def solve(self, rhs, **kwargs):
"""
Solve a linear system with `rhs` as right-hand side by the CG method.
The vector `rhs` should be a Numpy array.
:Keywords:
:guess: Initial guess (Numpy array). Default: 0.
:matvec_max: Max. number of operator-vector produts. Default: 2n.
:check_symmetric: Ensure operator is symmetric. Default: False.
:check_curvature: Ensure operator is positive definite. Default: True.
:store_resids: Store full residual vector history. Default: False.
:store_iterates: Store full iterate history. Default: False.
"""
n = rhs.shape[0]
nMatvec = 0
definite = True
check_sym = kwargs.get('check_symmetric', False)
check_curvature = kwargs.get('check_curvature', True)
store_resids = kwargs.get('store_resids', False)
store_iterates = kwargs.get('store_iterates', False)
if check_sym:
if not check_symmetric(self.op):
self.logger.error('Coefficient operator is not symmetric')
return
# Initial guess
result_type = np.result_type(self.op.dtype, rhs.dtype)
guess_supplied = 'guess' in kwargs.keys()
x = kwargs.get('guess', np.zeros(n)).astype(result_type)
if store_iterates:
self.iterates.append(x.copy())
matvec_max = kwargs.get('matvec_max', 2*n)
# Initial residual vector
r = -rhs
if guess_supplied:
r += self.op * x
nMatvec += 1
# Initial preconditioned residual vector
if self.precon is not None:
y = self.precon * r
else:
y = r
if store_resids:
self.resids.append(y.copy())
ry = np.dot(r,y)
self.residNorm0 = residNorm = np.abs(np.sqrt(ry))
self.residHistory.append(self.residNorm0)
threshold = max(self.abstol, self.reltol * self.residNorm0)
p = -r # Initial search direction (copy not to overwrite rhs if x=0)
hdr_fmt = '%6s %7s %8s'
hdr = hdr_fmt % ('Matvec', 'Resid', 'Curv')
self.logger.info(hdr)
self.logger.info('-' * len(hdr))
info = '%6d %7.1e' % (nMatvec, residNorm)
self.logger.info(info)
while residNorm > threshold and nMatvec < matvec_max and definite:
Ap = self.op * p
nMatvec += 1
pAp = np.dot(p, Ap)
if check_curvature:
if np.imag(pAp) > 1.0e-8 * np.abs(pAp) or np.real(pAp) <= 0:
self.logger.error('Coefficient operator is not positive definite')
self.infiniteDescent = p
definite = False
continue
# Compute step length
alpha = ry/pAp
# Update estimate and residual
x += alpha * p
r += alpha * Ap
if store_iterates:
self.iterates.append(x.copy())
# Compute preconditioned residual
if self.precon is not None:
y = self.precon * r
else:
y = r
if store_resids:
self.resids.append(y.copy())
# Update preconditioned residual norm
ry_next = np.dot(r,y)
# Update search direction
beta = ry_next/ry
p *= beta
p -= r
ry = ry_next
residNorm = np.abs(np.sqrt(ry))
self.residHistory.append(residNorm)
info = '%6d %7.1e %8.1e' % (nMatvec, residNorm, np.real(pAp))
self.logger.info(info)
self.converged = residNorm <= threshold
self.definite = definite
self.nMatvec = nMatvec
self.bestSolution = self.x = x
self.residNorm = residNorm
def solve(self, b, **kwargs):
A = self.op
n = b.shape[0]
# Read keyword arguments
precon = kwargs.get('precon', None)
shift = kwargs.get('shift', 0.0)
show = kwargs.get('show', True)
check = kwargs.get('check', True)
itnlim = kwargs.get('itnlim', 5*n)
rtol = kwargs.get('rtol', 1.0e-12)
etol = kwargs.get('etol', 1.0e-6)
store_resids = kwargs.get('store_resids', False)
store_iterates = kwargs.get('store_iterates', False)
window = kwargs.get('window', 5)
self.dir_errors_window = [] # Direct error estimates.
self.iterates = []
result_type = np.result_type(A.dtype, b.dtype)
x = zeros(n, dtype=result_type)
xNrgNorm2 = 0.0 # Squared energy norm of final solution.
dErr = zeros(window) # Truncated direct error terms.
trncDirErr = 0 # Truncated direct error.
