+++ /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"
-)
-
-// Dsytd2 reduces a symmetric n×n matrix A to symmetric tridiagonal form T by an
-// orthogonal similarity transformation
-// Q^T * A * Q = T
-// On entry, the matrix is contained in the specified triangle of a. On exit,
-// if uplo == blas.Upper, the diagonal and first super-diagonal of a are
-// overwritten with the elements of T. The elements above the first super-diagonal
-// are overwritten with the the elementary reflectors that are used with the
-// elements written to tau in order to construct Q. If uplo == blas.Lower, the
-// elements are written in the lower triangular region.
-//
-// d must have length at least n. e and tau must have length at least n-1. Dsytd2
-// will panic if these sizes are not met.
-//
-// Q is represented as a product of elementary reflectors.
-// If uplo == blas.Upper
-// Q = H_{n-2} * ... * H_1 * H_0
-// and if uplo == blas.Lower
-// Q = H_0 * H_1 * ... * H_{n-2}
-// where
-// H_i = I - tau * v * v^T
-// where tau is stored in tau[i], and v is stored in a.
-//
-// If uplo == blas.Upper, v[0:i-1] is stored in A[0:i-1,i+1], v[i] = 1, and
-// v[i+1:] = 0. The elements of a are
-// [ d e v2 v3 v4]
-// [ d e v3 v4]
-// [ d e v4]
-// [ d e]
-// [ d]
-// If uplo == blas.Lower, v[0:i+1] = 0, v[i+1] = 1, and v[i+2:] is stored in
-// A[i+2:n,i].
-// The elements of a are
-// [ d ]
-// [ e d ]
-// [v1 e d ]
-// [v1 v2 e d ]
-// [v1 v2 v3 e d]
-//
-// Dsytd2 is an internal routine. It is exported for testing purposes.
-func (impl Implementation) Dsytd2(uplo blas.Uplo, n int, a []float64, lda int, d, e, tau []float64) {
- checkMatrix(n, n, a, lda)
- if len(d) < n {
- panic(badD)
- }
- 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 {
- // Reduce the upper triangle of A.
- for i := n - 2; i >= 0; i-- {
- // Generate elementary reflector H_i = I - tau * v * v^T to
- // annihilate A[i:i-1, i+1].
- var taui float64
- a[i*lda+i+1], taui = impl.Dlarfg(i+1, a[i*lda+i+1], a[i+1:], lda)
- e[i] = a[i*lda+i+1]
- if taui != 0 {
- // Apply H_i from both sides to A[0:i,0:i].
- a[i*lda+i+1] = 1
-
- // Compute x := tau * A * v storing x in tau[0:i].
- bi.Dsymv(uplo, i+1, taui, a, lda, a[i+1:], lda, 0, tau, 1)
-
- // Compute w := x - 1/2 * tau * (x^T * v) * v.
- alpha := -0.5 * taui * bi.Ddot(i+1, tau, 1, a[i+1:], lda)
- bi.Daxpy(i+1, alpha, a[i+1:], lda, tau, 1)
-
- // Apply the transformation as a rank-2 update
- // A = A - v * w^T - w * v^T.
- bi.Dsyr2(uplo, i+1, -1, a[i+1:], lda, tau, 1, a, lda)
- a[i*lda+i+1] = e[i]
- }
- d[i+1] = a[(i+1)*lda+i+1]
- tau[i] = taui
- }
- d[0] = a[0]
- return
- }
- // Reduce the lower triangle of A.
- for i := 0; i < n-1; i++ {
- // Generate elementary reflector H_i = I - tau * v * v^T to
- // annihilate A[i+2:n, i].
- var taui float64
- a[(i+1)*lda+i], taui = impl.Dlarfg(n-i-1, a[(i+1)*lda+i], a[min(i+2, n-1)*lda+i:], lda)
- e[i] = a[(i+1)*lda+i]
- if taui != 0 {
- // Apply H_i from both sides to A[i+1:n, i+1:n].
- a[(i+1)*lda+i] = 1
-
- // Compute x := tau * A * v, storing y in tau[i:n-1].
- bi.Dsymv(uplo, n-i-1, taui, a[(i+1)*lda+i+1:], lda, a[(i+1)*lda+i:], lda, 0, tau[i:], 1)
-
- // Compute w := x - 1/2 * tau * (x^T * v) * v.
- alpha := -0.5 * taui * bi.Ddot(n-i-1, tau[i:], 1, a[(i+1)*lda+i:], lda)
- bi.Daxpy(n-i-1, alpha, a[(i+1)*lda+i:], lda, tau[i:], 1)
-
- // Apply the transformation as a rank-2 update
- // A = A - v * w^T - w * v^T.
- bi.Dsyr2(uplo, n-i-1, -1, a[(i+1)*lda+i:], lda, tau[i:], 1, a[(i+1)*lda+i+1:], lda)
- a[(i+1)*lda+i] = e[i]
- }
- d[i] = a[i*lda+i]
- tau[i] = taui
- }
- d[n-1] = a[(n-1)*lda+n-1]
-}
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