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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 (
"math"
"gonum.org/v1/gonum/blas/blas64"
)
// Dlasy2 solves the Sylvester matrix equation where the matrices are of order 1
// or 2. It computes the unknown n1×n2 matrix X so that
// TL*X + sgn*X*TR = scale*B if tranl == false and tranr == false,
// TLᵀ*X + sgn*X*TR = scale*B if tranl == true and tranr == false,
// TL*X + sgn*X*TRᵀ = scale*B if tranl == false and tranr == true,
// TLᵀ*X + sgn*X*TRᵀ = scale*B if tranl == true and tranr == true,
// where TL is n1×n1, TR is n2×n2, B is n1×n2, and 1 <= n1,n2 <= 2.
//
// isgn must be 1 or -1, and n1 and n2 must be 0, 1, or 2, but these conditions
// are not checked.
//
// Dlasy2 returns three values, a scale factor that is chosen less than or equal
// to 1 to prevent the solution overflowing, the infinity norm of the solution,
// and an indicator of success. If ok is false, TL and TR have eigenvalues that
// are too close, so TL or TR is perturbed to get a non-singular equation.
//
// Dlasy2 is an internal routine. It is exported for testing purposes.
func (impl Implementation) Dlasy2(tranl, tranr bool, isgn, n1, n2 int, tl []float64, ldtl int, tr []float64, ldtr int, b []float64, ldb int, x []float64, ldx int) (scale, xnorm float64, ok bool) {
// TODO(vladimir-ch): Add input validation checks conditionally skipped
// using the build tag mechanism.
ok = true
// Quick return if possible.
if n1 == 0 || n2 == 0 {
return scale, xnorm, ok
}
// Set constants to control overflow.
eps := dlamchP
smlnum := dlamchS / eps
sgn := float64(isgn)
if n1 == 1 && n2 == 1 {
// 1×1 case: TL11*X + sgn*X*TR11 = B11.
tau1 := tl[0] + sgn*tr[0]
bet := math.Abs(tau1)
if bet <= smlnum {
tau1 = smlnum
bet = smlnum
ok = false
}
scale = 1
gam := math.Abs(b[0])
if smlnum*gam > bet {
scale = 1 / gam
}
x[0] = b[0] * scale / tau1
xnorm = math.Abs(x[0])
return scale, xnorm, ok
}
if n1+n2 == 3 {
// 1×2 or 2×1 case.
var (
smin float64
tmp [4]float64 // tmp is used as a 2×2 row-major matrix.
btmp [2]float64
)
if n1 == 1 && n2 == 2 {
// 1×2 case: TL11*[X11 X12] + sgn*[X11 X12]*op[TR11 TR12] = [B11 B12].
// [TR21 TR22]
smin = math.Abs(tl[0])
smin = math.Max(smin, math.Max(math.Abs(tr[0]), math.Abs(tr[1])))
smin = math.Max(smin, math.Max(math.Abs(tr[ldtr]), math.Abs(tr[ldtr+1])))
smin = math.Max(eps*smin, smlnum)
tmp[0] = tl[0] + sgn*tr[0]
tmp[3] = tl[0] + sgn*tr[ldtr+1]
if tranr {
tmp[1] = sgn * tr[1]
tmp[2] = sgn * tr[ldtr]
} else {
tmp[1] = sgn * tr[ldtr]
tmp[2] = sgn * tr[1]
}
btmp[0] = b[0]
btmp[1] = b[1]
} else {
// 2×1 case: op[TL11 TL12]*[X11] + sgn*[X11]*TR11 = [B11].
