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lmpbvp.py
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#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Fri Apr 6 14:09:52 2018
@author: levi
"""
import numpy
import matplotlib.pyplot as plt
from utils import simp
from scipy.linalg import expm
class LMPBVPhelp():
"""Class for processing the Linear Multipoint Boundary Value Problem.
The biggest advantage in using an object is that the parallelization of
the computations for each solution is much easier."""
def __init__(self,sol,rho):
"""Initialization method. Comprises all the "common" calculations for
each independent solution to be added over.
According to the Miele and Wang (2003) convention,
rho = 0 for rest, and rho = 1 for grad.
"""
# debug options...
self.dbugOptGrad = sol.dbugOptGrad
self.dbugOptRest = sol.dbugOptRest
self.t = sol.t
# get sizes
Ns,N,m,n,p,q,s = sol.Ns,sol.N,sol.m,sol.n,sol.p,sol.q,sol.s
self.Ns,self.N,self.m,self.n,self.p,self.q,self.s = Ns,N,m,n,p,q,s
self.dt = 1.0/(N-1)
self.rho = rho
# get omission status from sol
self.omit = sol.omit
if self.omit:
self.omitEqMat = sol.omitEqMat
self.omitVarList = sol.omitVarList
# calculate psi
self.psi = sol.calcPsi()
if rho < .5:
# Restoration. Override the omit
self.omit = False
# calculate integration error (only if necessary)
err = sol.calcErr()
else:
err = numpy.zeros((N,n,s))
self.err = err
# Solver
# TODO: put the solver as an external option
self.solver = 'trap'#sol.solver
#######################################################################
if rho < 0.5 and self.dbugOptRest['plotErr']:
print("\nThis is err:")
for arc in range(s):
plt.plot(self.t,err[:,0,arc])
plt.ylabel("errPos")
plt.grid(True)
plt.show()
plt.clf()
plt.close('all')
if n>1:
plt.plot(self.t,err[:,1,arc])
plt.ylabel("errVel")
plt.grid(True)
plt.show()
plt.clf()
plt.close('all')
#######################################################################
# Get gradients
Grads = sol.calcGrads(calcCostTerm=(rho>0.5))
#dt6 = dt/6
phix = Grads['phix']
phiu = Grads['phiu']
phip = Grads['phip']
psiy = Grads['psiy']
psip = Grads['psip']
fx = Grads['fx']
fu = Grads['fu']
fp = Grads['fp']
self.phip = phip
self.psiy = psiy
self.psip = psip
self.fx = fx
self.fu = fu
self.fp = fp
# Prepare matrices with derivatives:
phixTr = numpy.empty_like(phix)
phiuTr = numpy.empty((N,m,n,s))
phipTr = numpy.empty((N,p,n,s))
phiuFu = numpy.empty((N,n,s))
for arc in range(s):
for k in range(N):
phixTr[k,:,:,arc] = phix[k,:,:,arc].transpose()
phiuTr[k,:,:,arc] = phiu[k,:,:,arc].transpose()
phipTr[k,:,:,arc] = phip[k,:,:,arc].transpose()
phiuFu[k,:,arc] = phiu[k,:,:,arc].dot(fu[k,:,arc])
self.phiuFu = phiuFu
self.phiuTr = phiuTr
self.phipTr = phipTr
InitCondMat = numpy.eye(Ns,Ns+1)
self.InitCondMat = InitCondMat
# Dynamics matrix for propagating the LSODE:
DynMat = numpy.zeros((N,2*n,2*n,s))
for arc in range(s):
for k in range(N):
DynMat[k,:n,:n,arc] = phix[k,:,:,arc]
DynMat[k,:n,n:,arc] = phiu[k,:,:,arc].dot(phiuTr[k,:,:,arc])
DynMat[k,n:,n:,arc] = -phixTr[k,:,:,arc]
self.DynMat = DynMat
# This is a strategy for lowering the cost of the trapezoidal solver.
# Instead of solving (N-1) * s * (2ns+p) linear systems of order 2n, resulting in a
# cost in the order of (N-1) * s * (2ns+p) * 4n² ; it is better to pre-invert
# (N-1) * s matrices of order 2n, resulting in a cost in the order of
# (N-1) * s * 8n³. This should always result in an increased performance because
# (N-1) * s * 8n³ < (N-1) * s * (2ns+p) * 4n²
if self.solver == 'trap':
I = numpy.eye(2 * n)
InvDynMat = numpy.zeros((N, 2 * n, 2 * n, s))
mhdt = -.5 * self.dt
for arc in range(s):
for k in range(1,N):
InvDynMat[k, :, :, arc] = numpy.linalg.inv(
I + mhdt * DynMat[k, :, :, arc])
self.InvDynMat = InvDynMat
#self.showEig(N,n,s)
def showEig(self,N,n,s,mustShow=False):
#print("\nLá vem os autovalores!")
