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// Copyright ©2015 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"
)
const noSVDO = "dgesvd: not coded for overwrite"
// Dgesvd computes the singular value decomposition of the input matrix A.
//
// The singular value decomposition is
// A = U * Sigma * Vᵀ
// where Sigma is an m×n diagonal matrix containing the singular values of A,
// U is an m×m orthogonal matrix and V is an n×n orthogonal matrix. The first
// min(m,n) columns of U and V are the left and right singular vectors of A
// respectively.
//
// jobU and jobVT are options for computing the singular vectors. The behavior
// is as follows
// jobU == lapack.SVDAll All m columns of U are returned in u
// jobU == lapack.SVDStore The first min(m,n) columns are returned in u
// jobU == lapack.SVDOverwrite The first min(m,n) columns of U are written into a
// jobU == lapack.SVDNone The columns of U are not computed.
// The behavior is the same for jobVT and the rows of Vᵀ. At most one of jobU
// and jobVT can equal lapack.SVDOverwrite, and Dgesvd will panic otherwise.
//
// On entry, a contains the data for the m×n matrix A. During the call to Dgesvd
// the data is overwritten. On exit, A contains the appropriate singular vectors
// if either job is lapack.SVDOverwrite.
//
// s is a slice of length at least min(m,n) and on exit contains the singular
// values in decreasing order.
//
// u contains the left singular vectors on exit, stored column-wise. If
// jobU == lapack.SVDAll, u is of size m×m. If jobU == lapack.SVDStore u is
// of size m×min(m,n). If jobU == lapack.SVDOverwrite or lapack.SVDNone, u is
// not used.
//
// vt contains the left singular vectors on exit, stored row-wise. If
// jobV == lapack.SVDAll, vt is of size n×n. If jobVT == lapack.SVDStore vt is
// of size min(m,n)×n. If jobVT == lapack.SVDOverwrite or lapack.SVDNone, vt is
// not used.
//
// work is a slice for storing temporary memory, and lwork is the usable size of
// the slice. lwork must be at least max(5*min(m,n), 3*min(m,n)+max(m,n)).
// If lwork == -1, instead of performing Dgesvd, the optimal work length will be
// stored into work[0]. Dgesvd will panic if the working memory has insufficient
// storage.
//
// Dgesvd returns whether the decomposition successfully completed.
func (impl Implementation) Dgesvd(jobU, jobVT lapack.SVDJob, m, n int, a []float64, lda int, s, u []float64, ldu int, vt []float64, ldvt int, work []float64, lwork int) (ok bool) {
if jobU == lapack.SVDOverwrite || jobVT == lapack.SVDOverwrite {
panic(noSVDO)
}
wantua := jobU == lapack.SVDAll
wantus := jobU == lapack.SVDStore
wantuas := wantua || wantus
wantuo := jobU == lapack.SVDOverwrite
wantun := jobU == lapack.SVDNone
if !(wantua || wantus || wantuo || wantun) {
panic(badSVDJob)
}
wantva := jobVT == lapack.SVDAll
wantvs := jobVT == lapack.SVDStore
wantvas := wantva || wantvs
wantvo := jobVT == lapack.SVDOverwrite
wantvn := jobVT == lapack.SVDNone
if !(wantva || wantvs || wantvo || wantvn) {
panic(badSVDJob)
}
if wantuo && wantvo {
panic(bothSVDOver)
}
minmn := min(m, n)
minwork := 1
if minmn > 0 {
minwork = max(3*minmn+max(m, n), 5*minmn)
}
switch {
case m < 0:
panic(mLT0)
case n < 0:
panic(nLT0)
case lda < max(1, n):
panic(badLdA)
case ldu < 1, wantua && ldu < m, wantus && ldu < minmn:
panic(badLdU)
case ldvt < 1 || (wantvas && ldvt < n):
panic(badLdVT)
case lwork < minwork && lwork != -1:
panic(badLWork)
case len(work) < max(1, lwork):
panic(shortWork)
}
// Quick return if possible.
if minmn == 0 {
work[0] = 1
return true
}
// Compute optimal workspace size for subroutines.
opts := string(jobU) + string(jobVT)
mnthr := impl.Ilaenv(6, "DGESVD", opts, m, n, 0, 0)
maxwrk := 1
var wrkbl, bdspac int
if m >= n {
bdspac = 5 * n
impl.Dgeqrf(m, n, a, lda, nil, work, -1)
lwork_dgeqrf := int(work[0])
impl.Dorgqr(m, n, n, a, lda, nil, work, -1)
lwork_dorgqr_n := int(work[0])
impl.Dorgqr(m, m, n, a, lda, nil, work, -1)
lwork_dorgqr_m := int(work[0])
impl.Dgebrd(n, n, a, lda, s, nil, nil, nil, work, -1)
lwork_dgebrd := int(work[0])
impl.Dorgbr(lapack.GeneratePT, n, n, n, a, lda, nil, work, -1)
lwork_dorgbr_p := int(work[0])
impl.Dorgbr(lapack.GenerateQ, n, n, n, a, lda, nil, work, -1)
lwork_dorgbr_q := int(work[0])
if m >= mnthr {
if wantun {
// Path 1 (m much larger than n, jobU == None)
maxwrk = n + lwork_dgeqrf
maxwrk = max(maxwrk, 3*n+lwork_dgebrd)
if wantvo || wantvas {
maxwrk = max(maxwrk, 3*n+lwork_dorgbr_p)
}
maxwrk = max(maxwrk, bdspac)
} else if wantuo && wantvn {
// Path 2 (m much larger than n, jobU == Overwrite, jobVT == None)
wrkbl = n + lwork_dgeqrf
wrkbl = max(wrkbl, n+lwork_dorgqr_n)
wrkbl = max(wrkbl, 3*n+lwork_dgebrd)
wrkbl = max(wrkbl, 3*n+lwork_dorgbr_q)
wrkbl = max(wrkbl, bdspac)
maxwrk = max(n*n+wrkbl, n*n+m*n+n)
} else if wantuo && wantvas {