// [TL21 TL22]*[X21] [X21] [B21]
smin = math.Abs(tr[0])
smin = math.Max(smin, math.Max(math.Abs(tl[0]), math.Abs(tl[1])))
smin = math.Max(smin, math.Max(math.Abs(tl[ldtl]), math.Abs(tl[ldtl+1])))
smin = math.Max(eps*smin, smlnum)
tmp[0] = tl[0] + sgn*tr[0]
tmp[3] = tl[ldtl+1] + sgn*tr[0]
if tranl {
tmp[1] = tl[ldtl]
tmp[2] = tl[1]
} else {
tmp[1] = tl[1]
tmp[2] = tl[ldtl]
}
btmp[0] = b[0]
btmp[1] = b[ldb]
}
// Solve 2×2 system using complete pivoting.
// Set pivots less than smin to smin.
bi := blas64.Implementation()
ipiv := bi.Idamax(len(tmp), tmp[:], 1)
// Compute the upper triangular matrix [u11 u12].
// [ 0 u22]
u11 := tmp[ipiv]
if math.Abs(u11) <= smin {
ok = false
u11 = smin
}
locu12 := [4]int{1, 0, 3, 2} // Index in tmp of the element on the same row as the pivot.
u12 := tmp[locu12[ipiv]]
locl21 := [4]int{2, 3, 0, 1} // Index in tmp of the element on the same column as the pivot.
l21 := tmp[locl21[ipiv]] / u11
locu22 := [4]int{3, 2, 1, 0} // Index in tmp of the remaining element.
u22 := tmp[locu22[ipiv]] - l21*u12
if math.Abs(u22) <= smin {
ok = false
u22 = smin
}
if ipiv&0x2 != 0 { // true for ipiv equal to 2 and 3.
// The pivot was in the second row, swap the elements of
// the right-hand side.
btmp[0], btmp[1] = btmp[1], btmp[0]-l21*btmp[1]
} else {
btmp[1] -= l21 * btmp[0]
}
scale = 1
if 2*smlnum*math.Abs(btmp[1]) > math.Abs(u22) || 2*smlnum*math.Abs(btmp[0]) > math.Abs(u11) {
scale = 0.5 / math.Max(math.Abs(btmp[0]), math.Abs(btmp[1]))
btmp[0] *= scale
btmp[1] *= scale
}
// Solve the system [u11 u12] [x21] = [ btmp[0] ].
// [ 0 u22] [x22] [ btmp[1] ]
x22 := btmp[1] / u22
x21 := btmp[0]/u11 - (u12/u11)*x22
if ipiv&0x1 != 0 { // true for ipiv equal to 1 and 3.
// The pivot was in the second column, swap the elements
// of the solution.
x21, x22 = x22, x21
}
x[0] = x21
if n1 == 1 {
x[1] = x22
xnorm = math.Abs(x[0]) + math.Abs(x[1])
} else {
x[ldx] = x22
xnorm = math.Max(math.Abs(x[0]), math.Abs(x[ldx]))
}
return scale, xnorm, ok
}
// 2×2 case: op[TL11 TL12]*[X11 X12] + SGN*[X11 X12]*op[TR11 TR12] = [B11 B12].
// [TL21 TL22] [X21 X22] [X21 X22] [TR21 TR22] [B21 B22]
//
// Solve equivalent 4×4 system using complete pivoting.
// Set pivots less than smin to smin.
