/**
Purpose
-------
SLAEX3 finds the roots of the secular equation, as defined by the
values in D, W, and RHO, between 1 and K. It makes the
appropriate calls to SLAED4 and then updates the eigenvectors by
multiplying the matrix of eigenvectors of the pair of eigensystems
being combined by the matrix of eigenvectors of the K-by-K system
which is solved here.
It is used in the last step when only a part of the eigenvectors
is required.
It compute only the required part of the eigenvectors and the rest
is not used.
This code makes very mild assumptions about floating point
arithmetic. It will work on machines with a guard digit in
add/subtract, or on those binary machines without guard digits
which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
It could conceivably fail on hexadecimal or decimal machines
without guard digits, but we know of none.
Arguments
---------
@param[in]
ngpu INTEGER
Number of GPUs to use. ngpu > 0.
@param[in]
k INTEGER
The number of terms in the rational function to be solved by
SLAED4. K >= 0.
@param[in]
n INTEGER
The number of rows and columns in the Q matrix.
N >= K (deflation may result in N > K).
@param[in]
n1 INTEGER
The location of the last eigenvalue in the leading submatrix.
min(1,N) <= N1 <= N/2.
@param[out]
d REAL array, dimension (N)
D(I) contains the updated eigenvalues for
1 <= I <= K.
@param[out]
Q REAL array, dimension (LDQ,N)
Initially the first K columns are used as workspace.
On output the columns ??? to ??? contain
the updated eigenvectors.
@param[in]
ldq INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
@param[in]
rho REAL
The value of the parameter in the rank one update equation.
RHO >= 0 required.
@param[in,out]
dlamda REAL array, dimension (K)
The first K elements of this array contain the old roots
of the deflated updating problem. These are the poles
of the secular equation. May be changed on output by
having lowest order bit set to zero on Cray X-MP, Cray Y-MP,
Cray-2, or Cray C-90, as described above.
@param[in]
Q2 REAL array, dimension (LDQ2, N)
The first K columns of this matrix contain the non-deflated
eigenvectors for the split problem.
@param[in]
indx INTEGER array, dimension (N)
The permutation used to arrange the columns of the deflated
Q matrix into three groups (see SLAED2).
The rows of the eigenvectors found by SLAED4 must be likewise
permuted before the matrix multiply can take place.
@param[in]
ctot INTEGER array, dimension (4)
A count of the total number of the various types of columns
in Q, as described in INDX. The fourth column type is any
column which has been deflated.
@param[in,out]
w REAL array, dimension (K)
The first K elements of this array contain the components
of the deflation-adjusted updating vector. Destroyed on
output.
@param
s (workspace) REAL array, dimension (N1 + 1)*K
Will contain the eigenvectors of the repaired matrix which
will be multiplied by the previously accumulated eigenvectors
to update the system.
@param[out]
indxq INTEGER array, dimension (N)
On exit, the permutation which will reintegrate the
subproblems back into sorted order,
i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.
@param
dwork (devices workspaces) REAL array of arrays,
dimension NRGPU.
if NRGPU = 1 the dimension of the first workspace
should be (3*N*N/2+3*N)
otherwise the NRGPU workspaces should have the size
ceil((N-N1) * (N-N1) / floor(ngpu/2)) +
NB * ((N-N1) + (N-N1) / floor(ngpu/2))
@param
queues (device queues) magma_queue_t array,
dimension (MagmaMaxGPUs,2)
@param[in]
range magma_range_t
- = MagmaRangeAll: all eigenvalues will be found.
- = MagmaRangeV: all eigenvalues in the half-open interval (VL,VU]
will be found.
- = MagmaRangeI: the IL-th through IU-th eigenvalues will be found.
TODO verify range, vl, vu, il, iu -- copied from slaex1.
@param[in]
vl REAL
@param[in]
vu REAL
if RANGE=MagmaRangeV, the lower and upper bounds of the interval to
be searched for eigenvalues. VL < VU.
Not referenced if RANGE = MagmaRangeAll or MagmaRangeI.
@param[in]
il INTEGER
@param[in]
iu INTEGER
if RANGE=MagmaRangeI, the indices (in ascending order) of the
smallest and largest eigenvalues to be returned.
1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
Not referenced if RANGE = MagmaRangeAll or MagmaRangeV.
@param[out]
info INTEGER
- = 0: successful exit.
- < 0: if INFO = -i, the i-th argument had an illegal value.
- > 0: if INFO = 1, an eigenvalue did not converge
Further Details
---------------
Based on contributions by
Jeff Rutter, Computer Science Division, University of California
at Berkeley, USA
Modified by Francoise Tisseur, University of Tennessee.
@ingroup magma_ssyev_aux
********************************************************************/
extern "C" magma_int_t
magma_slaex3_m(
magma_int_t ngpu,
magma_int_t k, magma_int_t n, magma_int_t n1, float *d,
float *Q, magma_int_t ldq, float rho,
float *dlamda, float *Q2, magma_int_t *indx,
magma_int_t *ctot, float *w, float *s, magma_int_t *indxq,
magmaFloat_ptr dwork[],
magma_queue_t queues[MagmaMaxGPUs][2],
magma_range_t range, float vl, float vu, magma_int_t il, magma_int_t iu,
magma_int_t *info )
{
#define Q(i_,j_) (Q + (i_) + (j_)*ldq)
#define dQ2(id) (dwork[id])
#define dS(id, ii) (dwork[id] + n2*n2_loc + (ii)*(n2*nb))
#define dQ(id, ii) (dwork[id] + n2*n2_loc + 2*(n2*nb) + (ii)*(n2_loc*nb))
if (ngpu == 1) {
magma_setdevice(0);
magma_slaex3(k, n, n1, d, Q, ldq, rho,
dlamda, Q2, indx, ctot, w, s, indxq,
*dwork, range, vl, vu, il, iu, info );
return *info;
}
float d_one = 1.;
float d_zero = 0.;
magma_int_t ione = 1;
magma_int_t ineg_one = -1;
magma_int_t iil, iiu, rk;
magma_int_t n1_loc, n2_loc, ib, nb, ib2, igpu;
magma_int_t ni_loc[MagmaMaxGPUs];
magma_int_t i, ind, iq2, j, n12, n2, n23, tmp;
float temp;
magma_int_t alleig, valeig, indeig;
alleig = (range == MagmaRangeAll);
valeig = (range == MagmaRangeV);
indeig = (range == MagmaRangeI);
*info = 0;
if (k < 0)
*info=-1;
else if (n < k)
*info=-2;
else if (ldq < max(1,n))
*info=-6;
else if (! (alleig || valeig || indeig))
*info = -15;
else {
if (valeig) {
if (n > 0 && vu <= vl)
*info = -17;
}
else if (indeig) {
if (il < 1 || il > max(1,n))
*info = -18;
else if (iu < min(n,il) || iu > n)
*info = -19;
}
}
if (*info != 0) {
magma_xerbla(__func__, -(*info));
return *info;
}
// Quick return if possible
if (k == 0)
return *info;
magma_device_t orig_dev;
magma_getdevice( &orig_dev );
magma_queue_t orig_stream;
magmablasGetKernelStream( &orig_stream );
/*
Modify values DLAMDA(i) to make sure all DLAMDA(i)-DLAMDA(j) can
be computed with high relative accuracy (barring over/underflow).
This is a problem on machines without a guard digit in
add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2).
The following code replaces DLAMDA(I) by 2*DLAMDA(I)-DLAMDA(I),
which on any of these machines zeros out the bottommost
bit of DLAMDA(I) if it is 1; this makes the subsequent
subtractions DLAMDA(I)-DLAMDA(J) unproblematic when cancellation
occurs. On binary machines with a guard digit (almost all
machines) it does not change DLAMDA(I) at all. On hexadecimal
and decimal machines with a guard digit, it slightly
changes the bottommost bits of DLAMDA(I). It does not account
for hexadecimal or decimal machines without guard digits
(we know of none). We use a subroutine call to compute
2*DLAMBDA(I) to prevent optimizing compilers from eliminating
this code.*/
//#define CHECK_CPU
#ifdef CHECK_CPU
float *hwS[2][MagmaMaxGPUs], *hwQ[2][MagmaMaxGPUs], *hwQ2[MagmaMaxGPUs];
#define hQ2(id) (hwQ2[id])
#define hS(id, ii) (hwS[ii][id])
#define hQ(id, ii) (hwQ[ii][id])
#endif
n2 = n - n1;
n12 = ctot[0] + ctot[1];
n23 = ctot[1] + ctot[2];
iq2 = n1 * n12;
//lq2 = iq2 + n2 * n23;
n1_loc = (n1-1) / (ngpu/2) + 1;
n2_loc = (n2-1) / (ngpu/2) + 1;
nb = magma_get_slaex3_m_nb();
if (n1 >= magma_get_slaex3_m_k()) {
#ifdef CHECK_CPU
for (igpu = 0; igpu < ngpu; ++igpu) {
magma_smalloc_pinned( &(hwS[0][igpu]), n2*nb );
magma_smalloc_pinned( &(hwS[1][igpu]), n2*nb );
magma_smalloc_pinned( &(hwQ2[igpu]), n2*n2_loc );
magma_smalloc_pinned( &(hwQ[0][igpu]), n2_loc*nb );
magma_smalloc_pinned( &(hwQ[1][igpu]), n2_loc*nb );
}
#endif
for (igpu = 0; igpu < ngpu-1; igpu += 2) {
ni_loc[igpu] = min(n1_loc, n1 - igpu/2 * n1_loc);
#ifdef CHECK_CPU
lapackf77_slacpy("A", &ni_loc[igpu], &n12, Q2+n1_loc*(igpu/2), &n1, hQ2(igpu), &n1_loc);
#endif
magma_setdevice(igpu);
magma_ssetmatrix_async( ni_loc[igpu], n12,
Q2+n1_loc*(igpu/2), n1,
dQ2(igpu), n1_loc, queues[igpu][0] );
ni_loc[igpu+1] = min(n2_loc, n2 - igpu/2 * n2_loc);
#ifdef CHECK_CPU
lapackf77_slacpy("A", &ni_loc[igpu+1], &n23, Q2+iq2+n2_loc*(igpu/2), &n2, hQ2(igpu+1), &n2_loc);
#endif
magma_setdevice(igpu+1);
magma_ssetmatrix_async( ni_loc[igpu+1], n23,
Q2+iq2+n2_loc*(igpu/2), n2,
dQ2(igpu+1), n2_loc, queues[igpu+1][0] );
}
}
//
#ifdef _OPENMP
/////////////////////////////////////////////////////////////////////////////////
//openmp implementation
/////////////////////////////////////////////////////////////////////////////////
magma_timer_t time=0;
timer_start( time );
#pragma omp parallel private(i, j, tmp, temp)
{
magma_int_t id = omp_get_thread_num();
magma_int_t tot = omp_get_num_threads();
magma_int_t ib = ( id * k) / tot; //start index of local loop
magma_int_t ie = ((id+1) * k) / tot; //end index of local loop
magma_int_t ik = ie - ib; //number of local indices
for (i = ib; i < ie; ++i)
dlamda[i]=lapackf77_slamc3(&dlamda[i], &dlamda[i]) - dlamda[i];
for (j = ib; j < ie; ++j) {
magma_int_t tmpp=j+1;
magma_int_t iinfo = 0;
lapackf77_slaed4(&k, &tmpp, dlamda, w, Q(0,j), &rho, &d[j], &iinfo);
// If the zero finder fails, the computation is terminated.
