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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 (
"math"
"gonum.org/v1/gonum/blas/blas64"
)
// Dlahqr computes the eigenvalues and Schur factorization of a block of an n×n
// upper Hessenberg matrix H, using the double-shift/single-shift QR algorithm.
//
// h and ldh represent the matrix H. Dlahqr works primarily with the Hessenberg
// submatrix H[ilo:ihi+1,ilo:ihi+1], but applies transformations to all of H if
// wantt is true. It is assumed that H[ihi+1:n,ihi+1:n] is already upper
// quasi-triangular, although this is not checked.
//
// It must hold that
// 0 <= ilo <= max(0,ihi), and ihi < n,
// and that
// H[ilo,ilo-1] == 0, if ilo > 0,
// otherwise Dlahqr will panic.
//
// If unconverged is zero on return, wr[ilo:ihi+1] and wi[ilo:ihi+1] will contain
// respectively the real and imaginary parts of the computed eigenvalues ilo
// to ihi. If two eigenvalues are computed as a complex conjugate pair, they are
// stored in consecutive elements of wr and wi, say the i-th and (i+1)th, with
// wi[i] > 0 and wi[i+1] < 0. If wantt is true, the eigenvalues are stored in
// the same order as on the diagonal of the Schur form returned in H, with
// wr[i] = H[i,i], and, if H[i:i+2,i:i+2] is a 2×2 diagonal block,
// wi[i] = sqrt(abs(H[i+1,i]*H[i,i+1])) and wi[i+1] = -wi[i].
//
// wr and wi must have length ihi+1.
//
// z and ldz represent an n×n matrix Z. If wantz is true, the transformations
// will be applied to the submatrix Z[iloz:ihiz+1,ilo:ihi+1] and it must hold that
// 0 <= iloz <= ilo, and ihi <= ihiz < n.
// If wantz is false, z is not referenced.
//
// unconverged indicates whether Dlahqr computed all the eigenvalues ilo to ihi
// in a total of 30 iterations per eigenvalue.
//
// If unconverged is zero, all the eigenvalues ilo to ihi have been computed and
// will be stored on return in wr[ilo:ihi+1] and wi[ilo:ihi+1].
//
// If unconverged is zero and wantt is true, H[ilo:ihi+1,ilo:ihi+1] will be
// overwritten on return by upper quasi-triangular full Schur form with any
// 2×2 diagonal blocks in standard form.
//
// If unconverged is zero and if wantt is false, the contents of h on return is
// unspecified.
//
// If unconverged is positive, some eigenvalues have not converged, and
// wr[unconverged:ihi+1] and wi[unconverged:ihi+1] contain those eigenvalues
// which have been successfully computed.
//
// If unconverged is positive and wantt is true, then on return
// (initial H)*U = U*(final H), (*)
// where U is an orthogonal matrix. The final H is upper Hessenberg and
// H[unconverged:ihi+1,unconverged:ihi+1] is upper quasi-triangular.
//
// If unconverged is positive and wantt is false, on return the remaining
// unconverged eigenvalues are the eigenvalues of the upper Hessenberg matrix
// H[ilo:unconverged,ilo:unconverged].
//
// If unconverged is positive and wantz is true, then on return
// (final Z) = (initial Z)*U,
// where U is the orthogonal matrix in (*) regardless of the value of wantt.
//
// Dlahqr is an internal routine. It is exported for testing purposes.
func (impl Implementation) Dlahqr(wantt, wantz bool, n, ilo, ihi int, h []float64, ldh int, wr, wi []float64, iloz, ihiz int, z []float64, ldz int) (unconverged int) {
switch {
case n < 0:
panic(nLT0)
case ilo < 0, max(0, ihi) < ilo:
panic(badIlo)
case ihi >= n:
panic(badIhi)
case ldh < max(1, n):
panic(badLdH)
case wantz && (iloz < 0 || ilo < iloz):
panic(badIloz)
case wantz && (ihiz < ihi || n <= ihiz):
panic(badIhiz)
case ldz < 1, wantz && ldz < n:
panic(badLdZ)
}
// Quick return if possible.