// Path 3 (m much larger than n, jobU == Overwrite, jobVT == Store or All)
wrkbl = n + lwork_dgeqrf
wrkbl = max(wrkbl, n+lwork_dorgqr_n)
wrkbl = max(wrkbl, 3*n+lwork_dgebrd)
wrkbl = max(wrkbl, 3*n+lwork_dorgbr_q)
wrkbl = max(wrkbl, 3*n+lwork_dorgbr_p)
wrkbl = max(wrkbl, bdspac)
maxwrk = max(n*n+wrkbl, n*n+m*n+n)
} else if wantus && wantvn {
// Path 4 (m much larger than n, jobU == Store, jobVT == None)
wrkbl = n + lwork_dgeqrf
wrkbl = max(wrkbl, n+lwork_dorgqr_n)
wrkbl = max(wrkbl, 3*n+lwork_dgebrd)
wrkbl = max(wrkbl, 3*n+lwork_dorgbr_q)
wrkbl = max(wrkbl, bdspac)
maxwrk = n*n + wrkbl
} else if wantus && wantvo {
// Path 5 (m much larger than n, jobU == Store, jobVT == Overwrite)
wrkbl = n + lwork_dgeqrf
wrkbl = max(wrkbl, n+lwork_dorgqr_n)
wrkbl = max(wrkbl, 3*n+lwork_dgebrd)
wrkbl = max(wrkbl, 3*n+lwork_dorgbr_q)
wrkbl = max(wrkbl, 3*n+lwork_dorgbr_p)
wrkbl = max(wrkbl, bdspac)
maxwrk = 2*n*n + wrkbl
} else if wantus && wantvas {
// Path 6 (m much larger than n, jobU == Store, jobVT == Store or All)
wrkbl = n + lwork_dgeqrf
wrkbl = max(wrkbl, n+lwork_dorgqr_n)
wrkbl = max(wrkbl, 3*n+lwork_dgebrd)
wrkbl = max(wrkbl, 3*n+lwork_dorgbr_q)
wrkbl = max(wrkbl, 3*n+lwork_dorgbr_p)
wrkbl = max(wrkbl, bdspac)
maxwrk = n*n + wrkbl
} else if wantua && wantvn {
// Path 7 (m much larger than n, jobU == All, jobVT == None)
wrkbl = n + lwork_dgeqrf
wrkbl = max(wrkbl, n+lwork_dorgqr_m)
wrkbl = max(wrkbl, 3*n+lwork_dgebrd)
wrkbl = max(wrkbl, 3*n+lwork_dorgbr_q)
wrkbl = max(wrkbl, bdspac)
maxwrk = n*n + wrkbl
} else if wantua && wantvo {
// Path 8 (m much larger than n, jobU == All, jobVT == Overwrite)
wrkbl = n + lwork_dgeqrf
wrkbl = max(wrkbl, n+lwork_dorgqr_m)
wrkbl = max(wrkbl, 3*n+lwork_dgebrd)
wrkbl = max(wrkbl, 3*n+lwork_dorgbr_q)
wrkbl = max(wrkbl, 3*n+lwork_dorgbr_p)
wrkbl = max(wrkbl, bdspac)
maxwrk = 2*n*n + wrkbl
} else if wantua && wantvas {
// Path 9 (m much larger than n, jobU == All, jobVT == Store or All)
wrkbl = n + lwork_dgeqrf
wrkbl = max(wrkbl, n+lwork_dorgqr_m)
wrkbl = max(wrkbl, 3*n+lwork_dgebrd)
wrkbl = max(wrkbl, 3*n+lwork_dorgbr_q)
wrkbl = max(wrkbl, 3*n+lwork_dorgbr_p)
wrkbl = max(wrkbl, bdspac)
maxwrk = n*n + wrkbl
}
} else {
// Path 10 (m at least n, but not much larger)
impl.Dgebrd(m, n, a, lda, s, nil, nil, nil, work, -1)
lwork_dgebrd := int(work[0])
maxwrk = 3*n + lwork_dgebrd
if wantus || wantuo {
impl.Dorgbr(lapack.GenerateQ, m, n, n, a, lda, nil, work, -1)
lwork_dorgbr_q = int(work[0])
maxwrk = max(maxwrk, 3*n+lwork_dorgbr_q)
}
if wantua {
impl.Dorgbr(lapack.GenerateQ, m, m, n, a, lda, nil, work, -1)
lwork_dorgbr_q := int(work[0])
maxwrk = max(maxwrk, 3*n+lwork_dorgbr_q)
}
if !wantvn {
maxwrk = max(maxwrk, 3*n+lwork_dorgbr_p)
}
maxwrk = max(maxwrk, bdspac)
}
} else {
bdspac = 5 * m
impl.Dgelqf(m, n, a, lda, nil, work, -1)
lwork_dgelqf := int(work[0])
impl.Dorglq(n, n, m, nil, n, nil, work, -1)
lwork_dorglq_n := int(work[0])
impl.Dorglq(m, n, m, a, lda, nil, work, -1)
lwork_dorglq_m := int(work[0])
impl.Dgebrd(m, m, a, lda, s, nil, nil, nil, work, -1)
lwork_dgebrd := int(work[0])
impl.Dorgbr(lapack.GeneratePT, m, m, m, a, n, nil, work, -1)
lwork_dorgbr_p := int(work[0])
impl.Dorgbr(lapack.GenerateQ, m, m, m, a, n, nil, work, -1)
lwork_dorgbr_q := int(work[0])
if n >= mnthr {
if wantvn {
// Path 1t (n much larger than m, jobVT == None)
maxwrk = m + lwork_dgelqf
maxwrk = max(maxwrk, 3*m+lwork_dgebrd)
if wantuo || wantuas {
maxwrk = max(maxwrk, 3*m+lwork_dorgbr_q)
}
maxwrk = max(maxwrk, bdspac)
} else if wantvo && wantun {
// Path 2t (n much larger than m, jobU == None, jobVT == Overwrite)
wrkbl = m + lwork_dgelqf
wrkbl = max(wrkbl, m+lwork_dorglq_m)
wrkbl = max(wrkbl, 3*m+lwork_dgebrd)
wrkbl = max(wrkbl, 3*m+lwork_dorgbr_p)
wrkbl = max(wrkbl, bdspac)
maxwrk = max(m*m+wrkbl, m*m+m*n+m)
} else if wantvo && wantuas {
// Path 3t (n much larger than m, jobU == Store or All, jobVT == Overwrite)
wrkbl = m + lwork_dgelqf
wrkbl = max(wrkbl, m+lwork_dorglq_m)
wrkbl = max(wrkbl, 3*m+lwork_dgebrd)
wrkbl = max(wrkbl, 3*m+lwork_dorgbr_p)
wrkbl = max(wrkbl, 3*m+lwork_dorgbr_q)
wrkbl = max(wrkbl, bdspac)
maxwrk = max(m*m+wrkbl, m*m+m*n+m)
} else if wantvs && wantun {
// Path 4t (n much larger than m, jobU == None, jobVT == Store)
wrkbl = m + lwork_dgelqf
wrkbl = max(wrkbl, m+lwork_dorglq_m)
wrkbl = max(wrkbl, 3*m+lwork_dgebrd)
wrkbl = max(wrkbl, 3*m+lwork_dorgbr_p)
wrkbl = max(wrkbl, bdspac)
maxwrk = m*m + wrkbl
} else if wantvs && wantuo {
// Path 5t (n much larger than m, jobU == Overwrite, jobVT == Store)
wrkbl = m + lwork_dgelqf
wrkbl = max(wrkbl, m+lwork_dorglq_m)
wrkbl = max(wrkbl, 3*m+lwork_dgebrd)
wrkbl = max(wrkbl, 3*m+lwork_dorgbr_p)
wrkbl = max(wrkbl, 3*m+lwork_dorgbr_q)
wrkbl = max(wrkbl, bdspac)
maxwrk = 2*m*m + wrkbl
} else if wantvs && wantuas {
// Path 6t (n much larger than m, jobU == Store or All, jobVT == Store)
wrkbl = m + lwork_dgelqf
