--- /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
+}