+++ /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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