def Fkkt(W):
# Returns a function f(x, y, z) that solves
#
# [ 0 0 P' -P' ] [ x[:n] ] [ bx[:n] ]
# [ 0 0 -I -I ] [ x[n:] ] [ bx[n:] ]
# [ P -I -D1^{-1} 0 ] [ z[:m] ] = [ bz[:m] ]
# [-P -I 0 -D2^{-1} ] [ z[m:] ] [ bz[m:] ]
#
# where D1 = diag(di[:m])^2, D2 = diag(di[m:])^2 and di = W['di'].
#
# On entry bx, bz are stored in x, z.
# On exit x, z contain the solution, with z scaled (di .* z is
# returned instead of z).
# Factor A = 4*P'*D*P where D = d1.*d2 ./(d1+d2) and
# d1 = d[:m].^2, d2 = d[m:].^2.
di = W['di']
d1, d2 = di[:m]**2, di[m:]**2
D = div(mul(d1, d2), d1 + d2)
Ds = spdiag(2 * sqrt(D))
base.gemm(Ds, P, Ps)
blas.syrk(Ps, A, trans='T')
lapack.potrf(A)
def f(x, y, z):
# Solve for x[:n]:
#
# A*x[:n] = bx[:n] + P' * ( ((D1-D2)*(D1+D2)^{-1})*bx[n:]
# + (2*D1*D2*(D1+D2)^{-1}) * (bz[:m] - bz[m:]) ).
blas.copy((mul(div(d1 - d2, d1 + d2), x[n:]) +
mul(2 * D, z[:m] - z[m:])), u)
blas.gemv(P, u, x, beta=1.0, trans='T')
lapack.potrs(A, x)
# x[n:] := (D1+D2)^{-1} * (bx[n:] - D1*bz[:m] - D2*bz[m:]
# + (D1-D2)*P*x[:n])
base.gemv(P, x, u)
x[n:] = div(
x[n:] - mul(d1, z[:m]) - mul(d2, z[m:]) + mul(d1 - d2, u),
d1 + d2)
# z[:m] := d1[:m] .* ( P*x[:n] - x[n:] - bz[:m])
# z[m:] := d2[m:] .* (-P*x[:n] - x[n:] - bz[m:])
z[:m] = mul(di[:m], u - x[n:] - z[:m])
z[m:] = mul(di[m:], -u - x[n:] - z[m:])
return f
def Fkkt(W):
# Returns a function f(x, y, z) that solves
#
# [ 0 0 P' -P' ] [ x[:n] ] [ bx[:n] ]
# [ 0 0 -I -I ] [ x[n:] ] [ bx[n:] ]
# [ P -I -D1^{-1} 0 ] [ z[:m] ] = [ bz[:m] ]
# [-P -I 0 -D2^{-1} ] [ z[m:] ] [ bz[m:] ]
#
# where D1 = diag(di[:m])^2, D2 = diag(di[m:])^2 and di = W['di'].
#
# On entry bx, bz are stored in x, z.
# On exit x, z contain the solution, with z scaled (di .* z is
# returned instead of z).
# Factor A = 4*P'*D*P where D = d1.*d2 ./(d1+d2) and
# d1 = d[:m].^2, d2 = d[m:].^2.
di = W['di']
d1, d2 = di[:m]**2, di[m:]**2
D = div( mul(d1,d2), d1+d2 )
Ds = spdiag(2 * sqrt(D))
base.gemm(Ds, P, Ps)
blas.syrk(Ps, A, trans = 'T')
lapack.potrf(A)
def f(x, y, z):
