def makefig1():
pylab.figure(1, facecolor='w', figsize=(6,6))
pylab.plot(V[0,:].T, V[1,:].T, 'b-')
nopts = 1000
angles = matrix( [a*2.0*pi/nopts for a in range(nopts) ],
(1,nopts) )
circle = matrix(0.0, (2,nopts))
circle[0,:], circle[1,:] = cos(angles), sin(angles)
for k in range(len(C)):
c = C[k]
pylab.plot([c[0]], [c[1]], 'og')
pylab.text(c[0], c[1], "s%d" %k)
pylab.plot(c[0] + circle[0,:].T, c[1]+circle[1,:].T, 'g:')
if k >= 1:
v = V[:,k-1]
if k==1:
dir = 0.5 * (C[k] + C[-1]) - v
else:
dir = 0.5 * (C[k] + C[k-1]) - v
pylab.plot([v[0], v[0] + 5*dir[0]],
[v[1], v[1] + 5*dir[1]], 'b-')
ellipse = +circle
blas.trsm(L, ellipse, transA='T')
pylab.plot(xc[0] + ellipse[0,:].T, xc[1]+ellipse[1,:].T, 'r-')
for Xk in X:
pylab.plot([Xk[0]], [Xk[1]], 'ro')
pylab.axis([-5, 5, -5, 5])
pylab.title('Geometrical interpretation of Chebyshev bound (fig. 7.7)')
pylab.axis('off')
def makefig1():
pylab.figure(1, facecolor='w', figsize=(6, 6))
pylab.plot(V[0, :].T, V[1, :].T, 'b-')
nopts = 1000
angles = matrix([a * 2.0 * pi / nopts for a in range(nopts)],
(1, nopts))
circle = matrix(0.0, (2, nopts))
circle[0, :], circle[1, :] = cos(angles), sin(angles)
for k in range(len(C)):
c = C[k]
pylab.plot([c[0]], [c[1]], 'ow')
pylab.text(c[0], c[1], "s%d" % k)
pylab.plot(c[0] + circle[0, :].T, c[1] + circle[1, :].T, 'g:')
if k >= 1:
v = V[:, k - 1]
if k == 1:
dir = 0.5 * (C[k] + C[-1]) - v
else:
dir = 0.5 * (C[k] + C[k - 1]) - v
pylab.plot([v[0], v[0] + 5 * dir[0]],
[v[1], v[1] + 5 * dir[1]], 'b-')
ellipse = +circle
blas.trsm(L, ellipse, transA='T')
pylab.plot(xc[0] + ellipse[0, :].T, xc[1] + ellipse[1, :].T, 'r-')
for Xk in X:
pylab.plot([Xk[0]], [Xk[1]], 'ro')
pylab.axis([-5, 5, -5, 5])
pylab.title('Geometrical interpretation of Chebyshev bound (fig. 7.7)')
pylab.axis('off')
def F(x=None, z=None):
if x is None: return 0, matrix(1.0, (n,1))
X = V * spdiag(x) * V.T
L = +X
try: lapack.potrf(L)
except ArithmeticError: return None
f = - 2.0 * (log(L[0,0]) + log(L[1,1]))
W = +V
blas.trsm(L, W)
gradf = matrix(-1.0, (1,2)) * W**2
if z is None: return f, gradf
H = matrix(0.0, (n,n))
blas.syrk(W, H, trans='T')
return f, gradf, z[0] * H**2
def test_trsm(self):
L = cp.cspmatrix(self.symb) + self.A
cp.cholesky(L)
B = cp.eye(self.symb.n)
Bt = matrix(B)[self.symb.p, :]
Lt = matrix(L.spmatrix(reordered=True, symmetric=False))
cp.trsm(L, B)
blas.trsm(Lt, Bt)
diff = list((B - Bt[self.symb.ip, :])[:])
self.assertAlmostEqualLists(diff, len(diff) * [0.0])
B = cp.eye(self.symb.n)
Bt = matrix(B)[self.symb.p, :]
Lt = matrix(L.spmatrix(reordered=True, symmetric=False))
cp.trsm(L, B, trans='T')
blas.trsm(Lt, Bt, transA='T')
diff = list(B - Bt[self.symb.ip, :])[:]
self.assertAlmostEqualLists(diff, len(diff) * [0.0])
def test_trsm(self):
L = cp.cspmatrix(self.symb) + self.A
cp.cholesky(L)
B = cp.eye(self.symb.n)
Bt = matrix(B)[self.symb.p,:]
Lt = matrix(L.spmatrix(reordered=True,symmetric=False))
cp.trsm(L, B)
blas.trsm(Lt,Bt)
diff = list((B-Bt[self.symb.ip,:])[:])
self.assertAlmostEqualLists(diff, len(diff)*[0.0])
B = cp.eye(self.symb.n)
Bt = matrix(B)[self.symb.p,:]
Lt = matrix(L.spmatrix(reordered=True,symmetric=False))
cp.trsm(L, B, trans = 'T')
blas.trsm(Lt,Bt,transA='T')
diff = list(B-Bt[self.symb.ip,:])[:]
self.assertAlmostEqualLists(diff, len(diff)*[0.0])
def factor(W, H=None, Df=None):
if F['firstcall']:
if type(G) is matrix:
F['Gs'] = matrix(0.0, G.size)
else:
F['Gs'] = spmatrix(0.0, G.I, G.J, G.size)
if mnl:
if type(Df) is matrix:
F['Dfs'] = matrix(0.0, Df.size)
else:
F['Dfs'] = spmatrix(0.0, Df.I, Df.J, Df.size)
if (mnl and type(Df) is matrix) or type(G) is matrix or \
type(H) is matrix:
F['S'] = matrix(0.0, (n, n))
F['K'] = matrix(0.0, (p, p))
else:
F['S'] = spmatrix([], [], [], (n, n), 'd')
F['Sf'] = None
if type(A) is matrix:
F['K'] = matrix(0.0, (p, p))
else:
F['K'] = spmatrix([], [], [], (p, p), 'd')
# Dfs = Wnl^{-1} * Df
if mnl:
base.gemm(spmatrix(W['dnli'], list(range(mnl)),
list(range(mnl))), Df, F['Dfs'], partial=True)
# Gs = Wl^{-1} * G.
base.gemm(spmatrix(W['di'], list(range(ml)), list(range(ml))),
G, F['Gs'], partial=True)
if F['firstcall']:
base.syrk(F['Gs'], F['S'], trans='T')
if mnl:
base.syrk(F['Dfs'], F['S'], trans='T', beta=1.0)
if H is not None:
F['S'] += H
try:
if type(F['S']) is matrix:
lapack.potrf(F['S'])
else:
F['Sf'] = cholmod.symbolic(F['S'])
cholmod.numeric(F['S'], F['Sf'])
except ArithmeticError:
F['singular'] = True
if type(A) is matrix and type(F['S']) is spmatrix:
F['S'] = matrix(0.0, (n, n))
base.syrk(F['Gs'], F['S'], trans='T')
if mnl:
base.syrk(F['Dfs'], F['S'], trans='T', beta=1.0)
base.syrk(A, F['S'], trans='T', beta=1.0)
if H is not None:
F['S'] += H
if type(F['S']) is matrix:
lapack.potrf(F['S'])
else:
F['Sf'] = cholmod.symbolic(F['S'])
cholmod.numeric(F['S'], F['Sf'])
F['firstcall'] = False
else:
base.syrk(F['Gs'], F['S'], trans='T', partial=True)
if mnl:
base.syrk(F['Dfs'], F['S'], trans='T', beta=1.0,
partial=True)
if H is not None:
F['S'] += H
if F['singular']:
base.syrk(A, F['S'], trans='T', beta=1.0, partial=True)
if type(F['S']) is matrix:
lapack.potrf(F['S'])
else:
cholmod.numeric(F['S'], F['Sf'])
if type(F['S']) is matrix:
# Asct := L^{-1}*A'. Factor K = Asct'*Asct.
if type(A) is matrix:
Asct = A.T
else:
Asct = matrix(A.T)
blas.trsm(F['S'], Asct)
blas.syrk(Asct, F['K'], trans='T')
lapack.potrf(F['K'])
else:
# Asct := L^{-1}*P*A'. Factor K = Asct'*Asct.
if type(A) is matrix:
Asct = A.T
cholmod.solve(F['Sf'], Asct, sys=7)
cholmod.solve(F['Sf'], Asct, sys=4)
blas.syrk(Asct, F['K'], trans='T')
lapack.potrf(F['K'])
else:
Asct = cholmod.spsolve(F['Sf'], A.T, sys=7)
Asct = cholmod.spsolve(F['Sf'], Asct, sys=4)
base.syrk(Asct, F['K'], trans='T')
Kf = cholmod.symbolic(F['K'])
cholmod.numeric(F['K'], Kf)
def solve(x, y, z):
# Solve
#
# [ H A' GG'*W^{-1} ] [ ux ] [ bx ]
# [ A 0 0 ] * [ uy ] = [ by ]
# [ W^{-T}*GG 0 -I ] [ W*uz ] [ W^{-T}*bz ]
#
# and return ux, uy, W*uz.
#
# If not F['singular']:
#
# K*uy = A * S^{-1} * ( bx + GG'*W^{-1}*W^{-T}*bz ) - by
# S*ux = bx + GG'*W^{-1}*W^{-T}*bz - A'*uy
# W*uz = W^{-T} * ( GG*ux - bz ).
#
# If F['singular']:
#
# K*uy = A * S^{-1} * ( bx + GG'*W^{-1}*W^{-T}*bz + A'*by )
# - by
# S*ux = bx + GG'*W^{-1}*W^{-T}*bz + A'*by - A'*y.
# W*uz = W^{-T} * ( GG*ux - bz ).
# z := W^{-1} * z = W^{-1} * bz
scale(z, W, trans='T', inverse='I')
# If not F['singular']:
# x := L^{-1} * P * (x + GGs'*z)
# = L^{-1} * P * (x + GG'*W^{-1}*W^{-T}*bz)
#
# If F['singular']:
# x := L^{-1} * P * (x + GGs'*z + A'*y))
# = L^{-1} * P * (x + GG'*W^{-1}*W^{-T}*bz + A'*y)
if mnl:
base.gemv(F['Dfs'], z, x, trans='T', beta=1.0)
base.gemv(F['Gs'], z, x, offsetx=mnl, trans='T',
beta=1.0)
if F['singular']:
base.gemv(A, y, x, trans='T', beta=1.0)
if type(F['S']) is matrix:
blas.trsv(F['S'], x)
else:
cholmod.solve(F['Sf'], x, sys=7)
cholmod.solve(F['Sf'], x, sys=4)
# y := K^{-1} * (Asc*x - y)
# = K^{-1} * (A * S^{-1} * (bx + GG'*W^{-1}*W^{-T}*bz) - by)
# (if not F['singular'])
# = K^{-1} * (A * S^{-1} * (bx + GG'*W^{-1}*W^{-T}*bz +
# A'*by) - by)
# (if F['singular']).
base.gemv(Asct, x, y, trans='T', beta=-1.0)
if type(F['K']) is matrix:
lapack.potrs(F['K'], y)
else:
cholmod.solve(Kf, y)
# x := P' * L^{-T} * (x - Asc'*y)
# = S^{-1} * (bx + GG'*W^{-1}*W^{-T}*bz - A'*y)
# (if not F['singular'])
# = S^{-1} * (bx + GG'*W^{-1}*W^{-T}*bz + A'*by - A'*y)
# (if F['singular'])
base.gemv(Asct, y, x, alpha=-1.0, beta=1.0)
if type(F['S']) is matrix:
blas.trsv(F['S'], x, trans='T')
else:
cholmod.solve(F['Sf'], x, sys=5)
cholmod.solve(F['Sf'], x, sys=8)
# W*z := GGs*x - z = W^{-T} * (GG*x - bz)
if mnl:
base.gemv(F['Dfs'], x, z, beta=-1.0)
base.gemv(F['Gs'], x, z, beta=-1.0, offsety=mnl)
return solve
pylab.plot(X[:, 0], X[:, 1], 'ko', X[:, 0], X[:, 1], '-k')
# Ellipsoid in the form { x | || L' * (x-c) ||_2 <= 1 }
L = +A
lapack.potrf(L)
c = +b
lapack.potrs(L, c)
# 1000 points on the unit circle
nopts = 1000
angles = matrix([a * 2.0 * pi / nopts for a in range(nopts)], (1, nopts))
circle = matrix(0.0, (2, nopts))
circle[0, :], circle[1, :] = cos(angles), sin(angles)
# ellipse = L^-T * circle + c
blas.trsm(L, circle, transA='T')
ellipse = circle + c[:, nopts * [0]]
ellipse2 = 0.5 * circle + c[:, nopts * [0]]
pylab.plot(ellipse[0, :].T, ellipse[1, :].T, 'k-')
pylab.fill(ellipse2[0, :].T, ellipse2[1, :].T, facecolor='#F0F0F0')
pylab.title('Loewner-John ellipsoid (fig 8.3)')
pylab.axis('equal')
pylab.axis('off')
# Maximum volume enclosed ellipsoid
#
# minimize -log det B
# subject to ||B * gk||_2 + gk'*c <= hk, k=1,...,m
#
# with variables B and c.
def F(W):
"""
Create a solver for the linear equations
C * ux + G' * uzl - 2*A'(uzs21) = bx
-uzs11 = bX1
-uzs22 = bX2
G * ux - Dl^2 * uzl = bzl
[ -uX1 -A(ux)' ] [ uzs11 uzs21' ]
[ ] - r*r' * [ ] * r*r' = bzs
[ -A(ux) -uX2 ] [ uzs21 uzs22 ]
where Dl = diag(W['l']), r = W['r'][0].