plt.subplots_adjust(0.0125,0.0,0.9,2.5,0.2,0.2)
#plt.subplot2grid((s,1),(0,0))
eigen = numpy.empty((N,n,s),dtype=complex)
for arc in range(s):
eigen[:,:,arc] = numpy.linalg.eigvals(self.DynMat[:,:n,:n,arc])
plt.subplot2grid((s,1),(arc,0))
for i in range(n):
plt.plot(eigen[:,i,arc].real,eigen[:,i,arc].imag,'o-',\
label='eig #' + str(i+1) + ", start @(" + \
str(eigen[0,i,arc])+")")
#
plt.grid(True)
plt.legend()
plt.xlabel('Real')
plt.ylabel('Imag')
if arc == 0:
plt.title('Eigenvalues for each arc')
if mustShow:
plt.show()
def propagate(self,j):
"""This method computes each solution, via propagation of the
applicable Linear System of Ordinary Differential Equations."""
# Load data (sizes, common matrices, etc)
rho = self.rho
rho1 = self.rho-1.0
Ns,N,n,m,p,s = self.Ns,self.N,self.n,self.m,self.p,self.s
dt = self.dt
InitCondMat = self.InitCondMat
phip = self.phip
err = self.err
phiuFu = self.phiuFu
fx = self.fx
if rho > .5:
rhoFu = self.fu
else:
rhoFu = numpy.zeros((N,m,s))
phiuTr = self.phiuTr
phipTr = self.phipTr
DynMat = self.DynMat
I = numpy.eye(2*n)
# Declare matrices for corrections
phiLamIntCol = numpy.zeros(p)
DtCol = numpy.empty(2*n*s)
EtCol = numpy.empty(2*n*s)
A = numpy.zeros((N,n,s))
B = numpy.zeros((N,m,s))
C = numpy.zeros((p,1))
lam = numpy.zeros((N,n,s))
# the vector that will be integrated is Xi = [A; lam]
Xi = numpy.zeros((N,2*n,s))
# Initial conditions for the LSODE:
for arc in range(s):
A[0,:,arc] = InitCondMat[2*n*arc:(2*n*arc+n) , j]
lam[0,:,arc] = InitCondMat[(2*n*arc+n):(2*n*(arc+1)) , j]
Xi[0,:n,arc],Xi[0,n:,arc] = A[0,:,arc],lam[0,:,arc]
C = InitCondMat[(2*n*s):,j]
# Non-homogeneous terms for the LSODE:
nonHom = numpy.empty((N,2*n,s))
for arc in range(s):
for k in range(N):
# minus sign in rho1 (rho-1) is on purpose!
nonHA = phip[k,:,:,arc].dot(C) + \
-rho1*err[k,:,arc] - rho*phiuFu[k,:,arc]
nonHL = rho * fx[k,:,arc]
nonHom[k,:n,arc] = nonHA#.copy()
nonHom[k,n:,arc] = nonHL#.copy()