if (iinfo != 0) {
#pragma omp critical (info)
*info = iinfo;
break;
}
}
#pragma omp barrier
if (*info == 0) {
#pragma omp single
{
//Prepare the INDXQ sorting permutation.
magma_int_t nk = n - k;
lapackf77_slamrg( &k, &nk, d, &ione, &ineg_one, indxq);
//compute the lower and upper bound of the non-deflated eigenvectors
if (valeig)
magma_svrange(k, d, &iil, &iiu, vl, vu);
else if (indeig)
magma_sirange(k, indxq, &iil, &iiu, il, iu);
else {
iil = 1;
iiu = k;
}
rk = iiu - iil + 1;
}
if (k == 2) {
#pragma omp single
{
for (j = 0; j < k; ++j) {
w[0] = *Q(0,j);
w[1] = *Q(1,j);
i = indx[0] - 1;
*Q(0,j) = w[i];
i = indx[1] - 1;
*Q(1,j) = w[i];
}
}
}
else if (k != 1) {
// Compute updated W.
blasf77_scopy( &ik, &w[ib], &ione, &s[ib], &ione);
// Initialize W(I) = Q(I,I)
tmp = ldq + 1;
blasf77_scopy( &ik, Q(ib,ib), &tmp, &w[ib], &ione);
for (j = 0; j < k; ++j) {
magma_int_t i_tmp = min(j, ie);
for (i = ib; i < i_tmp; ++i)
w[i] = w[i] * ( *Q(i, j) / ( dlamda[i] - dlamda[j] ) );
i_tmp = max(j+1, ib);
for (i = i_tmp; i < ie; ++i)
w[i] = w[i] * ( *Q(i, j) / ( dlamda[i] - dlamda[j] ) );
}
for (i = ib; i < ie; ++i)
w[i] = copysign( sqrt( -w[i] ), s[i]);
#pragma omp barrier
//reduce the number of used threads to have enough S workspace
tot = min(n1, omp_get_num_threads());
if (id < tot) {
ib = ( id * rk) / tot + iil - 1;
ie = ((id+1) * rk) / tot + iil - 1;
ik = ie - ib;
}
else {
ib = -1;
ie = -1;
ik = -1;
}
// Compute eigenvectors of the modified rank-1 modification.
for (j = ib; j < ie; ++j) {
for (i = 0; i < k; ++i)
s[id*k + i] = w[i] / *Q(i,j);
temp = magma_cblas_snrm2( k, s+id*k, 1 );
for (i = 0; i < k; ++i) {
magma_int_t iii = indx[i] - 1;
*Q(i,j) = s[id*k + iii] / temp;
}
}
}
}
}
if (*info != 0)
return *info;
timer_stop( time );
timer_printf( "eigenvalues/vector D+zzT = %6.2f\n", time );
#else
/////////////////////////////////////////////////////////////////////////////////
// Non openmp implementation
/////////////////////////////////////////////////////////////////////////////////
magma_timer_t time=0;
timer_start( time );
for (i = 0; i < k; ++i)
dlamda[i]=lapackf77_slamc3(&dlamda[i], &dlamda[i]) - dlamda[i];
for (j = 0; j < k; ++j) {
magma_int_t tmpp=j+1;
magma_int_t iinfo = 0;
lapackf77_slaed4(&k, &tmpp, dlamda, w, Q(0,j), &rho, &d[j], &iinfo);
// If the zero finder fails, the computation is terminated.
if (iinfo != 0)
*info=iinfo;
}
if (*info != 0)
return *info;
//Prepare the INDXQ sorting permutation.
magma_int_t nk = n - k;
lapackf77_slamrg( &k, &nk, d, &ione, &ineg_one, indxq);
//compute the lower and upper bound of the non-deflated eigenvectors
if (valeig)
magma_svrange(k, d, &iil, &iiu, vl, vu);
else if (indeig)
magma_sirange(k, indxq, &iil, &iiu, il, iu);
else {
iil = 1;
iiu = k;
}
rk = iiu - iil + 1;
if (k == 2) {
for (j = 0; j < k; ++j) {
w[0] = *Q(0,j);
w[1] = *Q(1,j);
i = indx[0] - 1;
*Q(0,j) = w[i];
i = indx[1] - 1;
*Q(1,j) = w[i];
}
}
else if (k != 1) {
// Compute updated W.
blasf77_scopy( &k, w, &ione, s, &ione);
// Initialize W(I) = Q(I,I)
tmp = ldq + 1;
blasf77_scopy( &k, Q, &tmp, w, &ione);
for (j = 0; j < k; ++j) {
for (i = 0; i < j; ++i)
w[i] = w[i] * ( *Q(i, j) / ( dlamda[i] - dlamda[j] ) );
for (i = j+1; i < k; ++i)
w[i] = w[i] * ( *Q(i, j) / ( dlamda[i] - dlamda[j] ) );
}
for (i = 0; i < k; ++i)
w[i] = copysign( sqrt( -w[i] ), s[i]);
// Compute eigenvectors of the modified rank-1 modification.
for (j = iil-1; j < iiu; ++j) {
for (i = 0; i < k; ++i)
s[i] = w[i] / *Q(i,j);
temp = magma_cblas_snrm2( k, s, 1 );
for (i = 0; i < k; ++i) {
magma_int_t iii = indx[i] - 1;
*Q(i,j) = s[iii] / temp;
}
}
}
timer_stop( time );
timer_printf( "eigenvalues/vector D+zzT = %6.2f\n", time );
#endif //_OPENMP
// Compute the updated eigenvectors.
timer_start( time );
if (rk > 0) {
if (n1 < magma_get_slaex3_m_k()) {
// stay on the CPU
if ( n23 != 0 ) {
lapackf77_slacpy("A", &n23, &rk, Q(ctot[0],iil-1), &ldq, s, &n23);
blasf77_sgemm("N", "N", &n2, &rk, &n23, &d_one, &Q2[iq2], &n2,
s, &n23, &d_zero, Q(n1,iil-1), &ldq );
}
else
lapackf77_slaset("A", &n2, &rk, &d_zero, &d_zero, Q(n1,iil-1), &ldq);
if ( n12 != 0 ) {
lapackf77_slacpy("A", &n12, &rk, Q(0,iil-1), &ldq, s, &n12);
blasf77_sgemm("N", "N", &n1, &rk, &n12, &d_one, Q2, &n1,
s, &n12, &d_zero, Q(0,iil-1), &ldq);
}
else
lapackf77_slaset("A", &n1, &rk, &d_zero, &d_zero, Q(0,iil-1), &ldq);
}
else {
//use the gpus
ib = min(nb, rk);
for (igpu = 0; igpu < ngpu-1; igpu += 2) {
if (n23 != 0) {
magma_setdevice(igpu+1);
magma_ssetmatrix_async( n23, ib,
Q(ctot[0],iil-1), ldq,
dS(igpu+1,0), n23, queues[igpu+1][0] );
}
if (n12 != 0) {
magma_setdevice(igpu);
magma_ssetmatrix_async( n12, ib,
Q(0,iil-1), ldq,
dS(igpu,0), n12, queues[igpu][0] );
}
}
for (i = 0; i < rk; i += nb) {
ib = min(nb, rk - i);
ind = (i/nb)%2;
if (i+nb < rk) {
ib2 = min(nb, rk - i - nb);
for (igpu = 0; igpu < ngpu-1; igpu += 2) {
if (n23 != 0) {
magma_setdevice(igpu+1);
magma_ssetmatrix_async( n23, ib2,
Q(ctot[0],iil-1+i+nb), ldq,
dS(igpu+1,(ind+1)%2), n23, queues[igpu+1][(ind+1)%2] );
}
if (n12 != 0) {
magma_setdevice(igpu);
magma_ssetmatrix_async( n12, ib2,
Q(0,iil-1+i+nb), ldq,
dS(igpu,(ind+1)%2), n12, queues[igpu][(ind+1)%2] );
}
}
}
// Ensure that the data is copied on gpu since we will overwrite it.
