+++ /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"
- "gonum.org/v1/gonum/lapack"
-)
-
-// Dlaexc swaps two adjacent diagonal blocks of order 1 or 2 in an n×n upper
-// quasi-triangular matrix T by an orthogonal similarity transformation.
-//
-// T must be in Schur canonical form, that is, block upper triangular with 1×1
-// and 2×2 diagonal blocks; each 2×2 diagonal block has its diagonal elements
-// equal and its off-diagonal elements of opposite sign. On return, T will
-// contain the updated matrix again in Schur canonical form.
-//
-// If wantq is true, the transformation is accumulated in the n×n matrix Q,
-// otherwise Q is not referenced.
-//
-// j1 is the index of the first row of the first block. n1 and n2 are the order
-// of the first and second block, respectively.
-//
-// work must have length at least n, otherwise Dlaexc will panic.
-//
-// If ok is false, the transformed matrix T would be too far from Schur form.
-// The blocks are not swapped, and T and Q are not modified.
-//
-// If n1 and n2 are both equal to 1, Dlaexc will always return true.
-//
-// Dlaexc is an internal routine. It is exported for testing purposes.
-func (impl Implementation) Dlaexc(wantq bool, n int, t []float64, ldt int, q []float64, ldq int, j1, n1, n2 int, work []float64) (ok bool) {
- checkMatrix(n, n, t, ldt)
- if wantq {
- checkMatrix(n, n, q, ldq)
- }
- if j1 < 0 || n <= j1 {
- panic("lapack: index j1 out of range")
- }
- if len(work) < n {
- panic(badWork)
- }
- if n1 < 0 || 2 < n1 {
- panic("lapack: invalid value of n1")
- }
- if n2 < 0 || 2 < n2 {
- panic("lapack: invalid value of n2")
- }
-
- if n == 0 || n1 == 0 || n2 == 0 {
- return true
- }
- if j1+n1 >= n {
- // TODO(vladimir-ch): Reference LAPACK does this check whether
- // the start of the second block is in the matrix T. It returns
- // true if it is not and moreover it does not check whether the
- // whole second block fits into T. This does not feel
- // satisfactory. The only caller of Dlaexc is Dtrexc, so if the
- // caller makes sure that this does not happen, we could be
- // stricter here.
- return true
- }
-
- j2 := j1 + 1
- j3 := j1 + 2
-
- bi := blas64.Implementation()
-
- if n1 == 1 && n2 == 1 {
- // Swap two 1×1 blocks.
- t11 := t[j1*ldt+j1]
- t22 := t[j2*ldt+j2]
-
- // Determine the transformation to perform the interchange.
- cs, sn, _ := impl.Dlartg(t[j1*ldt+j2], t22-t11)
-
- // Apply transformation to the matrix T.
- if n-j3 > 0 {
- bi.Drot(n-j3, t[j1*ldt+j3:], 1, t[j2*ldt+j3:], 1, cs, sn)
- }
- if j1 > 0 {
- bi.Drot(j1, t[j1:], ldt, t[j2:], ldt, cs, sn)
- }
-
- t[j1*ldt+j1] = t22
- t[j2*ldt+j2] = t11
-
- if wantq {
- // Accumulate transformation in the matrix Q.
- bi.Drot(n, q[j1:], ldq, q[j2:], ldq, cs, sn)
- }
-
- return true
- }
-
- // Swapping involves at least one 2×2 block.
- //
- // Copy the diagonal block of order n1+n2 to the local array d and
- // compute its norm.
- nd := n1 + n2
- var d [16]float64
- const ldd = 4
- impl.Dlacpy(blas.All, nd, nd, t[j1*ldt+j1:], ldt, d[:], ldd)
- dnorm := impl.Dlange(lapack.MaxAbs, nd, nd, d[:], ldd, work)
-
- // Compute machine-dependent threshold for test for accepting swap.