# This command probably broke the compatibility with other integration
# methods. They weren't working anyway, so...
coefList = simp([],N,onlyCoef=True)
for arc in range(s):
if self.solver == 'heun':
###############################################################################
# Integrate the LSODE (by Heun's method):
B[0,:,arc] = -rhoFu[0,:,arc] + \
phiuTr[0,:,:,arc].dot(lam[0,:,arc])
phiLamIntCol += .5 * (phipTr[0,:,:,arc].dot(lam[0,:,arc]))
# First point: simple propagation
derXik = DynMat[0,:,:,arc].dot(Xi[0,:,arc]) + \
nonHom[0,:,arc]
Xi[1,:,arc] = Xi[0,:,arc] + dt * derXik
#A[1,:,arc] = Xi[1,:n,arc]
lam[1,:,arc] = Xi[1,n:,arc]
B[1,:,arc] = -rhoFu[1,:,arc] + \
phiuTr[1,:,:,arc].dot(lam[1,:,arc])
phiLamIntCol += phipTr[1,:,:,arc].dot(lam[1,:,arc])
# "Middle" points: original Heun propagation
for k in range(1,N-2):
derXik = DynMat[k,:,:,arc].dot(Xi[k,:,arc]) + \
nonHom[k,:,arc]
aux = Xi[k,:,arc] + dt * derXik
Xi[k+1,:,arc] = Xi[k,:,arc] + .5 * dt * (derXik + \
DynMat[k+1,:,:,arc].dot(aux) + \
nonHom[k+1,:,arc])
#A[k+1,:,arc] = Xi[k+1,:n,arc]
lam[k+1,:,arc] = Xi[k+1,n:,arc]
B[k+1,:,arc] = -rhoFu[k+1,:,arc] + \
phiuTr[k+1,:,:,arc].dot(lam[k+1,:,arc])
phiLamIntCol += phipTr[k+1,:,:,arc].dot(lam[k+1,:,arc])
#
# Last point: simple propagation, but based on the last point
derXik = DynMat[N-1,:,:,arc].dot(Xi[N-2,:,arc]) + \
nonHom[N-1,:,arc]
Xi[N-1,:,arc] = Xi[N-2,:,arc] + dt * derXik
#A[N-1,:,arc] = Xi[N-1,:n,arc]
lam[N-1,:,arc] = Xi[N-1,n:,arc]
B[N-1,:,arc] = -rhoFu[N-1,:,arc] + \
phiuTr[N-1,:,:,arc].dot(lam[N-1,:,arc])
phiLamIntCol += .5*phipTr[N-1,:,:,arc].dot(lam[N-1,:,arc])
###############################################################################
elif self.solver == 'trap':
# Integrate the LSODE by trapezoidal (implicit) method
B[0,:,arc] = -rhoFu[0,:,arc] + \
phiuTr[0,:,:,arc].dot(lam[0,:,arc])
phiLamIntCol += coefList[0] * \
(phipTr[0,:,:,arc].dot(lam[0,:,arc]))
for k in range(N-1):
Xi[k+1,:,arc] = self.InvDynMat[k+1,:,:,arc].dot(
(I + .5 * dt * DynMat[k,:,:,arc]).dot(Xi[k,:,arc]) + \
.5 * dt * (nonHom[k+1,:,arc]+nonHom[k,:,arc]))
lam[k+1,:,arc] = Xi[k+1,n:,arc]
B[k+1,:,arc] = -rhoFu[k+1,:,arc] + \
phiuTr[k+1,:,:,arc].dot(lam[k+1,:,arc])
phiLamIntCol += coefList[k+1] * \
phipTr[k+1,:,:,arc].dot(lam[k+1,:,arc])
#
###############################################################################
elif self.solver == 'BEI':
# Integrate the LSODE by "original" Euler Backwards implicit
B[0,:,arc] = -rhoFu[0,:,arc] + \
phiuTr[0,:,:,arc].dot(lam[0,:,arc])
phiLamIntCol += .5*(phipTr[0,:,:,arc].dot(lam[0,:,arc]))
for k in range(N-1):
Xi[k+1,:,arc] = numpy.linalg.solve(I - dt*DynMat[k+1,:,:,arc],\
Xi[k,:,arc] + dt*nonHom[k+1,:,arc])
lam[k+1,:,arc] = Xi[k+1,n:,arc]
B[k+1,:,arc] = -rhoFu[k+1,:,arc] + \
phiuTr[k+1,:,:,arc].dot(lam[k+1,:,arc])
phiLamIntCol += phipTr[k+1,:,:,arc].dot(lam[k+1,:,arc])
phiLamIntCol -= .5*phipTr[N-1,:,:,arc].dot(lam[N-1,:,arc])
###############################################################################
if self.solver == 'leapfrog':
# Integrate the LSODE by "leapfrog" with special start and end
# with special 1st step...
B[0,:,arc] = -rhoFu[0,:,arc] + \
phiuTr[0,:,:,arc].dot(lam[0,:,arc])
phiLamIntCol += coefList[0] * (phipTr[0,:,:,arc].dot(lam[0,:,arc]))
Xi[1,:,arc] = Xi[0,:,arc] + dt * \
(DynMat[0,:,:,arc].dot(Xi[0,:,arc])+nonHom[0,:,arc])
lam[1,:,arc] = Xi[1,n:,arc]
B[1,:,arc] = -rhoFu[1,:,arc] + \
phiuTr[1,:,:,arc].dot(lam[1,:,arc])
phiLamIntCol += coefList[1] * \
phipTr[1,:,:,arc].dot(lam[1,:,arc])
for k in range(1,N-2):
Xi[k+1,:,arc] = Xi[k-1,:,arc] + 2. * dt * \
(DynMat[k,:,:,arc].dot(Xi[k,:,arc]) + nonHom[k,:,arc])
lam[k+1,:,arc] = Xi[k+1,n:,arc]
B[k+1,:,arc] = -rhoFu[k+1,:,arc] + \
phiuTr[k+1,:,:,arc].dot(lam[k+1,:,arc])
phiLamIntCol += coefList[k+1] * \
phipTr[k+1,:,:,arc].dot(lam[k+1,:,arc])
