// replaces x with Q.x
func (f LQFactor) applyQTo(x *Dense, trans bool) {
nh, nc := f.LQ.Dims()
m, n := x.Dims()
if m != nc {
panic(ErrShape)
}
proj := make([]float64, n)
if trans {
for k := nh - 1; k >= 0; k-- {
hh := f.LQ.RawRowView(k)[k:]
sub := x.View(k, 0, m-k, n).(*Dense)
blas64.Gemv(blas.Trans,
1, sub.mat, blas64.Vector{Inc: 1, Data: hh},
0, blas64.Vector{Inc: 1, Data: proj},
)
for i := k; i < m; i++ {
row := x.RawRowView(i)
blas64.Axpy(n, -hh[i-k],
blas64.Vector{Inc: 1, Data: proj},
blas64.Vector{Inc: 1, Data: row},
)
}
}
} else {
for k := 0; k < nh; k++ {
hh := f.LQ.RawRowView(k)[k:]
sub := x.View(k, 0, m-k, n).(*Dense)
blas64.Gemv(blas.Trans,
1, sub.mat, blas64.Vector{Inc: 1, Data: hh},
0, blas64.Vector{Inc: 1, Data: proj},
)
for i := k; i < m; i++ {
row := x.RawRowView(i)
blas64.Axpy(n, -hh[i-k],
blas64.Vector{Inc: 1, Data: proj},
blas64.Vector{Inc: 1, Data: row},
)
}
}
}
}
// LQ computes an LQ Decomposition for an m-by-n matrix a with m <= n by Householder
// reflections. The LQ decomposition is an m-by-n orthogonal matrix q and an m-by-m
// lower triangular matrix l so that a = l.q. LQ will panic with ErrShape if m > n.
//
// The LQ decomposition always exists, even if the matrix does not have full rank,
// so LQ will never fail unless m > n. The primary use of the LQ decomposition is
// in the least squares solution of non-square systems of simultaneous linear equations.
// This will fail if LQIsFullRank() returns false. The matrix a is overwritten by the
// decomposition.
func LQ(a *Dense) LQFactor {
// Initialize.
m, n := a.Dims()
if m > n {
panic(ErrShape)
}
lq := *a
lDiag := make([]float64, m)
projs := NewVector(m, nil)
// Main loop.
for k := 0; k < m; k++ {
hh := lq.RawRowView(k)[k:]
norm := blas64.Nrm2(len(hh), blas64.Vector{Inc: 1, Data: hh})
lDiag[k] = norm
if norm != 0 {
hhNorm := (norm * math.Sqrt(1-hh[0]/norm))
if hhNorm == 0 {
hh[0] = 0
} else {
// Form k-th Householder vector.
s := 1 / hhNorm
hh[0] -= norm
blas64.Scal(len(hh), s, blas64.Vector{Inc: 1, Data: hh})
// Apply transformation to remaining columns.
if k < m-1 {
a = lq.View(k+1, k, m-k-1, n-k).(*Dense)
projs = projs.ViewVec(0, m-k-1)
projs.MulVec(a, false, NewVector(len(hh), hh))
for j := 0; j < m-k-1; j++ {
dst := a.RawRowView(j)
blas64.Axpy(len(dst), -projs.at(j),
blas64.Vector{Inc: 1, Data: hh},
blas64.Vector{Inc: 1, Data: dst},
)
}
}
}
}
}
*a = lq
return LQFactor{a, lDiag}
}