magma_int_t magma_ztrevc3_mt(
magma_side_t side, magma_vec_t howmany,
magma_int_t *select, // logical in Fortran
magma_int_t n,
magmaDoubleComplex *T, magma_int_t ldt,
magmaDoubleComplex *VL, magma_int_t ldvl,
magmaDoubleComplex *VR, magma_int_t ldvr,
magma_int_t mm, magma_int_t *mout,
magmaDoubleComplex *work, magma_int_t lwork,
#ifdef COMPLEX
double *rwork,
#endif
magma_int_t *info )
{
#define T(i,j) ( T + (i) + (j)*ldt )
#define VL(i,j) (VL + (i) + (j)*ldvl)
#define VR(i,j) (VR + (i) + (j)*ldvr)
#define work(i,j) (work + (i) + (j)*n)
// .. Parameters ..
const magmaDoubleComplex c_zero = MAGMA_Z_ZERO;
const magmaDoubleComplex c_one = MAGMA_Z_ONE;
const magma_int_t nbmin = 16, nbmax = 128;
const magma_int_t ione = 1;
// .. Local Scalars ..
magma_int_t allv, bothv, leftv, over, rightv, somev;
magma_int_t i, ii, is, j, k, ki, iv, n2, nb, nb2, version;
double ovfl, remax, unfl; //smlnum, smin, ulp
// Decode and test the input parameters
bothv = (side == MagmaBothSides);
rightv = (side == MagmaRight) || bothv;
leftv = (side == MagmaLeft ) || bothv;
allv = (howmany == MagmaAllVec);
over = (howmany == MagmaBacktransVec);
somev = (howmany == MagmaSomeVec);
// Set mout to the number of columns required to store the selected
// eigenvectors.
if ( somev ) {
*mout = 0;
for( j=0; j < n; ++j ) {
if ( select[j] ) {
*mout += 1;
}
}
}
else {
*mout = n;
}
*info = 0;
if ( ! rightv && ! leftv )
*info = -1;
else if ( ! allv && ! over && ! somev )
*info = -2;
else if ( n < 0 )
*info = -4;
else if ( ldt < max( 1, n ) )
*info = -6;
else if ( ldvl < 1 || ( leftv && ldvl < n ) )
*info = -8;
else if ( ldvr < 1 || ( rightv && ldvr < n ) )
*info = -10;
else if ( mm < *mout )
*info = -11;
else if ( lwork < max( 1, 2*n ) )
*info = -14;
if ( *info != 0 ) {
magma_xerbla( __func__, -(*info) );
return *info;
}
// Quick return if possible.
if ( n == 0 ) {
return *info;
}
// Use blocked version (2) if sufficient workspace.
// Requires 1 vector to save diagonal elements, and 2*nb vectors for x and Q*x.
// (Compared to dtrevc3, rwork stores 1-norms.)
// Zero-out the workspace to avoid potential NaN propagation.
nb = 2;
if ( lwork >= n + 2*n*nbmin ) {
version = 2;
nb = (lwork - n) / (2*n);
nb = min( nb, nbmax );
nb2 = 1 + 2*nb;
lapackf77_zlaset( "F", &n, &nb2, &c_zero, &c_zero, work, &n );
}
else {
version = 1;
}
// Set the constants to control overflow.
unfl = lapackf77_dlamch( "Safe minimum" );
ovfl = 1. / unfl;
lapackf77_dlabad( &unfl, &ovfl );
//ulp = lapackf77_dlamch( "Precision" );
//smlnum = unfl*( n / ulp );
// Store the diagonal elements of T in working array work.
for( i=0; i < n; ++i ) {
*work(i,0) = *T(i,i);
}
// Compute 1-norm of each column of strictly upper triangular
// part of T to control overflow in triangular solver.
rwork[0] = 0.;
for( j=1; j < n; ++j ) {
rwork[j] = magma_cblas_dzasum( j, T(0,j), ione );
}
// launch threads -- each single-threaded MKL
magma_int_t nthread = magma_get_parallel_numthreads();
magma_int_t lapack_nthread = magma_get_lapack_numthreads();
magma_set_lapack_numthreads( 1 );
magma_thread_queue queue;
queue.launch( nthread );
//printf( "nthread %d, %d\n", nthread, lapack_nthread );
// gemm_nb = N/thread, rounded up to multiple of 16,
// but avoid multiples of page size, e.g., 512*8 bytes = 4096.
magma_int_t gemm_nb = magma_int_t( ceil( ceil( ((double)n) / nthread ) / 16. ) * 16. );
if ( gemm_nb % 512 == 0 ) {
gemm_nb += 32;
}
magma_timer_t time_total=0, time_trsv=0, time_gemm=0, time_gemv=0, time_trsv_sum=0, time_gemm_sum=0, time_gemv_sum=0;
timer_start( time_total );
if ( rightv ) {
// ============================================================
// Compute right eigenvectors.
// iv is index of column in current block.
// Non-blocked version always uses iv=1;
// blocked version starts with iv=nb, goes down to 1.
// (Note the "0-th" column is used to store the original diagonal.)
iv = 1;
if ( version == 2 ) {
iv = nb;
}
timer_start( time_trsv );
is = *mout - 1;
for( ki=n-1; ki >= 0; --ki ) {
if ( somev ) {
if ( ! select[ki] ) {
continue;
}
}
//smin = max( ulp*MAGMA_Z_ABS1( *T(ki,ki) ), smlnum );
// --------------------------------------------------------
// Complex right eigenvector
*work(ki,iv) = c_one;
// Form right-hand side.
for( k=0; k < ki; ++k ) {
*work(k,iv) = -(*T(k,ki));
}
// Solve upper triangular system:
// [ T(1:ki-1,1:ki-1) - T(ki,ki) ]*X = scale*work.
if ( ki > 0 ) {
queue.push_task( new magma_zlatrsd_task(
MagmaUpper, MagmaNoTrans, MagmaNonUnit, MagmaTrue,
ki, T, ldt, *T(ki,ki),
work(0,iv), work(ki,iv), rwork ));
}
// Copy the vector x or Q*x to VR and normalize.
if ( ! over ) {
// ------------------------------
// no back-transform: copy x to VR and normalize
queue.sync();
n2 = ki+1;
blasf77_zcopy( &n2, work(0,iv), &ione, VR(0,is), &ione );
ii = blasf77_izamax( &n2, VR(0,is), &ione ) - 1;
remax = 1. / MAGMA_Z_ABS1( *VR(ii,is) );
blasf77_zdscal( &n2, &remax, VR(0,is), &ione );
for( k=ki+1; k < n; ++k ) {
*VR(k,is) = c_zero;
}
}
else if ( version == 1 ) {
// ------------------------------
// version 1: back-transform each vector with GEMV, Q*x.
queue.sync();
time_trsv_sum += timer_stop( time_trsv );
timer_start( time_gemv );
if ( ki > 0 ) {
blasf77_zgemv( "n", &n, &ki, &c_one,
VR, &ldvr,
work(0, iv), &ione,
work(ki,iv), VR(0,ki), &ione );
}
time_gemv_sum += timer_stop( time_gemv );
ii = blasf77_izamax( &n, VR(0,ki), &ione ) - 1;
remax = 1. / MAGMA_Z_ABS1( *VR(ii,ki) );
blasf77_zdscal( &n, &remax, VR(0,ki), &ione );
timer_start( time_trsv );
}
else if ( version == 2 ) {
// ------------------------------
// version 2: back-transform block of vectors with GEMM
// zero out below vector
for( k=ki+1; k < n; ++k ) {
*work(k,iv) = c_zero;
}
// Columns iv:nb of work are valid vectors.
// When the number of vectors stored reaches nb,
// or if this was last vector, do the GEMM
if ( (iv == 1) || (ki == 0) ) {
queue.sync();
time_trsv_sum += timer_stop( time_trsv );
timer_start( time_gemm );
nb2 = nb-iv+1;
n2 = ki+nb-iv+1;
// split gemm into multiple tasks, each doing one block row
for( i=0; i < n; i += gemm_nb ) {
magma_int_t ib = min( gemm_nb, n-i );
queue.push_task( new zgemm_task(
MagmaNoTrans, MagmaNoTrans, ib, nb2, n2, c_one,
VR(i,0), ldvr,
work(0,iv ), n, c_zero,
work(i,nb+iv), n ));
}
queue.sync();
time_gemm_sum += timer_stop( time_gemm );
// normalize vectors
// TODO if somev, should copy vectors individually to correct location.
for( k = iv; k <= nb; ++k ) {
ii = blasf77_izamax( &n, work(0,nb+k), &ione ) - 1;
remax = 1. / MAGMA_Z_ABS1( *work(ii,nb+k) );
blasf77_zdscal( &n, &remax, work(0,nb+k), &ione );
}
lapackf77_zlacpy( "F", &n, &nb2, work(0,nb+iv), &n, VR(0,ki), &ldvr );
iv = nb;
timer_start( time_trsv );
}
else {
iv -= 1;
}
} // blocked back-transform
is -= 1;
}
}
timer_stop( time_trsv );
timer_stop( time_total );
timer_printf( "trevc trsv %.4f, gemm %.4f, gemv %.4f, total %.4f\n",
time_trsv_sum, time_gemm_sum, time_gemv_sum, time_total );
if ( leftv ) {
// ============================================================
// Compute left eigenvectors.
// iv is index of column in current block.
// Non-blocked version always uses iv=1;
// blocked version starts with iv=1, goes up to nb.
// (Note the "0-th" column is used to store the original diagonal.)
iv = 1;
is = 0;
for( ki=0; ki < n; ++ki ) {
if ( somev ) {
if ( ! select[ki] ) {
continue;
}
}
//smin = max( ulp*MAGMA_Z_ABS1( *T(ki,ki) ), smlnum );
// --------------------------------------------------------
// Complex left eigenvector
*work(ki,iv) = c_one;
// Form right-hand side.
for( k = ki + 1; k < n; ++k ) {
*work(k,iv) = -MAGMA_Z_CONJ( *T(ki,k) );
}
// Solve conjugate-transposed triangular system:
// [ T(ki+1:n,ki+1:n) - T(ki,ki) ]**H * X = scale*work.
