extern "C" magma_int_t
magma_dgeev(
char jobvl, char jobvr, magma_int_t n,
double *A, magma_int_t lda,
double *WR, double *WI,
double *vl, magma_int_t ldvl,
double *vr, magma_int_t ldvr,
double *work, magma_int_t lwork,
magma_int_t *info )
{
/* -- MAGMA (version 1.4.1) --
Univ. of Tennessee, Knoxville
Univ. of California, Berkeley
Univ. of Colorado, Denver
December 2013
Purpose
=======
DGEEV computes for an N-by-N real nonsymmetric matrix A, the
eigenvalues and, optionally, the left and/or right eigenvectors.
The right eigenvector v(j) of A satisfies
A * v(j) = lambda(j) * v(j)
where lambda(j) is its eigenvalue.
The left eigenvector u(j) of A satisfies
u(j)**T * A = lambda(j) * u(j)**T
where u(j)**T denotes the transpose of u(j).
The computed eigenvectors are normalized to have Euclidean norm
equal to 1 and largest component real.
Arguments
=========
JOBVL (input) CHARACTER*1
= 'N': left eigenvectors of A are not computed;
= 'V': left eigenvectors of are computed.
JOBVR (input) CHARACTER*1
= 'N': right eigenvectors of A are not computed;
= 'V': right eigenvectors of A are computed.
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the N-by-N matrix A.
On exit, A has been overwritten.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
WR (output) DOUBLE PRECISION array, dimension (N)
WI (output) DOUBLE PRECISION array, dimension (N)
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.
VL (output) DOUBLE PRECISION array, dimension (LDVL,N)
If JOBVL = 'V', the left eigenvectors u(j) are stored one
after another in the columns of VL, in the same order
as their eigenvalues.
If JOBVL = 'N', VL is not referenced.
u(j) = VL(:,j), the j-th column of VL.
LDVL (input) INTEGER
The leading dimension of the array VL. LDVL >= 1; if
JOBVL = 'V', LDVL >= N.
VR (output) DOUBLE PRECISION array, dimension (LDVR,N)
If JOBVR = 'V', the right eigenvectors v(j) are stored one
after another in the columns of VR, in the same order
as their eigenvalues.
If JOBVR = 'N', VR is not referenced.
v(j) = VR(:,j), the j-th column of VR.
LDVR (input) INTEGER
The leading dimension of the array VR. LDVR >= 1; if
JOBVR = 'V', LDVR >= N.
WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= (2+nb)*N.
For optimal performance, LWORK >= (2+2*nb)*N.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, the QR algorithm failed to compute all the
eigenvalues, and no eigenvectors have been computed;
elements and i+1:N of W contain eigenvalues which have
converged.
===================================================================== */
#define vl(i,j) (vl + (i) + (j)*ldvl)
#define vr(i,j) (vr + (i) + (j)*ldvr)
magma_int_t c_one = 1;
magma_int_t c_zero = 0;
double d__1, d__2;
double r, cs, sn, scl;
double dum[1], eps;
double anrm, cscale, bignum, smlnum;
magma_int_t i, k, ilo, ihi;
magma_int_t ibal, ierr, itau, iwrk, nout, liwrk, i__1, i__2, nb;
magma_int_t scalea, minwrk, lquery, wantvl, wantvr, select[1];
char side[2] = {0, 0};
char jobvl_[2] = {jobvl, 0};
char jobvr_[2] = {jobvr, 0};
*info = 0;
lquery = lwork == -1;
wantvl = lapackf77_lsame( jobvl_, "V" );