if n == 0 {
return 0
}
switch {
case len(h) < (n-1)*ldh+n:
panic(shortH)
case len(wr) != ihi+1:
panic(shortWr)
case len(wi) != ihi+1:
panic(shortWi)
case wantz && len(z) < (n-1)*ldz+n:
panic(shortZ)
case ilo > 0 && h[ilo*ldh+ilo-1] != 0:
panic(notIsolated)
}
if ilo == ihi {
wr[ilo] = h[ilo*ldh+ilo]
wi[ilo] = 0
return 0
}
// Clear out the trash.
for j := ilo; j < ihi-2; j++ {
h[(j+2)*ldh+j] = 0
h[(j+3)*ldh+j] = 0
}
if ilo <= ihi-2 {
h[ihi*ldh+ihi-2] = 0
}
nh := ihi - ilo + 1
nz := ihiz - iloz + 1
// Set machine-dependent constants for the stopping criterion.
ulp := dlamchP
smlnum := float64(nh) / ulp * dlamchS
// i1 and i2 are the indices of the first row and last column of H to
// which transformations must be applied. If eigenvalues only are being
// computed, i1 and i2 are set inside the main loop.
var i1, i2 int
if wantt {
i1 = 0
i2 = n - 1
}
itmax := 30 * max(10, nh) // Total number of QR iterations allowed.
// The main loop begins here. i is the loop index and decreases from ihi
// to ilo in steps of 1 or 2. Each iteration of the loop works with the
// active submatrix in rows and columns l to i. Eigenvalues i+1 to ihi
// have already converged. Either l = ilo or H[l,l-1] is negligible so
// that the matrix splits.
bi := blas64.Implementation()
i := ihi
for i >= ilo {
l := ilo
// Perform QR iterations on rows and columns ilo to i until a
// submatrix of order 1 or 2 splits off at the bottom because a
// subdiagonal element has become negligible.
converged := false
for its := 0; its <= itmax; its++ {
// Look for a single small subdiagonal element.
var k int
for k = i; k > l; k-- {
if math.Abs(h[k*ldh+k-1]) <= smlnum {
break
}
tst := math.Abs(h[(k-1)*ldh+k-1]) + math.Abs(h[k*ldh+k])
if tst == 0 {
if k-2 >= ilo {
tst += math.Abs(h[(k-1)*ldh+k-2])
}
if k+1 <= ihi {
tst += math.Abs(h[(k+1)*ldh+k])
}
}
// The following is a conservative small
// subdiagonal deflation criterion due to Ahues
// & Tisseur (LAWN 122, 1997). It has better
// mathematical foundation and improves accuracy
// in some cases.
if math.Abs(h[k*ldh+k-1]) <= ulp*tst {
ab := math.Max(math.Abs(h[k*ldh+k-1]), math.Abs(h[(k-1)*ldh+k]))
ba := math.Min(math.Abs(h[k*ldh+k-1]), math.Abs(h[(k-1)*ldh+k]))
aa := math.Max(math.Abs(h[k*ldh+k]), math.Abs(h[(k-1)*ldh+k-1]-h[k*ldh+k]))
bb := math.Min(math.Abs(h[k*ldh+k]), math.Abs(h[(k-1)*ldh+k-1]-h[k*ldh+k]))
s := aa + ab
if ab/s*ba <= math.Max(smlnum, aa/s*bb*ulp) {
break
}
}
}
l = k
if l > ilo {
// H[l,l-1] is negligible.
h[l*ldh+l-1] = 0
}
if l >= i-1 {
// Break the loop because a submatrix of order 1
// or 2 has split off.
converged = true
break
}
// Now the active submatrix is in rows and columns l to
// i. If eigenvalues only are being computed, only the
// active submatrix need be transformed.
if !wantt {
i1 = l
i2 = i
}
const (
dat1 = 3.0
dat2 = -0.4375
)
var h11, h21, h12, h22 float64
switch its {
case 10: // Exceptional shift.
