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617 lines
20 KiB
Go
617 lines
20 KiB
Go
// Copyright ©2016 The gonum Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package native
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import (
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"math"
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"gonum.org/v1/gonum/blas"
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"gonum.org/v1/gonum/blas/blas64"
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)
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// Dlaqr5 performs a single small-bulge multi-shift QR sweep on an isolated
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// block of a Hessenberg matrix.
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//
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// wantt and wantz determine whether the quasi-triangular Schur factor and the
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// orthogonal Schur factor, respectively, will be computed.
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//
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// kacc22 specifies the computation mode of far-from-diagonal orthogonal
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// updates. Permitted values are:
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// 0: Dlaqr5 will not accumulate reflections and will not use matrix-matrix
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// multiply to update far-from-diagonal matrix entries.
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// 1: Dlaqr5 will accumulate reflections and use matrix-matrix multiply to
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// update far-from-diagonal matrix entries.
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// 2: Dlaqr5 will accumulate reflections, use matrix-matrix multiply to update
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// far-from-diagonal matrix entries, and take advantage of 2×2 block
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// structure during matrix multiplies.
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// For other values of kacc2 Dlaqr5 will panic.
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//
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// n is the order of the Hessenberg matrix H.
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//
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// ktop and kbot are indices of the first and last row and column of an isolated
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// diagonal block upon which the QR sweep will be applied. It must hold that
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// ktop == 0, or 0 < ktop <= n-1 and H[ktop, ktop-1] == 0, and
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// kbot == n-1, or 0 <= kbot < n-1 and H[kbot+1, kbot] == 0,
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// otherwise Dlaqr5 will panic.
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//
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// nshfts is the number of simultaneous shifts. It must be positive and even,
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// otherwise Dlaqr5 will panic.
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//
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// sr and si contain the real and imaginary parts, respectively, of the shifts
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// of origin that define the multi-shift QR sweep. On return both slices may be
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// reordered by Dlaqr5. Their length must be equal to nshfts, otherwise Dlaqr5
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// will panic.
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//
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// h and ldh represent the Hessenberg matrix H of size n×n. On return
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// multi-shift QR sweep with shifts sr+i*si has been applied to the isolated
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// diagonal block in rows and columns ktop through kbot, inclusive.
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//
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// iloz and ihiz specify the rows of Z to which transformations will be applied
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// if wantz is true. It must hold that 0 <= iloz <= ihiz < n, otherwise Dlaqr5
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// will panic.
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//
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// z and ldz represent the matrix Z of size n×n. If wantz is true, the QR sweep
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// orthogonal similarity transformation is accumulated into
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// z[iloz:ihiz,iloz:ihiz] from the right, otherwise z not referenced.
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//
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// v and ldv represent an auxiliary matrix V of size (nshfts/2)×3. Note that V
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// is transposed with respect to the reference netlib implementation.
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//
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// u and ldu represent an auxiliary matrix of size (3*nshfts-3)×(3*nshfts-3).
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//
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// wh and ldwh represent an auxiliary matrix of size (3*nshfts-3)×nh.
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//
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// wv and ldwv represent an auxiliary matrix of size nv×(3*nshfts-3).
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//
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// Dlaqr5 is an internal routine. It is exported for testing purposes.
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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) {
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checkMatrix(n, n, h, ldh)
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if ktop < 0 || n <= ktop {
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panic("lapack: invalid value of ktop")
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}
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if ktop > 0 && h[ktop*ldh+ktop-1] != 0 {
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panic("lapack: diagonal block is not isolated")
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}
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if kbot < 0 || n <= kbot {
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panic("lapack: invalid value of kbot")
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}
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if kbot < n-1 && h[(kbot+1)*ldh+kbot] != 0 {
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panic("lapack: diagonal block is not isolated")
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}
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if nshfts < 0 || nshfts&0x1 != 0 {
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panic("lapack: invalid number of shifts")
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}
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if len(sr) != nshfts || len(si) != nshfts {
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panic(badSlice) // TODO(vladimir-ch) Another message?
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}
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if wantz {
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if ihiz >= n {
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panic("lapack: invalid value of ihiz")
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}
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if iloz < 0 || ihiz < iloz {
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panic("lapack: invalid value of iloz")
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}
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checkMatrix(n, n, z, ldz)
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}
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checkMatrix(nshfts/2, 3, v, ldv) // Transposed w.r.t. lapack.
