def __cngrnc(r, x, alpha = 1.0, trans = 'N', offsetx = 0):
"""
In-place congruence transformation
x := alpha * r * x * r' if trans = 'N'.
x := alpha * r' * x * r if trans = 'T'.
r is a square n x n-matrix.
x is a square n x n-matrix or n^2-vector.
"""
n = r.size[0]
# Scale diagonal of x by 1/2.
blas.scal(0.5, x, inc = n+1, offset = offsetx)
# a := r*tril(x) if trans is 'N', a := tril(x)*r if trans is 'T'.
a = +r
if trans == 'N':
blas.trmm(x, a, side = 'R', m = n, n = n, ldA = n, ldB = n,
offsetA = offsetx)
else:
blas.trmm(x, a, side = 'L', m = n, n = n, ldA = n, ldB = n,
offsetA = offsetx)
# x := alpha * (a * r' + r * a')
# = alpha * r * (tril(x) + tril(x)') * r' if trans is 'N'.
# x := alpha * (a' * r + r' * a)
# = alpha * r' * (tril(x)' + tril(x)) * r if trans = 'T'.
blas.syr2k(r, a, x, trans = trans, alpha = alpha, n = n, k = n,
ldB = n, ldC = n, offsetC = offsetx)
def cngrnc(r, x, alpha = 1.0):
"""
Congruence transformation
x := alpha * r'*x*r.
r and x are square matrices.
"""
# Scale diagonal of x by 1/2.
x[::n+1] *= 0.5
# a := tril(x)*r
a = +r
tx = matrix(x, (n,n))
blas.trmm(tx, a, side='L')
# x := alpha*(a*r' + r*a')
blas.syr2k(r, a, tx, trans = 'T', alpha = alpha)
x[:] = tx[:]
def cngrnc(r, x, alpha = 1.0):
"""
Congruence transformation
x := alpha * r'*x*r.
r and x are square matrices.
"""
# Scale diagonal of x by 1/2.
x[::n+1] *= 0.5
# a := tril(x)*r
a = +r
tx = matrix(x, (n,n))
blas.trmm(tx, a, side = 'L')
# x := alpha*(a*r' + r*a')
blas.syr2k(r, a, tx, trans = 'T', alpha = alpha)
x[:] = tx[:]
def __cngrnc(r, x, alpha=1.0, trans='N', offsetx=0):
"""
In-place congruence transformation
x := alpha * r * x * r' if trans = 'N'.
x := alpha * r' * x * r if trans = 'T'.
r is a square n x n-matrix.
x is a square n x n-matrix or n^2-vector.
"""
n = r.size[0]
# Scale diagonal of x by 1/2.
blas.scal(0.5, x, inc=n + 1, offset=offsetx)
# a := r*tril(x) if trans is 'N', a := tril(x)*r if trans is 'T'.
a = +r
if trans == 'N':
blas.trmm(x, a, side='R', m=n, n=n, ldA=n, ldB=n, offsetA=offsetx)
else:
blas.trmm(x, a, side='L', m=n, n=n, ldA=n, ldB=n, offsetA=offsetx)
# x := alpha * (a * r' + r * a')
# = alpha * r * (tril(x) + tril(x)') * r' if trans is 'N'.
# x := alpha * (a' * r + r' * a)
# = alpha * r' * (tril(x)' + tril(x)) * r if trans = 'T'.
blas.syr2k(r,
a,
x,
trans=trans,
alpha=alpha,
n=n,
k=n,
ldB=n,
ldC=n,
offsetC=offsetx)
def cngrnc(r, x, n, xstart=0, alpha=1.0):
"""
Congruence transformation
x := alpha * r'*x*r.
r is a square matrix, and x is a symmetric matrix.
"""
# return alpha*r.T*matrix(x, (n,n))*r
# Scale diagonal of x by 1/2.
nsqr = n * n
xend = xstart + nsqr
x[xstart:xend:n + 1] *= 0.5
# a := tril(x)*r
a = +r
tx = matrix(x[xstart:xend], (n, n))
blas.trmm(tx, a, side='L')
# x := alpha*(a*r' + r*a')
blas.syr2k(r, a, tx, trans='T', alpha=alpha)
x[xstart:xend] = tx[:]
def fsolve(x, y, z):
"""
Solves the system of equations
[ 0 G'*W^{-1} ] [ ux ] = [ bx ]
[ G -W' ] [ uz ] [ bz ]
"""
# Compute bx := bx + G'*W^{-1}*W^{-T}*bz
v = matrix(0., (N, 1))
for i in range(N):
blas.symv(z, rr, v, ldA=N, offsetA=1, n=N, offsetx=N * i)
x[i] += blas.dot(rr, v, n=N, offsetx=N * i)
blas.symv(z, q, v, ldA=N, offsetA=1, n=N)
x[N] += blas.dot(q, v) + z[0] * W['di'][0]**2
# Solve G'*W^{-1}*W^{-T}*G*ux = bx
lapack.potrs(H, x)
# Compute bz := -W^{-T}*(bz-G*ux)
# z -= G*x
z[1::N + 1] -= x[:-1]
z -= x[-1]
# Apply scaling
z[0] *= -W['di'][0]
blas.scal(0.5, z, n=N, offset=1, inc=N + 1)
tmp = +r
blas.trmm(z, tmp, ldA=N, offsetA=1, n=N, m=N)
blas.syr2k(r,
tmp,
z,
trans='T',
offsetC=1,
ldC=N,
n=N,
k=N,
alpha=-1.0)
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)
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)
def scale(x, W, trans='N', inverse='N'):
"""
Applies Nesterov-Todd scaling or its inverse.
