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// 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 Qᵀ * A * Q. Dlahr2 returns the
// matrices V and T which determine Q as a block reflector I - V*T*Vᵀ, 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 * vᵀ,
//
// 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*Vᵀ) * (A - Y*Vᵀ).
//
// 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) {
switch {
case n < 0:
panic(nLT0)
case k < 0:
panic(kLT0)
case nb < 0:
panic(nbLT0)
case nb > n:
panic(nbGTN)
case lda < max(1, n-k+1):
panic(badLdA)
case ldt < max(1, nb):
panic(badLdT)
case ldy < max(1, nb):
panic(badLdY)
}
// Quick return if possible.
if n < 0 {
return
}
switch {
case len(a) < (n-1)*lda+n-k+1:
panic(shortA)
case len(tau) < nb:
panic(shortTau)
case len(t) < (nb-1)*ldt+nb:
panic(shortT)
case len(y) < (n-1)*ldy+nb:
panic(shortY)
}
// 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 * Vᵀ.
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 * Tᵀ * Vᵀ 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 := V1ᵀ * 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 + V2ᵀ * 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 := Tᵀ * 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)
}
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