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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"
)
// Dgeev computes the eigenvalues and, optionally, the left and/or right
// eigenvectors for an n×n real nonsymmetric matrix A.
//
// The right eigenvector v_j of A corresponding to an eigenvalue λ_j
// is defined by
// A v_j = λ_j v_j,
// and the left eigenvector u_j corresponding to an eigenvalue λ_j is defined by
// u_jᴴ A = λ_j u_jᴴ,
// where u_jᴴ is the conjugate transpose of u_j.
//
// On return, A will be overwritten and the left and right eigenvectors will be
// stored, respectively, in the columns of the n×n matrices VL and VR in the
// same order as their eigenvalues. If the j-th eigenvalue is real, then
// u_j = VL[:,j],
// v_j = VR[:,j],
// and if it is not real, then j and j+1 form a complex conjugate pair and the
// eigenvectors can be recovered as
// u_j = VL[:,j] + i*VL[:,j+1],
// u_{j+1} = VL[:,j] - i*VL[:,j+1],
// v_j = VR[:,j] + i*VR[:,j+1],
// v_{j+1} = VR[:,j] - i*VR[:,j+1],
// where i is the imaginary unit. The computed eigenvectors are normalized to
// have Euclidean norm equal to 1 and largest component real.
//
// Left eigenvectors will be computed only if jobvl == lapack.LeftEVCompute,
// otherwise jobvl must be lapack.LeftEVNone.
// Right eigenvectors will be computed only if jobvr == lapack.RightEVCompute,
// otherwise jobvr must be lapack.RightEVNone.
// For other values of jobvl and jobvr Dgeev will panic.
//
// wr and wi contain the real and imaginary parts, respectively, of the computed
// eigenvalues. Complex conjugate pairs of eigenvalues appear consecutively with
// the eigenvalue having the positive imaginary part first.
// wr and wi must have length n, and Dgeev will panic otherwise.
//
// work must have length at least lwork and lwork must be at least max(1,4*n) if
// the left or right eigenvectors are computed, and at least max(1,3*n) if no
// eigenvectors are computed. For good performance, lwork must generally be
// larger. On return, optimal value of lwork will be stored in work[0].
//
// If lwork == -1, instead of performing Dgeev, the function only calculates the
// optimal value of lwork and stores it into work[0].
//
// On return, first is the index of the first valid eigenvalue. If first == 0,
// all eigenvalues and eigenvectors have been computed. If first is positive,
// Dgeev failed to compute all the eigenvalues, no eigenvectors have been
// computed and wr[first:] and wi[first:] contain those eigenvalues which have
// converged.
func (impl Implementation) Dgeev(jobvl lapack.LeftEVJob, jobvr lapack.RightEVJob, n int, a []float64, lda int, wr, wi []float64, vl []float64, ldvl int, vr []float64, ldvr int, work []float64, lwork int) (first int) {
wantvl := jobvl == lapack.LeftEVCompute
wantvr := jobvr == lapack.RightEVCompute
var minwrk int
if wantvl || wantvr {
minwrk = max(1, 4*n)
} else {
minwrk = max(1, 3*n)
}
switch {
case jobvl != lapack.LeftEVCompute && jobvl != lapack.LeftEVNone:
panic(badLeftEVJob)
case jobvr != lapack.RightEVCompute && jobvr != lapack.RightEVNone:
panic(badRightEVJob)
case n < 0:
panic(nLT0)
case lda < max(1, n):
panic(badLdA)
case ldvl < 1 || (ldvl < n && wantvl):
panic(badLdVL)
case ldvr < 1 || (ldvr < n && wantvr):
panic(badLdVR)
case lwork < minwrk && lwork != -1:
panic(badLWork)
case len(work) < lwork:
panic(shortWork)
}
// Quick return if possible.