// TODO what happens with T(k,k) - lambda is small? Used to have < smin test.
if ( ki < n-1 ) {
n2 = n-ki-1;
queue.push_task( new magma_zlatrsd_task(
MagmaUpper, MagmaConjTrans, MagmaNonUnit, MagmaTrue,
n2, T(ki+1,ki+1), ldt, *T(ki,ki),
work(ki+1,iv), work(ki,iv), rwork ));
}
// Copy the vector x or Q*x to VL and normalize.
if ( ! over ) {
// ------------------------------
// no back-transform: copy x to VL and normalize
queue.sync();
n2 = n-ki;
blasf77_zcopy( &n2, work(ki,iv), &ione, VL(ki,is), &ione );
ii = blasf77_izamax( &n2, VL(ki,is), &ione ) + ki - 1;
remax = 1. / MAGMA_Z_ABS1( *VL(ii,is) );
blasf77_zdscal( &n2, &remax, VL(ki,is), &ione );
for( k=0; k < ki; ++k ) {
*VL(k,is) = c_zero;
}
}
else if ( version == 1 ) {
// ------------------------------
// version 1: back-transform each vector with GEMV, Q*x.
queue.sync();
if ( ki < n-1 ) {
n2 = n-ki-1;
blasf77_zgemv( "n", &n, &n2, &c_one,
VL(0,ki+1), &ldvl,
work(ki+1,iv), &ione,
work(ki, iv), VL(0,ki), &ione );
}
ii = blasf77_izamax( &n, VL(0,ki), &ione ) - 1;
remax = 1. / MAGMA_Z_ABS1( *VL(ii,ki) );
blasf77_zdscal( &n, &remax, VL(0,ki), &ione );
}
else if ( version == 2 ) {
// ------------------------------
// version 2: back-transform block of vectors with GEMM
// zero out above vector
// could go from (ki+1)-NV+1 to ki
for( k=0; k < ki; ++k ) {
*work(k,iv) = c_zero;
}
// Columns 1:iv of work are valid vectors.
// When the number of vectors stored reaches nb,
// or if this was last vector, do the GEMM
if ( (iv == nb) || (ki == n-1) ) {
queue.sync();
n2 = n-(ki+1)+iv;
// split gemm into multiple tasks, each doing one block row
for( i=0; i < n; i += gemm_nb ) {
magma_int_t ib = min( gemm_nb, n-i );
queue.push_task( new zgemm_task(
MagmaNoTrans, MagmaNoTrans, ib, iv, n2, c_one,
VL(i,ki-iv+1), ldvl,
work(ki-iv+1,1), n, c_zero,
work(i,nb+1), n ));
}
queue.sync();
// normalize vectors
for( k=1; k <= iv; ++k ) {
ii = blasf77_izamax( &n, work(0,nb+k), &ione ) - 1;
remax = 1. / MAGMA_Z_ABS1( *work(ii,nb+k) );
blasf77_zdscal( &n, &remax, work(0,nb+k), &ione );
}
lapackf77_zlacpy( "F", &n, &iv, work(0,nb+1), &n, VL(0,ki-iv+1), &ldvl );
iv = 1;
}
else {
iv += 1;
}
} // blocked back-transform
is += 1;
}
}
// close down threads
queue.quit();
magma_set_lapack_numthreads( lapack_nthread );
return *info;
} // End of ZTREVC
magma_int_t magma_ztrevc3(
magma_side_t side, magma_vec_t howmany,
magma_int_t *select, // logical in Fortran
magma_int_t n,
magmaDoubleComplex *T, magma_int_t ldt,
magmaDoubleComplex *VL, magma_int_t ldvl,
magmaDoubleComplex *VR, magma_int_t ldvr,
magma_int_t mm, magma_int_t *mout,
magmaDoubleComplex *work, magma_int_t lwork,
double *rwork, magma_int_t *info )
{
#define T(i,j) ( T + (i) + (j)*ldt )
#define VL(i,j) (VL + (i) + (j)*ldvl)
#define VR(i,j) (VR + (i) + (j)*ldvr)
#define work(i,j) (work + (i) + (j)*n)
// .. Parameters ..
const magmaDoubleComplex c_zero = MAGMA_Z_ZERO;
const magmaDoubleComplex c_one = MAGMA_Z_ONE;
const magma_int_t nbmin = 16, nbmax = 128;
const magma_int_t ione = 1;
// .. Local Scalars ..
magma_int_t allv, bothv, leftv, over, rightv, somev;
magma_int_t i, ii, is, j, k, ki, iv, n2, nb, nb2, version;
double ovfl, remax, scale, smin, smlnum, ulp, unfl;
// Decode and test the input parameters
bothv = (side == MagmaBothSides);
rightv = (side == MagmaRight) || bothv;
leftv = (side == MagmaLeft ) || bothv;
allv = (howmany == MagmaAllVec);
over = (howmany == MagmaBacktransVec);
somev = (howmany == MagmaSomeVec);
// Set mout to the number of columns required to store the selected
// eigenvectors.
if ( somev ) {
*mout = 0;
for( j=0; j < n; ++j ) {
if ( select[j] ) {
*mout += 1;
}
}
}
else {
*mout = n;
}
*info = 0;
if ( ! rightv && ! leftv )
*info = -1;
else if ( ! allv && ! over && ! somev )
*info = -2;
else if ( n < 0 )
*info = -4;
else if ( ldt < max( 1, n ) )
*info = -6;
else if ( ldvl < 1 || ( leftv && ldvl < n ) )
*info = -8;
else if ( ldvr < 1 || ( rightv && ldvr < n ) )
*info = -10;
else if ( mm < *mout )
*info = -11;
else if ( lwork < max( 1, 2*n ) )
*info = -14;
if ( *info != 0 ) {
magma_xerbla( __func__, -(*info) );
return *info;
}
// Quick return if possible.
if ( n == 0 ) {
return *info;
}
// Use blocked version (2) if sufficient workspace.
// Requires 1 vector to save diagonal elements, and 2*nb vectors for x and Q*x.
// (Compared to dtrevc3, rwork stores 1-norms.)
// Zero-out the workspace to avoid potential NaN propagation.
nb = 2;
if ( lwork >= n + 2*n*nbmin ) {
version = 2;
nb = (lwork - n) / (2*n);
nb = min( nb, nbmax );
nb2 = 1 + 2*nb;
lapackf77_zlaset( "F", &n, &nb2, &c_zero, &c_zero, work, &n );
}
else {
version = 1;
}
// Set the constants to control overflow.
unfl = lapackf77_dlamch( "Safe minimum" );
ovfl = 1. / unfl;
lapackf77_dlabad( &unfl, &ovfl );
ulp = lapackf77_dlamch( "Precision" );
smlnum = unfl*( n / ulp );
// Store the diagonal elements of T in working array work.
for( i=0; i < n; ++i ) {
*work(i,0) = *T(i,i);
}
// Compute 1-norm of each column of strictly upper triangular
// part of T to control overflow in triangular solver.
rwork[0] = 0.;
for( j=1; j < n; ++j ) {
rwork[j] = cblas_dzasum( j, T(0,j), ione );
}
magma_timer_t time_total=0, time_trsv=0, time_gemm=0, time_gemv=0, time_trsv_sum=0, time_gemm_sum=0, time_gemv_sum=0;
timer_start( time_total );
if ( rightv ) {
// ============================================================
// Compute right eigenvectors.
// iv is index of column in current block.
// Non-blocked version always uses iv=1;
// blocked version starts with iv=nb, goes down to 1.
// (Note the "0-th" column is used to store the original diagonal.)
iv = 1;
if ( version == 2 ) {
iv = nb;
}
timer_start( time_trsv );
is = *mout - 1;
for( ki=n-1; ki >= 0; --ki ) {
if ( somev ) {
if ( ! select[ki] ) {
continue;
}
}
smin = max( ulp*( MAGMA_Z_ABS1( *T(ki,ki) ) ), smlnum );
// --------------------------------------------------------
// Complex right eigenvector
*work(ki,iv) = c_one;
// Form right-hand side.
for( k=0; k < ki; ++k ) {
*work(k,iv) = -(*T(k,ki));
}
// Solve upper triangular system:
// [ T(1:ki-1,1:ki-1) - T(ki,ki) ]*X = scale*work.
for( k=0; k < ki; ++k ) {
*T(k,k) -= *T(ki,ki);
if ( MAGMA_Z_ABS1( *T(k,k) ) < smin ) {
*T(k,k) = MAGMA_Z_MAKE( smin, 0. );
}
}
if ( ki > 0 ) {
lapackf77_zlatrs( "Upper", "No transpose", "Non-unit", "Y",
&ki, T, &ldt,
work(0,iv), &scale, rwork, info );
*work(ki,iv) = MAGMA_Z_MAKE( scale, 0. );
}
// Copy the vector x or Q*x to VR and normalize.
if ( ! over ) {
// ------------------------------
// no back-transform: copy x to VR and normalize
n2 = ki+1;
blasf77_zcopy( &n2, work(0,iv), &ione, VR(0,is), &ione );
ii = blasf77_izamax( &n2, VR(0,is), &ione ) - 1;
remax = 1. / MAGMA_Z_ABS1( *VR(ii,is) );
blasf77_zdscal( &n2, &remax, VR(0,is), &ione );
for( k=ki+1; k < n; ++k ) {
*VR(k,is) = c_zero;
}
}
else if ( version == 1 ) {
// ------------------------------
// version 1: back-transform each vector with GEMV, Q*x.
time_trsv_sum += timer_stop( time_trsv );
timer_start( time_gemv );
if ( ki > 0 ) {
blasf77_zgemv( "n", &n, &ki, &c_one,
VR, &ldvr,
work(0, iv), &ione,
work(ki,iv), VR(0,ki), &ione );
}
time_gemv_sum += timer_stop( time_gemv );
ii = blasf77_izamax( &n, VR(0,ki), &ione ) - 1;
remax = 1. / MAGMA_Z_ABS1( *VR(ii,ki) );
blasf77_zdscal( &n, &remax, VR(0,ki), &ione );
timer_start( time_trsv );
}
else if ( version == 2 ) {
// ------------------------------
// version 2: back-transform block of vectors with GEMM
// zero out below vector
for( k=ki+1; k < n; ++k ) {
*work(k,iv) = c_zero;
}
// Columns iv:nb of work are valid vectors.
// When the number of vectors stored reaches nb,
// or if this was last vector, do the GEMM
if ( (iv == 1) || (ki == 0) ) {
time_trsv_sum += timer_stop( time_trsv );
timer_start( time_gemm );
nb2 = nb-iv+1;
n2 = ki+nb-iv+1;
blasf77_zgemm( "n", "n", &n, &nb2, &n2, &c_one,
VR, &ldvr,
work(0,iv ), &n, &c_zero,
work(0,nb+iv), &n );
time_gemm_sum += timer_stop( time_gemm );
// normalize vectors
// TODO if somev, should copy vectors individually to correct location.
for( k = iv; k <= nb; ++k ) {
ii = blasf77_izamax( &n, work(0,nb+k), &ione ) - 1;
remax = 1. / MAGMA_Z_ABS1( *work(ii,nb+k) );
blasf77_zdscal( &n, &remax, work(0,nb+k), &ione );
}
lapackf77_zlacpy( "F", &n, &nb2, work(0,nb+iv), &n, VR(0,ki), &ldvr );
iv = nb;
timer_start( time_trsv );
}
else {
iv -= 1;
}
} // blocked back-transform
// Restore the original diagonal elements of T.
for( k=0; k <= ki - 1; ++k ) {
*T(k,k) = *work(k,0);
}
is -= 1;
}
}
timer_stop( time_trsv );
timer_stop( time_total );
timer_printf( "trevc trsv %.4f, gemm %.4f, gemv %.4f, total %.4f\n",
time_trsv_sum, time_gemm_sum, time_gemv_sum, time_total );
if ( leftv ) {
// ============================================================
// Compute left eigenvectors.
// iv is index of column in current block.
// Non-blocked version always uses iv=1;
// blocked version starts with iv=1, goes up to nb.