wantvr = lapackf77_lsame( jobvr_, "V" );
if (! wantvl && ! lapackf77_lsame( jobvl_, "N" )) {
*info = -1;
} else if (! wantvr && ! lapackf77_lsame( jobvr_, "N" )) {
*info = -2;
} else if (n < 0) {
*info = -3;
} else if (lda < max(1,n)) {
*info = -5;
} else if ( (ldvl < 1) || (wantvl && (ldvl < n))) {
*info = -9;
} else if ( (ldvr < 1) || (wantvr && (ldvr < n))) {
*info = -11;
}
/* Compute workspace */
nb = magma_get_dgehrd_nb( n );
if (*info == 0) {
minwrk = (2+nb)*n;
work[0] = MAGMA_D_MAKE( (double) minwrk, 0. );
if (lwork < minwrk && ! lquery) {
*info = -13;
}
}
if (*info != 0) {
magma_xerbla( __func__, -(*info) );
return *info;
}
else if (lquery) {
return *info;
}
/* Quick return if possible */
if (n == 0) {
return *info;
}
#if defined(VERSION3)
double *dT;
if (MAGMA_SUCCESS != magma_dmalloc( &dT, nb*n )) {
*info = MAGMA_ERR_DEVICE_ALLOC;
return *info;
}
#endif
/* Get machine constants */
eps = lapackf77_dlamch( "P" );
smlnum = lapackf77_dlamch( "S" );
bignum = 1. / smlnum;
lapackf77_dlabad( &smlnum, &bignum );
smlnum = magma_dsqrt( smlnum ) / eps;
bignum = 1. / smlnum;
/* Scale A if max element outside range [SMLNUM,BIGNUM] */
anrm = lapackf77_dlange( "M", &n, &n, A, &lda, dum );
scalea = 0;
if (anrm > 0. && anrm < smlnum) {
scalea = 1;
cscale = smlnum;
} else if (anrm > bignum) {
scalea = 1;
cscale = bignum;
}
if (scalea) {
lapackf77_dlascl( "G", &c_zero, &c_zero, &anrm, &cscale, &n, &n, A, &lda, &ierr );
}
/* Balance the matrix
* (Workspace: need N) */
ibal = 0;
lapackf77_dgebal( "B", &n, A, &lda, &ilo, &ihi, &work[ibal], &ierr );
/* Reduce to upper Hessenberg form
* (Workspace: need 3*N, prefer 2*N + N*NB) */
itau = ibal + n;
iwrk = itau + n;
liwrk = lwork - iwrk;
#if defined(VERSION1)
// Version 1 - LAPACK
lapackf77_dgehrd( &n, &ilo, &ihi, A, &lda,
&work[itau], &work[iwrk], &liwrk, &ierr );
#elif defined(VERSION2)
// Version 2 - LAPACK consistent HRD
magma_dgehrd2( n, ilo, ihi, A, lda,
&work[itau], &work[iwrk], liwrk, &ierr );
#elif defined(VERSION3)
// Version 3 - LAPACK consistent MAGMA HRD + T matrices stored,
magma_dgehrd( n, ilo, ihi, A, lda,
&work[itau], &work[iwrk], liwrk, dT, &ierr );
#endif
if (wantvl) {
/* Want left eigenvectors
* Copy Householder vectors to VL */
side[0] = 'L';
lapackf77_dlacpy( MagmaLowerStr, &n, &n,
A, &lda, vl, &ldvl );
/* Generate orthogonal matrix in VL
* (Workspace: need 3*N-1, prefer 2*N + (N-1)*NB) */
#if defined(VERSION1) || defined(VERSION2)
// Version 1 & 2 - LAPACK
lapackf77_dorghr( &n, &ilo, &ihi, vl, &ldvl, &work[itau],
&work[iwrk], &liwrk, &ierr );
#elif defined(VERSION3)
// Version 3 - LAPACK consistent MAGMA HRD + T matrices stored
magma_dorghr( n, ilo, ihi, vl, ldvl, &work[itau], dT, nb, &ierr );
#endif
/* Perform QR iteration, accumulating Schur vectors in VL
* (Workspace: need N+1, prefer N+HSWORK (see comments) ) */
iwrk = itau;
liwrk = lwork - iwrk;
lapackf77_dhseqr( "S", "V", &n, &ilo, &ihi, A, &lda, WR, WI,
vl, &ldvl, &work[iwrk], &liwrk, info );
if (wantvr) {
/* Want left and right eigenvectors
* Copy Schur vectors to VR */
side[0] = 'B';
lapackf77_dlacpy( "F", &n, &n, vl, &ldvl, vr, &ldvr );