On entry, x = (bx, bX1, bX2) and z = [ bzl; bzs[:] ].
On exit, x = (ux, uX1, uX2) and z = [ Dl*uzl; (r'*uzs*r)[:] ].
1. Compute matrices V1, V2 such that (with T = r*r')
[ V1 0 ] [ T11 T21' ] [ V1' 0 ] [ I S' ]
[ ] [ ] [ ] = [ ]
[ 0 V2' ] [ T21 T22 ] [ 0 V2 ] [ S I ]
and S = [ diag(s); 0 ], s a positive q-vector.
2. Factor the mapping X -> X + S * X' * S:
X + S * X' * S = L( L'( X )).
3. Compute scaled mappings: a matrix As with as its columns the
coefficients of the scaled mapping
L^-1( V2' * A() * V1' )
and the matrix Gs = Dl^-1 * G.
4. Cholesky factorization of H = C + Gs'*Gs + 2*As'*As.
"""
# 1. Compute V1, V2, s.
r = W['r'][0]
# LQ factorization R[:q, :] = L1 * Q1.
lapack.lacpy(r, Q1, m = q)
lapack.gelqf(Q1, tau1)
lapack.lacpy(Q1, L1, n = q, uplo = 'L')
lapack.orglq(Q1, tau1)
# LQ factorization R[q:, :] = L2 * Q2.
lapack.lacpy(r, Q2, m = p, offsetA = q)
lapack.gelqf(Q2, tau2)
lapack.lacpy(Q2, L2, n = p, uplo = 'L')
lapack.orglq(Q2, tau2)
# V2, V1, s are computed from an SVD: if
#
# Q2 * Q1' = U * diag(s) * V',
#
# then V1 = V' * L1^-1 and V2 = L2^-T * U.
# T21 = Q2 * Q1.T
blas.gemm(Q2, Q1, T21, transB = 'T')
# SVD T21 = U * diag(s) * V'. Store U in V2 and V' in V1.
lapack.gesvd(T21, s, jobu = 'A', jobvt = 'A', U = V2, Vt = V1)
# # Q2 := Q2 * Q1' without extracting Q1; store T21 in Q2
# this will requires lapack.ormlq or lapack.unmlq
# V2 = L2^-T * U
blas.trsm(L2, V2, transA = 'T')
# V1 = V' * L1^-1
blas.trsm(L1, V1, side = 'R')
# 2. Factorization X + S * X' * S = L( L'( X )).
#
# The factor L is stored as a diagonal matrix D and a sparse lower
# triangular matrix P, such that
#
# L(X)[:] = D**-1 * (I + P) * X[:]
# L^-1(X)[:] = D * (I - P) * X[:].
# SS is q x q with SS[i,j] = si*sj.
blas.scal(0.0, SS)
blas.syr(s, SS)
# For a p x q matrix X, P*X[:] is Y[:] where
#
# Yij = si * sj * Xji if i < j
# = 0 otherwise.
#
P.V = SS[Itril2]
# For a p x q matrix X, D*X[:] is Y[:] where
#
# Yij = Xij / sqrt( 1 - si^2 * sj^2 ) if i < j
# = Xii / sqrt( 1 + si^2 ) if i = j
# = Xij otherwise.
#
DV[Idiag] = sqrt(1.0 + SS[::q+1])
DV[Itriu] = sqrt(1.0 - SS[Itril3]**2)
D.V = DV**-1
# 3. Scaled linear mappings
# Ask := V2' * Ask * V1'
blas.scal(0.0, As)
base.axpy(A, As)
for i in xrange(n):
# tmp := V2' * As[i, :]
blas.gemm(V2, As, tmp, transA = 'T', m = p, n = q, k = p,
ldB = p, offsetB = i*p*q)
# As[:,i] := tmp * V1'
blas.gemm(tmp, V1, As, transB = 'T', m = p, n = q, k = q,
ldC = p, offsetC = i*p*q)
# As := D * (I - P) * As
# = L^-1 * As.
blas.copy(As, As2)
base.gemm(P, As, As2, alpha = -1.0, beta = 1.0)
base.gemm(D, As2, As)
# Gs := Dl^-1 * G
blas.scal(0.0, Gs)
base.axpy(G, Gs)
for k in xrange(n):
blas.tbmv(W['di'], Gs, n = m, k = 0, ldA = 1, offsetx = k*m)
# 4. Cholesky factorization of H = C + Gs' * Gs + 2 * As' * As.
blas.syrk(As, H, trans = 'T', alpha = 2.0)
blas.syrk(Gs, H, trans = 'T', beta = 1.0)
base.axpy(C, H)
lapack.potrf(H)
def f(x, y, z):
"""
Solve
C * ux + G' * uzl - 2*A'(uzs21) = bx
-uzs11 = bX1
-uzs22 = bX2
G * ux - D^2 * uzl = bzl
[ -uX1 -A(ux)' ] [ uzs11 uzs21' ]
[ ] - T * [ ] * T = bzs.
[ -A(ux) -uX2 ] [ uzs21 uzs22 ]
On entry, x = (bx, bX1, bX2) and z = [ bzl; bzs[:] ].
On exit, x = (ux, uX1, uX2) and z = [ D*uzl; (r'*uzs*r)[:] ].
Define X = uzs21, Z = T * uzs * T:
C * ux + G' * uzl - 2*A'(X) = bx
[ 0 X' ] [ bX1 0 ]
T * [ ] * T - Z = T * [ ] * T
[ X 0 ] [ 0 bX2 ]
G * ux - D^2 * uzl = bzl
[ -uX1 -A(ux)' ] [ Z11 Z21' ]
[ ] - [ ] = bzs
[ -A(ux) -uX2 ] [ Z21 Z22 ]
Return x = (ux, uX1, uX2), z = [ D*uzl; (rti'*Z*rti)[:] ].
We use the congruence transformation
[ V1 0 ] [ T11 T21' ] [ V1' 0 ] [ I S' ]
[ ] [ ] [ ] = [ ]
[ 0 V2' ] [ T21 T22 ] [ 0 V2 ] [ S I ]
and the factorization
X + S * X' * S = L( L'(X) )
to write this as
C * ux + G' * uzl - 2*A'(X) = bx
L'(V2^-1 * X * V1^-1) - L^-1(V2' * Z21 * V1') = bX
G * ux - D^2 * uzl = bzl
[ -uX1 -A(ux)' ] [ Z11 Z21' ]
[ ] - [ ] = bzs,
[ -A(ux) -uX2 ] [ Z21 Z22 ]
or
C * ux + Gs' * uuzl - 2*As'(XX) = bx
XX - ZZ21 = bX
Gs * ux - uuzl = D^-1 * bzl
-As(ux) - ZZ21 = bbzs_21
-uX1 - Z11 = bzs_11
-uX2 - Z22 = bzs_22
if we introduce scaled variables
uuzl = D * uzl
XX = L'(V2^-1 * X * V1^-1)
= L'(V2^-1 * uzs21 * V1^-1)
ZZ21 = L^-1(V2' * Z21 * V1')
and define
bbzs_21 = L^-1(V2' * bzs_21 * V1')
[ bX1 0 ]
bX = L^-1( V2' * (T * [ ] * T)_21 * V1').
[ 0 bX2 ]
Eliminating Z21 gives
C * ux + Gs' * uuzl - 2*As'(XX) = bx
Gs * ux - uuzl = D^-1 * bzl
-As(ux) - XX = bbzs_21 - bX
-uX1 - Z11 = bzs_11
-uX2 - Z22 = bzs_22
and eliminating uuzl and XX gives
H * ux = bx + Gs' * D^-1 * bzl + 2*As'(bX - bbzs_21)
Gs * ux - uuzl = D^-1 * bzl
-As(ux) - XX = bbzs_21 - bX
-uX1 - Z11 = bzs_11
-uX2 - Z22 = bzs_22.
In summary, we can use the following algorithm:
1. bXX := bX - bbzs21
[ bX1 0 ]
= L^-1( V2' * ((T * [ ] * T)_21 - bzs_21) * V1')
[ 0 bX2 ]
2. Solve H * ux = bx + Gs' * D^-1 * bzl + 2*As'(bXX).
3. From ux, compute
uuzl = Gs*ux - D^-1 * bzl and
X = V2 * L^-T(-As(ux) + bXX) * V1.
4. Return ux, uuzl,
rti' * Z * rti = r' * [ -bX1, X'; X, -bX2 ] * r
and uX1 = -Z11 - bzs_11, uX2 = -Z22 - bzs_22.
"""
# Save bzs_11, bzs_22, bzs_21.
lapack.lacpy(z, bz11, uplo = 'L', m = q, n = q, ldA = p+q,
offsetA = m)
lapack.lacpy(z, bz21, m = p, n = q, ldA = p+q, offsetA = m+q)
lapack.lacpy(z, bz22, uplo = 'L', m = p, n = p, ldA = p+q,
offsetA = m + (p+q+1)*q)
# zl := D^-1 * zl
# = D^-1 * bzl
blas.tbmv(W['di'], z, n = m, k = 0, ldA = 1)
# zs := r' * [ bX1, 0; 0, bX2 ] * r.
# zs := [ bX1, 0; 0, bX2 ]
blas.scal(0.0, z, offset = m)
lapack.lacpy(x[1], z, uplo = 'L', m = q, n = q, ldB = p+q,
offsetB = m)
lapack.lacpy(x[2], z, uplo = 'L', m = p, n = p, ldB = p+q,
offsetB = m + (p+q+1)*q)
# scale diagonal of zs by 1/2
blas.scal(0.5, z, inc = p+q+1, offset = m)
# a := tril(zs)*r
blas.copy(r, a)
blas.trmm(z, a, side = 'L', m = p+q, n = p+q, ldA = p+q, ldB =
p+q, offsetA = m)
# zs := a'*r + r'*a
blas.syr2k(r, a, z, trans = 'T', n = p+q, k = p+q, ldB = p+q,
ldC = p+q, offsetC = m)
# bz21 := L^-1( V2' * ((r * zs * r')_21 - bz21) * V1')
#
# [ bX1 0 ]
# = L^-1( V2' * ((T * [ ] * T)_21 - bz21) * V1').
# [ 0 bX2 ]
# a = [ r21 r22 ] * z
# = [ r21 r22 ] * r' * [ bX1, 0; 0, bX2 ] * r
# = [ T21 T22 ] * [ bX1, 0; 0, bX2 ] * r
blas.symm(z, r, a, side = 'R', m = p, n = p+q, ldA = p+q,
ldC = p+q, offsetB = q)
# bz21 := -bz21 + a * [ r11, r12 ]'
# = -bz21 + (T * [ bX1, 0; 0, bX2 ] * T)_21
blas.gemm(a, r, bz21, transB = 'T', m = p, n = q, k = p+q,
beta = -1.0, ldA = p+q, ldC = p)
# bz21 := V2' * bz21 * V1'
# = V2' * (-bz21 + (T*[bX1, 0; 0, bX2]*T)_21) * V1'
blas.gemm(V2, bz21, tmp, transA = 'T', m = p, n = q, k = p,
ldB = p)
blas.gemm(tmp, V1, bz21, transB = 'T', m = p, n = q, k = q,
ldC = p)
# bz21[:] := D * (I-P) * bz21[:]
# = L^-1 * bz21[:]
# = bXX[:]
blas.copy(bz21, tmp)
base.gemv(P, bz21, tmp, alpha = -1.0, beta = 1.0)
base.gemv(D, tmp, bz21)
# Solve H * ux = bx + Gs' * D^-1 * bzl + 2*As'(bXX).
# x[0] := x[0] + Gs'*zl + 2*As'(bz21)
# = bx + G' * D^-1 * bzl + 2 * As'(bXX)
blas.gemv(Gs, z, x[0], trans = 'T', alpha = 1.0, beta = 1.0)
blas.gemv(As, bz21, x[0], trans = 'T', alpha = 2.0, beta = 1.0)
# x[0] := H \ x[0]
# = ux
lapack.potrs(H, x[0])
# uuzl = Gs*ux - D^-1 * bzl
blas.gemv(Gs, x[0], z, alpha = 1.0, beta = -1.0)
# bz21 := V2 * L^-T(-As(ux) + bz21) * V1
# = X
blas.gemv(As, x[0], bz21, alpha = -1.0, beta = 1.0)
blas.tbsv(DV, bz21, n = p*q, k = 0, ldA = 1)
blas.copy(bz21, tmp)
base.gemv(P, tmp, bz21, alpha = -1.0, beta = 1.0, trans = 'T')
blas.gemm(V2, bz21, tmp)
blas.gemm(tmp, V1, bz21)
# zs := -zs + r' * [ 0, X'; X, 0 ] * r
# = r' * [ -bX1, X'; X, -bX2 ] * r.
# a := bz21 * [ r11, r12 ]
# = X * [ r11, r12 ]
blas.gemm(bz21, r, a, m = p, n = p+q, k = q, ldA = p, ldC = p+q)
# z := -z + [ r21, r22 ]' * a + a' * [ r21, r22 ]
# = rti' * uzs * rti
blas.syr2k(r, a, z, trans = 'T', beta = -1.0, n = p+q, k = p,
offsetA = q, offsetC = m, ldB = p+q, ldC = p+q)
# uX1 = -Z11 - bzs_11
# = -(r*zs*r')_11 - bzs_11
# uX2 = -Z22 - bzs_22
# = -(r*zs*r')_22 - bzs_22
blas.copy(bz11, x[1])
blas.copy(bz22, x[2])
# scale diagonal of zs by 1/2
blas.scal(0.5, z, inc = p+q+1, offset = m)
# a := r*tril(zs)
blas.copy(r, a)
blas.trmm(z, a, side = 'R', m = p+q, n = p+q, ldA = p+q, ldB =
p+q, offsetA = m)
# x[1] := -x[1] - a[:q,:] * r[:q, :]' - r[:q,:] * a[:q,:]'
# = -bzs_11 - (r*zs*r')_11
blas.syr2k(a, r, x[1], n = q, alpha = -1.0, beta = -1.0)
# x[2] := -x[2] - a[q:,:] * r[q:, :]' - r[q:,:] * a[q:,:]'
# = -bzs_22 - (r*zs*r')_22
blas.syr2k(a, r, x[2], n = p, alpha = -1.0, beta = -1.0,
offsetA = q, offsetB = q)
# scale diagonal of zs by 1/2
blas.scal(2.0, z, inc = p+q+1, offset = m)
return f
def F(W):
"""
Create a solver for the linear equations
C * ux + G' * uzl - 2*A'(uzs21) = bx
-uzs11 = bX1
-uzs22 = bX2
G * ux - Dl^2 * uzl = bzl
[ -uX1 -A(ux)' ] [ uzs11 uzs21' ]
[ ] - r*r' * [ ] * r*r' = bzs
[ -A(ux) -uX2 ] [ uzs21 uzs22 ]
where Dl = diag(W['l']), r = W['r'][0].
