```from copy import deepcopy
from positive_definite_system import positive_definite_system
from utilities import mat_mult, mat_minus

def additional_algorithm(verbose, a, rhs, scaling_limit, lam_iters, mat_iters):
a_copy, rhs_copy = deepcopy(a), deepcopy(rhs)
sol1 = positive_definite_system(verbose, a, rhs, scaling_limit, lam_iters, mat_iters)
rhs_arr = [[el] for el in rhs_copy]
sol1_arr = [[el] for el in sol1]
res_arr = mat_mult(a_copy, sol1_arr)
res_arr = mat_minus(rhs_arr, res_arr)
res = [el[0] for el in res_arr]

a_copy2, res_copy = deepcopy(a_copy), deepcopy(res)
sol2 = positive_definite_system(verbose, a_copy, res, scaling_limit, lam_iters, mat_iters)
rhs_arr = [[el] for el in res_copy]
sol2_arr = [[el] for el in sol2]
res_arr = mat_mult(a_copy2, sol2_arr)
res_arr = mat_minus(rhs_arr, res_arr)
res = [el[0] for el in res_arr]

sol3 = positive_definite_system(verbose, a_copy2, res, scaling_limit, lam_iters, mat_iters)

for i in range(len(a)):
sol3[i] += sol1[i] + sol2[i]

return sol3

if __name__ == "__main__":
import sys
a, rhs = [[3.0, 3.0, 5.0], [3.0, 5.0, 9.0], [5.0, 9.0, 17.0]], [24.0, 40.0, 74.0]
x = [1.0, 2.0, 3.0]
lam_iters, mat_iters, scaling_limit = 30, 25, 10
verbose = False
sol = additional_algorithm(verbose, a, rhs, scaling_limit, lam_iters, mat_iters)
print('\n')
for i in range(len(a)):
print('x{0:d} = {1:7.4f} (exact {2:7.4f})'.format((i + 1), sol[i], x[i]))

# Typical results:
#
# x1 =  1.0009 (exact  1.0000)
# x2 =  1.9964 (exact  2.0000)
# x3 =  3.0017 (exact  3.0000)
```