wrkbl = max(wrkbl, m+lwork_dorglq_m)
wrkbl = max(wrkbl, 3*m+lwork_dgebrd)
wrkbl = max(wrkbl, 3*m+lwork_dorgbr_p)
wrkbl = max(wrkbl, 3*m+lwork_dorgbr_q)
wrkbl = max(wrkbl, bdspac)
maxwrk = m*m + wrkbl
} else if wantva && wantun {
// Path 7t (n much larger than m, jobU== None, jobVT == All)
wrkbl = m + lwork_dgelqf
wrkbl = max(wrkbl, m+lwork_dorglq_n)
wrkbl = max(wrkbl, 3*m+lwork_dgebrd)
wrkbl = max(wrkbl, 3*m+lwork_dorgbr_p)
wrkbl = max(wrkbl, bdspac)
maxwrk = m*m + wrkbl
} else if wantva && wantuo {
// Path 8t (n much larger than m, jobU == Overwrite, jobVT == All)
wrkbl = m + lwork_dgelqf
wrkbl = max(wrkbl, m+lwork_dorglq_n)
wrkbl = max(wrkbl, 3*m+lwork_dgebrd)
wrkbl = max(wrkbl, 3*m+lwork_dorgbr_p)
wrkbl = max(wrkbl, 3*m+lwork_dorgbr_q)
wrkbl = max(wrkbl, bdspac)
maxwrk = 2*m*m + wrkbl
} else if wantva && wantuas {
// Path 9t (n much larger than m, jobU == Store or All, jobVT == All)
wrkbl = m + lwork_dgelqf
wrkbl = max(wrkbl, m+lwork_dorglq_n)
wrkbl = max(wrkbl, 3*m+lwork_dgebrd)
wrkbl = max(wrkbl, 3*m+lwork_dorgbr_p)
wrkbl = max(wrkbl, 3*m+lwork_dorgbr_q)
wrkbl = max(wrkbl, bdspac)
maxwrk = m*m + wrkbl
}
} else {
// Path 10t (n greater than m, but not much larger)
impl.Dgebrd(m, n, a, lda, s, nil, nil, nil, work, -1)
lwork_dgebrd = int(work[0])
maxwrk = 3*m + lwork_dgebrd
if wantvs || wantvo {
impl.Dorgbr(lapack.GeneratePT, m, n, m, a, n, nil, work, -1)
lwork_dorgbr_p = int(work[0])
maxwrk = max(maxwrk, 3*m+lwork_dorgbr_p)
}
if wantva {
impl.Dorgbr(lapack.GeneratePT, n, n, m, a, n, nil, work, -1)
lwork_dorgbr_p = int(work[0])
maxwrk = max(maxwrk, 3*m+lwork_dorgbr_p)
}
if !wantun {
maxwrk = max(maxwrk, 3*m+lwork_dorgbr_q)
}
maxwrk = max(maxwrk, bdspac)
}
}
maxwrk = max(maxwrk, minwork)
if lwork == -1 {
work[0] = float64(maxwrk)
return true
}
if len(a) < (m-1)*lda+n {
panic(shortA)
}
if len(s) < minmn {
panic(shortS)
}
if (len(u) < (m-1)*ldu+m && wantua) || (len(u) < (m-1)*ldu+minmn && wantus) {
panic(shortU)
}
if (len(vt) < (n-1)*ldvt+n && wantva) || (len(vt) < (minmn-1)*ldvt+n && wantvs) {
panic(shortVT)
}
// Perform decomposition.
eps := dlamchE
smlnum := math.Sqrt(dlamchS) / eps
bignum := 1 / smlnum
// Scale A if max element outside range [smlnum, bignum].
anrm := impl.Dlange(lapack.MaxAbs, m, n, a, lda, nil)
var iscl bool
if anrm > 0 && anrm < smlnum {
iscl = true
impl.Dlascl(lapack.General, 0, 0, anrm, smlnum, m, n, a, lda)
} else if anrm > bignum {
iscl = true
impl.Dlascl(lapack.General, 0, 0, anrm, bignum, m, n, a, lda)
}
bi := blas64.Implementation()
var ie int
if m >= n {
// If A has sufficiently more rows than columns, use the QR decomposition.
if m >= mnthr {
// m >> n
if wantun {
// Path 1.
itau := 0
iwork := itau + n
// Compute A = Q * R.
impl.Dgeqrf(m, n, a, lda, work[itau:], work[iwork:], lwork-iwork)
// Zero out below R.
impl.Dlaset(blas.Lower, n-1, n-1, 0, 0, a[lda:], lda)
ie = 0
itauq := ie + n
itaup := itauq + n
iwork = itaup + n
// Bidiagonalize R in A.
impl.Dgebrd(n, n, a, lda, s, work[ie:], work[itauq:],
work[itaup:], work[iwork:], lwork-iwork)
ncvt := 0
if wantvo || wantvas {
impl.Dorgbr(lapack.GeneratePT, n, n, n, a, lda, work[itaup:],
work[iwork:], lwork-iwork)
ncvt = n
}
iwork = ie + n
// Perform bidiagonal QR iteration computing right singular vectors
// of A in A if desired.
ok = impl.Dbdsqr(blas.Upper, n, ncvt, 0, 0, s, work[ie:],
a, lda, work, 1, work, 1, work[iwork:])
// If right singular vectors desired in VT, copy them there.
if wantvas {
impl.Dlacpy(blas.All, n, n, a, lda, vt, ldvt)
}
} else if wantuo && wantvn {
// Path 2
panic(noSVDO)
} else if wantuo && wantvas {
// Path 3
panic(noSVDO)
} else if wantus {
if wantvn {
// Path 4
if lwork >= n*n+max(4*n, bdspac) {
// Sufficient workspace for a fast algorithm.
ir := 0
var ldworkr int
if lwork >= wrkbl+lda*n {
ldworkr = lda
} else {
ldworkr = n
}
itau := ir + ldworkr*n
iwork := itau + n
// Compute A = Q * R.
impl.Dgeqrf(m, n, a, lda, work[itau:], work[iwork:], lwork-iwork)
// Copy R to work[ir:], zeroing out below it.
impl.Dlacpy(blas.Upper, n, n, a, lda, work[ir:], ldworkr)
impl.Dlaset(blas.Lower, n-1, n-1, 0, 0, work[ir+ldworkr:], ldworkr)
// Generate Q in A.
impl.Dorgqr(m, n, n, a, lda, work[itau:], work[iwork:], lwork-iwork)
ie := itau
itauq := ie + n
itaup := itauq + n
iwork = itaup + n
// Bidiagonalize R in work[ir:].
impl.Dgebrd(n, n, work[ir:], ldworkr, s, work[ie:],
work[itauq:], work[itaup:], work[iwork:], lwork-iwork)
// Generate left vectors bidiagonalizing R in work[ir:].
impl.Dorgbr(lapack.GenerateQ, n, n, n, work[ir:], ldworkr,
work[itauq:], work[iwork:], lwork-iwork)
iwork = ie + n
// Perform bidiagonal QR iteration, compuing left singular
// vectors of R in work[ir:].
ok = impl.Dbdsqr(blas.Upper, n, 0, n, 0, s, work[ie:], work, 1,
work[ir:], ldworkr, work, 1, work[iwork:])
// Multiply Q in A by left singular vectors of R in
// work[ir:], storing result in U.