if store_iterates:
self.iterates.append(x.copy())
# Transfer some pointers for readability
eps = self.eps
if show:
print self.first + 'Solution of symmetric Ax = b'
print 'n = %3d precon = %4s shift = %23.14e'\
% (n, (precon != None), shift)
print 'itnlim = %3d rtol = %11.2e\n' % (itnlim, rtol)
istop = 0; itn = 0; Anorm = 0.0; Acond = 0.0;
rnorm = 0.0; ynorm = 0.0; done = False;
#------------------------------------------------------------------
# Set up y and v for the first Lanczos vector v1.
# y = beta1 P' v1, where P = C**(-1).
# v is really P' v1.
#------------------------------------------------------------------
r1 = b
if precon is not None:
y = precon * b
else:
y = b.copy()
beta1 = dot(b,y)
# Test for an indefinite preconditioner.
# If b = 0 exactly, stop with x = 0.
if beta1 < 0:
istop = 9
self.show = True
done = True
if beta1 == 0.0:
self.show = True
done = True
if beta1 > 0:
beta1 = sqrt(beta1); # Normalize y to get v1 later.
self.residNorm0 = beta1
self.residNorm0 = beta1 # Initial residual norm.
# See if A is symmetric.
if check:
if not check_symmetric(A):
istop = 7
done = True
self.show = True
# See if preconditioner is symmetric.
if check and (precon is not None):
if not check_symmetric(precon):
istop = 8
show = True
done = True
# -------------------------------------------------------------------
# Initialize other quantities.
# ------------------------------------------------------------------
oldb = 0.0; beta = beta1; dbar = 0.0; epsln = 0.0
qrnorm = beta1; phibar = beta1; rhs1 = beta1; Arnorm = 0.0
rhs2 = 0.0; tnorm2 = 0.0; ynorm2 = 0.0
cs = -1.0; sn = 0.0
w = zeros(n, dtype=result_type)
w2 = zeros(n, dtype=result_type)
r2 = r1.copy()
if show:
print ' '*2
head1 = ' Itn x[0] Compatible LS'
head2 = ' norm(A) cond(A) gbar/|A|' ###### Check gbar
print head1 + head2
# ---------------------------------------------------------------------
# Main iteration loop.
# --------------------------------------------------------------------
if not done: # k = itn = 1 first time through
while itn < itnlim:
itn = itn + 1
# -------------------------------------------------------------
# Obtain quantities for the next Lanczos vector vk+1, k=1,2,...
# The general iteration is similar to the case k=1 with v0 = 0:
#
# p1 = Operator * v1 - beta1 * v0,
# alpha1 = v1'p1,
# q2 = p2 - alpha1 * v1,
# beta2^2 = q2'q2,
# v2 = (1/beta2) q2.
#
# Again, y = betak P vk, where P = C**(-1).
# .... more description needed.
# -------------------------------------------------------------
s = 1.0/beta # Normalize previous vector (in y).
v = s*y # v = vk if P = I
y = A * v
y -= shift*v
if itn >= 2:
y = y - (beta/oldb)*r1
alfa = dot(v,y)
y = (- alfa/beta)*r2 + y
r1 = r2.copy()
r2 = y.copy()
if precon is not None: y = precon * r2
oldb = beta
beta = dot(r2,y)
if beta < 0:
istop = 6
break
beta = sqrt(beta)
tnorm2 = tnorm2 + alfa**2 + oldb**2 + beta**2
if itn==1: # Initialize a few things.
if beta/beta1 <= 10*eps: # beta2 = 0 or ~ 0.
istop = -1 # Terminate later.
# tnorm2 = alfa**2 ??
gmax = abs(alfa) # alpha1
gmin = gmax # alpha1
# Apply previous rotation Qk-1 to get
# [deltak epslnk+1] = [cs sn][dbark 0 ]
# [gbar k dbar k+1] [sn -cs][alfak betak+1].
oldeps = epsln
delta = cs * dbar + sn * alfa # delta1 = 0 deltak
gbar = sn * dbar - cs * alfa # gbar 1 = alfa1 gbar k
# Note: There is severe cancellation in the computation of gbar
#print ' sn = %21.15e\n dbar = %21.15e\n cs = %21.15e\n alfa = %21.15e\n sn*dbar-cs*alfa = %21.15e\n gbar =%21.15e' % (sn, dbar, cs, alfa, sn*dbar-cs*alfa, gbar)
epsln = sn * beta # epsln2 = 0 epslnk+1
dbar = - cs * beta # dbar 2 = beta2 dbar k+1
root = self.normof2(gbar, dbar)
Arnorm = phibar * root
# Compute the next plane rotation Qk
gamma = self.normof2(gbar, beta) # gammak
gamma = max(gamma, eps)
