Exemple #1
0
// 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},
				)
			}
		}
	}
}
Exemple #2
0
// 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}
}