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// 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 "gonum.org/v1/gonum/lapack"
// Dtrexc reorders the real Schur factorization of a n×n real matrix
// A = Q*T*Qᵀ
// so that the diagonal block of T with row index ifst is moved to row ilst.
//
// On entry, 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 be reordered by an orthogonal similarity transformation Z
// as Zᵀ*T*Z, and will be again in Schur canonical form.
//
// If compq is lapack.UpdateSchur, on return the matrix Q of Schur vectors will be
// updated by post-multiplying it with Z.
// If compq is lapack.UpdateSchurNone, the matrix Q is not referenced and will not be
// updated.
// For other values of compq Dtrexc will panic.
//
// ifst and ilst specify the reordering of the diagonal blocks of T. The block
// with row index ifst is moved to row ilst, by a sequence of transpositions
// between adjacent blocks.
//
// If ifst points to the second row of a 2×2 block, ifstOut will point to the
// first row, otherwise it will be equal to ifst.
//
// ilstOut will point to the first row of the block in its final position. If ok
// is true, ilstOut may differ from ilst by +1 or -1.
//
// It must hold that
// 0 <= ifst < n, and 0 <= ilst < n,
// otherwise Dtrexc will panic.
//
// If ok is false, two adjacent blocks were too close to swap because the
// problem is very ill-conditioned. T may have been partially reordered, and
// ilstOut will point to the first row of the block at the position to which it
// has been moved.
//
// work must have length at least n, otherwise Dtrexc will panic.
//
// Dtrexc is an internal routine. It is exported for testing purposes.
func (impl Implementation) Dtrexc(compq lapack.UpdateSchurComp, n int, t []float64, ldt int, q []float64, ldq int, ifst, ilst int, work []float64) (ifstOut, ilstOut int, ok bool) {
switch {
case compq != lapack.UpdateSchur && compq != lapack.UpdateSchurNone:
panic(badUpdateSchurComp)
case n < 0:
panic(nLT0)
case ldt < max(1, n):
panic(badLdT)
case ldq < 1, compq == lapack.UpdateSchur && ldq < n:
panic(badLdQ)
case (ifst < 0 || n <= ifst) && n > 0:
panic(badIfst)
case (ilst < 0 || n <= ilst) && n > 0:
panic(badIlst)
}
// Quick return if possible.
if n == 0 {
return ifst, ilst, true
}
switch {
case len(t) < (n-1)*ldt+n:
panic(shortT)
case compq == lapack.UpdateSchur && len(q) < (n-1)*ldq+n:
panic(shortQ)
case len(work) < n:
panic(shortWork)
}
// Quick return if possible.
if n == 1 {
return ifst, ilst, true
}
// Determine the first row of specified block
// and find out it is 1×1 or 2×2.
if ifst > 0 && t[ifst*ldt+ifst-1] != 0 {
ifst--
}
nbf := 1 // Size of the first block.
if ifst+1 < n && t[(ifst+1)*ldt+ifst] != 0 {
nbf = 2
}
// Determine the first row of the final block
// and find out it is 1×1 or 2×2.
if ilst > 0 && t[ilst*ldt+ilst-1] != 0 {
ilst--
}
nbl := 1 // Size of the last block.
if ilst+1 < n && t[(ilst+1)*ldt+ilst] != 0 {
nbl = 2
}
ok = true
wantq := compq == lapack.UpdateSchur
switch {
case ifst == ilst:
return ifst, ilst, true
case ifst < ilst:
// Update ilst.
switch {
case nbf == 2 && nbl == 1:
ilst--
case nbf == 1 && nbl == 2:
ilst++
}
here := ifst
for here < ilst {
// Swap block with next one below.
if nbf == 1 || nbf == 2 {
// Current block either 1×1 or 2×2.
nbnext := 1 // Size of the next block.
if here+nbf+1 < n && t[(here+nbf+1)*ldt+here+nbf] != 0 {
nbnext = 2
}
ok = impl.Dlaexc(wantq, n, t, ldt, q, ldq, here, nbf, nbnext, work)
if !ok {
return ifst, here, false
}
here += nbnext
// Test if 2×2 block breaks into two 1×1 blocks.
if nbf == 2 && t[(here+1)*ldt+here] == 0 {
nbf = 3
}
continue
}
// Current block consists of two 1×1 blocks each of
// which must be swapped individually.
nbnext := 1 // Size of the next block.
if here+3 < n && t[(here+3)*ldt+here+2] != 0 {
nbnext = 2
}
ok = impl.Dlaexc(wantq, n, t, ldt, q, ldq, here+1, 1, nbnext, work)
if !ok {
return ifst, here, false
}
if nbnext == 1 {
// Swap two 1×1 blocks, no problems possible.
impl.Dlaexc(wantq, n, t, ldt, q, ldq, here, 1, nbnext, work)
here++
continue
}
// Recompute nbnext in case 2×2 split.
if t[(here+2)*ldt+here+1] == 0 {
nbnext = 1
}
if nbnext == 2 {
// 2×2 block did not split.
ok = impl.Dlaexc(wantq, n, t, ldt, q, ldq, here, 1, nbnext, work)
if !ok {
return ifst, here, false
}
} else {
// 2×2 block did split.
impl.Dlaexc(wantq, n, t, ldt, q, ldq, here, 1, 1, work)
impl.Dlaexc(wantq, n, t, ldt, q, ldq, here+1, 1, 1, work)
}
here += 2
}
return ifst, here, true
default: // ifst > ilst
here := ifst
for here > ilst {
// Swap block with next one above.
if nbf == 1 || nbf == 2 {
// Current block either 1×1 or 2×2.
nbnext := 1
if here-2 >= 0 && t[(here-1)*ldt+here-2] != 0 {
nbnext = 2
}
ok = impl.Dlaexc(wantq, n, t, ldt, q, ldq, here-nbnext, nbnext, nbf, work)
if !ok {
return ifst, here, false
}
here -= nbnext
// Test if 2×2 block breaks into two 1×1 blocks.
if nbf == 2 && t[(here+1)*ldt+here] == 0 {
nbf = 3
}
continue
}
// Current block consists of two 1×1 blocks each of
// which must be swapped individually.
nbnext := 1
if here-2 >= 0 && t[(here-1)*ldt+here-2] != 0 {
nbnext = 2
}
ok = impl.Dlaexc(wantq, n, t, ldt, q, ldq, here-nbnext, nbnext, 1, work)
if !ok {
return ifst, here, false
}
if nbnext == 1 {
// Swap two 1×1 blocks, no problems possible.
impl.Dlaexc(wantq, n, t, ldt, q, ldq, here, nbnext, 1, work)
here--
continue
}
// Recompute nbnext in case 2×2 split.
if t[here*ldt+here-1] == 0 {
nbnext = 1
}
if nbnext == 2 {
// 2×2 block did not split.
ok = impl.Dlaexc(wantq, n, t, ldt, q, ldq, here-1, 2, 1, work)
if !ok {
return ifst, here, false
}
} else {
// 2×2 block did split.
impl.Dlaexc(wantq, n, t, ldt, q, ldq, here, 1, 1, work)
impl.Dlaexc(wantq, n, t, ldt, q, ldq, here-1, 1, 1, work)
}
here -= 2
}
return ifst, here, true
}
}