}
}
else if (wantvr) {
/* Want right eigenvectors
* Copy Householder vectors to VR */
side[0] = 'R';
lapackf77_dlacpy( "L", &n, &n, A, &lda, vr, &ldvr );
/* Generate orthogonal matrix in VR
* (Workspace: need 3*N-1, prefer 2*N + (N-1)*NB) */
#if defined(VERSION1) || defined(VERSION2)
// Version 1 & 2 - LAPACK
lapackf77_dorghr( &n, &ilo, &ihi, vr, &ldvr, &work[itau],
&work[iwrk], &liwrk, &ierr );
#elif defined(VERSION3)
// Version 3 - LAPACK consistent MAGMA HRD + T matrices stored
magma_dorghr( n, ilo, ihi, vr, ldvr, &work[itau], dT, nb, &ierr );
#endif
/* Perform QR iteration, accumulating Schur vectors in VR
* (Workspace: need N+1, prefer N+HSWORK (see comments) ) */
iwrk = itau;
liwrk = lwork - iwrk;
lapackf77_dhseqr( "S", "V", &n, &ilo, &ihi, A, &lda, WR, WI,
vr, &ldvr, &work[iwrk], &liwrk, info );
}
else {
/* Compute eigenvalues only
* (Workspace: need N+1, prefer N+HSWORK (see comments) ) */
iwrk = itau;
liwrk = lwork - iwrk;
lapackf77_dhseqr( "E", "N", &n, &ilo, &ihi, A, &lda, WR, WI,
vr, &ldvr, &work[iwrk], &liwrk, info );
}
/* If INFO > 0 from DHSEQR, then quit */
if (*info > 0) {
goto CLEANUP;
}
if (wantvl || wantvr) {
/* Compute left and/or right eigenvectors
* (Workspace: need 4*N) */
liwrk = lwork - iwrk;
#if TREVC_VERSION == 1
lapackf77_dtrevc( side, "B", select, &n, A, &lda, vl, &ldvl,
vr, &ldvr, &n, &nout, &work[iwrk], &ierr );
#elif TREVC_VERSION == 2
lapackf77_dtrevc3( side, "B", select, &n, A, &lda, vl, &ldvl,
vr, &ldvr, &n, &nout, &work[iwrk], &liwrk, &ierr );
#endif
}
if (wantvl) {
/* Undo balancing of left eigenvectors
* (Workspace: need N) */
lapackf77_dgebak( "B", "L", &n, &ilo, &ihi, &work[ibal], &n,
vl, &ldvl, &ierr );
/* Normalize left eigenvectors and make largest component real */
for (i = 0; i < n; ++i) {
if ( WI[i] == 0. ) {
scl = 1. / cblas_dnrm2( n, vl(0,i), 1 );
cblas_dscal( n, scl, vl(0,i), 1 );
}
else if ( WI[i] > 0. ) {
d__1 = cblas_dnrm2( n, vl(0,i), 1 );
d__2 = cblas_dnrm2( n, vl(0,i+1), 1 );
scl = 1. / lapackf77_dlapy2( &d__1, &d__2 );
cblas_dscal( n, scl, vl(0,i), 1 );
cblas_dscal( n, scl, vl(0,i+1), 1 );
for (k = 0; k < n; ++k) {
/* Computing 2nd power */
d__1 = *vl(k,i);
d__2 = *vl(k,i+1);
work[iwrk + k] = d__1*d__1 + d__2*d__2;
}
k = cblas_idamax( n, &work[iwrk], 1 );
lapackf77_dlartg( vl(k,i), vl(k,i+1), &cs, &sn, &r );
cblas_drot( n, vl(0,i), 1, vl(0,i+1), 1, cs, sn );
*vl(k,i+1) = 0.;
}
}
}
if (wantvr) {
/* Undo balancing of right eigenvectors
* (Workspace: need N) */
lapackf77_dgebak( "B", "R", &n, &ilo, &ihi, &work[ibal], &n,
vr, &ldvr, &ierr );
/* Normalize right eigenvectors and make largest component real */
for (i = 0; i < n; ++i) {
if ( WI[i] == 0. ) {
scl = 1. / cblas_dnrm2( n, vr(0,i), 1 );
cblas_dscal( n, scl, vr(0,i), 1 );
}
else if ( WI[i] > 0. ) {
d__1 = cblas_dnrm2( n, vr(0,i), 1 );
d__2 = cblas_dnrm2( n, vr(0,i+1), 1 );
scl = 1. / lapackf77_dlapy2( &d__1, &d__2 );
cblas_dscal( n, scl, vr(0,i), 1 );
cblas_dscal( n, scl, vr(0,i+1), 1 );
for (k = 0; k < n; ++k) {
/* Computing 2nd power */
d__1 = *vr(k,i);
d__2 = *vr(k,i+1);
work[iwrk + k] = d__1*d__1 + d__2*d__2;