# # with special last step...
Xi[N-1,:,arc] = numpy.linalg.solve(I - dt*DynMat[N-1,:,:,arc],\
Xi[N-2,:,arc] + dt*nonHom[N-1,:,arc])
# # with special last step...
# Xi[N-1,:,arc] = Xi[N-2,:,arc] + dt * \
# (DynMat[N-1,:,:,arc].dot(Xi[N-2,:,arc])+nonHom[N-1,:,arc])
# with special last step...
# derXik = DynMat[N-2,:,:,arc].dot(Xi[N-2,:,arc]) + \
# nonHom[N-2,:,arc]
# aux = Xi[N-2,:,arc] + dt * derXik
# Xi[N-1,:,arc] = Xi[N-2,:,arc] + .5 * dt * (derXik + \
# DynMat[N-1,:,:,arc].dot(aux) + \
# nonHom[N-1,:,arc])
lam[N-1,:,arc] = Xi[N-1,n:,arc]
B[N-1,:,arc] = -rhoFu[N-1,:,arc] + \
phiuTr[N-1,:,:,arc].dot(lam[N-1,:,arc])
phiLamIntCol += coefList[N-1] * \
phipTr[N-1,:,:,arc].dot(lam[N-1,:,arc])
###############################################################################
if self.solver == 'BEI_spec':
# Integrate the LSODE by Euler Backwards implicit,
# with special 1st step...
B[0,:,arc] = -rhoFu[0,:,arc] + \
phiuTr[0,:,:,arc].dot(lam[0,:,arc])
phiLamIntCol += coefList[0] * (phipTr[0,:,:,arc].dot(lam[0,:,arc]))
Xi[1,:,arc] = Xi[0,:,arc] + dt * \
(DynMat[0,:,:,arc].dot(Xi[0,:,arc])+nonHom[0,:,arc])
lam[1,:,arc] = Xi[1,n:,arc]
B[1,:,arc] = -rhoFu[1,:,arc] + \
phiuTr[1,:,:,arc].dot(lam[1,:,arc])
phiLamIntCol += coefList[1] * \
phipTr[1,:,:,arc].dot(lam[1,:,arc])
for k in range(1,N-1):
Xi[k+1,:,arc] = numpy.linalg.solve(I - dt*DynMat[k+1,:,:,arc],\
Xi[k,:,arc] + dt*nonHom[k+1,:,arc])
lam[k+1,:,arc] = Xi[k+1,n:,arc]
B[k+1,:,arc] = -rhoFu[k+1,:,arc] + \
phiuTr[k+1,:,:,arc].dot(lam[k+1,:,arc])
phiLamIntCol += coefList[k+1] * \
phipTr[k+1,:,:,arc].dot(lam[k+1,:,arc])
###############################################################################
if self.solver == 'hamming_mod':
# Integrate the LSODE by Hamming's mod predictor-corrector method
B[0,:,arc] = -rhoFu[0,:,arc] + \
phiuTr[0,:,:,arc].dot(lam[0,:,arc])
phiLamIntCol += coefList[0] * (phipTr[0,:,:,arc].dot(lam[0,:,arc]))
# First points: RKF4
Xi[1,:,arc] = Xi[0,:,arc] + dt * \
(DynMat[0,:,:,arc].dot(Xi[0,:,arc]) + nonHom[0,:,arc])
lam[1,:,arc] = Xi[1,n:,arc]
B[1,:,arc] = -rhoFu[1,:,arc] + \
phiuTr[1,:,:,arc].dot(lam[1,:,arc])
phiLamIntCol += coefList[1] * \
phipTr[1,:,:,arc].dot(lam[1,:,arc])
for k in range(1,3):
Xik = Xi[k,:,arc]
DM13 = DynMat[k,:,:,arc]*(2./3.) + DynMat[k+1,:,:,arc]*(1./3.)
