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124 lines
3.9 KiB
Go
124 lines
3.9 KiB
Go
// Copyright ©2016 The gonum Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package native
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import (
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"gonum.org/v1/gonum/blas"
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"gonum.org/v1/gonum/blas/blas64"
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)
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// Dsytd2 reduces a symmetric n×n matrix A to symmetric tridiagonal form T by an
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// orthogonal similarity transformation
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// Q^T * A * Q = T
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// On entry, the matrix is contained in the specified triangle of a. On exit,
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// if uplo == blas.Upper, the diagonal and first super-diagonal of a are
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// overwritten with the elements of T. The elements above the first super-diagonal
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// are overwritten with the the elementary reflectors that are used with the
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// elements written to tau in order to construct Q. If uplo == blas.Lower, the
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// elements are written in the lower triangular region.
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//
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// d must have length at least n. e and tau must have length at least n-1. Dsytd2
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// will panic if these sizes are not met.
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//
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// Q is represented as a product of elementary reflectors.
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// If uplo == blas.Upper
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// Q = H_{n-2} * ... * H_1 * H_0
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// and if uplo == blas.Lower
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// Q = H_0 * H_1 * ... * H_{n-2}
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// where
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// H_i = I - tau * v * v^T
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// where tau is stored in tau[i], and v is stored in a.
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//
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// If uplo == blas.Upper, v[0:i-1] is stored in A[0:i-1,i+1], v[i] = 1, and
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// v[i+1:] = 0. The elements of a are
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// [ d e v2 v3 v4]
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// [ d e v3 v4]
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// [ d e v4]
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// [ d e]
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// [ d]
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// If uplo == blas.Lower, v[0:i+1] = 0, v[i+1] = 1, and v[i+2:] is stored in
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// A[i+2:n,i].
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// The elements of a are
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// [ d ]
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// [ e d ]
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// [v1 e d ]
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// [v1 v2 e d ]
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// [v1 v2 v3 e d]
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//
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// Dsytd2 is an internal routine. It is exported for testing purposes.
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func (impl Implementation) Dsytd2(uplo blas.Uplo, n int, a []float64, lda int, d, e, tau []float64) {
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checkMatrix(n, n, a, lda)
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if len(d) < n {
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panic(badD)
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}
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if len(e) < n-1 {
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panic(badE)
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}
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if len(tau) < n-1 {
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panic(badTau)
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}
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if n <= 0 {
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return
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}
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bi := blas64.Implementation()
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if uplo == blas.Upper {
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// Reduce the upper triangle of A.
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for i := n - 2; i >= 0; i-- {
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// Generate elementary reflector H_i = I - tau * v * v^T to
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// annihilate A[i:i-1, i+1].
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var taui float64
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a[i*lda+i+1], taui = impl.Dlarfg(i+1, a[i*lda+i+1], a[i+1:], lda)
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e[i] = a[i*lda+i+1]
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if taui != 0 {
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// Apply H_i from both sides to A[0:i,0:i].
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a[i*lda+i+1] = 1
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// Compute x := tau * A * v storing x in tau[0:i].
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bi.Dsymv(uplo, i+1, taui, a, lda, a[i+1:], lda, 0, tau, 1)
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// Compute w := x - 1/2 * tau * (x^T * v) * v.
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alpha := -0.5 * taui * bi.Ddot(i+1, tau, 1, a[i+1:], lda)
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bi.Daxpy(i+1, alpha, a[i+1:], lda, tau, 1)
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// Apply the transformation as a rank-2 update
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// A = A - v * w^T - w * v^T.
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bi.Dsyr2(uplo, i+1, -1, a[i+1:], lda, tau, 1, a, lda)
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a[i*lda+i+1] = e[i]
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}
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d[i+1] = a[(i+1)*lda+i+1]
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tau[i] = taui
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}
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d[0] = a[0]
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return
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}
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// Reduce the lower triangle of A.
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for i := 0; i < n-1; i++ {
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// Generate elementary reflector H_i = I - tau * v * v^T to
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// annihilate A[i+2:n, i].
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var taui float64
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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)
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e[i] = a[(i+1)*lda+i]
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if taui != 0 {
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// Apply H_i from both sides to A[i+1:n, i+1:n].
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a[(i+1)*lda+i] = 1
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// Compute x := tau * A * v, storing y in tau[i:n-1].
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bi.Dsymv(uplo, n-i-1, taui, a[(i+1)*lda+i+1:], lda, a[(i+1)*lda+i:], lda, 0, tau[i:], 1)
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// Compute w := x - 1/2 * tau * (x^T * v) * v.
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alpha := -0.5 * taui * bi.Ddot(n-i-1, tau[i:], 1, a[(i+1)*lda+i:], lda)
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bi.Daxpy(n-i-1, alpha, a[(i+1)*lda+i:], lda, tau[i:], 1)
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// Apply the transformation as a rank-2 update
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// A = A - v * w^T - w * v^T.
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bi.Dsyr2(uplo, n-i-1, -1, a[(i+1)*lda+i:], lda, tau[i:], 1, a[(i+1)*lda+i+1:], lda)
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a[(i+1)*lda+i] = e[i]
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}
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d[i] = a[i*lda+i]
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tau[i] = taui
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}
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d[n-1] = a[(n-1)*lda+n-1]
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}
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