}
k = cblas_idamax( n, &work[iwrk], 1 );
lapackf77_dlartg( vr(k,i), vr(k,i+1), &cs, &sn, &r );
cblas_drot( n, vr(0,i), 1, vr(0,i+1), 1, cs, sn );
*vr(k,i+1) = 0.;
}
}
}
CLEANUP:
/* Undo scaling if necessary */
if (scalea) {
i__1 = n - (*info);
i__2 = max( n - (*info), 1 );
lapackf77_dlascl( "G", &c_zero, &c_zero, &cscale, &anrm, &i__1, &c_one,
WR + (*info), &i__2, &ierr );
lapackf77_dlascl( "G", &c_zero, &c_zero, &cscale, &anrm, &i__1, &c_one,
WI + (*info), &i__2, &ierr );
if (*info > 0) {
i__1 = ilo - 1;
lapackf77_dlascl( "G", &c_zero, &c_zero, &cscale, &anrm, &i__1, &c_one,
WR, &n, &ierr );
lapackf77_dlascl( "G", &c_zero, &c_zero, &cscale, &anrm, &i__1, &c_one,
WI, &n, &ierr );
}
}
#if defined(VERSION3)
magma_free( dT );
#endif
return *info;
} /* magma_dgeev */
/**
Purpose
-------
DGEEV computes for an N-by-N real nonsymmetric matrix A, the
eigenvalues and, optionally, the left and/or right eigenvectors.
The right eigenvector v(j) of A satisfies
A * v(j) = lambda(j) * v(j)
where lambda(j) is its eigenvalue.
The left eigenvector u(j) of A satisfies
u(j)**T * A = lambda(j) * u(j)**T
where u(j)**T denotes the transpose of u(j).
The computed eigenvectors are normalized to have Euclidean norm
equal to 1 and largest component real.
Arguments
---------
@param[in]
jobvl magma_vec_t
- = MagmaNoVec: left eigenvectors of A are not computed;
- = MagmaVec: left eigenvectors of are computed.
@param[in]
jobvr magma_vec_t
- = MagmaNoVec: right eigenvectors of A are not computed;
- = MagmaVec: right eigenvectors of A are computed.
@param[in]
n INTEGER
The order of the matrix A. N >= 0.
@param[in,out]
A DOUBLE PRECISION array, dimension (LDA,N)
On entry, the N-by-N matrix A.
On exit, A has been overwritten.
@param[in]
lda INTEGER
The leading dimension of the array A. LDA >= max(1,N).
@param[out]
wr DOUBLE PRECISION array, dimension (N)
@param[out]
wi DOUBLE PRECISION array, dimension (N)
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.
@param[out]
VL DOUBLE PRECISION array, dimension (LDVL,N)
If JOBVL = MagmaVec, the left eigenvectors u(j) are stored one
after another in the columns of VL, in the same order
as their eigenvalues.
If JOBVL = MagmaNoVec, VL is not referenced.
u(j) = VL(:,j), the j-th column of VL.
@param[in]
ldvl INTEGER
The leading dimension of the array VL. LDVL >= 1; if
JOBVL = MagmaVec, LDVL >= N.
@param[out]
VR DOUBLE PRECISION array, dimension (LDVR,N)
If JOBVR = MagmaVec, the right eigenvectors v(j) are stored one
after another in the columns of VR, in the same order
as their eigenvalues.
If JOBVR = MagmaNoVec, VR is not referenced.
v(j) = VR(:,j), the j-th column of VR.
@param[in]
ldvr INTEGER
The leading dimension of the array VR. LDVR >= 1; if
JOBVR = MagmaVec, LDVR >= N.
@param[out]
work (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
@param[in]
lwork INTEGER
The dimension of the array WORK. LWORK >= (2+nb)*N.
For optimal performance, LWORK >= (2+2*nb)*N.
\n
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
@param[out]
info INTEGER
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value.
