def ConjugatedGradient(m,A,a,epsilon,p):
'''
The following 5 lines are not part of the algorithm per se. They are helpers.
'''
VECTOR = 1 # See above. Here, it is 1 because we use math notation
nOptim = 0
xOptim = Matrix(m,m)
mathP = p+1 #We use mathP (p+1) because the algorithm is written in mathematical notation (k = 1,p). By employing the matrix API, we can directly use the mathematical notation
mathM = m+1 #We use mathM for the same reason we use mathP. It is used only for iterations, not defining sizes
X = Matrix(m,VECTOR)
Y = Matrix(m,VECTOR)
r = Matrix(m,VECTOR)
aux = Matrix(m,VECTOR)
v = Matrix(m,VECTOR)
for i in range(1,mathM):
X.mathInsert(i,VECTOR,1)
aux = copy.deepcopy(A.multiplyMatrix(X))
r = copy.deepcopy(a.substractMatrix(aux))
v = copy.deepcopy(r)
for i in range(1,mathM):
sum1 = 0
for j in range(1,mathM):
sum1 = sum1 + r.mathAt(j,1)**2
av = Matrix(m,VECTOR)
av = copy.deepcopy(A.multiplyMatrix(v))
sum2 = 0
for j in range(1,mathM):
sum2 = sum2 + av.mathAt(j,1) * v.mathAt(j,1)
ai = 0
ai = sum1 / sum2+(10**(-10))
aux = copy.deepcopy(v.scalarMultiplication(ai))
aux = copy.deepcopy(aux.addMatrix(X))
Y = copy.deepcopy(aux)
aux = copy.deepcopy(A.multiplyMatrix(Y))
r = copy.deepcopy(a.substractMatrix(aux))
sum3 = 0
ci = 0
for j in range(1,mathM):
sum3 = sum3 + r.mathAt(j,1)**2
ci = sum3 / sum1
aux = copy.deepcopy(v.scalarMultiplication(ci))
aux = copy.deepcopy(r.addMatrix(aux))
v = copy.deepcopy(aux)
X = copy.deepcopy(Y)
print("===== Conjugated Gradient =====")
print("Optim solution (x):")
X.display()
print("Test:")
result = A.multiplyMatrix(X)
result.display()
print("====================")
def GaussSiedel(m,A,a,epsilon,p):
'''
The following 5 lines are not part of the algorithm per se. They are helpers.
'''
VECTOR = 1 # See above. Here, it is 1 because we use math notation
nOptim = 0
xOptim = Matrix(m,m)
mathP = p+1 #We use mathP (p+1) because the algorithm is written in mathematical notation (k = 1,p). By employing the matrix API, we can directly use the mathematical notation
mathM = m+1 #We use mathM for the same reason we use mathP. It is used only for iterations, not defining sizes
for k in range(1,mathP):
sigma = ((2*k)/(p+1))
n = 0
x = Matrix(m,VECTOR)
condition = True
while condition:
n = n+1
y = Matrix(m,1)
for i in range(1,mathM):
yi = ((1-sigma) * x.mathAt(i,VECTOR)) + (sigma/A.mathAt(i,i)*(a.mathAt(i,VECTOR)-computeAijYjSum(A,y,i) - computeAijXjSum(A,x,i,mathM)))
y.mathInsert(i,VECTOR,yi)
err = copy.deepcopy(sqrt(abs(computeGaussSiedelErrSum(A,y,x,mathM))))
for i in range(1,mathM):
x.mathInsert(i,VECTOR,(copy.deepcopy(y.mathAt(i,VECTOR))))
condition = err < epsilon
if k == 1:
nOptim = copy.deepcopy(n)
xOptim = copy.deepcopy(x)
elif k>1:
if n < nOptim:
nOptim = copy.deepcopy(n)
xOptim = copy.deepcopy(x)
else:
print("This should never be seen. If you see this, something is very, very wrong ...")
print("===== Gauss Siedel =====")
print("Optim n:",nOptim)
print("Optim solution (x):")
x.display()
print("Test:")
result = A.multiplyMatrix(x)
result.display()
print("====================")
def Jacobi(m,A,a,epsilon,p):
'''
The following 5 lines are not part of the algorithm per se. They are helpers.