smin := math.Max(math.Abs(tr[0]), math.Abs(tr[1]))
smin = math.Max(smin, math.Max(math.Abs(tr[ldtr]), math.Abs(tr[ldtr+1])))
smin = math.Max(smin, math.Max(math.Abs(tl[0]), math.Abs(tl[1])))
smin = math.Max(smin, math.Max(math.Abs(tl[ldtl]), math.Abs(tl[ldtl+1])))
smin = math.Max(eps*smin, smlnum)
var t [4][4]float64
t[0][0] = tl[0] + sgn*tr[0]
t[1][1] = tl[0] + sgn*tr[ldtr+1]
t[2][2] = tl[ldtl+1] + sgn*tr[0]
t[3][3] = tl[ldtl+1] + sgn*tr[ldtr+1]
if tranl {
t[0][2] = tl[ldtl]
t[1][3] = tl[ldtl]
t[2][0] = tl[1]
t[3][1] = tl[1]
} else {
t[0][2] = tl[1]
t[1][3] = tl[1]
t[2][0] = tl[ldtl]
t[3][1] = tl[ldtl]
}
if tranr {
t[0][1] = sgn * tr[1]
t[1][0] = sgn * tr[ldtr]
t[2][3] = sgn * tr[1]
t[3][2] = sgn * tr[ldtr]
} else {
t[0][1] = sgn * tr[ldtr]
t[1][0] = sgn * tr[1]
t[2][3] = sgn * tr[ldtr]
t[3][2] = sgn * tr[1]
}
var btmp [4]float64
btmp[0] = b[0]
btmp[1] = b[1]
btmp[2] = b[ldb]
btmp[3] = b[ldb+1]
// Perform elimination.
var jpiv [4]int // jpiv records any column swaps for pivoting.
for i := 0; i < 3; i++ {
var (
xmax float64
ipsv, jpsv int
)
for ip := i; ip < 4; ip++ {
for jp := i; jp < 4; jp++ {
if math.Abs(t[ip][jp]) >= xmax {
xmax = math.Abs(t[ip][jp])
ipsv = ip
jpsv = jp
}
}
}
if ipsv != i {
// The pivot is not in the top row of the unprocessed
// block, swap rows ipsv and i of t and btmp.
t[ipsv], t[i] = t[i], t[ipsv]
btmp[ipsv], btmp[i] = btmp[i], btmp[ipsv]
}
if jpsv != i {
// The pivot is not in the left column of the
// unprocessed block, swap columns jpsv and i of t.
for k := 0; k < 4; k++ {
t[k][jpsv], t[k][i] = t[k][i], t[k][jpsv]
}
}
jpiv[i] = jpsv
if math.Abs(t[i][i]) < smin {
ok = false
t[i][i] = smin
}
for k := i + 1; k < 4; k++ {
t[k][i] /= t[i][i]
btmp[k] -= t[k][i] * btmp[i]
for j := i + 1; j < 4; j++ {
t[k][j] -= t[k][i] * t[i][j]
}
}
}
if math.Abs(t[3][3]) < smin {
ok = false
t[3][3] = smin
}
scale = 1
if 8*smlnum*math.Abs(btmp[0]) > math.Abs(t[0][0]) ||
8*smlnum*math.Abs(btmp[1]) > math.Abs(t[1][1]) ||
8*smlnum*math.Abs(btmp[2]) > math.Abs(t[2][2]) ||
8*smlnum*math.Abs(btmp[3]) > math.Abs(t[3][3]) {
maxbtmp := math.Max(math.Abs(btmp[0]), math.Abs(btmp[1]))
maxbtmp = math.Max(maxbtmp, math.Max(math.Abs(btmp[2]), math.Abs(btmp[3])))
scale = 1 / 8 / maxbtmp
btmp[0] *= scale
btmp[1] *= scale
btmp[2] *= scale
btmp[3] *= scale
}
// Compute the solution of the upper triangular system t * tmp = btmp.
var tmp [4]float64
for i := 3; i >= 0; i-- {
temp := 1 / t[i][i]
tmp[i] = btmp[i] * temp
for j := i + 1; j < 4; j++ {
tmp[i] -= temp * t[i][j] * tmp[j]
}
}
for i := 2; i >= 0; i-- {
if jpiv[i] != i {
tmp[i], tmp[jpiv[i]] = tmp[jpiv[i]], tmp[i]
}
}
x[0] = tmp[0]
x[1] = tmp[1]
x[ldx] = tmp[2]
x[ldx+1] = tmp[3]
xnorm = math.Max(math.Abs(tmp[0])+math.Abs(tmp[1]), math.Abs(tmp[2])+math.Abs(tmp[3]))
return scale, xnorm, ok
}