--- /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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