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systems_qr.py
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from rot_givens import *
######################################################################
# Description: single system solution wx = b based on QR Factorization
# Usage: single_system(w, b)
# Dependencies: rot_givens.py
#
# Pre-Condition: w: n x m matrix,
# b: n x 1 matrix
# Post-Condition: returns x satisfying wx = b
#
# Authors: Carlo Bellinati & Rafael Badain @ University of Sao Paulo
######################################################################
def single_system(w, b):
# Matrix Shape
n = w.shape[0]
m = w.shape[1]
# QR factorization
for k in range(m): # percorre horizontalmente
for j in range((n - 1), k, -1): # percorre verticalmente, de baixo para cima
i = j - 1 # se o elemento é != aplica rot_givens
if abs(w[j][k]) > 1e-09: # verifica se w[j][k] é nulo usando intervalo de erro (10^-9)
angles = rotation_angle_for_zero(w[i][k], w[j][k])
w = rot_givens(w, m, i, j, k, angles["c"], angles["s"])
b = rot_givens_unopt(b, 1, i, j, angles["c"], angles["s"])
# Generating x solution vector
x = np.zeros((m,1))
x[m - 1,0] = b[m - 1,0] / w[m - 1,m - 1]
# Back substitution
for k in range((m - 2), -1, -1):
s = 0
for j in range((k + 1), m):
s += w[k][j] * x[j]
x[k][0] = (b[k][0] - s) / w[k][k]
return x
########################################################################
# Description: multiple system solution wh = a based on QR Factorization
# Usage: single_system(w, a)
# Dependencies: rot_givens.py
#
# Pre-Condition: w: n x m matrix,
# a: n x p matrix
# Post-Condition: returns h satisfying wh = a
#
# Authors: Carlo Bellinati & Rafael Badain @ University of Sao Paulo
########################################################################
def multiple_system(w, a):
# Matrix Shapes
m = a.shape[1]
n = w.shape[0]
p = w.shape[1]
# QR factorization
for k in range(p): # percorre horizontalmente
for j in range((n - 1), k, -1): # percorre verticalmente, de baixo para cima
i = j - 1 # se o elemento é != aplica rot_givens
if abs(w[j][k]) > 1e-09: # verifica se w[j][k] é nulo usando intervalo de erro (10^-9)
angles = rotation_angle_for_zero(w[i][k], w[j][k])
w = rot_givens(w, p, i, j, k, angles["c"], angles["s"])
a = rot_givens_unopt(a, m, i, j, angles["c"], angles["s"])
# H solution matrix
h = np.zeros((p,m))
for j in range(m):
h[p - 1][j] = a[p - 1][j] / w[p - 1][p - 1]
# Back substitution
for k in range(p - 2, -1, -1):
for j in range(m):
s = 0
for i in range((k + 1), p):
s += w[k][i] * h[i][j]
h[k][j] = (a[k][j] - s) / w[k][k]
return h
########################################################################
# Description: Error of the linear sistem solution
# Authors: Carlo Bellinati & Rafael Badain @ University of Sao Paulo
########################################################################
def erro(W,H,A):
WH = np.matmul(W,H)
err = np.subtract(WH,A)
erro = math.sqrt(np.sum(pow(err[:,:],2)))
return erro
########################################################################
# Description: System resolution based on QR Factoration Validation
# Authors: Carlo Bellinati & Rafael Badain @ University of Sao Paulo
########################################################################
# generates A matrix for examples c and d
def create_A_matrix(n, m):
A = np.zeros((n,m))
for j in range(m):
for i in range(n):
if (j == 0):
A[i][j] = 1
elif (j == 1):
A[i][j] = i + 1
else:
A[i][j] = 2 * (i + 1) - 1
return A
def main():
# Output
f = open("Relatório/testes_primeira_tarefa.txt", "w")
# a) Single system Wx = b: n = m = 64; W = wa, b = ba
wa = np.zeros((64,64))
for i in range(64):
for j in range(64):
if (i == j):
wa[i][j] = 2
elif (abs(i - j) == 1):
wa[i][j] = 1
ba = np.ones((64, 1))
wa_copy = wa.copy()
wc = wa.copy()
xa = single_system(wa, ba)
f.write("Teste A:")
f.write("Erro = " + str(erro(wa_copy, xa, np.ones((64,1)))))
np.savetxt("Relatório/Teste_A.txt", xa)
# b) Single system Wx = b: n = 20, m = 17; W = wb, b = bb
wb = np.zeros((20,17))
bb = np.zeros((20,1))
for i in range(20):
bb[i] = i + 1
for j in range(17):
if (abs(i - j) <= 4):
wb[i][j] = 1 / (i + j + 1)
wb_copy = wb.copy()
bb_copy = bb.copy()
wd = wb.copy()
xb = single_system(wb, bb)
f.write("Teste B:")
f.write("Erro = " + str(erro(wb_copy, xb, bb_copy)))
np.savetxt("Relatório/Teste_B.txt", xb)
# c) Multiple systems WH = A; n = p = 63, m = 3, W = wc, A = Ac
Ac = create_A_matrix(64, 3)
Ac_copy = Ac.copy()
wc_copy = wc.copy()
hc = multiple_system(wc, Ac)
f.write("Teste C:")
f.write("Erro = " + str(erro(wc_copy, hc, Ac_copy)))
np.savetxt("Relatório/Teste_C.txt", hc)
# d) Multiple systems WH = A; n = 20, p = 17; m = 3, W = wd, A = Ad
Ad = create_A_matrix(20, 3)
Ad_copy = Ad.copy()
wd_copy = wd.copy()
hd = multiple_system(wd, Ad)
f.write("Teste D:")
f.write("Erro = " + str(erro(wd_copy, hd, Ad_copy)))
np.savetxt("Relatório/Teste_D.txt", hd)
return
validation = False
if (validation): main()