/**
Purpose
-------
CGEEV computes for an N-by-N complex 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)**H * A = lambda(j) * u(j)**H
where u(j)**H denotes the conjugate 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 COMPLEX 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]
W COMPLEX array, dimension (N)
W contains the computed eigenvalues.
@param[out]
VL COMPLEX 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 COMPLEX 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) COMPLEX array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
@param[in]
lwork INTEGER
The dimension of the array WORK. LWORK >= (1+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
rwork (workspace) REAL array, dimension (2*N)
@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_cgeev_driver
********************************************************************/
extern "C" magma_int_t
magma_cgeev(
magma_vec_t jobvl, magma_vec_t jobvr, magma_int_t n,
magmaFloatComplex *A, magma_int_t lda,
magmaFloatComplex *W,
magmaFloatComplex *VL, magma_int_t ldvl,
magmaFloatComplex *VR, magma_int_t ldvr,
magmaFloatComplex *work, magma_int_t lwork,
float *rwork, magma_int_t *info )
{
#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;
float d__1, d__2;
magmaFloatComplex tmp;
float scl;
float dum[1], eps;
float 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, irwork, lquery, wantvl, wantvr, select[1];
magma_side_t side = MagmaRight;
irwork = 0;
*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 = -8;
} else if ( (ldvr < 1) || (wantvr && (ldvr < n))) {
*info = -10;
}
/* Compute workspace */
nb = magma_get_cgehrd_nb( n );
if (*info == 0) {
minwrk = (1+nb)*n;
work[0] = MAGMA_C_MAKE( minwrk, 0 );
if (lwork < minwrk && ! lquery) {
*info = -12;
}
}
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)
magmaFloatComplex *dT;
if (MAGMA_SUCCESS != magma_cmalloc( &dT, nb*n )) {
*info = MAGMA_ERR_DEVICE_ALLOC;
return *info;
}
#endif
/* Get machine constants */
eps = lapackf77_slamch( "P" );
smlnum = lapackf77_slamch( "S" );
bignum = 1. / smlnum;
lapackf77_slabad( &smlnum, &bignum );
smlnum = magma_ssqrt( smlnum ) / eps;
bignum = 1. / smlnum;
/* Scale A if max element outside range [SMLNUM,BIGNUM] */
anrm = lapackf77_clange( "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_clascl( "G", &c_zero, &c_zero, &anrm, &cscale, &n, &n, A, &lda, &ierr );
}
/* Balance the matrix
* (CWorkspace: none)
* (RWorkspace: need N)
* - this space is reserved until after gebak */
ibal = 0;
lapackf77_cgebal( "B", &n, A, &lda, &ilo, &ihi, &rwork[ibal], &ierr );
/* Reduce to upper Hessenberg form
* (CWorkspace: need 2*N, prefer N + N*NB)
* (RWorkspace: N)
* - including N reserved for gebal/gebak, unused by cgehrd */
itau = 0;
iwrk = itau + n;
liwrk = lwork - iwrk;
#if defined(VERSION1)
// Version 1 - LAPACK
lapackf77_cgehrd( &n, &ilo, &ihi, A, &lda,
&work[itau], &work[iwrk], &liwrk, &ierr );
#elif defined(VERSION2)
// Version 2 - LAPACK consistent HRD
magma_cgehrd2( n, ilo, ihi, A, lda,
&work[itau], &work[iwrk], liwrk, &ierr );
#elif defined(VERSION3)