cs = gbar / gamma # ck
sn = beta / gamma # sk
phi = cs * phibar # phik
phibar = sn * phibar # phibark+1
# Update x.
denom = 1.0/gamma
w1 = w2.copy()
w2 = w.copy()
w = (v - oldeps*w1 - delta*w2) * denom
x += phi*w #x = x + phi*w
if store_iterates:
self.iterates.append(x.copy())
# Update energy norm of x.
xNrgNorm2 += phi*phi
dErr[itn % window] = phi
if itn > window:
trncDirErr = norm(dErr)
xNrgNorm = sqrt(xNrgNorm2)
self.dir_errors_window.append(trncDirErr / xNrgNorm)
if trncDirErr < etol * xNrgNorm:
istop = 10
# Go round again.
gmax = max(gmax, gamma)
gmin = min(gmin, gamma)
z = rhs1 / gamma
ynorm2 = z**2 + ynorm2
rhs1 = rhs2 - delta*z
rhs2 = - epsln*z
# Estimate various norms and test for convergence.
Anorm = sqrt(tnorm2)
ynorm = sqrt(ynorm2)
epsa = Anorm * eps
epsx = Anorm * ynorm * eps
epsr = Anorm * ynorm * rtol
diag = gbar
if diag==0: diag = epsa
qrnorm = phibar
rnorm = qrnorm
test1 = rnorm / (Anorm*ynorm) # ||r|| / (||A|| ||x||)
test2 = root / Anorm # ||Ar|| / (||A|| ||r||)
self.residHistory.append(rnorm)
# Estimate cond(A).
# In this version we look at the diagonals of R in the
# factorization of the lower Hessenberg matrix, Q * H = R,
# where H is the tridiagonal matrix from Lanczos with one
# extra row, beta(k+1) e_k^T.
Acond = gmax/gmin
# See if any of the stopping criteria are satisfied.
# In rare cases istop is already -1 from above (Abar = const*I)
if istop==0:
t1 = 1 + test1 # These tests work if rtol < eps
t2 = 1 + test2
if t2 <= 1: istop = 2
if t1 <= 1: istop = 1
if itn >= itnlim: istop = 6
if Acond >= 0.1/eps: istop = 4
if epsx >= beta1: istop = 3
# if rnorm <= epsx: istop = 2
# if rnorm <= epsr: istop = 1
if test2 <= rtol: istop = 2
if test1 <= rtol: istop = 1
# See if it is time to print something.
prnt = False
if n <= 40: prnt = True
if itn <= 10: prnt = True
if itn >= itnlim-10: prnt = True
if (itn % 10)==0: prnt = True
if qrnorm <= 10*epsx: prnt = True
if qrnorm <= 10*epsr: prnt = True
if Acond <= 1e-2/eps: prnt = True
if istop != 0: prnt = True
if show and prnt:
str1 = '%6g %12.5e %10.3e' % (itn, x[0], test1)
str2 = ' %10.3e' % test2
str3 = ' %8.1e %8.1e %8.1e' % (Anorm, Acond, gbar/Anorm)
print str1 + str2 + str3
if istop > 0: break
if (itn % 10)==0: print ' '
# Display final status.
if show:
last = self.last
print last+' istop = %3g itn =%5g' % (istop,itn)
print last+' Anorm = %12.4e Acond = %12.4e' %(Anorm,Acond)
print last+' rnorm = %12.4e ynorm = %12.4e' % (rnorm,ynorm)
print last+' Arnorm = %12.4e' % Arnorm
print last+self.msg[istop+1]
self.converged = istop in [1,2,3,4,10]
if istop == 10: self.status = 'direct error small'
self.bestSolution = self.x = x
self.istop = istop
self.itn = itn
self.residNorm = self.rnorm = rnorm
self.x = self.bestSolution = x
self.istop = istop
self.itn = self.nMatvec = itn
self.rnorm = self.residNorm = rnorm
self.Arnorm = Arnorm
self.Anorm = Anorm
self.Acond = Acond
self.ynorm = ynorm
return
def solve(self, b, **kwargs):
A = self.op
n = b.shape[0]
# Read keyword arguments
precon = kwargs.get('precon', None)
shift = kwargs.get('shift', 0.0)
show = kwargs.get('show', True)
check = kwargs.get('check', True)
itnlim = kwargs.get('itnlim', 5 * n)
rtol = kwargs.get('rtol', 1.0e-12)
etol = kwargs.get('etol', 1.0e-6)
store_resids = kwargs.get('store_resids', False)
store_iterates = kwargs.get('store_iterates', False)
window = kwargs.get('window', 5)
self.dir_errors_window = [] # Direct error estimates.