- > 0: if INFO = i, the QR algorithm failed to compute all the
eigenvalues, and no eigenvectors have been computed;
elements and i+1:N of W contain eigenvalues which have
converged.
@ingroup magma_dgeev_driver
********************************************************************/
extern "C" magma_int_t
magma_dgeev_m(
magma_vec_t jobvl, magma_vec_t jobvr, magma_int_t n,
double *A, magma_int_t lda,
double *wr, double *wi,
double *VL, magma_int_t ldvl,
double *VR, magma_int_t ldvr,
double *work, magma_int_t lwork,
magma_int_t *info )
{
#define VL(i,j) (VL + (i) + (j)*ldvl)
#define VR(i,j) (VR + (i) + (j)*ldvr)
const magma_int_t ione = 1;
const magma_int_t izero = 0;
double d__1, d__2;
double r, cs, sn, scl;
double dum[1], eps;
double anrm, cscale, bignum, smlnum;
magma_int_t i, k, ilo, ihi;
magma_int_t ibal, ierr, itau, iwrk, nout, liwrk, nb;
magma_int_t scalea, minwrk, optwrk, lquery, wantvl, wantvr, select[1];
magma_side_t side = MagmaRight;
magma_timer_t time_total=0, time_gehrd=0, time_unghr=0, time_hseqr=0, time_trevc=0, time_sum=0;
magma_flops_t flop_total=0, flop_gehrd=0, flop_unghr=0, flop_hseqr=0, flop_trevc=0, flop_sum=0;
timer_start( time_total );
flops_start( flop_total );
*info = 0;
lquery = (lwork == -1);
wantvl = (jobvl == MagmaVec);
wantvr = (jobvr == MagmaVec);
if (! wantvl && jobvl != MagmaNoVec) {
*info = -1;
} else if (! wantvr && jobvr != MagmaNoVec) {
*info = -2;
} else if (n < 0) {
*info = -3;
} else if (lda < max(1,n)) {
*info = -5;
} else if ( (ldvl < 1) || (wantvl && (ldvl < n))) {
*info = -9;
} else if ( (ldvr < 1) || (wantvr && (ldvr < n))) {
*info = -11;
}
/* Compute workspace */
nb = magma_get_dgehrd_nb( n );
if (*info == 0) {
minwrk = (2 + nb)*n;
optwrk = (2 + 2*nb)*n;
work[0] = MAGMA_D_MAKE( (double) optwrk, 0. );
if (lwork < minwrk && ! lquery) {
*info = -13;
}
}
if (*info != 0) {
magma_xerbla( __func__, -(*info) );
return *info;
}
else if (lquery) {
return *info;
}
/* Quick return if possible */
if (n == 0) {
return *info;
}
#if defined(Version3) || defined(Version4) || defined(Version5)
double *dT;
if (MAGMA_SUCCESS != magma_dmalloc( &dT, nb*n )) {
*info = MAGMA_ERR_DEVICE_ALLOC;
return *info;
}
#endif
#if defined(Version4) || defined(Version5)
double *T;
if (MAGMA_SUCCESS != magma_dmalloc_cpu( &T, nb*n )) {
magma_free( dT );
*info = MAGMA_ERR_HOST_ALLOC;
return *info;
}
#endif
/* Get machine constants */
eps = lapackf77_dlamch( "P" );
smlnum = lapackf77_dlamch( "S" );
bignum = 1. / smlnum;
lapackf77_dlabad( &smlnum, &bignum );
smlnum = magma_dsqrt( smlnum ) / eps;
bignum = 1. / smlnum;
/* Scale A if max element outside range [SMLNUM,BIGNUM] */
anrm = lapackf77_dlange( "M", &n, &n, A, &lda, dum );
scalea = 0;
if (anrm > 0. && anrm < smlnum) {
scalea = 1;
cscale = smlnum;
} else if (anrm > bignum) {
scalea = 1;
cscale = bignum;
}
if (scalea) {
lapackf77_dlascl( "G", &izero, &izero, &anrm, &cscale, &n, &n, A, &lda, &ierr );
}
/* Balance the matrix
* (Workspace: need N)
