+++ /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"
-)
-
-// Dlahr2 reduces the first nb columns of a real general n×(n-k+1) matrix A so
-// that elements below the k-th subdiagonal are zero. The reduction is performed
-// by an orthogonal similarity transformation Q^T * A * Q. Dlahr2 returns the
-// matrices V and T which determine Q as a block reflector I - V*T*V^T, and
-// also the matrix Y = A * V * T.
-//
-// The matrix Q is represented as a product of nb elementary reflectors
-// Q = H_0 * H_1 * ... * H_{nb-1}.
-// Each H_i has the form
-// H_i = I - tau[i] * v * v^T,
-// where v is a real vector with v[0:i+k-1] = 0 and v[i+k-1] = 1. v[i+k:n] is
-// stored on exit in A[i+k+1:n,i].
-//
-// The elements of the vectors v together form the (n-k+1)×nb matrix
-// V which is needed, with T and Y, to apply the transformation to the
-// unreduced part of the matrix, using an update of the form
-// A = (I - V*T*V^T) * (A - Y*V^T).
-//
-// On entry, a contains the n×(n-k+1) general matrix A. On return, the elements
-// on and above the k-th subdiagonal in the first nb columns are overwritten
-// with the corresponding elements of the reduced matrix; the elements below the
-// k-th subdiagonal, with the slice tau, represent the matrix Q as a product of
-// elementary reflectors. The other columns of A are unchanged.
-//
-// The contents of A on exit are illustrated by the following example
-// with n = 7, k = 3 and nb = 2:
-// [ a a a a a ]
-// [ a a a a a ]
-// [ a a a a a ]
-// [ h h a a a ]
-// [ v0 h a a a ]
-// [ v0 v1 a a a ]
-// [ v0 v1 a a a ]
-// where a denotes an element of the original matrix A, h denotes a
-// modified element of the upper Hessenberg matrix H, and vi denotes an
-// element of the vector defining H_i.
-//
-// k is the offset for the reduction. Elements below the k-th subdiagonal in the
-// first nb columns are reduced to zero.
-//
-// nb is the number of columns to be reduced.
-//
-// On entry, a represents the n×(n-k+1) matrix A. On return, the elements on and
-// above the k-th subdiagonal in the first nb columns are overwritten with the
-// corresponding elements of the reduced matrix. The elements below the k-th
-// subdiagonal, with the slice tau, represent the matrix Q as a product of
-// elementary reflectors. The other columns of A are unchanged.
-//
-// tau will contain the scalar factors of the elementary reflectors. It must
-// have length at least nb.
-//
-// t and ldt represent the nb×nb upper triangular matrix T, and y and ldy
-// represent the n×nb matrix Y.
-//
-// Dlahr2 is an internal routine. It is exported for testing purposes.
-func (impl Implementation) Dlahr2(n, k, nb int, a []float64, lda int, tau, t []float64, ldt int, y []float64, ldy int) {
- checkMatrix(n, n-k+1, a, lda)
- if len(tau) < nb {
- panic(badTau)
- }
- checkMatrix(nb, nb, t, ldt)
- checkMatrix(n, nb, y, ldy)
-
- // Quick return if possible.
- if n <= 1 {
- return
- }
-
- bi := blas64.Implementation()
- var ei float64
- for i := 0; i < nb; i++ {
- if i > 0 {
- // Update A[k:n,i].
-
- // Update i-th column of A - Y * V^T.
- bi.Dgemv(blas.NoTrans, n-k, i,
- -1, y[k*ldy:], ldy,
- a[(k+i-1)*lda:], 1,
- 1, a[k*lda+i:], lda)
-
- // Apply I - V * T^T * V^T to this column (call it b)
- // from the left, using the last column of T as
- // workspace.
- // Let V = [ V1 ] and b = [ b1 ] (first i rows)
- // [ V2 ] [ b2 ]
- // where V1 is unit lower triangular.
- //
- // w := V1^T * b1.
- bi.Dcopy(i, a[k*lda+i:], lda, t[nb-1:], ldt)
- bi.Dtrmv(blas.Lower, blas.Trans, blas.Unit, i,
- a[k*lda:], lda, t[nb-1:], ldt)
-
- // w := w + V2^T * b2.
- bi.Dgemv(blas.Trans, n-k-i, i,
- 1, a[(k+i)*lda:], lda,
- a[(k+i)*lda+i:], lda,
- 1, t[nb-1:], ldt)
-
- // w := T^T * w.
- bi.Dtrmv(blas.Upper, blas.Trans, blas.NonUnit, i,
- t, ldt, t[nb-1:], ldt)
-
- // b2 := b2 - V2*w.
- bi.Dgemv(blas.NoTrans, n-k-i, i,
- -1, a[(k+i)*lda:], lda,
- t[nb-1:], ldt,
- 1, a[(k+i)*lda+i:], lda)
-
- // b1 := b1 - V1*w.
- bi.Dtrmv(blas.Lower, blas.NoTrans, blas.Unit, i,
- a[k*lda:], lda, t[nb-1:], ldt)
- bi.Daxpy(i, -1, t[nb-1:], ldt, a[k*lda+i:], lda)
-
- a[(k+i-1)*lda+i-1] = ei
- }
-
- // Generate the elementary reflector H_i to annihilate
- // A[k+i+1:n,i].
- ei, tau[i] = impl.Dlarfg(n-k-i, a[(k+i)*lda+i], a[min(k+i+1, n-1)*lda+i:], lda)
- a[(k+i)*lda+i] = 1
-
- // Compute Y[k:n,i].
- bi.Dgemv(blas.NoTrans, n-k, n-k-i,
- 1, a[k*lda+i+1:], lda,
- a[(k+i)*lda+i:], lda,
- 0, y[k*ldy+i:], ldy)
- bi.Dgemv(blas.Trans, n-k-i, i,
- 1, a[(k+i)*lda:], lda,
- a[(k+i)*lda+i:], lda,
- 0, t[i:], ldt)
- bi.Dgemv(blas.NoTrans, n-k, i,
- -1, y[k*ldy:], ldy,
- t[i:], ldt,
- 1, y[k*ldy+i:], ldy)
- bi.Dscal(n-k, tau[i], y[k*ldy+i:], ldy)
-
- // Compute T[0:i,i].
- bi.Dscal(i, -tau[i], t[i:], ldt)
- bi.Dtrmv(blas.Upper, blas.NoTrans, blas.NonUnit, i,
- t, ldt, t[i:], ldt)
-
- t[i*ldt+i] = tau[i]
- }
- a[(k+nb-1)*lda+nb-1] = ei
-
- // Compute Y[0:k,0:nb].
- impl.Dlacpy(blas.All, k, nb, a[1:], lda, y, ldy)
- bi.Dtrmm(blas.Right, blas.Lower, blas.NoTrans, blas.Unit, k, nb,
- 1, a[k*lda:], lda, y, ldy)
- if n > k+nb {
- bi.Dgemm(blas.NoTrans, blas.NoTrans, k, nb, n-k-nb,
- 1, a[1+nb:], lda,
- a[(k+nb)*lda:], lda,
- 1, y, ldy)
- }
- bi.Dtrmm(blas.Right, blas.Upper, blas.NoTrans, blas.NonUnit, k, nb,
- 1, t, ldt, y, ldy)
-}
OSDN Git Service