// Version 3 - LAPACK consistent MAGMA HRD + T matrices stored,
magma_cgehrd( n, ilo, ihi, A, lda,
&work[itau], &work[iwrk], liwrk, dT, &ierr );
#endif
if (wantvl) {
/* Want left eigenvectors
* Copy Householder vectors to VL */
side = MagmaLeft;
lapackf77_clacpy( MagmaLowerStr, &n, &n, A, &lda, VL, &ldvl );
/* Generate unitary matrix in VL
* (CWorkspace: need 2*N-1, prefer N + (N-1)*NB)
* (RWorkspace: N)
* - including N reserved for gebal/gebak, unused by cunghr */
#if defined(VERSION1) || defined(VERSION2)
// Version 1 & 2 - LAPACK
lapackf77_cunghr( &n, &ilo, &ihi, VL, &ldvl, &work[itau],
&work[iwrk], &liwrk, &ierr );
#elif defined(VERSION3)
// Version 3 - LAPACK consistent MAGMA HRD + T matrices stored
magma_cunghr( n, ilo, ihi, VL, ldvl, &work[itau], dT, nb, &ierr );
#endif
/* Perform QR iteration, accumulating Schur vectors in VL
* (CWorkspace: need 1, prefer HSWORK (see comments) )
* (RWorkspace: N)
* - including N reserved for gebal/gebak, unused by chseqr */
iwrk = itau;
liwrk = lwork - iwrk;
lapackf77_chseqr( "S", "V", &n, &ilo, &ihi, A, &lda, W,
VL, &ldvl, &work[iwrk], &liwrk, info );
if (wantvr) {
/* Want left and right eigenvectors
* Copy Schur vectors to VR */
side = MagmaBothSides;
lapackf77_clacpy( "F", &n, &n, VL, &ldvl, VR, &ldvr );
}
}
else if (wantvr) {
/* Want right eigenvectors
* Copy Householder vectors to VR */
side = MagmaRight;
lapackf77_clacpy( "L", &n, &n, A, &lda, VR, &ldvr );
/* Generate unitary matrix in VR
* (CWorkspace: need 2*N-1, prefer N + (N-1)*NB)
* (RWorkspace: N)
* - including N reserved for gebal/gebak, unused by cunghr */
#if defined(VERSION1) || defined(VERSION2)
// Version 1 & 2 - LAPACK
lapackf77_cunghr( &n, &ilo, &ihi, VR, &ldvr, &work[itau],
&work[iwrk], &liwrk, &ierr );
#elif defined(VERSION3)
// Version 3 - LAPACK consistent MAGMA HRD + T matrices stored
magma_cunghr( n, ilo, ihi, VR, ldvr, &work[itau], dT, nb, &ierr );
#endif
/* Perform QR iteration, accumulating Schur vectors in VR
* (CWorkspace: need 1, prefer HSWORK (see comments) )
* (RWorkspace: N)
* - including N reserved for gebal/gebak, unused by chseqr */
iwrk = itau;
liwrk = lwork - iwrk;
lapackf77_chseqr( "S", "V", &n, &ilo, &ihi, A, &lda, W,
VR, &ldvr, &work[iwrk], &liwrk, info );
}
else {
/* Compute eigenvalues only
* (CWorkspace: need 1, prefer HSWORK (see comments) )
* (RWorkspace: N)
* - including N reserved for gebal/gebak, unused by chseqr */
iwrk = itau;
liwrk = lwork - iwrk;
lapackf77_chseqr( "E", "N", &n, &ilo, &ihi, A, &lda, W,
VR, &ldvr, &work[iwrk], &liwrk, info );
}
/* If INFO > 0 from CHSEQR, then quit */
if (*info > 0) {
goto CLEANUP;
}
if (wantvl || wantvr) {
/* Compute left and/or right eigenvectors
* (CWorkspace: need 2*N)
* (RWorkspace: need 2*N)
* - including N reserved for gebal/gebak, unused by ctrevc */
irwork = ibal + n;
#if TREVC_VERSION == 1
lapackf77_ctrevc( lapack_side_const(side), "B", select, &n, A, &lda, VL, &ldvl,
VR, &ldvr, &n, &nout, &work[iwrk], &rwork[irwork], &ierr );
#elif TREVC_VERSION == 2
liwrk = lwork - iwrk;
lapackf77_ctrevc3( lapack_side_const(side), "B", select, &n, A, &lda, VL, &ldvl,
VR, &ldvr, &n, &nout, &work[iwrk], &liwrk, &rwork[irwork], &ierr );