bi.Dgemm(blas.NoTrans, blas.NoTrans, m, n, n, 1, a, lda,
work[ir:], ldworkr, 0, u, ldu)
} else {
// Insufficient workspace for a fast algorithm.
itau := 0
iwork := itau + n
// Compute A = Q*R, copying result to U.
impl.Dgeqrf(m, n, a, lda, work[itau:], work[iwork:], lwork-iwork)
impl.Dlacpy(blas.Lower, m, n, a, lda, u, ldu)
// Generate Q in U.
impl.Dorgqr(m, n, n, u, ldu, work[itau:], work[iwork:], lwork-iwork)
ie := itau
itauq := ie + n
itaup := itauq + n
iwork = itaup + n
// Zero out below R in A.
impl.Dlaset(blas.Lower, n-1, n-1, 0, 0, a[lda:], lda)
// Bidiagonalize R in A.
impl.Dgebrd(n, n, a, lda, s, work[ie:],
work[itauq:], work[itaup:], work[iwork:], lwork-iwork)
// Multiply Q in U by left vectors bidiagonalizing R.
impl.Dormbr(lapack.ApplyQ, blas.Right, blas.NoTrans, m, n, n,
a, lda, work[itauq:], u, ldu, work[iwork:], lwork-iwork)
iwork = ie + n
// Perform bidiagonal QR iteration, computing left
// singular vectors of A in U.
ok = impl.Dbdsqr(blas.Upper, n, 0, m, 0, s, work[ie:], work, 1,
u, ldu, work, 1, work[iwork:])
}
} else if wantvo {
// Path 5
panic(noSVDO)
} else if wantvas {
// Path 6
if lwork >= n*n+max(4*n, bdspac) {
// Sufficient workspace for a fast algorithm.
iu := 0
var ldworku int
if lwork >= wrkbl+lda*n {
ldworku = lda
} else {
ldworku = n
}
itau := iu + ldworku*n
iwork := itau + n
// Compute A = Q * R.
impl.Dgeqrf(m, n, a, lda, work[itau:], work[iwork:], lwork-iwork)
// Copy R to work[iu:], zeroing out below it.
impl.Dlacpy(blas.Upper, n, n, a, lda, work[iu:], ldworku)
impl.Dlaset(blas.Lower, n-1, n-1, 0, 0, work[iu+ldworku:], ldworku)
// Generate Q in A.
impl.Dorgqr(m, n, n, a, lda, work[itau:], work[iwork:], lwork-iwork)
ie := itau
itauq := ie + n
itaup := itauq + n
iwork = itaup + n
// Bidiagonalize R in work[iu:], copying result to VT.
impl.Dgebrd(n, n, work[iu:], ldworku, s, work[ie:],
work[itauq:], work[itaup:], work[iwork:], lwork-iwork)
impl.Dlacpy(blas.Upper, n, n, work[iu:], ldworku, vt, ldvt)
// Generate left bidiagonalizing vectors in work[iu:].
impl.Dorgbr(lapack.GenerateQ, n, n, n, work[iu:], ldworku,
work[itauq:], work[iwork:], lwork-iwork)
// Generate right bidiagonalizing vectors in VT.
impl.Dorgbr(lapack.GeneratePT, n, n, n, vt, ldvt,
work[itaup:], work[iwork:], lwork-iwork)
iwork = ie + n
// Perform bidiagonal QR iteration, computing left singular
// vectors of R in work[iu:], and computing right singular
// vectors of R in VT.
ok = impl.Dbdsqr(blas.Upper, n, n, n, 0, s, work[ie:],
vt, ldvt, work[iu:], ldworku, work, 1, work[iwork:])
// Multiply Q in A by left singular vectors of R in
// work[iu:], storing result in U.
bi.Dgemm(blas.NoTrans, blas.NoTrans, m, n, n, 1, a, lda,
work[iu:], ldworku, 0, u, ldu)
} else {
// Insufficient workspace for a fast algorithm.
itau := 0
iwork := itau + n
// Compute A = Q * R, copying result to U.
impl.Dgeqrf(m, n, a, lda, work[itau:], work[iwork:], lwork-iwork)
impl.Dlacpy(blas.Lower, m, n, a, lda, u, ldu)
// Generate Q in U.
impl.Dorgqr(m, n, n, u, ldu, work[itau:], work[iwork:], lwork-iwork)
// Copy R to VT, zeroing out below it.
impl.Dlacpy(blas.Upper, n, n, a, lda, vt, ldvt)
impl.Dlaset(blas.Lower, n-1, n-1, 0, 0, vt[ldvt:], ldvt)
ie := itau
itauq := ie + n
itaup := itauq + n
iwork = itaup + n
// Bidiagonalize R in VT.
impl.Dgebrd(n, n, vt, ldvt, s, work[ie:],
work[itauq:], work[itaup:], work[iwork:], lwork-iwork)
// Multiply Q in U by left bidiagonalizing vectors in VT.
impl.Dormbr(lapack.ApplyQ, blas.Right, blas.NoTrans, m, n, n,
vt, ldvt, work[itauq:], u, ldu, work[iwork:], lwork-iwork)
// Generate right bidiagonalizing vectors in VT.
impl.Dorgbr(lapack.GeneratePT, n, n, n, vt, ldvt,
work[itaup:], work[iwork:], lwork-iwork)
iwork = ie + n
// Perform bidiagonal QR iteration, computing left singular
// vectors of A in U and computing right singular vectors
// of A in VT.
ok = impl.Dbdsqr(blas.Upper, n, n, m, 0, s, work[ie:],
vt, ldvt, u, ldu, work, 1, work[iwork:])
}
}
} else if wantua {
if wantvn {
// Path 7
if lwork >= n*n+max(max(n+m, 4*n), bdspac) {
// Sufficient workspace for a fast algorithm.
ir := 0
var ldworkr int
if lwork >= wrkbl+lda*n {
ldworkr = lda
} else {
ldworkr = n
}
itau := ir + ldworkr*n
iwork := itau + n
// Compute A = Q*R, copying result to U.
impl.Dgeqrf(m, n, a, lda, work[itau:], work[iwork:], lwork-iwork)
impl.Dlacpy(blas.Lower, m, n, a, lda, u, ldu)
// Copy R to work[ir:], zeroing out below it.
impl.Dlacpy(blas.Upper, n, n, a, lda, work[ir:], ldworkr)
impl.Dlaset(blas.Lower, n-1, n-1, 0, 0, work[ir+ldworkr:], ldworkr)
// Generate Q in U.
impl.Dorgqr(m, m, n, u, ldu, work[itau:], work[iwork:], lwork-iwork)
ie := itau
itauq := ie + n
itaup := itauq + n
iwork = itaup + n
// Bidiagonalize R in work[ir:].
impl.Dgebrd(n, n, work[ir:], ldworkr, s, work[ie:],
work[itauq:], work[itaup:], work[iwork:], lwork-iwork)
// Generate left bidiagonalizing vectors in work[ir:].
impl.Dorgbr(lapack.GenerateQ, n, n, n, work[ir:], ldworkr,
work[itauq:], work[iwork:], lwork-iwork)
iwork = ie + n
// Perform bidiagonal QR iteration, computing left singular
// vectors of R in work[ir:].