for (igpu = 0; igpu < ngpu-1; igpu += 2) {
if (n23 != 0) {
#ifdef CHECK_CPU
lapackf77_slacpy("A", &n23, &ib, Q(ctot[0],iil-1+i), &ldq, hS(igpu+1,ind), &n23);
#endif
magma_setdevice(igpu+1);
magma_queue_sync( queues[igpu+1][ind] );
}
if (n12 != 0) {
#ifdef CHECK_CPU
lapackf77_slacpy("A", &n12, &ib, Q(0,iil-1+i), &ldq, hS(igpu,ind), &n12);
#endif
magma_setdevice(igpu);
magma_queue_sync( queues[igpu][ind] );
}
}
for (igpu = 0; igpu < ngpu-1; igpu += 2) {
if (n23 != 0) {
#ifdef CHECK_CPU
blasf77_sgemm("N", "N", &ni_loc[igpu+1], &ib, &n23, &d_one, hQ2(igpu+1), &n2_loc,
hS(igpu+1,ind), &n23, &d_zero, hQ(igpu+1, ind), &n2_loc);
#endif
magma_setdevice(igpu+1);
magmablasSetKernelStream(queues[igpu+1][ind]);
magma_sgemm(MagmaNoTrans, MagmaNoTrans, ni_loc[igpu+1], ib, n23, d_one, dQ2(igpu+1), n2_loc,
dS(igpu+1, ind), n23, d_zero, dQ(igpu+1, ind), n2_loc);
#ifdef CHECK_CPU
printf("norm Q %d: %f\n", igpu+1, cpu_gpu_sdiff(ni_loc[igpu+1], ib, hQ(igpu+1, ind), n2_loc, dQ(igpu+1, ind), n2_loc));
#endif
}
if (n12 != 0) {
#ifdef CHECK_CPU
blasf77_sgemm("N", "N", &ni_loc[igpu], &ib, &n12, &d_one, hQ2(igpu), &n1_loc,
hS(igpu,ind%2), &n12, &d_zero, hQ(igpu, ind%2), &n1_loc);
#endif
magma_setdevice(igpu);
magmablasSetKernelStream(queues[igpu][ind]);
magma_sgemm(MagmaNoTrans, MagmaNoTrans, ni_loc[igpu], ib, n12, d_one, dQ2(igpu), n1_loc,
dS(igpu, ind), n12, d_zero, dQ(igpu, ind), n1_loc);
#ifdef CHECK_CPU
printf("norm Q %d: %f\n", igpu, cpu_gpu_sdiff(ni_loc[igpu], ib, hQ(igpu, ind), n1_loc, dQ(igpu, ind), n1_loc));
#endif
}
}
for (igpu = 0; igpu < ngpu-1; igpu += 2) {
if (n23 != 0) {
magma_setdevice(igpu+1);
magma_sgetmatrix( ni_loc[igpu+1], ib, dQ(igpu+1, ind), n2_loc,
Q(n1+n2_loc*(igpu/2),iil-1+i), ldq );
// magma_sgetmatrix_async( ni_loc[igpu+1], ib, dQ(igpu+1, ind), n2_loc,
// Q(n1+n2_loc*(igpu/2),iil-1+i), ldq, queues[igpu+1][ind] );
}
if (n12 != 0) {
magma_setdevice(igpu);
magma_sgetmatrix( ni_loc[igpu], ib, dQ(igpu, ind), n1_loc,
Q(n1_loc*(igpu/2),iil-1+i), ldq );
// magma_sgetmatrix_async( ni_loc[igpu], ib, dQ(igpu, ind), n1_loc,
// Q(n1_loc*(igpu/2),iil-1+i), ldq, queues[igpu][ind] );
}
}
}
for (igpu = 0; igpu < ngpu; ++igpu) {
#ifdef CHECK_CPU
magma_free_pinned( hwS[1][igpu] );
magma_free_pinned( hwS[0][igpu] );
magma_free_pinned( hwQ2[igpu] );
magma_free_pinned( hwQ[1][igpu] );
magma_free_pinned( hwQ[0][igpu] );
#endif
magma_setdevice(igpu);
magma_queue_sync( queues[igpu][0] );
magma_queue_sync( queues[igpu][1] );
}
if ( n23 == 0 )
lapackf77_slaset("A", &n2, &rk, &d_zero, &d_zero, Q(n1,iil-1), &ldq);
if ( n12 == 0 )
lapackf77_slaset("A", &n1, &rk, &d_zero, &d_zero, Q(0,iil-1), &ldq);
}
}
timer_stop( time );
timer_printf( "gemms = %6.2f\n", time );
magma_setdevice( orig_dev );
magmablasSetKernelStream( orig_stream );
return *info;
} /* magma_slaed3_m */
/**
Purpose
-------
SGEEV computes for an N-by-N real nonsymmetric matrix A, the
eigenvalues and, optionally, the left and/or right eigenvectors.
The right eigenvector v(j) of A satisfies
A * v(j) = lambda(j) * v(j)
where lambda(j) is its eigenvalue.
The left eigenvector u(j) of A satisfies
u(j)**T * A = lambda(j) * u(j)**T
where u(j)**T denotes the transpose of u(j).
The computed eigenvectors are normalized to have Euclidean norm
equal to 1 and largest component real.
Arguments
---------
@param[in]
jobvl magma_vec_t
- = MagmaNoVec: left eigenvectors of A are not computed;
- = MagmaVec: left eigenvectors of are computed.
@param[in]
jobvr magma_vec_t
- = MagmaNoVec: right eigenvectors of A are not computed;
- = MagmaVec: right eigenvectors of A are computed.
@param[in]
n INTEGER
The order of the matrix A. N >= 0.
@param[in,out]
A REAL array, dimension (LDA,N)
On entry, the N-by-N matrix A.
On exit, A has been overwritten.
@param[in]
lda INTEGER
The leading dimension of the array A. LDA >= max(1,N).
@param[out]
wr REAL array, dimension (N)
@param[out]
wi REAL array, dimension (N)
WR and WI contain the real and imaginary parts,
respectively, of the computed eigenvalues. Complex
conjugate pairs of eigenvalues appear consecutively
with the eigenvalue having the positive imaginary part
first.
@param[out]
VL REAL 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 REAL 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) REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
@param[in]
lwork INTEGER
The dimension of the array WORK. LWORK >= (2 + nb + nb*ngpu)*N.
For optimal performance, LWORK >= (2 + 2*nb + nb*ngpu)*N.
\n
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
@param[out]
info INTEGER
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value.
- > 0: if INFO = i, the QR algorithm failed to compute all the
eigenvalues, and no eigenvectors have been computed;
elements and i+1:N of W contain eigenvalues which have
converged.
@ingroup magma_sgeev_driver
********************************************************************/
extern "C" magma_int_t
magma_sgeev_m(
magma_vec_t jobvl, magma_vec_t jobvr, magma_int_t n,
float *A, magma_int_t lda,
#ifdef COMPLEX
float *w,
#else
float *wr, float *wi,
#endif
float *VL, magma_int_t ldvl,
float *VR, magma_int_t ldvr,
float *work, magma_int_t lwork,
#ifdef COMPLEX
float *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;
float d__1, d__2;
float r, cs, sn, 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, optwrk, lquery, wantvl, wantvr, select[1];
magma_side_t side = MagmaRight;
magma_int_t ngpu = magma_num_gpus();
magma_timer_t time_total=0, time_gehrd=0, time_unghr=0, time_hseqr=0, time_trevc=0, time_sum=0;
magma_flops_t flop_total=0, flop_gehrd=0, flop_unghr=0, flop_hseqr=0, flop_trevc=0, flop_sum=0;
timer_start( time_total );
flops_start( flop_total );
*info = 0;
lquery = (lwork == -1);
wantvl = (jobvl == MagmaVec);
wantvr = (jobvr == MagmaVec);
if (! wantvl && jobvl != MagmaNoVec) {
*info = -1;
} else if (! wantvr && jobvr != MagmaNoVec) {
*info = -2;
} else if (n < 0) {
*info = -3;
} else if (lda < max(1,n)) {
*info = -5;
} else if ( (ldvl < 1) || (wantvl && (ldvl < n))) {
*info = -9;
} else if ( (ldvr < 1) || (wantvr && (ldvr < n))) {
*info = -11;
}
/* Compute workspace */
nb = magma_get_sgehrd_nb( n );
if (*info == 0) {
minwrk = (2 + nb + nb*ngpu)*n;
optwrk = (2 + 2*nb + nb*ngpu)*n;
work[0] = magma_smake_lwork( optwrk );
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)
float *dT;
if (MAGMA_SUCCESS != magma_smalloc( &dT, nb*n )) {
*info = MAGMA_ERR_DEVICE_ALLOC;
return *info;
}
#endif
#if defined(Version5)
float *T;
if (MAGMA_SUCCESS != magma_smalloc_cpu( &T, nb*n )) {
*info = MAGMA_ERR_HOST_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_slange( "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_slascl( "G", &izero, &izero, &anrm, &cscale, &n, &n, A, &lda, &ierr );
}
/* Balance the matrix
* (Workspace: need N)
* - this space is reserved until after gebak */
ibal = 0;
lapackf77_sgebal( "B", &n, A, &lda, &ilo, &ihi, &work[ibal], &ierr );
/* Reduce to upper Hessenberg form
* (Workspace: need 3*N, prefer 2*N + N*NB + NB*NGPU)
* - added NB*NGPU needed for multi-GPU magma_sgehrd_m
* - including N reserved for gebal/gebak, unused by sgehrd */
itau = ibal + n;
iwrk = itau + n;
liwrk = lwork - iwrk;
timer_start( time_gehrd );
flops_start( flop_gehrd );
#if defined(Version1)
// Version 1 - LAPACK
lapackf77_sgehrd( &n, &ilo, &ihi, A, &lda,
&work[itau], &work[iwrk], &liwrk, &ierr );
#elif defined(Version2)
// Version 2 - LAPACK consistent HRD
magma_sgehrd2( n, ilo, ihi, A, lda,
&work[itau], &work[iwrk], liwrk, &ierr );
#elif defined(Version3)
// Version 3 - LAPACK consistent MAGMA HRD + T matrices stored,
magma_sgehrd( n, ilo, ihi, A, lda,
&work[itau], &work[iwrk], liwrk, dT, &ierr );
#elif defined(Version5)
// Version 4 - Multi-GPU, T on host
magma_sgehrd_m( n, ilo, ihi, A, lda,
