+++ /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"
- "gonum.org/v1/gonum/blas/blas64"
-)
-
-// Dlaqr5 performs a single small-bulge multi-shift QR sweep on an isolated
-// block of a Hessenberg matrix.
-//
-// wantt and wantz determine whether the quasi-triangular Schur factor and the
-// orthogonal Schur factor, respectively, will be computed.
-//
-// kacc22 specifies the computation mode of far-from-diagonal orthogonal
-// updates. Permitted values are:
-// 0: Dlaqr5 will not accumulate reflections and will not use matrix-matrix
-// multiply to update far-from-diagonal matrix entries.
-// 1: Dlaqr5 will accumulate reflections and use matrix-matrix multiply to
-// update far-from-diagonal matrix entries.
-// 2: Dlaqr5 will accumulate reflections, use matrix-matrix multiply to update
-// far-from-diagonal matrix entries, and take advantage of 2×2 block
-// structure during matrix multiplies.
-// For other values of kacc2 Dlaqr5 will panic.
-//
-// n is the order of the Hessenberg matrix H.
-//
-// ktop and kbot are indices of the first and last row and column of an isolated
-// diagonal block upon which the QR sweep will be applied. It must hold that
-// ktop == 0, or 0 < ktop <= n-1 and H[ktop, ktop-1] == 0, and
-// kbot == n-1, or 0 <= kbot < n-1 and H[kbot+1, kbot] == 0,
-// otherwise Dlaqr5 will panic.
-//
-// nshfts is the number of simultaneous shifts. It must be positive and even,
-// otherwise Dlaqr5 will panic.
-//
-// sr and si contain the real and imaginary parts, respectively, of the shifts
-// of origin that define the multi-shift QR sweep. On return both slices may be
-// reordered by Dlaqr5. Their length must be equal to nshfts, otherwise Dlaqr5
-// will panic.
-//
-// h and ldh represent the Hessenberg matrix H of size n×n. On return
-// multi-shift QR sweep with shifts sr+i*si has been applied to the isolated
-// diagonal block in rows and columns ktop through kbot, inclusive.
-//
-// iloz and ihiz specify the rows of Z to which transformations will be applied
-// if wantz is true. It must hold that 0 <= iloz <= ihiz < n, otherwise Dlaqr5
-// will panic.
-//
-// z and ldz represent the matrix Z of size n×n. If wantz is true, the QR sweep
-// orthogonal similarity transformation is accumulated into
-// z[iloz:ihiz,iloz:ihiz] from the right, otherwise z not referenced.
-//
-// v and ldv represent an auxiliary matrix V of size (nshfts/2)×3. Note that V
-// is transposed with respect to the reference netlib implementation.
-//
-// u and ldu represent an auxiliary matrix of size (3*nshfts-3)×(3*nshfts-3).
-//
-// wh and ldwh represent an auxiliary matrix of size (3*nshfts-3)×nh.
-//
-// wv and ldwv represent an auxiliary matrix of size nv×(3*nshfts-3).
-//
-// Dlaqr5 is an internal routine. It is exported for testing purposes.
-func (impl Implementation) Dlaqr5(wantt, wantz bool, kacc22 int, n, ktop, kbot, nshfts int, sr, si []float64, h []float64, ldh int, iloz, ihiz int, z []float64, ldz int, v []float64, ldv int, u []float64, ldu int, nv int, wv []float64, ldwv int, nh int, wh []float64, ldwh int) {
- checkMatrix(n, n, h, ldh)
- if ktop < 0 || n <= ktop {
- panic("lapack: invalid value of ktop")
- }
- if ktop > 0 && h[ktop*ldh+ktop-1] != 0 {
- panic("lapack: diagonal block is not isolated")
- }
- if kbot < 0 || n <= kbot {
- panic("lapack: invalid value of kbot")
- }
- if kbot < n-1 && h[(kbot+1)*ldh+kbot] != 0 {
- panic("lapack: diagonal block is not isolated")
- }
- if nshfts < 0 || nshfts&0x1 != 0 {
- panic("lapack: invalid number of shifts")
- }
- if len(sr) != nshfts || len(si) != nshfts {
- panic(badSlice) // TODO(vladimir-ch) Another message?