s := math.Abs(h[(l+1)*ldh+l]) + math.Abs(h[(l+2)*ldh+l+1])
h11 = dat1*s + h[l*ldh+l]
h12 = dat2 * s
h21 = s
h22 = h11
case 20: // Exceptional shift.
s := math.Abs(h[i*ldh+i-1]) + math.Abs(h[(i-1)*ldh+i-2])
h11 = dat1*s + h[i*ldh+i]
h12 = dat2 * s
h21 = s
h22 = h11
default: // Prepare to use Francis' double shift (i.e.,
// 2nd degree generalized Rayleigh quotient).
h11 = h[(i-1)*ldh+i-1]
h21 = h[i*ldh+i-1]
h12 = h[(i-1)*ldh+i]
h22 = h[i*ldh+i]
}
s := math.Abs(h11) + math.Abs(h12) + math.Abs(h21) + math.Abs(h22)
var (
rt1r, rt1i float64
rt2r, rt2i float64
)
if s != 0 {
h11 /= s
h21 /= s
h12 /= s
h22 /= s
tr := (h11 + h22) / 2
det := (h11-tr)*(h22-tr) - h12*h21
rtdisc := math.Sqrt(math.Abs(det))
if det >= 0 {
// Complex conjugate shifts.
rt1r = tr * s
rt2r = rt1r
rt1i = rtdisc * s
rt2i = -rt1i
} else {
// Real shifts (use only one of them).
rt1r = tr + rtdisc
rt2r = tr - rtdisc
if math.Abs(rt1r-h22) <= math.Abs(rt2r-h22) {
rt1r *= s
rt2r = rt1r
} else {
rt2r *= s
rt1r = rt2r
}
rt1i = 0
rt2i = 0
}
}
// Look for two consecutive small subdiagonal elements.
var m int
var v [3]float64
for m = i - 2; m >= l; m-- {
// Determine the effect of starting the
// double-shift QR iteration at row m, and see
// if this would make H[m,m-1] negligible. The
// following uses scaling to avoid overflows and
// most underflows.
h21s := h[(m+1)*ldh+m]
s := math.Abs(h[m*ldh+m]-rt2r) + math.Abs(rt2i) + math.Abs(h21s)
h21s /= s
v[0] = h21s*h[m*ldh+m+1] + (h[m*ldh+m]-rt1r)*((h[m*ldh+m]-rt2r)/s) - rt2i/s*rt1i
v[1] = h21s * (h[m*ldh+m] + h[(m+1)*ldh+m+1] - rt1r - rt2r)
v[2] = h21s * h[(m+2)*ldh+m+1]
s = math.Abs(v[0]) + math.Abs(v[1]) + math.Abs(v[2])
v[0] /= s
v[1] /= s
v[2] /= s
if m == l {
break
}
dsum := math.Abs(h[(m-1)*ldh+m-1]) + math.Abs(h[m*ldh+m]) + math.Abs(h[(m+1)*ldh+m+1])
if math.Abs(h[m*ldh+m-1])*(math.Abs(v[1])+math.Abs(v[2])) <= ulp*math.Abs(v[0])*dsum {
break
}
}
// Double-shift QR step.
for k := m; k < i; k++ {
// The first iteration of this loop determines a
// reflection G from the vector V and applies it
// from left and right to H, thus creating a
// non-zero bulge below the subdiagonal.
//
// Each subsequent iteration determines a
// reflection G to restore the Hessenberg form
// in the (k-1)th column, and thus chases the
// bulge one step toward the bottom of the
// active submatrix. nr is the order of G.
nr := min(3, i-k+1)
if k > m {
bi.Dcopy(nr, h[k*ldh+k-1:], ldh, v[:], 1)
}
var t0 float64
v[0], t0 = impl.Dlarfg(nr, v[0], v[1:], 1)
if k > m {
h[k*ldh+k-1] = v[0]
h[(k+1)*ldh+k-1] = 0
if k < i-1 {
h[(k+2)*ldh+k-1] = 0
}
} else if m > l {
// Use the following instead of H[k,k-1] = -H[k,k-1]
// to avoid a bug when v[1] and v[2] underflow.