On entry, x = (bx, bX1, bX2) and z = [ bzl; bzs[:] ].
On exit, x = (ux, uX1, uX2) and z = [ Dl*uzl; (r'*uzs*r)[:] ].
1. Compute matrices V1, V2 such that (with T = r*r')
[ V1 0 ] [ T11 T21' ] [ V1' 0 ] [ I S' ]
[ ] [ ] [ ] = [ ]
[ 0 V2' ] [ T21 T22 ] [ 0 V2 ] [ S I ]
and S = [ diag(s); 0 ], s a positive q-vector.
2. Factor the mapping X -> X + S * X' * S:
X + S * X' * S = L( L'( X )).
3. Compute scaled mappings: a matrix As with as its columns the
coefficients of the scaled mapping
L^-1( V2' * A() * V1' )
and the matrix Gs = Dl^-1 * G.
4. Cholesky factorization of H = C + Gs'*Gs + 2*As'*As.
"""
# 1. Compute V1, V2, s.
r = W['r'][0]
# LQ factorization R[:q, :] = L1 * Q1.
lapack.lacpy(r, Q1, m=q)
lapack.gelqf(Q1, tau1)
lapack.lacpy(Q1, L1, n=q, uplo='L')
lapack.orglq(Q1, tau1)
# LQ factorization R[q:, :] = L2 * Q2.
lapack.lacpy(r, Q2, m=p, offsetA=q)
lapack.gelqf(Q2, tau2)
lapack.lacpy(Q2, L2, n=p, uplo='L')
lapack.orglq(Q2, tau2)
# V2, V1, s are computed from an SVD: if
#
# Q2 * Q1' = U * diag(s) * V',
#
# then V1 = V' * L1^-1 and V2 = L2^-T * U.
# T21 = Q2 * Q1.T
blas.gemm(Q2, Q1, T21, transB='T')
# SVD T21 = U * diag(s) * V'. Store U in V2 and V' in V1.
lapack.gesvd(T21, s, jobu='A', jobvt='A', U=V2, Vt=V1)
# # Q2 := Q2 * Q1' without extracting Q1; store T21 in Q2
# this will requires lapack.ormlq or lapack.unmlq
# V2 = L2^-T * U
blas.trsm(L2, V2, transA='T')
# V1 = V' * L1^-1
blas.trsm(L1, V1, side='R')
# 2. Factorization X + S * X' * S = L( L'( X )).
#
# The factor L is stored as a diagonal matrix D and a sparse lower
# triangular matrix P, such that
#
# L(X)[:] = D**-1 * (I + P) * X[:]
# L^-1(X)[:] = D * (I - P) * X[:].
# SS is q x q with SS[i,j] = si*sj.
blas.scal(0.0, SS)
blas.syr(s, SS)
# For a p x q matrix X, P*X[:] is Y[:] where
#
# Yij = si * sj * Xji if i < j
# = 0 otherwise.
#
P.V = SS[Itril2]
# For a p x q matrix X, D*X[:] is Y[:] where
#
# Yij = Xij / sqrt( 1 - si^2 * sj^2 ) if i < j
# = Xii / sqrt( 1 + si^2 ) if i = j
# = Xij otherwise.
#
DV[Idiag] = sqrt(1.0 + SS[::q + 1])
DV[Itriu] = sqrt(1.0 - SS[Itril3]**2)
D.V = DV**-1
# 3. Scaled linear mappings
# Ask := V2' * Ask * V1'
blas.scal(0.0, As)
base.axpy(A, As)
for i in xrange(n):
# tmp := V2' * As[i, :]
blas.gemm(V2,
As,
tmp,
transA='T',
m=p,
n=q,
k=p,
ldB=p,
offsetB=i * p * q)
# As[:,i] := tmp * V1'
blas.gemm(tmp,
V1,
As,
transB='T',
m=p,
n=q,
k=q,
ldC=p,
offsetC=i * p * q)
# As := D * (I - P) * As
# = L^-1 * As.
blas.copy(As, As2)
base.gemm(P, As, As2, alpha=-1.0, beta=1.0)
base.gemm(D, As2, As)
# Gs := Dl^-1 * G
blas.scal(0.0, Gs)
base.axpy(G, Gs)
for k in xrange(n):
blas.tbmv(W['di'], Gs, n=m, k=0, ldA=1, offsetx=k * m)
# 4. Cholesky factorization of H = C + Gs' * Gs + 2 * As' * As.
blas.syrk(As, H, trans='T', alpha=2.0)
blas.syrk(Gs, H, trans='T', beta=1.0)
base.axpy(C, H)
lapack.potrf(H)
def f(x, y, z):
"""
Solve
C * ux + G' * uzl - 2*A'(uzs21) = bx
-uzs11 = bX1
-uzs22 = bX2
G * ux - D^2 * uzl = bzl
[ -uX1 -A(ux)' ] [ uzs11 uzs21' ]
[ ] - T * [ ] * T = bzs.
[ -A(ux) -uX2 ] [ uzs21 uzs22 ]
On entry, x = (bx, bX1, bX2) and z = [ bzl; bzs[:] ].
On exit, x = (ux, uX1, uX2) and z = [ D*uzl; (r'*uzs*r)[:] ].
Define X = uzs21, Z = T * uzs * T:
C * ux + G' * uzl - 2*A'(X) = bx
[ 0 X' ] [ bX1 0 ]
T * [ ] * T - Z = T * [ ] * T
[ X 0 ] [ 0 bX2 ]
G * ux - D^2 * uzl = bzl
[ -uX1 -A(ux)' ] [ Z11 Z21' ]
[ ] - [ ] = bzs
[ -A(ux) -uX2 ] [ Z21 Z22 ]
Return x = (ux, uX1, uX2), z = [ D*uzl; (rti'*Z*rti)[:] ].
We use the congruence transformation
[ V1 0 ] [ T11 T21' ] [ V1' 0 ] [ I S' ]
[ ] [ ] [ ] = [ ]
[ 0 V2' ] [ T21 T22 ] [ 0 V2 ] [ S I ]
and the factorization
X + S * X' * S = L( L'(X) )
to write this as
C * ux + G' * uzl - 2*A'(X) = bx
L'(V2^-1 * X * V1^-1) - L^-1(V2' * Z21 * V1') = bX
G * ux - D^2 * uzl = bzl
[ -uX1 -A(ux)' ] [ Z11 Z21' ]
[ ] - [ ] = bzs,
[ -A(ux) -uX2 ] [ Z21 Z22 ]
or
C * ux + Gs' * uuzl - 2*As'(XX) = bx
XX - ZZ21 = bX
Gs * ux - uuzl = D^-1 * bzl
-As(ux) - ZZ21 = bbzs_21
-uX1 - Z11 = bzs_11
-uX2 - Z22 = bzs_22
if we introduce scaled variables
uuzl = D * uzl
XX = L'(V2^-1 * X * V1^-1)
= L'(V2^-1 * uzs21 * V1^-1)
ZZ21 = L^-1(V2' * Z21 * V1')
and define
bbzs_21 = L^-1(V2' * bzs_21 * V1')
[ bX1 0 ]
bX = L^-1( V2' * (T * [ ] * T)_21 * V1').
[ 0 bX2 ]
Eliminating Z21 gives
C * ux + Gs' * uuzl - 2*As'(XX) = bx
Gs * ux - uuzl = D^-1 * bzl
-As(ux) - XX = bbzs_21 - bX
-uX1 - Z11 = bzs_11
-uX2 - Z22 = bzs_22
and eliminating uuzl and XX gives
H * ux = bx + Gs' * D^-1 * bzl + 2*As'(bX - bbzs_21)
Gs * ux - uuzl = D^-1 * bzl
-As(ux) - XX = bbzs_21 - bX
-uX1 - Z11 = bzs_11
-uX2 - Z22 = bzs_22.
In summary, we can use the following algorithm:
1. bXX := bX - bbzs21
[ bX1 0 ]
= L^-1( V2' * ((T * [ ] * T)_21 - bzs_21) * V1')
[ 0 bX2 ]
2. Solve H * ux = bx + Gs' * D^-1 * bzl + 2*As'(bXX).
3. From ux, compute
uuzl = Gs*ux - D^-1 * bzl and
X = V2 * L^-T(-As(ux) + bXX) * V1.
4. Return ux, uuzl,
rti' * Z * rti = r' * [ -bX1, X'; X, -bX2 ] * r
and uX1 = -Z11 - bzs_11, uX2 = -Z22 - bzs_22.
"""
# Save bzs_11, bzs_22, bzs_21.
lapack.lacpy(z, bz11, uplo='L', m=q, n=q, ldA=p + q, offsetA=m)
lapack.lacpy(z, bz21, m=p, n=q, ldA=p + q, offsetA=m + q)
lapack.lacpy(z,
bz22,
uplo='L',
m=p,
n=p,
ldA=p + q,
offsetA=m + (p + q + 1) * q)
# zl := D^-1 * zl
# = D^-1 * bzl
blas.tbmv(W['di'], z, n=m, k=0, ldA=1)
# zs := r' * [ bX1, 0; 0, bX2 ] * r.
# zs := [ bX1, 0; 0, bX2 ]
blas.scal(0.0, z, offset=m)
lapack.lacpy(x[1], z, uplo='L', m=q, n=q, ldB=p + q, offsetB=m)
lapack.lacpy(x[2],
z,
uplo='L',
m=p,
n=p,
ldB=p + q,
offsetB=m + (p + q + 1) * q)
# scale diagonal of zs by 1/2
blas.scal(0.5, z, inc=p + q + 1, offset=m)
# a := tril(zs)*r
blas.copy(r, a)
blas.trmm(z,
a,
side='L',
m=p + q,
n=p + q,
ldA=p + q,
ldB=p + q,
offsetA=m)
# zs := a'*r + r'*a
blas.syr2k(r,
a,
z,
trans='T',
n=p + q,
k=p + q,
ldB=p + q,
ldC=p + q,
offsetC=m)
# bz21 := L^-1( V2' * ((r * zs * r')_21 - bz21) * V1')
#
# [ bX1 0 ]
# = L^-1( V2' * ((T * [ ] * T)_21 - bz21) * V1').
# [ 0 bX2 ]
# a = [ r21 r22 ] * z
# = [ r21 r22 ] * r' * [ bX1, 0; 0, bX2 ] * r
# = [ T21 T22 ] * [ bX1, 0; 0, bX2 ] * r
blas.symm(z,
r,
a,
side='R',
m=p,
n=p + q,
ldA=p + q,
ldC=p + q,
offsetB=q)
# bz21 := -bz21 + a * [ r11, r12 ]'
# = -bz21 + (T * [ bX1, 0; 0, bX2 ] * T)_21
blas.gemm(a,
r,
bz21,
transB='T',
m=p,
n=q,
k=p + q,
beta=-1.0,
ldA=p + q,
ldC=p)
# bz21 := V2' * bz21 * V1'
# = V2' * (-bz21 + (T*[bX1, 0; 0, bX2]*T)_21) * V1'
blas.gemm(V2, bz21, tmp, transA='T', m=p, n=q, k=p, ldB=p)
blas.gemm(tmp, V1, bz21, transB='T', m=p, n=q, k=q, ldC=p)
# bz21[:] := D * (I-P) * bz21[:]
# = L^-1 * bz21[:]
# = bXX[:]
blas.copy(bz21, tmp)
base.gemv(P, bz21, tmp, alpha=-1.0, beta=1.0)
base.gemv(D, tmp, bz21)