self.iterates = []
result_type = np.result_type(A.dtype, b.dtype)
x = zeros(n, dtype=result_type)
xNrgNorm2 = 0.0 # Squared energy norm of final solution.
dErr = zeros(window) # Truncated direct error terms.
trncDirErr = 0 # Truncated direct error.
if store_iterates:
self.iterates.append(x.copy())
# Transfer some pointers for readability
eps = self.eps
if show:
print self.first + 'Solution of symmetric Ax = b'
print 'n = %3d precon = %4s shift = %23.14e'\
% (n, (precon != None), shift)
print 'itnlim = %3d rtol = %11.2e\n' % (itnlim, rtol)
istop = 0
itn = 0
Anorm = 0.0
Acond = 0.0
rnorm = 0.0
ynorm = 0.0
done = False
#------------------------------------------------------------------
# Set up y and v for the first Lanczos vector v1.
# y = beta1 P' v1, where P = C**(-1).
# v is really P' v1.
#------------------------------------------------------------------
r1 = b
if precon is not None:
y = precon * b
else:
y = b.copy()
beta1 = dot(b, y)
# Test for an indefinite preconditioner.
# If b = 0 exactly, stop with x = 0.
if beta1 < 0:
istop = 9
self.show = True
done = True
if beta1 == 0.0:
self.show = True
done = True
if beta1 > 0:
beta1 = sqrt(beta1)
# Normalize y to get v1 later.
self.residNorm0 = beta1
self.residNorm0 = beta1 # Initial residual norm.
# See if A is symmetric.
if check:
if not check_symmetric(A):
istop = 7
done = True
self.show = True
# See if preconditioner is symmetric.
if check and (precon is not None):
if not check_symmetric(precon):
istop = 8
show = True
done = True
# -------------------------------------------------------------------
# Initialize other quantities.
# ------------------------------------------------------------------
oldb = 0.0
beta = beta1
dbar = 0.0
epsln = 0.0
qrnorm = beta1
phibar = beta1
rhs1 = beta1
Arnorm = 0.0
rhs2 = 0.0
tnorm2 = 0.0
ynorm2 = 0.0
cs = -1.0
sn = 0.0
w = zeros(n, dtype=result_type)
w2 = zeros(n, dtype=result_type)
r2 = r1.copy()
if show:
print ' ' * 2
head1 = ' Itn x[0] Compatible LS'
head2 = ' norm(A) cond(A) gbar/|A|' ###### Check gbar
print head1 + head2
# ---------------------------------------------------------------------
# Main iteration loop.
# --------------------------------------------------------------------
if not done: # k = itn = 1 first time through
while itn < itnlim:
itn = itn + 1
# -------------------------------------------------------------
# Obtain quantities for the next Lanczos vector vk+1, k=1,2,...
# The general iteration is similar to the case k=1 with v0 = 0:
#
# p1 = Operator * v1 - beta1 * v0,
# alpha1 = v1'p1,
# q2 = p2 - alpha1 * v1,
# beta2^2 = q2'q2,
# v2 = (1/beta2) q2.
#
# Again, y = betak P vk, where P = C**(-1).
# .... more description needed.
# -------------------------------------------------------------
s = 1.0 / beta # Normalize previous vector (in y).
v = s * y # v = vk if P = I
y = A * v
y -= shift * v
if itn >= 2:
y = y - (beta / oldb) * r1
alfa = dot(v, y)
y = (-alfa / beta) * r2 + y
r1 = r2.copy()
r2 = y.copy()
if precon is not None: y = precon * r2
oldb = beta
beta = dot(r2, y)
if beta < 0:
istop = 6
break
beta = sqrt(beta)
tnorm2 = tnorm2 + alfa**2 + oldb**2 + beta**2
if itn == 1: # Initialize a few things.
if beta / beta1 <= 10 * eps: # beta2 = 0 or ~ 0.
istop = -1 # Terminate later.