'''
VECTOR = 1 # See above. Here, it is 1 because we use math notation
nOptim = 0
xOptim = Matrix(m,m)
mathP = p+1 #We use mathP (p+1) because the algorithm is written in mathematical notation (k = 1,p). By employing the matrix API, we can directly use the mathematical notation
mathM = m+1 #We use mathM for the same reason we use mathP. It is used only for iterations, not defining sizes
ni = copy.deepcopy(A.infiniteNorm())
for k in range(1,mathP):
sigma = ((2*k)/((mathP+1)*ni))
Bsigma = Matrix(m,m)
'''
Compute Bsigma Matrix
'''
for i in range(1,mathM):
for j in range(1,mathM):
if i == j:
Bsigma.mathInsert(i,j,1-sigma) #(1-siga*A.mathAt(i,i)))
else:
Bsigma.mathInsert(i,j,-sigma*(A.mathAt(i,j)/A.mathAt(i,i))) #(-sigma*A.mathAt(i,j)))
'''
Compute bsig vector
'''
bsig = Matrix(m,1)
for i in range(1,mathM):
bsig.mathInsert(i,VECTOR,(sigma*A.mathAt(i,VECTOR)))
'''
Initialize
'''
n = 0
x = Matrix(m,1)
'''
Do while loop
'''
condition = True
while condition:
n = n + 1
y = Matrix(m,1)
for i in range(1,mathM):
yi = copy.deepcopy(computeYiSum(Bsigma,bsig,x,i,mathM))
y.mathInsert(i,VECTOR,yi)
err = copy.deepcopy(sqrt( abs(computeErrSum(A,y,x,i,mathM)) ))
for i in range(1,mathM):
x.mathInsert(i,VECTOR,copy.deepcopy(y.mathAt(i,VECTOR)))
condition = err < epsilon
if k == 1:
nOptim = copy.deepcopy(n)
xOptim = copy.deepcopy(x)
elif k>1:
if n < nOptim:
nOptim = copy.deepcopy(n)
xOptim = copy.deepcopy(x)
else:
print("This should never be seen. If you see this, something is very, very wrong ...")
print("===== Jacobi =====")
print("Optim n:",nOptim)
print("Optim solution (x):")
x.display()
print("Test:")
result = A.multiplyMatrix(x)
result.display()
print("====================")
def otherGaussSiedel(m,A,a,epsilon,p):
VECTOR = 1 # See above. Here, it is 1 because we use math notation
nOptim = 0
xOptim = Matrix(m,m)
mathP = p+1 #We use mathP (p+1) because the algorithm is written in mathematical notation (k = 1,p). By employing the matrix API, we can directly use the mathematical notation
mathM = m+1 #We use mathM for the same reason we use mathP. It is used only for iterations, not defining sizes
B = Matrix(m,m)
B = copy.deepcopy(otherGaussSiedelB(A,m))
q0 = B.normOne()
q1 = B.infiniteNorm()
q = -1
norm = -1
if q0 < 1 or q1 < 1:
if q0 < 1:
q = q0
norm = 1
elif q1 < 1:
q = q1
norm = 999
#Initialize x
x = Matrix(m,VECTOR)
#print("X:")
#x.display()
x.mathInsert(random.randrange(1,mathM),VECTOR,random.randrange(1,5))
#print("X after first insertion:")
#x.display()
x.mathInsert(random.randrange(1,mathM)-1,VECTOR,random.randrange(1,5)+2)
#print("X after second insertion:")
#x.display()
x.mathInsert(random.randrange(1,mathM)-1,VECTOR,random.randrange(1,5)+7)
#print("X after third insertion:")
#x.display()
newX = copy.deepcopy(Matrix(m,VECTOR))
condition = True
iteration = 0
while condition:
iteration += 1
newX = copy.deepcopy(Matrix(m,VECTOR))
for i in range(1,mathM):
newElementOfX = otherGaussSiedelSum(B,x,a,i,mathM)
#print(">!< Haha")
newX.mathInsert(i,VECTOR,newElementOfX)
conditionValue = ( (q/(1-q))*computeNorm(x,newX,norm) )
condition = conditionValue > epsilon
'''
print("Old x:")
x.display()
print("New x:")
newX.display()
'''
for xIndex in range(1,mathM):
x.mathInsert(xIndex,VECTOR,newX.mathAt(xIndex,VECTOR))
'''
print("X after reasignment:")
x.display()
'''
#print("Iteration:",iteration," - ",conditionValue,"out of",epsilon,".")
print("===== Other Gauss Siedel =====")
print("Optim solution (x):")
x.display()
print("Test:")
result = A.multiplyMatrix(newX)
result.display()
print("====================")
else:
print("Else Other Gauss Siedel.")