- }
- if wantz {
- if ihiz >= n {
- panic("lapack: invalid value of ihiz")
- }
- if iloz < 0 || ihiz < iloz {
- panic("lapack: invalid value of iloz")
- }
- checkMatrix(n, n, z, ldz)
- }
- checkMatrix(nshfts/2, 3, v, ldv) // Transposed w.r.t. lapack.
- checkMatrix(3*nshfts-3, 3*nshfts-3, u, ldu)
- checkMatrix(nv, 3*nshfts-3, wv, ldwv)
- checkMatrix(3*nshfts-3, nh, wh, ldwh)
- if kacc22 != 0 && kacc22 != 1 && kacc22 != 2 {
- panic("lapack: invalid value of kacc22")
- }
-
- // If there are no shifts, then there is nothing to do.
- if nshfts < 2 {
- return
- }
- // If the active block is empty or 1×1, then there is nothing to do.
- if ktop >= kbot {
- return
- }
-
- // Shuffle shifts into pairs of real shifts and pairs of complex
- // conjugate shifts assuming complex conjugate shifts are already
- // adjacent to one another.
- for i := 0; i < nshfts-2; i += 2 {
- if si[i] == -si[i+1] {
- continue
- }
- sr[i], sr[i+1], sr[i+2] = sr[i+1], sr[i+2], sr[i]
- si[i], si[i+1], si[i+2] = si[i+1], si[i+2], si[i]
- }
-
- // Note: lapack says that nshfts must be even but allows it to be odd
- // anyway. We panic above if nshfts is not even, so reducing it by one
- // is unnecessary. The only caller Dlaqr04 uses only even nshfts.
- //
- // The original comment and code from lapack-3.6.0/SRC/dlaqr5.f:341:
- // * ==== NSHFTS is supposed to be even, but if it is odd,
- // * . then simply reduce it by one. The shuffle above
- // * . ensures that the dropped shift is real and that
- // * . the remaining shifts are paired. ====
- // *
- // NS = NSHFTS - MOD( NSHFTS, 2 )
- ns := nshfts
-
- safmin := dlamchS
- ulp := dlamchP
- smlnum := safmin * float64(n) / ulp
-
- // Use accumulated reflections to update far-from-diagonal entries?
- accum := kacc22 == 1 || kacc22 == 2
- // If so, exploit the 2×2 block structure?
- blk22 := ns > 2 && kacc22 == 2
-
- // Clear trash.
- if ktop+2 <= kbot {
- h[(ktop+2)*ldh+ktop] = 0
- }
-
- // nbmps = number of 2-shift bulges in the chain.
- nbmps := ns / 2
-
- // kdu = width of slab.
- kdu := 6*nbmps - 3
-
- // Create and chase chains of nbmps bulges.
- for incol := 3*(1-nbmps) + ktop - 1; incol <= kbot-2; incol += 3*nbmps - 2 {
- ndcol := incol + kdu
- if accum {
- impl.Dlaset(blas.All, kdu, kdu, 0, 1, u, ldu)
- }
-
- // Near-the-diagonal bulge chase. The following loop performs
- // the near-the-diagonal part of a small bulge multi-shift QR
- // sweep. Each 6*nbmps-2 column diagonal chunk extends from
- // column incol to column ndcol (including both column incol and
- // column ndcol). The following loop chases a 3*nbmps column
- // long chain of nbmps bulges 3*nbmps-2 columns to the right.
- // (incol may be less than ktop and ndcol may be greater than
- // kbot indicating phantom columns from which to chase bulges
- // before they are actually introduced or to which to chase
- // bulges beyond column kbot.)
- for krcol := incol; krcol <= min(incol+3*nbmps-3, kbot-2); krcol++ {
- // Bulges number mtop to mbot are active double implicit
- // shift bulges. There may or may not also be small 2×2
- // bulge, if there is room. The inactive bulges (if any)
- // must wait until the active bulges have moved down the
- // diagonal to make room. The phantom matrix paradigm
- // described above helps keep track.