# Solve H * ux = bx + Gs' * D^-1 * bzl + 2*As'(bXX).
# x[0] := x[0] + Gs'*zl + 2*As'(bz21)
# = bx + G' * D^-1 * bzl + 2 * As'(bXX)
blas.gemv(Gs, z, x[0], trans='T', alpha=1.0, beta=1.0)
blas.gemv(As, bz21, x[0], trans='T', alpha=2.0, beta=1.0)
# x[0] := H \ x[0]
# = ux
lapack.potrs(H, x[0])
# uuzl = Gs*ux - D^-1 * bzl
blas.gemv(Gs, x[0], z, alpha=1.0, beta=-1.0)
# bz21 := V2 * L^-T(-As(ux) + bz21) * V1
# = X
blas.gemv(As, x[0], bz21, alpha=-1.0, beta=1.0)
blas.tbsv(DV, bz21, n=p * q, k=0, ldA=1)
blas.copy(bz21, tmp)
base.gemv(P, tmp, bz21, alpha=-1.0, beta=1.0, trans='T')
blas.gemm(V2, bz21, tmp)
blas.gemm(tmp, V1, bz21)
# zs := -zs + r' * [ 0, X'; X, 0 ] * r
# = r' * [ -bX1, X'; X, -bX2 ] * r.
# a := bz21 * [ r11, r12 ]
# = X * [ r11, r12 ]
blas.gemm(bz21, r, a, m=p, n=p + q, k=q, ldA=p, ldC=p + q)
# z := -z + [ r21, r22 ]' * a + a' * [ r21, r22 ]
# = rti' * uzs * rti
blas.syr2k(r,
a,
z,
trans='T',
beta=-1.0,
n=p + q,
k=p,
offsetA=q,
offsetC=m,
ldB=p + q,
ldC=p + q)
# uX1 = -Z11 - bzs_11
# = -(r*zs*r')_11 - bzs_11
# uX2 = -Z22 - bzs_22
# = -(r*zs*r')_22 - bzs_22
blas.copy(bz11, x[1])
blas.copy(bz22, x[2])
# scale diagonal of zs by 1/2
blas.scal(0.5, z, inc=p + q + 1, offset=m)
# a := r*tril(zs)
blas.copy(r, a)
blas.trmm(z,
a,
side='R',
m=p + q,
n=p + q,
ldA=p + q,
ldB=p + q,
offsetA=m)
# x[1] := -x[1] - a[:q,:] * r[:q, :]' - r[:q,:] * a[:q,:]'
# = -bzs_11 - (r*zs*r')_11
blas.syr2k(a, r, x[1], n=q, alpha=-1.0, beta=-1.0)
# x[2] := -x[2] - a[q:,:] * r[q:, :]' - r[q:,:] * a[q:,:]'
# = -bzs_22 - (r*zs*r')_22
blas.syr2k(a,
r,
x[2],
n=p,
alpha=-1.0,
beta=-1.0,
offsetA=q,
offsetB=q)
# scale diagonal of zs by 1/2
blas.scal(2.0, z, inc=p + q + 1, offset=m)
return f
Hac = G.T * spdiag((h-G*xac)**-1) * G
if pylab_installed:
pylab.figure(3, facecolor='w')
# polyhedron
for k in range(m):
edge = X[[k,k+1],:] + 0.1 * matrix([1., 0., 0., -1.], (2,2)) * \
(X[2*[k],:] - X[2*[k+1],:])
pylab.plot(edge[:,0], edge[:,1], 'k')
# 1000 points on the unit circle
nopts = 1000
angles = matrix( [ a*2.0*pi/nopts for a in range(nopts) ], (1,nopts) )
circle = matrix(0.0, (2,nopts))
circle[0,:], circle[1,:] = cos(angles), sin(angles)
# ellipse = L^-T * circle + xc where Hac = L*L'
lapack.potrf(Hac)
ellipse = +circle
blas.trsm(Hac, ellipse, transA='T')
ellipse += xac[:, nopts*[0]]
pylab.fill(ellipse[0,:].T, ellipse[1,:].T, facecolor = '#F0F0F0')
pylab.plot([xac[0]], [xac[1]], 'ko')
pylab.title('Analytic center (fig 8.7)')
pylab.axis('equal')
pylab.axis('off')
pylab.show()
def F(W):
"""
Returns a function f(x, y, z) that solves the KKT conditions
"""
U = +W['d'][0:n*r]
U = U ** 2
V = +W['d'][n*r:2*n*r]
V = V ** 2
Wd = +W['d']
if DEBUG > 0: print "# Computing L22"
L22 = []
for i in range(r):
BBTi = BBT + spmatrix(U[i*n:(i+1)*n] ,range(n),range(n))
cholesky(BBTi)
L22.append(BBTi)
if DEBUG > 0: print "# Computing L32"
L32 = []
for i in range(r):
# Solves L32 . L22 = - B . B^\top
C = -BBT
blas.trsm(L22[i], C, side='R', transA='T')
L32.append(C)
if DEBUG > 0: print "# Computing L33"
L33 = []
for i in range(r):
A = spmatrix(V[i*n:(i+1)*n] ,range(n),range(n)) + BBT - L32[i] * L32[i].T
cholesky(A)
L33.append(A)
if DEBUG > 0: print "# Computing L42"
L42 = []
for i in range(r):
A = +L22[i].T
lapack.trtri(A, uplo='U')
L42.append(A)
if DEBUG > 0: print "# Computing L43"
L43 = []
for i in range(r):
A = id_n - L42[i] * L32[i].T
blas.trsm(L33[i], A, side='R', transA='T')
L43.append(A)
if DEBUG > 0: print "# Computing L44 and D4"
# The indices for the diagonal of a dense matrix
L44 = matrix(0, (n,n))
for i in range(r):
L44 = L44 + L43[i] * L43[i].T + L42[i] * L42[i].T
cholesky(L44)
# WARNING: y, z and t have been permuted (LD33 and LD44piv)
if DEBUG > 0: print "## PRE-COMPUTATION DONE ##"
# Checking the decomposition
if DEBUG > 1:
if DEBUG > 2:
mA = []
mB = []
mD = []
for i in range(3*r+1):
m = []
for j in range(3*r+1): m.append(zero_n)
mA.append(m)
for i in range(r):
mA[i][i] = cholK
mA[i+r][i+r] = L22[i]
mA[i][i+r] = -B
mA[i][i+2*r] = B
mA[i+r][i+2*r] = L32[i]
mA[i+2*r][i+2*r] = L33[i]
mA[i+r][3*r] = L42[i]
mA[i+2*r][3*r] = L43[i]
mD.append(id_n)
mA[3*r][3*r] = L44
for i in range(2*r): mD.append(-id_n)
mD.append(id_n)
printing.options['width'] = 30
mA = sparse(mA)
mD = spdiag(mD)
print "LL^T =\n%s" % (mA * mD * mA.T),
print "P =\n%s" % P,
print g
print "W = %s" % W["d"].T,
print "U^2 = %s" % U.T,
print "V^2 = %s" % V.T,
print "### Pre-compute for check"
solve = chol2(W,P)
def f(x, y, z):
"""
On entry bx, bz are stored in x, z. On exit x, z contain the solution,
with z scaled: z./di is returned instead of z.
"""
# Maps to our variables x,y,z and t
if DEBUG > 0:
print "... Computing ..."
print "bx = %sbz = %s" % (x.T, z.T),
a = []
b = x[n*r:n*r + n]
c = []
d = []
for i in range(r):
a.append(x[i*n:(i+1)*n])
c.append(z[i*n:(i+1)*n])
d.append(z[(i+r)*n:(i+r+1)*n])
if DEBUG:
# Now solves using cvxopt
xp = +x
zp = +z
solve(xp,y,zp)
# First phase
for i in range(r):
blas.trsm(cholK, a[i])
Bai = B * a[i]
c[i] = - Bai - c[i]
blas.trsm(L22[i], c[i])
d[i] = Bai - L32[i] * c[i] - d[i]
blas.trsm(L33[i], d[i])
b = b + L42[i] * c[i] + L43[i] * d[i]
blas.trsm(L44, b)
# Second phase
blas.trsm(L44, b, transA='T')
for i in range(r):
d[i] = d[i] - L43[i].T * b
blas.trsm(L33[i], d[i], transA='T')
c[i] = c[i] - L32[i].T * d[i] - L42[i].T * b
blas.trsm(L22[i], c[i], transA='T')
a[i] = a[i] + B.T * (c[i] - d[i])
blas.trsm(cholK, a[i], transA='T')
# Store in vectors and scale
x[n*r:n*r + n] = b
for i in range(r):
x[i*n:(i+1)*n] = a[i]
z[i*n:(i+1)*n] = c[i]
z[(i+r)*n:(i+r+1)*n] = d[i]
z[:] = mul( Wd, z)
if DEBUG:
print "x = %s" % x.T,
print "z = %s" % z.T,
print "Delta(x) = %s" % (x - xp).T,
print "Delta(z) = %s" % (z - zp).T,
delta= blas.nrm2(x-xp) + blas.nrm2(z-zp)
if (delta > 1e-8):
print "--- DELTA TOO HIGH = %.3e ---" % delta
return f
def __scale(L, Y, U, adj=False, inv=False, factored_updates=True):
n = L.symb.n
snpost = L.symb.snpost
snptr = L.symb.snptr
chptr = L.symb.chptr
chidx = L.symb.chidx
relptr = L.symb.relptr
relidx = L.symb.relidx
blkptr = L.symb.blkptr
stack = []
for k in reversed(list(snpost)):
nn = snptr[k + 1] - snptr[k] # |Nk|
na = relptr[k + 1] - relptr[k] # |Ak|
nj = na + nn
F = matrix(0.0, (nj, nj))
lapack.lacpy(Y.blkval,
F,
m=nj,
n=nn,
ldA=nj,
offsetA=blkptr[k],
uplo='L')
# if supernode k is not a root node:
if na > 0:
# copy Vk to 2,2 block of F
Vk = stack.pop()
lapack.lacpy(Vk,
F,
ldB=nj,
offsetB=nn * (nj + 1),
m=na,
n=na,
uplo='L')
# if supernode k has any children:
for ii in range(chptr[k], chptr[k + 1]):
i = chidx[ii]
if factored_updates:
r = relidx[relptr[i]:relptr[i + 1]]
stack.append(frontal_get_update_factor(F, r, nn, na))
else:
stack.append(frontal_get_update(F, relidx, relptr, i))
# if supernode k is not a root node:
if na > 0:
if factored_updates:
# In this case we have Vk = Lk'*Lk
if adj is False: trns = 'N'
elif adj is True: trns = 'T'
else:
# factorize Vk
lapack.potrf(F, offsetA=nj * nn + nn, n=na, ldA=nj)
# In this case we have Vk = Lk*Lk'
if adj is False: trns = 'T'
elif adj is True: trns = 'N'
if adj is False: tr = ['T', 'N']
elif adj is True: tr = ['N', 'T']
if inv is False:
for Ut in U:
# symmetrize (1,1) block of Ut_{k} and scale
U11 = matrix(0.0, (nn, nn))
lapack.lacpy(Ut.blkval,
U11,
offsetA=blkptr[k],
m=nn,
n=nn,
ldA=nj,
uplo='L')
U11 += U11.T
U11[::nn + 1] *= 0.5
lapack.lacpy(U11,
Ut.blkval,
offsetB=blkptr[k],
m=nn,
n=nn,
ldB=nj,
uplo='N')
blas.trsm(L.blkval, Ut.blkval, side = 'R', transA = tr[0],\
m = nj, n = nn, offsetA = blkptr[k], ldA = nj,\
offsetB = blkptr[k], ldB = nj)
blas.trsm(L.blkval, Ut.blkval, m = nn, n = nn, transA = tr[1],\
offsetA = blkptr[k], offsetB = blkptr[k],\
ldA = nj, ldB = nj)
# zero-out strict upper triangular part of {Nj,Nj} block
for i in range(1, nn):
blas.scal(0.0, Ut.blkval, offset=blkptr[k] + nj * i, n=i)
if na > 0: blas.trmm(F, Ut.blkval, m = na, n = nn, transA = trns,\
offsetA = nj*nn+nn, ldA = nj,\
offsetB = blkptr[k]+nn, ldB = nj)
else: # inv is True
for Ut in U:
# symmetrize (1,1) block of Ut_{k} and scale
U11 = matrix(0.0, (nn, nn))
lapack.lacpy(Ut.blkval,
U11,
offsetA=blkptr[k],
m=nn,
n=nn,
ldA=nj,
uplo='L')
U11 += U11.T
U11[::nn + 1] *= 0.5
lapack.lacpy(U11,
Ut.blkval,
offsetB=blkptr[k],
m=nn,
n=nn,
ldB=nj,
uplo='N')
blas.trmm(L.blkval, Ut.blkval, side = 'R', transA = tr[0],\
m = nj, n = nn, offsetA = blkptr[k], ldA = nj,\
offsetB = blkptr[k], ldB = nj)
blas.trmm(L.blkval, Ut.blkval, m = nn, n = nn, transA = tr[1],\
offsetA = blkptr[k], offsetB = blkptr[k],\
ldA = nj, ldB = nj)
# zero-out strict upper triangular part of {Nj,Nj} block
for i in range(1, nn):
blas.scal(0.0, Ut.blkval, offset=blkptr[k] + nj * i, n=i)
if na > 0: blas.trsm(F, Ut.blkval, m = na, n = nn, transA = trns,\
offsetA = nj*nn+nn, ldA = nj,\
offsetB = blkptr[k]+nn, ldB = nj)
return
def F(W):
"""
Generate a solver for
A'(uz0) = bx[0]
-uz0 - uz1 = bx[1]
A(ux[0]) - ux[1] - r0*r0' * uz0 * r0*r0' = bz0
- ux[1] - r1*r1' * uz1 * r1*r1' = bz1.
uz0, uz1, bz0, bz1 are symmetric m x m-matrices.
ux[0], bx[0] are n-vectors.
ux[1], bx[1] are symmetric m x m-matrices.
