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Fastest way to solve non-negative linear diophantine equations

Let A be a list of n lists of m non-negative integers, such that for all j there is i with A[i][j] nonzero. Let V be a list of m positive integers.
Question: What is the fastest way to find all the lists X of n non-negative integers such that for all i then sum_j A[i][j] X[j] = V[i]?
The assumptions implies that the number of solutions is finite. See below an example of A and V, with 5499 solutions X.
Let me reformulate the problem using matrix and vector. Let A be a n-by-m matrix with non-negative integral entries and without zero column. Let V be a vector with positive integral entries. What is the fastest way to find all the vectors X, with non-negative integral entries, such that AX=V?
The usual functions for solving such a system may underuse the non-negativity. To prove so, I wrote a brute-force code finding all the solutions of such a system and applied it to an example (see below, and these crossposts on mathoverflow and on ask.sagemath), but I'm still looking for something significantly faster than this; in fact I'm looking for the fastest way.
Example
Here is the kind of system I need to solve (where x_i is non-negative integral), but with possibly more equations and variables.
[
5*x0 + 5*x1 + 5*x2 + 6*x3 + 7*x4 + 7*x5 == 24,
5*x1 + 7*x10 + 5*x6 + 5*x7 + 6*x8 + 7*x9 == 25,
5*x11 + 6*x12 + 7*x13 + 7*x14 + 5*x2 + 5*x7 == 25,
5*x12 + 6*x15 + 7*x16 + 7*x17 + 5*x3 + 5*x8 == 30,
5*x13 + 6*x16 + 7*x18 + 7*x19 + 5*x4 + 5*x9 == 35,
5*x10 + 5*x14 + 6*x17 + 7*x19 + 7*x20 + 5*x5 == 35,
5*x1 + 7*x10 + 5*x6 + 5*x7 + 6*x8 + 7*x9 == 25,
5*x21 + 5*x22 + 6*x23 + 7*x24 + 7*x25 + 5*x6 == 24,
5*x22 + 5*x26 + 6*x27 + 7*x28 + 7*x29 + 5*x7 == 25,
5*x23 + 5*x27 + 6*x30 + 7*x31 + 7*x32 + 5*x8 == 30,
5*x24 + 5*x28 + 6*x31 + 7*x33 + 7*x34 + 5*x9 == 35,
5*x10 + 5*x25 + 5*x29 + 6*x32 + 7*x34 + 7*x35 == 35,
5*x11 + 6*x12 + 7*x13 + 7*x14 + 5*x2 + 5*x7 == 25,
5*x22 + 5*x26 + 6*x27 + 7*x28 + 7*x29 + 5*x7 == 25,
5*x11 + 5*x26 + 5*x36 + 6*x37 + 7*x38 + 7*x39 == 24,
5*x12 + 5*x27 + 5*x37 + 6*x40 + 7*x41 + 7*x42 == 30,
5*x13 + 5*x28 + 5*x38 + 6*x41 + 7*x43 + 7*x44 == 35,
5*x14 + 5*x29 + 5*x39 + 6*x42 + 7*x44 + 7*x45 == 35,
5*x12 + 6*x15 + 7*x16 + 7*x17 + 5*x3 + 5*x8 == 30,
5*x23 + 5*x27 + 6*x30 + 7*x31 + 7*x32 + 5*x8 == 30,
5*x12 + 5*x27 + 5*x37 + 6*x40 + 7*x41 + 7*x42 == 30,
5*x15 + 5*x30 + 5*x40 + 6*x46 + 7*x47 + 7*x48 == 35,
5*x16 + 5*x31 + 5*x41 + 6*x47 + 7*x49 + 7*x50 == 42,
5*x17 + 5*x32 + 5*x42 + 6*x48 + 7*x50 + 7*x51 == 42,
5*x13 + 6*x16 + 7*x18 + 7*x19 + 5*x4 + 5*x9 == 35,
5*x24 + 5*x28 + 6*x31 + 7*x33 + 7*x34 + 5*x9 == 35,
5*x13 + 5*x28 + 5*x38 + 6*x41 + 7*x43 + 7*x44 == 35,
5*x16 + 5*x31 + 5*x41 + 6*x47 + 7*x49 + 7*x50 == 42,
5*x18 + 5*x33 + 5*x43 + 6*x49 + 7*x52 + 7*x53 == 48,
5*x19 + 5*x34 + 5*x44 + 6*x50 + 7*x53 + 7*x54 == 49,
5*x10 + 5*x14 + 6*x17 + 7*x19 + 7*x20 + 5*x5 == 35,
5*x10 + 5*x25 + 5*x29 + 6*x32 + 7*x34 + 7*x35 == 35,
5*x14 + 5*x29 + 5*x39 + 6*x42 + 7*x44 + 7*x45 == 35,
5*x17 + 5*x32 + 5*x42 + 6*x48 + 7*x50 + 7*x51 == 42,
5*x19 + 5*x34 + 5*x44 + 6*x50 + 7*x53 + 7*x54 == 49,
5*x20 + 5*x35 + 5*x45 + 6*x51 + 7*x54 + 7*x55 == 48
]
Here are explicit A and V from above system (in list form):
A=[
[5,5,5,6,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,5,0,0,0,0,5,5,6,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,5,0,0,0,0,5,0,0,0,5,6,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,5,0,0,0,0,5,0,0,0,5,0,0,6,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,5,0,0,0,0,5,0,0,0,5,0,0,6,0,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,5,0,0,0,0,5,0,0,0,5,0,0,6,0,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,5,0,0,0,0,5,5,6,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,5,6,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,5,6,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,5,0,0,6,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,5,0,0,6,0,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,5,0,0,6,0,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,5,0,0,0,0,5,0,0,0,5,6,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,5,6,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,5,6,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,6,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,6,0,7,7,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,6,0,7,7,0,0,0,0,0,0,0,0,0,0],
[0,0,0,5,0,0,0,0,5,0,0,0,5,0,0,6,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,5,0,0,6,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,6,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,6,7,7,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,6,0,7,7,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,6,0,7,7,0,0,0,0],
[0,0,0,0,5,0,0,0,0,5,0,0,0,5,0,0,6,0,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,5,0,0,6,0,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,6,0,7,7,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,6,0,7,7,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,6,0,0,7,7,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,6,0,0,7,7,0],
[0,0,0,0,0,5,0,0,0,0,5,0,0,0,5,0,0,6,0,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,5,0,0,6,0,7,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,6,0,7,7,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,6,0,7,7,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,6,0,0,7,7,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,6,0,0,7,7]
]
V=[24,25,25,30,35,35,25,24,25,30,35,35,25,25,24,30,35,35,30,30,30,35,42,42,35,35,35,42,48,49,35,35,35,42,49,48]
Computation
I wrote a brute-force code finding all the solutions of such a system, then applied it to A, V above. It took 12 seconds to find all 5499 solutions. I'm looking for something significantly faster than this.
sage: %time LX=NonnegativeSolverPartition(A,V)
CPU times: user 11.8 s, sys: 0 ns, total: 11.8 s
Wall time: 11.8 s
sage: len(LX)
5499
Note that the time reduces to 3 seconds with PyPy3, but it is still too slow for other (bigger) such systems I need to solve.