-
- mtop := max(0, ((ktop-1)-krcol+2)/3)
- mbot := min(nbmps, (kbot-krcol)/3) - 1
- m22 := mbot + 1
- bmp22 := (mbot < nbmps-1) && (krcol+3*m22 == kbot-2)
-
- // Generate reflections to chase the chain right one
- // column. (The minimum value of k is ktop-1.)
- for m := mtop; m <= mbot; m++ {
- k := krcol + 3*m
- if k == ktop-1 {
- impl.Dlaqr1(3, h[ktop*ldh+ktop:], ldh,
- sr[2*m], si[2*m], sr[2*m+1], si[2*m+1],
- v[m*ldv:m*ldv+3])
- alpha := v[m*ldv]
- _, v[m*ldv] = impl.Dlarfg(3, alpha, v[m*ldv+1:m*ldv+3], 1)
- continue
- }
- beta := h[(k+1)*ldh+k]
- v[m*ldv+1] = h[(k+2)*ldh+k]
- v[m*ldv+2] = h[(k+3)*ldh+k]
- beta, v[m*ldv] = impl.Dlarfg(3, beta, v[m*ldv+1:m*ldv+3], 1)
-
- // A bulge may collapse because of vigilant deflation or
- // destructive underflow. In the underflow case, try the
- // two-small-subdiagonals trick to try to reinflate the
- // bulge.
- if h[(k+3)*ldh+k] != 0 || h[(k+3)*ldh+k+1] != 0 || h[(k+3)*ldh+k+2] == 0 {
- // Typical case: not collapsed (yet).
- h[(k+1)*ldh+k] = beta
- h[(k+2)*ldh+k] = 0
- h[(k+3)*ldh+k] = 0
- continue
- }
-
- // Atypical case: collapsed. Attempt to reintroduce
- // ignoring H[k+1,k] and H[k+2,k]. If the fill
- // resulting from the new reflector is too large,
- // then abandon it. Otherwise, use the new one.
- var vt [3]float64
- impl.Dlaqr1(3, h[(k+1)*ldh+k+1:], ldh, sr[2*m],
- si[2*m], sr[2*m+1], si[2*m+1], vt[:])
- alpha := vt[0]
- _, vt[0] = impl.Dlarfg(3, alpha, vt[1:3], 1)
- refsum := vt[0] * (h[(k+1)*ldh+k] + vt[1]*h[(k+2)*ldh+k])
-
- dsum := math.Abs(h[k*ldh+k]) + math.Abs(h[(k+1)*ldh+k+1]) + math.Abs(h[(k+2)*ldh+k+2])
- if math.Abs(h[(k+2)*ldh+k]-refsum*vt[1])+math.Abs(refsum*vt[2]) > ulp*dsum {
- // Starting a new bulge here would create
- // non-negligible fill. Use the old one with
- // trepidation.
- h[(k+1)*ldh+k] = beta
- h[(k+2)*ldh+k] = 0
- h[(k+3)*ldh+k] = 0
- continue
- } else {
- // Starting a new bulge here would create
- // only negligible fill. Replace the old
- // reflector with the new one.
- h[(k+1)*ldh+k] -= refsum
- h[(k+2)*ldh+k] = 0
- h[(k+3)*ldh+k] = 0
- v[m*ldv] = vt[0]
- v[m*ldv+1] = vt[1]
- v[m*ldv+2] = vt[2]
- }
- }
-
- // Generate a 2×2 reflection, if needed.
- if bmp22 {
- k := krcol + 3*m22
- if k == ktop-1 {
- impl.Dlaqr1(2, h[(k+1)*ldh+k+1:], ldh,
- sr[2*m22], si[2*m22], sr[2*m22+1], si[2*m22+1],
- v[m22*ldv:m22*ldv+2])
- beta := v[m22*ldv]
- _, v[m22*ldv] = impl.Dlarfg(2, beta, v[m22*ldv+1:m22*ldv+2], 1)
- } else {
- beta := h[(k+1)*ldh+k]
- v[m22*ldv+1] = h[(k+2)*ldh+k]
- beta, v[m22*ldv] = impl.Dlarfg(2, beta, v[m22*ldv+1:m22*ldv+2], 1)
- h[(k+1)*ldh+k] = beta
- h[(k+2)*ldh+k] = 0
- }
- }
-
- // Multiply H by reflections from the left.
