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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" "gonum.org/v1/gonum/blas/blas64" ) // Dlatrd reduces nb rows and columns of a real n×n symmetric matrix A to symmetric // tridiagonal form. It computes the orthonormal similarity transformation // Qᵀ * A * Q // and returns the matrices V and W to apply to the unreduced part of A. If // uplo == blas.Upper, the upper triangle is supplied and the last nb rows are // reduced. If uplo == blas.Lower, the lower triangle is supplied and the first // nb rows are reduced. // // a contains the symmetric matrix on entry with active triangular half specified // by uplo. On exit, the nb columns have been reduced to tridiagonal form. The // diagonal contains the diagonal of the reduced matrix, the off-diagonal is // set to 1, and the remaining elements contain the data to construct Q. // // If uplo == blas.Upper, with n = 5 and nb = 2 on exit a is // [ a a a v4 v5] // [ a a v4 v5] // [ a 1 v5] // [ d 1] // [ d] // // If uplo == blas.Lower, with n = 5 and nb = 2, on exit a is // [ d ] // [ 1 d ] // [v1 1 a ] // [v1 v2 a a ] // [v1 v2 a a a] // // e contains the superdiagonal elements of the reduced matrix. If uplo == blas.Upper, // e[n-nb:n-1] contains the last nb columns of the reduced matrix, while if // uplo == blas.Lower, e[:nb] contains the first nb columns of the reduced matrix. // e must have length at least n-1, and Dlatrd will panic otherwise. // // tau contains the scalar factors of the elementary reflectors needed to construct Q. // The reflectors are stored in tau[n-nb:n-1] if uplo == blas.Upper, and in // tau[:nb] if uplo == blas.Lower. tau must have length n-1, and Dlatrd will panic // otherwise. // // w is an n×nb matrix. On exit it contains the data to update the unreduced part // of A. // // The matrix Q is represented as a product of elementary reflectors. Each reflector // H has the form // I - tau * v * vᵀ // If uplo == blas.Upper, // Q = H_{n-1} * H_{n-2} * ... * H_{n-nb} // where v[:i-1] is stored in A[:i-1,i], v[i-1] = 1, and v[i:n] = 0. // // If uplo == blas.Lower, // Q = H_0 * H_1 * ... * H_{nb-1} // where v[:i+1] = 0, v[i+1] = 1, and v[i+2:n] is stored in A[i+2:n,i]. // // The vectors v form the n×nb matrix V which is used with W to apply a // symmetric rank-2 update to the unreduced part of A // A = A - V * Wᵀ - W * Vᵀ // // Dlatrd is an internal routine. It is exported for testing purposes. func (impl Implementation) Dlatrd(uplo blas.Uplo, n, nb int, a []float64, lda int, e, tau, w []float64, ldw int) { switch { case uplo != blas.Upper && uplo != blas.Lower: panic(badUplo) case n < 0: panic(nLT0) case nb < 0: panic(nbLT0) case nb > n: panic(nbGTN) case lda < max(1, n): panic(badLdA) case ldw < max(1, nb): panic(badLdW) } if n == 0 { return } switch { case len(a) < (n-1)*lda+n: panic(shortA) case len(w) < (n-1)*ldw+nb: panic(shortW) case len(e) < n-1: panic(shortE) case len(tau) < n-1: panic(shortTau) } bi := blas64.Implementation() if uplo == blas.Upper { for i := n - 1; i >= n-nb; i-- { iw := i - n + nb if i < n-1 { // Update A(0:i, i). bi.Dgemv(blas.NoTrans, i+1, n-i-1, -1, a[i+1:], lda, w[i*ldw+iw+1:], 1, 1, a[i:], lda) bi.Dgemv(blas.NoTrans, i+1, n-i-1, -1, w[iw+1:], ldw, a[i*lda+i+1:], 1, 1, a[i:], lda) } if i > 0 { // Generate elementary reflector H_i to annihilate A(0:i-2,i). e[i-1], tau[i-1] = impl.Dlarfg(i, a[(i-1)*lda+i], a[i:], lda) a[(i-1)*lda+i] = 1 // Compute W(0:i-1, i). bi.Dsymv(blas.Upper, i, 1, a, lda, a[i:], lda, 0, w[iw:], ldw) if i < n-1 { bi.Dgemv(blas.Trans, i, n-i-1, 1, w[iw+1:], ldw, a[i:], lda, 0, w[(i+1)*ldw+iw:], ldw) bi.Dgemv(blas.NoTrans, i, n-i-1, -1, a[i+1:], lda, w[(i+1)*ldw+iw:], ldw, 1, w[iw:], ldw) bi.Dgemv(blas.Trans, i, n-i-1, 1, a[i+1:], lda, a[i:], lda, 0, w[(i+1)*ldw+iw:], ldw) bi.Dgemv(blas.NoTrans, i, n-i-1, -1, w[iw+1:], ldw, w[(i+1)*ldw+iw:], ldw, 1, w[iw:], ldw) } bi.Dscal(i, tau[i-1], w[iw:], ldw) alpha := -0.5 * tau[i-1] * bi.Ddot(i, w[iw:], ldw, a[i:], lda) bi.Daxpy(i, alpha, a[i:], lda, w[iw:], ldw) } } } else { // Reduce first nb columns of lower triangle. for i := 0; i < nb; i++ { // Update A(i:n, i) bi.Dgemv(blas.NoTrans, n-i, i, -1, a[i*lda:], lda, w[i*ldw:], 1, 1, a[i*lda+i:], lda) bi.Dgemv(blas.NoTrans, n-i, i, -1, w[i*ldw:], ldw, a[i*lda:], 1, 1, a[i*lda+i:], lda) if i < n-1 { // Generate elementary reflector H_i to annihilate A(i+2:n,i). e[i], tau[i] = impl.Dlarfg(n-i-1, a[(i+1)*lda+i], a[min(i+2, n-1)*lda+i:], lda) a[(i+1)*lda+i] = 1 // Compute W(i+1:n,i). bi.Dsymv(blas.Lower, n-i-1, 1, a[(i+1)*lda+i+1:], lda, a[(i+1)*lda+i:], lda, 0, w[(i+1)*ldw+i:], ldw) bi.Dgemv(blas.Trans, n-i-1, i, 1, w[(i+1)*ldw:], ldw, a[(i+1)*lda+i:], lda, 0, w[i:], ldw) bi.Dgemv(blas.NoTrans, n-i-1, i, -1, a[(i+1)*lda:], lda, w[i:], ldw, 1, w[(i+1)*ldw+i:], ldw) bi.Dgemv(blas.Trans, n-i-1, i, 1, a[(i+1)*lda:], lda, a[(i+1)*lda+i:], lda, 0, w[i:], ldw) bi.Dgemv(blas.NoTrans, n-i-1, i, -1, w[(i+1)*ldw:], ldw, w[i:], ldw, 1, w[(i+1)*ldw+i:], ldw) bi.Dscal(n-i-1, tau[i], w[(i+1)*ldw+i:], ldw) alpha := -0.5 * tau[i] * bi.Ddot(n-i-1, w[(i+1)*ldw+i:], ldw, a[(i+1)*lda+i:], lda) bi.Daxpy(n-i-1, alpha, a[(i+1)*lda+i:], lda, w[(i+1)*ldw+i:], ldw) } } } }