// (Note the "0-th" column is used to store the original diagonal.)
iv = 1;
is = 0;
for( ki=0; ki < n; ++ki ) {
if ( somev ) {
if ( ! select[ki] ) {
continue;
}
}
smin = max( ulp*MAGMA_Z_ABS1( *T(ki,ki) ), smlnum );
// --------------------------------------------------------
// Complex left eigenvector
*work(ki,iv) = c_one;
// Form right-hand side.
for( k = ki + 1; k < n; ++k ) {
*work(k,iv) = -MAGMA_Z_CNJG( *T(ki,k) );
}
// Solve conjugate-transposed triangular system:
// [ T(ki+1:n,ki+1:n) - T(ki,ki) ]**H * X = scale*work.
for( k = ki + 1; k < n; ++k ) {
*T(k,k) -= *T(ki,ki);
if ( MAGMA_Z_ABS1( *T(k,k) ) < smin ) {
*T(k,k) = MAGMA_Z_MAKE( smin, 0. );
}
}
if ( ki < n-1 ) {
n2 = n-ki-1;
lapackf77_zlatrs( "Upper", "Conjugate transpose", "Non-unit", "Y",
&n2, T(ki+1,ki+1), &ldt,
work(ki+1,iv), &scale, rwork, info );
*work(ki,iv) = MAGMA_Z_MAKE( scale, 0. );
}
// Copy the vector x or Q*x to VL and normalize.
if ( ! over ) {
// ------------------------------
// no back-transform: copy x to VL and normalize
n2 = n-ki;
blasf77_zcopy( &n2, work(ki,iv), &ione, VL(ki,is), &ione );
ii = blasf77_izamax( &n2, VL(ki,is), &ione ) + ki - 1;
remax = 1. / MAGMA_Z_ABS1( *VL(ii,is) );
blasf77_zdscal( &n2, &remax, VL(ki,is), &ione );
for( k=0; k < ki; ++k ) {
*VL(k,is) = c_zero;
}
}
else if ( version == 1 ) {
// ------------------------------
// version 1: back-transform each vector with GEMV, Q*x.
if ( ki < n-1 ) {
n2 = n-ki-1;
blasf77_zgemv( "n", &n, &n2, &c_one,
VL(0,ki+1), &ldvl,
work(ki+1,iv), &ione,
work(ki, iv), VL(0,ki), &ione );
}
ii = blasf77_izamax( &n, VL(0,ki), &ione ) - 1;
remax = 1. / MAGMA_Z_ABS1( *VL(ii,ki) );
blasf77_zdscal( &n, &remax, VL(0,ki), &ione );
}
else if ( version == 2 ) {
// ------------------------------
// version 2: back-transform block of vectors with GEMM
// zero out above vector
// could go from (ki+1)-NV+1 to ki
for( k=0; k < ki; ++k ) {
*work(k,iv) = c_zero;
}
// Columns 1:iv of work are valid vectors.
// When the number of vectors stored reaches nb,
// or if this was last vector, do the GEMM
if ( (iv == nb) || (ki == n-1) ) {
n2 = n-(ki+1)+iv;
blasf77_zgemm( "n", "n", &n, &iv, &n2, &c_one,
VL(0,ki-iv+1), &ldvl,
work(ki-iv+1,1 ), &n, &c_zero,
work(0, nb+1), &n );
// normalize vectors
for( k=1; k <= iv; ++k ) {
ii = blasf77_izamax( &n, work(0,nb+k), &ione ) - 1;
remax = 1. / MAGMA_Z_ABS1( *work(ii,nb+k) );
blasf77_zdscal( &n, &remax, work(0,nb+k), &ione );
}
lapackf77_zlacpy( "F", &n, &iv, work(0,nb+1), &n, VL(0,ki-iv+1), &ldvl );
iv = 1;
}
else {
iv += 1;
}
} // blocked back-transform
// Restore the original diagonal elements of T.
for( k = ki + 1; k < n; ++k ) {
*T(k,k) = *work(k,0);
}
is += 1;
}
}
return *info;
} // End of ZTREVC
extern "C" magma_int_t
magma_zgeev(magma_vec_t jobvl, magma_vec_t jobvr, magma_int_t n,
magmaDoubleComplex *a, magma_int_t lda,
magmaDoubleComplex *geev_w_array,
magmaDoubleComplex *vl, magma_int_t ldvl,
magmaDoubleComplex *vr, magma_int_t ldvr,
magmaDoubleComplex *work, magma_int_t lwork,
double *rwork, magma_int_t *info, magma_queue_t queue)
{
/* -- clMAGMA (version 1.0.0) --
Univ. of Tennessee, Knoxville
Univ. of California, Berkeley
Univ. of Colorado, Denver
September 2012
Purpose
=======
ZGEEV 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
=========
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) COMPLEX*16 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).
W (output) COMPLEX*16 array, dimension (N)
W contains the computed eigenvalues.
VL (output) COMPLEX*16 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) COMPLEX*16 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) COMPLEX*16 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 >= (1+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.
RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
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.
===================================================================== */
magma_int_t c__1 = 1;
magma_int_t c__0 = 0;
magma_int_t a_dim1, a_offset, vl_dim1, vl_offset, vr_dim1, vr_offset, i__1,
i__2, i__3;
double d__1, d__2;
magmaDoubleComplex z__1, z__2;
magma_int_t i__, k, ihi;
double scl;
magma_int_t ilo;
double dum[1], eps;
magmaDoubleComplex tmp;
magma_int_t ibal;
double anrm;
magma_int_t ierr, itau, iwrk, nout;
magma_int_t scalea;
double cscale;
magma_int_t select[1];
double bignum;
magma_int_t minwrk;
magma_int_t wantvl;
double smlnum;
magma_int_t irwork;
magma_int_t lquery, wantvr;
magma_int_t nb = 0;
magmaDoubleComplex_ptr dT;
//magma_timestr_t start, end;
char side[2] = {0, 0};
magma_vec_t jobvl_ = jobvl;
magma_vec_t jobvr_ = jobvr;
*info = 0;
lquery = lwork == -1;
wantvl = lapackf77_lsame(lapack_const(jobvl_), "V");
wantvr = lapackf77_lsame(lapack_const(jobvr_), "V");
if (! wantvl && ! lapackf77_lsame(lapack_const(jobvl_), "N")) {
*info = -1;
} else if (! wantvr && ! lapackf77_lsame(lapack_const(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 = -8;
} else if ( (ldvr < 1) || (wantvr && (ldvr < n))) {
*info = -10;
}
/* Compute workspace */
if (*info == 0) {
nb = magma_get_zgehrd_nb(n);
minwrk = (1+nb)*n;
work[0] = MAGMA_Z_MAKE((double) 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 eigenvectors are needed
#if defined(VERSION3)
if (MAGMA_SUCCESS != magma_malloc(&dT, nb*n*sizeof(magmaDoubleComplex) )) {
*info = MAGMA_ERR_DEVICE_ALLOC;
return *info;
}
#endif
a_dim1 = lda;
a_offset = 1 + a_dim1;
a -= a_offset;
vl_dim1 = ldvl;
vl_offset = 1 + vl_dim1;
vl -= vl_offset;
vr_dim1 = ldvr;
vr_offset = 1 + vr_dim1;
vr -= vr_offset;
--work;
--rwork;
/* 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_zlange("M", &n, &n, &a[a_offset], &lda, dum);
scalea = 0;
if (anrm > 0. && anrm < smlnum) {
scalea = 1;
cscale = smlnum;
} else if (anrm > bignum) {
scalea = 1;
cscale = bignum;
}
if (scalea) {
lapackf77_zlascl("G", &c__0, &c__0, &anrm, &cscale, &n, &n, &a[a_offset], &lda, &
ierr);
}
/* Balance the matrix
(CWorkspace: none)
(RWorkspace: need N) */
ibal = 1;
lapackf77_zgebal("B", &n, &a[a_offset], &lda, &ilo, &ihi, &rwork[ibal], &ierr);
/* Reduce to upper Hessenberg form
(CWorkspace: need 2*N, prefer N+N*NB)
(RWorkspace: none) */
itau = 1;
iwrk = itau + n;
i__1 = lwork - iwrk + 1;
//start = get_current_time();
#if defined(VERSION1)
/*
* Version 1 - LAPACK
*/
lapackf77_zgehrd(&n, &ilo, &ihi, &a[a_offset], &lda,
&work[itau], &work[iwrk], &i__1, &ierr);
#elif defined(VERSION2)
/*
* Version 2 - LAPACK consistent HRD
*/
magma_zgehrd2(n, ilo, ihi, &a[a_offset], lda,
&work[itau], &work[iwrk], &i__1, &ierr);
#elif defined(VERSION3)
/*
* Version 3 - LAPACK consistent MAGMA HRD + matrices T stored,
*/
magma_zgehrd(n, ilo, ihi, &a[a_offset], lda,
&work[itau], &work[iwrk], i__1, dT, 0, &ierr, queue);
#endif
//end = get_current_time();
//printf(" Time for zgehrd = %5.2f sec\n", GetTimerValue(start,end)/1000.);
if (wantvl) {
/* Want left eigenvectors
Copy Householder vectors to VL */
side[0] = 'L';
lapackf77_zlacpy(MagmaLowerStr, &n, &n,
&a[a_offset], &lda, &vl[vl_offset], &ldvl);
/* Generate unitary matrix in VL
(CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
(RWorkspace: none) */
i__1 = lwork - iwrk + 1;
//start = get_current_time();
#if defined(VERSION1) || defined(VERSION2)
/*
* Version 1 & 2 - LAPACK
*/
lapackf77_zunghr(&n, &ilo, &ihi, &vl[vl_offset], &ldvl,
&work[itau], &work[iwrk], &i__1, &ierr);
#elif defined(VERSION3)
/*
* Version 3 - LAPACK consistent MAGMA HRD + matrices T stored
*/
magma_zunghr(n, ilo, ihi, &vl[vl_offset], ldvl, &work[itau],
dT, 0, nb, &ierr, queue);
#endif
//end = get_current_time();
//printf(" Time for zunghr = %5.2f sec\n", GetTimerValue(start,end)/1000.);
/* Perform QR iteration, accumulating Schur vectors in VL
(CWorkspace: need 1, prefer HSWORK (see comments) )
(RWorkspace: none) */
iwrk = itau;
i__1 = lwork - iwrk + 1;
lapackf77_zhseqr("S", "V", &n, &ilo, &ihi, &a[a_offset], &lda, geev_w_array,
&vl[vl_offset], &ldvl, &work[iwrk], &i__1, info);
if (wantvr)
{
/* Want left and right eigenvectors
Copy Schur vectors to VR */
side[0] = 'B';
lapackf77_zlacpy("F", &n, &n, &vl[vl_offset], &ldvl, &vr[vr_offset], &ldvr);
}
} else if (wantvr) {
/* Want right eigenvectors
Copy Householder vectors to VR */
side[0] = 'R';
lapackf77_zlacpy("L", &n, &n, &a[a_offset], &lda, &vr[vr_offset], &ldvr);
/* Generate unitary matrix in VR
(CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
(RWorkspace: none) */
i__1 = lwork - iwrk + 1;