NH13 = nonHom[k,:,arc] * (2./3.) + nonHom[k+1,:,arc] * (1./3.)
DM23 = DynMat[k,:,:,arc]*(1./3.) + DynMat[k+1,:,:,arc]*(2./3.)
NH23 = nonHom[k,:,arc] * (1./3.) + nonHom[k+1,:,arc] * (2./3.)
f1 = DynMat[k,:,:,arc].dot(Xi[k,:,arc]) + nonHom[k,:,arc]
f2 = DM13.dot(Xik+(1./3.)*dt*f1) + NH13
f3 = DM23.dot(Xik + dt*(-(1./3.)*f1 + f2)) + NH23
f4 = DynMat[k+1,:,:,arc].dot(Xik + dt*(f1-f2+f3)) + nonHom[k+1,:,arc]
Xi[k+1,:,arc] = Xik + dt * (f1 + 3.*f2 + 3.*f3 + f4)/8.
# Xik = Xi[k,:,arc]
# DM14 = DynMat[k,:,:,arc]*.75 + DynMat[k+1,:,:,arc]*.25
# NH14 = nonHom[k,:,arc] * .75 + nonHom[k+1,:,arc] * .25
# DM38 = DynMat[k,:,:,arc]*.625 + DynMat[k+1,:,:,arc]*.375
# NH38 = nonHom[k,:,arc] * .625 + nonHom[k+1,:,arc] * .375
# DM12 = DynMat[k,:,:,arc]*.5 + DynMat[k+1,:,:,arc]*.5
# NH12 = nonHom[k,:,arc] * .5 + nonHom[k+1,:,arc] * .5
# DM1213 = DynMat[k,:,:,arc]*(1./13.) + DynMat[k+1,:,:,arc]*(12./13.)
# NH1213 = nonHom[k,:,arc] * (1./13.) + nonHom[k+1,:,arc] * (1./13.)
# f1 = DynMat[k,:,:,arc].dot(Xi[k,:,arc]) + nonHom[k,:,arc]
# f2 = DM14.dot(Xik+.25*dt*f1) + NH14
# f3 = DM38.dot(Xik + dt*(3.*f1 + 9.*f2)/32.) + NH38
# f4 = DM1213.dot(Xik + dt*(1932.*f1 - 7200.*f2 + 7296.)/2197.) + NH1213
# f5 = DynMat[k+1,:,:,arc].dot(Xik + dt*((439./216.)*f1-8.*f2+(3680./513.)*f3-(845./4104.)*f4)) + nonHom[k+1,:,arc]
# f6 = DM12.dot(Xik+dt*(-(8./27.)*f1 + 2.*f2 -(3544./2565.)*f3 +(1859./4104.)*f4 -(11./40.)*f5)) + NH12
#
# Xi[k+1,:,arc] = Xik + dt * ((16./135.)*f1 + \
# (6656./12825.)*f3 + \
# (28561./56430.)*f4 + \
# -(9./50.)*f5 + \
# (2./55.)*f6)
lam[k+1,:,arc] = Xi[k+1,n:,arc]
B[k+1,:,arc] = -rhoFu[k+1,:,arc] + \
phiuTr[k+1,:,:,arc].dot(lam[k+1,:,arc])
phiLamIntCol += coefList[k+1] * \
phipTr[k+1,:,:,arc].dot(lam[k+1,:,arc])
#
# Now, Hamming's...
pk = numpy.zeros_like(Xi[0,:,arc])
ck = pk
for k in range(3,N-1):
fk = DynMat[k,:,:,arc].dot(Xi[k,:,arc]) + nonHom[k,:,arc]
fkm1 = DynMat[k-1,:,:,arc].dot(Xi[k-1,:,arc]) + nonHom[k-1,:,arc]
fkm2 = DynMat[k-2,:,:,arc].dot(Xi[k-2,:,arc]) + nonHom[k-2,:,arc]
pkp1 = Xi[k-3,:,arc] + (4.0*dt/3.0) * \
(2.0 * fk - fkm1 + 2.0 * fkm2)
mkp1 = pkp1 + (112.0/121.0) * (ck - pk)
rdkp1 = DynMat[k+1,:,:,arc].dot(mkp1)
ckp1 = (9.0/8.0) * Xi[k,:,arc] -(1.0/8.0) * Xi[k-2,:,arc] + \
(3.0*dt/8.0) * (rdkp1 + 2.0 * fk - fkm1)
Xi[k+1,:,arc] = ckp1 + (9.0/121.0) * (pkp1 - ckp1)
lam[k+1,:,arc] = Xi[k+1,n:,arc]
B[k+1,:,arc] = -rhoFu[k+1,:,arc] + \
phiuTr[k+1,:,:,arc].dot(lam[k+1,:,arc])
phiLamIntCol += coefList[k+1] * \
phipTr[k+1,:,:,arc].dot(lam[k+1,:,arc])
pk, ck = pkp1, ckp1
#
###############################################################################
if self.solver == 'hamming':
# Integrate the LSODE by Hamming's predictor-corrector method
B[0,:,arc] = -rhoFu[0,:,arc] + \
phiuTr[0,:,:,arc].dot(lam[0,:,arc])
phiLamIntCol += coefList[0] * (phipTr[0,:,:,arc].dot(lam[0,:,arc]))