# Solve for x[:n]:
#
# A*x[:n] = bx[:n] + P' * ( ((D1-D2)*(D1+D2)^{-1})*bx[n:]
# + (2*D1*D2*(D1+D2)^{-1}) * (bz[:m] - bz[m:]) ).
blas.copy(( mul( div(d1-d2, d1+d2), x[n:]) +
mul( 2*D, z[:m]-z[m:] ) ), u)
blas.gemv(P, u, x, beta = 1.0, trans = 'T')
lapack.potrs(A, x)
# x[n:] := (D1+D2)^{-1} * (bx[n:] - D1*bz[:m] - D2*bz[m:]
# + (D1-D2)*P*x[:n])
base.gemv(P, x, u)
x[n:] = div( x[n:] - mul(d1, z[:m]) - mul(d2, z[m:]) +
mul(d1-d2, u), d1+d2 )
# z[:m] := d1[:m] .* ( P*x[:n] - x[n:] - bz[:m])
# z[m:] := d2[m:] .* (-P*x[:n] - x[n:] - bz[m:])
z[:m] = mul(di[:m], u-x[n:]-z[:m])
z[m:] = mul(di[m:], -u-x[n:]-z[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
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 sysid(y, u, vsig, svth=None):
"""
System identification using the subspace method and nuclear norm
optimization. Estimate a linear time-invariant state-space model
given inputs and outputs. The algorithm is described in [1].
INPUT
y 'd' matrix of size (p, N). y are the measured outputs, p is
the number of outputs, and N is the number of data points
measured.
u 'd' matrix of size (m, N). u are the inputs, m is the number
of inputs, and N is the number of data points.
vsig a weighting parameter in the nuclear norm optimization, its
value is approximately the 1-sigma output noise level
svth an optional parameter, if specified, the model order is
determined as the number of singular values greater than svth
times the maximum singular value. The default value is 1E-3
OUTPUT
sol a dictionary with the following words
-- 'A', 'B', 'C', 'D' are the state-space matrices
-- 'svN', the original singular values of the Hankel matrix
-- 'sv', the optimized singular values of the Hankel matrix
-- 'x0', the initial state x(0)
-- 'n', the model order
[1] Zhang Liu and Lieven Vandenberghe. "Interior-point method for
nuclear norm approximation with application to system
identification."
"""
m, N, p = u.size[0], u.size[1], y.size[0]
if y.size[1] != N:
raise ValueError, "y and u must have the same length"
# Y = G*X + H*U + V, Y has size a x b, U has size c x b, Un has b x d
r = min(int(30 / p), int((N + 1.0) / (p + m + 1) + 1.0))
a = r * p
c = r * m
b = N - r + 1
d = b - c
# construct Hankel matrix Y
Y = Hankel(y, r, b, p=p, q=1)
# construct Hankel matrix U
U = Hankel(u, r, b, p=m, q=1)
# compute Un = null(U) and YUn = Y*Un
Vt = matrix(0.0, (b, b))
Stemp = matrix(0.0, (c, 1))
Un = matrix(0.0, (b, d))
YUn = matrix(0.0, (a, d))
lapack.gesvd(U, Stemp, jobvt='A', Vt=Vt)
Un[:, :] = Vt.T[:, c:]
blas.gemm(Y, Un, YUn)
# compute original singular values
svN = matrix(0.0, (min(a, d), 1))
lapack.gesvd(YUn, svN)
# variable, [y(1);...;y(N)]
# form the coefficient matrices for the nuclear norm optimization
# minimize | Yh * Un |_* + alpha * | y - yh |_F
AA = Hankel_basis(r, b, p=p, q=1)
A = matrix(0.0, (a * d, p * N))
temp = spmatrix([], [], [], (a, b), 'd')
temp2 = matrix(0.0, (a, d))
for ii in xrange(p * N):
temp[:] = AA[:, ii]
base.gemm(temp, Un, temp2)
A[:, ii] = temp2[:]
B = matrix(0.0, (a, d))
# flip the matrix if columns is more than rows
if a < d:
Itrans = [i + j * a for i in xrange(a) for j in xrange(d)]
B[:] = B[Itrans]
B.size = (d, a)
for ii in xrange(p * N):
A[:, ii] = A[Itrans, ii]
# regularized term
x0 = y[:]
Qd = matrix(2.0 * svN[0] / p / N / (vsig**2), (p * N, 1))
# solve the nuclear norm optimization
sol = nrmapp(A, B, C=base.spdiag(Qd), d=-base.mul(x0, Qd))
status = sol['status']
x = sol['x']
# construct YhUn and take the svd
YhUn = matrix(B)