//start = get_current_time();
#if defined(VERSION1) || defined(VERSION2)
/*
* Version 1 & 2 - LAPACK
*/
lapackf77_zunghr(&n, &ilo, &ihi, &vr[vr_offset], &ldvr,
&work[itau], &work[iwrk], &i__1, &ierr);
#elif defined(VERSION3)
/*
* Version 3 - LAPACK consistent MAGMA HRD + matrices T stored
*/
magma_zunghr(n, ilo, ihi, &vr[vr_offset], ldvr,
&work[itau], dT, 0, nb, &ierr, queue);
#endif
//end = get_current_time();
//printf(" Time for zunghr = %5.2f sec\n", GetTimerValue(start,end)/1000.);
/* Perform QR iteration, accumulating Schur vectors in VR
(CWorkspace: need 1, prefer HSWORK (see comments) )
(RWorkspace: none) */
iwrk = itau;
i__1 = lwork - iwrk + 1;
lapackf77_zhseqr("S", "V", &n, &ilo, &ihi, &a[a_offset], &lda, geev_w_array,
&vr[vr_offset], &ldvr, &work[iwrk], &i__1, info);
} else {
/* Compute eigenvalues only
(CWorkspace: need 1, prefer HSWORK (see comments) )
(RWorkspace: none) */
iwrk = itau;
i__1 = lwork - iwrk + 1;
lapackf77_zhseqr("E", "N", &n, &ilo, &ihi, &a[a_offset], &lda, geev_w_array,
&vr[vr_offset], &ldvr, &work[iwrk], &i__1, info);
}
/* If INFO > 0 from ZHSEQR, then quit */
if (*info > 0) {
goto L50;
}
if (wantvl || wantvr) {
/* Compute left and/or right eigenvectors
(CWorkspace: need 2*N)
(RWorkspace: need 2*N) */
irwork = ibal + n;
lapackf77_ztrevc(side, "B", select, &n, &a[a_offset], &lda, &vl[vl_offset], &ldvl,
&vr[vr_offset], &ldvr, &n, &nout, &work[iwrk], &rwork[irwork],
&ierr);
}
if (wantvl) {
/* Undo balancing of left eigenvectors
(CWorkspace: none)
(RWorkspace: need N) */
lapackf77_zgebak("B", "L", &n, &ilo, &ihi, &rwork[ibal], &n,
&vl[vl_offset], &ldvl, &ierr);
/* Normalize left eigenvectors and make largest component real */
for (i__ = 1; i__ <= n; ++i__) {
scl = 1. / cblas_dznrm2(n, &vl[i__ * vl_dim1 + 1], 1);
cblas_zdscal(n, scl, &vl[i__ * vl_dim1 + 1], 1);
i__2 = n;
for (k = 1; k <= i__2; ++k)
{
i__3 = k + i__ * vl_dim1;
/* Computing 2nd power */
d__1 = MAGMA_Z_REAL(vl[i__3]);
/* Computing 2nd power */
d__2 = MAGMA_Z_IMAG(vl[k + i__ * vl_dim1]);
rwork[irwork + k - 1] = d__1 * d__1 + d__2 * d__2;
}
/* Comment:
Fortran BLAS does not have to add 1
C BLAS must add one to cblas_idamax */
k = cblas_idamax(n, &rwork[irwork], 1)+1;
z__2 = MAGMA_Z_CNJG(vl[k + i__ * vl_dim1]);
d__1 = magma_dsqrt(rwork[irwork + k - 1]);
MAGMA_Z_DSCALE(z__1, z__2, d__1);
tmp = z__1;
cblas_zscal(n, CBLAS_SADDR(tmp), &vl[i__ * vl_dim1 + 1], 1);
i__2 = k + i__ * vl_dim1;
i__3 = k + i__ * vl_dim1;
d__1 = MAGMA_Z_REAL(vl[i__3]);
MAGMA_Z_SET2REAL(z__1, d__1);
vl[i__2] = z__1;
}
}
if (wantvr) {
/* Undo balancing of right eigenvectors
(CWorkspace: none)
(RWorkspace: need N) */
lapackf77_zgebak("B", "R", &n, &ilo, &ihi, &rwork[ibal], &n,
&vr[vr_offset], &ldvr, &ierr);
/* Normalize right eigenvectors and make largest component real */
for (i__ = 1; i__ <= n; ++i__) {
scl = 1. / cblas_dznrm2(n, &vr[i__ * vr_dim1 + 1], 1);
cblas_zdscal(n, scl, &vr[i__ * vr_dim1 + 1], 1);
i__2 = n;
for (k = 1; k <= i__2; ++k) {
i__3 = k + i__ * vr_dim1;
/* Computing 2nd power */
d__1 = MAGMA_Z_REAL(vr[i__3]);
/* Computing 2nd power */
d__2 = MAGMA_Z_IMAG(vr[k + i__ * vr_dim1]);
rwork[irwork + k - 1] = d__1 * d__1 + d__2 * d__2;
}
/* Comment:
Fortran BLAS does not have to add 1
C BLAS must add one to cblas_idamax */
k = cblas_idamax(n, &rwork[irwork], 1)+1;
z__2 = MAGMA_Z_CNJG(vr[k + i__ * vr_dim1]);
d__1 = magma_dsqrt(rwork[irwork + k - 1]);
MAGMA_Z_DSCALE(z__1, z__2, d__1);
tmp = z__1;
cblas_zscal(n, CBLAS_SADDR(tmp), &vr[i__ * vr_dim1 + 1], 1);
i__2 = k + i__ * vr_dim1;
i__3 = k + i__ * vr_dim1;
d__1 = MAGMA_Z_REAL(vr[i__3]);
MAGMA_Z_SET2REAL(z__1, d__1);
vr[i__2] = z__1;
}
}
/* Undo scaling if necessary */
L50:
if (scalea) {
i__1 = n - *info;
/* Computing MAX */
i__3 = n - *info;
i__2 = max(i__3,1);
lapackf77_zlascl("G", &c__0, &c__0, &cscale, &anrm, &i__1, &c__1,
geev_w_array + *info, &i__2, &ierr);
if (*info > 0) {
i__1 = ilo - 1;
lapackf77_zlascl("G", &c__0, &c__0, &cscale, &anrm, &i__1, &c__1,
geev_w_array, &n, &ierr);
}
}
#if defined(VERSION3)
magma_free( dT );
#endif
return *info;
} /* magma_zgeev */
extern "C" magma_int_t
magma_zgeev_m(
char jobvl, char jobvr, magma_int_t n,
magmaDoubleComplex *A, magma_int_t lda,
magmaDoubleComplex *W,
magmaDoubleComplex *vl, magma_int_t ldvl,
magmaDoubleComplex *vr, magma_int_t ldvr,
magmaDoubleComplex *work, magma_int_t lwork,
double *rwork, magma_int_t *info )
{
/* -- MAGMA (version 1.4.1) --
Univ. of Tennessee, Knoxville
Univ. of California, Berkeley
Univ. of Colorado, Denver
December 2013
Purpose
=======
ZGEEV 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
=========
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) COMPLEX*16 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).
W (output) COMPLEX*16 array, dimension (N)
W contains the computed eigenvalues.
VL (output) COMPLEX*16 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) COMPLEX*16 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) COMPLEX*16 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 >= (1+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.
RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
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;
magmaDoubleComplex z__1, z__2;
magmaDoubleComplex tmp;
double 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, irwork, lquery, wantvl, wantvr, select[1];
char side[2] = {0, 0};
char jobvl_[2] = {jobvl, 0};
char jobvr_[2] = {jobvr, 0};
irwork = 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 = -8;
} else if ( (ldvr < 1) || (wantvr && (ldvr < n))) {
*info = -10;
}
/* Compute workspace */
nb = magma_get_zgehrd_nb( n );
if (*info == 0) {
minwrk = (1+nb)*n;
work[0] = MAGMA_Z_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) || defined(Version4) || defined(Version5)
magmaDoubleComplex *dT;
if (MAGMA_SUCCESS != magma_zmalloc( &dT, nb*n )) {
*info = MAGMA_ERR_DEVICE_ALLOC;
return *info;
}
#endif
#if defined(Version4) || defined(Version5)
magmaDoubleComplex *T;
if (MAGMA_SUCCESS != magma_zmalloc_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_zlange( "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_zlascl( "G", &c_zero, &c_zero, &anrm, &cscale, &n, &n, A, &lda, &ierr );
}
/* Balance the matrix
* (CWorkspace: none)
* (RWorkspace: need N) */
ibal = 0;
lapackf77_zgebal( "B", &n, A, &lda, &ilo, &ihi, &rwork[ibal], &ierr );
/* Reduce to upper Hessenberg form
* (CWorkspace: need 2*N, prefer N + N*NB)
* (RWorkspace: none) */
itau = 0;
iwrk = itau + n;
liwrk = lwork - iwrk;
#if defined(Version1)
// Version 1 - LAPACK
lapackf77_zgehrd( &n, &ilo, &ihi, A, &lda,
&work[itau], &work[iwrk], &liwrk, &ierr );
#elif defined(Version2)
// Version 2 - LAPACK consistent HRD
magma_zgehrd2( n, ilo, ihi, A, lda,
&work[itau], &work[iwrk], &liwrk, &ierr );
#elif defined(Version3)
// Version 3 - LAPACK consistent MAGMA HRD + matrices T stored,
magma_zgehrd( 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_zgehrd_m( n, ilo, ihi, A, lda,
&work[itau], &work[iwrk], liwrk, T, &ierr );
magma_zsetmatrix( nb, n, T, nb, dT, nb );
#endif
if (wantvl) {
/* Want left eigenvectors
* Copy Householder vectors to VL */
side[0] = 'L';
lapackf77_zlacpy( MagmaLowerStr, &n, &n, A, &lda, vl, &ldvl );
/* Generate unitary matrix in VL
* (CWorkspace: need 2*N-1, prefer N + (N-1)*NB)
* (RWorkspace: none) */
#if defined(Version1) || defined(Version2)
// Version 1 & 2 - LAPACK
lapackf77_zunghr( &n, &ilo, &ihi, vl, &ldvl, &work[itau],
&work[iwrk], &liwrk, &ierr );
#elif defined(Version3) || defined(Version4)
// Version 3 - LAPACK consistent MAGMA HRD + matrices T stored
magma_zunghr( n, ilo, ihi, vl, ldvl, &work[itau], dT, nb, &ierr );
#elif defined(Version5)
// Version 5 - Multi-GPU, T on host
magma_zunghr_m( n, ilo, ihi, vl, ldvl, &work[itau], T, nb, &ierr );
#endif
/* Perform QR iteration, accumulating Schur vectors in VL
* (CWorkspace: need 1, prefer HSWORK (see comments) )
* (RWorkspace: none) */
iwrk = itau;
liwrk = lwork - iwrk;
lapackf77_zhseqr( "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[0] = 'B';
lapackf77_zlacpy( "F", &n, &n, vl, &ldvl, vr, &ldvr );
}
}
else if (wantvr) {
/* Want right eigenvectors
* Copy Householder vectors to VR */
side[0] = 'R';
lapackf77_zlacpy( "L", &n, &n, A, &lda, vr, &ldvr );
/* Generate unitary matrix in VR
* (CWorkspace: need 2*N-1, prefer N + (N-1)*NB)
* (RWorkspace: none) */
#if defined(Version1) || defined(Version2)
// Version 1 & 2 - LAPACK