&work[itau], &work[iwrk], liwrk, T, &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_slacpy( MagmaLowerStr, &n, &n, A, &lda, VL, &ldvl );
/* Generate orthogonal matrix in VL
* (Workspace: need 3*N-1, prefer 2*N + (N-1)*NB)
* - including N reserved for gebal/gebak, unused by sorghr */
timer_start( time_unghr );
flops_start( flop_unghr );
#if defined(Version1) || defined(Version2)
// Version 1 & 2 - LAPACK
lapackf77_sorghr( &n, &ilo, &ihi, VL, &ldvl, &work[itau],
&work[iwrk], &liwrk, &ierr );
#elif defined(Version3)
// Version 3 - LAPACK consistent MAGMA HRD + T matrices stored
magma_sorghr( n, ilo, ihi, VL, ldvl, &work[itau], dT, nb, &ierr );
#elif defined(Version5)
// Version 5 - Multi-GPU, T on host
magma_sorghr_m( n, ilo, ihi, VL, ldvl, &work[itau], T, nb, &ierr );
#endif
time_sum += timer_stop( time_unghr );
flop_sum += flops_stop( flop_unghr );
timer_start( time_hseqr );
flops_start( flop_hseqr );
/* Perform QR iteration, accumulating Schur vectors in VL
* (Workspace: need N+1, prefer N+HSWORK (see comments) )
* - including N reserved for gebal/gebak, unused by shseqr */
iwrk = itau;
liwrk = lwork - iwrk;
lapackf77_shseqr( "S", "V", &n, &ilo, &ihi, A, &lda, wr, wi,
VL, &ldvl, &work[iwrk], &liwrk, info );
time_sum += timer_stop( time_hseqr );
flop_sum += flops_stop( flop_hseqr );
if (wantvr) {
/* Want left and right eigenvectors
* Copy Schur vectors to VR */
side = MagmaBothSides;
lapackf77_slacpy( "F", &n, &n, VL, &ldvl, VR, &ldvr );
}
}
else if (wantvr) {
/* Want right eigenvectors
* Copy Householder vectors to VR */
side = MagmaRight;
lapackf77_slacpy( "L", &n, &n, A, &lda, VR, &ldvr );
/* Generate orthogonal matrix in VR
* (Workspace: need 3*N-1, prefer 2*N + (N-1)*NB)
* - including N reserved for gebal/gebak, unused by sorghr */
timer_start( time_unghr );
flops_start( flop_unghr );
#if defined(Version1) || defined(Version2)
// Version 1 & 2 - LAPACK
lapackf77_sorghr( &n, &ilo, &ihi, VR, &ldvr, &work[itau],
&work[iwrk], &liwrk, &ierr );
#elif defined(Version3)
// Version 3 - LAPACK consistent MAGMA HRD + T matrices stored
magma_sorghr( n, ilo, ihi, VR, ldvr, &work[itau], dT, nb, &ierr );
#elif defined(Version5)
// Version 5 - Multi-GPU, T on host
magma_sorghr_m( n, ilo, ihi, VR, ldvr, &work[itau], T, nb, &ierr );
#endif
time_sum += timer_stop( time_unghr );
flop_sum += flops_stop( flop_unghr );
/* Perform QR iteration, accumulating Schur vectors in VR
* (Workspace: need N+1, prefer N+HSWORK (see comments) )
* - including N reserved for gebal/gebak, unused by shseqr */
timer_start( time_hseqr );
flops_start( flop_hseqr );
iwrk = itau;
liwrk = lwork - iwrk;
lapackf77_shseqr( "S", "V", &n, &ilo, &ihi, A, &lda, wr, wi,
VR, &ldvr, &work[iwrk], &liwrk, info );
time_sum += timer_stop( time_hseqr );
flop_sum += flops_stop( flop_hseqr );
}
else {
/* Compute eigenvalues only
* (Workspace: need N+1, prefer N+HSWORK (see comments) )
* - including N reserved for gebal/gebak, unused by shseqr */
timer_start( time_hseqr );
flops_start( flop_hseqr );
iwrk = itau;
liwrk = lwork - iwrk;
lapackf77_shseqr( "E", "N", &n, &ilo, &ihi, A, &lda, wr, wi,
VR, &ldvr, &work[iwrk], &liwrk, info );
time_sum += timer_stop( time_hseqr );
flop_sum += flops_stop( flop_hseqr );
}
/* If INFO > 0 from SHSEQR, then quit */
if (*info > 0) {
goto CLEANUP;
}
timer_start( time_trevc );
flops_start( flop_trevc );
if (wantvl || wantvr) {
/* Compute left and/or right eigenvectors
* (Workspace: need 4*N, prefer (2 + 2*nb)*N)
* - including N reserved for gebal/gebak, unused by strevc */
liwrk = lwork - iwrk;
#if TREVC_VERSION == 1
lapackf77_strevc( lapack_side_const(side), "B", select, &n, A, &lda, VL, &ldvl,
VR, &ldvr, &n, &nout, &work[iwrk], &ierr );
#elif TREVC_VERSION == 2
lapackf77_strevc3( lapack_side_const(side), "B", select, &n, A, &lda, VL, &ldvl,
VR, &ldvr, &n, &nout, &work[iwrk], &liwrk, &ierr );
#elif TREVC_VERSION == 3
magma_strevc3( side, MagmaBacktransVec, select, n, A, lda, VL, ldvl,
VR, ldvr, n, &nout, &work[iwrk], liwrk, &ierr );
#elif TREVC_VERSION == 4
magma_strevc3_mt( side, MagmaBacktransVec, select, n, A, lda, VL, ldvl,
VR, ldvr, n, &nout, &work[iwrk], liwrk, &ierr );
#elif TREVC_VERSION == 5
magma_strevc3_mt_gpu( side, MagmaBacktransVec, select, n, A, lda, VL, ldvl,
VR, ldvr, n, &nout, &work[iwrk], liwrk, &ierr );
#else
#error Unknown TREVC_VERSION
#endif
}
time_sum += timer_stop( time_trevc );
flop_sum += flops_stop( flop_trevc );
if (wantvl) {
/* Undo balancing of left eigenvectors
* (Workspace: need N) */
lapackf77_sgebak( "B", "L", &n, &ilo, &ihi, &work[ibal], &n,
VL, &ldvl, &ierr );
/* Normalize left eigenvectors and make largest component real */
for (i = 0; i < n; ++i) {
if ( wi[i] == 0. ) {
scl = 1. / magma_cblas_snrm2( n, VL(0,i), 1 );
blasf77_sscal( &n, &scl, VL(0,i), &ione );
}
else if ( wi[i] > 0. ) {
d__1 = magma_cblas_snrm2( n, VL(0,i), 1 );
d__2 = magma_cblas_snrm2( n, VL(0,i+1), 1 );
scl = 1. / lapackf77_slapy2( &d__1, &d__2 );
blasf77_sscal( &n, &scl, VL(0,i), &ione );
blasf77_sscal( &n, &scl, VL(0,i+1), &ione );
for (k = 0; k < n; ++k) {
/* Computing 2nd power */
d__1 = *VL(k,i);
d__2 = *VL(k,i+1);
work[iwrk + k] = d__1*d__1 + d__2*d__2;
}
k = blasf77_isamax( &n, &work[iwrk], &ione ) - 1; // subtract 1; k is 0-based
lapackf77_slartg( VL(k,i), VL(k,i+1), &cs, &sn, &r );
blasf77_srot( &n, VL(0,i), &ione, VL(0,i+1), &ione, &cs, &sn );
*VL(k,i+1) = 0.;
}
}
}
if (wantvr) {
/* Undo balancing of right eigenvectors
* (Workspace: need N) */
lapackf77_sgebak( "B", "R", &n, &ilo, &ihi, &work[ibal], &n,
VR, &ldvr, &ierr );
/* Normalize right eigenvectors and make largest component real */
for (i = 0; i < n; ++i) {
if ( wi[i] == 0. ) {
scl = 1. / magma_cblas_snrm2( n, VR(0,i), 1 );
blasf77_sscal( &n, &scl, VR(0,i), &ione );
}
else if ( wi[i] > 0. ) {
d__1 = magma_cblas_snrm2( n, VR(0,i), 1 );
d__2 = magma_cblas_snrm2( n, VR(0,i+1), 1 );
scl = 1. / lapackf77_slapy2( &d__1, &d__2 );
blasf77_sscal( &n, &scl, VR(0,i), &ione );
blasf77_sscal( &n, &scl, VR(0,i+1), &ione );
for (k = 0; k < n; ++k) {
/* Computing 2nd power */
d__1 = *VR(k,i);
d__2 = *VR(k,i+1);
work[iwrk + k] = d__1*d__1 + d__2*d__2;
}
k = blasf77_isamax( &n, &work[iwrk], &ione ) - 1; // subtract 1; k is 0-based
lapackf77_slartg( VR(k,i), VR(k,i+1), &cs, &sn, &r );
blasf77_srot( &n, VR(0,i), &ione, VR(0,i+1), &ione, &cs, &sn );
*VR(k,i+1) = 0.;
}
}
}
CLEANUP:
/* Undo scaling if necessary */
if (scalea) {
// converged eigenvalues, stored in wr[i+1:n] and wi[i+1:n] for i = INFO
magma_int_t nval = n - (*info);
magma_int_t ld = max( nval, 1 );
lapackf77_slascl( "G", &izero, &izero, &cscale, &anrm, &nval, &ione, wr + (*info), &ld, &ierr );
lapackf77_slascl( "G", &izero, &izero, &cscale, &anrm, &nval, &ione, wi + (*info), &ld, &ierr );
if (*info > 0) {
// first ilo columns were already upper triangular,
// so the corresponding eigenvalues are also valid.
nval = ilo - 1;
lapackf77_slascl( "G", &izero, &izero, &cscale, &anrm, &nval, &ione, wr, &n, &ierr );
lapackf77_slascl( "G", &izero, &izero, &cscale, &anrm, &nval, &ione, wi, &n, &ierr );
}
}
#if defined(Version3)
magma_free( dT );
#endif
#if defined(Version5)
magma_free_cpu( T );
#endif
timer_stop( time_total );
flops_stop( flop_total );
timer_printf( "sgeev 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( "sgeev 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_smake_lwork( optwrk );
return *info;
} /* magma_sgeev */
/**
Purpose
-------
SLAEX3 finds the roots of the secular equation, as defined by the
values in D, W, and RHO, between 1 and K. It makes the
appropriate calls to SLAED4 and then updates the eigenvectors by
multiplying the matrix of eigenvectors of the pair of eigensystems
being combined by the matrix of eigenvectors of the K-by-K system
which is solved here.