blas.gemv(A, x, YhUn, beta=1.0)
if a < d:
YhUn = YhUn.T
Uh = matrix(0.0, (a, d))
sv = matrix(0.0, (d, 1))
lapack.gesvd(YhUn, sv, jobu='S', U=Uh)
# determine model order
if svth is None:
svth = 1E-3
svthn = sv[0] * svth
n = 1
while sv[n] >= svthn and n < 10:
n = n + 1
# estimate A, C
Uhn = Uh[:, :n]
for ii in xrange(n):
blas.scal(sv[ii], Uhn, n=a, offset=ii * a)
syseC = Uhn[:p, :]
Als = Uhn[:-p, :]
Bls = Uhn[p:, :]
lapack.gels(Als, Bls)
syseA = Bls[:n, :]
Als[:, :] = Uhn[:-p, :]
Bls[:, :] = Uhn[p:, :]
blas.gemm(Als, syseA, Bls, beta=-1.0)
Aerr = blas.nrm2(Bls)
# stabilize A
Sc = matrix(0.0, (n, n), 'z')
w = matrix(0.0, (n, 1), 'z')
Vs = matrix(0.0, (n, n), 'z')
def F(w):
return (abs(w) < 1.0)
Sc[:, :] = syseA
ns = lapack.gees(Sc, w, Vs, select=F)
while ns < n:
#print "stabilize matrix A"
w[ns:] = w[ns:]**-1
Sc[::n + 1] = w
Sc = Vs * Sc * Vs.H
syseA[:, :] = Sc.real()
Sc[:, :] = syseA
ns = lapack.gees(Sc, w, Vs, select=F)
# estimate B,D,x0 stored in vector [x0; vec(D); vec(B)]
F1 = matrix(0.0, (p * N, n))
F1[:p, :] = syseC
for ii in xrange(1, N):
F1[ii * p:(ii + 1) * p, :] = F1[(ii - 1) * p:ii * p, :] * syseA
F2 = matrix(0.0, (p * N, p * m))
ut = u.T
for ii in xrange(p):
F2[ii::p, ii::p] = ut
F3 = matrix(0.0, (p * N, n * m))
F3t = matrix(0.0, (p * (N - 1), n * m))
for ii in xrange(1, N):
for jj in xrange(p):
for kk in xrange(n):
F3t[jj:jj + (N - ii) * p:p,
kk::n] = ut[:N - ii, :] * F1[(ii - 1) * p + jj, kk]
F3[ii * p:, :] = F3[ii * p:, :] + F3t[:(N - ii) * p, :]
F = matrix([[F1], [F2], [F3]])
yls = y[:]
Sls = matrix(0.0, (F.size[1], 1))
Uls = matrix(0.0, (F.size[0], F.size[1]))
Vtls = matrix(0.0, (F.size[1], F.size[1]))
lapack.gesvd(F, Sls, jobu='S', jobvt='S', U=Uls, Vt=Vtls)
Frank = len([ii for ii in xrange(Sls.size[0]) if Sls[ii] >= 1E-6])
#print 'Rank deficiency = ', F.size[1] - Frank
xx = matrix(0.0, (F.size[1], 1))
xx[:Frank] = Uls.T[:Frank, :] * yls
xx[:Frank] = base.mul(xx[:Frank], Sls[:Frank]**-1)
xx[:] = Vtls.T[:, :Frank] * xx[:Frank]
blas.gemv(F, xx, yls, beta=-1.0)
xxerr = blas.nrm2(yls)
x0 = xx[:n]
syseD = xx[n:n + p * m]
syseD.size = (p, m)
syseB = xx[n + p * m:]
syseB.size = (n, m)
return {'A': syseA, 'B': syseB, 'C': syseC, 'D': syseD, 'svN': svN, 'sv': \
sv, 'x0': x0, 'n': n, 'Aerr': Aerr, 'xxerr': xxerr}
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
def sysid(y, u, vsig, svth = None):
"""
System identification using the subspace method and nuclear norm
optimization. Estimate a linear time-invariant state-space model
given inputs and outputs. The algorithm is described in [1].
INPUT
y 'd' matrix of size (p, N). y are the measured outputs, p is
the number of outputs, and N is the number of data points
measured.
u 'd' matrix of size (m, N). u are the inputs, m is the number
of inputs, and N is the number of data points.
vsig a weighting parameter in the nuclear norm optimization, its
value is approximately the 1-sigma output noise level
svth an optional parameter, if specified, the model order is
determined as the number of singular values greater than svth
times the maximum singular value. The default value is 1E-3
OUTPUT
sol a dictionary with the following words
-- 'A', 'B', 'C', 'D' are the state-space matrices
-- 'svN', the original singular values of the Hankel matrix
-- 'sv', the optimized singular values of the Hankel matrix
-- 'x0', the initial state x(0)
-- 'n', the model order
[1] Zhang Liu and Lieven Vandenberghe. "Interior-point method for
nuclear norm approximation with application to system
identification."