#elif TREVC_VERSION == 3
magma_ctrevc3( side, MagmaBacktransVec, select, n, A, lda, VL, ldvl,
VR, ldvr, n, &nout, &work[iwrk], liwrk, &rwork[irwork], &ierr );
#elif TREVC_VERSION == 4
magma_ctrevc3_mt( side, MagmaBacktransVec, select, n, A, lda, VL, ldvl,
VR, ldvr, n, &nout, &work[iwrk], liwrk, &rwork[irwork], &ierr );
#elif TREVC_VERSION == 5
magma_ctrevc3_mt_gpu( side, MagmaBacktransVec, select, n, A, lda, VL, ldvl,
VR, ldvr, n, &nout, &work[iwrk], liwrk, &rwork[irwork], &ierr );
#else
#error Unknown TREVC_VERSION
#endif
}
if (wantvl) {
/* Undo balancing of left eigenvectors
* (CWorkspace: none)
* (RWorkspace: need N) */
lapackf77_cgebak( "B", "L", &n, &ilo, &ihi, &rwork[ibal], &n,
VL, &ldvl, &ierr );
/* Normalize left eigenvectors and make largest component real */
for (i = 0; i < n; ++i) {
scl = 1. / cblas_scnrm2( n, VL(0,i), 1 );
cblas_csscal( n, scl, VL(0,i), 1 );
for (k = 0; k < n; ++k) {
/* Computing 2nd power */
d__1 = MAGMA_C_REAL( *VL(k,i) );
d__2 = MAGMA_C_IMAG( *VL(k,i) );
rwork[irwork + k] = d__1*d__1 + d__2*d__2;
}
k = cblas_isamax( n, &rwork[irwork], 1 );
tmp = MAGMA_C_CNJG( *VL(k,i) ) / magma_ssqrt( rwork[irwork + k] );
cblas_cscal( n, CBLAS_SADDR(tmp), VL(0,i), 1 );
*VL(k,i) = MAGMA_C_MAKE( MAGMA_C_REAL( *VL(k,i) ), 0. );
}
}
if (wantvr) {
/* Undo balancing of right eigenvectors
* (CWorkspace: none)
* (RWorkspace: need N) */
lapackf77_cgebak( "B", "R", &n, &ilo, &ihi, &rwork[ibal], &n,
VR, &ldvr, &ierr );
/* Normalize right eigenvectors and make largest component real */
for (i = 0; i < n; ++i) {
scl = 1. / cblas_scnrm2( n, VR(0,i), 1 );
cblas_csscal( n, scl, VR(0,i), 1 );
for (k = 0; k < n; ++k) {
/* Computing 2nd power */
d__1 = MAGMA_C_REAL( *VR(k,i) );
d__2 = MAGMA_C_IMAG( *VR(k,i) );
rwork[irwork + k] = d__1*d__1 + d__2*d__2;
}
k = cblas_isamax( n, &rwork[irwork], 1 );
tmp = MAGMA_C_CNJG( *VR(k,i) ) / magma_ssqrt( rwork[irwork + k] );
cblas_cscal( n, CBLAS_SADDR(tmp), VR(0,i), 1 );
*VR(k,i) = MAGMA_C_MAKE( MAGMA_C_REAL( *VR(k,i) ), 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_clascl( "G", &c_zero, &c_zero, &cscale, &anrm, &nval, &c_one, W + (*info), &ld, &ierr );
if (*info > 0) {
// first ilo columns were already upper triangular,
// so the corresponding eigenvalues are also valid.
nval = ilo - 1;
lapackf77_clascl( "G", &c_zero, &c_zero, &cscale, &anrm, &nval, &c_one, W, &n, &ierr );
}
}
#if defined(VERSION3)
magma_free( dT );
#endif
work[0] = MAGMA_C_MAKE( (float) minwrk, 0. ); // TODO use optwrk as in dgeev
return *info;
} /* magma_cgeev */
void ATL_larfg(ATL_CINT N, TYPE *ALPHA, TYPE *X, ATL_CINT INCX, TYPE *TAU)
{
#ifdef TREAL
TYPE ONE=1.0, ZERO=0.0, BETA, BETAp, RSAFMN, SAFMAX, XNORM;
int j, KNT;
if (N < 1)
{
*TAU = ZERO;
return;
}
XNORM = cblas_nrm2(N-1, X, INCX); /* Get the norm2 . */
if (XNORM == ZERO)
{
/*
* H = I
*/
*TAU = ZERO;
} else
{
BETAp = ATL_lapy2((*ALPHA), XNORM);/* Get sqrt(a^2+b^2) */
BETA = BETAp; /* Assume ALPHA < 0 */
if ((*ALPHA) > 0) BETA = 0.-BETAp; /* Change if assumed wrong. */
if (BETAp < ATL_laSAFMIN)
{
/*
* XNORM, BETA may be inaccurate; scale X and recompute them