It is used in the last step when only a part of the eigenvectors
is required.
It compute only the required part of the eigenvectors and the rest
is not used.
This code makes very mild assumptions about floating point
arithmetic. It will work on machines with a guard digit in
add/subtract, or on those binary machines without guard digits
which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
It could conceivably fail on hexadecimal or decimal machines
without guard digits, but we know of none.
Arguments
---------
@param[in]
k INTEGER
The number of terms in the rational function to be solved by
SLAED4. K >= 0.
@param[in]
n INTEGER
The number of rows and columns in the Q matrix.
N >= K (deflation may result in N > K).
@param[in]
n1 INTEGER
The location of the last eigenvalue in the leading submatrix.
min(1,N) <= N1 <= N/2.
@param[out]
d REAL array, dimension (N)
D(I) contains the updated eigenvalues for
1 <= I <= K.
@param[out]
Q REAL array, dimension (LDQ,N)
Initially the first K columns are used as workspace.
On output the columns ??? to ??? contain
the updated eigenvectors.
@param[in]
ldq INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
@param[in]
rho REAL
The value of the parameter in the rank one update equation.
RHO >= 0 required.
@param[in,out]
dlamda REAL array, dimension (K)
The first K elements of this array contain the old roots
of the deflated updating problem. These are the poles
of the secular equation. May be changed on output by
having lowest order bit set to zero on Cray X-MP, Cray Y-MP,
Cray-2, or Cray C-90, as described above.
@param[in]
Q2 REAL array, dimension (LDQ2, N)
The first K columns of this matrix contain the non-deflated
eigenvectors for the split problem.
TODO what is LDQ2?
@param[in]
indx INTEGER array, dimension (N)
The permutation used to arrange the columns of the deflated
Q matrix into three groups (see SLAED2).
The rows of the eigenvectors found by SLAED4 must be likewise
permuted before the matrix multiply can take place.
@param[in]
ctot INTEGER array, dimension (4)
A count of the total number of the various types of columns
in Q, as described in INDX. The fourth column type is any
column which has been deflated.
@param[in,out]
w REAL array, dimension (K)
The first K elements of this array contain the components
of the deflation-adjusted updating vector. Destroyed on
output.
@param
s (workspace) REAL array, dimension (N1 + 1)*K
Will contain the eigenvectors of the repaired matrix which
will be multiplied by the previously accumulated eigenvectors
to update the system.
@param[out]
indxq INTEGER array, dimension (N)
On exit, the permutation which will reintegrate the
subproblems back into sorted order,
i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.
@param
dwork (workspace) REAL array, dimension (3*N*N/2+3*N)
@param[in]
range magma_range_t
- = MagmaRangeAll: all eigenvalues will be found.
- = MagmaRangeV: all eigenvalues in the half-open interval (VL,VU]
will be found.
- = MagmaRangeI: the IL-th through IU-th eigenvalues will be found.
TODO verify range, vl, vu, il, iu -- copied from slaex1.
@param[in]
vl REAL
@param[in]
vu REAL
if RANGE=MagmaRangeV, the lower and upper bounds of the interval to
be searched for eigenvalues. VL < VU.
Not referenced if RANGE = MagmaRangeAll or MagmaRangeI.
@param[in]
il INTEGER
@param[in]
iu INTEGER
if RANGE=MagmaRangeI, the indices (in ascending order) of the
smallest and largest eigenvalues to be returned.
1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
Not referenced if RANGE = MagmaRangeAll or MagmaRangeV.
@param[out]
info INTEGER
- = 0: successful exit.
- < 0: if INFO = -i, the i-th argument had an illegal value.
- > 0: if INFO = 1, an eigenvalue did not converge
Further Details
---------------
Based on contributions by
Jeff Rutter, Computer Science Division, University of California
at Berkeley, USA
Modified by Francoise Tisseur, University of Tennessee.
@ingroup magma_ssyev_aux
********************************************************************/
extern "C" magma_int_t
magma_slaex3(magma_int_t k, magma_int_t n, magma_int_t n1, float* d,
float* Q, magma_int_t ldq, float rho,
float* dlamda, float* Q2, magma_int_t* indx,
magma_int_t* ctot, float* w, float* s, magma_int_t* indxq,
float* dwork,
magma_range_t range, float vl, float vu, magma_int_t il, magma_int_t iu,
magma_int_t* info )
{
#define Q(i_,j_) (Q + (i_) + (j_)*ldq)
float d_one = 1.;
float d_zero = 0.;
magma_int_t ione = 1;
magma_int_t ineg_one = -1;
magma_int_t iil, iiu, rk;
float* dq2= dwork;
float* ds = dq2 + n*(n/2+1);
float* dq = ds + n*(n/2+1);
magma_int_t lddq = n/2 + 1;
magma_int_t i, iq2, j, n12, n2, n23, tmp, lq2;
float temp;
magma_int_t alleig, valeig, indeig;
alleig = (range == MagmaRangeAll);
valeig = (range == MagmaRangeV);
indeig = (range == MagmaRangeI);
*info = 0;
if (k < 0)
*info=-1;
else if (n < k)
*info=-2;
else if (ldq < max(1,n))
*info=-6;
else if (! (alleig || valeig || indeig))
*info = -15;
else {
if (valeig) {
if (n > 0 && vu <= vl)
*info = -17;
}
else if (indeig) {
if (il < 1 || il > max(1,n))
*info = -18;
else if (iu < min(n,il) || iu > n)
*info = -19;
}
}
if (*info != 0) {
magma_xerbla(__func__, -(*info));
return *info;
}
// Quick return if possible
if (k == 0)
return *info;
/*
Modify values DLAMDA(i) to make sure all DLAMDA(i)-DLAMDA(j) can
be computed with high relative accuracy (barring over/underflow).
This is a problem on machines without a guard digit in
add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2).
The following code replaces DLAMDA(I) by 2*DLAMDA(I)-DLAMDA(I),
which on any of these machines zeros out the bottommost
bit of DLAMDA(I) if it is 1; this makes the subsequent
subtractions DLAMDA(I)-DLAMDA(J) unproblematic when cancellation
occurs. On binary machines with a guard digit (almost all
machines) it does not change DLAMDA(I) at all. On hexadecimal
and decimal machines with a guard digit, it slightly
changes the bottommost bits of DLAMDA(I). It does not account
for hexadecimal or decimal machines without guard digits
(we know of none). We use a subroutine call to compute
2*DLAMBDA(I) to prevent optimizing compilers from eliminating
this code.*/
n2 = n - n1;
n12 = ctot[0] + ctot[1];
n23 = ctot[1] + ctot[2];
iq2 = n1 * n12;
lq2 = iq2 + n2 * n23;
magma_ssetvector_async( lq2, Q2, 1, dq2, 1, NULL );
#ifdef _OPENMP
/////////////////////////////////////////////////////////////////////////////////
//openmp implementation
/////////////////////////////////////////////////////////////////////////////////
magma_timer_t time=0;
timer_start( time );
#pragma omp parallel private(i, j, tmp, temp)
{
magma_int_t id = omp_get_thread_num();
magma_int_t tot = omp_get_num_threads();
magma_int_t ib = ( id * k) / tot; //start index of local loop
magma_int_t ie = ((id+1) * k) / tot; //end index of local loop
magma_int_t ik = ie - ib; //number of local indices
for (i = ib; i < ie; ++i)
dlamda[i]=lapackf77_slamc3(&dlamda[i], &dlamda[i]) - dlamda[i];
for (j = ib; j < ie; ++j) {
magma_int_t tmpp=j+1;
magma_int_t iinfo = 0;
lapackf77_slaed4(&k, &tmpp, dlamda, w, Q(0,j), &rho, &d[j], &iinfo);
// If the zero finder fails, the computation is terminated.
if (iinfo != 0) {
#pragma omp critical (info)
*info=iinfo;
break;
}
}
#pragma omp barrier
if (*info == 0) {
#pragma omp single
{
//Prepare the INDXQ sorting permutation.
magma_int_t nk = n - k;
lapackf77_slamrg( &k, &nk, d, &ione, &ineg_one, indxq);
//compute the lower and upper bound of the non-deflated eigenvectors
if (valeig)
magma_svrange(k, d, &iil, &iiu, vl, vu);
else if (indeig)
magma_sirange(k, indxq, &iil, &iiu, il, iu);
else {
iil = 1;
iiu = k;
}
rk = iiu - iil + 1;
}
if (k == 2) {
#pragma omp single
{
for (j = 0; j < k; ++j) {
w[0] = *Q(0,j);
w[1] = *Q(1,j);
i = indx[0] - 1;
*Q(0,j) = w[i];
i = indx[1] - 1;
*Q(1,j) = w[i];
}
}
}
else if (k != 1) {
// Compute updated W.
blasf77_scopy( &ik, &w[ib], &ione, &s[ib], &ione);
// Initialize W(I) = Q(I,I)
tmp = ldq + 1;
blasf77_scopy( &ik, Q(ib,ib), &tmp, &w[ib], &ione);
for (j = 0; j < k; ++j) {
magma_int_t i_tmp = min(j, ie);
for (i = ib; i < i_tmp; ++i)
w[i] = w[i] * ( *Q(i, j) / ( dlamda[i] - dlamda[j] ) );
i_tmp = max(j+1, ib);
for (i = i_tmp; i < ie; ++i)
w[i] = w[i] * ( *Q(i, j) / ( dlamda[i] - dlamda[j] ) );
}
for (i = ib; i < ie; ++i)
w[i] = copysign( sqrt( -w[i] ), s[i]);
#pragma omp barrier
//reduce the number of used threads to have enough S workspace
tot = min(n1, omp_get_num_threads());
if (id < tot) {
ib = ( id * rk) / tot + iil - 1;
ie = ((id+1) * rk) / tot + iil - 1;
ik = ie - ib;
}
else {
ib = -1;
ie = -1;
ik = -1;
}
// Compute eigenvectors of the modified rank-1 modification.
