+++ /dev/null
-// 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
-// Q^T * 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 * v^T
-// 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 * W^T - W * V^T
-//
-// 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) {
- checkMatrix(n, n, a, lda)
- checkMatrix(n, nb, w, ldw)
- if len(e) < n-1 {
- panic(badE)
- }
- if len(tau) < n-1 {
- panic(badTau)
- }
- if n <= 0 {
- return
- }
- 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)
- }
- }
- }
-}
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