Computes
x := W*x (trans is 'N', inverse = 'N')
x := W^T*x (trans is 'T', inverse = 'N')
x := W^{-1}*x (trans is 'N', inverse = 'I')
x := W^{-T}*x (trans is 'T', inverse = 'I').
x is a dense 'd' matrix.
W is a dictionary with entries:
- W['dnl']: positive vector
- W['dnli']: componentwise inverse of W['dnl']
- W['d']: positive vector
- W['di']: componentwise inverse of W['d']
- W['v']: lists of 2nd order cone vectors with unit hyperbolic norms
- W['beta']: list of positive numbers
- W['r']: list of square matrices
- W['rti']: list of square matrices. rti[k] is the inverse transpose
of r[k].
The 'dnl' and 'dnli' entries are optional, and only present when the
function is called from the nonlinear solver.
"""
from cvxopt import blas
ind = 0
# Scaling for nonlinear component xk is xk := dnl .* xk; inverse
# scaling is xk ./ dnl = dnli .* xk, where dnl = W['dnl'],
# dnli = W['dnli'].
if 'dnl' in W:
if inverse == 'N':
w = W['dnl']
else:
w = W['dnli']
for k in range(x.size[1]):
blas.tbmv(w, x, n=w.size[0], k=0, ldA=1, offsetx=k * x.size[0])
ind += w.size[0]
# Scaling for linear 'l' component xk is xk := d .* xk; inverse
# scaling is xk ./ d = di .* xk, where d = W['d'], di = W['di'].
if inverse == 'N':
w = W['d']
else:
w = W['di']
for k in range(x.size[1]):
blas.tbmv(w, x, n=w.size[0], k=0, ldA=1, offsetx=k * x.size[0] + ind)
ind += w.size[0]
# Scaling for 'q' component is
#
# xk := beta * (2*v*v' - J) * xk
# = beta * (2*v*(xk'*v)' - J*xk)
#
# where beta = W['beta'][k], v = W['v'][k], J = [1, 0; 0, -I].
#
# Inverse scaling is
#
# xk := 1/beta * (2*J*v*v'*J - J) * xk
# = 1/beta * (-J) * (2*v*((-J*xk)'*v)' + xk).
w = matrix(0.0, (x.size[1], 1))
for k in range(len(W['v'])):
v = W['v'][k]
m = v.size[0]
if inverse == 'I':
blas.scal(-1.0, x, offset=ind, inc=x.size[0])
blas.gemv(x, v, w, trans='T', m=m, n=x.size[1], offsetA=ind, ldA=x.size[0])
blas.scal(-1.0, x, offset=ind, inc=x.size[0])
blas.ger(v, w, x, alpha=2.0, m=m, n=x.size[1], ldA=x.size[0], offsetA=ind)
if inverse == 'I':
blas.scal(-1.0, x, offset=ind, inc=x.size[0])
a = 1.0 / W['beta'][k]
else:
a = W['beta'][k]
for i in range(x.size[1]):
blas.scal(a, x, n=m, offset=ind + i * x.size[0])
ind += m
# Scaling for 's' component xk is
#
# xk := vec( r' * mat(xk) * r ) if trans = 'N'
# xk := vec( r * mat(xk) * r' ) if trans = 'T'.
#
# r is kth element of W['r'].
#
# Inverse scaling is
#
# xk := vec( rti * mat(xk) * rti' ) if trans = 'N'
# xk := vec( rti' * mat(xk) * rti ) if trans = 'T'.
#
# rti is kth element of W['rti'].
maxn = max([0] + [r.size[0] for r in W['r']])
a = matrix(0.0, (maxn, maxn))
for k in range(len(W['r'])):
if inverse == 'N':
r = W['r'][k]
if trans == 'N':
t = 'T'
else:
t = 'N'
else:
r = W['rti'][k]
t = trans
n = r.size[0]
for i in range(x.size[1]):
# scale diagonal of xk by 0.5
blas.scal(0.5, x, offset=ind + i * x.size[0], inc=n + 1, n=n)
# a = r*tril(x) (t is 'N') or a = tril(x)*r (t is 'T')
blas.copy(r, a)
if t == 'N':
blas.trmm(x, a, side='R', m=n, n=n, ldA=n, ldB=n,
offsetA=ind + i * x.size[0])
else:
blas.trmm(x, a, side='L', m=n, n=n, ldA=n, ldB=n,
offsetA=ind + i * x.size[0])
# x := (r*a' + a*r') if t is 'N'
# x := (r'*a + a'*r) if t is 'T'
blas.syr2k(r, a, x, trans=t, n=n, k=n, ldB=n, ldC=n,
offsetC=ind + i * x.size[0])
ind += n ** 2
