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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"
)
// Dsytrd reduces a symmetric n×n matrix A to symmetric tridiagonal form by an
// orthogonal similarity transformation
//
// Qᵀ * A * Q = T
//
// where Q is an orthonormal matrix and T is symmetric and tridiagonal.
//
// On entry, a contains the elements of the input matrix in the triangle specified
// by uplo. On exit, the diagonal and sub/super-diagonal are overwritten by the
// corresponding elements of the tridiagonal matrix T. The remaining elements in
// the triangle, along with the array tau, contain the data to construct Q as
// the product of elementary reflectors.
//
// If uplo == blas.Upper, Q is constructed with
//
// Q = H_{n-2} * ... * H_1 * H_0
//
// where
//
// H_i = I - tau_i * v * vᵀ
//
// v is constructed as v[i+1:n] = 0, v[i] = 1, v[0:i-1] is stored in A[0:i-1, i+1].
// The elements of A are
//
// [ d e v1 v2 v3]
// [ d e v2 v3]
// [ d e v3]
// [ d e]
// [ e]
//
// If uplo == blas.Lower, Q is constructed with
//
// Q = H_0 * H_1 * ... * H_{n-2}
//
// where
//
// H_i = I - tau_i * v * vᵀ
//
// v is constructed as v[0:i+1] = 0, v[i+1] = 1, v[i+2:n] is stored in A[i+2:n, i].
// The elements of A are
//
// [ d ]
// [ e d ]
// [v0 e d ]
// [v0 v1 e d ]
// [v0 v1 v2 e d]
//
// d must have length n, and e and tau must have length n-1. Dsytrd will panic if
// these conditions are not met.
//
// work is temporary storage, and lwork specifies the usable memory length. At minimum,
// lwork >= 1, and Dsytrd will panic otherwise. The amount of blocking is
// limited by the usable length.
// If lwork == -1, instead of computing Dsytrd the optimal work length is stored
// into work[0].
//
// Dsytrd is an internal routine. It is exported for testing purposes.
func (impl Implementation) Dsytrd(uplo blas.Uplo, n int, a []float64, lda int, d, e, tau, work []float64, lwork int) {
switch {
case uplo != blas.Upper && uplo != blas.Lower:
panic(badUplo)
case n < 0:
panic(nLT0)
case lda < max(1, n):
panic(badLdA)
case lwork < 1 && lwork != -1:
panic(badLWork)
case len(work) < max(1, lwork):
panic(shortWork)
}
// Quick return if possible.
if n == 0 {
work[0] = 1
return
}
nb := impl.Ilaenv(1, "DSYTRD", string(uplo), n, -1, -1, -1)
lworkopt := n * nb
if lwork == -1 {
work[0] = float64(lworkopt)
return
}
switch {
case len(a) < (n-1)*lda+n:
panic(shortA)
case len(d) < n:
panic(shortD)
case len(e) < n-1:
panic(shortE)
case len(tau) < n-1:
panic(shortTau)
}
bi := blas64.Implementation()
nx := n
iws := 1
var ldwork int
if 1 < nb && nb < n {
// Determine when to cross over from blocked to unblocked code. The last
// block is always handled by unblocked code.
nx = max(nb, impl.Ilaenv(3, "DSYTRD", string(uplo), n, -1, -1, -1))
if nx < n {
// Determine if workspace is large enough for blocked code.
ldwork = nb
iws = n * ldwork
if lwork < iws {
// Not enough workspace to use optimal nb: determine the minimum
// value of nb and reduce nb or force use of unblocked code by
// setting nx = n.
nb = max(lwork/n, 1)
nbmin := impl.Ilaenv(2, "DSYTRD", string(uplo), n, -1, -1, -1)
if nb < nbmin {
nx = n
}
}
} else {
nx = n
}
} else {
nb = 1
}
ldwork = nb
if uplo == blas.Upper {
// Reduce the upper triangle of A. Columns 0:kk are handled by the
// unblocked method.
var i int
kk := n - ((n-nx+nb-1)/nb)*nb
for i = n - nb; i >= kk; i -= nb {
// Reduce columns i:i+nb to tridiagonal form and form the matrix W
// which is needed to update the unreduced part of the matrix.
impl.Dlatrd(uplo, i+nb, nb, a, lda, e, tau, work, ldwork)
// Update the unreduced submatrix A[0:i-1,0:i-1], using an update
// of the form A = A - V*Wᵀ - W*Vᵀ.
bi.Dsyr2k(uplo, blas.NoTrans, i, nb, -1, a[i:], lda, work, ldwork, 1, a, lda)
// Copy superdiagonal elements back into A, and diagonal elements into D.
for j := i; j < i+nb; j++ {
a[(j-1)*lda+j] = e[j-1]
d[j] = a[j*lda+j]
}
}
// Use unblocked code to reduce the last or only block
// check that i == kk.
impl.Dsytd2(uplo, kk, a, lda, d, e, tau)
} else {
var i int
// Reduce the lower triangle of A.
for i = 0; i < n-nx; i += nb {
// Reduce columns 0:i+nb to tridiagonal form and form the matrix W
// which is needed to update the unreduced part of the matrix.
impl.Dlatrd(uplo, n-i, nb, a[i*lda+i:], lda, e[i:], tau[i:], work, ldwork)
// Update the unreduced submatrix A[i+ib:n, i+ib:n], using an update
// of the form A = A + V*Wᵀ - W*Vᵀ.
bi.Dsyr2k(uplo, blas.NoTrans, n-i-nb, nb, -1, a[(i+nb)*lda+i:], lda,
work[nb*ldwork:], ldwork, 1, a[(i+nb)*lda+i+nb:], lda)
// Copy subdiagonal elements back into A, and diagonal elements into D.
for j := i; j < i+nb; j++ {
a[(j+1)*lda+j] = e[j]
d[j] = a[j*lda+j]
}
}
// Use unblocked code to reduce the last or only block.
impl.Dsytd2(uplo, n-i, a[i*lda+i:], lda, d[i:], e[i:], tau[i:])
}
work[0] = float64(iws)
}
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