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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 } }