# first point: simple propagation?
# Xi[1,:,arc] = numpy.linalg.solve(I - dt*DynMat[1,:,:,arc],\
# Xi[0,:,arc] + dt*nonHom[1,:,arc])
# First points: Heun...
for k in range(3):
Xi[k+1,:,arc] = numpy.linalg.solve(I - .5*dt*DynMat[k+1,:,:,arc],\
Xi[k,:,arc] + .5 * dt * \
(DynMat[k,:,:,arc].dot(Xi[k,:,arc]) + \
nonHom[k,:,arc] + nonHom[k+1,:,arc]))
# derXik = DynMat[k,:,:,arc].dot(Xi[k,:,arc]) + \
# nonHom[k,:,arc]
# aux = Xi[k,:,arc] + dt * derXik
# Xi[k+1,:,arc] = Xi[k,:,arc] + .5 * dt * (derXik + \
# DynMat[k+1,:,:,arc].dot(aux) + \
# nonHom[k+1,:,arc])
lam[k+1,:,arc] = Xi[k+1,n:,arc]
B[k+1,:,arc] = -rhoFu[k+1,:,arc] + \
phiuTr[k+1,:,:,arc].dot(lam[k+1,:,arc])
phiLamIntCol += coefList[k+1] * \
phipTr[k+1,:,:,arc].dot(lam[k+1,:,arc])
#
# Now, Hamming's...
for k in range(3,N-1):
fk = DynMat[k,:,:,arc].dot(Xi[k,:,arc]) + nonHom[k,:,arc]
fkm1 = DynMat[k-1,:,:,arc].dot(Xi[k-1,:,arc]) + nonHom[k-1,:,arc]
fkm2 = DynMat[k-2,:,:,arc].dot(Xi[k-2,:,arc]) + nonHom[k-2,:,arc]
Xikp1 = Xi[k-3,:,arc] + (4.0*dt/3.0) * \
(2.0 * fk - fkm1 + 2.0 * fkm2)
fkp1 = DynMat[k+1,:,:,arc].dot(Xikp1) + nonHom[k+1,:,arc]
Xi[k+1,:,arc] = (9.0/8.0) * Xi[k,:,arc] + \
-(1.0/8.0) * Xi[k-2,:,arc] + (3.0*dt/8.0) * \
(fkp1 - fkm1 + 2.0 * fk)
lam[k+1,:,arc] = Xi[k+1,n:,arc]
B[k+1,:,arc] = -rhoFu[k+1,:,arc] + \
phiuTr[k+1,:,:,arc].dot(lam[k+1,:,arc])
phiLamIntCol += coefList[k+1] * \
phipTr[k+1,:,:,arc].dot(lam[k+1,:,arc])
#
###############################################################################
if self.solver == 'expm':
# Integrate the LSODE by matrix exponentiation
B[0,:,arc] = -rhoFu[0,:,arc] + \
phiuTr[0,:,:,arc].dot(lam[0,:,arc])
phiLamIntCol += .5 * (phipTr[0,:,:,arc].dot(lam[0,:,arc]))
for k in range(N-1):
expDM = expm(DynMat[k,:,:,arc]*dt)
NHterm = expDM.dot(nonHom[k,:,arc]) - nonHom[k,:,arc]
Xi[k+1,:,arc] = expDM.dot(Xi[k,:,arc]) + \
numpy.linalg.solve(DynMat[k,:,:,arc], NHterm)
lam[k+1,:,arc] = Xi[k+1,n:,arc]
B[k+1,:,arc] = -rhoFu[k+1,:,arc] + \
phiuTr[k+1,:,:,arc].dot(lam[k+1,:,arc])
phiLamIntCol += phipTr[k+1,:,:,arc].dot(lam[k+1,:,arc])
#
phiLamIntCol -= .5*phipTr[N-1,:,:,arc].dot(lam[N-1,:,arc])
###############################################################################
# Get the A values from Xi
A[:,:,arc] = Xi[:,:n,arc]
# Put initial and final conditions of A and Lambda into matrices
# DtCol and EtCol, which represent the columns of Dtilde(Dt) and
# Etilde(Et)
DtCol[(2*arc)*n : (2*arc+1)*n] = A[0,:,arc] # eq (32a)
DtCol[(2*arc+1)*n : (2*arc+2)*n] = A[N-1,:,arc] # eq (32a)
EtCol[(2*arc)*n : (2*arc+1)*n] = -lam[0,:,arc] # eq (32b)
EtCol[(2*arc+1)*n : (2*arc+2)*n] = lam[N-1,:,arc] # eq (32b)