* - this space is reserved until after gebak */
ibal = 0;
lapackf77_dgebal( "B", &n, A, &lda, &ilo, &ihi, &work[ibal], &ierr );
/* Reduce to upper Hessenberg form
* (Workspace: need 3*N, prefer 2*N + N*NB)
* - including N reserved for gebal/gebak, unused by dgehrd */
itau = ibal + n;
iwrk = itau + n;
liwrk = lwork - iwrk;
timer_start( time_gehrd );
flops_start( flop_gehrd );
#if defined(Version1)
// Version 1 - LAPACK
lapackf77_dgehrd( &n, &ilo, &ihi, A, &lda,
&work[itau], &work[iwrk], &liwrk, &ierr );
#elif defined(Version2)
// Version 2 - LAPACK consistent HRD
magma_dgehrd2( n, ilo, ihi, A, lda,
&work[itau], &work[iwrk], liwrk, &ierr );
#elif defined(Version3)
// Version 3 - LAPACK consistent MAGMA HRD + T matrices stored,
magma_dgehrd( n, ilo, ihi, A, lda,
&work[itau], &work[iwrk], liwrk, dT, &ierr );
#elif defined(Version4) || defined(Version5)
// Version 4 - Multi-GPU, T on host
magma_dgehrd_m( n, ilo, ihi, A, lda,
&work[itau], &work[iwrk], liwrk, T, &ierr );
magma_dsetmatrix( nb, n, T, nb, dT, nb );
#endif
time_sum += timer_stop( time_gehrd );
flop_sum += flops_stop( flop_gehrd );
if (wantvl) {
/* Want left eigenvectors
* Copy Householder vectors to VL */
side = MagmaLeft;
lapackf77_dlacpy( MagmaLowerStr, &n, &n, A, &lda, VL, &ldvl );
/* Generate orthogonal matrix in VL
* (Workspace: need 3*N-1, prefer 2*N + (N-1)*NB)
* - including N reserved for gebal/gebak, unused by dorghr */
timer_start( time_unghr );
flops_start( flop_unghr );
#if defined(Version1) || defined(Version2)
// Version 1 & 2 - LAPACK
lapackf77_dorghr( &n, &ilo, &ihi, VL, &ldvl, &work[itau],
&work[iwrk], &liwrk, &ierr );
#elif defined(Version3) || defined(Version4)
// Version 3 - LAPACK consistent MAGMA HRD + T matrices stored
magma_dorghr( n, ilo, ihi, VL, ldvl, &work[itau], dT, nb, &ierr );
#elif defined(Version5)
// Version 5 - Multi-GPU, T on host
magma_dorghr_m( n, ilo, ihi, VL, ldvl, &work[itau], T, nb, &ierr );
#endif
time_sum += timer_stop( time_unghr );
flop_sum += flops_stop( flop_unghr );
timer_start( time_hseqr );
flops_start( flop_hseqr );
/* Perform QR iteration, accumulating Schur vectors in VL
* (Workspace: need N+1, prefer N+HSWORK (see comments) )
* - including N reserved for gebal/gebak, unused by dhseqr */
iwrk = itau;
liwrk = lwork - iwrk;
lapackf77_dhseqr( "S", "V", &n, &ilo, &ihi, A, &lda, wr, wi,
VL, &ldvl, &work[iwrk], &liwrk, info );
time_sum += timer_stop( time_hseqr );
flop_sum += flops_stop( flop_hseqr );
if (wantvr) {
/* Want left and right eigenvectors
* Copy Schur vectors to VR */
side = MagmaBothSides;
lapackf77_dlacpy( "F", &n, &n, VL, &ldvl, VR, &ldvr );
}
}
else if (wantvr) {
/* Want right eigenvectors
* Copy Householder vectors to VR */
side = MagmaRight;
lapackf77_dlacpy( "L", &n, &n, A, &lda, VR, &ldvr );
/* Generate orthogonal matrix in VR
* (Workspace: need 3*N-1, prefer 2*N + (N-1)*NB)
* - including N reserved for gebal/gebak, unused by dorghr */
timer_start( time_unghr );
flops_start( flop_unghr );