We first calculate a congruence that diagonalizes r0*r0' and r1*r1':
U' * r0 * r0' * U = I, U' * r1 * r1' * U = S.
We then make a change of variables
usx[0] = ux[0],
usx[1] = U' * ux[1] * U
usz0 = U^-1 * uz0 * U^-T
usz1 = U^-1 * uz1 * U^-T
and define
As() = U' * A() * U'
bsx[1] = U^-1 * bx[1] * U^-T
bsz0 = U' * bz0 * U
bsz1 = U' * bz1 * U.
This gives
As'(usz0) = bx[0]
-usz0 - usz1 = bsx[1]
As(usx[0]) - usx[1] - usz0 = bsz0
-usx[1] - S * usz1 * S = bsz1.
1. Eliminate usz0, usz1 using equations 3 and 4,
usz0 = As(usx[0]) - usx[1] - bsz0
usz1 = -S^-1 * (usx[1] + bsz1) * S^-1.
This gives two equations in usx[0] an usx[1].
As'(As(usx[0]) - usx[1]) = bx[0] + As'(bsz0)
-As(usx[0]) + usx[1] + S^-1 * usx[1] * S^-1
= bsx[1] - bsz0 - S^-1 * bsz1 * S^-1.
2. Eliminate usx[1] using equation 2:
usx[1] + S * usx[1] * S
= S * ( As(usx[0]) + bsx[1] - bsz0 ) * S - bsz1
i.e., with Gamma[i,j] = 1.0 + S[i,i] * S[j,j],
usx[1] = ( S * As(usx[0]) * S ) ./ Gamma
+ ( S * ( bsx[1] - bsz0 ) * S - bsz1 ) ./ Gamma.
This gives an equation in usx[0].
As'( As(usx[0]) ./ Gamma )
= bx0 + As'(bsz0) +
As'( (S * ( bsx[1] - bsz0 ) * S - bsz1) ./ Gamma )
= bx0 + As'( ( bsz0 - bsz1 + S * bsx[1] * S ) ./ Gamma ).
"""
# Calculate U s.t.
#
# U' * r0*r0' * U = I, U' * r1*r1' * U = diag(s).
# Cholesky factorization r0 * r0' = L * L'
blas.syrk(W['r'][0], L)
lapack.potrf(L)
# SVD L^-1 * r1 = U * diag(s) * V'
blas.copy(W['r'][1], U)
blas.trsm(L, U)
lapack.gesvd(U, s, jobu='O')
# s := s**2
s[:] = s**2
# Uti := U
blas.copy(U, Uti)
# U := L^-T * U
blas.trsm(L, U, transA='T')
# Uti := L * Uti = U^-T
blas.trmm(L, Uti)
# Us := U * diag(s)^-1
blas.copy(U, Us)
for i in range(m):
blas.tbsv(s, Us, n=m, k=0, ldA=1, incx=m, offsetx=i)
# S is m x m with lower triangular entries s[i] * s[j]
# sqrtG is m x m with lower triangular entries sqrt(1.0 + s[i]*s[j])
# Upper triangular entries are undefined but nonzero.
blas.scal(0.0, S)
blas.syrk(s, S)
Gamma = 1.0 + S
sqrtG = sqrt(Gamma)
# Asc[i] = (U' * Ai * * U ) ./ sqrtG, for i = 1, ..., n
# = Asi ./ sqrt(Gamma)
blas.copy(A, Asc)
misc.scale(
Asc, # only 'r' part of the dictionary is used
{
'dnl': matrix(0.0, (0, 1)),
'dnli': matrix(0.0, (0, 1)),
'd': matrix(0.0, (0, 1)),
'di': matrix(0.0, (0, 1)),
'v': [],
'beta': [],
'r': [U],
'rti': [U]
})
for i in range(n):
blas.tbsv(sqrtG, Asc, n=msq, k=0, ldA=1, offsetx=i * msq)
# Convert columns of Asc to packed storage
misc.pack2(Asc, {'l': 0, 'q': [], 's': [m]})
# Cholesky factorization of Asc' * Asc.
H = matrix(0.0, (n, n))
blas.syrk(Asc, H, trans='T', k=mpckd)
lapack.potrf(H)
def solve(x, y, z):
"""
1. Solve for usx[0]:
Asc'(Asc(usx[0]))
= bx0 + Asc'( ( bsz0 - bsz1 + S * bsx[1] * S ) ./ sqrtG)
= bx0 + Asc'( ( bsz0 + S * ( bsx[1] - bssz1) S )
./ sqrtG)
where bsx[1] = U^-1 * bx[1] * U^-T, bsz0 = U' * bz0 * U,
bsz1 = U' * bz1 * U, bssz1 = S^-1 * bsz1 * S^-1
2. Solve for usx[1]:
usx[1] + S * usx[1] * S
= S * ( As(usx[0]) + bsx[1] - bsz0 ) * S - bsz1
usx[1]
= ( S * (As(usx[0]) + bsx[1] - bsz0) * S - bsz1) ./ Gamma
= -bsz0 + (S * As(usx[0]) * S) ./ Gamma
+ (bsz0 - bsz1 + S * bsx[1] * S ) . / Gamma
= -bsz0 + (S * As(usx[0]) * S) ./ Gamma
+ (bsz0 + S * ( bsx[1] - bssz1 ) * S ) . / Gamma
Unscale ux[1] = Uti * usx[1] * Uti'
3. Compute usz0, usz1
r0' * uz0 * r0 = r0^-1 * ( A(ux[0]) - ux[1] - bz0 ) * r0^-T
r1' * uz1 * r1 = r1^-1 * ( -ux[1] - bz1 ) * r1^-T
"""
# z0 := U' * z0 * U
# = bsz0
__cngrnc(U, z, trans='T')
# z1 := Us' * bz1 * Us
# = S^-1 * U' * bz1 * U * S^-1
# = S^-1 * bsz1 * S^-1
__cngrnc(Us, z, trans='T', offsetx=msq)
# x[1] := Uti' * x[1] * Uti
# = bsx[1]
__cngrnc(Uti, x[1], trans='T')
# x[1] := x[1] - z[msq:]
# = bsx[1] - S^-1 * bsz1 * S^-1
blas.axpy(z, x[1], alpha=-1.0, offsetx=msq)
# x1 = (S * x[1] * S + z[:msq] ) ./ sqrtG
# = (S * ( bsx[1] - S^-1 * bsz1 * S^-1) * S + bsz0 ) ./ sqrtG
# = (S * bsx[1] * S - bsz1 + bsz0 ) ./ sqrtG
# in packed storage
blas.copy(x[1], x1)
blas.tbmv(S, x1, n=msq, k=0, ldA=1)
blas.axpy(z, x1, n=msq)
blas.tbsv(sqrtG, x1, n=msq, k=0, ldA=1)
misc.pack2(x1, {'l': 0, 'q': [], 's': [m]})
# x[0] := x[0] + Asc'*x1
# = bx0 + Asc'( ( bsz0 - bsz1 + S * bsx[1] * S ) ./ sqrtG)
# = bx0 + As'( ( bz0 - bz1 + S * bx[1] * S ) ./ Gamma )
blas.gemv(Asc, x1, x[0], m=mpckd, trans='T', beta=1.0)
# x[0] := H^-1 * x[0]
# = ux[0]
lapack.potrs(H, x[0])
# x1 = Asc(x[0]) .* sqrtG (unpacked)
# = As(x[0])
blas.gemv(Asc, x[0], tmp, m=mpckd)
misc.unpack(tmp, x1, {'l': 0, 'q': [], 's': [m]})
blas.tbmv(sqrtG, x1, n=msq, k=0, ldA=1)
# usx[1] = (x1 + (x[1] - z[:msq])) ./ sqrtG**2
# = (As(ux[0]) + bsx[1] - bsz0 - S^-1 * bsz1 * S^-1)
# ./ Gamma
# x[1] := x[1] - z[:msq]
# = bsx[1] - bsz0 - S^-1 * bsz1 * S^-1
blas.axpy(z, x[1], -1.0, n=msq)
# x[1] := x[1] + x1
# = As(ux) + bsx[1] - bsz0 - S^-1 * bsz1 * S^-1
blas.axpy(x1, x[1])
# x[1] := x[1] / Gammma
# = (As(ux) + bsx[1] - bsz0 + S^-1 * bsz1 * S^-1 ) / Gamma
# = S^-1 * usx[1] * S^-1
blas.tbsv(Gamma, x[1], n=msq, k=0, ldA=1)
# z[msq:] := r1' * U * (-z[msq:] - x[1]) * U * r1
# := -r1' * U * S^-1 * (bsz1 + ux[1]) * S^-1 * U * r1
# := -r1' * uz1 * r1
blas.axpy(x[1], z, n=msq, offsety=msq)
blas.scal(-1.0, z, offset=msq)
__cngrnc(U, z, offsetx=msq)
__cngrnc(W['r'][1], z, trans='T', offsetx=msq)
# x[1] := S * x[1] * S
# = usx1
blas.tbmv(S, x[1], n=msq, k=0, ldA=1)
# z[:msq] = r0' * U' * ( x1 - x[1] - z[:msq] ) * U * r0
# = r0' * U' * ( As(ux) - usx1 - bsz0 ) * U * r0
# = r0' * U' * usz0 * U * r0
# = r0' * uz0 * r0
blas.axpy(x1, z, -1.0, n=msq)
blas.scal(-1.0, z, n=msq)
blas.axpy(x[1], z, -1.0, n=msq)
__cngrnc(U, z)
__cngrnc(W['r'][0], z, trans='T')
# x[1] := Uti * x[1] * Uti'
# = ux[1]
__cngrnc(Uti, x[1])
return solve
def cholesky(X):
"""
Supernodal multifrontal Cholesky factorization:
.. math::
X = LL^T
where :math:`L` is lower-triangular. On exit, the argument :math:`X`
contains the Cholesky factor :math:`L`.
:param X: :py:class:`cspmatrix`
"""
assert isinstance(X, cspmatrix) and X.is_factor is False, "X must be a cspmatrix"
n = X.symb.n
snpost = X.symb.snpost
snptr = X.symb.snptr
chptr = X.symb.chptr
chidx = X.symb.chidx
relptr = X.symb.relptr
relidx = X.symb.relidx
blkptr = X.symb.blkptr
blkval = X.blkval
stack = []
for k in snpost:
nn = snptr[k+1]-snptr[k] # |Nk|
na = relptr[k+1]-relptr[k] # |Ak|
nj = na + nn
# build frontal matrix
F = matrix(0.0, (nj, nj))
lapack.lacpy(blkval, F, offsetA = blkptr[k], m = nj, n = nn, ldA = nj, uplo = 'L')
# add update matrices from children to frontal matrix
for i in range(chptr[k+1]-1,chptr[k]-1,-1):
Ui = stack.pop()
frontal_add_update(F, Ui, relidx, relptr, chidx[i])
# factor L_{Nk,Nk}
lapack.potrf(F, n = nn, ldA = nj)
# if supernode k is not a root node, compute and push update matrix onto stack
if na > 0:
# compute L_{Ak,Nk} := A_{Ak,Nk}*inv(L_{Nk,Nk}')
blas.trsm(F, F, m = na, n = nn, ldA = nj,
ldB = nj, offsetB = nn, transA = 'T', side = 'R')
# compute Uk = Uk - L_{Ak,Nk}*inv(D_{Nk,Nk})*L_{Ak,Nk}'
if nn == 1:
blas.syr(F, F, n = na, offsetx = nn, \
offsetA = nn*nj+nn, ldA = nj, alpha = -1.0)
else:
blas.syrk(F, F, k = nn, n = na, offsetA = nn, ldA = nj,
offsetC = nn*nj+nn, ldC = nj, alpha = -1.0, beta = 1.0)
# compute L_{Ak,Nk} := L_{Ak,Nk}*inv(L_{Nk,Nk})
blas.trsm(F, F, m = na, n = nn,\
ldA = nj, ldB = nj, offsetB = nn, side = 'R')
# add Uk to stack
Uk = matrix(0.0,(na,na))
lapack.lacpy(F, Uk, m = na, n = na, uplo = 'L', offsetA = nn*nj+nn, ldA = nj)
stack.append(Uk)
# copy the leading Nk columns of frontal matrix to blkval
lapack.lacpy(F, blkval, uplo = "L", offsetB = blkptr[k], m = nj, n = nn, ldB = nj)
X.is_factor = True
return
def F(W):
"""
Generate a solver for
A'(uz0) = bx[0]
-uz0 - uz1 = bx[1]
A(ux[0]) - ux[1] - r0*r0' * uz0 * r0*r0' = bz0
- ux[1] - r1*r1' * uz1 * r1*r1' = bz1.
uz0, uz1, bz0, bz1 are symmetric m x m-matrices.
ux[0], bx[0] are n-vectors.
ux[1], bx[1] are symmetric m x m-matrices.
We first calculate a congruence that diagonalizes r0*r0' and r1*r1':
U' * r0 * r0' * U = I, U' * r1 * r1' * U = S.
We then make a change of variables
usx[0] = ux[0],
usx[1] = U' * ux[1] * U
usz0 = U^-1 * uz0 * U^-T
usz1 = U^-1 * uz1 * U^-T
and define
As() = U' * A() * U'
bsx[1] = U^-1 * bx[1] * U^-T
bsz0 = U' * bz0 * U
bsz1 = U' * bz1 * U.
