def HausQR(a): #another qr decomposition, but now using Hausholder reflections
c = mh.MatrixTrans(a)
q = HausholderRef(c[0])
c = mh.MatrixTrans(mh.MatrixMinor(mh.MatrixMulti(q, a), 0, 0))
for _ in range(1, min(len(a) - 1, len(a[0]))):
t = fill(len(a), HausholderRef(c[0]))
q = mh.MatrixMulti(t, q)
c = mh.MatrixTrans(mh.MatrixMinor(mh.MatrixMulti(q, a), 0, 0))
r = mh.MatrixMulti(q, a)
q = mh.MatrixTrans(q)
return q, r
def QR(a): #Gram-Schmidt's proccess
c = mh.MatrixTrans(a) #easy acces to columns
u = [c[0].copy()
] #first iteration eliminates problem of none return of vecSigma
e = [list(map(lambda k: k / magn(u[0]), u[0]))]
for i in range(1, len(a[0])):
#print(vecSigma( lambda k: ortProj(u[k],c[i]),0,i-1 ))
u.append(
vecSub(c[i], vecSigma(lambda k: ortProj(u[k], c[i]), 0, i - 1)))
e.append(list(map(lambda k: k / magn(u[i]), u[i])))
q = mh.MatrixTrans(e)
r = mh.MatrixMulti(e, a) #QR = A => (Q^-1)QR = (Q^-1)A => R = (Qt)A
return q, r
def SVD(a):
aat = mh.MatrixMulti(a, mh.MatrixTrans(a))
ata = mh.MatrixMulti(mh.MatrixTrans(a), a)
mh.MatrixPrint(aat, "aat")
mh.MatrixPrint(ata, "ata")
eigenvalues_aat = list(map(lambda x: round(x, ndigits=6), Francis(aat)))
singularvalues_aat = list(map(sqrt, eigenvalues_aat))
mh.MatrixPrint([eigenvalues_aat], "eigen aat")
mh.MatrixPrint([singularvalues_aat], "singular aat")
eigenvalues_ata = list(map(lambda x: round(x, ndigits=6), Francis(ata)))
singularvalues_ata = list(map(sqrt, eigenvalues_ata))
mh.MatrixPrint([eigenvalues_ata], "eigen ata")
mh.MatrixPrint([singularvalues_ata], "singular ata")
S = mh.MatrixMake(len(a), len(a[0]))
for i in range(min(len(a), len(a[0]))):
S[i][i] = singularvalues_aat[i]
mh.MatrixPrint(S, msg="Σ")
def HausholderRef(
a
): #funtion for returning matrics that makes hauholder refletion based on a vector
v = a.copy()
v[0] -= magn(a)
I = mh.MatrixIdentity(len(a))
return mh.MatrixAdd(
I,
mh.MatrixScale(mh.MatrixMulti(mh.MatrixTrans([v]), [v]),
-2 / dotProd(v, v)))
def Francis(a, bl=0.00000000001, c=1000):
'''
Francis algorithm of finding eigenvalues using Hausholder reflections.
bl = computational squared error,
c = abort counter - after c iterations func just returns currently computed values
'''
i = 0
while not (triU(a, bl) | triU(mh.MatrixTrans(a), bl) | (i == c)):
#print("i =",i)
q, r = HausQR(a)
i += 1
a = mh.MatrixMulti(r, q)
#mh.MatrixPrint(a)
#print("end of algorithm after",i,"iterations")
return list(map(lambda x: a[x][x], range(len(a))))
mh.MatrixPrint(aat, "aat")
mh.MatrixPrint(ata, "ata")
eigenvalues_aat = list(map(lambda x: round(x, ndigits=6), Francis(aat)))
singularvalues_aat = list(map(sqrt, eigenvalues_aat))
mh.MatrixPrint([eigenvalues_aat], "eigen aat")
mh.MatrixPrint([singularvalues_aat], "singular aat")
eigenvalues_ata = list(map(lambda x: round(x, ndigits=6), Francis(ata)))
singularvalues_ata = list(map(sqrt, eigenvalues_ata))
mh.MatrixPrint([eigenvalues_ata], "eigen ata")
mh.MatrixPrint([singularvalues_ata], "singular ata")
S = mh.MatrixMake(len(a), len(a[0]))
for i in range(min(len(a), len(a[0]))):
S[i][i] = singularvalues_aat[i]
mh.MatrixPrint(S, msg="Σ")
A = [[3, 2, 2], [2, 3, -2]]
C = [[2, 4], [1, 3], [0, 0], [0, 0]]
cU = [[0.82, -0.58, 0, 0], [0.58, 0.82, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]
cS = [[5.47, 0], [0, 0.37], [0, 0], [0, 0]]
cV = [[0.40, -0.91], [0.91, 0.40]]
mh.MatrixPrint(mh.MatrixMulti(C, mh.MatrixTrans(C)))
mh.MatrixPrint(mh.MatrixMulti(mh.MatrixTrans(C), C))
SVD(A)
mh.MatrixPrint(mh.MatrixMulti(mh.MatrixMulti(cU, cS), mh.MatrixTrans(cV)),
msg="USVt")