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 // Copyright ©2017 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" // Dgerq2 computes an RQ factorization of the m×n matrix A, // A = R * Q. // On exit, if m <= n, the upper triangle of the subarray // A[0:m, n-m:n] contains the m×m upper triangular matrix R. // If m >= n, the elements on and above the (m-n)-th subdiagonal // contain the m×n upper trapezoidal matrix R. // The remaining elements, with tau, represent the // orthogonal matrix Q as a product of min(m,n) elementary // reflectors. // // The matrix Q is represented as a product of elementary reflectors // Q = H_0 H_1 . . . H_{min(m,n)-1}. // Each H(i) has the form // H_i = I - tau_i * v * vᵀ // where v is a vector with v[0:n-k+i-1] stored in A[m-k+i, 0:n-k+i-1], // v[n-k+i:n] = 0 and v[n-k+i] = 1. // // tau must have length min(m,n) and work must have length m, otherwise // Dgerq2 will panic. // // Dgerq2 is an internal routine. It is exported for testing purposes. func (impl Implementation) Dgerq2(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) case len(work) < m: 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 := k - 1; i >= 0; i-- { // Generate elementary reflector H[i] to annihilate // A[m-k+i, 0:n-k+i-1]. mki := m - k + i nki := n - k + i var aii float64 aii, tau[i] = impl.Dlarfg(nki+1, a[mki*lda+nki], a[mki*lda:], 1) // Apply H[i] to A[0:m-k+i-1, 0:n-k+i] from the right. a[mki*lda+nki] = 1 impl.Dlarf(blas.Right, mki, nki+1, a[mki*lda:], 1, tau[i], a, lda, work) a[mki*lda+nki] = aii } }