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// Copyright ©2015 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"
// Dgeqr2 computes a QR factorization of the m×n matrix A.
//
// In a QR factorization, Q is an m×m orthonormal matrix, and R is an
// upper triangular m×n matrix.
//
// A is modified to contain the information to construct Q and R.
// The upper triangle of a contains the matrix R. The lower triangular elements
// (not including the diagonal) contain the elementary reflectors. tau is modified
// to contain the reflector scales. tau must have length at least min(m,n), and
// this function will panic otherwise.
//
// The ith elementary reflector can be explicitly constructed by first extracting
// the
// v[j] = 0 j < i
// v[j] = 1 j == i
// v[j] = a[j*lda+i] j > i
// and computing H_i = I - tau[i] * v * vᵀ.
//
// The orthonormal matrix Q can be constructed from a product of these elementary
// reflectors, Q = H_0 * H_1 * ... * H_{k-1}, where k = min(m,n).
//
// work is temporary storage of length at least n and this function will panic otherwise.
//
// Dgeqr2 is an internal routine. It is exported for testing purposes.
func (impl Implementation) Dgeqr2(m, n int, a []float64, lda int, tau, work []float64) {
// TODO(btracey): This is oriented such that columns of a are eliminated.
// This likely could be re-arranged to take better advantage of row-major
// storage.
switch {
case m < 0:
panic(mLT0)
case n < 0:
panic(nLT0)
case lda < max(1, n):
panic(badLdA)
case len(work) < n:
panic(shortWork)
}
// 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)
}
for i := 0; i < k; i++ {
// Generate elementary reflector H_i.
a[i*lda+i], tau[i] = impl.Dlarfg(m-i, a[i*lda+i], a[min((i+1), m-1)*lda+i:], lda)
if i < n-1 {
aii := a[i*lda+i]
a[i*lda+i] = 1
impl.Dlarf(blas.Left, m-i, n-i-1,
a[i*lda+i:], lda,
tau[i],
a[i*lda+i+1:], lda,
work)
a[i*lda+i] = aii
}
}
}