- var jbot int
- switch {
- case accum:
- jbot = min(ndcol, kbot)
- case wantt:
- jbot = n - 1
- default:
- jbot = kbot
- }
- for j := max(ktop, krcol); j <= jbot; j++ {
- mend := min(mbot+1, (j-krcol+2)/3) - 1
- for m := mtop; m <= mend; m++ {
- k := krcol + 3*m
- refsum := v[m*ldv] * (h[(k+1)*ldh+j] +
- v[m*ldv+1]*h[(k+2)*ldh+j] + v[m*ldv+2]*h[(k+3)*ldh+j])
- h[(k+1)*ldh+j] -= refsum
- h[(k+2)*ldh+j] -= refsum * v[m*ldv+1]
- h[(k+3)*ldh+j] -= refsum * v[m*ldv+2]
- }
- }
- if bmp22 {
- k := krcol + 3*m22
- for j := max(k+1, ktop); j <= jbot; j++ {
- refsum := v[m22*ldv] * (h[(k+1)*ldh+j] + v[m22*ldv+1]*h[(k+2)*ldh+j])
- h[(k+1)*ldh+j] -= refsum
- h[(k+2)*ldh+j] -= refsum * v[m22*ldv+1]
- }
- }
-
- // Multiply H by reflections from the right. Delay filling in the last row
- // until the vigilant deflation check is complete.
- var jtop int
- switch {
- case accum:
- jtop = max(ktop, incol)
- case wantt:
- jtop = 0
- default:
- jtop = ktop
- }
- for m := mtop; m <= mbot; m++ {
- if v[m*ldv] == 0 {
- continue
- }
- k := krcol + 3*m
- for j := jtop; j <= min(kbot, k+3); j++ {
- refsum := v[m*ldv] * (h[j*ldh+k+1] +
- v[m*ldv+1]*h[j*ldh+k+2] + v[m*ldv+2]*h[j*ldh+k+3])
- h[j*ldh+k+1] -= refsum
- h[j*ldh+k+2] -= refsum * v[m*ldv+1]
- h[j*ldh+k+3] -= refsum * v[m*ldv+2]
- }
- if accum {
- // Accumulate U. (If necessary, update Z later with with an
- // efficient matrix-matrix multiply.)
- kms := k - incol
- for j := max(0, ktop-incol-1); j < kdu; j++ {
- refsum := v[m*ldv] * (u[j*ldu+kms] +
- v[m*ldv+1]*u[j*ldu+kms+1] + v[m*ldv+2]*u[j*ldu+kms+2])
- u[j*ldu+kms] -= refsum
- u[j*ldu+kms+1] -= refsum * v[m*ldv+1]
- u[j*ldu+kms+2] -= refsum * v[m*ldv+2]
- }
- } else if wantz {
- // U is not accumulated, so update Z now by multiplying by
- // reflections from the right.
- for j := iloz; j <= ihiz; j++ {
- refsum := v[m*ldv] * (z[j*ldz+k+1] +
- v[m*ldv+1]*z[j*ldz+k+2] + v[m*ldv+2]*z[j*ldz+k+3])
- z[j*ldz+k+1] -= refsum
- z[j*ldz+k+2] -= refsum * v[m*ldv+1]
- z[j*ldz+k+3] -= refsum * v[m*ldv+2]
- }
- }
- }
-
- // Special case: 2×2 reflection (if needed).
- if bmp22 && v[m22*ldv] != 0 {
- k := krcol + 3*m22
- for j := jtop; j <= min(kbot, k+3); j++ {
- refsum := v[m22*ldv] * (h[j*ldh+k+1] + v[m22*ldv+1]*h[j*ldh+k+2])
- h[j*ldh+k+1] -= refsum
- h[j*ldh+k+2] -= refsum * v[m22*ldv+1]
- }
- if accum {
- kms := k - incol
- for j := max(0, ktop-incol-1); j < kdu; j++ {
- refsum := v[m22*ldv] * (u[j*ldu+kms] + v[m22*ldv+1]*u[j*ldu+kms+1])
- u[j*ldu+kms] -= refsum
- u[j*ldu+kms+1] -= refsum * v[m22*ldv+1]
- }
- } else if wantz {
- for j := iloz; j <= ihiz; j++ {
- refsum := v[m22*ldv] * (z[j*ldz+k+1] + v[m22*ldv+1]*z[j*ldz+k+2])
- z[j*ldz+k+1] -= refsum
- z[j*ldz+k+2] -= refsum * v[m22*ldv+1]
- }
- }
- }
-
- // Vigilant deflation check.
