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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 }