for (j = ib; j < ie; ++j) {
for (i = 0; i < k; ++i)
s[id*k + i] = w[i] / *Q(i,j);
temp = magma_cblas_snrm2( k, s+id*k, 1 );
for (i = 0; i < k; ++i) {
magma_int_t iii = indx[i] - 1;
*Q(i,j) = s[id*k + iii] / temp;
}
}
}
}
}
if (*info != 0)
return *info;
timer_stop( time );
timer_printf( "eigenvalues/vector D+zzT = %6.2f\n", time );
#else
/////////////////////////////////////////////////////////////////////////////////
// Non openmp implementation
/////////////////////////////////////////////////////////////////////////////////
magma_timer_t time=0;
timer_start( time );
for (i = 0; i < k; ++i)
dlamda[i]=lapackf77_slamc3(&dlamda[i], &dlamda[i]) - dlamda[i];
for (j = 0; j < k; ++j) {
magma_int_t tmpp=j+1;
magma_int_t iinfo = 0;
lapackf77_slaed4(&k, &tmpp, dlamda, w, Q(0,j), &rho, &d[j], &iinfo);
// If the zero finder fails, the computation is terminated.
if (iinfo != 0)
*info=iinfo;
}
if (*info != 0)
return *info;
//Prepare the INDXQ sorting permutation.
magma_int_t nk = n - k;
lapackf77_slamrg( &k, &nk, d, &ione, &ineg_one, indxq);
//compute the lower and upper bound of the non-deflated eigenvectors
if (valeig)
magma_svrange(k, d, &iil, &iiu, vl, vu);
else if (indeig)
magma_sirange(k, indxq, &iil, &iiu, il, iu);
else {
iil = 1;
iiu = k;
}
rk = iiu - iil + 1;
if (k == 2) {
for (j = 0; j < k; ++j) {
w[0] = *Q(0,j);
w[1] = *Q(1,j);
i = indx[0] - 1;
*Q(0,j) = w[i];
i = indx[1] - 1;
*Q(1,j) = w[i];
}
}
else if (k != 1) {
// Compute updated W.
blasf77_scopy( &k, w, &ione, s, &ione);
// Initialize W(I) = Q(I,I)
tmp = ldq + 1;
blasf77_scopy( &k, Q, &tmp, w, &ione);
for (j = 0; j < k; ++j) {
for (i = 0; i < j; ++i)
w[i] = w[i] * ( *Q(i, j) / ( dlamda[i] - dlamda[j] ) );
for (i = j+1; i < k; ++i)
w[i] = w[i] * ( *Q(i, j) / ( dlamda[i] - dlamda[j] ) );
}
for (i = 0; i < k; ++i)
w[i] = copysign( sqrt( -w[i] ), s[i]);
// Compute eigenvectors of the modified rank-1 modification.
for (j = iil-1; j < iiu; ++j) {
for (i = 0; i < k; ++i)
s[i] = w[i] / *Q(i,j);
temp = magma_cblas_snrm2( k, s, 1 );
for (i = 0; i < k; ++i) {
magma_int_t iii = indx[i] - 1;
*Q(i,j) = s[iii] / temp;
}
}
}
timer_stop( time );
timer_printf( "eigenvalues/vector D+zzT = %6.2f\n", time );
#endif //_OPENMP
// Compute the updated eigenvectors.
timer_start( time );
magma_queue_sync( NULL );
if (rk != 0) {
if ( n23 != 0 ) {
if (rk < magma_get_slaed3_k()) {
lapackf77_slacpy("A", &n23, &rk, Q(ctot[0],iil-1), &ldq, s, &n23);
blasf77_sgemm("N", "N", &n2, &rk, &n23, &d_one, &Q2[iq2], &n2,
s, &n23, &d_zero, Q(n1,iil-1), &ldq );
} else {
magma_ssetmatrix( n23, rk, Q(ctot[0],iil-1), ldq, ds, n23 );
magma_sgemm( MagmaNoTrans, MagmaNoTrans, n2, rk, n23, d_one, &dq2[iq2], n2, ds, n23, d_zero, dq, lddq);
magma_sgetmatrix( n2, rk, dq, lddq, Q(n1,iil-1), ldq );
}
} else
lapackf77_slaset("A", &n2, &rk, &d_zero, &d_zero, Q(n1,iil-1), &ldq);
if ( n12 != 0 ) {
if (rk < magma_get_slaed3_k()) {
lapackf77_slacpy("A", &n12, &rk, Q(0,iil-1), &ldq, s, &n12);
blasf77_sgemm("N", "N", &n1, &rk, &n12, &d_one, Q2, &n1,
s, &n12, &d_zero, Q(0,iil-1), &ldq);
} else {
magma_ssetmatrix( n12, rk, Q(0,iil-1), ldq, ds, n12 );
magma_sgemm( MagmaNoTrans, MagmaNoTrans, n1, rk, n12, d_one, dq2, n1, ds, n12, d_zero, dq, lddq);
magma_sgetmatrix( n1, rk, dq, lddq, Q(0,iil-1), ldq );
}
} else
lapackf77_slaset("A", &n1, &rk, &d_zero, &d_zero, Q(0,iil-1), &ldq);
}
timer_stop( time );
timer_printf( "gemms = %6.2f\n", time );
return *info;
} /* magma_slaex3 */
/**
Purpose
-------
SLAQPS computes a step of QR factorization with column pivoting
of a real M-by-N matrix A by using Blas-3. It tries to factorize
NB columns from A starting from the row OFFSET+1, and updates all
of the matrix with Blas-3 xGEMM.
In some cases, due to catastrophic cancellations, it cannot
factorize NB columns. Hence, the actual number of factorized
columns is returned in KB.
Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.
Arguments
---------
@param[in]
m INTEGER
The number of rows of the matrix A. M >= 0.
@param[in]
n INTEGER
The number of columns of the matrix A. N >= 0
@param[in]
offset INTEGER
The number of rows of A that have been factorized in
previous steps.
@param[in]
nb INTEGER
The number of columns to factorize.
@param[out]
kb INTEGER
The number of columns actually factorized.
@param[in,out]
A REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, block A(OFFSET+1:M,1:KB) is the triangular
factor obtained and block A(1:OFFSET,1:N) has been
accordingly pivoted, but no factorized.
The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has
been updated.
@param[in]
lda INTEGER
The leading dimension of the array A. LDA >= max(1,M).
@param[in,out]
jpvt INTEGER array, dimension (N)
JPVT(I) = K <==> Column K of the full matrix A has been
permuted into position I in AP.
@param[out]
tau REAL array, dimension (KB)
The scalar factors of the elementary reflectors.
@param[in,out]
vn1 REAL array, dimension (N)
The vector with the partial column norms.
@param[in,out]
vn2 REAL array, dimension (N)
The vector with the exact column norms.
@param[in,out]
auxv REAL array, dimension (NB)
Auxiliar vector.
@param[in,out]
F REAL array, dimension (LDF,NB)
Matrix F' = L*Y'*A.
@param[in]
ldf INTEGER
The leading dimension of the array F. LDF >= max(1,N).
@ingroup magma_sgeqp3_aux
********************************************************************/
extern "C" magma_int_t
magma_slaqps(magma_int_t m, magma_int_t n, magma_int_t offset,
magma_int_t nb, magma_int_t *kb,
float *A, magma_int_t lda,
float *dA, magma_int_t ldda,
magma_int_t *jpvt, float *tau, float *vn1, float *vn2,
float *auxv,
float *F, magma_int_t ldf,
float *dF, magma_int_t lddf)
{
#define A(i, j) (A + (i) + (j)*(lda ))
#define dA(i, j) (dA + (i) + (j)*(ldda))
#define F(i, j) (F + (i) + (j)*(ldf ))
#define dF(i, j) (dF + (i) + (j)*(lddf))
float c_zero = MAGMA_S_MAKE( 0.,0.);
float c_one = MAGMA_S_MAKE( 1.,0.);
float c_neg_one = MAGMA_S_MAKE(-1.,0.);
magma_int_t ione = 1;
magma_int_t i__1, i__2;
float d__1;
float z__1;
magma_int_t j, k, rk;
float Akk;
magma_int_t pvt;
float temp, temp2, tol3z;
magma_int_t itemp;
magma_int_t lsticc;
magma_int_t lastrk;
lastrk = min( m, n + offset );
tol3z = magma_ssqrt( lapackf77_slamch("Epsilon"));
magma_queue_t stream;
magma_queue_create( &stream );
lsticc = 0;
k = 0;
while( k < nb && lsticc == 0 ) {
rk = offset + k;
/* Determine ith pivot column and swap if necessary */
// subtract 1 from Fortran isamax; pvt, k are 0-based.
i__1 = n-k;
pvt = k + blasf77_isamax( &i__1, &vn1[k], &ione ) - 1;
if (pvt != k) {
if (pvt >= nb) {
/* 1. Start copy from GPU */
magma_sgetmatrix_async( m - offset - nb, 1,
dA(offset + nb, pvt), ldda,
A (offset + nb, pvt), lda, stream );
}
/* F gets swapped so F must be sent at the end to GPU */
i__1 = k;
blasf77_sswap( &i__1, F(pvt,0), &ldf, F(k,0), &ldf );
itemp = jpvt[pvt];
jpvt[pvt] = jpvt[k];
jpvt[k] = itemp;
vn1[pvt] = vn1[k];
vn2[pvt] = vn2[k];
if (pvt < nb) {
/* no need of transfer if pivot is within the panel */
blasf77_sswap( &m, A(0, pvt), &ione, A(0, k), &ione );
}
else {
/* 1. Finish copy from GPU */
magma_queue_sync( stream );
/* 2. Swap as usual on CPU */
blasf77_sswap(&m, A(0, pvt), &ione, A(0, k), &ione);
/* 3. Restore the GPU */
magma_ssetmatrix_async( m - offset - nb, 1,
A (offset + nb, pvt), lda,
dA(offset + nb, pvt), ldda, stream);
}
}
/* Apply previous Householder reflectors to column K:
A(RK:M,K) := A(RK:M,K) - A(RK:M,1:K-1)*F(K,1:K-1)'.