This gives
As'(usz0) = bx[0]
-usz0 - usz1 = bsx[1]
As(usx[0]) - usx[1] - usz0 = bsz0
-usx[1] - S * usz1 * S = bsz1.
1. Eliminate usz0, usz1 using equations 3 and 4,
usz0 = As(usx[0]) - usx[1] - bsz0
usz1 = -S^-1 * (usx[1] + bsz1) * S^-1.
This gives two equations in usx[0] an usx[1].
As'(As(usx[0]) - usx[1]) = bx[0] + As'(bsz0)
-As(usx[0]) + usx[1] + S^-1 * usx[1] * S^-1
= bsx[1] - bsz0 - S^-1 * bsz1 * S^-1.
2. Eliminate usx[1] using equation 2:
usx[1] + S * usx[1] * S
= S * ( As(usx[0]) + bsx[1] - bsz0 ) * S - bsz1
i.e., with Gamma[i,j] = 1.0 + S[i,i] * S[j,j],
usx[1] = ( S * As(usx[0]) * S ) ./ Gamma
+ ( S * ( bsx[1] - bsz0 ) * S - bsz1 ) ./ Gamma.
This gives an equation in usx[0].
As'( As(usx[0]) ./ Gamma )
= bx0 + As'(bsz0) +
As'( (S * ( bsx[1] - bsz0 ) * S - bsz1) ./ Gamma )
= bx0 + As'( ( bsz0 - bsz1 + S * bsx[1] * S ) ./ Gamma ).
"""
# Calculate U s.t.
#
# U' * r0*r0' * U = I, U' * r1*r1' * U = diag(s).
# Cholesky factorization r0 * r0' = L * L'
blas.syrk(W['r'][0], L)
lapack.potrf(L)
# SVD L^-1 * r1 = U * diag(s) * V'
blas.copy(W['r'][1], U)
blas.trsm(L, U)
lapack.gesvd(U, s, jobu = 'O')
# s := s**2
s[:] = s**2
# Uti := U
blas.copy(U, Uti)
# U := L^-T * U
blas.trsm(L, U, transA = 'T')
# Uti := L * Uti = U^-T
blas.trmm(L, Uti)
# Us := U * diag(s)^-1
blas.copy(U, Us)
for i in range(m):
blas.tbsv(s, Us, n = m, k = 0, ldA = 1, incx = m, offsetx = i)
# S is m x m with lower triangular entries s[i] * s[j]
# sqrtG is m x m with lower triangular entries sqrt(1.0 + s[i]*s[j])
# Upper triangular entries are undefined but nonzero.
blas.scal(0.0, S)
blas.syrk(s, S)
Gamma = 1.0 + S
sqrtG = sqrt(Gamma)
# Asc[i] = (U' * Ai * * U ) ./ sqrtG, for i = 1, ..., n
# = Asi ./ sqrt(Gamma)
blas.copy(A, Asc)
misc.scale(Asc, # only 'r' part of the dictionary is used
{'dnl': matrix(0.0, (0, 1)), 'dnli': matrix(0.0, (0, 1)),
'd': matrix(0.0, (0, 1)), 'di': matrix(0.0, (0, 1)),
'v': [], 'beta': [], 'r': [ U ], 'rti': [ U ]})
for i in range(n):
blas.tbsv(sqrtG, Asc, n = msq, k = 0, ldA = 1, offsetx = i*msq)
# Convert columns of Asc to packed storage
misc.pack2(Asc, {'l': 0, 'q': [], 's': [ m ]})
# Cholesky factorization of Asc' * Asc.
H = matrix(0.0, (n, n))
blas.syrk(Asc, H, trans = 'T', k = mpckd)
lapack.potrf(H)
def solve(x, y, z):
"""
1. Solve for usx[0]:
Asc'(Asc(usx[0]))
= bx0 + Asc'( ( bsz0 - bsz1 + S * bsx[1] * S ) ./ sqrtG)
= bx0 + Asc'( ( bsz0 + S * ( bsx[1] - bssz1) S )
./ sqrtG)
where bsx[1] = U^-1 * bx[1] * U^-T, bsz0 = U' * bz0 * U,
bsz1 = U' * bz1 * U, bssz1 = S^-1 * bsz1 * S^-1
2. Solve for usx[1]:
usx[1] + S * usx[1] * S
= S * ( As(usx[0]) + bsx[1] - bsz0 ) * S - bsz1
usx[1]
= ( S * (As(usx[0]) + bsx[1] - bsz0) * S - bsz1) ./ Gamma
= -bsz0 + (S * As(usx[0]) * S) ./ Gamma
+ (bsz0 - bsz1 + S * bsx[1] * S ) . / Gamma
= -bsz0 + (S * As(usx[0]) * S) ./ Gamma
+ (bsz0 + S * ( bsx[1] - bssz1 ) * S ) . / Gamma
Unscale ux[1] = Uti * usx[1] * Uti'
3. Compute usz0, usz1
r0' * uz0 * r0 = r0^-1 * ( A(ux[0]) - ux[1] - bz0 ) * r0^-T
r1' * uz1 * r1 = r1^-1 * ( -ux[1] - bz1 ) * r1^-T
"""
# z0 := U' * z0 * U
# = bsz0
__cngrnc(U, z, trans = 'T')
# z1 := Us' * bz1 * Us
# = S^-1 * U' * bz1 * U * S^-1
# = S^-1 * bsz1 * S^-1
__cngrnc(Us, z, trans = 'T', offsetx = msq)
# x[1] := Uti' * x[1] * Uti
# = bsx[1]
__cngrnc(Uti, x[1], trans = 'T')
# x[1] := x[1] - z[msq:]
# = bsx[1] - S^-1 * bsz1 * S^-1
blas.axpy(z, x[1], alpha = -1.0, offsetx = msq)
# x1 = (S * x[1] * S + z[:msq] ) ./ sqrtG
# = (S * ( bsx[1] - S^-1 * bsz1 * S^-1) * S + bsz0 ) ./ sqrtG
# = (S * bsx[1] * S - bsz1 + bsz0 ) ./ sqrtG
# in packed storage
blas.copy(x[1], x1)
blas.tbmv(S, x1, n = msq, k = 0, ldA = 1)
blas.axpy(z, x1, n = msq)
blas.tbsv(sqrtG, x1, n = msq, k = 0, ldA = 1)
misc.pack2(x1, {'l': 0, 'q': [], 's': [m]})
# x[0] := x[0] + Asc'*x1
# = bx0 + Asc'( ( bsz0 - bsz1 + S * bsx[1] * S ) ./ sqrtG)
# = bx0 + As'( ( bz0 - bz1 + S * bx[1] * S ) ./ Gamma )
blas.gemv(Asc, x1, x[0], m = mpckd, trans = 'T', beta = 1.0)
# x[0] := H^-1 * x[0]
# = ux[0]
lapack.potrs(H, x[0])
# x1 = Asc(x[0]) .* sqrtG (unpacked)
# = As(x[0])
blas.gemv(Asc, x[0], tmp, m = mpckd)
misc.unpack(tmp, x1, {'l': 0, 'q': [], 's': [m]})
blas.tbmv(sqrtG, x1, n = msq, k = 0, ldA = 1)
# usx[1] = (x1 + (x[1] - z[:msq])) ./ sqrtG**2
# = (As(ux[0]) + bsx[1] - bsz0 - S^-1 * bsz1 * S^-1)
# ./ Gamma
# x[1] := x[1] - z[:msq]
# = bsx[1] - bsz0 - S^-1 * bsz1 * S^-1
blas.axpy(z, x[1], -1.0, n = msq)
# x[1] := x[1] + x1
# = As(ux) + bsx[1] - bsz0 - S^-1 * bsz1 * S^-1
blas.axpy(x1, x[1])
# x[1] := x[1] / Gammma
# = (As(ux) + bsx[1] - bsz0 + S^-1 * bsz1 * S^-1 ) / Gamma
# = S^-1 * usx[1] * S^-1
blas.tbsv(Gamma, x[1], n = msq, k = 0, ldA = 1)
# z[msq:] := r1' * U * (-z[msq:] - x[1]) * U * r1
# := -r1' * U * S^-1 * (bsz1 + ux[1]) * S^-1 * U * r1
# := -r1' * uz1 * r1
blas.axpy(x[1], z, n = msq, offsety = msq)
blas.scal(-1.0, z, offset = msq)
__cngrnc(U, z, offsetx = msq)
__cngrnc(W['r'][1], z, trans = 'T', offsetx = msq)
# x[1] := S * x[1] * S
# = usx1
blas.tbmv(S, x[1], n = msq, k = 0, ldA = 1)
# z[:msq] = r0' * U' * ( x1 - x[1] - z[:msq] ) * U * r0
# = r0' * U' * ( As(ux) - usx1 - bsz0 ) * U * r0
# = r0' * U' * usz0 * U * r0
# = r0' * uz0 * r0
blas.axpy(x1, z, -1.0, n = msq)
blas.scal(-1.0, z, n = msq)
blas.axpy(x[1], z, -1.0, n = msq)
__cngrnc(U, z)
__cngrnc(W['r'][0], z, trans = 'T')
# x[1] := Uti * x[1] * Uti'
# = ux[1]
__cngrnc(Uti, x[1])
return solve
if pylab_installed:
pylab.figure(2, facecolor='w', figsize=(6,6))
pylab.plot(V[0,:], V[1,:],'ow', mec = 'k')
pylab.plot([0], [0], 'k+')
I = [ k for k in range(n) if xe[k] > 1e-5 ]
pylab.plot(V[0,I], V[1,I],'or')
# Enclosing ellipse follows from the solution of the dual problem:
#
# minimize mu
# subject to diag(V'*Z*V) <= mu*1
# Z >= 0
lapack.potrf(Z)
ellipse = sqrt(mu) * circle
blas.trsm(Z, ellipse, transA='T')
if pylab_installed:
pylab.plot(ellipse[0,:].T, ellipse[1,:].T, 'k--')
pylab.axis([-5, 5, -5, 5])
pylab.title('E-optimal design (fig. 7.10)')
pylab.axis('off')
# A-design.
#
# minimize tr (V*diag(x)*V')^{-1}
# subject to x >= 0
# sum(x) = 1
#
# minimize tr Y
# subject to [ V*diag(x)*V', I ]
def kkt(W):
"""
KKT solver for
X*X' * ux + uy * 1_m' + mat(uz) = bx
ux * 1_m = by
ux - d.^2 .* mat(uz) = mat(bz).
ux and bx are N x m matrices.
uy and by are N-vectors.
uz and bz are N*m-vectors. mat(uz) is the N x m matrix that
satisfies mat(uz)[:] = uz.
d = mat(W['d']) a positive N x m matrix.
If we eliminate uz from the last equation using
mat(uz) = (ux - mat(bz)) ./ d.^2
we get two equations in ux, uy:
X*X' * ux + ux ./ d.^2 + uy * 1_m' = bx + mat(bz) ./ d.^2
ux * 1_m = by.
From the 1st equation,
uxk = (X*X' + Dk^-2)^-1 * (-uy + bxk + Dk^-2 * bzk)
= Dk * (I + Xk*Xk')^-1 * Dk * (-uy + bxk + Dk^-2 * bzk)
for k = 1, ..., m, where Dk = diag(d[:,k]), Xk = Dk * X,
uxk is column k of ux, and bzk is column k of mat(bz).
We use the matrix inversion lemma
( I + Xk * Xk' )^-1 = I - Xk * (I + Xk' * Xk)^-1 * Xk'
= I - Xk * Hk^-1 * Xk'
= I - Xk * Lk^-T * Lk^-1 * Xk'
where Hk = I + Xk' * Xk = Lk * Lk' to write this as
uxk = Dk * (I - Xk * Hk^-1 * Xk') * Dk *
(-uy + bxk + Dk^-2 * bzk)
= (Dk^2 - Dk^2 * X * Hk^-1 * X' * Dk^2) *
(-uy + bxk + Dk^-2 * bzk).
Substituting this in the second equation gives an equation
for uy:
sum_k (Dk^2 - Dk^2 * X * Hk^-1 * X' * Dk^2 ) * uy
= -by + sum_k (Dk^2 - Dk^2 * X * Hk^-1 * X' * Dk^2) *
( bxk + Dk^-2 * bzk ),
i.e., with D = (sum_k Dk^2)^1/2, Yk = D^-1 * Dk^2 * X * Lk^-T,
D * ( I - sum_k Yk * Yk' ) * D * uy
= -by + sum_k (Dk^2 - Dk^2 * X * Hk^-1 * X' * Dk^2) *
( bxk + Dk^-2 *bzk ).
Another application of the matrix inversion lemma gives
uy = D^-1 * (I + Y * S^-1 * Y') * D^-1 *
( -by + sum_k ( Dk^2 - Dk^2 * X * Hk^-1 * X' * Dk^2 ) *
( bxk + Dk^-2 *bzk ) )
with S = I - Y' * Y, Y = [ Y1 ... Ym ].
Summary:
1. Compute
uy = D^-1 * (I + Y * S^-1 * Y') * D^-1 *
( -by + sum_k (Dk^2 - Dk^2 * X * Hk^-1 * X' * Dk^2)
* ( bxk + Dk^-2 *bzk ) )
2. For k = 1, ..., m:
uxk = (Dk^2 - Dk^2 * X * Hk^-1 * X' * Dk^2) *
(-uy + bxk + Dk^-2 * bzk)
3. Solve for uz
d .* uz = ( ux - mat(bz) ) ./ d.
Return ux, uy, d .* uz.