Code
Here is my (python) code, improved by Peter Taylor (see his comment):
def NonnegativeSolverPartition(A,V):
WB = WeakBasis(A)
VB = VarBound(A,V)
PP = []
for i, ll in WB:
L = tuple(A[i][j] for j in ll)
B = tuple(VB[j] for j in ll)
PP.append(WeightedPartitionSolver(L, B, V[i]))
return list(NonnegativeSolverPartitionInter(A, V, PP, WB, [-1] * len(A[0])))
def NonnegativeSolverPartitionInter(A, V, PP, WB, X):
if any(len(P) > 1 for P in PP):
_, ii = min((len(P), i) for i, P in enumerate(PP) if len(P) > 1)
for p in PP[ii]:
PPP = PP[:ii] + [[p]] + PP[ii+1:]
Fi = Filter(PPP, list(X), WB)
if Fi:
PPPP, XX = Fi
yield from NonnegativeSolverPartitionInter(A, V, PPPP, WB, XX)
else:
assert -1 not in X
yield X
def WeakBasis(A):
return tuple(enumerate([j for j, tmp in enumerate(row) if tmp] for row in A))
def WeightedPartitions(ws, n):
def inner(ws, n):
if n == 0:
yield (0,) * len(ws)
elif ws:
w = ws[0]
lim = n // w
ws = ws[1:]
for i in range(lim + 1):
for tl in inner(ws, n - w * i):
yield (i,) + tl
return list(inner(ws, n))
def VarBound(A,V):
nvars = len(A[0])
# Solve the individual constraints and then intersect the solutions.
possible_values = [None] * nvars
row_solns = []
for row, v in zip(A, V):
lut = []
ws = []
var_assignments = []
for j, val in enumerate(row):
if val:
lut.append(j)
ws.append(val)
var_assignments.append(set())
for soln in WeightedPartitions(ws, v):
for i, w in enumerate(soln):
var_assignments[i].add(w)
for j, assignments in zip(lut, var_assignments):
if possible_values[j] is None:
possible_values[j] = assignments
else:
possible_values[j] &= assignments
return tuple(frozenset(x) for x in possible_values)
def WeightedPartitionSolver(L, B, n):
# the entries of L must be non-negative
# B gives valid values coming from other equations (see VarBound)
def inner(L, B, n):
if n == 0:
yield (0,) * len(L)
elif L:
w, allowed = L[0], B[0]
L, B = L[1:], B[1:]
for i in range(n // w + 1):
if i in allowed:
for tl in inner(L, B, n - w * i):
yield (i,) + tl
return list(inner(L, B, n))
def Filter(PP, X, W):
if [] in PP:
return None
while True:
for Wi, P in zip(W, PP):
F = FixedPoints(P)
for j in F:
P0j = P[0][j]
Wij = Wi[1][j]
if X[Wij] == -1:
X[Wij] = P0j
elif X[Wij] != P0j:
return None
LL=[]
for Wi, P in zip(W, PP):
LL.append([p for p in P if not any(X[idx] not in (-1, pval) for idx, pval in zip(Wi[1], p))])
if not LL[-1]:
return None
if PP == LL:
return LL, X
PP = LL
def FixedPoints(P):
# This would prefer P to be transposed
m=len(P)
n=len(P[0])
return tuple(i for i in range(n) if all(P[j][i] == P[0][i] for j in range(m)))
A simpler brute-force code by Max Alekseyev is available in this answer.

IndexError: invalid index to scalar variable.- PSO algorithm

While trying to solve this problem using PSO algorithm I got the error message as "IndexError: invalid index to scalar variable". Since I am new to PSO algorithm, I am unable to figure out the error. Kindly help me to solve the issue.
import random
import numpy as np
import math
import matplotlib.pyplot as plt
def fitness_function(position):
s1=0.014109786*position[0] + 0.004596846*position[1] + 0.01603721*position[2] + 0.029275618*position[3] + 0.007085358*position[4] + 0.013234328*position[5] + 0.012554958*position[6] + 0.012447232*position[7] + 0.007867602*position[8] + 0.011312568*position[9] + 0.003087858*position[10] + 0.016566954*position[11] + 0.008428942*position[12] + 0.008477444*position[13] + 0.004357354*position[14]
s2=0.016566954*position[0] + 0.00585045*position[1] + 0.053172638*position[2] + 0.113404042*position[3] + 0.028190744*position[4] + 0.046330688*position[5] + 0.05629084*position[6] + 0.047796486*position[7] + 0.025793022*position[8] + 0.046164518*position[9] + 0.026696192*position[10] + 0.080422654*position[11] + 0.029074508*position[12] + 0.039611624*position[13] + 0.044835566*position[14]
return s1+s2
#Some variables to calculate the velocity
W = 0.5
c1 = 0.8
c2 = 0.9
target = 1
n_iterations = int(input("Inform the number of iterations: "))
target_error = float(input("Inform the target error: "))
n_particles = int(input("Inform the number of particles: "))
particle_position_vector=np.array([np.random.uniform(low=0.025, high=0.035) for i in range (n_particles)])
print(particle_position_vector)
pbest_position = particle_position_vector
pbest_fitness_value = float('inf')
gbest_fitness_value = float('inf')
gbest_position = np.array([float('inf')for _ in range(n_particles)])
velocity_vector =np.array([0,0,0,0,0,0,0,0,0,0,0,0,0,0])
print(velocity_vector)
iteration = 0
while iteration < n_iterations:
for i in range(n_particles):
fitness_cadidate = fitness_function(particle_position_vector[i])
print(fitness_cadidate, ' ', particle_position_vector[i])
