def sqrtm(A,disp=1):
"""Matrix square root
If disp is non-zero display warning if singular matrix.
If disp is zero then return residual ||A-X*X||_F / ||A||_F
Uses algorithm by Nicholas J. Higham
"""
A = asarray(A)
if len(A.shape)!=2:
raise ValueError, "Non-matrix input to matrix function."
T, Z = schur(A)
T, Z = rsf2csf(T,Z)
n,n = T.shape
R = sb.zeros((n,n),T.dtype.char)
for j in range(n):
R[j,j] = sqrt(T[j,j])
for i in range(j-1,-1,-1):
s = 0
for k in range(i+1,j):
s = s + R[i,k]*R[k,j]
R[i,j] = (T[i,j] - s)/(R[i,i] + R[j,j])
R, Z = all_mat(R,Z)
X = (Z * R * Z.H)
if disp:
nzeig = sb.any(sb.diag(T)==0)
if nzeig:
print "Matrix is singular and may not have a square root."
return X.A
else:
arg2 = norm(X*X - A,'fro')**2 / norm(A,'fro')
return X.A, arg2
def funm(A,func,disp=1):
"""matrix function for arbitrary callable object func.
"""
# func should take a vector of arguments (see vectorize if
# it needs wrapping.
# Perform Shur decomposition (lapack ?gees)
A = asarray(A)
if len(A.shape)!=2:
raise ValueError, "Non-matrix input to matrix function."
if A.dtype.char in ['F', 'D', 'G']:
cmplx_type = 1
else:
cmplx_type = 0
T, Z = schur(A)
T, Z = rsf2csf(T,Z)
n,n = T.shape
F = diag(func(diag(T))) # apply function to diagonal elements
F = F.astype(T.dtype.char) # e.g. when F is real but T is complex
minden = abs(T[0,0])
# implement Algorithm 11.1.1 from Golub and Van Loan
# "matrix Computations."
for p in range(1,n):
for i in range(1,n-p+1):
j = i + p
s = T[i-1,j-1] * (F[j-1,j-1] - F[i-1,i-1])
ksl = slice(i,j-1)
val = dot(T[i-1,ksl],F[ksl,j-1]) - dot(F[i-1,ksl],T[ksl,j-1])
s = s + val
den = T[j-1,j-1] - T[i-1,i-1]
if den != 0.0:
s = s / den
F[i-1,j-1] = s
minden = min(minden,abs(den))
F = dot(dot(Z, F),transpose(conjugate(Z)))
if not cmplx_type:
F = toreal(F)
tol = {0:feps, 1:eps}[_array_precision[F.dtype.char]]
if minden == 0.0:
minden = tol
err = min(1, max(tol,(tol/minden)*norm(triu(T,1),1)))
if product(ravel(logical_not(isfinite(F))),axis=0):
err = Inf
if disp:
if err > 1000*tol:
print "Result may be inaccurate, approximate err =", err
return F
else:
return F, err
def sqrtm(A,disp=1):
"""Matrix square root.
Parameters
----------
A : array, shape(M,M)
Matrix whose square root to evaluate
disp : boolean
Print warning if error in the result is estimated large
instead of returning estimated error. (Default: True)
Returns
-------
sgnA : array, shape(M,M)
Value of the sign function at A
(if disp == False)
errest : float
Frobenius norm of the estimated error, ||err||_F / ||A||_F
Notes
-----
Uses algorithm by Nicholas J. Higham
"""
A = asarray(A)
if len(A.shape)!=2:
raise ValueError, "Non-matrix input to matrix function."
T, Z = schur(A)
T, Z = rsf2csf(T,Z)
n,n = T.shape
R = np.zeros((n,n),T.dtype.char)
for j in range(n):
R[j,j] = sqrt(T[j,j])
for i in range(j-1,-1,-1):
s = 0
for k in range(i+1,j):
s = s + R[i,k]*R[k,j]
R[i,j] = (T[i,j] - s)/(R[i,i] + R[j,j])
R, Z = all_mat(R,Z)
X = (Z * R * Z.H)
if disp:
nzeig = np.any(diag(T)==0)
if nzeig:
print "Matrix is singular and may not have a square root."
return X.A
else:
arg2 = norm(X*X - A,'fro')**2 / norm(A,'fro')
return X.A, arg2
def funm(A,func,disp=1):
"""Evaluate a matrix function specified by a callable.
Returns the value of matrix-valued function f at A. The function f
is an extension of the scalar-valued function func to matrices.
Parameters
----------
A : array, shape(M,M)
Matrix at which to evaluate the function
func : callable
Callable object that evaluates a scalar function f.
Must be vectorized (eg. using vectorize).
disp : boolean
Print warning if error in the result is estimated large
instead of returning estimated error. (Default: True)
Returns
-------
fA : array, shape(M,M)
Value of the matrix function specified by func evaluated at A
(if disp == False)
errest : float
1-norm of the estimated error, ||err||_1 / ||A||_1
"""
# Perform Shur decomposition (lapack ?gees)
A = asarray(A)
if len(A.shape)!=2:
raise ValueError, "Non-matrix input to matrix function."
if A.dtype.char in ['F', 'D', 'G']:
cmplx_type = 1
else:
cmplx_type = 0
T, Z = schur(A)
T, Z = rsf2csf(T,Z)
n,n = T.shape
F = diag(func(diag(T))) # apply function to diagonal elements
F = F.astype(T.dtype.char) # e.g. when F is real but T is complex
minden = abs(T[0,0])
# implement Algorithm 11.1.1 from Golub and Van Loan
# "matrix Computations."
for p in range(1,n):
for i in range(1,n-p+1):
j = i + p
s = T[i-1,j-1] * (F[j-1,j-1] - F[i-1,i-1])
ksl = slice(i,j-1)
val = dot(T[i-1,ksl],F[ksl,j-1]) - dot(F[i-1,ksl],T[ksl,j-1])
s = s + val
den = T[j-1,j-1] - T[i-1,i-1]
if den != 0.0:
s = s / den
F[i-1,j-1] = s
minden = min(minden,abs(den))
F = dot(dot(Z, F),transpose(conjugate(Z)))
if not cmplx_type:
F = toreal(F)
tol = {0:feps, 1:eps}[_array_precision[F.dtype.char]]
if minden == 0.0:
minden = tol
err = min(1, max(tol,(tol/minden)*norm(triu(T,1),1)))
if product(ravel(logical_not(isfinite(F))),axis=0):
err = Inf
if disp:
if err > 1000*tol:
print "Result may be inaccurate, approximate err =", err
return F
else:
return F, err