#if defined(Version1) || defined(Version2)
// Version 1 & 2 - LAPACK
lapackf77_dorghr( &n, &ilo, &ihi, VR, &ldvr, &work[itau],
&work[iwrk], &liwrk, &ierr );
#elif defined(Version3) || defined(Version4)
// Version 3 - LAPACK consistent MAGMA HRD + T matrices stored
magma_dorghr( n, ilo, ihi, VR, ldvr, &work[itau], dT, nb, &ierr );
#elif defined(Version5)
// Version 5 - Multi-GPU, T on host
magma_dorghr_m( n, ilo, ihi, VR, ldvr, &work[itau], T, nb, &ierr );
#endif
time_sum += timer_stop( time_unghr );
flop_sum += flops_stop( flop_unghr );
/* Perform QR iteration, accumulating Schur vectors in VR
* (Workspace: need N+1, prefer N+HSWORK (see comments) )
* - including N reserved for gebal/gebak, unused by dhseqr */
timer_start( time_hseqr );
flops_start( flop_hseqr );
iwrk = itau;
liwrk = lwork - iwrk;
lapackf77_dhseqr( "S", "V", &n, &ilo, &ihi, A, &lda, wr, wi,
VR, &ldvr, &work[iwrk], &liwrk, info );
time_sum += timer_stop( time_hseqr );
flop_sum += flops_stop( flop_hseqr );
}
else {
/* Compute eigenvalues only
* (Workspace: need N+1, prefer N+HSWORK (see comments) )
* - including N reserved for gebal/gebak, unused by dhseqr */
timer_start( time_hseqr );
flops_start( flop_hseqr );
iwrk = itau;
liwrk = lwork - iwrk;
lapackf77_dhseqr( "E", "N", &n, &ilo, &ihi, A, &lda, wr, wi,
VR, &ldvr, &work[iwrk], &liwrk, info );
time_sum += timer_stop( time_hseqr );
flop_sum += flops_stop( flop_hseqr );
}
/* If INFO > 0 from DHSEQR, then quit */
if (*info > 0) {
goto CLEANUP;
}
timer_start( time_trevc );
flops_start( flop_trevc );
if (wantvl || wantvr) {
/* Compute left and/or right eigenvectors
* (Workspace: need 4*N, prefer (2 + 2*nb)*N)
* - including N reserved for gebal/gebak, unused by dtrevc */
liwrk = lwork - iwrk;
#if TREVC_VERSION == 1
lapackf77_dtrevc( lapack_side_const(side), "B", select, &n, A, &lda, VL, &ldvl,
VR, &ldvr, &n, &nout, &work[iwrk], &ierr );
#elif TREVC_VERSION == 2
lapackf77_dtrevc3( lapack_side_const(side), "B", select, &n, A, &lda, VL, &ldvl,
VR, &ldvr, &n, &nout, &work[iwrk], &liwrk, &ierr );
#elif TREVC_VERSION == 3
magma_dtrevc3( side, MagmaBacktransVec, select, n, A, lda, VL, ldvl,
VR, ldvr, n, &nout, &work[iwrk], liwrk, &ierr );
#elif TREVC_VERSION == 4
magma_dtrevc3_mt( side, MagmaBacktransVec, select, n, A, lda, VL, ldvl,
VR, ldvr, n, &nout, &work[iwrk], liwrk, &ierr );
#elif TREVC_VERSION == 5
magma_dtrevc3_mt_gpu( side, MagmaBacktransVec, select, n, A, lda, VL, ldvl,
VR, ldvr, n, &nout, &work[iwrk], liwrk, &ierr );
#else
#error Unknown TREVC_VERSION
#endif
}
time_sum += timer_stop( time_trevc );
flop_sum += flops_stop( flop_trevc );
if (wantvl) {
/* Undo balancing of left eigenvectors
* (Workspace: need N) */
lapackf77_dgebak( "B", "L", &n, &ilo, &ihi, &work[ibal], &n,
VL, &ldvl, &ierr );
/* Normalize left eigenvectors and make largest component real */
for (i = 0; i < n; ++i) {
if ( wi[i] == 0. ) {
scl = 1. / magma_cblas_dnrm2( n, VL(0,i), 1 );
blasf77_dscal( &n, &scl, VL(0,i), &ione );