ok = impl.Dbdsqr(blas.Upper, n, 0, n, 0, s, work[ie:], work, 1,
work[ir:], ldworkr, work, 1, work[iwork:])
// Multiply Q in U by left singular vectors of R in
// work[ir:], storing result in A.
bi.Dgemm(blas.NoTrans, blas.NoTrans, m, n, n, 1, u, ldu,
work[ir:], ldworkr, 0, a, lda)
// Copy left singular vectors of A from A to U.
impl.Dlacpy(blas.All, m, n, a, lda, u, ldu)
} else {
// Insufficient workspace for a fast algorithm.
itau := 0
iwork := itau + n
// Compute A = Q*R, copying result to U.
impl.Dgeqrf(m, n, a, lda, work[itau:], work[iwork:], lwork-iwork)
impl.Dlacpy(blas.Lower, m, n, a, lda, u, ldu)
// Generate Q in U.
impl.Dorgqr(m, m, n, u, ldu, work[itau:], work[iwork:], lwork-iwork)
ie := itau
itauq := ie + n
itaup := itauq + n
iwork = itaup + n
// Zero out below R in A.
impl.Dlaset(blas.Lower, n-1, n-1, 0, 0, a[lda:], lda)
// Bidiagonalize R in A.
impl.Dgebrd(n, n, a, lda, s, work[ie:],
work[itauq:], work[itaup:], work[iwork:], lwork-iwork)
// Multiply Q in U by left bidiagonalizing vectors in A.
impl.Dormbr(lapack.ApplyQ, blas.Right, blas.NoTrans, m, n, n,
a, lda, work[itauq:], u, ldu, work[iwork:], lwork-iwork)
iwork = ie + n
// Perform bidiagonal QR iteration, computing left
// singular vectors of A in U.
ok = impl.Dbdsqr(blas.Upper, n, 0, m, 0, s, work[ie:],
work, 1, u, ldu, work, 1, work[iwork:])
}
} else if wantvo {
// Path 8.
panic(noSVDO)
} else if wantvas {
// Path 9.
if lwork >= n*n+max(max(n+m, 4*n), bdspac) {
// Sufficient workspace for a fast algorithm.
iu := 0
var ldworku int
if lwork >= wrkbl+lda*n {
ldworku = lda
} else {
ldworku = n
}
itau := iu + ldworku*n
iwork := itau + n
// Compute A = Q * R, copying result to U.
impl.Dgeqrf(m, n, a, lda, work[itau:], work[iwork:], lwork-iwork)
impl.Dlacpy(blas.Lower, m, n, a, lda, u, ldu)
// Generate Q in U.
impl.Dorgqr(m, m, n, u, ldu, work[itau:], work[iwork:], lwork-iwork)
// Copy R to work[iu:], zeroing out below it.
impl.Dlacpy(blas.Upper, n, n, a, lda, work[iu:], ldworku)
impl.Dlaset(blas.Lower, n-1, n-1, 0, 0, work[iu+ldworku:], ldworku)
ie = itau
itauq := ie + n
itaup := itauq + n
iwork = itaup + n
// Bidiagonalize R in work[iu:], copying result to VT.
impl.Dgebrd(n, n, work[iu:], ldworku, s, work[ie:],
work[itauq:], work[itaup:], work[iwork:], lwork-iwork)
impl.Dlacpy(blas.Upper, n, n, work[iu:], ldworku, vt, ldvt)
// Generate left bidiagonalizing vectors in work[iu:].
impl.Dorgbr(lapack.GenerateQ, n, n, n, work[iu:], ldworku,
work[itauq:], work[iwork:], lwork-iwork)
// Generate right bidiagonalizing vectors in VT.
impl.Dorgbr(lapack.GeneratePT, n, n, n, vt, ldvt,
work[itaup:], work[iwork:], lwork-iwork)
iwork = ie + n
// Perform bidiagonal QR iteration, computing left singular
// vectors of R in work[iu:] and computing right
// singular vectors of R in VT.
ok = impl.Dbdsqr(blas.Upper, n, n, n, 0, s, work[ie:],
vt, ldvt, work[iu:], ldworku, work, 1, work[iwork:])
// Multiply Q in U by left singular vectors of R in
// work[iu:], storing result in A.
bi.Dgemm(blas.NoTrans, blas.NoTrans, m, n, n, 1,
u, ldu, work[iu:], ldworku, 0, a, lda)
// Copy left singular vectors of A from A to U.
impl.Dlacpy(blas.All, m, n, a, lda, u, ldu)
/*
// Bidiagonalize R in VT.
impl.Dgebrd(n, n, vt, ldvt, s, work[ie:],
work[itauq:], work[itaup:], work[iwork:], lwork-iwork)
// Multiply Q in U by left bidiagonalizing vectors in VT.
impl.Dormbr(lapack.ApplyQ, blas.Right, blas.NoTrans,
m, n, n, vt, ldvt, work[itauq:], u, ldu, work[iwork:], lwork-iwork)
// Generate right bidiagonalizing vectors in VT.
impl.Dorgbr(lapack.GeneratePT, n, n, n, vt, ldvt,
work[itaup:], work[iwork:], lwork-iwork)
iwork = ie + n
// Perform bidiagonal QR iteration, computing left singular
// vectors of A in U and computing right singular vectors
// of A in VT.
ok = impl.Dbdsqr(blas.Upper, n, n, m, 0, s, work[ie:],
vt, ldvt, u, ldu, work, 1, work[iwork:])
*/
} else {
// Insufficient workspace for a fast algorithm.
itau := 0
iwork := itau + n
// Compute A = Q*R, copying result to U.
impl.Dgeqrf(m, n, a, lda, work[itau:], work[iwork:], lwork-iwork)
impl.Dlacpy(blas.Lower, m, n, a, lda, u, ldu)
// Generate Q in U.
impl.Dorgqr(m, m, n, u, ldu, work[itau:], work[iwork:], lwork-iwork)
// Copy R from A to VT, zeroing out below it.
impl.Dlacpy(blas.Upper, n, n, a, lda, vt, ldvt)
if n > 1 {
impl.Dlaset(blas.Lower, n-1, n-1, 0, 0, vt[ldvt:], ldvt)
}
ie := itau
itauq := ie + n
itaup := itauq + n
iwork = itaup + n
// Bidiagonalize R in VT.
impl.Dgebrd(n, n, vt, ldvt, s, work[ie:],
work[itauq:], work[itaup:], work[iwork:], lwork-iwork)
// Multiply Q in U by left bidiagonalizing vectors in VT.
impl.Dormbr(lapack.ApplyQ, blas.Right, blas.NoTrans,
m, n, n, vt, ldvt, work[itauq:], u, ldu, work[iwork:], lwork-iwork)
// Generate right bidiagonizing vectors in VT.
impl.Dorgbr(lapack.GeneratePT, n, n, n, vt, ldvt,
work[itaup:], work[iwork:], lwork-iwork)
iwork = ie + n
// Perform bidiagonal QR iteration, computing left singular
// vectors of A in U and computing right singular vectors
// of A in VT.
ok = impl.Dbdsqr(blas.Upper, n, n, m, 0, s, work[ie:],
vt, ldvt, u, ldu, work, 1, work[iwork:])
}
}
}
} else {
// Path 10.