# tnorm2 = alfa**2 ??
gmax = abs(alfa) # alpha1
gmin = gmax # alpha1
# Apply previous rotation Qk-1 to get
# [deltak epslnk+1] = [cs sn][dbark 0 ]
# [gbar k dbar k+1] [sn -cs][alfak betak+1].
oldeps = epsln
delta = cs * dbar + sn * alfa # delta1 = 0 deltak
gbar = sn * dbar - cs * alfa # gbar 1 = alfa1 gbar k
# Note: There is severe cancellation in the computation of gbar
#print ' sn = %21.15e\n dbar = %21.15e\n cs = %21.15e\n alfa = %21.15e\n sn*dbar-cs*alfa = %21.15e\n gbar =%21.15e' % (sn, dbar, cs, alfa, sn*dbar-cs*alfa, gbar)
epsln = sn * beta # epsln2 = 0 epslnk+1
dbar = -cs * beta # dbar 2 = beta2 dbar k+1
root = self.normof2(gbar, dbar)
Arnorm = phibar * root
# Compute the next plane rotation Qk
gamma = self.normof2(gbar, beta) # gammak
gamma = max(gamma, eps)
cs = gbar / gamma # ck
sn = beta / gamma # sk
phi = cs * phibar # phik
phibar = sn * phibar # phibark+1
# Update x.
denom = 1.0 / gamma
w1 = w2.copy()
w2 = w.copy()
w = (v - oldeps * w1 - delta * w2) * denom
x += phi * w #x = x + phi*w
if store_iterates:
self.iterates.append(x.copy())
# Update energy norm of x.
xNrgNorm2 += phi * phi
dErr[itn % window] = phi
if itn > window:
trncDirErr = norm(dErr)
xNrgNorm = sqrt(xNrgNorm2)
self.dir_errors_window.append(trncDirErr / xNrgNorm)
if trncDirErr < etol * xNrgNorm:
istop = 10
# Go round again.
gmax = max(gmax, gamma)
gmin = min(gmin, gamma)
z = rhs1 / gamma
ynorm2 = z**2 + ynorm2
rhs1 = rhs2 - delta * z
rhs2 = -epsln * z
# Estimate various norms and test for convergence.
Anorm = sqrt(tnorm2)
ynorm = sqrt(ynorm2)
epsa = Anorm * eps
epsx = Anorm * ynorm * eps
epsr = Anorm * ynorm * rtol
diag = gbar
if diag == 0: diag = epsa
qrnorm = phibar
rnorm = qrnorm
test1 = rnorm / (Anorm * ynorm) # ||r|| / (||A|| ||x||)
test2 = root / Anorm # ||Ar|| / (||A|| ||r||)
self.residHistory.append(rnorm)
# Estimate cond(A).
# In this version we look at the diagonals of R in the
# factorization of the lower Hessenberg matrix, Q * H = R,
# where H is the tridiagonal matrix from Lanczos with one
# extra row, beta(k+1) e_k^T.
Acond = gmax / gmin
# See if any of the stopping criteria are satisfied.
# In rare cases istop is already -1 from above (Abar = const*I)
if istop == 0:
t1 = 1 + test1 # These tests work if rtol < eps
t2 = 1 + test2
if t2 <= 1: istop = 2
if t1 <= 1: istop = 1
if itn >= itnlim: istop = 6
if Acond >= 0.1 / eps: istop = 4
if epsx >= beta1: istop = 3
# if rnorm <= epsx: istop = 2
# if rnorm <= epsr: istop = 1
if test2 <= rtol: istop = 2
if test1 <= rtol: istop = 1
# See if it is time to print something.
prnt = False
if n <= 40: prnt = True
if itn <= 10: prnt = True
if itn >= itnlim - 10: prnt = True
if (itn % 10) == 0: prnt = True
if qrnorm <= 10 * epsx: prnt = True
if qrnorm <= 10 * epsr: prnt = True
if Acond <= 1e-2 / eps: prnt = True
if istop != 0: prnt = True
if show and prnt:
str1 = '%6g %12.5e %10.3e' % (itn, x[0], test1)
str2 = ' %10.3e' % test2
str3 = ' %8.1e %8.1e %8.1e' % (Anorm, Acond, gbar / Anorm)
print str1 + str2 + str3
if istop > 0: break
if (itn % 10) == 0: print ' '