h[k*ldh+k-1] *= 1 - t0
}
t1 := t0 * v[1]
if nr == 3 {
t2 := t0 * v[2]
// Apply G from the left to transform
// the rows of the matrix in columns k
// to i2.
for j := k; j <= i2; j++ {
sum := h[k*ldh+j] + v[1]*h[(k+1)*ldh+j] + v[2]*h[(k+2)*ldh+j]
h[k*ldh+j] -= sum * t0
h[(k+1)*ldh+j] -= sum * t1
h[(k+2)*ldh+j] -= sum * t2
}
// Apply G from the right to transform
// the columns of the matrix in rows i1
// to min(k+3,i).
for j := i1; j <= min(k+3, i); j++ {
sum := h[j*ldh+k] + v[1]*h[j*ldh+k+1] + v[2]*h[j*ldh+k+2]
h[j*ldh+k] -= sum * t0
h[j*ldh+k+1] -= sum * t1
h[j*ldh+k+2] -= sum * t2
}
if wantz {
// Accumulate transformations in the matrix Z.
for j := iloz; j <= ihiz; j++ {
sum := z[j*ldz+k] + v[1]*z[j*ldz+k+1] + v[2]*z[j*ldz+k+2]
z[j*ldz+k] -= sum * t0
z[j*ldz+k+1] -= sum * t1
z[j*ldz+k+2] -= sum * t2
}
}
} else if nr == 2 {
// Apply G from the left to transform
// the rows of the matrix in columns k
// to i2.
for j := k; j <= i2; j++ {
sum := h[k*ldh+j] + v[1]*h[(k+1)*ldh+j]
h[k*ldh+j] -= sum * t0
h[(k+1)*ldh+j] -= sum * t1
}
// Apply G from the right to transform
// the columns of the matrix in rows i1
// to min(k+3,i).
for j := i1; j <= i; j++ {
sum := h[j*ldh+k] + v[1]*h[j*ldh+k+1]
h[j*ldh+k] -= sum * t0
h[j*ldh+k+1] -= sum * t1
}
if wantz {
// Accumulate transformations in the matrix Z.
for j := iloz; j <= ihiz; j++ {
sum := z[j*ldz+k] + v[1]*z[j*ldz+k+1]
z[j*ldz+k] -= sum * t0
z[j*ldz+k+1] -= sum * t1
}
}
}
}
}
if !converged {
// The QR iteration finished without splitting off a
// submatrix of order 1 or 2.
return i + 1
}
if l == i {
// H[i,i-1] is negligible: one eigenvalue has converged.
wr[i] = h[i*ldh+i]
wi[i] = 0
} else if l == i-1 {
// H[i-1,i-2] is negligible: a pair of eigenvalues have converged.
// Transform the 2×2 submatrix to standard Schur form,
// and compute and store the eigenvalues.
var cs, sn float64
a, b := h[(i-1)*ldh+i-1], h[(i-1)*ldh+i]
c, d := h[i*ldh+i-1], h[i*ldh+i]
a, b, c, d, wr[i-1], wi[i-1], wr[i], wi[i], cs, sn = impl.Dlanv2(a, b, c, d)
h[(i-1)*ldh+i-1], h[(i-1)*ldh+i] = a, b
h[i*ldh+i-1], h[i*ldh+i] = c, d
if wantt {
// Apply the transformation to the rest of H.
if i2 > i {
bi.Drot(i2-i, h[(i-1)*ldh+i+1:], 1, h[i*ldh+i+1:], 1, cs, sn)
}
bi.Drot(i-i1-1, h[i1*ldh+i-1:], ldh, h[i1*ldh+i:], ldh, cs, sn)
}
if wantz {
// Apply the transformation to Z.
bi.Drot(nz, z[iloz*ldz+i-1:], ldz, z[iloz*ldz+i:], ldz, cs, sn)
}
}
// Return to start of the main loop with new value of i.
i = l - 1
}
return 0
}