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// Copyright ©2016 The Gonum Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package gonum
import "gonum.org/v1/gonum/blas"
// Dgeql2 computes the QL factorization of the m×n matrix A. That is, Dgeql2
// computes Q and L such that
//
// A = Q * L
//
// where Q is an m×m orthonormal matrix and L is a lower trapezoidal matrix.
//
// Q is represented as a product of elementary reflectors,
//
// Q = H_{k-1} * ... * H_1 * H_0
//
// where k = min(m,n) and each H_i has the form
//
// H_i = I - tau[i] * v_i * v_iᵀ
//
// Vector v_i has v[m-k+i+1:m] = 0, v[m-k+i] = 1, and v[:m-k+i+1] is stored on
// exit in A[0:m-k+i-1, n-k+i].
//
// tau must have length at least min(m,n), and Dgeql2 will panic otherwise.
//
// work is temporary memory storage and must have length at least n.
//
// Dgeql2 is an internal routine. It is exported for testing purposes.
func (impl Implementation) Dgeql2(m, n int, a []float64, lda int, tau, work []float64) {
switch {
case m < 0:
panic(mLT0)
case n < 0:
panic(nLT0)
case lda < max(1, n):
panic(badLdA)
}
// Quick return if possible.
k := min(m, n)
if k == 0 {
return
}
switch {
case len(a) < (m-1)*lda+n:
panic(shortA)
case len(tau) < k:
panic(shortTau)
case len(work) < n:
panic(shortWork)
}
var aii float64
for i := k - 1; i >= 0; i-- {
// Generate elementary reflector H_i to annihilate A[0:m-k+i-1, n-k+i].
aii, tau[i] = impl.Dlarfg(m-k+i+1, a[(m-k+i)*lda+n-k+i], a[n-k+i:], lda)
// Apply H_i to A[0:m-k+i, 0:n-k+i-1] from the left.
a[(m-k+i)*lda+n-k+i] = 1
impl.Dlarf(blas.Left, m-k+i+1, n-k+i, a[n-k+i:], lda, tau[i], a, lda, work)
a[(m-k+i)*lda+n-k+i] = aii
}
}
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