// M at least N, but not much larger.
ie = 0
itauq := ie + n
itaup := itauq + n
iwork := itaup + n
// Bidiagonalize A.
impl.Dgebrd(m, n, a, lda, s, work[ie:], work[itauq:],
work[itaup:], work[iwork:], lwork-iwork)
if wantuas {
// Left singular vectors are desired in U. Copy result to U and
// generate left biadiagonalizing vectors in U.
impl.Dlacpy(blas.Lower, m, n, a, lda, u, ldu)
var ncu int
if wantus {
ncu = n
}
if wantua {
ncu = m
}
impl.Dorgbr(lapack.GenerateQ, m, ncu, n, u, ldu, work[itauq:], work[iwork:], lwork-iwork)
}
if wantvas {
// Right singular vectors are desired in VT. Copy result to VT and
// generate left biadiagonalizing vectors in VT.
impl.Dlacpy(blas.Upper, n, n, a, lda, vt, ldvt)
impl.Dorgbr(lapack.GeneratePT, n, n, n, vt, ldvt, work[itaup:], work[iwork:], lwork-iwork)
}
if wantuo {
panic(noSVDO)
}
if wantvo {
panic(noSVDO)
}
iwork = ie + n
var nru, ncvt int
if wantuas || wantuo {
nru = m
}
if wantun {
nru = 0
}
if wantvas || wantvo {
ncvt = n
}
if wantvn {
ncvt = 0
}
if !wantuo && !wantvo {
// Perform bidiagonal QR iteration, if desired, computing left
// singular vectors in U and right singular vectors in VT.
ok = impl.Dbdsqr(blas.Upper, n, ncvt, nru, 0, s, work[ie:],
vt, ldvt, u, ldu, work, 1, work[iwork:])
} else {
// There will be two branches when the implementation is complete.
panic(noSVDO)
}
}
} else {
// A has more columns than rows. If A has sufficiently more columns than
// rows, first reduce using the LQ decomposition.
if n >= mnthr {
// n >> m.
if wantvn {
// Path 1t.
itau := 0
iwork := itau + m
// Compute A = L*Q.
impl.Dgelqf(m, n, a, lda, work[itau:], work[iwork:], lwork-iwork)
// Zero out above L.
impl.Dlaset(blas.Upper, m-1, m-1, 0, 0, a[1:], lda)
ie := 0
itauq := ie + m
itaup := itauq + m
iwork = itaup + m
// Bidiagonalize L in A.
impl.Dgebrd(m, m, a, lda, s, work[ie:itauq],
work[itauq:itaup], work[itaup:iwork], work[iwork:], lwork-iwork)
if wantuo || wantuas {
impl.Dorgbr(lapack.GenerateQ, m, m, m, a, lda,
work[itauq:], work[iwork:], lwork-iwork)
}
iwork = ie + m
nru := 0
if wantuo || wantuas {
nru = m
}
// Perform bidiagonal QR iteration, computing left singular vectors
// of A in A if desired.
ok = impl.Dbdsqr(blas.Upper, m, 0, nru, 0, s, work[ie:],
work, 1, a, lda, work, 1, work[iwork:])
// If left singular vectors desired in U, copy them there.
if wantuas {
impl.Dlacpy(blas.All, m, m, a, lda, u, ldu)
}
} else if wantvo && wantun {
// Path 2t.
panic(noSVDO)
} else if wantvo && wantuas {
// Path 3t.
panic(noSVDO)
} else if wantvs {
if wantun {
// Path 4t.
if lwork >= m*m+max(4*m, bdspac) {
// Sufficient workspace for a fast algorithm.
ir := 0
var ldworkr int
if lwork >= wrkbl+lda*m {
ldworkr = lda
} else {
ldworkr = m
}
itau := ir + ldworkr*m
iwork := itau + m
// Compute A = L*Q.
impl.Dgelqf(m, n, a, lda, work[itau:], work[iwork:], lwork-iwork)
// Copy L to work[ir:], zeroing out above it.
impl.Dlacpy(blas.Lower, m, m, a, lda, work[ir:], ldworkr)
impl.Dlaset(blas.Upper, m-1, m-1, 0, 0, work[ir+1:], ldworkr)
// Generate Q in A.
impl.Dorglq(m, n, m, a, lda, work[itau:], work[iwork:], lwork-iwork)
ie := itau
itauq := ie + m
itaup := itauq + m
iwork = itaup + m
// Bidiagonalize L in work[ir:].
impl.Dgebrd(m, m, work[ir:], ldworkr, s, work[ie:],
work[itauq:], work[itaup:], work[iwork:], lwork-iwork)
// Generate right vectors bidiagonalizing L in work[ir:].
impl.Dorgbr(lapack.GeneratePT, m, m, m, work[ir:], ldworkr,
work[itaup:], work[iwork:], lwork-iwork)
iwork = ie + m
// Perform bidiagonal QR iteration, computing right singular
// vectors of L in work[ir:].
ok = impl.Dbdsqr(blas.Upper, m, m, 0, 0, s, work[ie:],
work[ir:], ldworkr, work, 1, work, 1, work[iwork:])
// Multiply right singular vectors of L in work[ir:] by
// Q in A, storing result in VT.
bi.Dgemm(blas.NoTrans, blas.NoTrans, m, n, m, 1,
work[ir:], ldworkr, a, lda, 0, vt, ldvt)
} else {
// Insufficient workspace for a fast algorithm.
itau := 0
iwork := itau + m
// Compute A = L*Q.
impl.Dgelqf(m, n, a, lda, work[itau:], work[iwork:], lwork-iwork)
// Copy result to VT.
impl.Dlacpy(blas.Upper, m, n, a, lda, vt, ldvt)
// Generate Q in VT.
impl.Dorglq(m, n, m, vt, ldvt, work[itau:], work[iwork:], lwork-iwork)
ie := itau
itauq := ie + m
itaup := itauq + m
iwork = itaup + m
// Zero out above L in A.
impl.Dlaset(blas.Upper, m-1, m-1, 0, 0, a[1:], lda)
// Bidiagonalize L in A.
impl.Dgebrd(m, m, a, lda, s, work[ie:],
work[itauq:], work[itaup:], work[iwork:], lwork-iwork)
// Multiply right vectors bidiagonalizing L by Q in VT.
impl.Dormbr(lapack.ApplyP, blas.Left, blas.Trans, m, n, m,
a, lda, work[itaup:], vt, ldvt, work[iwork:], lwork-iwork)
iwork = ie + m
// Perform bidiagonal QR iteration, computing right
// singular vectors of A in VT.
ok = impl.Dbdsqr(blas.Upper, m, n, 0, 0, s, work[ie:],
vt, ldvt, work, 1, work, 1, work[iwork:])
}
} else if wantuo {
// Path 5t.
panic(noSVDO)
} else if wantuas {
// Path 6t.
if lwork >= m*m+max(4*m, bdspac) {
// Sufficient workspace for a fast algorithm.