Optimization: multiply with beta=0; wait for vector and subtract */
if (k > 0) {
#if defined(PRECISION_c) || defined(PRECISION_z)
for (j = 0; j < k; ++j) {
*F(k,j) = MAGMA_S_CNJG( *F(k,j) );
}
#endif
i__1 = m - rk;
i__2 = k;
blasf77_sgemv( MagmaNoTransStr, &i__1, &i__2,
&c_neg_one, A(rk, 0), &lda,
F(k, 0), &ldf,
&c_one, A(rk, k), &ione );
#if defined(PRECISION_c) || defined(PRECISION_z)
for (j = 0; j < k; ++j) {
*F(k,j) = MAGMA_S_CNJG( *F(k,j) );
}
#endif
}
/* Generate elementary reflector H(k). */
if (rk < m-1) {
i__1 = m - rk;
lapackf77_slarfg( &i__1, A(rk, k), A(rk + 1, k), &ione, &tau[k] );
} else {
lapackf77_slarfg( &ione, A(rk, k), A(rk, k), &ione, &tau[k] );
}
Akk = *A(rk, k);
*A(rk, k) = c_one;
/* Compute Kth column of F:
Compute F(K+1:N,K) := tau(K)*A(RK:M,K+1:N)'*A(RK:M,K) on the GPU */
if (k < n-1) {
i__1 = m - rk;
i__2 = n - k - 1;
/* Send the vector to the GPU */
magma_ssetmatrix( i__1, 1, A(rk, k), lda, dA(rk,k), ldda );
/* Multiply on GPU */
// was CALL SGEMV( 'Conjugate transpose', M-RK+1, N-K,
// TAU( K ), A( RK, K+1 ), LDA,
// A( RK, K ), 1,
// CZERO, F( K+1, K ), 1 )
magma_int_t i__3 = nb-k-1;
magma_int_t i__4 = i__2 - i__3;
magma_int_t i__5 = nb-k;
magma_sgemv( MagmaConjTrans, i__1 - i__5, i__2 - i__3,
tau[k], dA(rk +i__5, k+1+i__3), ldda,
dA(rk +i__5, k ), ione,
c_zero, dF(k+1+i__3, k ), ione );
magma_sgetmatrix_async( i__2-i__3, 1,
dF(k + 1 +i__3, k), i__2,
F (k + 1 +i__3, k), i__2, stream );
blasf77_sgemv( MagmaConjTransStr, &i__1, &i__3,
&tau[k], A(rk, k+1), &lda,
A(rk, k ), &ione,
&c_zero, F(k+1, k ), &ione );
magma_queue_sync( stream );
blasf77_sgemv( MagmaConjTransStr, &i__5, &i__4,
&tau[k], A(rk, k+1+i__3), &lda,
A(rk, k ), &ione,
&c_one, F(k+1+i__3, k ), &ione );
}
/* Padding F(1:K,K) with zeros. */
for (j = 0; j < k; ++j) {
*F(j, k) = c_zero;
}
/* Incremental updating of F:
F(1:N,K) := F(1:N,K) - tau(K)*F(1:N,1:K-1)*A(RK:M,1:K-1)'*A(RK:M,K). */
if (k > 0) {
i__1 = m - rk;
i__2 = k;
z__1 = MAGMA_S_NEGATE( tau[k] );
blasf77_sgemv( MagmaConjTransStr, &i__1, &i__2,
&z__1, A(rk, 0), &lda,
A(rk, k), &ione,
&c_zero, auxv, &ione );
i__1 = k;
blasf77_sgemv( MagmaNoTransStr, &n, &i__1,
&c_one, F(0,0), &ldf,
auxv, &ione,
&c_one, F(0,k), &ione );
}
/* Optimization: On the last iteration start sending F back to the GPU */
/* Update the current row of A:
A(RK,K+1:N) := A(RK,K+1:N) - A(RK,1:K)*F(K+1:N,1:K)'. */
if (k < n-1) {
i__1 = n - k - 1;
i__2 = k + 1;
blasf77_sgemm( MagmaNoTransStr, MagmaConjTransStr, &ione, &i__1, &i__2,
&c_neg_one, A(rk, 0 ), &lda,
F(k+1,0 ), &ldf,
&c_one, A(rk, k+1), &lda );
}
/* Update partial column norms. */
if (rk < lastrk) {
for (j = k + 1; j < n; ++j) {
if (vn1[j] != 0.) {
/* NOTE: The following 4 lines follow from the analysis in
Lapack Working Note 176. */
temp = MAGMA_S_ABS( *A(rk,j) ) / vn1[j];
temp = max( 0., ((1. + temp) * (1. - temp)) );
d__1 = vn1[j] / vn2[j];
temp2 = temp * (d__1 * d__1);
if (temp2 <= tol3z) {
vn2[j] = (float) lsticc;
lsticc = j;
} else {
vn1[j] *= magma_ssqrt(temp);
}
}
}
}
*A(rk, k) = Akk;
++k;
}
// leave k as the last column done
--k;
*kb = k + 1;
rk = offset + *kb - 1;
/* Apply the block reflector to the rest of the matrix:
A(OFFSET+KB+1:M,KB+1:N) := A(OFFSET+KB+1:M,KB+1:N) - A(OFFSET+KB+1:M,1:KB)*F(KB+1:N,1:KB)' */
if (*kb < min(n, m - offset)) {
i__1 = m - rk - 1;
i__2 = n - *kb;
/* Send F to the GPU */
magma_ssetmatrix( i__2, *kb,
F (*kb, 0), ldf,
dF(*kb, 0), i__2 );
magma_sgemm( MagmaNoTrans, MagmaConjTrans, i__1, i__2, *kb,
c_neg_one, dA(rk+1, 0 ), ldda,
dF(*kb, 0 ), i__2,
c_one, dA(rk+1, *kb), ldda );
}
/* Recomputation of difficult columns. */
while( lsticc > 0 ) {
itemp = (magma_int_t)(vn2[lsticc] >= 0. ? floor(vn2[lsticc] + .5) : -floor(.5 - vn2[lsticc]));
i__1 = m - rk - 1;
if (lsticc <= nb)
vn1[lsticc] = magma_cblas_snrm2( i__1, A(rk+1,lsticc), ione );
else {
/* Where is the data, CPU or GPU ? */
float r1, r2;
r1 = magma_cblas_snrm2( nb-k, A(rk+1,lsticc), ione );
r2 = magma_snrm2(m-offset-nb, dA(offset + nb + 1, lsticc), ione);
//vn1[lsticc] = magma_snrm2(i__1, dA(rk + 1, lsticc), ione);
vn1[lsticc] = magma_ssqrt(r1*r1 + r2*r2);
}
/* NOTE: The computation of VN1( LSTICC ) relies on the fact that
SNRM2 does not fail on vectors with norm below the value of SQRT(SLAMCH('S')) */
vn2[lsticc] = vn1[lsticc];
lsticc = itemp;
}
magma_queue_destroy( stream );
return MAGMA_SUCCESS;
} /* magma_slaqps */
/**
Purpose
-------
SGEQP3 computes a QR factorization with column pivoting of a
matrix A: A*P = Q*R using Level 3 BLAS.
Arguments
---------
@param[in]
m INTEGER
The number of rows of the matrix A. M >= 0.
@param[in]
n INTEGER
The number of columns of the matrix A. N >= 0.
@param[in,out]
A REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the upper triangle of the array contains the
min(M,N)-by-N upper trapezoidal matrix R; the elements below
the diagonal, together with the array TAU, represent the
unitary matrix Q as a product of min(M,N) elementary
reflectors.
@param[in]
lda INTEGER
The leading dimension of the array A. LDA >= max(1,M).
@param[in,out]
jpvt INTEGER array, dimension (N)
On entry, if JPVT(J).ne.0, the J-th column of A is permuted
to the front of A*P (a leading column); if JPVT(J)=0,
the J-th column of A is a free column.
On exit, if JPVT(J)=K, then the J-th column of A*P was the
the K-th column of A.
@param[out]
tau REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors.
@param[out]
work (workspace) REAL 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.
For [sd]geqp3, LWORK >= (N+1)*NB + 2*N;
for [cz]geqp3, LWORK >= (N+1)*NB,
where NB is the optimal blocksize.
\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, for [cz]geqp3 only) REAL array, dimension (2*N)
@param[out]
info INTEGER
- = 0: successful exit.
- < 0: if INFO = -i, the i-th argument had an illegal value.
Further Details
---------------
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v'
where tau is a real scalar, and v is a real vector
with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
A(i+1:m,i), and tau in TAU(i).
@ingroup magma_sgeqp3_comp
********************************************************************/
extern "C" magma_int_t
magma_sgeqp3(
magma_int_t m, magma_int_t n,
float *A, magma_int_t lda,
magma_int_t *jpvt, float *tau,
float *work, magma_int_t lwork,
#ifdef COMPLEX
float *rwork,
#endif
magma_int_t *info )
{
#define A(i, j) (A + (i) + (j)*(lda ))
#define dA(i, j) (dwork + (i) + (j)*(ldda))
float *dwork, *df;
magma_int_t ione = 1;
magma_int_t n_j, ldda, ldwork;
magma_int_t j, jb, na, nb, sm, sn, fjb, nfxd, minmn;
magma_int_t topbmn, sminmn, lwkopt=0, lquery;
*info = 0;
lquery = (lwork == -1);
if (m < 0) {
*info = -1;
} else if (n < 0) {
*info = -2;
} else if (lda < max(1,m)) {
*info = -4;
}
nb = magma_get_sgeqp3_nb(min(m, n));
minmn = min(m,n);
if (*info == 0) {
if (minmn == 0) {
lwkopt = 1;
} else {
lwkopt = (n + 1)*nb;
#ifdef REAL
lwkopt += 2*n;
#endif
}
work[0] = MAGMA_S_MAKE( lwkopt, 0. );
if (lwork < lwkopt && ! lquery) {
*info = -8;
}
}
if (*info != 0) {
magma_xerbla( __func__, -(*info) );
return *info;
} else if (lquery) {
return *info;
}
if (minmn == 0)
return *info;
#ifdef REAL
float *rwork = work + (n + 1)*nb;
#endif
ldda = ((m+31)/32)*32;
ldwork = n*ldda + (n+1)*nb;
if (MAGMA_SUCCESS != magma_smalloc( &dwork, ldwork )) {
*info = MAGMA_ERR_DEVICE_ALLOC;
return *info;
}
df = dwork + n*ldda;
// dwork used for dA
magma_queue_t stream;
magma_queue_create( &stream );
/* Move initial columns up front.