*/
RSAFMN = ONE / ATL_laSAFMIN; /* Set a maximum */
KNT = 0;
while (BETAp < ATL_laSAFMIN)
{
KNT++;
cblas_scal(N-1, RSAFMN, X, INCX);
BETA *= RSAFMN;
BETAp *= RSAFMN;
*ALPHA *= RSAFMN;
}
/*
* New BETA is at most 1, at least SAFMIN
*/
XNORM = cblas_nrm2(N-1, X, INCX);
BETA = ATL_lapy2((*ALPHA), XNORM); /* Will always be positive */
if ((*ALPHA) > 0) BETA = -BETA; /* -SIGN(BETA, ALPHA) */
*TAU = (BETA-(*ALPHA)) / BETA;
cblas_scal(N-1, ONE/((*ALPHA)-BETA), X, INCX);
/*
* If ALPHA is subnormal, it may lose relative accuracy
*/
*ALPHA = BETA;
for (j=0; j<KNT; j++)
{
(*ALPHA) *= ATL_laSAFMIN;
}
} else /* General case */
{
*TAU = (BETA-(*ALPHA)) / BETA;
cblas_scal(N-1, ONE / ((*ALPHA)-BETA), X, INCX);
*ALPHA = BETA;
}
}
return;
/*
* End of Real Precision ATL_larfg
*/
#else
/*
* Beginning of Complex Precision ATL_larfg
*/
TYPE ONE=1.0, ZERO=0.0, BETA, BETAp, RSAFMN, SAFMAX, XNORM, ALPHI, ALPHR;
TYPE ONEVAL[2] = {ATL_rone, ATL_rzero};
int j, KNT;
if ( N < 0)
{
/*
* H = I
*/
*(TAU) = 0.0;
*(TAU + 1) = 0.0;
return;
}
XNORM = cblas_nrm2(N-1, X, INCX); /* Get the nrm2 */
ALPHR = *( ALPHA) ;
ALPHI = *( ALPHA + 1) ;
if ( (XNORM == ZERO) && (ALPHI == ZERO) )
{
/*
* H = I
*/
*(TAU) = 0.0;
*(TAU + 1) = 0.0;
}
else
{
BETAp = ATL_lapy3(ALPHR, ALPHI, XNORM); /* Get sqrt(a^2 + b^2 + c^2) */
BETA = BETAp; /* Assume ALPHA < 0 */
if ( (*ALPHA) > 0) BETA = 0. - BETAp; /* Change if assumed wrong */
RSAFMN = ONE / ATL_laSAFMIN ;
if ( BETAp < ATL_laSAFMIN )
{
/*
* XNORM, BETA may be inaccurate; scale X and recompute them
*/
KNT = 0;
while ( BETAp < ATL_laSAFMIN )
{
KNT++;
#ifdef DCPLX
cblas_zdscal(N-1, RSAFMN, X, INCX);
#else
cblas_csscal(N-1, RSAFMN, X, INCX);
#endif
BETA *= RSAFMN;
BETAp *= RSAFMN;
ALPHI = ALPHI*RSAFMN;
ALPHR = ALPHR*RSAFMN;
}
/*
* New BETA is at most 1, at least SAFMIN
*/
XNORM = cblas_nrm2(N-1, X, INCX);
*(ALPHA) = ALPHR;
*(ALPHA + 1) = ALPHI;
BETA = ATL_lapy3(ALPHR, ALPHI,
XNORM);/* Will always be positive */
if (ALPHR > 0) BETA = -BETA; /* -SIGN(BETA, ALPHR) */
*(TAU) = ( BETA-ALPHR ) / BETA ;
*(TAU + 1) = (-1.0 * ALPHI) / BETA ;
/* Modify alpha to alpha - beta, which is equal to alphar -beta */
*(ALPHA) = *(ALPHA) - BETA;
/* Perform complex division before scaling the X vector */
ATL_ladiv( ONEVAL, ALPHA, ALPHA); /* ALPHA will have the result*/
cblas_scal(N-1, ALPHA, X, INCX);
/*
* If ALPHA is subnormal, it may lose relative accuracy
*/
*(ALPHA) = BETA; /* Real Part of alpha */
for (j=0; j<KNT; j++)
{
(*ALPHA) *= ATL_laSAFMIN;
}
*(ALPHA + 1) = 0.0; /* Set Imaginary part to Zero */
}
else /* BETA > SAFMIN */
{
*(TAU) = ( BETA-ALPHR ) / BETA ;
*(TAU + 1) = (-1.0 * ALPHI) / BETA ;
/* Modify alpha to alpha - beta, which is equal to alphar -beta */
*(ALPHA) = *(ALPHA) - BETA ;
/* Perform complex division before scaling the X vector */
ATL_ladiv( ONEVAL, ALPHA, ALPHA);
cblas_scal(N-1, ALPHA, X, INCX);
*(ALPHA) = BETA; /* Real Part of alpha */
*(ALPHA + 1) = 0.0; /* Set Imaginary part to Zero */
}
}
return;
#endif
} /* END ATL_larfg */
void F77_csscal(const int *N, const float *alpha, void *X,
const int *incX)
{
cblas_csscal(*N, *alpha, X, *incX);
return;
}