lapackf77_zunghr( &n, &ilo, &ihi, vr, &ldvr, &work[itau],
&work[iwrk], &liwrk, &ierr );
#elif defined(Version3) || defined(Version4)
// Version 3 - LAPACK consistent MAGMA HRD + matrices T stored
magma_zunghr( n, ilo, ihi, vr, ldvr, &work[itau], dT, nb, &ierr );
#elif defined(Version5)
// Version 5 - Multi-GPU, T on host
magma_zunghr_m( n, ilo, ihi, vr, ldvr, &work[itau], T, nb, &ierr );
#endif
/* Perform QR iteration, accumulating Schur vectors in VR
* (CWorkspace: need 1, prefer HSWORK (see comments) )
* (RWorkspace: none) */
iwrk = itau;
liwrk = lwork - iwrk;
lapackf77_zhseqr( "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: none) */
iwrk = itau;
liwrk = lwork - iwrk;
lapackf77_zhseqr( "E", "N", &n, &ilo, &ihi, A, &lda, W,
vr, &ldvr, &work[iwrk], &liwrk, info );
}
/* If INFO > 0 from ZHSEQR, then quit */
if (*info > 0) {
goto CLEANUP;
}
if (wantvl || wantvr) {
/* Compute left and/or right eigenvectors
* (CWorkspace: need 2*N)
* (RWorkspace: need 2*N) */
irwork = ibal + n;
lapackf77_ztrevc( side, "B", select, &n, A, &lda, vl, &ldvl,
vr, &ldvr, &n, &nout, &work[iwrk], &rwork[irwork], &ierr );
}
if (wantvl) {
/* Undo balancing of left eigenvectors
* (CWorkspace: none)
* (RWorkspace: need N) */
lapackf77_zgebak( "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_dznrm2( n, vl(0,i), 1 );
cblas_zdscal( n, scl, vl(0,i), 1 );
for (k = 0; k < n; ++k) {
/* Computing 2nd power */
d__1 = MAGMA_Z_REAL( *vl(k,i) );
d__2 = MAGMA_Z_IMAG( *vl(k,i) );
rwork[irwork + k] = d__1*d__1 + d__2*d__2;
}
k = cblas_idamax( n, &rwork[irwork], 1 );
z__2 = MAGMA_Z_CNJG( *vl(k,i) );
d__1 = magma_dsqrt( rwork[irwork + k] );
MAGMA_Z_DSCALE( z__1, z__2, d__1 );
tmp = z__1;
cblas_zscal( n, CBLAS_SADDR(tmp), vl(0,i), 1 );
d__1 = MAGMA_Z_REAL( *vl(k,i) );
z__1 = MAGMA_Z_MAKE( d__1, 0 );
*vl(k,i) = z__1;
}
}
if (wantvr) {
/* Undo balancing of right eigenvectors
* (CWorkspace: none)
* (RWorkspace: need N) */
lapackf77_zgebak( "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_dznrm2( n, vr(0,i), 1 );
cblas_zdscal( n, scl, vr(0,i), 1 );
for (k = 0; k < n; ++k) {
/* Computing 2nd power */
d__1 = MAGMA_Z_REAL( *vr(k,i) );
d__2 = MAGMA_Z_IMAG( *vr(k,i) );
rwork[irwork + k] = d__1*d__1 + d__2*d__2;
}
k = cblas_idamax( n, &rwork[irwork], 1 );
z__2 = MAGMA_Z_CNJG( *vr(k,i) );
d__1 = magma_dsqrt( rwork[irwork + k] );
MAGMA_Z_DSCALE( z__1, z__2, d__1 );
tmp = z__1;
cblas_zscal( n, CBLAS_SADDR(tmp), vr(0,i), 1 );
d__1 = MAGMA_Z_REAL( *vr(k,i) );
z__1 = MAGMA_Z_MAKE( d__1, 0 );
*vr(k,i) = z__1;
}
}
CLEANUP:
/* Undo scaling if necessary */
if (scalea) {
i__1 = n - (*info);
i__2 = max( n - (*info), 1 );
lapackf77_zlascl( "G", &c_zero, &c_zero, &cscale, &anrm, &i__1, &c_one,
W + (*info), &i__2, &ierr );
if (*info > 0) {
i__1 = ilo - 1;
lapackf77_zlascl( "G", &c_zero, &c_zero, &cscale, &anrm, &i__1, &c_one,
W, &n, &ierr );
}
}
#if defined(Version3) || defined(Version4) || defined(Version5)
magma_free( dT );
#endif
#if defined(Version4) || defined(Version5)
magma_free_cpu( T );
#endif
return *info;
} /* magma_zgeev */
extern "C" magma_int_t
magma_dgeev(
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_queue_t queue,
magma_int_t *info)
{
/* -- clMAGMA (version 1.3.0) --
Univ. of Tennessee, Knoxville
Univ. of California, Berkeley
Univ. of Colorado, Denver
@date November 2014
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 >= (1+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.
===================================================================== */
magma_int_t ione = 1;
magma_int_t c__1 = 1;
magma_int_t c__0 = 0;
magma_int_t c_n1 = -1;
magma_int_t a_dim1, a_offset, vl_dim1, vl_offset, vr_dim1, vr_offset, i__1,
i__2, i__3;
double d__1, d__2;
magma_int_t i__, k, ihi, ilo;
double r__, cs, sn, scl;
double dum[1], eps;
magma_int_t ibal;
double anrm;
magma_int_t ierr, itau, iwrk, nout;
magma_int_t scalea;
double cscale;
double bignum;
magma_int_t minwrk;
magma_int_t wantvl;
double smlnum;
magma_int_t lquery, wantvr, select[1];
magma_int_t nb = 0;
magmaDouble_ptr dT;
//magma_timestr_t start, end;
const char* side_ = NULL;
*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 */
if (*info == 0) {
nb = magma_get_dgehrd_nb(n);
minwrk = (2+nb)*n;
work[0] = (double) minwrk;
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 eigenvectors are needed
#if defined(VERSION3)
if (MAGMA_SUCCESS != magma_dmalloc( &dT, nb*n )) {
*info = MAGMA_ERR_DEVICE_ALLOC;
return *info;
}
#endif
// subtract row and col for 1-based indexing
a_dim1 = lda;
a_offset = 1 + a_dim1;
a -= a_offset;
vl_dim1 = ldvl;
vl_offset = 1 + vl_dim1;
vl -= vl_offset;
vr_dim1 = ldvr;
vr_offset = 1 + vr_dim1;
vr -= vr_offset;
--work;
/* 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[a_offset], &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__0, &c__0, &anrm, &cscale, &n, &n,
&a[a_offset], &lda, &ierr);
}
/* Balance the matrix
(Workspace: need N) */
ibal = 1;
lapackf77_dgebal("B", &n, &a[a_offset], &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;
i__1 = lwork - iwrk + 1;
//start = get_current_time();
#if defined(VERSION1)
/*
* Version 1 - LAPACK
*/
lapackf77_dgehrd(&n, &ilo, &ihi, &a[a_offset], &lda,
&work[itau], &work[iwrk], &i__1, &ierr);
#elif defined(VERSION2)
/*
* Version 2 - LAPACK consistent HRD
*/
magma_dgehrd2(n, ilo, ihi, &a[a_offset], lda,
&work[itau], &work[iwrk], &i__1, &ierr);
#elif defined(VERSION3)
/*
* Version 3 - LAPACK consistent MAGMA HRD + matrices T stored,
*/
magma_dgehrd(n, ilo, ihi, &a[a_offset], lda,
&work[itau], &work[iwrk], i__1, dT, 0, queue, &ierr);
#endif
//end = get_current_time();
//printf(" Time for dgehrd = %5.2f sec\n", GetTimerValue(start,end)/1000.);
if (wantvl) {
/* Want left eigenvectors
Copy Householder vectors to VL */
side_ = "Left";
lapackf77_dlacpy(MagmaLowerStr, &n, &n,
&a[a_offset], &lda, &vl[vl_offset], &ldvl);
/*
* Generate orthogonal matrix in VL
* (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB)
*/
i__1 = lwork - iwrk + 1;
//start = get_current_time();
#if defined(VERSION1) || defined(VERSION2)
/*
* Version 1 & 2 - LAPACK
*/
lapackf77_dorghr(&n, &ilo, &ihi, &vl[vl_offset], &ldvl,
&work[itau], &work[iwrk], &i__1, &ierr);
#elif defined(VERSION3)
/*
* Version 3 - LAPACK consistent MAGMA HRD + matrices T stored
*/
magma_dorghr(n, ilo, ihi, &vl[vl_offset], ldvl, &work[itau],
dT, 0, nb, queue, &ierr);
#endif
//end = get_current_time();
//printf(" Time for dorghr = %5.2f sec\n", GetTimerValue(start,end)/1000.);
/*
* Perform QR iteration, accumulating Schur vectors in VL
* (Workspace: need N+1, prefer N+HSWORK (see comments) )
*/
iwrk = itau;
i__1 = lwork - iwrk + 1;
lapackf77_dhseqr("S", "V", &n, &ilo, &ihi, &a[a_offset], &lda, WR, WI,
&vl[vl_offset], &ldvl, &work[iwrk], &i__1, info);
if (wantvr) {
/* Want left and right eigenvectors
Copy Schur vectors to VR */
side_ = "Both";
lapackf77_dlacpy("F", &n, &n, &vl[vl_offset], &ldvl, &vr[vr_offset], &ldvr);
}
} else if (wantvr) {
/* Want right eigenvectors
Copy Householder vectors to VR */
side_ = "Right";
lapackf77_dlacpy("L", &n, &n, &a[a_offset], &lda, &vr[vr_offset], &ldvr);
/*
* Generate orthogonal matrix in VR
* (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB)
*/
i__1 = lwork - iwrk + 1;
//start = get_current_time();
#if defined(VERSION1) || defined(VERSION2)
/*
* Version 1 & 2 - LAPACK
*/
lapackf77_dorghr(&n, &ilo, &ihi, &vr[vr_offset], &ldvr,
&work[itau], &work[iwrk], &i__1, &ierr);
#elif defined(VERSION3)
/*
* Version 3 - LAPACK consistent MAGMA HRD + matrices T stored
*/
magma_dorghr(n, ilo, ihi, &vr[vr_offset], ldvr,
&work[itau], dT, 0, nb, queue, &ierr);
#endif
//end = get_current_time();
//printf(" Time for dorghr = %5.2f sec\n", GetTimerValue(start,end)/1000.);
/*
* Perform QR iteration, accumulating Schur vectors in VR
* (Workspace: need N+1, prefer N+HSWORK (see comments) )
*/
iwrk = itau;
i__1 = lwork - iwrk + 1;
lapackf77_dhseqr("S", "V", &n, &ilo, &ihi, &a[a_offset], &lda, WR, WI,
&vr[vr_offset], &ldvr, &work[iwrk], &i__1, info);
} else {
/*
* Compute eigenvalues only
* (Workspace: need N+1, prefer N+HSWORK (see comments) )
*/
iwrk = itau;
i__1 = lwork - iwrk + 1;
lapackf77_dhseqr("E", "N", &n, &ilo, &ihi, &a[a_offset], &lda, WR, WI,
&vr[vr_offset], &ldvr, &work[iwrk], &i__1, info);
}
/* If INFO > 0 from DHSEQR, then quit */
if (*info > 0) {
fprintf(stderr, "DHSEQR returned with info = %d\n", (int) *info);
goto L50;
}
if (wantvl || wantvr) {
/*