- mstart := mtop
- if krcol+3*mstart < ktop {
- mstart++
- }
- mend := mbot
- if bmp22 {
- mend++
- }
- if krcol == kbot-2 {
- mend++
- }
- for m := mstart; m <= mend; m++ {
- k := min(kbot-1, krcol+3*m)
-
- // The following convergence test requires that the tradition
- // small-compared-to-nearby-diagonals criterion and the Ahues &
- // Tisseur (LAWN 122, 1997) criteria both be satisfied. The latter
- // improves accuracy in some examples. Falling back on an alternate
- // convergence criterion when tst1 or tst2 is zero (as done here) is
- // traditional but probably unnecessary.
-
- if h[(k+1)*ldh+k] == 0 {
- continue
- }
- tst1 := math.Abs(h[k*ldh+k]) + math.Abs(h[(k+1)*ldh+k+1])
- if tst1 == 0 {
- if k >= ktop+1 {
- tst1 += math.Abs(h[k*ldh+k-1])
- }
- if k >= ktop+2 {
- tst1 += math.Abs(h[k*ldh+k-2])
- }
- if k >= ktop+3 {
- tst1 += math.Abs(h[k*ldh+k-3])
- }
- if k <= kbot-2 {
- tst1 += math.Abs(h[(k+2)*ldh+k+1])
- }
- if k <= kbot-3 {
- tst1 += math.Abs(h[(k+3)*ldh+k+1])
- }
- if k <= kbot-4 {
- tst1 += math.Abs(h[(k+4)*ldh+k+1])
- }
- }
- if math.Abs(h[(k+1)*ldh+k]) <= math.Max(smlnum, ulp*tst1) {
- h12 := math.Max(math.Abs(h[(k+1)*ldh+k]), math.Abs(h[k*ldh+k+1]))
- h21 := math.Min(math.Abs(h[(k+1)*ldh+k]), math.Abs(h[k*ldh+k+1]))
- h11 := math.Max(math.Abs(h[(k+1)*ldh+k+1]), math.Abs(h[k*ldh+k]-h[(k+1)*ldh+k+1]))
- h22 := math.Min(math.Abs(h[(k+1)*ldh+k+1]), math.Abs(h[k*ldh+k]-h[(k+1)*ldh+k+1]))
- scl := h11 + h12
- tst2 := h22 * (h11 / scl)
- if tst2 == 0 || h21*(h12/scl) <= math.Max(smlnum, ulp*tst2) {
- h[(k+1)*ldh+k] = 0
- }
- }
- }
-
- // Fill in the last row of each bulge.
- mend = min(nbmps, (kbot-krcol-1)/3) - 1
- for m := mtop; m <= mend; m++ {
- k := krcol + 3*m
- refsum := v[m*ldv] * v[m*ldv+2] * h[(k+4)*ldh+k+3]
- h[(k+4)*ldh+k+1] = -refsum
- h[(k+4)*ldh+k+2] = -refsum * v[m*ldv+1]
- h[(k+4)*ldh+k+3] -= refsum * v[m*ldv+2]
- }
- }
-
- // Use U (if accumulated) to update far-from-diagonal entries in H.
- // If required, use U to update Z as well.
- if !accum {
- continue
- }
- var jtop, jbot int
- if wantt {
- jtop = 0
- jbot = n - 1
- } else {
- jtop = ktop
- jbot = kbot
- }
- bi := blas64.Implementation()
- if !blk22 || incol < ktop || kbot < ndcol || ns <= 2 {
- // Updates not exploiting the 2×2 block structure of U. k0 and nu keep track
- // of the location and size of U in the special cases of introducing bulges
- // and chasing bulges off the bottom. In these special cases and in case the
- // number of shifts is ns = 2, there is no 2×2 block structure to exploit.
