--- /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/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,
+// TL^T*X + sgn*X*TR = scale*B, if tranl == true and tranr == false,
+// TL*X + sgn*X*TR^T = scale*B, if tranl == false and tranr == true,
+// TL^T*X + sgn*X*TR^T = 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
+}
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