* Compute left and/or right eigenvectors
* (Workspace: need 4*N)
*/
lapackf77_dtrevc(side_, "B", select, &n, &a[a_offset], &lda, &vl[vl_offset], &ldvl,
&vr[vr_offset], &ldvr, &n, &nout, &work[iwrk], &ierr);
}
if (wantvl) {
/*
* Undo balancing of left eigenvectors
* (Workspace: need N)
*/
lapackf77_dgebak("B", "L", &n, &ilo, &ihi,
&work[ibal], &n, &vl[vl_offset], &ldvl, &ierr);
/* Normalize left eigenvectors and make largest component real */
for (i__ = 1; i__ <= n; ++i__) {
if ( WI[i__-1] == 0.) {
scl = magma_cblas_dnrm2(n, &vl[i__ * vl_dim1 + 1], 1);
scl = 1. / scl;
blasf77_dscal( &n, &scl, &vl[i__ * vl_dim1 + 1], &ione );
} else if (WI[i__-1] > 0.) {
d__1 = magma_cblas_dnrm2(n, &vl[ i__ * vl_dim1 + 1], 1);
d__2 = magma_cblas_dnrm2(n, &vl[(i__ + 1) * vl_dim1 + 1], 1);
scl = lapackf77_dlapy2(&d__1, &d__2);
scl = 1. / scl;
blasf77_dscal( &n, &scl, &vl[ i__ * vl_dim1 + 1], &ione );
blasf77_dscal( &n, &scl, &vl[(i__ + 1) * vl_dim1 + 1], &ione );
i__2 = n;
for (k = 1; k <= i__2; ++k) {
/* Computing 2nd power */
d__1 = vl[k + i__ * vl_dim1];
/* Computing 2nd power */
d__2 = vl[k + (i__ + 1) * vl_dim1];
work[iwrk + k - 1] = d__1 * d__1 + d__2 * d__2;
}
/* Comment:
Fortran BLAS does not have to add 1
C BLAS must add one to cblas_idamax */
k = blasf77_idamax( &n, &work[iwrk], &ione ); //+1;
lapackf77_dlartg(&vl[k + i__ * vl_dim1],
&vl[k + (i__ + 1) * vl_dim1], &cs, &sn, &r__);
blasf77_drot( &n, &vl[ i__ * vl_dim1 + 1], &ione,
&vl[(i__ + 1) * vl_dim1 + 1], &ione, &cs, &sn );
vl[k + (i__ + 1) * vl_dim1] = 0.;
}
}
}
if (wantvr) {
/*
* Undo balancing of right eigenvectors
* (Workspace: need N)
*/
lapackf77_dgebak("B", "R", &n, &ilo, &ihi, &work[ibal], &n,
&vr[vr_offset], &ldvr, &ierr);
/* Normalize right eigenvectors and make largest component real */
for (i__ = 1; i__ <= n; ++i__) {
if (WI[i__-1] == 0.) {
scl = 1. / magma_cblas_dnrm2(n, &vr[i__ * vr_dim1 + 1], 1);
blasf77_dscal( &n, &scl, &vr[i__ * vr_dim1 + 1], &ione );
} else if (WI[i__-1] > 0.) {
d__1 = magma_cblas_dnrm2(n, &vr[ i__ * vr_dim1 + 1], 1);
d__2 = magma_cblas_dnrm2(n, &vr[(i__ + 1) * vr_dim1 + 1], 1);
scl = lapackf77_dlapy2(&d__1, &d__2);
scl = 1. / scl;
blasf77_dscal( &n, &scl, &vr[ i__ * vr_dim1 + 1], &ione );
blasf77_dscal( &n, &scl, &vr[(i__ + 1) * vr_dim1 + 1], &ione );
i__2 = n;
for (k = 1; k <= i__2; ++k) {
/* Computing 2nd power */
d__1 = vr[k + i__ * vr_dim1];
/* Computing 2nd power */
d__2 = vr[k + (i__ + 1) * vr_dim1];
work[iwrk + k - 1] = d__1 * d__1 + d__2 * d__2;
}
/* Comment:
Fortran BLAS does not have to add 1
C BLAS must add one to cblas_idamax */
k = blasf77_idamax( &n, &work[iwrk], &ione ); //+1;
lapackf77_dlartg(&vr[k + i__ * vr_dim1], &vr[k + (i__ + 1) * vr_dim1],
&cs, &sn, &r__);
blasf77_drot( &n, &vr[ i__ * vr_dim1 + 1], &ione,
&vr[(i__ + 1) * vr_dim1 + 1], &ione, &cs, &sn );
vr[k + (i__ + 1) * vr_dim1] = 0.;
}
}
}
/* Undo scaling if necessary */
L50:
if (scalea) {
i__1 = n - *info;
/* Computing MAX */
i__3 = n - *info;
i__2 = max(i__3,1);
lapackf77_dlascl("G", &c__0, &c__0, &cscale, &anrm, &i__1, &c__1,
WR + (*info), &i__2, &ierr);
i__1 = n - *info;
/* Computing MAX */
i__3 = n - *info;
i__2 = max(i__3,1);
lapackf77_dlascl("G", &c__0, &c__0, &cscale, &anrm, &i__1, &c__1,
WI + (*info), &i__2, &ierr);
if (*info > 0) {
i__1 = ilo - 1;
lapackf77_dlascl("G", &c__0, &c__0, &cscale, &anrm, &i__1, &c__1,
WR, &n, &ierr);
i__1 = ilo - 1;
lapackf77_dlascl("G", &c__0, &c__0, &cscale, &anrm, &i__1, &c__1,
WI, &n, &ierr);
}
}
#if defined(VERSION3)
magma_free( dT );
#endif
return *info;
} /* magma_dgeev */
magma_int_t magma_zlatrsd(
magma_uplo_t uplo, magma_trans_t trans, magma_diag_t diag, magma_bool_t normin,
magma_int_t n, const magmaDoubleComplex *A, magma_int_t lda,
magmaDoubleComplex lambda,
magmaDoubleComplex *x,
double *scale, double *cnorm,
magma_int_t *info)
{
#define A(i,j) (A + (i) + (j)*lda)
/* constants */
const magma_int_t ione = 1;
const double d_half = 0.5;
const magmaDoubleComplex c_zero = MAGMA_Z_ZERO;
const magmaDoubleComplex c_one = MAGMA_Z_ONE;
/* System generated locals */
magma_int_t len;
magmaDoubleComplex ztmp;
/* Local variables */
magma_int_t i, j;
double xj, rec, tjj;
magma_int_t jinc;
double xbnd;
magma_int_t imax;
double tmax;
magmaDoubleComplex tjjs;
double xmax, grow;
double tscal;
magmaDoubleComplex uscal;
magma_int_t jlast;
magmaDoubleComplex csumj;
double bignum;
magma_int_t jfirst;
double smlnum;
/* Function Body */
*info = 0;
magma_int_t upper = (uplo == MagmaUpper);
magma_int_t notran = (trans == MagmaNoTrans);
magma_int_t nounit = (diag == MagmaNonUnit);
/* Test the input parameters. */
if ( ! upper && uplo != MagmaLower ) {
*info = -1;
}
else if (! notran &&
trans != MagmaTrans &&
trans != MagmaConjTrans) {
*info = -2;
}
else if ( ! nounit && diag != MagmaUnit ) {
*info = -3;
}
else if ( ! (normin == MagmaTrue) &&
! (normin == MagmaFalse) ) {
*info = -4;
}
else if ( n < 0 ) {
*info = -5;
}
else if ( lda < max(1,n) ) {
*info = -7;
}
if ( *info != 0 ) {
magma_xerbla( __func__, -(*info) );
return *info;
}
/* Quick return if possible */
if ( n == 0 ) {
return *info;
}
/* Determine machine dependent parameters to control overflow. */
smlnum = lapackf77_dlamch( "Safe minimum" );
bignum = 1. / smlnum;
lapackf77_dlabad( &smlnum, &bignum );
smlnum /= lapackf77_dlamch( "Precision" );
bignum = 1. / smlnum;
*scale = 1.;
if ( normin == MagmaFalse ) {
/* Compute the 1-norm of each column, not including the diagonal. */
if ( upper ) {
/* A is upper triangular. */
cnorm[0] = 0.;
for( j = 1; j < n; ++j ) {
cnorm[j] = magma_cblas_dzasum( j, A(0,j), ione );
}
}
else {
/* A is lower triangular. */
for( j = 0; j < n-1; ++j ) {
cnorm[j] = magma_cblas_dzasum( n-(j+1), A(j+1,j), ione );
}
cnorm[n-1] = 0.;
}
}
/* Scale the column norms by TSCAL if the maximum element in CNORM is */
/* greater than BIGNUM/2. */
imax = blasf77_idamax( &n, &cnorm[0], &ione ) - 1;
tmax = cnorm[imax];
if ( tmax <= bignum * 0.5 ) {
tscal = 1.;
}
else {
tscal = 0.5 / (smlnum * tmax);
blasf77_dscal( &n, &tscal, &cnorm[0], &ione );
}
/* ================================================================= */
/* Compute a bound on the computed solution vector to see if the */
/* Level 2 BLAS routine ZTRSV can be used. */
xmax = 0.;
for( j = 0; j < n; ++j ) {
xmax = max( xmax, 0.5*MAGMA_Z_ABS1( x[j] ));
}
xbnd = xmax;
if ( notran ) {
/* ---------------------------------------- */
/* Compute the growth in A * x = b. */
if ( upper ) {
jfirst = n-1;
jlast = 0;
jinc = -1;
}
else {
jfirst = 0;
jlast = n;
jinc = 1;
}
if ( tscal != 1. ) {
grow = 0.;
goto L60;
}
/* A is non-unit triangular. */
/* Compute GROW = 1/G(j) and XBND = 1/M(j). */
/* Initially, G(0) = max{x(i), i=1,...,n}. */
grow = 0.5 / max( xbnd, smlnum );
xbnd = grow;
for( j = jfirst; (jinc < 0 ? j >= jlast : j < jlast); j += jinc ) {
/* Exit the loop if the growth factor is too small. */
if ( grow <= smlnum ) {
goto L60;
}
if ( nounit ) {
tjjs = *A(j,j) - lambda;
}
else {
tjjs = c_one - lambda;
}
tjj = MAGMA_Z_ABS1( tjjs );
if ( tjj >= smlnum ) {
/* M(j) = G(j-1) / abs(A(j,j)) */
xbnd = min( xbnd, min(1.,tjj)*grow );
}
else {
/* M(j) could overflow, set XBND to 0. */
xbnd = 0.;
}
if ( tjj + cnorm[j] >= smlnum ) {
/* G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) ) */
grow *= (tjj / (tjj + cnorm[j]));
}
else {
/* G(j) could overflow, set GROW to 0. */
grow = 0.;
}
}
grow = xbnd;
L60:
;
}
else {
/* ---------------------------------------- */
/* Compute the growth in A**T * x = b or A**H * x = b. */
if ( upper ) {
jfirst = 0;
jlast = n;
jinc = 1;
}
else {
jfirst = n-1;
jlast = 0;
jinc = -1;
}
if ( tscal != 1. ) {
grow = 0.;
goto L90;
}
/* A is non-unit triangular. */
/* Compute GROW = 1/G(j) and XBND = 1/M(j). */
/* Initially, M(0) = max{x(i), i=1,...,n}. */
grow = 0.5 / max( xbnd, smlnum );
xbnd = grow;
for( j = jfirst; (jinc < 0 ? j >= jlast : j < jlast); j += jinc ) {
/* Exit the loop if the growth factor is too small. */
if ( grow <= smlnum ) {
goto L90;
}
/* G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) ) */
xj = 1. + cnorm[j];
grow = min( grow, xbnd / xj );
if ( nounit ) {
tjjs = *A(j,j) - lambda;
}
else {
tjjs = c_one - lambda;
}
tjj = MAGMA_Z_ABS1( tjjs );
if ( tjj >= smlnum ) {
/* M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j)) */
if ( xj > tjj ) {
xbnd *= (tjj / xj);
}
}
else {
/* M(j) could overflow, set XBND to 0. */
xbnd = 0.;
}
}
grow = min( grow, xbnd );
L90:
;
}
