+++ /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 (
- "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) {
- checkMatrix(n, n, h, ldh)
- switch {
- case ilo < 0 || max(0, ihi) < ilo:
- panic(badIlo)
- case n <= ihi:
- panic(badIhi)
- case len(wr) != ihi+1:
- panic("lapack: bad length of wr")
- case len(wi) != ihi+1:
- panic("lapack: bad length of wi")
- case ilo > 0 && h[ilo*ldh+ilo-1] != 0:
- panic("lapack: block is not isolated")
- }
- if wantz {
- checkMatrix(n, n, z, ldz)
- switch {
- case iloz < 0 || ilo < iloz:
- panic("lapack: iloz out of range")
- case ihiz < ihi || n <= ihiz:
- panic("lapack: ihiz out of range")
- }
- }
-
- // Quick return if possible.
- if n == 0 {
- return 0
- }
- 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
-}
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