--- /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