# Display final status.
if show:
last = self.last
print last + ' istop = %3g itn =%5g' % (istop,
itn)
print last + ' Anorm = %12.4e Acond = %12.4e' % (Anorm,
Acond)
print last + ' rnorm = %12.4e ynorm = %12.4e' % (rnorm,
ynorm)
print last + ' Arnorm = %12.4e' % Arnorm
print last + self.msg[istop + 1]
self.converged = istop in [1, 2, 3, 4, 10]
if istop == 10: self.status = 'direct error small'
self.bestSolution = self.x = x
self.istop = istop
self.itn = itn
self.residNorm = self.rnorm = rnorm
self.x = self.bestSolution = x
self.istop = istop
self.itn = self.nMatvec = itn
self.rnorm = self.residNorm = rnorm
self.Arnorm = Arnorm
self.Anorm = Anorm
self.Acond = Acond
self.ynorm = ynorm
return
def solve(self, rhs, **kwargs):
"""
Solve a linear system with `rhs` as right-hand side by the CG method.
The vector `rhs` should be a Numpy array.
:Keywords:
:guess: Initial guess (Numpy array). Default: 0.
:matvec_max: Max. number of operator-vector produts. Default: 2n.
:check_symmetric: Ensure operator is symmetric. Default: False.
:check_curvature: Ensure operator is positive definite. Default: True.
:store_resids: Store full residual vector history. Default: False.
:store_iterates: Store full iterate history. Default: False.
"""
n = rhs.shape[0]
nMatvec = 0
definite = True
check_sym = kwargs.get('check_symmetric', False)
check_curvature = kwargs.get('check_curvature', True)
store_resids = kwargs.get('store_resids', False)
store_iterates = kwargs.get('store_iterates', False)
if check_sym:
if not check_symmetric(self.op):
self.logger.error('Coefficient operator is not symmetric')
return
# Initial guess
result_type = np.result_type(self.op.dtype, rhs.dtype)
guess_supplied = 'guess' in kwargs.keys()
x = kwargs.get('guess', np.zeros(n)).astype(result_type)
if store_iterates:
self.iterates.append(x.copy())
matvec_max = kwargs.get('matvec_max', 2 * n)
# Initial residual vector
r = -rhs
if guess_supplied:
r += self.op * x
nMatvec += 1
# Initial preconditioned residual vector
if self.precon is not None:
y = self.precon * r
else:
y = r
if store_resids:
self.resids.append(y.copy())
ry = np.dot(r, y)
self.residNorm0 = residNorm = np.abs(np.sqrt(ry))
self.residHistory.append(self.residNorm0)
threshold = max(self.abstol, self.reltol * self.residNorm0)
p = -r # Initial search direction (copy not to overwrite rhs if x=0)
hdr_fmt = '%6s %7s %8s'
hdr = hdr_fmt % ('Matvec', 'Resid', 'Curv')
self.logger.info(hdr)
self.logger.info('-' * len(hdr))
info = '%6d %7.1e' % (nMatvec, residNorm)
self.logger.info(info)
while residNorm > threshold and nMatvec < matvec_max and definite:
Ap = self.op * p
nMatvec += 1
pAp = np.dot(p, Ap)
if check_curvature:
if np.imag(pAp) > 1.0e-8 * np.abs(pAp) or np.real(pAp) <= 0:
self.logger.error(
'Coefficient operator is not positive definite')
self.infiniteDescent = p
definite = False
continue
# Compute step length
alpha = ry / pAp
# Update estimate and residual
x += alpha * p
r += alpha * Ap
if store_iterates:
self.iterates.append(x.copy())
# Compute preconditioned residual
if self.precon is not None:
y = self.precon * r
else:
y = r
if store_resids:
self.resids.append(y.copy())
# Update preconditioned residual norm
ry_next = np.dot(r, y)
# Update search direction
beta = ry_next / ry
p *= beta
p -= r
ry = ry_next
residNorm = np.abs(np.sqrt(ry))
self.residHistory.append(residNorm)
info = '%6d %7.1e %8.1e' % (nMatvec, residNorm, np.real(pAp))
self.logger.info(info)
self.converged = residNorm <= threshold
self.definite = definite
self.nMatvec = nMatvec
self.bestSolution = self.x = x
self.residNorm = residNorm