if(pbest_fitness_value[i] > fitness_cadidate):
pbest_fitness_value[i] = fitness_cadidate
pbest_position[i] = particle_position_vector[i]
if(gbest_fitness_value > fitness_cadidate):
gbest_fitness_value = fitness_cadidate
gbest_position = particle_position_vector[i]
if(abs(gbest_fitness_value - target) < target_error):
break
for i in range(n_particles):
new_velocity = (W*velocity_vector[i]) + (c1*random.random()) * (pbest_position[i] - particle_position_vector[i]) + (c2*random.random()) * (gbest_position-particle_position_vector[i])
new_position = new_velocity + particle_position_vector[i]
particle_position_vector[i] = new_position
iteration = iteration + 1
print("The best position is ", gbest_position, "in iteration number ", iteration)```
```IndexError Traceback (most recent call last)
<ipython-input-35-5610603d3302> in <module>
32 while iteration < n_iterations:
33 for i in range(n_particles):
---> 34 fitness_cadidate = fitness_function(particle_position_vector[i])
35 print(fitness_cadidate, ' ', particle_position_vector[i])
36
<ipython-input-35-5610603d3302> in fitness_function(position)
5
6 def fitness_function(position):
----> 7 s1=0.014109786*position[0] + 0.004596846*position[1] + 0.01603721*position[2] + 0.029275618*position[3] + 0.007085358*position[4] + 0.013234328*position[5] + 0.012554958*position[6] + 0.012447232*position[7] + 0.007867602*position[8] + 0.011312568*position[9] + 0.003087858*position[10] + 0.016566954*position[11] + 0.008428942*position[12] + 0.008477444*position[13] + 0.004357354*position[14]
8 s2=0.016566954*position[0] + 0.00585045*position[1] + 0.053172638*position[2] + 0.113404042*position[3] + 0.028190744*position[4] + 0.046330688*position[5] + 0.05629084*position[6] + 0.047796486*position[7] + 0.025793022*position[8] + 0.046164518*position[9] + 0.026696192*position[10] + 0.080422654*position[11] + 0.029074508*position[12] + 0.039611624*position[13] + 0.044835566*position[14]
9 return s1+s2
IndexError: invalid index to scalar variable.
Moreover, I want to plot a graph of fitness value with the iteration numbers. Please help me.

numpy.ndarray lambda is not callable within loop when using scipy.minimize_scalar, works fine single-run

I'm working on a piece of code that calculates surface wave velocities. When I run this as a single-shot in my Jupyter notebook it works fine, however when I try to start looping over the phi1 angle, I end up with the 'numpy.ndarray is not callable error' after the zero iteration of the loop.
It seems to be focused on the lambda (g) I am using, but I am not sure the best way to go about debugging it. Any help/review would be greatly appreciated as Google searches aren't helping too much.
I do apologize for the novel of code, but this is trimmed down to where it should be easily reproducible.
## This code is an implementation of the framework described by M. Cherry et al. in
## Cherry, Matthew R., Shamachary Sathish, and Ramana Grandhi. "A numerical method for predicting Rayleigh surface wave
## velocity in anisotropic crystals." Journal of Computational Physics 351 (2017): 108-120.
## The logic of this code is as follows.
## (1) It then calculates R, T, and Qv (which is the velocity independent Q, needed to determine the velocity range
## over which to search.
## (2) It then iterates to determine the velocity range by changing a variable called 'angle'.
## (3) It then finds a minimum vL, and solves for Q (with velocity as a term), N, A, B, and Z matrices.
## (4) It stores the three Euler angles (p1, P, p2) and RSW_vel
## This does not deal with pseudowaves yet.
import matplotlib.pyplot as plt
import numpy as np
import scipy as sp
import scipy.optimize as spo
import scipy.linalg as la
from scipy.linalg import eig, eigvals
import math
from decimal import *
import time
ts1 = time.time()
# Setup material properties
# Setup the Cij matrix of Ni (units in Pa)
C11 = C22 = C33 = 2.41e11
C12 = C13 = C23 = C21 = C31 = C32 = 1.47e11
C14 = C15 = C16 = C24 = C25 = C26 = C34 = C35 = C36 = C41 = C42 = C43 = C51 = C52 = C53 = C61 = C62 = C63 = 0
C44 = C55 = C66 = 1.26e11
C45 = C46 = C56 = C54 = C64 = C65 = 0
Cij_list = np.array([C11,C12,C13,C14,C15,C16,C21,C22,C23,C24,C25,C26,C31,C32,C33,C34,C35,C36,C41,C42,C43,C44,C45,C46,C51,C52,C53,C54,C55,C56,C61,C62,C63,C64,C65,C66])
Cij=Cij_list.reshape(6,6)
print('Cij =')
print(Cij)
# Setup density of Ni (units are in kg/m3)
rho = 8910
RSW_matrix = np.array([0,0,0,0])
## This is where the setting up of the loop for p1, P, and p2 should be.
# Setup Euler Angles in Bunge's active notation (one time only - demo, will need to be iterated over), angles in degrees. p1 = phi1, P = Phi, p2 = phi2
#p1 = 15
P = 45
p2 = 8
for rotation in range(0, 180):
