def cond(A, norm=lambda x: mnorm(x,1)):
"""
Calculate the condition number of a matrix using a specified matrix norm.
The condition number estimates the sensitivity of a matrix to errors.
Example: small input errors for ill-conditionded coefficient matrices
alter the solution of the system dramatically.
For ill-conditioned matrices it's recommended to use qr_solve() instead
of lu_solve(). This does not help with input errors however, it just avoids
to add additional errors.
Definition: cond(A) = ||A|| * ||A**-1||
"""
return norm(A) * norm(inverse(A))
def cond(A, norm=lambda x: mnorm(x,1)):
"""
Calculate the condition number of a matrix using a specified matrix norm.
The condition number estimates the sensitivity of a matrix to errors.
Example: small input errors for ill-conditioned coefficient matrices
alter the solution of the system dramatically.
For ill-conditioned matrices it's recommended to use qr_solve() instead
of lu_solve(). This does not help with input errors however, it just avoids
to add additional errors.
Definition: cond(A) = ||A|| * ||A**-1||
"""
return norm(A) * norm(inverse(A))
def qr_solve(A, b, norm=norm, **kwargs):
"""
Ax = b => x, ||Ax - b||
Solve a determined or overdetermined linear equations system and
calculate the norm of the residual (error).
QR decompostion using Householder factorization is applied, which gives very
accurate results even for ill-conditioned matrices. qr_solve is twice as
efficient.
"""
# do not overwrite A nor b
A, b = matrix(A, **kwargs).copy(), matrix(b, **kwargs).copy()
if A.rows < A.cols:
raise ValueError('cannot solve underdetermined system')
H, p, x, r = householder(extend(A, b))
res = norm(r)
# calculate residual "manually" for determined systems
if res == 0:
res = norm(residual(A, x, b))
return matrix(x, **kwargs), res
def qr_solve(A, b, norm=norm, **kwargs):
"""
Ax = b => x, ||Ax - b||
Solve a determined or overdetermined linear equations system and
calculate the norm of the residual (error).
QR decomposition using Householder factorization is applied, which gives very
accurate results even for ill-conditioned matrices. qr_solve is twice as
efficient.
"""
# do not overwrite A nor b
A, b = matrix(A, **kwargs).copy(), matrix(b, **kwargs).copy()
if A.rows < A.cols:
raise ValueError('cannot solve underdetermined system')
H, p, x, r = householder(extend(A, b))
res = norm(r)
# calculate residual "manually" for determined systems
if res == 0:
res = norm(residual(A, x, b))
return matrix(x, **kwargs), res
def improve_solution(A, x, b, maxsteps=1):
"""
Improve a solution to a linear equation system iteratively.
This re-uses the LU decomposition and is thus cheap.
Usually 3 up to 4 iterations are giving the maximal improvement.
"""
assert A.rows == A.cols, 'need n*n matrix' # TODO: really?
for _ in xrange(maxsteps):
r = residual(A, x, b)
if norm(r, 2) < 10*eps:
break
# this uses cached LU decomposition and is thus cheap
dx = lu_solve(A, -r)
x += dx
return x
def improve_solution(A, x, b, maxsteps=1):
"""
Improve a solution to a linear equation system iteratively.
This re-uses the LU decomposition and is thus cheap.
Usually 3 up to 4 iterations are giving the maximal improvement.
"""
assert A.rows == A.cols, 'need n*n matrix' # TODO: really?
for _ in xrange(maxsteps):
r = residual(A, x, b)
if norm(r, 2) < 10*eps:
break
# this uses cached LU decomposition and is thus cheap
dx = lu_solve(A, -r)
x += dx
return x