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checkMatrix(3*nshfts-3, 3*nshfts-3, u, ldu)
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checkMatrix(nv, 3*nshfts-3, wv, ldwv)
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checkMatrix(3*nshfts-3, nh, wh, ldwh)
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if kacc22 != 0 && kacc22 != 1 && kacc22 != 2 {
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panic("lapack: invalid value of kacc22")
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}
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// If there are no shifts, then there is nothing to do.
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if nshfts < 2 {
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return
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}
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// If the active block is empty or 1×1, then there is nothing to do.
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if ktop >= kbot {
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return
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}
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// Shuffle shifts into pairs of real shifts and pairs of complex
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// conjugate shifts assuming complex conjugate shifts are already
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// adjacent to one another.
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for i := 0; i < nshfts-2; i += 2 {
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if si[i] == -si[i+1] {
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continue
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}
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sr[i], sr[i+1], sr[i+2] = sr[i+1], sr[i+2], sr[i]
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si[i], si[i+1], si[i+2] = si[i+1], si[i+2], si[i]
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}
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// Note: lapack says that nshfts must be even but allows it to be odd
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// anyway. We panic above if nshfts is not even, so reducing it by one
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// is unnecessary. The only caller Dlaqr04 uses only even nshfts.
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//
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// The original comment and code from lapack-3.6.0/SRC/dlaqr5.f:341:
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// * ==== NSHFTS is supposed to be even, but if it is odd,
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// * . then simply reduce it by one. The shuffle above
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// * . ensures that the dropped shift is real and that
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// * . the remaining shifts are paired. ====
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// *
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// NS = NSHFTS - MOD( NSHFTS, 2 )
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ns := nshfts
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safmin := dlamchS
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ulp := dlamchP
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smlnum := safmin * float64(n) / ulp
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// Use accumulated reflections to update far-from-diagonal entries?
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accum := kacc22 == 1 || kacc22 == 2
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// If so, exploit the 2×2 block structure?
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blk22 := ns > 2 && kacc22 == 2
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// Clear trash.
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if ktop+2 <= kbot {
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h[(ktop+2)*ldh+ktop] = 0
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}
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// nbmps = number of 2-shift bulges in the chain.
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nbmps := ns / 2
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// kdu = width of slab.
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kdu := 6*nbmps - 3
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// Create and chase chains of nbmps bulges.
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for incol := 3*(1-nbmps) + ktop - 1; incol <= kbot-2; incol += 3*nbmps - 2 {
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ndcol := incol + kdu
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if accum {
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impl.Dlaset(blas.All, kdu, kdu, 0, 1, u, ldu)
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}
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// Near-the-diagonal bulge chase. The following loop performs
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// the near-the-diagonal part of a small bulge multi-shift QR
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// sweep. Each 6*nbmps-2 column diagonal chunk extends from
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// column incol to column ndcol (including both column incol and
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// column ndcol). The following loop chases a 3*nbmps column
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// long chain of nbmps bulges 3*nbmps-2 columns to the right.
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// (incol may be less than ktop and ndcol may be greater than
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// kbot indicating phantom columns from which to chase bulges
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// before they are actually introduced or to which to chase
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// bulges beyond column kbot.)
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for krcol := incol; krcol <= min(incol+3*nbmps-3, kbot-2); krcol++ {
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// Bulges number mtop to mbot are active double implicit
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// shift bulges. There may or may not also be small 2×2
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// bulge, if there is room. The inactive bulges (if any)
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// must wait until the active bulges have moved down the
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// diagonal to make room. The phantom matrix paradigm
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// described above helps keep track.
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mtop := max(0, ((ktop-1)-krcol+2)/3)
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mbot := min(nbmps, (kbot-krcol)/3) - 1
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m22 := mbot + 1
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bmp22 := (mbot < nbmps-1) && (krcol+3*m22 == kbot-2)
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// Generate reflections to chase the chain right one
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// column. (The minimum value of k is ktop-1.)
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for m := mtop; m <= mbot; m++ {
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k := krcol + 3*m
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if k == ktop-1 {
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impl.Dlaqr1(3, h[ktop*ldh+ktop:], ldh,
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sr[2*m], si[2*m], sr[2*m+1], si[2*m+1],
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v[m*ldv:m*ldv+3])
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alpha := v[m*ldv]
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_, v[m*ldv] = impl.Dlarfg(3, alpha, v[m*ldv+1:m*ldv+3], 1)
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continue
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}
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beta := h[(k+1)*ldh+k]
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v[m*ldv+1] = h[(k+2)*ldh+k]
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v[m*ldv+2] = h[(k+3)*ldh+k]
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beta, v[m*ldv] = impl.Dlarfg(3, beta, v[m*ldv+1:m*ldv+3], 1)
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// A bulge may collapse because of vigilant deflation or
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// destructive underflow. In the underflow case, try the
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// two-small-subdiagonals trick to try to reinflate the
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// bulge.