## Below this point, everything must be done for every Euler calculation.
p1_rad = np.radians(rotation)
P_rad = np.radians(P)
p2_rad = np.radians(p2)
Eul_Ang_deg = np.array([p1,P,p2])
Eul_Ang_rad = np.array([p1_rad,P_rad,p2_rad])
c1 = np.cos(p1_rad)
c2 = np.cos(P_rad)
c3 = np.cos(p2_rad)
s1 = np.sin(p1_rad)
s2 = np.sin(P_rad)
s3 = np.sin(p2_rad)
# Solve Rotation Matrix from Bunge's active notation, and m and n (m is wave propagation vector, n is plane normal)
Rot_mat = np.array([[c1*c3-c2*s1*s3,-c1*s3-c2*c3*s1,s1*s2],
[c3*s1+c1*c2*s3,c1*c2*c3-s1*s3,-c1*s2],
[s2*s3,c3*s2,c2]]) #- could fill in the rest here and just extract col 1 and col 3
print('Rotation Matrix =')
print(Rot_mat)
m_vec = Rot_mat[[0],:]
m1 = Rot_mat[[0],[0]]
m2 = Rot_mat[[0],[1]]
m3 = Rot_mat[[0],[2]]
n_vec = Rot_mat[[2],:]
n1 = Rot_mat[[2],[0]]
n2 = Rot_mat[[2],[1]]
n3 = Rot_mat[[2],[2]]
# Below, we calculate R, T and Q matrices
# CORRECT R matrix calculations
Rij = np.ndarray(shape=(3,3))
Rij[0,0] = C11*m1*n1 + C61*m2*n1 + C51*m3*n1 + C16*m1*n2 + C66*m2*n2 + C56*m3*n2 + C15*m1*n3 + C65*m2*n3 + C55*m3*n3
Rij[0,1] = C16*m1*n1 + C66*m2*n1 + C56*m3*n1 + C12*m1*n2 + C62*m2*n2 + C52*m3*n2 + C14*m1*n3 + C64*m2*n3 + C54*m3*n3
Rij[0,2] = C15*m1*n1 + C65*m2*n1 + C55*m3*n1 + C14*m1*n2 + C64*m2*n2 + C54*m3*n2 + C13*m1*n3 + C63*m2*n3 + C53*m3*n3
Rij[1,0] = C61*m1*n1 + C21*m2*n1 + C41*m3*n1 + C66*m1*n2 + C26*m2*n2 + C46*m3*n2 + C65*m1*n3 + C25*m2*n3 + C45*m3*n3
Rij[1,1] = C66*m1*n1 + C26*m2*n1 + C46*m3*n1 + C62*m1*n2 + C22*m2*n2 + C42*m3*n2 + C64*m1*n3 + C24*m2*n3 + C44*m3*n3
Rij[1,2] = C65*m1*n1 + C25*m2*n1 + C45*m3*n1 + C64*m1*n2 + C24*m2*n2 + C44*m3*n2 + C63*m1*n3 + C23*m2*n3 + C43*m3*n3
Rij[2,0] = C51*m1*n1 + C41*m2*n1 + C31*m3*n1 + C56*m1*n2 + C46*m2*n2 + C36*m3*n2 + C55*m1*n3 + C45*m2*n3 + C35*m3*n3
Rij[2,1] = C56*m1*n1 + C46*m2*n1 + C36*m3*n1 + C52*m1*n2 + C42*m2*n2 + C32*m3*n2 + C54*m1*n3 + C44*m2*n3 + C34*m3*n3
Rij[2,2] = C55*m1*n1 + C45*m2*n1 + C35*m3*n1 + C54*m1*n2 + C44*m2*n2 + C34*m3*n2 + C53*m1*n3 + C43*m2*n3 + C33*m3*n3
Rij_trans = np.transpose(Rij)
Rij_inv = np.linalg.inv(Rij)
# CORRECT T matrix calculations
Tij = np.ndarray(shape=(3,3))
Tij[0,0] = C11*n1*n1 + C16*n2*n1 + C51*n3*n1 + C16*n1*n2 + C66*n2*n2 + C56*n3*n2 + C15*n1*n3 + C65*n2*n3 + C55*n3*n3
Tij[0,1] = C16*n1*n1 + C66*n2*n1 + C56*n3*n1 + C12*n1*n2 + C62*n2*n2 + C52*n3*n2 + C14*n1*n3 + C64*n2*n3 + C54*n3*n3
Tij[0,2] = C15*n1*n1 + C65*n2*n1 + C55*n3*n1 + C14*n1*n2 + C64*n2*n2 + C54*n3*n2 + C13*n1*n3 + C63*n2*n3 + C53*n3*n3
Tij[1,0] = C61*n1*n1 + C21*n2*n1 + C41*n3*n1 + C66*n1*n2 + C26*n2*n2 + C46*n3*n2 + C65*n1*n3 + C25*n2*n3 + C45*n3*n3
Tij[1,1] = C66*n1*n1 + C26*n2*n1 + C46*n3*n1 + C62*n1*n2 + C22*n2*n2 + C42*n3*n2 + C64*n1*n3 + C24*n2*n3 + C44*n3*n3
Tij[1,2] = C65*n1*n1 + C25*n2*n1 + C45*n3*n1 + C64*n1*n2 + C24*n2*n2 + C44*n3*n2 + C63*n1*n3 + C23*n2*n3 + C43*n3*n3
Tij[2,0] = C51*n1*n1 + C41*n2*n1 + C31*n3*n1 + C56*n1*n2 + C46*n2*n2 + C36*n3*n2 + C55*n1*n3 + C45*n2*n3 + C35*n3*n3
Tij[2,1] = C56*n1*n1 + C46*n2*n1 + C36*n3*n1 + C52*n1*n2 + C42*n2*n2 + C32*n3*n2 + C54*n1*n3 + C44*n2*n3 + C34*n3*n3
Tij[2,2] = C55*n1*n1 + C45*n2*n1 + C35*n3*n1 + C54*n1*n2 + C44*n2*n2 + C34*n3*n2 + C53*n1*n3 + C43*n2*n3 + C33*n3*n3
Tij_trans = np.transpose(Tij)
Tij_inv = np.linalg.inv(Tij)
## R and T defined as above. Q needs to be new - and not remove the rv2. Thus, we introduce Qv for "Q to determine
## critical velocity"
Qv = np.ndarray(shape=(3,3))
Qv[0,0] = C11*m1*m1 + C61*m2*m1 + C51*m3*m1 + C16*m1*m2 + C66*m2*m2 + C56*m3*m2 + C15*m1*m3 + C65*m2*m3 + C55*m3*m3
Qv[0,1] = C16*m1*m1 + C66*m2*m1 + C56*m3*m1 + C12*m1*m2 + C62*m2*m2 + C52*m3*m2 + C14*m1*m3 + C64*m2*m3 + C54*m3*m3
Qv[0,2] = C15*m1*m1 + C65*m2*m1 + C55*m3*m1 + C14*m1*m2 + C64*m2*m2 + C54*m3*m2 + C13*m1*m3 + C63*m2*m3 + C53*m3*m3
Qv[1,0] = C61*m1*m1 + C21*m2*m1 + C41*m3*m1 + C66*m1*m2 + C26*m2*m2 + C46*m3*m2 + C65*m1*m3 + C25*m2*m3 + C45*m3*m3
Qv[1,1] = C66*m1*m1 + C26*m2*m1 + C46*m3*m1 + C62*m1*m2 + C22*m2*m2 + C42*m3*m2 + C64*m1*m3 + C24*m2*m3 + C44*m3*m3