"""
###
utime0, stime0 = cputime()
###
d = matrix(W['d'], (N, m))
dsq = matrix(W['d']**2, (N, m))
# Factor the matrices
#
# H[k] = I + Xk' * Xk
# = I + X' * Dk^2 * X.
#
# Dk = diag(d[:,k]).
for k in range(m):
# H[k] = I
blas.scal(0.0, H[k])
H[k][::n + 1] = 1.0
# Xs = Dk * X
# = diag(d[:,k]]) * X
blas.copy(X, Xs)
for j in range(n):
blas.tbmv(d,
Xs,
n=N,
k=0,
ldA=1,
offsetA=k * N,
offsetx=j * N)
# H[k] := H[k] + Xs' * Xs
# = I + Xk' * Xk
blas.syrk(Xs, H[k], trans='T', beta=1.0)
# Factorization H[k] = Lk * Lk'
lapack.potrf(H[k])
###
utime, stime = cputime()
print("Factor Hk's: utime = %.2f, stime = %.2f" \
%(utime-utime0, stime-stime0))
utime0, stime0 = cputime()
###
# diag(D) = ( sum_k d[:,k]**2 ) ** 1/2
# = ( sum_k Dk^2) ** 1/2.
blas.gemv(dsq, ones, D)
D[:] = sqrt(D)
###
# utime, stime = cputime()
# print("Compute D: utime = %.2f, stime = %.2f" \
# %(utime-utime0, stime-stime0))
utime0, stime0 = cputime()
###
# S = I - Y'* Y is an m x m block matrix.
# The i,j block of Y' * Y is
#
# Yi' * Yj = Li^-1 * X' * Di^2 * D^-1 * Dj^2 * X * Lj^-T.
#
# We compute only the lower triangular blocks in Y'*Y.
blas.scal(0.0, S)
for i in range(m):
for j in range(i + 1):
# Xs = Di * Dj * D^-1 * X
blas.copy(X, Xs)
blas.copy(d, wN, n=N, offsetx=i * N)
blas.tbmv(d, wN, n=N, k=0, ldA=1, offsetA=j * N)
blas.tbsv(D, wN, n=N, k=0, ldA=1)
for k in range(n):
blas.tbmv(wN, Xs, n=N, k=0, ldA=1, offsetx=k * N)
# block i, j of S is Xs' * Xs (as nonsymmetric matrix so we
# get the correct multiple after scaling with Li, Lj)
blas.gemm(Xs,
Xs,
S,
transA='T',
ldC=m * n,
offsetC=(j * n) * m * n + i * n)
###
utime, stime = cputime()
print("Form S: utime = %.2f, stime = %.2f" \
%(utime-utime0, stime-stime0))
utime0, stime0 = cputime()
###
for i in range(m):
# multiply block row i of S on the left with Li^-1
blas.trsm(H[i],
S,
m=n,
n=(i + 1) * n,
ldB=m * n,
offsetB=i * n)
# multiply block column i of S on the right with Li^-T
blas.trsm(H[i],
S,
side='R',
transA='T',
m=(m - i) * n,
n=n,
ldB=m * n,
offsetB=i * n * (m * n + 1))
blas.scal(-1.0, S)
S[::(m * n + 1)] += 1.0
###
utime, stime = cputime()
print("Form S (2): utime = %.2f, stime = %.2f" \
%(utime-utime0, stime-stime0))
utime0, stime0 = cputime()
###
# S = L*L'
lapack.potrf(S)
###
utime, stime = cputime()
print("Factor S: utime = %.2f, stime = %.2f" \
%(utime-utime0, stime-stime0))
utime0, stime0 = cputime()
###
def f(x, y, z):
"""
1. Compute
uy = D^-1 * (I + Y * S^-1 * Y') * D^-1 *
( -by + sum_k (Dk^2 - Dk^2 * X * Hk^-1 * X' * Dk^2)
* ( bxk + Dk^-2 *bzk ) )
2. For k = 1, ..., m:
uxk = (Dk^2 - Dk^2 * X * Hk^-1 * X' * Dk^2) *
(-uy + bxk + Dk^-2 * bzk)
3. Solve for uz
d .* uz = ( ux - mat(bz) ) ./ d.
Return ux, uy, d .* uz.
"""
###
utime0, stime0 = cputime()
###
# xk := Dk^2 * xk + zk
# = Dk^2 * bxk + bzk
blas.tbmv(dsq, x, n=N * m, k=0, ldA=1)
blas.axpy(z, x)
# y := -y + sum_k ( I - Dk^2 * X * Hk^-1 * X' ) * xk
# = -y + x*ones - sum_k Dk^2 * X * Hk^-1 * X' * xk
# y := -y + x*ones
blas.gemv(x, ones, y, alpha=1.0, beta=-1.0)
# wnm = X' * x (wnm interpreted as an n x m matrix)
blas.gemm(X, x, wnm, m=n, k=N, n=m, transA='T', ldB=N, ldC=n)
# wnm[:,k] = Hk \ wnm[:,k] (for wnm as an n x m matrix)
for k in range(m):
lapack.potrs(H[k], wnm, offsetB=k * n)
for k in range(m):
# wN = X * wnm[:,k]
blas.gemv(X, wnm, wN, offsetx=n * k)
# wN = Dk^2 * wN
blas.tbmv(dsq[:, k], wN, n=N, k=0, ldA=1)
# y := y - wN
blas.axpy(wN, y, -1.0)
# y = D^-1 * (I + Y * S^-1 * Y') * D^-1 * y
#
# Y = [Y1 ... Ym ], Yk = D^-1 * Dk^2 * X * Lk^-T.
# y := D^-1 * y
blas.tbsv(D, y, n=N, k=0, ldA=1)
# wnm = Y' * y (interpreted as an Nm vector)
# = [ L1^-1 * X' * D1^2 * D^-1 * y;
# L2^-1 * X' * D2^2 * D^-1 * y;
# ...
# Lm^-1 * X' * Dm^2 * D^-1 * y ]
for k in range(m):
# wN = D^-1 * Dk^2 * y
blas.copy(y, wN)
blas.tbmv(dsq, wN, n=N, k=0, ldA=1, offsetA=k * N)
blas.tbsv(D, wN, n=N, k=0, ldA=1)
# wnm[:,k] = X' * wN
blas.gemv(X, wN, wnm, trans='T', offsety=k * n)
# wnm[:,k] = Lk^-1 * wnm[:,k]
blas.trsv(H[k], wnm, offsetx=k * n)
# wnm := S^-1 * wnm (an mn-vector)
lapack.potrs(S, wnm)
# y := y + Y * wnm
# = y + D^-1 * [ D1^2 * X * L1^-T ... D2^k * X * Lk^-T]
# * wnm
for k in range(m):
# wnm[:,k] = Lk^-T * wnm[:,k]
blas.trsv(H[k], wnm, trans='T', offsetx=k * n)
# wN = X * wnm[:,k]
blas.gemv(X, wnm, wN, offsetx=k * n)
# wN = D^-1 * Dk^2 * wN
blas.tbmv(dsq, wN, n=N, k=0, ldA=1, offsetA=k * N)
blas.tbsv(D, wN, n=N, k=0, ldA=1)
# y += wN
blas.axpy(wN, y)
# y := D^-1 * y
blas.tbsv(D, y, n=N, k=0, ldA=1)
# For k = 1, ..., m:
#
# xk = (I - Dk^2 * X * Hk^-1 * X') * (-Dk^2 * y + xk)
# x = x - [ D1^2 * y ... Dm^2 * y] (as an N x m matrix)
for k in range(m):
blas.copy(y, wN)
blas.tbmv(dsq, wN, n=N, k=0, ldA=1, offsetA=k * N)
blas.axpy(wN, x, -1.0, offsety=k * N)
# wnm = X' * x (as an n x m matrix)
blas.gemm(X, x, wnm, transA='T', m=n, n=m, k=N, ldB=N, ldC=n)
# wnm[:,k] = Hk^-1 * wnm[:,k]
for k in range(m):
lapack.potrs(H[k], wnm, offsetB=n * k)
for k in range(m):
# wN = X * wnm[:,k]
blas.gemv(X, wnm, wN, offsetx=k * n)
# wN = Dk^2 * wN
blas.tbmv(dsq, wN, n=N, k=0, ldA=1, offsetA=k * N)
# x[:,k] := x[:,k] - wN
blas.axpy(wN, x, -1.0, n=N, offsety=k * N)
# z := ( x - z ) ./ d
blas.axpy(x, z, -1.0)
blas.scal(-1.0, z)
blas.tbsv(d, z, n=N * m, k=0, ldA=1)
###
utime, stime = cputime()
print("Solve: utime = %.2f, stime = %.2f" \
%(utime-utime0, stime-stime0))
###
return f
def trsm(L, B, alpha=1.0, trans='N', nrhs=None, offsetB=0, ldB=None):
r"""
Solves a triangular system of equations with multiple right-hand
sides. Computes
.. math::
B &:= \alpha L^{-1} B \text{ if trans is 'N'} \\
B &:= \alpha L^{-T} B \text{ if trans is 'T'}
where :math:`L` is a :py:class:`cspmatrix` factor.
:param L: :py:class:`cspmatrix` factor
:param B: matrix
:param alpha: float (default: 1.0)
:param trans: 'N' or 'T' (default: 'N')
:param nrhs: number of right-hand sides (default: number of columns in :math:`B`)
:param offsetB: integer (default: 0)
:param ldB: leading dimension of :math:`B` (default: number of rows in :math:`B`)
"""
assert isinstance(
L, cspmatrix) and L.is_factor is True, "L must be a cspmatrix factor"
assert isinstance(B, matrix), "B must be a matrix"
if ldB is None: ldB = B.size[0]
if nrhs is None: nrhs = B.size[1]
assert trans in ['N', 'T']
n = L.symb.n
snpost = L.symb.snpost
snptr = L.symb.snptr
snode = L.symb.snode
chptr = L.symb.chptr
chidx = L.symb.chidx
relptr = L.symb.relptr
relidx = L.symb.relidx
blkptr = L.symb.blkptr
blkval = L.blkval
p = L.symb.p
if p is None: p = range(n)
stack = []
if trans is 'N':
for k in snpost:
nn = snptr[k + 1] - snptr[k] # |Nk|
na = relptr[k + 1] - relptr[k] # |Ak|
nj = na + nn
# extract block from rhs
Uk = matrix(0.0, (nj, nrhs))
for j in range(nrhs):
for i, ir in enumerate(snode[snptr[k]:snptr[k + 1]]):
Uk[i, j] = alpha * B[offsetB + j * ldB + p[ir]]
# add contributions from children
for _ in range(chptr[k], chptr[k + 1]):
Ui, i = stack.pop()
r = relidx[relptr[i]:relptr[i + 1]]
Uk[r, :] += Ui
# if k is not a root node
if na > 0:
blas.gemm(blkval, Uk, Uk, alpha = -1.0, beta = 1.0, m = na, n = nrhs, k = nn,\
offsetA = blkptr[k]+nn, ldA = nj, offsetC = nn)
stack.append((Uk[nn:, :], k))
# scale and copy block to rhs
blas.trsm(blkval, Uk, m=nn, n=nrhs, offsetA=blkptr[k], ldA=nj)
for j in range(nrhs):
for i, ir in enumerate(snode[snptr[k]:snptr[k + 1]]):
B[offsetB + j * ldB + p[ir]] = Uk[i, j]
else: # trans is 'T'
for k in reversed(list(snpost)):
nn = snptr[k + 1] - snptr[k] # |Nk|
na = relptr[k + 1] - relptr[k] # |Ak|
nj = na + nn
# extract block from rhs and scale
Uk = matrix(0.0, (nj, nrhs))
for j in range(nrhs):
for i, ir in enumerate(snode[snptr[k]:snptr[k + 1]]):
Uk[i, j] = alpha * B[offsetB + j * ldB + p[ir]]
blas.trsm(blkval,
Uk,
transA='T',
m=nn,
n=nrhs,
offsetA=blkptr[k],
ldA=nj)
# if k is not a root node
if na > 0:
Uk[nn:, :] = stack.pop()
blas.gemm(blkval, Uk, Uk, alpha = -1.0, beta = 1.0, m = nn, n = nrhs, k = na,\
transA = 'T', offsetA = blkptr[k]+nn, ldA = nj, offsetB = nn)
# stack contributions for children
for ii in range(chptr[k], chptr[k + 1]):
i = chidx[ii]
stack.append(Uk[relidx[relptr[i]:relptr[i + 1]], :])
# copy block to rhs
for j in range(nrhs):
for i, ir in enumerate(snode[snptr[k]:snptr[k + 1]]):
B[offsetB + j * ldB + p[ir]] = Uk[i, j]
return
def f(x, y, z):
"""
On entry bx, bz are stored in x, z. On exit x, z contain the solution,
with z scaled: z./di is returned instead of z.