"""
m, N, p = u.size[0], u.size[1], y.size[0]
if y.size[1] != N:
raise ValueError, "y and u must have the same length"
# Y = G*X + H*U + V, Y has size a x b, U has size c x b, Un has b x d
r = min(int(30/p),int((N+1.0)/(p+m+1)+1.0))
a = r*p
c = r*m
b = N-r+1
d = b-c
# construct Hankel matrix Y
Y = Hankel(y,r,b,p=p,q=1)
# construct Hankel matrix U
U = Hankel(u,r,b,p=m,q=1)
# compute Un = null(U) and YUn = Y*Un
Vt = matrix(0.0,(b,b))
Stemp = matrix(0.0,(c,1))
Un = matrix(0.0,(b,d))
YUn = matrix(0.0,(a,d))
lapack.gesvd(U,Stemp,jobvt='A',Vt=Vt)
Un[:,:] = Vt.T[:,c:]
blas.gemm(Y,Un,YUn)
# compute original singular values
svN = matrix(0.0,(min(a,d),1))
lapack.gesvd(YUn,svN)
# variable, [y(1);...;y(N)]
# form the coefficient matrices for the nuclear norm optimization
# minimize | Yh * Un |_* + alpha * | y - yh |_F
AA = Hankel_basis(r,b,p=p,q=1)
A = matrix(0.0,(a*d,p*N))
temp = spmatrix([],[],[],(a,b),'d')
temp2 = matrix(0.0,(a,d))
for ii in xrange(p*N):
temp[:] = AA[:,ii]
base.gemm(temp,Un,temp2)
A[:,ii] = temp2[:]
B = matrix(0.0,(a,d))
# flip the matrix if columns is more than rows
if a < d:
Itrans = [i+j*a for i in xrange(a) for j in xrange(d)]
B[:] = B[Itrans]
B.size = (d,a)
for ii in xrange(p*N):
A[:,ii] = A[Itrans,ii]
# regularized term
x0 = y[:]
Qd = matrix(2.0*svN[0]/p/N/(vsig**2),(p*N,1))
# solve the nuclear norm optimization
sol = nrmapp(A, B, C = base.spdiag(Qd), d = -base.mul(x0, Qd))
status = sol['status']
x = sol['x']
# construct YhUn and take the svd
YhUn = matrix(B)
blas.gemv(A,x,YhUn,beta=1.0)
if a < d:
YhUn = YhUn.T
Uh = matrix(0.0,(a,d))
sv = matrix(0.0,(d,1))
lapack.gesvd(YhUn,sv,jobu='S',U=Uh)
# determine model order
if svth is None:
svth = 1E-3
svthn = sv[0]*svth
n=1
while sv[n] >= svthn and n < 10:
n=n+1
# estimate A, C
Uhn = Uh[:,:n]
for ii in xrange(n):
blas.scal(sv[ii],Uhn,n=a,offset=ii*a)
syseC = Uhn[:p,:]
Als = Uhn[:-p,:]
Bls = Uhn[p:,:]
lapack.gels(Als,Bls)
syseA = Bls[:n,:]
Als[:,:] = Uhn[:-p,:]
Bls[:,:] = Uhn[p:,:]
blas.gemm(Als,syseA,Bls,beta=-1.0)
Aerr = blas.nrm2(Bls)
# stabilize A
Sc = matrix(0.0,(n,n),'z')
w = matrix(0.0, (n,1), 'z')
Vs = matrix(0.0, (n,n), 'z')
def F(w):
return (abs(w) < 1.0)
Sc[:,:] = syseA
ns = lapack.gees(Sc, w, Vs, select = F)
while ns < n:
#print "stabilize matrix A"
w[ns:] = w[ns:]**-1
Sc[::n+1] = w
Sc = Vs*Sc*Vs.H
syseA[:,:] = Sc.real()
Sc[:,:] = syseA
ns = lapack.gees(Sc, w, Vs, select = F)
# estimate B,D,x0 stored in vector [x0; vec(D); vec(B)]
F1 = matrix(0.0,(p*N,n))