-
- k0 := max(0, ktop-incol-1)
- nu := kdu - max(0, ndcol-kbot) - k0
-
- // Horizontal multiply.
- for jcol := min(ndcol, kbot) + 1; jcol <= jbot; jcol += nh {
- jlen := min(nh, jbot-jcol+1)
- bi.Dgemm(blas.Trans, blas.NoTrans, nu, jlen, nu,
- 1, u[k0*ldu+k0:], ldu,
- h[(incol+k0+1)*ldh+jcol:], ldh,
- 0, wh, ldwh)
- impl.Dlacpy(blas.All, nu, jlen, wh, ldwh, h[(incol+k0+1)*ldh+jcol:], ldh)
- }
-
- // Vertical multiply.
- for jrow := jtop; jrow <= max(ktop, incol)-1; jrow += nv {
- jlen := min(nv, max(ktop, incol)-jrow)
- bi.Dgemm(blas.NoTrans, blas.NoTrans, jlen, nu, nu,
- 1, h[jrow*ldh+incol+k0+1:], ldh,
- u[k0*ldu+k0:], ldu,
- 0, wv, ldwv)
- impl.Dlacpy(blas.All, jlen, nu, wv, ldwv, h[jrow*ldh+incol+k0+1:], ldh)
- }
-
- // Z multiply (also vertical).
- if wantz {
- for jrow := iloz; jrow <= ihiz; jrow += nv {
- jlen := min(nv, ihiz-jrow+1)
- bi.Dgemm(blas.NoTrans, blas.NoTrans, jlen, nu, nu,
- 1, z[jrow*ldz+incol+k0+1:], ldz,
- u[k0*ldu+k0:], ldu,
- 0, wv, ldwv)
- impl.Dlacpy(blas.All, jlen, nu, wv, ldwv, z[jrow*ldz+incol+k0+1:], ldz)
- }
- }
-
- continue
- }
-
- // Updates exploiting U's 2×2 block structure.
-
- // i2, i4, j2, j4 are the last rows and columns of the blocks.
- i2 := (kdu + 1) / 2
- i4 := kdu
- j2 := i4 - i2
- j4 := kdu
-
- // kzs and knz deal with the band of zeros along the diagonal of one of the
- // triangular blocks.
- kzs := (j4 - j2) - (ns + 1)
- knz := ns + 1
-
- // Horizontal multiply.
- for jcol := min(ndcol, kbot) + 1; jcol <= jbot; jcol += nh {
- jlen := min(nh, jbot-jcol+1)
-
- // Copy bottom of H to top+kzs of scratch (the first kzs
- // rows get multiplied by zero).
- impl.Dlacpy(blas.All, knz, jlen, h[(incol+1+j2)*ldh+jcol:], ldh, wh[kzs*ldwh:], ldwh)
-
- // Multiply by U21^T.
- impl.Dlaset(blas.All, kzs, jlen, 0, 0, wh, ldwh)
- bi.Dtrmm(blas.Left, blas.Upper, blas.Trans, blas.NonUnit, knz, jlen,
- 1, u[j2*ldu+kzs:], ldu, wh[kzs*ldwh:], ldwh)
-
- // Multiply top of H by U11^T.
- bi.Dgemm(blas.Trans, blas.NoTrans, i2, jlen, j2,
- 1, u, ldu, h[(incol+1)*ldh+jcol:], ldh,
- 1, wh, ldwh)
-
- // Copy top of H to bottom of WH.
- impl.Dlacpy(blas.All, j2, jlen, h[(incol+1)*ldh+jcol:], ldh, wh[i2*ldwh:], ldwh)
-
- // Multiply by U21^T.
- bi.Dtrmm(blas.Left, blas.Lower, blas.Trans, blas.NonUnit, j2, jlen,
- 1, u[i2:], ldu, wh[i2*ldwh:], ldwh)
-
- // Multiply by U22.
- bi.Dgemm(blas.Trans, blas.NoTrans, i4-i2, jlen, j4-j2,
- 1, u[j2*ldu+i2:], ldu, h[(incol+1+j2)*ldh+jcol:], ldh,
- 1, wh[i2*ldwh:], ldwh)
-
- // Copy it back.