#
# All integrations ready!
# no longer used, because coefList from simp already includes dt
#phiLamIntCol *= dt
###############################################################################
if (rho > 0.5 and self.dbugOptGrad['plotCorr']) or \
(rho < 0.5 and self.dbugOptRest['plotCorr']):
print("\nHere are the corrections for iteration " + str(j+1) + \
" of " + str(Ns+1) + ":\n")
for arc in range(s):
print("> Corrections for arc =",arc)
plt.plot(self.t,A[:,0,arc])
plt.grid(True)
plt.ylabel('A: pos')
plt.show()
plt.clf()
plt.close('all')
plt.plot(self.t,lam[:,0,arc])
plt.grid(True)
plt.ylabel('lambda: pos')
plt.show()
plt.clf()
plt.close('all')
if n>1:
plt.plot(self.t,A[:,1,arc])
plt.grid(True)
plt.ylabel('A: vel')
plt.show()
plt.clf()
plt.close('all')
plt.plot(self.t,lam[:,1,arc])
plt.grid(True)
plt.ylabel('lambda: vel')
plt.show()
plt.clf()
plt.close('all')
if n>2:
plt.plot(self.t,A[:,2,arc])
plt.grid(True)
plt.ylabel('A: gama')
plt.show()
plt.clf()
plt.close('all')
plt.plot(self.t,lam[:,2,arc])
plt.grid(True)
plt.ylabel('lambda: gamma')
plt.show()
plt.clf()
plt.close('all')
if n>3:
plt.plot(self.t,A[:,3,arc])
plt.grid(True)
plt.ylabel('A: m')
plt.show()
plt.clf()
plt.close('all')
plt.plot(self.t,lam[:,3,arc])
plt.grid(True)
plt.ylabel('lambda: m')
plt.show()
plt.clf()
plt.close('all')
plt.plot(self.t,B[:,0,arc])
plt.grid(True)
plt.ylabel('B0')
plt.show()
plt.clf()
plt.close('all')
if m>1:
plt.plot(self.t,B[:,1,arc])
plt.grid(True)
plt.ylabel('B1')
plt.show()
plt.clf()
plt.close('all')
print("C[arc] =",C[arc])
#input(" > ")
###############################################################################
# All the outputs go to main output dictionary; the final solution is
# computed by the next method, 'getCorr'.
outp = {'A':A,'B':B,'C':C,'L':lam,'Dt':DtCol,'Et':EtCol,
'phiLam':phiLamIntCol}
return outp
def getCorr(self,res,log):
""" Computes the actual correction for this grad/rest step, by linear
combination of the solutions generated by method 'propagate'."""
# Get sizes
Ns,N,n,m,p,q,s = self.Ns,self.N,self.n,self.m,self.p,self.q,self.s
# Declare matrices Ctilde, Dtilde, Etilde, and the integral term
if self.omit:
# Grad and omit: proceed to omit
NsRed = len(self.omitVarList)
q_ = q + NsRed - Ns - 1
omit = self.omitEqMat
else:
# nothing is omitted
NsRed = Ns + 1
q_ = q
Ct = numpy.empty((p,NsRed))
Dt = numpy.empty((2*n*s,NsRed))
Et = numpy.empty((2*n*s,NsRed))
phiLamInt = numpy.empty((p,NsRed))