F1[:p,:] = syseC
for ii in xrange(1,N):
F1[ii*p:(ii+1)*p,:] = F1[(ii-1)*p:ii*p,:]*syseA
F2 = matrix(0.0,(p*N,p*m))
ut = u.T
for ii in xrange(p):
F2[ii::p,ii::p] = ut
F3 = matrix(0.0,(p*N,n*m))
F3t = matrix(0.0,(p*(N-1),n*m))
for ii in xrange(1,N):
for jj in xrange(p):
for kk in xrange(n):
F3t[jj:jj+(N-ii)*p:p,kk::n] = ut[:N-ii,:]*F1[(ii-1)*p+jj,kk]
F3[ii*p:,:] = F3[ii*p:,:] + F3t[:(N-ii)*p,:]
F = matrix([[F1],[F2],[F3]])
yls = y[:]
Sls = matrix(0.0,(F.size[1],1))
Uls = matrix(0.0,(F.size[0],F.size[1]))
Vtls = matrix(0.0,(F.size[1],F.size[1]))
lapack.gesvd(F, Sls, jobu='S', jobvt='S', U=Uls, Vt=Vtls)
Frank=len([ii for ii in xrange(Sls.size[0]) if Sls[ii] >= 1E-6])
#print 'Rank deficiency = ', F.size[1] - Frank
xx = matrix(0.0,(F.size[1],1))
xx[:Frank] = Uls.T[:Frank,:] * yls
xx[:Frank] = base.mul(xx[:Frank],Sls[:Frank]**-1)
xx[:] = Vtls.T[:,:Frank]*xx[:Frank]
blas.gemv(F,xx,yls,beta=-1.0)
xxerr = blas.nrm2(yls)
x0 = xx[:n]
syseD = xx[n:n+p*m]
syseD.size = (p,m)
syseB = xx[n+p*m:]
syseB.size = (n,m)
return {'A': syseA, 'B': syseB, 'C': syseC, 'D': syseD, 'svN': svN, 'sv': \
sv, 'x0': x0, 'n': n, 'Aerr': Aerr, 'xxerr': xxerr}
b = N-r+1
d = b-c
U = sysid.Hankel(u,r,b,p=m,q=1)
Vt = matrix(0.0,(b,b))
Stemp = matrix(0.0,(c,1))
Un = matrix(0.0,(b,d))
lapack.gesvd(U,Stemp,jobvt='A',Vt=Vt)
Un[:,:] = Vt.T[:,c:]
AA = sysid.Hankel_basis(r,b,p=p,q=1)
A = matrix(0.0,(a*d,p*N))
temp = spmatrix([],[],[],(a,b),'d')
temp2 = matrix(0.0,(a,d))
for ii in xrange(p*N):
temp[:] = AA[:,ii]
base.gemm(temp,Un,temp2)
A[:,ii] = temp2[:]
B = matrix(0.0,(a,d))
# flip the matrix if columns is more than rows
if a < d:
Itrans = [i+j*a for i in xrange(a) for j in xrange(d)]
B[:] = B[Itrans]
B.size = (d,a)
for ii in xrange(p*N):
A[:,ii] = A[Itrans,ii]
n = p*N
if a < d:
p, q = d, a
else:
b = N - r + 1
d = b - c
U = sysid.Hankel(u, r, b, p=m, q=1)
Vt = matrix(0.0, (b, b))
Stemp = matrix(0.0, (c, 1))
Un = matrix(0.0, (b, d))
lapack.gesvd(U, Stemp, jobvt='A', Vt=Vt)
Un[:, :] = Vt.T[:, c:]
AA = sysid.Hankel_basis(r, b, p=p, q=1)
A = matrix(0.0, (a * d, p * N))
temp = spmatrix([], [], [], (a, b), 'd')
temp2 = matrix(0.0, (a, d))
for ii in xrange(p * N):
temp[:] = AA[:, ii]
base.gemm(temp, Un, temp2)
A[:, ii] = temp2[:]
B = matrix(0.0, (a, d))
# flip the matrix if columns is more than rows
if a < d:
Itrans = [i + j * a for i in xrange(a) for j in xrange(d)]
B[:] = B[Itrans]
B.size = (d, a)
for ii in xrange(p * N):
A[:, ii] = A[Itrans, ii]
n = p * N
if a < d:
p, q = d, a
else:
