def P(x, y, alpha=1.0, beta=0.0):
"""
x and y are N x m matrices.
y = alpha * Q * x + beta * y.
"""
blas.symm(Q, x, y, alpha=alpha, beta=beta)
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 __M2T(L, U, inv=False):
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 = []
alpha = 1.0
if inv: alpha = -1.0
for Ut in U:
for k in reversed(list(snpost)):
nn = snptr[k + 1] - snptr[k] # |Nk|
na = relptr[k + 1] - relptr[k] # |Ak|
nj = na + nn
# allocate F and copy Ut_{Jk,Nk} to leading columns of F
F = matrix(0.0, (nj, nj))
lapack.lacpy(Ut.blkval,
F,
offsetA=blkptr[k],
ldA=nj,
m=nj,
n=nn,
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,
offsetB=nn * (nj + 1),
m=na,
n=na,
uplo='L')
## compute T_{Jk,Nk} (stored in leading columns of F)
if inv:
# if supernode k has any children:
for ii in range(chptr[k], chptr[k + 1]):
stack.append(
frontal_get_update(F, relidx, relptr, chidx[ii]))
# if supernode k is not a root node:
if na > 0:
# F_{Nk,Nk} := F_{Nk,Nk} - alpha*F_{Ak,Nk}'*L_{Ak,Nk}
blas.gemm(F, L.blkval, F, beta = 1.0, alpha = -alpha, m = nn, n = nn, k = na,\
transA = 'T', ldA = nj, ldB = nj, ldC = nj,\
offsetA = nn, offsetB = blkptr[k]+nn, offsetC = 0)
# F_{Ak,Nk} := F_{Ak,Nk} - alpha*F_{Ak,Ak}*L_{Ak,Nk}
blas.symm(F, L.blkval, F, side = 'L', beta = 1.0, alpha = -alpha,\
m = na, n = nn, ldA = nj, ldB = nj, ldC = nj,\
offsetA = (nj+1)*nn, offsetB = blkptr[k]+nn, offsetC = nn)
# F_{Nk,Nk} := F_{Nk,Nk} - alpha*L_{Ak,Nk}'*F_{Ak,Nk}
blas.gemm(L.blkval, F, F, beta = 1.0, alpha = -alpha, m = nn, n = nn, k = na,\
transA = 'T', ldA = nj, ldB = nj, ldC = nj,\
offsetA = blkptr[k]+nn, offsetB = nn, offsetC = 0)
# copy the leading Nk columns of frontal matrix to Ut
lapack.lacpy(F,
Ut.blkval,
offsetB=blkptr[k],
ldB=nj,
m=nj,
n=nn,
uplo='L')
if not inv:
# if supernode k has any children:
for ii in range(chptr[k], chptr[k + 1]):
stack.append(
frontal_get_update(F, relidx, relptr, chidx[ii]))
return
def __Y2K(L, U, inv=False):
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 = []
alpha = 1.0
if inv: alpha = -1.0
for Ut in U:
for k in snpost:
nn = snptr[k + 1] - snptr[k] # |Nk|
na = relptr[k + 1] - relptr[k] # |Ak|
nj = na + nn
# allocate F and copy Ut_{Jk,Nk} to leading columns of F
F = matrix(0.0, (nj, nj))
lapack.lacpy(Ut.blkval,
F,
offsetA=blkptr[k],
ldA=nj,
m=nj,
n=nn,
uplo='L')
if not inv:
# 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])
if na > 0:
# F_{Ak,Ak} := F_{Ak,Ak} - alpha*L_{Ak,Nk}*F_{Ak,Nk}'
blas.gemm(L.blkval, F, F, beta = 1.0, alpha = -alpha, m = na, n = na, k = nn,\
ldA = nj, ldB = nj, ldC = nj, transB = 'T',\
offsetA = blkptr[k]+nn, offsetB = nn, offsetC = nn*(nj+1))
# F_{Ak,Nk} := F_{Ak,Nk} - alpha*L_{Ak,Nk}*F_{Nk,Nk}
blas.symm(F, L.blkval, F, side = 'R', beta = 1.0, alpha = -alpha,\
m = na, n = nn, ldA = nj, ldB = nj, ldC = nj,\
offsetA = 0, offsetB = blkptr[k]+nn, offsetC = nn)
# F_{Ak,Ak} := F_{Ak,Ak} - alpha*F_{Ak,Nk}*L_{Ak,Nk}'
blas.gemm(F, L.blkval, F, beta = 1.0, alpha = -alpha, m = na, n = na, k = nn,\
ldA = nj, ldB = nj, ldC = nj, transB = 'T',\
offsetA = nn, offsetB = blkptr[k]+nn, offsetC = nn*(nj+1))
if inv:
# 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])
if na > 0:
# add Uk' to stack
Uk = matrix(0.0, (na, na))
lapack.lacpy(F,
Uk,
m=na,
n=na,
offsetA=nn * (nj + 1),
ldA=nj,
uplo='L')
stack.append(Uk)
# copy the leading Nk columns of frontal matrix to blkval
lapack.lacpy(F,
Ut.blkval,
uplo='L',
offsetB=blkptr[k],
m=nj,
n=nn,
ldB=nj)
return
def projected_inverse(L):
"""
Supernodal multifrontal projected inverse. The routine computes the projected inverse
.. math::
Y = P(L^{-T}L^{-1})
where :math:`L` is a Cholesky factor. On exit, the argument :math:`L` contains the
projected inverse :math:`Y`.
:param L: :py:class:`cspmatrix` (factor)
"""
assert isinstance(L, cspmatrix) and L.is_factor is True, "L must be a cspmatrix factor"
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
blkval = L.blkval
stack = []
for k in reversed(list(snpost)):
nn = snptr[k+1]-snptr[k] # |Nk|
na = relptr[k+1]-relptr[k] # |Ak|
nj = na + nn
# invert factor of D_{Nk,Nk}
lapack.trtri(blkval, offsetA = blkptr[k], ldA = nj, n = nn)
# zero-out strict upper triangular part of {Nj,Nj} block (just in case!)
for i in range(1,nn): blas.scal(0.0, blkval, offset = blkptr[k] + nj*i, n = i)
# compute inv(D_{Nk,Nk}) (store in 1,1 block of F)
F = matrix(0.0, (nj,nj))
blas.syrk(blkval, F, trans = 'T', offsetA = blkptr[k], ldA = nj, n = nn, k = nn)
# if supernode k is not a root node:
if na > 0:
# copy "update matrix" to 2,2 block of F
Vk = stack.pop()
lapack.lacpy(Vk, F, ldB = nj, offsetB = nn*nj+nn, m = na, n = na, uplo = 'L')
# compute S_{Ak,Nk} = -Vk*L_{Ak,Nk}; store in 2,1 block of F
blas.symm(Vk, blkval, F, m = na, n = nn, offsetB = blkptr[k]+nn,\
ldB = nj, offsetC = nn, ldC = nj, alpha = -1.0)
# compute S_nn = inv(D_{Nk,Nk}) - S_{Ak,Nk}'*L_{Ak,Nk}; store in 1,1 block of F
blas.gemm(F, blkval, F, transA = 'T', m = nn, n = nn, k = na,\
offsetA = nn, alpha = -1.0, beta = 1.0,\
offsetB = blkptr[k]+nn, ldB = nj)
# extract update matrices if supernode k has any children
for ii in range(chptr[k],chptr[k+1]):
i = chidx[ii]
stack.append(frontal_get_update(F, relidx, relptr, i))
# copy S_{Jk,Nk} (i.e., 1,1 and 2,1 blocks of F) to blkval
lapack.lacpy(F, blkval, m = nj, n = nn, offsetB = blkptr[k], ldB = nj, uplo = 'L')
L._is_factor = False
return