# Unpack outputs from 'propagate' into proper matrices Ct, Dt, etc.
for j in range(NsRed):
Ct[:,j] = res[j]['C']
Dt[:,j] = res[j]['Dt']
Et[:,j] = res[j]['Et']
phiLamInt[:,j] = res[j]['phiLam']
# Assembly of matrix M and column 'Col' for the linear system
# Matrix for linear system involving k's and mu's
M = numpy.zeros((NsRed+q,NsRed+q))
# from eq (34d) - k term
M[0,:NsRed] = numpy.ones(NsRed)
# from eq (34b) - mu term
M[(q_+1):(q_+1+p),NsRed:] = self.psip.transpose()
# from eq (34c) - mu term
M[(p+q_+1):,NsRed:] = self.psiy.transpose()
# from eq (34a) - k term
if self.omit:
# under omission, less equations are needed
M[1:(q_+1), :NsRed] = omit.dot(self.psiy.dot(Dt) + self.psip.dot(Ct))
else:
# no omission, all the equations are needed
M[1:(q_ + 1), :NsRed] = self.psiy.dot(Dt) + self.psip.dot(Ct)
# from eq (34b) - k term
M[(q_+1):(q_+p+1),:NsRed] = Ct - phiLamInt
# from eq (34c) - k term
M[(q_+p+1):,:NsRed] = Et
# column vector for linear system involving k's and mu's [eqs (34)]
col = numpy.zeros(NsRed+q)
col[0] = 1.0 # eq (34d)
# Integral term
if self.rho > 0.5:
# eq (34b) - only applicable for grad
# sumIntFpi = numpy.zeros(p)
# for arc in range(s):
# for ind in range(p):
# sumIntFpi[ind] += self.fp[:,ind,arc].sum()
# sumIntFpi[ind] -= .5 * ( self.fp[0,ind,arc] + \
# self.fp[-1,ind,arc])
# sumIntFpi *= self.dt
sumIntFpi = numpy.zeros(p)
for arc in range(s):
for ind in range(p):
sumIntFpi[ind] += simp(self.fp[:,ind,arc],N)
#
#
col[(q_+1):(q_+p+1)] = - sumIntFpi
else:
# eq (34a) - only applicable for rest
col[1:(q+1)] = - self.psi
# Calculations of weights k:
KMi = numpy.linalg.solve(M,col)
Res = M.dot(KMi)-col
log.printL("LMPBVP: Residual of the Linear System: " + \
str(Res.transpose().dot(Res)))
K,mu = KMi[:NsRed], KMi[NsRed:]
log.printL("LMPBVP: coefficients of particular solutions: " + \
str(K))
# summing up linear combinations
A = numpy.zeros((N,n,s))
B = numpy.zeros((N,m,s))
C = numpy.zeros(p)
lam = numpy.zeros((N,n,s))
for j in range(NsRed):
A += K[j] * res[j]['A']#self.arrayA[j,:,:,:]
B += K[j] * res[j]['B']#self.arrayB[j,:,:,:]
C += K[j] * res[j]['C']#self.arrayC[j,:]
lam += K[j] * res[j]['L']#self.arrayL[j,:,:,:]
###############################################################################
if (self.rho > 0.5 and self.dbugOptGrad['plotCorrFin']) or \
(self.rho < 0.5 and self.dbugOptRest['plotCorrFin']):
log.printL("\n------------------------------------------------------------")
log.printL("Final corrections:\n")
for arc in range(s):
log.printL("> Corrections for arc =",arc)
plt.plot(self.t,A[:,0,arc])
plt.grid(True)
plt.ylabel('A: pos')
plt.show()
plt.clf()
plt.close('all')
if n>1:
plt.plot(self.t,A[:,1,arc])
plt.grid(True)
plt.ylabel('A: vel')
plt.show()
plt.clf()
plt.close('all')
if n>2:
plt.plot(self.t,A[:,2,arc])
plt.grid(True)
plt.ylabel('A: gama')
plt.show()
plt.clf()
plt.close('all')
if n>3:
plt.plot(self.t,A[:,3,arc])
plt.grid(True)
plt.ylabel('A: m')
plt.show()
plt.clf()
plt.close('all')
plt.plot(self.t,B[:,0,arc])
plt.grid(True)
plt.ylabel('B0')
plt.show()
plt.clf()
plt.close('all')
if m>1:
plt.plot(self.t,B[:,1,arc])
plt.grid(True)
plt.ylabel('B1')
plt.show()
plt.clf()
plt.close('all')
log.printL("C[arc] =",C[arc])
#input(" > ")
###############################################################################
return A,B,C,lam,mu