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if h[(k+3)*ldh+k] != 0 || h[(k+3)*ldh+k+1] != 0 || h[(k+3)*ldh+k+2] == 0 {
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// Typical case: not collapsed (yet).
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h[(k+1)*ldh+k] = beta
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h[(k+2)*ldh+k] = 0
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h[(k+3)*ldh+k] = 0
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continue
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}
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// Atypical case: collapsed. Attempt to reintroduce
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// ignoring H[k+1,k] and H[k+2,k]. If the fill
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// resulting from the new reflector is too large,
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// then abandon it. Otherwise, use the new one.
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var vt [3]float64
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impl.Dlaqr1(3, h[(k+1)*ldh+k+1:], ldh, sr[2*m],
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si[2*m], sr[2*m+1], si[2*m+1], vt[:])
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alpha := vt[0]
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_, vt[0] = impl.Dlarfg(3, alpha, vt[1:3], 1)
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refsum := vt[0] * (h[(k+1)*ldh+k] + vt[1]*h[(k+2)*ldh+k])
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dsum := math.Abs(h[k*ldh+k]) + math.Abs(h[(k+1)*ldh+k+1]) + math.Abs(h[(k+2)*ldh+k+2])
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if math.Abs(h[(k+2)*ldh+k]-refsum*vt[1])+math.Abs(refsum*vt[2]) > ulp*dsum {
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// Starting a new bulge here would create
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// non-negligible fill. Use the old one with
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// trepidation.
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h[(k+1)*ldh+k] = beta
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h[(k+2)*ldh+k] = 0
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h[(k+3)*ldh+k] = 0
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continue
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} else {
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// Starting a new bulge here would create
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// only negligible fill. Replace the old
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// reflector with the new one.
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h[(k+1)*ldh+k] -= refsum
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h[(k+2)*ldh+k] = 0
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h[(k+3)*ldh+k] = 0
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v[m*ldv] = vt[0]
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v[m*ldv+1] = vt[1]
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v[m*ldv+2] = vt[2]
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}
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}
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// Generate a 2×2 reflection, if needed.
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if bmp22 {
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k := krcol + 3*m22
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if k == ktop-1 {
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impl.Dlaqr1(2, h[(k+1)*ldh+k+1:], ldh,
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sr[2*m22], si[2*m22], sr[2*m22+1], si[2*m22+1],
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v[m22*ldv:m22*ldv+2])
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beta := v[m22*ldv]
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_, v[m22*ldv] = impl.Dlarfg(2, beta, v[m22*ldv+1:m22*ldv+2], 1)
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} else {
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beta := h[(k+1)*ldh+k]
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v[m22*ldv+1] = h[(k+2)*ldh+k]
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beta, v[m22*ldv] = impl.Dlarfg(2, beta, v[m22*ldv+1:m22*ldv+2], 1)
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h[(k+1)*ldh+k] = beta
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h[(k+2)*ldh+k] = 0
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}
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}
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// Multiply H by reflections from the left.
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var jbot int
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switch {
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case accum:
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jbot = min(ndcol, kbot)
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case wantt:
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jbot = n - 1
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default:
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jbot = kbot
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}
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for j := max(ktop, krcol); j <= jbot; j++ {
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mend := min(mbot+1, (j-krcol+2)/3) - 1
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for m := mtop; m <= mend; m++ {
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k := krcol + 3*m
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refsum := v[m*ldv] * (h[(k+1)*ldh+j] +
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v[m*ldv+1]*h[(k+2)*ldh+j] + v[m*ldv+2]*h[(k+3)*ldh+j])
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h[(k+1)*ldh+j] -= refsum
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h[(k+2)*ldh+j] -= refsum * v[m*ldv+1]
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h[(k+3)*ldh+j] -= refsum * v[m*ldv+2]
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}
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}
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if bmp22 {
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k := krcol + 3*m22
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for j := max(k+1, ktop); j <= jbot; j++ {
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refsum := v[m22*ldv] * (h[(k+1)*ldh+j] + v[m22*ldv+1]*h[(k+2)*ldh+j])
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h[(k+1)*ldh+j] -= refsum
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h[(k+2)*ldh+j] -= refsum * v[m22*ldv+1]
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}
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}
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// Multiply H by reflections from the right. Delay filling in the last row
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// until the vigilant deflation check is complete.
