Example #1
0
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)
Example #2
0
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)
Example #3
0
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'
Example #4
0
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'
Example #5
0
#!/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")
Example #6
0
#!/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")