_dims[0] * mat_dims[1] and not is_output and ind not in additional_tmps) or \
(ind >= num_multiplies and is_output):
add += arith_expression(coeff, data_name, j)
else:
add += arith_expression(coeff, data_access(data_name),
j)
if is_output and handle_beta:
add += ' + beta * ' + data_access(subblock_name(i))
add += ';'
write_line(header, 3, add)
write_line(header, 2, '}') # end i loop
write_line(header, 1, '}') # end j loop
if not is_output:
inner_loop()
else:
write_line(header, 1, 'if (beta == Scalar(0.0)) {')
inner_loop()
write_line(header, 1, '} else {')
inner_loop(True)
write_line(header, 1, '}')