/* ================================================================= */
/* Due to modified diagonal, we can't use regular BLAS ztrsv. */
/* Use a Level 1 BLAS solve, scaling intermediate results. */
if ( xmax > bignum * 0.5 ) {
/* Scale X so that its components are less than or equal to */
/* BIGNUM in absolute value. */
*scale = (bignum * 0.5) / xmax;
blasf77_zdscal( &n, scale, &x[0], &ione );
xmax = bignum;
}
else {
xmax *= 2.;
}
if ( notran ) {
/* ---------------------------------------- */
/* Solve A * x = b */
for( j = jfirst; (jinc < 0 ? j >= jlast : j < jlast); j += jinc ) {
/* Compute x(j) = b(j) / A(j,j), scaling x if necessary. */
xj = MAGMA_Z_ABS1( x[j] );
if ( nounit ) {
tjjs = (*A(j,j) - lambda ) * tscal;
}
else {
tjjs = (c_one - lambda) * tscal;
if ( tscal == 1. ) {
goto L110;
}
}
tjj = MAGMA_Z_ABS1( tjjs );
if ( tjj > smlnum ) {
/* abs(A(j,j)) > SMLNUM: */
if ( tjj < 1. ) {
if ( xj > tjj * bignum ) {
/* Scale x by 1/b(j). */
rec = 1. / xj;
blasf77_zdscal( &n, &rec, &x[0], &ione );
*scale *= rec;
xmax *= rec;
}
}
x[j] = x[j] / tjjs;
xj = MAGMA_Z_ABS1( x[j] );
}
else if ( tjj > 0. ) {
/* 0 < abs(A(j,j)) <= SMLNUM: */
if ( xj > tjj * bignum ) {
/* Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM */
/* to avoid overflow when dividing by A(j,j). */
rec = (tjj * bignum) / xj;
if ( cnorm[j] > 1. ) {
/* Scale by 1/CNORM(j) to avoid overflow when */
/* multiplying x(j) times column j. */
rec /= cnorm[j];
}
blasf77_zdscal( &n, &rec, &x[0], &ione );
*scale *= rec;
xmax *= rec;
}
x[j] = x[j] / tjjs;
xj = MAGMA_Z_ABS1( x[j] );
}
else {
/* A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and */
/* scale = 0, and compute a solution to A*x = 0. */
for( i = 0; i < n; ++i ) {
x[i] = c_zero;
}
x[j] = c_one;
xj = 1.;
*scale = 0.;
xmax = 0.;
}
L110:
/* Scale x if necessary to avoid overflow when adding a */
/* multiple of column j of A. */
if ( xj > 1. ) {
rec = 1. / xj;
if ( cnorm[j] > (bignum - xmax) * rec ) {
/* Scale x by 1/(2*abs(x(j))). */
rec *= 0.5;
blasf77_zdscal( &n, &rec, &x[0], &ione );
*scale *= rec;
}
}
else if ( xj * cnorm[j] > bignum - xmax ) {
/* Scale x by 1/2. */
blasf77_zdscal( &n, &d_half, &x[0], &ione );
*scale *= 0.5;
}
if ( upper ) {
if ( j > 0 ) {
/* Compute the update */
/* x(1:j-1) := x(1:j-1) - x(j) * A(1:j-1,j) */
len = j;
ztmp = -tscal * x[j];
blasf77_zaxpy( &len, &ztmp, A(0,j), &ione, &x[0], &ione );
i = blasf77_izamax( &len, &x[0], &ione ) - 1;
xmax = MAGMA_Z_ABS1( x[i] );
}
}
else {
if ( j < n-1 ) {
/* Compute the update */
/* x(j+1:n) := x(j+1:n) - x(j) * A(j+1:n,j) */
len = n - (j+1);
ztmp = -tscal * x[j];
blasf77_zaxpy( &len, &ztmp, A(j+1,j), &ione, &x[j + 1], &ione );
i = j + blasf77_izamax( &len, &x[j + 1], &ione );
xmax = MAGMA_Z_ABS1( x[i] );
}
}
}
}
else if ( trans == MagmaTrans ) {
/* ---------------------------------------- */
/* Solve A**T * x = b */
for( j = jfirst; (jinc < 0 ? j >= jlast : j < jlast); j += jinc ) {
/* Compute x(j) = b(j) - sum A(k,j)*x(k). */
/* k<>j */
xj = MAGMA_Z_ABS1( x[j] );
uscal = MAGMA_Z_MAKE( tscal, 0. );
rec = 1. / max( xmax, 1. );
if ( cnorm[j] > (bignum - xj) * rec ) {
/* If x(j) could overflow, scale x by 1/(2*XMAX). */
rec *= 0.5;
if ( nounit ) {
tjjs = (*A(j,j) - lambda) * tscal;
}
else {
tjjs = (c_one - lambda) * tscal;
}
tjj = MAGMA_Z_ABS1( tjjs );
if ( tjj > 1. ) {
/* Divide by A(j,j) when scaling x if A(j,j) > 1. */
rec = min( 1., rec * tjj );
uscal = uscal / tjjs;
}
if ( rec < 1. ) {
blasf77_zdscal( &n, &rec, &x[0], &ione );
*scale *= rec;
xmax *= rec;
}
}
csumj = c_zero;
if ( uscal == c_one ) {
/* If the scaling needed for A in the dot product is 1, */
/* call ZDOTU to perform the dot product. */
if ( upper ) {
csumj = magma_cblas_zdotu( j, A(0,j), ione, &x[0], ione );
}
else if ( j < n-1 ) {
csumj = magma_cblas_zdotu( n-(j+1), A(j+1,j), ione, &x[j+1], ione );
}
}
else {
/* Otherwise, use in-line code for the dot product. */
if ( upper ) {
for( i = 0; i < j; ++i ) {
csumj += (*A(i,j) * uscal) * x[i];
}
}
else if ( j < n-1 ) {
for( i = j+1; i < n; ++i ) {
csumj += (*A(i,j) * uscal) * x[i];
}
}
}
if ( uscal == MAGMA_Z_MAKE( tscal, 0. )) {
/* Compute x(j) := ( x(j) - CSUMJ ) / A(j,j) if 1/A(j,j) */
/* was not used to scale the dotproduct. */
x[j] -= csumj;
xj = MAGMA_Z_ABS1( x[j] );
if ( nounit ) {
tjjs = (*A(j,j) - lambda) * tscal;
}
else {
tjjs = (c_one - lambda) * tscal;
if ( tscal == 1. ) {
goto L160;
}
}
/* Compute x(j) = x(j) / A(j,j), scaling if necessary. */
tjj = MAGMA_Z_ABS1( tjjs );
if ( tjj > smlnum ) {
/* abs(A(j,j)) > SMLNUM: */
if ( tjj < 1. ) {
if ( xj > tjj * bignum ) {
/* Scale X by 1/abs(x(j)). */
rec = 1. / xj;
blasf77_zdscal( &n, &rec, &x[0], &ione );
*scale *= rec;
xmax *= rec;
}
}
x[j] = x[j] / tjjs;
}
else if ( tjj > 0. ) {
/* 0 < abs(A(j,j)) <= SMLNUM: */
if ( xj > tjj * bignum ) {
/* Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM. */
rec = (tjj * bignum) / xj;
blasf77_zdscal( &n, &rec, &x[0], &ione );
*scale *= rec;
xmax *= rec;
}
x[j] = x[j] / tjjs;
}
else {
/* A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and */
/* scale = 0 and compute a solution to A**T *x = 0. */
for( i = 0; i < n; ++i ) {
x[i] = c_zero;
}
x[j] = c_one;
*scale = 0.;
xmax = 0.;
}
L160:
;
}
else {
/* Compute x(j) := x(j) / A(j,j) - CSUMJ if the dot */
/* product has already been divided by 1/A(j,j). */
x[j] = (x[j] / tjjs) - csumj;
}
xmax = max( xmax, MAGMA_Z_ABS1( x[j] ));
}
}
else {
/* ---------------------------------------- */
/* Solve A**H * x = b */
for( j = jfirst; (jinc < 0 ? j >= jlast : j < jlast); j += jinc ) {
/* Compute x(j) = b(j) - sum A(k,j)*x(k). */
/* k<>j */
xj = MAGMA_Z_ABS1( x[j] );
uscal = MAGMA_Z_MAKE( tscal, 0. );
rec = 1. / max(xmax, 1.);
if ( cnorm[j] > (bignum - xj) * rec ) {
/* If x(j) could overflow, scale x by 1/(2*XMAX). */
rec *= 0.5;
if ( nounit ) {
tjjs = MAGMA_Z_CONJ( *A(j,j) - lambda ) * tscal;
}
else {
tjjs = (c_one - lambda) * tscal;
}
tjj = MAGMA_Z_ABS1( tjjs );
if ( tjj > 1. ) {
/* Divide by A(j,j) when scaling x if A(j,j) > 1. */
rec = min( 1., rec * tjj );
uscal = uscal / tjjs;
}
if ( rec < 1. ) {
blasf77_zdscal( &n, &rec, &x[0], &ione );
*scale *= rec;
xmax *= rec;
}
}
csumj = c_zero;
if ( uscal == c_one ) {
/* If the scaling needed for A in the dot product is 1, */
/* call ZDOTC to perform the dot product. */
if ( upper ) {
csumj = magma_cblas_zdotc( j, A(0,j), ione, &x[0], ione );
}
else if ( j < n-1 ) {
csumj = magma_cblas_zdotc( n-(j+1), A(j+1,j), ione, &x[j+1], ione );
}
}
else {
/* Otherwise, use in-line code for the dot product. */
if ( upper ) {
for( i = 0; i < j; ++i ) {
csumj += (MAGMA_Z_CONJ( *A(i,j) ) * uscal) * x[i];
}
}
else if ( j < n-1 ) {
for( i = j + 1; i < n; ++i ) {
csumj += (MAGMA_Z_CONJ( *A(i,j) ) * uscal) * x[i];
}
}
}
if ( uscal == tscal ) {
/* Compute x(j) := ( x(j) - CSUMJ ) / A(j,j) if 1/A(j,j) */
/* was not used to scale the dotproduct. */
x[j] -= csumj;
xj = MAGMA_Z_ABS1( x[j] );
if ( nounit ) {
tjjs = MAGMA_Z_CONJ( *A(j,j) - lambda ) * tscal;
}
else {
tjjs = (c_one - lambda) * tscal;
if ( tscal == 1. ) {
goto L210;
}
}
/* Compute x(j) = x(j) / A(j,j), scaling if necessary. */
tjj = MAGMA_Z_ABS1( tjjs );
if ( tjj > smlnum ) {
/* abs(A(j,j)) > SMLNUM: */
if ( tjj < 1. ) {
if ( xj > tjj * bignum ) {
/* Scale X by 1/abs(x(j)). */
rec = 1. / xj;
blasf77_zdscal( &n, &rec, &x[0], &ione );
*scale *= rec;
xmax *= rec;
}
}
x[j] = x[j] / tjjs;
}
else if ( tjj > 0. ) {
/* 0 < abs(A(j,j)) <= SMLNUM: */
if ( xj > tjj * bignum ) {
/* Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM. */
rec = (tjj * bignum) / xj;
blasf77_zdscal( &n, &rec, &x[0], &ione );
*scale *= rec;
xmax *= rec;
}
x[j] = x[j] / tjjs;
}
else {
/* A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and */
/* scale = 0 and compute a solution to A**H *x = 0. */
for( i = 0; i < n; ++i ) {
x[i] = c_zero;
}
x[j] = c_one;
*scale = 0.;
xmax = 0.;
}
L210:
;
}
else {
/* Compute x(j) := x(j) / A(j,j) - CSUMJ if the dot */
/* product has already been divided by 1/A(j,j). */
x[j] = (x[j] / tjjs) - csumj;
}
xmax = max( xmax, MAGMA_Z_ABS1( x[j] ));
}
}
*scale /= tscal;
/* Scale the column norms by 1/TSCAL for return. */
if ( tscal != 1. ) {
double d = 1. / tscal;
blasf77_dscal( &n, &d, &cnorm[0], &ione );
}
return *info;
} /* end zlatrsd */
/**
Purpose
-------
ZGEEV 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_16 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_16 array, dimension (N)
w contains the computed eigenvalues.