iu := 0
var ldworku int
if lwork >= wrkbl+lda*m {
ldworku = lda
} else {
ldworku = m
}
itau := iu + ldworku*m
iwork := itau + m
// Compute A = L*Q.
impl.Dgelqf(m, n, a, lda, work[itau:], work[iwork:], lwork-iwork)
// Copy L to work[iu:], zeroing out above it.
impl.Dlacpy(blas.Lower, m, m, a, lda, work[iu:], ldworku)
impl.Dlaset(blas.Upper, m-1, m-1, 0, 0, work[iu+1:], ldworku)
// Generate Q in A.
impl.Dorglq(m, n, m, a, lda, work[itau:], work[iwork:], lwork-iwork)
ie := itau
itauq := ie + m
itaup := itauq + m
iwork = itaup + m
// Bidiagonalize L in work[iu:], copying result to U.
impl.Dgebrd(m, m, work[iu:], ldworku, s, work[ie:],
work[itauq:], work[itaup:], work[iwork:], lwork-iwork)
impl.Dlacpy(blas.Lower, m, m, work[iu:], ldworku, u, ldu)
// Generate right bidiagionalizing vectors in work[iu:].
impl.Dorgbr(lapack.GeneratePT, m, m, m, work[iu:], ldworku,
work[itaup:], work[iwork:], lwork-iwork)
// Generate left bidiagonalizing vectors in U.
impl.Dorgbr(lapack.GenerateQ, m, m, m, u, ldu, work[itauq:], work[iwork:], lwork-iwork)
iwork = ie + m
// Perform bidiagonal QR iteration, computing left singular
// vectors of L in U and computing right singular vectors of
// L in work[iu:].
ok = impl.Dbdsqr(blas.Upper, m, m, m, 0, s, work[ie:],
work[iu:], ldworku, u, ldu, work, 1, work[iwork:])
// Multiply right singular vectors of L in work[iu:] by
// Q in A, storing result in VT.
bi.Dgemm(blas.NoTrans, blas.NoTrans, m, n, m, 1,
work[iu:], ldworku, a, lda, 0, vt, ldvt)
} else {
// Insufficient workspace for a fast algorithm.
itau := 0
iwork := itau + m
// Compute A = L*Q, copying result to VT.
impl.Dgelqf(m, n, a, lda, work[itau:], work[iwork:], lwork-iwork)
impl.Dlacpy(blas.Upper, m, n, a, lda, vt, ldvt)
// Generate Q in VT.
impl.Dorglq(m, n, m, vt, ldvt, work[itau:], work[iwork:], lwork-iwork)
// Copy L to U, zeroing out above it.
impl.Dlacpy(blas.Lower, m, m, a, lda, u, ldu)
impl.Dlaset(blas.Upper, m-1, m-1, 0, 0, u[1:], ldu)
ie := itau
itauq := ie + m
itaup := itauq + m
iwork = itaup + m
// Bidiagonalize L in U.
impl.Dgebrd(m, m, u, ldu, s, work[ie:],
work[itauq:], work[itaup:], work[iwork:], lwork-iwork)
// Multiply right bidiagonalizing vectors in U by Q in VT.
impl.Dormbr(lapack.ApplyP, blas.Left, blas.Trans, m, n, m,
u, ldu, work[itaup:], vt, ldvt, work[iwork:], lwork-iwork)
// Generate left bidiagonalizing vectors in U.
impl.Dorgbr(lapack.GenerateQ, m, m, m, u, ldu, work[itauq:], work[iwork:], lwork-iwork)
iwork = ie + m
// Perform bidiagonal QR iteration, computing left singular
// vectors of A in U and computing right singular vectors
// of A in VT.
ok = impl.Dbdsqr(blas.Upper, m, n, m, 0, s, work[ie:], vt, ldvt,
u, ldu, work, 1, work[iwork:])
}
}
} else if wantva {
if wantun {
// Path 7t.
if lwork >= m*m+max(max(n+m, 4*m), bdspac) {
// Sufficient workspace for a fast algorithm.
ir := 0
var ldworkr int
if lwork >= wrkbl+lda*m {
ldworkr = lda
} else {
ldworkr = m
}
itau := ir + ldworkr*m
iwork := itau + m
// Compute A = L*Q, copying result to VT.
impl.Dgelqf(m, n, a, lda, work[itau:], work[iwork:], lwork-iwork)
impl.Dlacpy(blas.Upper, m, n, a, lda, vt, ldvt)
// Copy L to work[ir:], zeroing out above it.
impl.Dlacpy(blas.Lower, m, m, a, lda, work[ir:], ldworkr)
impl.Dlaset(blas.Upper, m-1, m-1, 0, 0, work[ir+1:], ldworkr)
// Generate Q in VT.
impl.Dorglq(n, n, m, vt, ldvt, work[itau:], work[iwork:], lwork-iwork)
ie := itau
itauq := ie + m
itaup := itauq + m
iwork = itaup + m
// Bidiagonalize L in work[ir:].
impl.Dgebrd(m, m, work[ir:], ldworkr, s, work[ie:],
work[itauq:], work[itaup:], work[iwork:], lwork-iwork)
// Generate right bidiagonalizing vectors in work[ir:].
impl.Dorgbr(lapack.GeneratePT, m, m, m, work[ir:], ldworkr,
work[itaup:], work[iwork:], lwork-iwork)
iwork = ie + m
// Perform bidiagonal QR iteration, computing right
// singular vectors of L in work[ir:].
ok = impl.Dbdsqr(blas.Upper, m, m, 0, 0, s, work[ie:],
work[ir:], ldworkr, work, 1, work, 1, work[iwork:])
// Multiply right singular vectors of L in work[ir:] by
// Q in VT, storing result in A.
bi.Dgemm(blas.NoTrans, blas.NoTrans, m, n, m, 1,
work[ir:], ldworkr, vt, ldvt, 0, a, lda)
// Copy right singular vectors of A from A to VT.
impl.Dlacpy(blas.All, m, n, a, lda, vt, ldvt)
} else {
// Insufficient workspace for a fast algorithm.
itau := 0
iwork := itau + m
// Compute A = L * Q, copying result to VT.
impl.Dgelqf(m, n, a, lda, work[itau:], work[iwork:], lwork-iwork)
impl.Dlacpy(blas.Upper, m, n, a, lda, vt, ldvt)
// Generate Q in VT.
impl.Dorglq(n, n, m, vt, ldvt, work[itau:], work[iwork:], lwork-iwork)
ie := itau
itauq := ie + m
itaup := itauq + m
iwork = itaup + m
// Zero out above L in A.
impl.Dlaset(blas.Upper, m-1, m-1, 0, 0, a[1:], lda)
// Bidiagonalize L in A.
impl.Dgebrd(m, m, a, lda, s, work[ie:], work[itauq:],
work[itaup:], work[iwork:], lwork-iwork)
// Multiply right bidiagonalizing vectors in A by Q in VT.
impl.Dormbr(lapack.ApplyP, blas.Left, blas.Trans, m, n, m,
a, lda, work[itaup:], vt, ldvt, work[iwork:], lwork-iwork)
iwork = ie + m
// Perform bidiagonal QR iteration, computing right singular
// vectors of A in VT.