Qv[1,2] = C65*m1*m1 + C25*m2*m1 + C45*m3*m1 + C64*m1*m2 + C24*m2*m2 + C44*m3*m2 + C63*m1*m3 + C23*m2*m3 + C43*m3*m3
Qv[2,0] = C51*m1*m1 + C41*m2*m1 + C31*m3*m1 + C56*m1*m2 + C46*m2*m2 + C36*m3*m2 + C55*m1*m3 + C45*m2*m3 + C35*m3*m3
Qv[2,1] = C56*m1*m1 + C46*m2*m1 + C36*m3*m1 + C52*m1*m2 + C42*m2*m2 + C32*m3*m2 + C54*m1*m3 + C44*m2*m3 + C34*m3*m3
Qv[2,2] = C55*m1*m1 + C45*m2*m1 + C35*m3*m1 + C54*m1*m2 + C44*m2*m2 + C34*m3*m2 + C53*m1*m3 + C43*m2*m3 + C33*m3*m3
res = []
res = spo.minimize_scalar(lambda x: ((min(eigvals(Qv * np.cos(np.radians(x)) * np.cos(np.radians(x)) + (Rij + Rij_trans) * np.cos(np.radians(x)) * np.sin(np.radians(x)) + Tij * np.sin(np.radians(x)) * np.sin(np.radians(x)))) / rho)**(0.5))*(1/(np.abs(np.cos(np.radians(x))))), bracket=(-90,0,90), method='Golden')
vel_L = res.fun.real
vel_L_int = int(vel_L)
print('velocity limit')
print(vel_L)
print(vel_L_int)
Det_B_try = 1
Qij = np.zeros(shape=(3,3))
Qij[0,1] = C16*m1*m1 + C66*m2*m1 + C56*m3*m1 + C12*m1*m2 + C62*m2*m2 + C52*m3*m2 + C14*m1*m3 + C64*m2*m3 + C54*m3*m3
Qij[0,2] = C15*m1*m1 + C65*m2*m1 + C55*m3*m1 + C14*m1*m2 + C64*m2*m2 + C54*m3*m2 + C13*m1*m3 + C63*m2*m3 + C53*m3*m3
Qij[1,0] = C61*m1*m1 + C21*m2*m1 + C41*m3*m1 + C66*m1*m2 + C26*m2*m2 + C46*m3*m2 + C65*m1*m3 + C25*m2*m3 + C45*m3*m3
Qij[1,2] = C65*m1*m1 + C25*m2*m1 + C45*m3*m1 + C64*m1*m2 + C24*m2*m2 + C44*m3*m2 + C63*m1*m3 + C23*m2*m3 + C43*m3*m3
Qij[2,0] = C51*m1*m1 + C41*m2*m1 + C31*m3*m1 + C56*m1*m2 + C46*m2*m2 + C36*m3*m2 + C55*m1*m3 + C45*m2*m3 + C35*m3*m3
Qij[2,1] = C56*m1*m1 + C46*m2*m1 + C36*m3*m1 + C52*m1*m2 + C42*m2*m2 + C32*m3*m2 + C54*m1*m3 + C44*m2*m3 + C34*m3*m3
for v in range(1, vel_L_int):
# Calculate rho vel-squared
rv2 = rho * v ** 2
# CORRECT Q matrix calculations
Qij[0,0] = C11*m1*m1 + C61*m2*m1 + C51*m3*m1 + C16*m1*m2 + C66*m2*m2 + C56*m3*m2 + C15*m1*m3 + C65*m2*m3 + C55*m3*m3 - rv2
Qij[1,1] = C66*m1*m1 + C26*m2*m1 + C46*m3*m1 + C62*m1*m2 + C22*m2*m2 + C42*m3*m2 + C64*m1*m3 + C24*m2*m3 + C44*m3*m3 - rv2
Qij[2,2] = C55*m1*m1 + C45*m2*m1 + C35*m3*m1 + C54*m1*m2 + C44*m2*m2 + C34*m3*m2 + C53*m1*m3 + C43*m2*m3 + C33*m3*m3 - rv2
Qij_trans = np.transpose(Qij)
Qij_inv = np.linalg.inv(Qij)
# Create a new Nik matrix
TR_ik = -1*np.matmul(Tij_inv, Rij_trans)
TI_ik = Tij_inv
QRTR_ik = -Qij + np.matmul(np.matmul(Rij, Tij_inv), Rij_trans)
RT_ik = -1*np.matmul(Rij, Tij_inv)
# Nik Creation
Nik_r1to3 = np.concatenate((TR_ik,TI_ik),axis=1)
Nik_r4to6 = np.concatenate((QRTR_ik,RT_ik),axis=1)
Nik_whole = np.concatenate((Nik_r1to3,Nik_r4to6))
# Eigenvalues and Eigenvectors of Nik
eigvals, eigvecs = eig(Nik_whole)
## The following code sorts the eigenvalues for those with negative imaginary numbers, and conducts the matrix
## operations to generate the B, A, and Z matrices from the COLUMNS in Pythons representation of eigenvectors
eigvecsT = np.transpose(eigvecs)
output_ab = output_ab_neg_imag = eigvecsT[eigvals.imag < 0]
output_abT = np.transpose(output_ab)
### This is the B matrix (bottom 3 vectors assumed as b)
B_matrix = output_b = np.delete(output_abT, [0,1,2], axis=0)
B_matrixT = np.transpose(B_matrix)
### This is the A matrix (these are the displacement vectors, and when A is combined with B, the surface impedance (Z) matrix can be found)
A_matrix = output_a = np.delete(output_abT, [3,4,5], axis=0)
det_B = np.linalg.det(B_matrix)
det_cc_B = det_B.conjugate()
det_B_x_det_cc_B = det_B*det_cc_B
v_real = v.real
det_B_x_det_cc_B_real = det_B_x_det_cc_B.real
if det_B_x_det_cc_B_real < Det_B_try:
RSW_vel = v_real
Det_B_try = det_B_x_det_cc_B_real
print('RSW =')
print(RSW_vel)
print('Min |B|x|B|* =')
print(Det_B_try)
col_to_be_added2 = np.array([p1, P, p2, RSW_vel])
RSW_matrix = np.column_stack((RSW_matrix,col_to_be_added2))
print('RSW vel matrix is')
print(RSW_matrix)

How to convert symbolic expressions to float in Python?

Now i m trying to solve 6th order of nonlinear equations.
For solving this problem, 'fsolve' is the best module for my situation.
But i have a problem for using this 'fsolve'.