- eps := dlamchP
- thresh := math.Max(10*eps*dnorm, dlamchS/eps)
-
- // Solve T11*X - X*T22 = scale*T12 for X.
- var x [4]float64
- const ldx = 2
- scale, _, _ := impl.Dlasy2(false, false, -1, n1, n2, d[:], ldd, d[n1*ldd+n1:], ldd, d[n1:], ldd, x[:], ldx)
-
- // Swap the adjacent diagonal blocks.
- switch {
- case n1 == 1 && n2 == 2:
- // Generate elementary reflector H so that
- // ( scale, X11, X12 ) H = ( 0, 0, * )
- u := [3]float64{scale, x[0], 1}
- _, tau := impl.Dlarfg(3, x[1], u[:2], 1)
- t11 := t[j1*ldt+j1]
-
- // Perform swap provisionally on diagonal block in d.
- impl.Dlarfx(blas.Left, 3, 3, u[:], tau, d[:], ldd, work)
- impl.Dlarfx(blas.Right, 3, 3, u[:], tau, d[:], ldd, work)
-
- // Test whether to reject swap.
- if math.Max(math.Abs(d[2*ldd]), math.Max(math.Abs(d[2*ldd+1]), math.Abs(d[2*ldd+2]-t11))) > thresh {
- return false
- }
-
- // Accept swap: apply transformation to the entire matrix T.
- impl.Dlarfx(blas.Left, 3, n-j1, u[:], tau, t[j1*ldt+j1:], ldt, work)
- impl.Dlarfx(blas.Right, j2+1, 3, u[:], tau, t[j1:], ldt, work)
-
- t[j3*ldt+j1] = 0
- t[j3*ldt+j2] = 0
- t[j3*ldt+j3] = t11
-
- if wantq {
- // Accumulate transformation in the matrix Q.
- impl.Dlarfx(blas.Right, n, 3, u[:], tau, q[j1:], ldq, work)
- }
-
- case n1 == 2 && n2 == 1:
- // Generate elementary reflector H so that:
- // H ( -X11 ) = ( * )
- // ( -X21 ) = ( 0 )
- // ( scale ) = ( 0 )
- u := [3]float64{1, -x[ldx], scale}
- _, tau := impl.Dlarfg(3, -x[0], u[1:], 1)
- t33 := t[j3*ldt+j3]
-
- // Perform swap provisionally on diagonal block in D.
- impl.Dlarfx(blas.Left, 3, 3, u[:], tau, d[:], ldd, work)
- impl.Dlarfx(blas.Right, 3, 3, u[:], tau, d[:], ldd, work)
-
- // Test whether to reject swap.
- if math.Max(math.Abs(d[ldd]), math.Max(math.Abs(d[2*ldd]), math.Abs(d[0]-t33))) > thresh {
- return false
- }
-
- // Accept swap: apply transformation to the entire matrix T.
- impl.Dlarfx(blas.Right, j3+1, 3, u[:], tau, t[j1:], ldt, work)
- impl.Dlarfx(blas.Left, 3, n-j1-1, u[:], tau, t[j1*ldt+j2:], ldt, work)
-
- t[j1*ldt+j1] = t33
- t[j2*ldt+j1] = 0
- t[j3*ldt+j1] = 0
-
- if wantq {
- // Accumulate transformation in the matrix Q.
- impl.Dlarfx(blas.Right, n, 3, u[:], tau, q[j1:], ldq, work)
- }
-
- default: // n1 == 2 && n2 == 2
- // Generate elementary reflectors H_1 and H_2 so that:
- // H_2 H_1 ( -X11 -X12 ) = ( * * )
- // ( -X21 -X22 ) ( 0 * )
- // ( scale 0 ) ( 0 0 )
- // ( 0 scale ) ( 0 0 )
- u1 := [3]float64{1, -x[ldx], scale}
- _, tau1 := impl.Dlarfg(3, -x[0], u1[1:], 1)
-
- temp := -tau1 * (x[1] + u1[1]*x[ldx+1])
- u2 := [3]float64{1, -temp * u1[2], scale}
- _, tau2 := impl.Dlarfg(3, -temp*u1[1]-x[ldx+1], u2[1:], 1)
-
- // Perform swap provisionally on diagonal block in D.