ok = impl.Dbdsqr(blas.Upper, m, n, 0, 0, s, work[ie:],
vt, ldvt, work, 1, work, 1, work[iwork:])
}
} else if wantuo {
panic(noSVDO)
} else if wantuas {
// Path 9t.
if lwork >= m*m+max(max(m+n, 4*m), bdspac) {
// Sufficient workspace for a fast algorithm.
iu := 0
var ldworku int
if lwork >= wrkbl+lda*m {
ldworku = lda
} else {
ldworku = m
}
itau := iu + ldworku*m
iwork := itau + m
// Generate A = L * Q copying result to VT.
impl.Dgelqf(m, n, a, lda, work[itau:], work[iwork:], lwork-iwork)
impl.Dlacpy(blas.Upper, m, n, a, lda, vt, ldvt)
// Generate Q in VT.
impl.Dorglq(n, n, m, vt, ldvt, work[itau:], work[iwork:], lwork-iwork)
// Copy L to work[iu:], zeroing out above it.
impl.Dlacpy(blas.Lower, m, m, a, lda, work[iu:], ldworku)
impl.Dlaset(blas.Upper, m-1, m-1, 0, 0, work[iu+1:], ldworku)
ie = itau
itauq := ie + m
itaup := itauq + m
iwork = itaup + m
// Bidiagonalize L in work[iu:], copying result to U.
impl.Dgebrd(m, m, work[iu:], ldworku, s, work[ie:],
work[itauq:], work[itaup:], work[iwork:], lwork-iwork)
impl.Dlacpy(blas.Lower, m, m, work[iu:], ldworku, u, ldu)
// Generate right bidiagonalizing vectors in work[iu:].
impl.Dorgbr(lapack.GeneratePT, m, m, m, work[iu:], ldworku,
work[itaup:], work[iwork:], lwork-iwork)
// Generate left bidiagonalizing vectors in U.
impl.Dorgbr(lapack.GenerateQ, m, m, m, u, ldu, work[itauq:], work[iwork:], lwork-iwork)
iwork = ie + m
// Perform bidiagonal QR iteration, computing left singular
// vectors of L in U and computing right singular vectors
// of L in work[iu:].
ok = impl.Dbdsqr(blas.Upper, m, m, m, 0, s, work[ie:],
work[iu:], ldworku, u, ldu, work, 1, work[iwork:])
// Multiply right singular vectors of L in work[iu:]
// Q in VT, storing result in A.
bi.Dgemm(blas.NoTrans, blas.NoTrans, m, n, m, 1,
work[iu:], ldworku, vt, ldvt, 0, a, lda)
// Copy right singular vectors of A from A to VT.
impl.Dlacpy(blas.All, m, n, a, lda, vt, ldvt)
} else {
// Insufficient workspace for a fast algorithm.
itau := 0
iwork := itau + m
// Compute A = L * Q, copying result to VT.
impl.Dgelqf(m, n, a, lda, work[itau:], work[iwork:], lwork-iwork)
impl.Dlacpy(blas.Upper, m, n, a, lda, vt, ldvt)
// Generate Q in VT.
impl.Dorglq(n, n, m, vt, ldvt, work[itau:], work[iwork:], lwork-iwork)
// Copy L to U, zeroing out above it.
impl.Dlacpy(blas.Lower, m, m, a, lda, u, ldu)
impl.Dlaset(blas.Upper, m-1, m-1, 0, 0, u[1:], ldu)
ie = itau
itauq := ie + m
itaup := itauq + m
iwork = itaup + m
// Bidiagonalize L in U.
impl.Dgebrd(m, m, u, ldu, s, work[ie:], work[itauq:],
work[itaup:], work[iwork:], lwork-iwork)
// Multiply right bidiagonalizing vectors in U by Q in VT.
impl.Dormbr(lapack.ApplyP, blas.Left, blas.Trans, m, n, m,
u, ldu, work[itaup:], vt, ldvt, work[iwork:], lwork-iwork)
// Generate left bidiagonalizing vectors in U.
impl.Dorgbr(lapack.GenerateQ, m, m, m, u, ldu, work[itauq:], work[iwork:], lwork-iwork)
iwork = ie + m
// Perform bidiagonal QR iteration, computing left singular
// vectors of A in U and computing right singular vectors
// of A in VT.
ok = impl.Dbdsqr(blas.Upper, m, n, m, 0, s, work[ie:],
vt, ldvt, u, ldu, work, 1, work[iwork:])
}
}
}
} else {
// Path 10t.
// N at least M, but not much larger.
ie = 0
itauq := ie + m
itaup := itauq + m
iwork := itaup + m
// Bidiagonalize A.
impl.Dgebrd(m, n, a, lda, s, work[ie:], work[itauq:], work[itaup:], work[iwork:], lwork-iwork)
if wantuas {
// If left singular vectors desired in U, copy result to U and
// generate left bidiagonalizing vectors in U.
impl.Dlacpy(blas.Lower, m, m, a, lda, u, ldu)
impl.Dorgbr(lapack.GenerateQ, m, m, n, u, ldu, work[itauq:], work[iwork:], lwork-iwork)
}
if wantvas {
// If right singular vectors desired in VT, copy result to VT
// and generate right bidiagonalizing vectors in VT.
impl.Dlacpy(blas.Upper, m, n, a, lda, vt, ldvt)
var nrvt int
if wantva {
nrvt = n
} else {
nrvt = m
}
impl.Dorgbr(lapack.GeneratePT, nrvt, n, m, vt, ldvt, work[itaup:], work[iwork:], lwork-iwork)
}
if wantuo {
panic(noSVDO)
}
if wantvo {
panic(noSVDO)
}
iwork = ie + m
var nru, ncvt int
if wantuas || wantuo {
nru = m
}
if wantvas || wantvo {
ncvt = n
}
if !wantuo && !wantvo {
// Perform bidiagonal QR iteration, if desired, computing left
// singular vectors in U and computing right singular vectors in
// VT.
ok = impl.Dbdsqr(blas.Lower, m, ncvt, nru, 0, s, work[ie:],
vt, ldvt, u, ldu, work, 1, work[iwork:])
} else {
// There will be two branches when the implementation is complete.
panic(noSVDO)
}
}
}
if !ok {
if ie > 1 {
for i := 0; i < minmn-1; i++ {
work[i+1] = work[i+ie]
}
}
if ie < 1 {
for i := minmn - 2; i >= 0; i-- {
work[i+1] = work[i+ie]
}
}
}
// Undo scaling if necessary.
if iscl {
if anrm > bignum {
impl.Dlascl(lapack.General, 0, 0, bignum, anrm, 1, minmn, s, minmn)
}
if !ok && anrm > bignum {
impl.Dlascl(lapack.General, 0, 0, bignum, anrm, 1, minmn-1, work[1:], minmn)
}
if anrm < smlnum {
impl.Dlascl(lapack.General, 0, 0, smlnum, anrm, 1, minmn, s, minmn)
}
if !ok && anrm < smlnum {
impl.Dlascl(lapack.General, 0, 0, smlnum, anrm, 1, minmn-1, work[1:], minmn)
}
}
work[0] = float64(maxwrk)
return ok
}