def householder(a):
n = len(a)
for k in range(n - 2):
u = a[k + 1:n, k]
uMag = sqrt(dot(u, u))
if u[0] < 0.0: uMag = -uMag
u[0] = u[0] + uMag
h = dot(u, u) / 2.0
v = matrixmultiply(a[k + 1:n, k + 1:n], u) / h
g = dot(u, v) / (2.0 * h)
v = v - g * u
a[k+1:n,k+1:n] = a[k+1:n,k+1:n] - outerproduct(v,u) \
- outerproduct(u,v)
a[k, k + 1] = -uMag
return diagonal(a), diagonal(a, 1)
def householder(a):
n = len(a)
for k in range(n-2):
u = a[k+1:n,k]
uMag = sqrt(dot(u,u))
if u[0] < 0.0: uMag = -uMag
u[0] = u[0] + uMag
h = dot(u,u)/2.0
v = matrixmultiply(a[k+1:n,k+1:n],u)/h
g = dot(u,v)/(2.0*h)
v = v - g*u
a[k+1:n,k+1:n] = a[k+1:n,k+1:n] - outerproduct(v,u) \
- outerproduct(u,v)
a[k,k+1] = -uMag
return diagonal(a),diagonal(a,1)
def jacobi(a, tol=1.0e-9): # Jacobi method
def maxElem(a): # Find largest off-diag. element a[k,l]
n = len(a)
aMax = 0.0
for i in range(n - 1):
for j in range(i + 1, n):
if abs(a[i, j]) >= aMax:
aMax = abs(a[i, j])
k = i
l = j
return aMax, k, l
def rotate(a, p, k, l): # Rotate to make a[k,l] = 0
n = len(a)
aDiff = a[l, l] - a[k, k]
if abs(a[k, l]) < abs(aDiff) * 1.0e-36: t = a[k, l] / aDiff
else:
phi = aDiff / (2.0 * a[k, l])
t = 1.0 / (abs(phi) + sqrt(phi**2 + 1.0))
if phi < 0.0: t = -t
c = 1.0 / sqrt(t**2 + 1.0)
s = t * c
tau = s / (1.0 + c)
temp = a[k, l]
a[k, l] = 0.0
a[k, k] = a[k, k] - t * temp
a[l, l] = a[l, l] + t * temp
for i in range(k): # Case of i < k
temp = a[i, k]
a[i, k] = temp - s * (a[i, l] + tau * temp)
a[i, l] = a[i, l] + s * (temp - tau * a[i, l])
for i in range(k + 1, l): # Case of k < i < l
temp = a[k, i]
a[k, i] = temp - s * (a[i, l] + tau * a[k, i])
a[i, l] = a[i, l] + s * (temp - tau * a[i, l])
for i in range(l + 1, n): # Case of i > l
temp = a[k, i]
a[k, i] = temp - s * (a[l, i] + tau * temp)
a[l, i] = a[l, i] + s * (temp - tau * a[l, i])
for i in range(n): # Update transformation matrix
temp = p[i, k]
p[i, k] = temp - s * (p[i, l] + tau * p[i, k])
p[i, l] = p[i, l] + s * (temp - tau * p[i, l])
n = len(a)
maxRot = 5 * (n**2) # Set limit on number of rotations
p = identity(n) * 1.0 # Initialize transformation matrix
for i in range(maxRot): # Jacobi rotation loop
aMax, k, l = maxElem(a)
if aMax < tol: return diagonal(a), p
rotate(a, p, k, l)
print 'Jacobi method did not converge'
def jacobi(a,tol = 1.0e-9): # Jacobi method
def maxElem(a): # Find largest off-diag. element a[k,l]
n = len(a)
aMax = 0.0
for i in range(n-1):
for j in range(i+1,n):
if abs(a[i,j]) >= aMax:
aMax = abs(a[i,j])
k = i; l = j
return aMax,k,l
def rotate(a,p,k,l): # Rotate to make a[k,l] = 0
n = len(a)
aDiff = a[l,l] - a[k,k]
if abs(a[k,l]) < abs(aDiff)*1.0e-36: t = a[k,l]/aDiff
else:
phi = aDiff/(2.0*a[k,l])
t = 1.0/(abs(phi) + sqrt(phi**2 + 1.0))
if phi < 0.0: t = -t
c = 1.0/sqrt(t**2 + 1.0); s = t*c
tau = s/(1.0 + c)
temp = a[k,l]
a[k,l] = 0.0
a[k,k] = a[k,k] - t*temp
a[l,l] = a[l,l] + t*temp
for i in range(k): # Case of i < k
temp = a[i,k]
a[i,k] = temp - s*(a[i,l] + tau*temp)
a[i,l] = a[i,l] + s*(temp - tau*a[i,l])
for i in range(k+1,l): # Case of k < i < l
temp = a[k,i]
a[k,i] = temp - s*(a[i,l] + tau*a[k,i])
a[i,l] = a[i,l] + s*(temp - tau*a[i,l])
for i in range(l+1,n): # Case of i > l
temp = a[k,i]
a[k,i] = temp - s*(a[l,i] + tau*temp)
a[l,i] = a[l,i] + s*(temp - tau*a[l,i])
for i in range(n): # Update transformation matrix
temp = p[i,k]
p[i,k] = temp - s*(p[i,l] + tau*p[i,k])
p[i,l] = p[i,l] + s*(temp - tau*p[i,l])
n = len(a)
maxRot = 5*(n**2) # Set limit on number of rotations
p = identity(n)*1.0 # Initialize transformation matrix
for i in range(maxRot): # Jacobi rotation loop
aMax,k,l = maxElem(a)
if aMax < tol: return diagonal(a),p
rotate(a,p,k,l)
print 'Jacobi method did not converge'
#!/usr/bin/python
## example2_7
from numarray import zeros, array, Float64, product, diagonal
from LUdecomp import *
a = array([[ 3.0, -1.0, 4.0], \
[-2.0, 0.0, 5.0], \
[ 7.0, 2.0, -2.0]])
b = array([[ 6.0, 3.0, 7.0], \
[-4.0, 2.0, -5.0]])
a = LUdecomp(a)
det = product(diagonal(a))
print "\nDeterminant =", det
for i in range(len(b)):
x = LUsolve(a, b[i])
print "x", i + 1, "=", x
raw_input("\nPress return to exit")
#!/usr/bin/python
## example2_7
from numarray import zeros,array,Float64,product,diagonal
from LUdecomp import *
a = array([[ 3.0, -1.0, 4.0], \
[-2.0, 0.0, 5.0], \
[ 7.0, 2.0, -2.0]])
b = array([[ 6.0, 3.0, 7.0], \
[-4.0, 2.0, -5.0]])
a = LUdecomp(a)
det = product(diagonal(a))
print "\nDeterminant =",det
for i in range(len(b)):
x = LUsolve(a,b[i])
print "x",i+1,"=",x
raw_input("\nPress return to exit")