My Equations below : eq1, eq2, eq3, eq4, eq5, eq6
U = (E*h/32)*(pi**4*K3+8*pi**2*K4+16*J2-pi**4/b*K1**2-8*pi**2/b*K1*J1)+pi**2/2*D*((K1*K2)**0.5+(1-v)*K5-v*K6)+F/(4*b)*pi**2*K1
eq1 = diff(U,b_1)
eq2 = diff(U,b_2)
eq3 = diff(U,b_3)
eq4 = diff(U,b_4)
eq5 = diff(U,b_5)
eq6 = diff(U,b_6)
Now i m gonna try to define functions:
def functions(v):
b_1 = v[0]
b_2 = v[1]
b_3 = v[2]
b_4 = v[3]
b_5 = v[4]
b_6 = v[5]
return eq1,eq2,eq3,eq4,eq5,eq6
Until now, all of code is perfectly completed.
But next, I got a error code for 'fsolve'
x0 = [0.1,0.1,0.1,0.1,0.1,0.1]
solutions = fsolve(functions,x0)
Traceback (most recent call last):
File "C:\Users\user\AppData\Roaming\Python\Python37\site-packages\sympy\core\expr.py", line 327, in __float__
raise TypeError("can't convert expression to float")
TypeError: can't convert expression to float
Traceback (most recent call last):
File "C:\Users\user\Desktop\-----\trial.py", line 97, in <module>
solutions = fsolve(functions,x0)
File "C:\Users\user\anaconda3\lib\site-packages\scipy\optimize\minpack.py", line 147, in fsolve
res = _root_hybr(func, x0, args, jac=fprime, **options)
File "C:\Users\user\anaconda3\lib\site-packages\scipy\optimize\minpack.py", line 225, in _root_hybr
ml, mu, epsfcn, factor, diag)
error: Result from function call is not a proper array of floats.
Actually i don't know exactly meaning about 'array of floats'.
The eq1~eq6 is really complicated expressions. For example (eq1):
519749583.393768*b_1**3 + 519749583.393768*b_1**2*b_2 + 311849750.036261*b_1**2*b_3 + 222749821.454472*b_1**2*b_4 + 173249861.131256*b_1**2*b_5 + 141749886.380119*b_1**2*b_6 + 589049527.846271*b_1*b_2**2 + 920699262.011818*b_1*b_2*b_3 + 742499404.84824*b_1*b_2*b_4 + 619499503.439037*b_1*b_2*b_5 + 530653420.807624*b_1*b_2*b_6 + 395009683.379264*b_1*b_3**2 + 672299461.117134*b_1*b_3*b_4 + 581053380.409443*b_1*b_3*b_5 + 510299590.968427*b_1*b_3*b_6 + 296183828.527375*b_1*b_4**2 + 524699579.42609*b_1*b_4*b_5 + 469323153.224931*b_1*b_4*b_6 + 236661575.009363*b_1*b_5**2 + 429394392.660243*b_1*b_5*b_6 + 196977114.839905*b_1*b_6**2 + 39642.9110110422*b_1 + 193049845.260542*b_2**3 + 482129613.548124*b_2**2*b_3 + 406949673.808743*b_2**2*b_4 + 350618949.729969*b_2**2*b_5 + 307488215.070719*b_2**2*b_6 + 420149663.228267*b_2*b_3**2 + 732095017.5835*b_2*b_3*b_4 + 645437944.186494*b_2*b_3*b_5 + 575987321.121736*b_2*b_3*b_6 + 326318969.207663*b_2*b_4**2 + 585464236.60242*b_2*b_4*b_5 + 529609601.687166*b_2*b_4*b_6 + 266189415.118358*b_2*b_5**2 + 486862510.0121*b_2*b_5*b_6 + 224616040.694387*b_2*b_6**2 - 29192.3833775011*b_2 + 125272976.510293*b_3**3 + 334349732.001359*b_3**2*b_4 + 299655189.675087*b_3**2*b_5 + 270932940.727941*b_3**2*b_6 + 302393649.018531*b_3*b_4**2 + 549176644.826696*b_3*b_4*b_5 + 501838635.183477*b_3*b_4*b_6 + 252001834.206563*b_3*b_5**2 + 464613445.578031*b_3*b_5*b_6 + 215729827.081362*b_3*b_6**2 - 28558.195613361*b_3 + 92286363.2746599*b_4**3 + 253972873.350655*b_4**2*b_5 + 234063996.922265*b_4**2*b_6 + 234970783.525745*b_4*b_5**2 + 436229928.976872*b_4*b_5*b_6 + 203716223.26551*b_4*b_6**2 - 24903.5955817128*b_4 + 72968765.0411644*b_5**3 + 204424758.743579*b_5**2*b_6 + 191897759.069662*b_5*b_6**2 - 21706.6337724143*b_5 + 60314467.7838601*b_6**3 - 19178.52782655*b_6 + 52.9828871992195*((35.720610813872*b_2**2 + 142.882443255488*b_2*b_3 + 214.323664883232*b_2*b_4 + 285.764886510976*b_2*b_5 + 357.20610813872*b_2*b_6 + 257.188397859878*b_3**2 + 918.529992356708*b_3*b_4 + 1333.56947038455*b_3*b_5 + 1753.55725813553*b_3*b_6 + 893.0152703468*b_4**2 + 2727.7557348775*b_4*b_5 + 3709.44804605594*b_4*b_6 + 2154.22760600582*b_5**2 + 6001.0626167305*b_5*b_6 + 4254.9551116524*b_6**2)*(0.482*b_1**2 + 0.321333333333333*b_1*b_2 + 0.1928*b_1*b_3 + 0.137714285714286*b_1*b_4 + 0.107111111111111*b_1*b_5 + 0.0876363636363637*b_1*b_6 + 0.0964*b_2**2 + 0.137714285714286*b_2*b_3 + 0.107111111111111*b_2*b_4 + 0.0876363636363637*b_2*b_5 + 0.0741538461538462*b_2*b_6 + 0.0535555555555556*b_3**2 + 0.0876363636363637*b_3*b_4 + 0.0741538461538462*b_3*b_5 + 0.0642666666666667*b_3*b_6 + 0.0370769230769231*b_4**2 + 0.0642666666666667*b_4*b_5 + 0.0567058823529412*b_4*b_6 + 0.0283529411764706*b_5**2 + 0.0507368421052632*b_5*b_6 + 0.022952380952381*b_6**2))**0.5*(0.964*b_1 + 0.321333333333333*b_2 + 0.1928*b_3 + 0.137714285714286*b_4 + 0.107111111111111*b_5 + 0.0876363636363637*b_6)/(0.482*b_1**2 + 0.321333333333333*b_1*b_2 + 0.1928*b_1*b_3 + 0.137714285714286*b_1*b_4 + 0.107111111111111*b_1*b_5 + 0.0876363636363637*b_1*b_6 + 0.0964*b_2**2 + 0.137714285714286*b_2*b_3 + 0.107111111111111*b_2*b_4 + 0.0876363636363637*b_2*b_5 + 0.0741538461538462*b_2*b_6 + 0.0535555555555556*b_3**2 + 0.0876363636363637*b_3*b_4 + 0.0741538461538462*b_3*b_5 + 0.0642666666666667*b_3*b_6 + 0.0370769230769231*b_4**2 + 0.0642666666666667*b_4*b_5 + 0.0567058823529412*b_4*b_6 + 0.0283529411764706*b_5**2 + 0.0507368421052632*b_5*b_6 + 0.022952380952381*b_6**2) + (-1.28102213260338e-7*b_2 - 3.20255533150846e-8*b_3 - 3.20255533150846e-8*b_4 - 1.60127766575423e-8*b_6)*(0.482*b_1**2 + 0.321333333333333*b_1*b_2 + 0.1928*b_1*b_3 + 0.137714285714286*b_1*b_4 + 0.107111111111111*b_1*b_5 + 0.0876363636363637*b_1*b_6 + 0.0964*b_2**2 + 0.137714285714286*b_2*b_3 + 0.107111111111111*b_2*b_4 + 0.0876363636363637*b_2*b_5 + 0.0741538461538462*b_2*b_6 + 0.0535555555555556*b_3**2 + 0.0876363636363637*b_3*b_4 + 0.0741538461538462*b_3*b_5 + 0.0642666666666667*b_3*b_6 + 0.0370769230769231*b_4**2 + 0.0642666666666667*b_4*b_5 + 0.0567058823529412*b_4*b_6 + 0.0283529411764706*b_5**2 + 0.0507368421052632*b_5*b_6 + 0.022952380952381*b_6**2) + (1.18477228028269e-9*b_1 + 6.14508444130024e-10*b_2 + 4.60881333097518e-10*b_3 + 1.53627111032506e-10*b_4 + 1.53627111032506e-10*b_5)*(78.9568352087149*b_1**2 + 52.6378901391433*b_1*b_2 + 31.582734083486*b_1*b_3 + 22.5590957739185*b_1*b_4 + 17.5459633797144*b_1*b_5 + 14.3557882197663*b_1*b_6 + 15.791367041743*b_2**2 + 22.5590957739185*b_2*b_3 + 17.5459633797144*b_2*b_4 + 14.3557882197663*b_2*b_5 + 12.1472054167254*b_2*b_6 + 8.77298168985721*b_3**2 + 14.3557882197663*b_3*b_4 + 12.1472054167254*b_3*b_5 + 10.5275780278287*b_3*b_6 + 6.07360270836269*b_4**2 + 10.5275780278287*b_4*b_5 + 9.2890394363194*b_4*b_6 + 4.6445197181597*b_5**2 + 8.31124581144368*b_5*b_6 + 3.75984929565309*b_6**2) + (0.964*b_1 + 0.321333333333333*b_2 + 0.1928*b_3 + 0.137714285714286*b_4 + 0.107111111111111*b_5 + 0.0876363636363637*b_6)*(-1.28102213260338e-7*b_1*b_2 - 3.20255533150846e-8*b_1*b_3 - 3.20255533150846e-8*b_1*b_4 - 1.60127766575423e-8*b_1*b_6 - 1.60127766575423e-8*b_2**2 - 3.20255533150846e-8*b_2*b_3 - 1.60127766575423e-8*b_2*b_5 - 8.00638832877115e-9*b_3**2 + 1.60127766575423e-8*b_3*b_4 - 8.00638832877115e-9*b_3*b_6 - 8.00638832877115e-9*b_4*b_5 + 8.00638832877115e-9*b_4*b_6 + 4.00319416438558e-9*b_5**2 - 8.00638832877115e-9*b_5*b_6 + 4.00319416438558e-9*b_6**2 + 41123.3516712057) - 129937395.848442*(4*b_1 + 1.33333333333333*b_2 + 0.8*b_3 + 0.571428571428572*b_4 + 0.444444444444445*b_5 + 0.363636363636364*b_6)*(b_1**2 + 0.666666666666667*b_1*b_2 + 0.4*b_1*b_3 + 0.285714285714286*b_1*b_4 + 0.222222222222222*b_1*b_5 + 0.181818181818182*b_1*b_6 + 0.2*b_2**2 + 0.285714285714286*b_2*b_3 + 0.222222222222222*b_2*b_4 + 0.181818181818182*b_2*b_5 + 0.153846153846154*b_2*b_6 + 0.111111111111111*b_3**2 + 0.181818181818182*b_3*b_4 + 0.153846153846154*b_3*b_5 + 0.133333333333333*b_3*b_6 + 0.076923076923077*b_4**2 + 0.133333333333333*b_4*b_5 + 0.117647058823529*b_4*b_6 + 0.0588235294117647*b_5**2 + 0.105263157894737*b_5*b_6 + 0.0476190476190477*b_6**2) + (157.91367041743*b_1 + 52.6378901391433*b_2 + 31.582734083486*b_3 + 22.5590957739185*b_4 + 17.5459633797144*b_5 + 14.3557882197663*b_6)*(5.92386140141343e-10*b_1**2 + 6.14508444130024e-10*b_1*b_2 + 4.60881333097518e-10*b_1*b_3 + 1.53627111032506e-10*b_1*b_4 + 1.53627111032506e-10*b_1*b_5 + 2.30440666548759e-10*b_2**2 + 1.53627111032506e-10*b_2*b_3 + 1.53627111032506e-10*b_2*b_4 + 1.53627111032506e-10*b_2*b_6 + 7.6813555516253e-11*b_3**2 - 7.6813555516253e-11*b_3*b_4 + 7.6813555516253e-11*b_3*b_6 + 7.6813555516253e-11*b_4*b_5 + 7.6813555516253e-11*b_4*b_6 + 3.84067777581265e-11*b_5**2 - 251.041666666667)
I must solve such theses 6 equations.
But " Result from function call is not a proper array of floats. " error message comes out.
Is there anyone who help my code?