@param[out]
VL COMPLEX_16 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_16 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_16 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 >= (1+nb)*N.
For optimal performance, LWORK >= (1+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
rwork (workspace) DOUBLE PRECISION 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_zgeev_driver
********************************************************************/
extern "C" magma_int_t
magma_zgeev(
magma_vec_t jobvl, magma_vec_t jobvr, magma_int_t n,
magmaDoubleComplex *A, magma_int_t lda,
#ifdef COMPLEX
magmaDoubleComplex *w,
#else
double *wr, double *wi,
#endif
magmaDoubleComplex *VL, magma_int_t ldvl,
magmaDoubleComplex *VR, magma_int_t ldvr,
magmaDoubleComplex *work, magma_int_t lwork,
#ifdef COMPLEX
double *rwork,
#endif
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;
magmaDoubleComplex tmp;
double 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, irwork, 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 );
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_zgehrd_nb( n );
if (*info == 0) {
minwrk = (1+ nb)*n;
optwrk = (1+2*nb)*n;
work[0] = MAGMA_Z_MAKE( optwrk, 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)
magmaDoubleComplex_ptr dT;
if (MAGMA_SUCCESS != magma_zmalloc( &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_zlange( "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_zlascl( "G", &izero, &izero, &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_zgebal( "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 zgehrd */
itau = 0;
iwrk = itau + n;
liwrk = lwork - iwrk;
timer_start( time_gehrd );
flops_start( flop_gehrd );
#if defined(VERSION1)
// Version 1 - LAPACK
lapackf77_zgehrd( &n, &ilo, &ihi, A, &lda,
&work[itau], &work[iwrk], &liwrk, &ierr );
#elif defined(VERSION2)
// Version 2 - LAPACK consistent HRD
magma_zgehrd2( n, ilo, ihi, A, lda,
&work[itau], &work[iwrk], liwrk, &ierr );
#elif defined(VERSION3)
// Version 3 - LAPACK consistent MAGMA HRD + T matrices stored,
magma_zgehrd( n, ilo, ihi, A, lda,
&work[itau], &work[iwrk], liwrk, dT, &ierr );
#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_zlacpy( 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 zunghr */
timer_start( time_unghr );
flops_start( flop_unghr );
#if defined(VERSION1) || defined(VERSION2)
// Version 1 & 2 - LAPACK
lapackf77_zunghr( &n, &ilo, &ihi, VL, &ldvl, &work[itau],
&work[iwrk], &liwrk, &ierr );
#elif defined(VERSION3)
// Version 3 - LAPACK consistent MAGMA HRD + T matrices stored
magma_zunghr( n, ilo, ihi, VL, ldvl, &work[itau], dT, 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
* (CWorkspace: need 1, prefer HSWORK (see comments) )
* (RWorkspace: N)
* - including N reserved for gebal/gebak, unused by zhseqr */
iwrk = itau;
liwrk = lwork - iwrk;
lapackf77_zhseqr( "S", "V", &n, &ilo, &ihi, A, &lda, w,
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_zlacpy( "F", &n, &n, VL, &ldvl, VR, &ldvr );
}
}
else if (wantvr) {
/* Want right eigenvectors
* Copy Householder vectors to VR */
side = MagmaRight;
lapackf77_zlacpy( "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 zunghr */
timer_start( time_unghr );
flops_start( flop_unghr );
#if defined(VERSION1) || defined(VERSION2)
// Version 1 & 2 - LAPACK
lapackf77_zunghr( &n, &ilo, &ihi, VR, &ldvr, &work[itau],
&work[iwrk], &liwrk, &ierr );
#elif defined(VERSION3)
// Version 3 - LAPACK consistent MAGMA HRD + T matrices stored
magma_zunghr( n, ilo, ihi, VR, ldvr, &work[itau], dT, nb, &ierr );
#endif
time_sum += timer_stop( time_unghr );
flop_sum += flops_stop( flop_unghr );
/* 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 zhseqr */
timer_start( time_hseqr );
flops_start( flop_hseqr );
iwrk = itau;
liwrk = lwork - iwrk;
lapackf77_zhseqr( "S", "V", &n, &ilo, &ihi, A, &lda, w,
VR, &ldvr, &work[iwrk], &liwrk, info );
time_sum += timer_stop( time_hseqr );
flop_sum += flops_stop( flop_hseqr );
}
else {
/* Compute eigenvalues only
* (CWorkspace: need 1, prefer HSWORK (see comments) )
* (RWorkspace: N)
* - including N reserved for gebal/gebak, unused by zhseqr */
timer_start( time_hseqr );
flops_start( flop_hseqr );
iwrk = itau;
liwrk = lwork - iwrk;
lapackf77_zhseqr( "E", "N", &n, &ilo, &ihi, A, &lda, w,
VR, &ldvr, &work[iwrk], &liwrk, info );
time_sum += timer_stop( time_hseqr );
flop_sum += flops_stop( flop_hseqr );
}
/* If INFO > 0 from ZHSEQR, then quit */
if (*info > 0) {
goto CLEANUP;
}
timer_start( time_trevc );
flops_start( flop_trevc );
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 ztrevc */
irwork = ibal + n;
#if TREVC_VERSION == 1
lapackf77_ztrevc( 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_ztrevc3( 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_ztrevc3( side, MagmaBacktransVec, select, n, A, lda, VL, ldvl,
VR, ldvr, n, &nout, &work[iwrk], liwrk, &rwork[irwork], &ierr );
#elif TREVC_VERSION == 4
magma_ztrevc3_mt( side, MagmaBacktransVec, select, n, A, lda, VL, ldvl,
VR, ldvr, n, &nout, &work[iwrk], liwrk, &rwork[irwork], &ierr );
#elif TREVC_VERSION == 5
magma_ztrevc3_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
}
time_sum += timer_stop( time_trevc );
flop_sum += flops_stop( flop_trevc );
if (wantvl) {
/* Undo balancing of left eigenvectors
* (CWorkspace: none)
* (RWorkspace: need N) */
lapackf77_zgebak( "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. / magma_cblas_dznrm2( n, VL(0,i), 1 );
blasf77_zdscal( &n, &scl, VL(0,i), &ione );
for (k = 0; k < n; ++k) {
/* Computing 2nd power */
d__1 = MAGMA_Z_REAL( *VL(k,i) );
d__2 = MAGMA_Z_IMAG( *VL(k,i) );
rwork[irwork + k] = d__1*d__1 + d__2*d__2;
}
k = blasf77_idamax( &n, &rwork[irwork], &ione ) - 1; // subtract 1; k is 0-based
tmp = MAGMA_Z_CNJG( *VL(k,i) ) / magma_dsqrt( rwork[irwork + k] );
blasf77_zscal( &n, &tmp, VL(0,i), &ione );
*VL(k,i) = MAGMA_Z_MAKE( MAGMA_Z_REAL( *VL(k,i) ), 0 );
}
}
if (wantvr) {
/* Undo balancing of right eigenvectors
* (CWorkspace: none)
* (RWorkspace: need N) */
lapackf77_zgebak( "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. / magma_cblas_dznrm2( n, VR(0,i), 1 );
blasf77_zdscal( &n, &scl, VR(0,i), &ione );
for (k = 0; k < n; ++k) {
/* Computing 2nd power */
d__1 = MAGMA_Z_REAL( *VR(k,i) );
d__2 = MAGMA_Z_IMAG( *VR(k,i) );
rwork[irwork + k] = d__1*d__1 + d__2*d__2;
}
k = blasf77_idamax( &n, &rwork[irwork], &ione ) - 1; // subtract 1; k is 0-based
tmp = MAGMA_Z_CNJG( *VR(k,i) ) / magma_dsqrt( rwork[irwork + k] );
blasf77_zscal( &n, &tmp, VR(0,i), &ione );
*VR(k,i) = MAGMA_Z_MAKE( MAGMA_Z_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_zlascl( "G", &izero, &izero, &cscale, &anrm, &nval, &ione, 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_zlascl( "G", &izero, &izero, &cscale, &anrm, &nval, &ione, w, &n, &ierr );
}
}
#if defined(VERSION3)
magma_free( dT );
#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_Z_MAKE( (double) optwrk, 0. );
return *info;
} /* magma_zgeev */
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 */