if n == 0 {
work[0] = 1
return 0
}
maxwrk := 2*n + n*impl.Ilaenv(1, "DGEHRD", " ", n, 1, n, 0)
if wantvl || wantvr {
maxwrk = max(maxwrk, 2*n+(n-1)*impl.Ilaenv(1, "DORGHR", " ", n, 1, n, -1))
impl.Dhseqr(lapack.EigenvaluesAndSchur, lapack.SchurOrig, n, 0, n-1,
a, lda, wr, wi, nil, n, work, -1)
maxwrk = max(maxwrk, max(n+1, n+int(work[0])))
side := lapack.EVLeft
if wantvr {
side = lapack.EVRight
}
impl.Dtrevc3(side, lapack.EVAllMulQ, nil, n, a, lda, vl, ldvl, vr, ldvr,
n, work, -1)
maxwrk = max(maxwrk, n+int(work[0]))
maxwrk = max(maxwrk, 4*n)
} else {
impl.Dhseqr(lapack.EigenvaluesOnly, lapack.SchurNone, n, 0, n-1,
a, lda, wr, wi, vr, ldvr, work, -1)
maxwrk = max(maxwrk, max(n+1, n+int(work[0])))
}
maxwrk = max(maxwrk, minwrk)
if lwork == -1 {
work[0] = float64(maxwrk)
return 0
}
switch {
case len(a) < (n-1)*lda+n:
panic(shortA)
case len(wr) != n:
panic(badLenWr)
case len(wi) != n:
panic(badLenWi)
case len(vl) < (n-1)*ldvl+n && wantvl:
panic(shortVL)
case len(vr) < (n-1)*ldvr+n && wantvr:
panic(shortVR)
}
// Get machine constants.
smlnum := math.Sqrt(dlamchS) / dlamchP
bignum := 1 / smlnum
// Scale A if max element outside range [smlnum,bignum].
anrm := impl.Dlange(lapack.MaxAbs, n, n, a, lda, nil)
var scalea bool
var cscale float64
if 0 < anrm && anrm < smlnum {
scalea = true
cscale = smlnum
} else if anrm > bignum {
scalea = true
cscale = bignum
}
if scalea {
impl.Dlascl(lapack.General, 0, 0, anrm, cscale, n, n, a, lda)
}
// Balance the matrix.
workbal := work[:n]
ilo, ihi := impl.Dgebal(lapack.PermuteScale, n, a, lda, workbal)
// Reduce to upper Hessenberg form.
iwrk := 2 * n
tau := work[n : iwrk-1]
impl.Dgehrd(n, ilo, ihi, a, lda, tau, work[iwrk:], lwork-iwrk)
var side lapack.EVSide
if wantvl {
side = lapack.EVLeft
// Copy Householder vectors to VL.
impl.Dlacpy(blas.Lower, n, n, a, lda, vl, ldvl)
// Generate orthogonal matrix in VL.
impl.Dorghr(n, ilo, ihi, vl, ldvl, tau, work[iwrk:], lwork-iwrk)
// Perform QR iteration, accumulating Schur vectors in VL.
iwrk = n
first = impl.Dhseqr(lapack.EigenvaluesAndSchur, lapack.SchurOrig, n, ilo, ihi,
a, lda, wr, wi, vl, ldvl, work[iwrk:], lwork-iwrk)
if wantvr {
// Want left and right eigenvectors.
// Copy Schur vectors to VR.
side = lapack.EVBoth
impl.Dlacpy(blas.All, n, n, vl, ldvl, vr, ldvr)
}
} else if wantvr {
side = lapack.EVRight
// Copy Householder vectors to VR.
impl.Dlacpy(blas.Lower, n, n, a, lda, vr, ldvr)
// Generate orthogonal matrix in VR.
impl.Dorghr(n, ilo, ihi, vr, ldvr, tau, work[iwrk:], lwork-iwrk)
// Perform QR iteration, accumulating Schur vectors in VR.
iwrk = n
first = impl.Dhseqr(lapack.EigenvaluesAndSchur, lapack.SchurOrig, n, ilo, ihi,
a, lda, wr, wi, vr, ldvr, work[iwrk:], lwork-iwrk)
} else {
// Compute eigenvalues only.