- impl.Dlarfx(blas.Left, 3, 4, u1[:], tau1, d[:], ldd, work)
- impl.Dlarfx(blas.Right, 4, 3, u1[:], tau1, d[:], ldd, work)
- impl.Dlarfx(blas.Left, 3, 4, u2[:], tau2, d[ldd:], ldd, work)
- impl.Dlarfx(blas.Right, 4, 3, u2[:], tau2, d[1:], ldd, work)
-
- // Test whether to reject swap.
- m1 := math.Max(math.Abs(d[2*ldd]), math.Abs(d[2*ldd+1]))
- m2 := math.Max(math.Abs(d[3*ldd]), math.Abs(d[3*ldd+1]))
- if math.Max(m1, m2) > thresh {
- return false
- }
-
- // Accept swap: apply transformation to the entire matrix T.
- j4 := j1 + 3
- impl.Dlarfx(blas.Left, 3, n-j1, u1[:], tau1, t[j1*ldt+j1:], ldt, work)
- impl.Dlarfx(blas.Right, j4+1, 3, u1[:], tau1, t[j1:], ldt, work)
- impl.Dlarfx(blas.Left, 3, n-j1, u2[:], tau2, t[j2*ldt+j1:], ldt, work)
- impl.Dlarfx(blas.Right, j4+1, 3, u2[:], tau2, t[j2:], ldt, work)
-
- t[j3*ldt+j1] = 0
- t[j3*ldt+j2] = 0
- t[j4*ldt+j1] = 0
- t[j4*ldt+j2] = 0
-
- if wantq {
- // Accumulate transformation in the matrix Q.
- impl.Dlarfx(blas.Right, n, 3, u1[:], tau1, q[j1:], ldq, work)
- impl.Dlarfx(blas.Right, n, 3, u2[:], tau2, q[j2:], ldq, work)
- }
- }
-
- if n2 == 2 {
- // Standardize new 2×2 block T11.
- a, b := t[j1*ldt+j1], t[j1*ldt+j2]
- c, d := t[j2*ldt+j1], t[j2*ldt+j2]
- var cs, sn float64
- t[j1*ldt+j1], t[j1*ldt+j2], t[j2*ldt+j1], t[j2*ldt+j2], _, _, _, _, cs, sn = impl.Dlanv2(a, b, c, d)
- if n-j1-2 > 0 {
- bi.Drot(n-j1-2, t[j1*ldt+j1+2:], 1, t[j2*ldt+j1+2:], 1, cs, sn)
- }
- if j1 > 0 {
- bi.Drot(j1, t[j1:], ldt, t[j2:], ldt, cs, sn)
- }
- if wantq {
- bi.Drot(n, q[j1:], ldq, q[j2:], ldq, cs, sn)
- }
- }
- if n1 == 2 {
- // Standardize new 2×2 block T22.
- j3 := j1 + n2
- j4 := j3 + 1
- a, b := t[j3*ldt+j3], t[j3*ldt+j4]
- c, d := t[j4*ldt+j3], t[j4*ldt+j4]
- var cs, sn float64
- t[j3*ldt+j3], t[j3*ldt+j4], t[j4*ldt+j3], t[j4*ldt+j4], _, _, _, _, cs, sn = impl.Dlanv2(a, b, c, d)
- if n-j3-2 > 0 {
- bi.Drot(n-j3-2, t[j3*ldt+j3+2:], 1, t[j4*ldt+j3+2:], 1, cs, sn)
- }
- bi.Drot(j3, t[j3:], ldt, t[j4:], ldt, cs, sn)
- if wantq {
- bi.Drot(n, q[j3:], ldq, q[j4:], ldq, cs, sn)
- }
- }
-
- return true
-}
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