"""
# Maps to our variables x,y,z and t
if DEBUG > 0:
print "... Computing ..."
print "bx = %sbz = %s" % (x.T, z.T),
a = []
b = x[n * r:n * r + n]
c = []
d = []
for i in range(r):
a.append(x[i * n:(i + 1) * n])
c.append(z[i * n:(i + 1) * n])
d.append(z[(i + r) * n:(i + r + 1) * n])
if DEBUG:
# Now solves using cvxopt
xp = +x
zp = +z
solve(xp, y, zp)
# First phase
for i in range(r):
blas.trsm(cholK, a[i])
Bai = B * a[i]
c[i] = -Bai - c[i]
blas.trsm(L22[i], c[i])
d[i] = Bai - L32[i] * c[i] - d[i]
blas.trsm(L33[i], d[i])
b = b + L42[i] * c[i] + L43[i] * d[i]
blas.trsm(L44, b)
# Second phase
blas.trsm(L44, b, transA='T')
for i in range(r):
d[i] = d[i] - L43[i].T * b
blas.trsm(L33[i], d[i], transA='T')
c[i] = c[i] - L32[i].T * d[i] - L42[i].T * b
blas.trsm(L22[i], c[i], transA='T')
a[i] = a[i] + B.T * (c[i] - d[i])
blas.trsm(cholK, a[i], transA='T')
# Store in vectors and scale
x[n * r:n * r + n] = b
for i in range(r):
x[i * n:(i + 1) * n] = a[i]
z[i * n:(i + 1) * n] = c[i]
z[(i + r) * n:(i + r + 1) * n] = d[i]
z[:] = mul(Wd, z)
if DEBUG:
print "x = %s" % x.T,
print "z = %s" % z.T,
print "Delta(x) = %s" % (x - xp).T,
print "Delta(z) = %s" % (z - zp).T,
delta = blas.nrm2(x - xp) + blas.nrm2(z - zp)
if (delta > 1e-8):
print "--- DELTA TOO HIGH = %.3e ---" % delta
def f(x, y, z):
"""
On entry bx, bz are stored in x, z. On exit x, z contain the solution,
with z scaled: z./di is returned instead of z.
"""
# Maps to our variables x,y,z and t
if DEBUG > 0:
print "... Computing ..."
print "bx = %sbz = %s" % (x.T, z.T),
a = []
b = x[n*r:n*r + n]
c = []
d = []
for i in range(r):
a.append(x[i*n:(i+1)*n])
c.append(z[i*n:(i+1)*n])
d.append(z[(i+r)*n:(i+r+1)*n])
if DEBUG:
# Now solves using cvxopt
xp = +x
zp = +z
solve(xp,y,zp)
# First phase
for i in range(r):
blas.trsm(cholK, a[i])
Bai = B * a[i]
c[i] = - Bai - c[i]
blas.trsm(L22[i], c[i])
d[i] = Bai - L32[i] * c[i] - d[i]
blas.trsm(L33[i], d[i])
b = b + L42[i] * c[i] + L43[i] * d[i]
blas.trsm(L44, b)
# Second phase
blas.trsm(L44, b, transA='T')
for i in range(r):
d[i] = d[i] - L43[i].T * b
blas.trsm(L33[i], d[i], transA='T')
c[i] = c[i] - L32[i].T * d[i] - L42[i].T * b
blas.trsm(L22[i], c[i], transA='T')
a[i] = a[i] + B.T * (c[i] - d[i])
blas.trsm(cholK, a[i], transA='T')
# Store in vectors and scale
x[n*r:n*r + n] = b
for i in range(r):
x[i*n:(i+1)*n] = a[i]
z[i*n:(i+1)*n] = c[i]
z[(i+r)*n:(i+r+1)*n] = d[i]
z[:] = mul( Wd, z)
if DEBUG:
print "x = %s" % x.T,
print "z = %s" % z.T,
print "Delta(x) = %s" % (x - xp).T,
print "Delta(z) = %s" % (z - zp).T,
delta= blas.nrm2(x-xp) + blas.nrm2(z-zp)
if (delta > 1e-8):
print "--- DELTA TOO HIGH = %.3e ---" % delta
def F(W):
"""
Returns a function f(x, y, z) that solves the KKT conditions
"""
U = +W['d'][0:n * r]
U = U**2
V = +W['d'][n * r:2 * n * r]
V = V**2
Wd = +W['d']
if DEBUG > 0: print "# Computing L22"
L22 = []
for i in range(r):
BBTi = BBT + spmatrix(U[i * n:(i + 1) * n], range(n), range(n))
cholesky(BBTi)
L22.append(BBTi)
if DEBUG > 0: print "# Computing L32"
L32 = []
for i in range(r):
# Solves L32 . L22 = - B . B^\top
C = -BBT
blas.trsm(L22[i], C, side='R', transA='T')
L32.append(C)
if DEBUG > 0: print "# Computing L33"
L33 = []
for i in range(r):
A = spmatrix(V[i * n:(i + 1) * n], range(n),
range(n)) + BBT - L32[i] * L32[i].T
cholesky(A)
L33.append(A)
if DEBUG > 0: print "# Computing L42"
L42 = []
for i in range(r):
A = +L22[i].T
lapack.trtri(A, uplo='U')
L42.append(A)
if DEBUG > 0: print "# Computing L43"
L43 = []
for i in range(r):
A = id_n - L42[i] * L32[i].T
blas.trsm(L33[i], A, side='R', transA='T')
L43.append(A)
if DEBUG > 0: print "# Computing L44 and D4"
# The indices for the diagonal of a dense matrix
L44 = matrix(0, (n, n))
for i in range(r):
L44 = L44 + L43[i] * L43[i].T + L42[i] * L42[i].T
cholesky(L44)
# WARNING: y, z and t have been permuted (LD33 and LD44piv)
if DEBUG > 0: print "## PRE-COMPUTATION DONE ##"
# Checking the decomposition
if DEBUG > 1:
if DEBUG > 2:
mA = []
mB = []
mD = []
for i in range(3 * r + 1):
m = []
for j in range(3 * r + 1):
m.append(zero_n)
mA.append(m)
for i in range(r):
mA[i][i] = cholK
mA[i + r][i + r] = L22[i]
mA[i][i + r] = -B
mA[i][i + 2 * r] = B
mA[i + r][i + 2 * r] = L32[i]
mA[i + 2 * r][i + 2 * r] = L33[i]
mA[i + r][3 * r] = L42[i]
mA[i + 2 * r][3 * r] = L43[i]
mD.append(id_n)
mA[3 * r][3 * r] = L44
for i in range(2 * r):
mD.append(-id_n)
mD.append(id_n)
printing.options['width'] = 30
mA = sparse(mA)
mD = spdiag(mD)
print "LL^T =\n%s" % (mA * mD * mA.T),
print "P =\n%s" % P,
print g
print "W = %s" % W["d"].T,
print "U^2 = %s" % U.T,
print "V^2 = %s" % V.T,
print "### Pre-compute for check"
solve = chol2(W, P)
def f(x, y, z):
"""
On entry bx, bz are stored in x, z. On exit x, z contain the solution,
with z scaled: z./di is returned instead of z.
"""
# Maps to our variables x,y,z and t
if DEBUG > 0:
print "... Computing ..."
print "bx = %sbz = %s" % (x.T, z.T),
a = []
b = x[n * r:n * r + n]
c = []
d = []
for i in range(r):
a.append(x[i * n:(i + 1) * n])
c.append(z[i * n:(i + 1) * n])
d.append(z[(i + r) * n:(i + r + 1) * n])
if DEBUG:
# Now solves using cvxopt
xp = +x
zp = +z
solve(xp, y, zp)
# First phase
for i in range(r):
blas.trsm(cholK, a[i])
Bai = B * a[i]
c[i] = -Bai - c[i]
blas.trsm(L22[i], c[i])
d[i] = Bai - L32[i] * c[i] - d[i]
blas.trsm(L33[i], d[i])
b = b + L42[i] * c[i] + L43[i] * d[i]
blas.trsm(L44, b)
# Second phase
blas.trsm(L44, b, transA='T')
for i in range(r):
d[i] = d[i] - L43[i].T * b
blas.trsm(L33[i], d[i], transA='T')
c[i] = c[i] - L32[i].T * d[i] - L42[i].T * b
blas.trsm(L22[i], c[i], transA='T')
a[i] = a[i] + B.T * (c[i] - d[i])
blas.trsm(cholK, a[i], transA='T')
# Store in vectors and scale
x[n * r:n * r + n] = b
for i in range(r):
x[i * n:(i + 1) * n] = a[i]
z[i * n:(i + 1) * n] = c[i]
z[(i + r) * n:(i + r + 1) * n] = d[i]
z[:] = mul(Wd, z)
if DEBUG:
print "x = %s" % x.T,
print "z = %s" % z.T,
print "Delta(x) = %s" % (x - xp).T,
print "Delta(z) = %s" % (z - zp).T,
delta = blas.nrm2(x - xp) + blas.nrm2(z - zp)
if (delta > 1e-8):
print "--- DELTA TOO HIGH = %.3e ---" % delta
return f
pylab.plot(X[:,0], X[:,1], 'ko', X[:,0], X[:,1], '-k')
# Ellipsoid in the form { x | || L' * (x-c) ||_2 <= 1 }
L = +A
lapack.potrf(L)
c = +b
lapack.potrs(L, c)
# 1000 points on the unit circle
nopts = 1000
angles = matrix( [ a*2.0*pi/nopts for a in range(nopts) ], (1,nopts) )
circle = matrix(0.0, (2,nopts))
circle[0,:], circle[1,:] = cos(angles), sin(angles)
# ellipse = L^-T * circle + c
blas.trsm(L, circle, transA='T')
ellipse = circle + c[:, nopts*[0]]
ellipse2 = 0.5 * circle + c[:, nopts*[0]]
pylab.plot(ellipse[0,:].T, ellipse[1,:].T, 'k-')
pylab.fill(ellipse2[0,:].T, ellipse2[1,:].T, facecolor = '#F0F0F0')
pylab.title('Loewner-John ellipsoid (fig 8.3)')
pylab.axis('equal')
pylab.axis('off')
# Maximum volume enclosed ellipsoid
#
# minimize -log det B
# subject to ||B * gk||_2 + gk'*c <= hk, k=1,...,m
#
def trsm(L, B, alpha = 1.0, trans = 'N', nrhs = None, offsetB = 0, ldB = None):
r"""
Solves a triangular system of equations with multiple right-hand
sides. Computes
.. math::
B &:= \alpha L^{-1} B \text{ if trans is 'N'} \\
B &:= \alpha L^{-T} B \text{ if trans is 'T'}
where :math:`L` is a :py:class:`cspmatrix` factor.
:param L: :py:class:`cspmatrix` factor
:param B: matrix
:param alpha: float (default: 1.0)
:param trans: 'N' or 'T' (default: 'N')
:param nrhs: number of right-hand sides (default: number of columns in :math:`B`)
:param offsetB: integer (default: 0)
:param ldB: leading dimension of :math:`B` (default: number of rows in :math:`B`)
"""
assert isinstance(L, cspmatrix) and L.is_factor is True, "L must be a cspmatrix factor"
assert isinstance(B, matrix), "B must be a matrix"
if ldB is None: ldB = B.size[0]
if nrhs is None: nrhs = B.size[1]
assert trans in ['N', 'T']
n = L.symb.n
snpost = L.symb.snpost
snptr = L.symb.snptr
snode = L.symb.snode
chptr = L.symb.chptr
chidx = L.symb.chidx
relptr = L.symb.relptr
relidx = L.symb.relidx
blkptr = L.symb.blkptr
blkval = L.blkval
p = L.symb.p
if p is None: p = range(n)
stack = []
if trans is 'N':
for k in snpost:
nn = snptr[k+1]-snptr[k] # |Nk|
na = relptr[k+1]-relptr[k] # |Ak|
nj = na + nn
# extract block from rhs
Uk = matrix(0.0,(nj,nrhs))
for j in range(nrhs):
for i,ir in enumerate(snode[snptr[k]:snptr[k+1]]):
Uk[i,j] = alpha*B[offsetB + j*ldB + p[ir]]
# add contributions from children
for _ in range(chptr[k],chptr[k+1]):
Ui, i = stack.pop()
r = relidx[relptr[i]:relptr[i+1]]
Uk[r,:] += Ui
# if k is not a root node
if na > 0:
blas.gemm(blkval, Uk, Uk, alpha = -1.0, beta = 1.0, m = na, n = nrhs, k = nn,\
offsetA = blkptr[k]+nn, ldA = nj, offsetC = nn)
stack.append((Uk[nn:,:],k))
# scale and copy block to rhs
blas.trsm(blkval, Uk, m = nn, n = nrhs, offsetA = blkptr[k], ldA = nj)
for j in range(nrhs):
for i,ir in enumerate(snode[snptr[k]:snptr[k+1]]):
B[offsetB + j*ldB + p[ir]] = Uk[i,j]
else: # trans is 'T'
for k in reversed(list(snpost)):
nn = snptr[k+1]-snptr[k] # |Nk|
na = relptr[k+1]-relptr[k] # |Ak|
nj = na + nn
# extract block from rhs and scale
Uk = matrix(0.0,(nj,nrhs))
for j in range(nrhs):
for i,ir in enumerate(snode[snptr[k]:snptr[k+1]]):
Uk[i,j] = alpha*B[offsetB + j*ldB + p[ir]]
blas.trsm(blkval, Uk, transA = 'T', m = nn, n = nrhs, offsetA = blkptr[k], ldA = nj)
# if k is not a root node
if na > 0:
Uk[nn:,:] = stack.pop()
blas.gemm(blkval, Uk, Uk, alpha = -1.0, beta = 1.0, m = nn, n = nrhs, k = na,\
transA = 'T', offsetA = blkptr[k]+nn, ldA = nj, offsetB = nn)
# stack contributions for children
for ii in range(chptr[k],chptr[k+1]):
i = chidx[ii]
stack.append(Uk[relidx[relptr[i]:relptr[i+1]],:])
# copy block to rhs
for j in range(nrhs):
for i,ir in enumerate(snode[snptr[k]:snptr[k+1]]):
B[offsetB + j*ldB + p[ir]] = Uk[i,j]
return