- impl.Dlacpy(blas.All, kdu, jlen, wh, ldwh, h[(incol+1)*ldh+jcol:], ldh)
- }
-
- // Vertical multiply.
- for jrow := jtop; jrow <= max(incol, ktop)-1; jrow += nv {
- jlen := min(nv, max(incol, ktop)-jrow)
-
- // Copy right of H to scratch (the first kzs columns get multiplied
- // by zero).
- impl.Dlacpy(blas.All, jlen, knz, h[jrow*ldh+incol+1+j2:], ldh, wv[kzs:], ldwv)
-
- // Multiply by U21.
- impl.Dlaset(blas.All, jlen, kzs, 0, 0, wv, ldwv)
- bi.Dtrmm(blas.Right, blas.Upper, blas.NoTrans, blas.NonUnit, jlen, knz,
- 1, u[j2*ldu+kzs:], ldu, wv[kzs:], ldwv)
-
- // Multiply by U11.
- bi.Dgemm(blas.NoTrans, blas.NoTrans, jlen, i2, j2,
- 1, h[jrow*ldh+incol+1:], ldh, u, ldu,
- 1, wv, ldwv)
-
- // Copy left of H to right of scratch.
- impl.Dlacpy(blas.All, jlen, j2, h[jrow*ldh+incol+1:], ldh, wv[i2:], ldwv)
-
- // Multiply by U21.
- bi.Dtrmm(blas.Right, blas.Lower, blas.NoTrans, blas.NonUnit, jlen, i4-i2,
- 1, u[i2:], ldu, wv[i2:], ldwv)
-
- // Multiply by U22.
- bi.Dgemm(blas.NoTrans, blas.NoTrans, jlen, i4-i2, j4-j2,
- 1, h[jrow*ldh+incol+1+j2:], ldh, u[j2*ldu+i2:], ldu,
- 1, wv[i2:], ldwv)
-
- // Copy it back.
- impl.Dlacpy(blas.All, jlen, kdu, wv, ldwv, h[jrow*ldh+incol+1:], ldh)
- }
-
- if !wantz {
- continue
- }
- // Multiply Z (also vertical).
- for jrow := iloz; jrow <= ihiz; jrow += nv {
- jlen := min(nv, ihiz-jrow+1)
-
- // Copy right of Z to left of scratch (first kzs columns get
- // multiplied by zero).
- impl.Dlacpy(blas.All, jlen, knz, z[jrow*ldz+incol+1+j2:], ldz, wv[kzs:], ldwv)
-
- // Multiply by U12.
- impl.Dlaset(blas.All, jlen, kzs, 0, 0, wv, ldwv)
- bi.Dtrmm(blas.Right, blas.Upper, blas.NoTrans, blas.NonUnit, jlen, knz,
- 1, u[j2*ldu+kzs:], ldu, wv[kzs:], ldwv)
-
- // Multiply by U11.
- bi.Dgemm(blas.NoTrans, blas.NoTrans, jlen, i2, j2,
- 1, z[jrow*ldz+incol+1:], ldz, u, ldu,
- 1, wv, ldwv)
-
- // Copy left of Z to right of scratch.
- impl.Dlacpy(blas.All, jlen, j2, z[jrow*ldz+incol+1:], ldz, wv[i2:], ldwv)
-
- // Multiply by U21.
- bi.Dtrmm(blas.Right, blas.Lower, blas.NoTrans, blas.NonUnit, jlen, i4-i2,
- 1, u[i2:], ldu, wv[i2:], ldwv)
-
- // Multiply by U22.
- bi.Dgemm(blas.NoTrans, blas.NoTrans, jlen, i4-i2, j4-j2,
- 1, z[jrow*ldz+incol+1+j2:], ldz, u[j2*ldu+i2:], ldu,
- 1, wv[i2:], ldwv)
-
- // Copy the result back to Z.
- impl.Dlacpy(blas.All, jlen, kdu, wv, ldwv, z[jrow*ldz+incol+1:], ldz)
- }
- }
-}
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