I'm Python beginning, so Plz understane me

integrate cos(x)*cos(2x)*...*cos(mx) via SAGE

I'm going to find $I_m=\int_0^{2\pi} \prod_{k=1}^m cos(kx){}dx$, where $m=1,2,3\ldots$
Simple SAGE code:
x=var('x')
f = lambda m,x : prod([cos(k*x) for k in range(1,m+1)])
for m in range(1,15+1):
print m, numerical_integral(f(m,x), 0, 2*pi)[0],integrate(f(m,x),x,0,2*pi).n()
Output:
1 -1.47676658757e-16 0.000000000000000
2 -5.27735962315e-16 0.000000000000000
3 1.57079632679 1.57079632679490
4 0.785398163397 0.785398163397448
5 -2.60536121164e-16 0.000000000000000
6 -1.81559273097e-16 0.000000000000000
7 0.392699081699 0.392699081698724
8 0.343611696486 0.147262155637022
9 -1.72448482421e-16 0.294524311274043
10 -1.8747663502e-16 0.196349540849362
11 0.214757310304 0.312932080728671
12 0.190213617698 0.177941771394734
13 -1.30355375996e-16 0.208621387152447
14 -1.25168280013e-16 0.0859029241215959
15 0.138441766107 0.134223318939994
As you can see numerical answer is right, but result of integrate(...) is right for $m=1,2,\ldots,7$ and then there is some bug.
We can print indefinite integral:
for m in range(7,11+1):
print 'm=',m
print 'Indef_I_m=',integrate(f(m,x),x)
And Output:
m = 7
Indef_I_m = 1/16*x + 1/16*sin(2*x) + 1/32*sin(4*x) + 7/384*sin(6*x) +
7/512*sin(8*x) + 3/320*sin(10*x) + 5/768*sin(12*x) + 5/896*sin(14*x) +
1/256*sin(16*x) + 1/384*sin(18*x) + 1/640*sin(20*x) + 1/704*sin(22*x) +
1/1536*sin(24*x) + 1/1664*sin(26*x) + 1/1792*sin(28*x)
m = 8
Indef_I_m = 3/128*x + 5/256*sin(2*x) + 1/32*sin(3*x) + 5/512*sin(4*x) +
5/768*sin(6*x) + 1/256*sin(8*x) + 1/256*sin(10*x) + 1/256*sin(12*x) +
1/256*sin(14*x) + 1/256*sin(16*x) + 7/2304*sin(18*x) + 3/1280*sin(20*x)
+ 5/2816*sin(22*x) + 1/768*sin(24*x) + 3/3328*sin(26*x) +
1/1792*sin(28*x) + 1/1920*sin(30*x) + 1/4096*sin(32*x) +
1/4352*sin(34*x) + 1/4608*sin(36*x) + 3/32*sin(x)
m = 9
Indef_I_m = 3/64*x + 3/128*sin(2*x) + 23/768*sin(3*x) + 3/256*sin(4*x) +
3/640*sin(5*x) + 1/128*sin(6*x) + 5/1792*sin(7*x) + 5/2304*sin(9*x) +
3/2816*sin(11*x) + 1/832*sin(13*x) + 1/1280*sin(15*x) + 3/4352*sin(17*x)
+ 5/4864*sin(19*x) + 1/1344*sin(21*x) + 3/2944*sin(23*x) +
7/6400*sin(25*x) + 1/1152*sin(27*x) + 3/3712*sin(29*x) +
5/7936*sin(31*x) + 1/2112*sin(33*x) + 3/8960*sin(35*x) +
1/4736*sin(37*x) + 1/4992*sin(39*x) + 1/10496*sin(41*x) +
1/11008*sin(43*x) + 1/11520*sin(45*x) + 23/256*sin(x)
m = 10
Indef_I_m = 1/32*x + 1/64*sin(2*x) + 17/512*sin(3*x) + 1/128*sin(4*x) +
7/2560*sin(5*x) + 1/192*sin(6*x) + 3/1792*sin(7*x) + 1/1152*sin(9*x) +
5/5632*sin(11*x) + 3/6656*sin(13*x) + 1/2560*sin(15*x) +
5/8704*sin(17*x) + 3/9728*sin(19*x) + 1/2688*sin(21*x) +
1/2944*sin(23*x) + 1/6400*sin(25*x) + 1/4608*sin(27*x) +
3/14848*sin(29*x) + 3/15872*sin(31*x) + 5/16896*sin(33*x) +
3/8960*sin(35*x) + 3/9472*sin(37*x) + 1/3328*sin(39*x) +
5/20992*sin(41*x) + 1/5504*sin(43*x) + 1/7680*sin(45*x) +
1/12032*sin(47*x) + 1/12544*sin(49*x) + 1/26112*sin(51*x) +
1/27136*sin(53*x) + 1/28160*sin(55*x) + 13/128*sin(x)
m = 11
Indef_I_m = 51/1024*x + 53/2048*sin(2*x) + 13/768*sin(3*x) + 53/4096*sin(4*x) +
13/1536*sin(6*x) + 1/2048*sin(8*x) + 1/2560*sin(10*x) + 1/3072*sin(12*x)
+ 5/14336*sin(14*x) + 1/4096*sin(16*x) + 5/18432*sin(18*x) +
1/4096*sin(20*x) + 1/5632*sin(22*x) + 5/24576*sin(24*x) +
5/26624*sin(26*x) + 5/28672*sin(28*x) + 1/5120*sin(30*x) +
3/16384*sin(32*x) + 5/34816*sin(34*x) + 1/9216*sin(36*x) +
5/38912*sin(38*x) + 1/10240*sin(40*x) + 1/10752*sin(42*x) +
3/22528*sin(44*x) + 3/23552*sin(46*x) + 1/8192*sin(48*x) +
3/25600*sin(50*x) + 5/53248*sin(52*x) + 1/13824*sin(54*x) +
3/57344*sin(56*x) + 1/29696*sin(58*x) + 1/30720*sin(60*x) +
1/63488*sin(62*x) + 1/65536*sin(64*x) + 1/67584*sin(66*x) +
13/256*sin(x)
so for $m=7$ answer is right compare with Indef_I_7 via WolframAlpha
and for $m=8$ answer is incorrect Indef_I_8 via WolframAlpha
There should be Indef_I_8=$\frac{7x}{128}+\ldots$ and no $\sin(x)$, $\sin(3x)$ in summation, only $\sin(2k)$ for $k=1,2,3,\ldots 18$
Sorry for volumetric calculations !
The question is - Am I right that it is the bug in the symbolic integration?

Well, apparently setting algorithm='mathematica_free' solved the issue; this is probably a bug in the default algorithm used bye SAGE ('maxima').

So the answer is - Yes. It is a bug in algorithm='maxima', so use algorithm='mathematica_free' (def new function to find definite integral) or simplify_full() for such product of cos(kx) and than integrate.
f(8,x).simplify_full().integral(x,0,2pi) == 7/64pi and
(7/64*pi).n() == 0.343611696486384 is correct