iwrk = n
first = impl.Dhseqr(lapack.EigenvaluesOnly, lapack.SchurNone, n, ilo, ihi,
a, lda, wr, wi, nil, 1, work[iwrk:], lwork-iwrk)
}
if first > 0 {
if scalea {
// Undo scaling.
impl.Dlascl(lapack.General, 0, 0, cscale, anrm, n-first, 1, wr[first:], 1)
impl.Dlascl(lapack.General, 0, 0, cscale, anrm, n-first, 1, wi[first:], 1)
impl.Dlascl(lapack.General, 0, 0, cscale, anrm, ilo, 1, wr, 1)
impl.Dlascl(lapack.General, 0, 0, cscale, anrm, ilo, 1, wi, 1)
}
work[0] = float64(maxwrk)
return first
}
if wantvl || wantvr {
// Compute left and/or right eigenvectors.
impl.Dtrevc3(side, lapack.EVAllMulQ, nil, n,
a, lda, vl, ldvl, vr, ldvr, n, work[iwrk:], lwork-iwrk)
}
bi := blas64.Implementation()
if wantvl {
// Undo balancing of left eigenvectors.
impl.Dgebak(lapack.PermuteScale, lapack.EVLeft, n, ilo, ihi, workbal, n, vl, ldvl)
// Normalize left eigenvectors and make largest component real.
for i, wii := range wi {
if wii < 0 {
continue
}
if wii == 0 {
scl := 1 / bi.Dnrm2(n, vl[i:], ldvl)
bi.Dscal(n, scl, vl[i:], ldvl)
continue
}
scl := 1 / impl.Dlapy2(bi.Dnrm2(n, vl[i:], ldvl), bi.Dnrm2(n, vl[i+1:], ldvl))
bi.Dscal(n, scl, vl[i:], ldvl)
bi.Dscal(n, scl, vl[i+1:], ldvl)
for k := 0; k < n; k++ {
vi := vl[k*ldvl+i]
vi1 := vl[k*ldvl+i+1]
work[iwrk+k] = vi*vi + vi1*vi1
}
k := bi.Idamax(n, work[iwrk:iwrk+n], 1)
cs, sn, _ := impl.Dlartg(vl[k*ldvl+i], vl[k*ldvl+i+1])
bi.Drot(n, vl[i:], ldvl, vl[i+1:], ldvl, cs, sn)
vl[k*ldvl+i+1] = 0
}
}
if wantvr {
// Undo balancing of right eigenvectors.
impl.Dgebak(lapack.PermuteScale, lapack.EVRight, n, ilo, ihi, workbal, n, vr, ldvr)
// Normalize right eigenvectors and make largest component real.
for i, wii := range wi {
if wii < 0 {
continue
}
if wii == 0 {
scl := 1 / bi.Dnrm2(n, vr[i:], ldvr)
bi.Dscal(n, scl, vr[i:], ldvr)
continue
}
scl := 1 / impl.Dlapy2(bi.Dnrm2(n, vr[i:], ldvr), bi.Dnrm2(n, vr[i+1:], ldvr))
bi.Dscal(n, scl, vr[i:], ldvr)
bi.Dscal(n, scl, vr[i+1:], ldvr)
for k := 0; k < n; k++ {
vi := vr[k*ldvr+i]
vi1 := vr[k*ldvr+i+1]
work[iwrk+k] = vi*vi + vi1*vi1
}
k := bi.Idamax(n, work[iwrk:iwrk+n], 1)
cs, sn, _ := impl.Dlartg(vr[k*ldvr+i], vr[k*ldvr+i+1])
bi.Drot(n, vr[i:], ldvr, vr[i+1:], ldvr, cs, sn)
vr[k*ldvr+i+1] = 0
}
}
if scalea {
// Undo scaling.
impl.Dlascl(lapack.General, 0, 0, cscale, anrm, n-first, 1, wr[first:], 1)
impl.Dlascl(lapack.General, 0, 0, cscale, anrm, n-first, 1, wi[first:], 1)
}
work[0] = float64(maxwrk)
return first
}