def inv_transform(xs, ys, zs, M):
iM = linear_algebra.inverse(M)
vec = vec_pad_ones(xs, ys, zs)
vecr = nx.matrixmultiply(iM, vec)
try:
vecr = vecr / vecr[3]
except OverflowError:
pass
return vecr[0], vecr[1], vecr[2]
def mfuncC(f, x):
"""
mfuncC(f, x) : matrix function with possibly complex eigenvalues.
Note: Numeric defines (v,u) = eig(x) => x*u.T = u.T * Diag(v)
This function is needed by sqrtm and allows further functions.
"""
x = array(x)
(v, u) = numerix.mlab.eig(x)
uT = transpose(u)
V = numerix.mlab.diag(f(v + 0j))
y = matrixmultiply(uT, matrixmultiply(V, linear_algebra.inverse(uT)))
return approx_real(y)
def mfuncC(f, x):
"""
mfuncC(f, x) : matrix function with possibly complex eigenvalues.
Note: Numeric defines (v,u) = eig(x) => x*u.T = u.T * Diag(v)
This function is needed by sqrtm and allows further functions.
"""
x = array(x)
(v, u) = numerix.mlab.eig(x)
uT = transpose(u)
V = numerix.mlab.diag(f(v+0j))
y = matrixmultiply(
uT, matrixmultiply(
V, linear_algebra.inverse(uT)))
return approx_real(y)
def polyfit(x,y,N):
"""
Do a best fit polynomial of order N of y to x. Return value is a
vector of polynomial coefficients [pk ... p1 p0]. Eg, for N=2
p2*x0^2 + p1*x0 + p0 = y1
p2*x1^2 + p1*x1 + p0 = y1
p2*x2^2 + p1*x2 + p0 = y2
.....
p2*xk^2 + p1*xk + p0 = yk
Method: if X is a the Vandermonde Matrix computed from x (see
http://mathworld.wolfram.com/VandermondeMatrix.html), then the
polynomial least squares solution is given by the 'p' in
X*p = y
where X is a len(x) x N+1 matrix, p is a N+1 length vector, and y
is a len(x) x 1 vector
This equation can be solved as
p = (XT*X)^-1 * XT * y
where XT is the transpose of X and -1 denotes the inverse.
For more info, see
http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html,
but note that the k's and n's in the superscripts and subscripts
on that page. The linear algebra is correct, however.
See also polyval
"""
x = asarray(x)+0.
y = asarray(y)+0.
y = reshape(y, (len(y),1))
X = Matrix(vander(x, N+1))
Xt = Matrix(transpose(X))
c = array(linear_algebra.inverse(Xt*X)*Xt*y) # convert back to array
c.shape = (N+1,)
return c
def polyfit(x, y, N):
"""
Do a best fit polynomial of order N of y to x. Return value is a
vector of polynomial coefficients [pk ... p1 p0]. Eg, for N=2
p2*x0^2 + p1*x0 + p0 = y1
p2*x1^2 + p1*x1 + p0 = y1
p2*x2^2 + p1*x2 + p0 = y2
.....
p2*xk^2 + p1*xk + p0 = yk
Method: if X is a the Vandermonde Matrix computed from x (see
http://mathworld.wolfram.com/VandermondeMatrix.html), then the
polynomial least squares solution is given by the 'p' in
X*p = y
where X is a len(x) x N+1 matrix, p is a N+1 length vector, and y
is a len(x) x 1 vector
This equation can be solved as
p = (XT*X)^-1 * XT * y
where XT is the transpose of X and -1 denotes the inverse.
For more info, see
http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html,
but note that the k's and n's in the superscripts and subscripts
on that page. The linear algebra is correct, however.
See also polyval
"""
x = asarray(x) + 0.
y = asarray(y) + 0.
y = reshape(y, (len(y), 1))
X = Matrix(vander(x, N + 1))
Xt = Matrix(transpose(X))
c = array(linear_algebra.inverse(Xt * X) * Xt * y) # convert back to array
c.shape = (N + 1, )
return c