}
else if ( wi[i] > 0. ) {
d__1 = magma_cblas_dnrm2( n, VL(0,i), 1 );
d__2 = magma_cblas_dnrm2( n, VL(0,i+1), 1 );
scl = 1. / lapackf77_dlapy2( &d__1, &d__2 );
blasf77_dscal( &n, &scl, VL(0,i), &ione );
blasf77_dscal( &n, &scl, VL(0,i+1), &ione );
for (k = 0; k < n; ++k) {
/* Computing 2nd power */
d__1 = *VL(k,i);
d__2 = *VL(k,i+1);
work[iwrk + k] = d__1*d__1 + d__2*d__2;
}
k = blasf77_idamax( &n, &work[iwrk], &ione ) - 1; // subtract 1; k is 0-based
lapackf77_dlartg( VL(k,i), VL(k,i+1), &cs, &sn, &r );
blasf77_drot( &n, VL(0,i), &ione, VL(0,i+1), &ione, &cs, &sn );
*VL(k,i+1) = 0.;
}
}
}
if (wantvr) {
/* Undo balancing of right eigenvectors
* (Workspace: need N) */
lapackf77_dgebak( "B", "R", &n, &ilo, &ihi, &work[ibal], &n,
VR, &ldvr, &ierr );
/* Normalize right eigenvectors and make largest component real */
for (i = 0; i < n; ++i) {
if ( wi[i] == 0. ) {
scl = 1. / magma_cblas_dnrm2( n, VR(0,i), 1 );
blasf77_dscal( &n, &scl, VR(0,i), &ione );
}
else if ( wi[i] > 0. ) {
d__1 = magma_cblas_dnrm2( n, VR(0,i), 1 );
d__2 = magma_cblas_dnrm2( n, VR(0,i+1), 1 );
scl = 1. / lapackf77_dlapy2( &d__1, &d__2 );
blasf77_dscal( &n, &scl, VR(0,i), &ione );
blasf77_dscal( &n, &scl, VR(0,i+1), &ione );
for (k = 0; k < n; ++k) {
/* Computing 2nd power */
d__1 = *VR(k,i);
d__2 = *VR(k,i+1);
work[iwrk + k] = d__1*d__1 + d__2*d__2;
}
k = blasf77_idamax( &n, &work[iwrk], &ione ) - 1; // subtract 1; k is 0-based
lapackf77_dlartg( VR(k,i), VR(k,i+1), &cs, &sn, &r );
blasf77_drot( &n, VR(0,i), &ione, VR(0,i+1), &ione, &cs, &sn );
*VR(k,i+1) = 0.;
}
}
}
CLEANUP:
/* Undo scaling if necessary */
if (scalea) {
// converged eigenvalues, stored in wr[i+1:n] and wi[i+1:n] for i = INFO
magma_int_t nval = n - (*info);
magma_int_t ld = max( nval, 1 );
lapackf77_dlascl( "G", &izero, &izero, &cscale, &anrm, &nval, &ione, wr + (*info), &ld, &ierr );
lapackf77_dlascl( "G", &izero, &izero, &cscale, &anrm, &nval, &ione, wi + (*info), &ld, &ierr );
if (*info > 0) {
// first ilo columns were already upper triangular,
// so the corresponding eigenvalues are also valid.
nval = ilo - 1;
lapackf77_dlascl( "G", &izero, &izero, &cscale, &anrm, &nval, &ione, wr, &n, &ierr );
lapackf77_dlascl( "G", &izero, &izero, &cscale, &anrm, &nval, &ione, wi, &n, &ierr );
}
}
#if defined(Version3) || defined(Version4) || defined(Version5)
magma_free( dT );
#endif
#if defined(Version4) || defined(Version5)
magma_free_cpu( T );
#endif
timer_stop( time_total );
flops_stop( flop_total );
timer_printf( "dgeev times n %5d, gehrd %7.3f, unghr %7.3f, hseqr %7.3f, trevc %7.3f, total %7.3f, sum %7.3f\n",
(int) n, time_gehrd, time_unghr, time_hseqr, time_trevc, time_total, time_sum );
timer_printf( "dgeev flops n %5d, gehrd %7lld, unghr %7lld, hseqr %7lld, trevc %7lld, total %7lld, sum %7lld\n",
(int) n, flop_gehrd, flop_unghr, flop_hseqr, flop_trevc, flop_total, flop_sum );
work[0] = MAGMA_D_MAKE( (double) optwrk, 0. );
return *info;
} /* magma_dgeev */