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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" "gonum.org/v1/gonum/blas/blas64" "gonum.org/v1/gonum/lapack" ) // Dlaexc swaps two adjacent diagonal blocks of order 1 or 2 in an n×n upper // quasi-triangular matrix T by an orthogonal similarity transformation. // // 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 // contain the updated matrix again in Schur canonical form. // // If wantq is true, the transformation is accumulated in the n×n matrix Q, // otherwise Q is not referenced. // // j1 is the index of the first row of the first block. n1 and n2 are the order // of the first and second block, respectively. // // work must have length at least n, otherwise Dlaexc will panic. // // If ok is false, the transformed matrix T would be too far from Schur form. // The blocks are not swapped, and T and Q are not modified. // // If n1 and n2 are both equal to 1, Dlaexc will always return true. // // Dlaexc is an internal routine. It is exported for testing purposes. func (impl Implementation) Dlaexc(wantq bool, n int, t []float64, ldt int, q []float64, ldq int, j1, n1, n2 int, work []float64) (ok bool) { switch { case n < 0: panic(nLT0) case ldt < max(1, n): panic(badLdT) case wantq && ldt < max(1, n): panic(badLdQ) case j1 < 0 || n <= j1: panic(badJ1) case len(work) < n: panic(shortWork) case n1 < 0 || 2 < n1: panic(badN1) case n2 < 0 || 2 < n2: panic(badN2) } if n == 0 || n1 == 0 || n2 == 0 { return true } switch { case len(t) < (n-1)*ldt+n: panic(shortT) case wantq && len(q) < (n-1)*ldq+n: panic(shortQ) } if j1+n1 >= n { // TODO(vladimir-ch): Reference LAPACK does this check whether // the start of the second block is in the matrix T. It returns // true if it is not and moreover it does not check whether the // whole second block fits into T. This does not feel // satisfactory. The only caller of Dlaexc is Dtrexc, so if the // caller makes sure that this does not happen, we could be // stricter here. return true } j2 := j1 + 1 j3 := j1 + 2 bi := blas64.Implementation() if n1 == 1 && n2 == 1 { // Swap two 1×1 blocks. t11 := t[j1*ldt+j1] t22 := t[j2*ldt+j2] // Determine the transformation to perform the interchange. cs, sn, _ := impl.Dlartg(t[j1*ldt+j2], t22-t11) // Apply transformation to the matrix T. if n-j3 > 0 { bi.Drot(n-j3, t[j1*ldt+j3:], 1, t[j2*ldt+j3:], 1, cs, sn) } if j1 > 0 { bi.Drot(j1, t[j1:], ldt, t[j2:], ldt, cs, sn) } t[j1*ldt+j1] = t22 t[j2*ldt+j2] = t11 if wantq { // Accumulate transformation in the matrix Q. bi.Drot(n, q[j1:], ldq, q[j2:], ldq, cs, sn) } return true } // Swapping involves at least one 2×2 block. // // Copy the diagonal block of order n1+n2 to the local array d and // compute its norm. nd := n1 + n2 var d [16]float64 const ldd = 4 impl.Dlacpy(blas.All, nd, nd, t[j1*ldt+j1:], ldt, d[:], ldd) dnorm := impl.Dlange(lapack.MaxAbs, nd, nd, d[:], ldd, work) // Compute machine-dependent threshold for test for accepting swap. eps := dlamchP thresh := math.Max(10*eps*dnorm, dlamchS/eps) // Solve T11*X - X*T22 = scale*T12 for X. var x [4]float64 const ldx = 2 scale, _, _ := impl.Dlasy2(false, false, -1, n1, n2, d[:], ldd, d[n1*ldd+n1:], ldd, d[n1:], ldd, x[:], ldx) // Swap the adjacent diagonal blocks. switch { case n1 == 1 && n2 == 2: // Generate elementary reflector H so that // ( scale, X11, X12 ) H = ( 0, 0, * ) u := [3]float64{scale, x[0], 1} _, tau := impl.Dlarfg(3, x[1], u[:2], 1) t11 := t[j1*ldt+j1] // Perform swap provisionally on diagonal block in d. impl.Dlarfx(blas.Left, 3, 3, u[:], tau, d[:], ldd, work) impl.Dlarfx(blas.Right, 3, 3, u[:], tau, d[:], ldd, work) // Test whether to reject swap. if math.Max(math.Abs(d[2*ldd]), math.Max(math.Abs(d[2*ldd+1]), math.Abs(d[2*ldd+2]-t11))) > thresh { return false } // Accept swap: apply transformation to the entire matrix T. impl.Dlarfx(blas.Left, 3, n-j1, u[:], tau, t[j1*ldt+j1:], ldt, work) impl.Dlarfx(blas.Right, j2+1, 3, u[:], tau, t[j1:], ldt, work) t[j3*ldt+j1] = 0 t[j3*ldt+j2] = 0 t[j3*ldt+j3] = t11 if wantq { // Accumulate transformation in the matrix Q. impl.Dlarfx(blas.Right, n, 3, u[:], tau, q[j1:], ldq, work) } case n1 == 2 && n2 == 1: // Generate elementary reflector H so that: // H ( -X11 ) = ( * ) // ( -X21 ) = ( 0 ) // ( scale ) = ( 0 ) u := [3]float64{1, -x[ldx], scale} _, tau := impl.Dlarfg(3, -x[0], u[1:], 1) t33 := t[j3*ldt+j3] // Perform swap provisionally on diagonal block in D. impl.Dlarfx(blas.Left, 3, 3, u[:], tau, d[:], ldd, work) impl.Dlarfx(blas.Right, 3, 3, u[:], tau, d[:], ldd, work) // Test whether to reject swap. if math.Max(math.Abs(d[ldd]), math.Max(math.Abs(d[2*ldd]), math.Abs(d[0]-t33))) > thresh { return false } // Accept swap: apply transformation to the entire matrix T. impl.Dlarfx(blas.Right, j3+1, 3, u[:], tau, t[j1:], ldt, work) impl.Dlarfx(blas.Left, 3, n-j1-1, u[:], tau, t[j1*ldt+j2:], ldt, work) t[j1*ldt+j1] = t33 t[j2*ldt+j1] = 0 t[j3*ldt+j1] = 0 if wantq { // Accumulate transformation in the matrix Q. impl.Dlarfx(blas.Right, n, 3, u[:], tau, q[j1:], ldq, work) } default: // n1 == 2 && n2 == 2 // Generate elementary reflectors H_1 and H_2 so that: // H_2 H_1 ( -X11 -X12 ) = ( * * ) // ( -X21 -X22 ) ( 0 * ) // ( scale 0 ) ( 0 0 ) // ( 0 scale ) ( 0 0 ) u1 := [3]float64{1, -x[ldx], scale} _, tau1 := impl.Dlarfg(3, -x[0], u1[1:], 1) temp := -tau1 * (x[1] + u1[1]*x[ldx+1]) u2 := [3]float64{1, -temp * u1[2], scale} _, tau2 := impl.Dlarfg(3, -temp*u1[1]-x[ldx+1], u2[1:], 1) // Perform swap provisionally on diagonal block in D. impl.Dlarfx(blas.Left, 3, 4, u1[:], tau1, d[:], ldd, work) impl.Dlarfx(blas.Right, 4, 3, u1[:], tau1, d[:], ldd, work) impl.Dlarfx(blas.Left, 3, 4, u2[:], tau2, d[ldd:], ldd, work) impl.Dlarfx(blas.Right, 4, 3, u2[:], tau2, d[1:], ldd, work) // Test whether to reject swap. m1 := math.Max(math.Abs(d[2*ldd]), math.Abs(d[2*ldd+1])) m2 := math.Max(math.Abs(d[3*ldd]), math.Abs(d[3*ldd+1])) if math.Max(m1, m2) > thresh { return false } // Accept swap: apply transformation to the entire matrix T. j4 := j1 + 3 impl.Dlarfx(blas.Left, 3, n-j1, u1[:], tau1, t[j1*ldt+j1:], ldt, work) impl.Dlarfx(blas.Right, j4+1, 3, u1[:], tau1, t[j1:], ldt, work) impl.Dlarfx(blas.Left, 3, n-j1, u2[:], tau2, t[j2*ldt+j1:], ldt, work) impl.Dlarfx(blas.Right, j4+1, 3, u2[:], tau2, t[j2:], ldt, work) t[j3*ldt+j1] = 0 t[j3*ldt+j2] = 0 t[j4*ldt+j1] = 0 t[j4*ldt+j2] = 0 if wantq { // Accumulate transformation in the matrix Q. impl.Dlarfx(blas.Right, n, 3, u1[:], tau1, q[j1:], ldq, work) impl.Dlarfx(blas.Right, n, 3, u2[:], tau2, q[j2:], ldq, work) } } if n2 == 2 { // Standardize new 2×2 block T11. a, b := t[j1*ldt+j1], t[j1*ldt+j2] c, d := t[j2*ldt+j1], t[j2*ldt+j2] var cs, sn float64 t[j1*ldt+j1], t[j1*ldt+j2], t[j2*ldt+j1], t[j2*ldt+j2], _, _, _, _, cs, sn = impl.Dlanv2(a, b, c, d) if n-j1-2 > 0 { bi.Drot(n-j1-2, t[j1*ldt+j1+2:], 1, t[j2*ldt+j1+2:], 1, cs, sn) } if j1 > 0 { bi.Drot(j1, t[j1:], ldt, t[j2:], ldt, cs, sn) } if wantq { bi.Drot(n, q[j1:], ldq, q[j2:], ldq, cs, sn) } } if n1 == 2 { // Standardize new 2×2 block T22. j3 := j1 + n2 j4 := j3 + 1 a, b := t[j3*ldt+j3], t[j3*ldt+j4] c, d := t[j4*ldt+j3], t[j4*ldt+j4] var cs, sn float64 t[j3*ldt+j3], t[j3*ldt+j4], t[j4*ldt+j3], t[j4*ldt+j4], _, _, _, _, cs, sn = impl.Dlanv2(a, b, c, d) if n-j3-2 > 0 { bi.Drot(n-j3-2, t[j3*ldt+j3+2:], 1, t[j4*ldt+j3+2:], 1, cs, sn) } bi.Drot(j3, t[j3:], ldt, t[j4:], ldt, cs, sn) if wantq { bi.Drot(n, q[j3:], ldq, q[j4:], ldq, cs, sn) } } return true }