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var jtop int
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switch {
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case accum:
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jtop = max(ktop, incol)
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case wantt:
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jtop = 0
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default:
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jtop = ktop
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}
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for m := mtop; m <= mbot; m++ {
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if v[m*ldv] == 0 {
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continue
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}
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k := krcol + 3*m
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for j := jtop; j <= min(kbot, k+3); j++ {
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refsum := v[m*ldv] * (h[j*ldh+k+1] +
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v[m*ldv+1]*h[j*ldh+k+2] + v[m*ldv+2]*h[j*ldh+k+3])
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h[j*ldh+k+1] -= refsum
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h[j*ldh+k+2] -= refsum * v[m*ldv+1]
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h[j*ldh+k+3] -= refsum * v[m*ldv+2]
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}
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if accum {
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// Accumulate U. (If necessary, update Z later with with an
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// efficient matrix-matrix multiply.)
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kms := k - incol
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for j := max(0, ktop-incol-1); j < kdu; j++ {
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refsum := v[m*ldv] * (u[j*ldu+kms] +
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v[m*ldv+1]*u[j*ldu+kms+1] + v[m*ldv+2]*u[j*ldu+kms+2])
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u[j*ldu+kms] -= refsum
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u[j*ldu+kms+1] -= refsum * v[m*ldv+1]
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u[j*ldu+kms+2] -= refsum * v[m*ldv+2]
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}
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} else if wantz {
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// U is not accumulated, so update Z now by multiplying by
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// reflections from the right.
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for j := iloz; j <= ihiz; j++ {
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refsum := v[m*ldv] * (z[j*ldz+k+1] +
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v[m*ldv+1]*z[j*ldz+k+2] + v[m*ldv+2]*z[j*ldz+k+3])
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z[j*ldz+k+1] -= refsum
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z[j*ldz+k+2] -= refsum * v[m*ldv+1]
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z[j*ldz+k+3] -= refsum * v[m*ldv+2]
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}
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}
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}
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// Special case: 2×2 reflection (if needed).
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if bmp22 && v[m22*ldv] != 0 {
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k := krcol + 3*m22
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for j := jtop; j <= min(kbot, k+3); j++ {
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refsum := v[m22*ldv] * (h[j*ldh+k+1] + v[m22*ldv+1]*h[j*ldh+k+2])
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h[j*ldh+k+1] -= refsum
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h[j*ldh+k+2] -= refsum * v[m22*ldv+1]
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}
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if accum {
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kms := k - incol
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for j := max(0, ktop-incol-1); j < kdu; j++ {
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refsum := v[m22*ldv] * (u[j*ldu+kms] + v[m22*ldv+1]*u[j*ldu+kms+1])
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u[j*ldu+kms] -= refsum
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u[j*ldu+kms+1] -= refsum * v[m22*ldv+1]
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}
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} else if wantz {
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for j := iloz; j <= ihiz; j++ {
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refsum := v[m22*ldv] * (z[j*ldz+k+1] + v[m22*ldv+1]*z[j*ldz+k+2])
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z[j*ldz+k+1] -= refsum
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z[j*ldz+k+2] -= refsum * v[m22*ldv+1]
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}
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}
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}
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// Vigilant deflation check.
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mstart := mtop
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if krcol+3*mstart < ktop {
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mstart++
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}
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mend := mbot
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if bmp22 {
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mend++
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}
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if krcol == kbot-2 {
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mend++
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}
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for m := mstart; m <= mend; m++ {
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k := min(kbot-1, krcol+3*m)
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// The following convergence test requires that the tradition
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// small-compared-to-nearby-diagonals criterion and the Ahues &
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// Tisseur (LAWN 122, 1997) criteria both be satisfied. The latter
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// improves accuracy in some examples. Falling back on an alternate
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// convergence criterion when tst1 or tst2 is zero (as done here) is
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// traditional but probably unnecessary.
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if h[(k+1)*ldh+k] == 0 {
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continue
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}
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tst1 := math.Abs(h[k*ldh+k]) + math.Abs(h[(k+1)*ldh+k+1])
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if tst1 == 0 {
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if k >= ktop+1 {
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tst1 += math.Abs(h[k*ldh+k-1])
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}
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if k >= ktop+2 {
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tst1 += math.Abs(h[k*ldh+k-2])
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}
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if k >= ktop+3 {
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tst1 += math.Abs(h[k*ldh+k-3])
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}
|
||
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)
|
||
}
|
||
}
|
||
}
|