* Note jpvt uses 1-based indices for historical compatibility. */
nfxd = 0;
for (j = 0; j < n; ++j) {
if (jpvt[j] != 0) {
if (j != nfxd) {
blasf77_sswap(&m, A(0, j), &ione, A(0, nfxd), &ione);
jpvt[j] = jpvt[nfxd];
jpvt[nfxd] = j + 1;
}
else {
jpvt[j] = j + 1;
}
++nfxd;
}
else {
jpvt[j] = j + 1;
}
}
/* Factorize fixed columns
=======================
Compute the QR factorization of fixed columns and update
remaining columns. */
if (nfxd > 0) {
na = min(m,nfxd);
lapackf77_sgeqrf(&m, &na, A, &lda, tau, work, &lwork, info);
if (na < n) {
n_j = n - na;
lapackf77_sormqr( MagmaLeftStr, MagmaConjTransStr, &m, &n_j, &na,
A, &lda, tau, A(0, na), &lda,
work, &lwork, info );
}
}
/* Factorize free columns */
if (nfxd < minmn) {
sm = m - nfxd;
sn = n - nfxd;
sminmn = minmn - nfxd;
if (nb < sminmn) {
j = nfxd;
// Set the original matrix to the GPU
magma_ssetmatrix_async( m, sn,
A (0,j), lda,
dA(0,j), ldda, stream );
}
/* Initialize partial column norms. */
for (j = nfxd; j < n; ++j) {
rwork[j] = magma_cblas_snrm2( sm, A(nfxd,j), ione );
rwork[n + j] = rwork[j];
}
j = nfxd;
if (nb < sminmn) {
/* Use blocked code initially. */
magma_queue_sync( stream );
/* Compute factorization: while loop. */
topbmn = minmn - nb;
while(j < topbmn) {
jb = min(nb, topbmn - j);
/* Factorize JB columns among columns J:N. */
n_j = n - j;
if (j > nfxd) {
// Get panel to the CPU
magma_sgetmatrix( m-j, jb,
dA(j,j), ldda,
A (j,j), lda );
// Get the rows
magma_sgetmatrix( jb, n_j - jb,
dA(j,j + jb), ldda,
A (j,j + jb), lda );
}
magma_slaqps( m, n_j, j, jb, &fjb,
A (0, j), lda,
dA(0, j), ldda,
&jpvt[j], &tau[j], &rwork[j], &rwork[n + j],
work,
&work[jb], n_j,
&df[jb], n_j );
j += fjb; /* fjb is actual number of columns factored */
}
}
/* Use unblocked code to factor the last or only block. */
if (j < minmn) {
n_j = n - j;
if (j > nfxd) {
magma_sgetmatrix( m-j, n_j,
dA(j,j), ldda,
A (j,j), lda );
}
lapackf77_slaqp2(&m, &n_j, &j, A(0, j), &lda, &jpvt[j],
&tau[j], &rwork[j], &rwork[n+j], work );
}
}
work[0] = MAGMA_S_MAKE( lwkopt, 0. );
magma_free( dwork );
magma_queue_destroy( stream );
return *info;
} /* magma_sgeqp3 */
/* ////////////////////////////////////////////////////////////////////////////
-- Testing sgeev
*/
int main( int argc, char** argv)
{
TESTING_INIT();
real_Double_t gpu_time, cpu_time;
float *h_A, *h_R, *VL, *VR, *h_work, *w1, *w2;
float *w1i, *w2i;
magmaFloatComplex *w1copy, *w2copy;
magmaFloatComplex c_neg_one = MAGMA_C_NEG_ONE;
float tnrm, result[9];
magma_int_t N, n2, lda, nb, lwork, info;
magma_int_t ione = 1;
magma_int_t ISEED[4] = {0,0,0,1};
float ulp, ulpinv, error;
magma_int_t status = 0;
ulp = lapackf77_slamch( "P" );
ulpinv = 1./ulp;
magma_opts opts;
parse_opts( argc, argv, &opts );
// need slightly looser bound (60*eps instead of 30*eps) for some tests
opts.tolerance = max( 60., opts.tolerance );
float tol = opts.tolerance * lapackf77_slamch("E");
float tolulp = opts.tolerance * lapackf77_slamch("P");
// enable at least some minimal checks, if requested
if ( opts.check && !opts.lapack && opts.jobvl == MagmaNoVec && opts.jobvr == MagmaNoVec ) {
fprintf( stderr, "NOTE: Some checks require vectors to be computed;\n"
" set jobvl=V (option -LV), or jobvr=V (option -RV), or both.\n"
" Some checks require running lapack (-l); setting lapack.\n\n");
opts.lapack = true;
}
printf(" N CPU Time (sec) GPU Time (sec) |W_magma - W_lapack| / |W_lapack|\n");
printf("===========================================================================\n");
for( int itest = 0; itest < opts.ntest; ++itest ) {
for( int iter = 0; iter < opts.niter; ++iter ) {
N = opts.nsize[itest];
lda = N;
n2 = lda*N;
nb = magma_get_sgehrd_nb(N);
lwork = N*(2 + nb);
// generous workspace - required by sget22
lwork = max( lwork, N*(5 + 2*N) );
TESTING_MALLOC_CPU( w1copy, magmaFloatComplex, N );
TESTING_MALLOC_CPU( w2copy, magmaFloatComplex, N );
TESTING_MALLOC_CPU( w1, float, N );
TESTING_MALLOC_CPU( w2, float, N );
TESTING_MALLOC_CPU( w1i, float, N );
TESTING_MALLOC_CPU( w2i, float, N );
TESTING_MALLOC_CPU( h_A, float, n2 );
TESTING_MALLOC_PIN( h_R, float, n2 );
TESTING_MALLOC_PIN( VL, float, n2 );
TESTING_MALLOC_PIN( VR, float, n2 );
TESTING_MALLOC_PIN( h_work, float, lwork );
/* Initialize the matrix */
lapackf77_slarnv( &ione, ISEED, &n2, h_A );
lapackf77_slacpy( MagmaUpperLowerStr, &N, &N, h_A, &lda, h_R, &lda );
/* ====================================================================
Performs operation using MAGMA
=================================================================== */
gpu_time = magma_wtime();
magma_sgeev( opts.jobvl, opts.jobvr,
N, h_R, lda, w1, w1i,
VL, lda, VR, lda,
h_work, lwork, &info );
gpu_time = magma_wtime() - gpu_time;
if (info != 0)
printf("magma_sgeev returned error %d: %s.\n",
(int) info, magma_strerror( info ));
/* =====================================================================
Check the result
=================================================================== */
if ( opts.check ) {
/* ===================================================================
* Check the result following LAPACK's [zcds]drvev routine.
* The following tests are performed:
* (1) | A * VR - VR * W | / ( n |A| )
*
* Here VR is the matrix of unit right eigenvectors.
* W is a diagonal matrix with diagonal entries W(j).
*
* (2) | |VR(i)| - 1 | and whether largest component real
*
* VR(i) denotes the i-th column of VR.
*
* (3) | A**T * VL - VL * W**T | / ( n |A| )
*
* Here VL is the matrix of unit left eigenvectors, A**T is the
* transpose of A, and W is as above.
*
* (4) | |VL(i)| - 1 | and whether largest component real
*
* VL(i) denotes the i-th column of VL.
*
* (5) W(full) = W(partial, W only) -- currently skipped
* (6) W(full) = W(partial, W and VR)
* (7) W(full) = W(partial, W and VL)
*
* W(full) denotes the eigenvalues computed when both VR and VL
* are also computed, and W(partial) denotes the eigenvalues
* computed when only W, only W and VR, or only W and VL are
* computed.
*
* (8) VR(full) = VR(partial, W and VR)
*
* VR(full) denotes the right eigenvectors computed when both VR
* and VL are computed, and VR(partial) denotes the result
* when only VR is computed.
*
* (9) VL(full) = VL(partial, W and VL)
*
* VL(full) denotes the left eigenvectors computed when both VR
* and VL are also computed, and VL(partial) denotes the result
* when only VL is computed.
*
* (1, 2) only if jobvr = V
* (3, 4) only if jobvl = V
* (5-9) only if check = 2 (option -c2)
================================================================= */
float vmx, vrmx, vtst;
// Initialize result. -1 indicates test was not run.
for( int j = 0; j < 9; ++j )
result[j] = -1.;
if ( opts.jobvr == MagmaVec ) {
// Do test 1: | A * VR - VR * W | / ( n |A| )
// Note this writes result[1] also
lapackf77_sget22( MagmaNoTransStr, MagmaNoTransStr, MagmaNoTransStr,
&N, h_A, &lda, VR, &lda, w1, w1i,
h_work, &result[0] );
result[0] *= ulp;
// Do test 2: | |VR(i)| - 1 | and whether largest component real
result[1] = -1.;
for( int j = 0; j < N; ++j ) {
tnrm = 1.;
if (w1i[j] == 0.)
tnrm = magma_cblas_snrm2( N, &VR[j*lda], ione );
else if (w1i[j] > 0.)
tnrm = magma_slapy2( magma_cblas_snrm2( N, &VR[j*lda], ione ),
magma_cblas_snrm2( N, &VR[(j+1)*lda], ione ));
result[1] = max( result[1], min( ulpinv, MAGMA_S_ABS(tnrm-1.)/ulp ));
if (w1i[j] > 0.) {
vmx = vrmx = 0.;
for( int jj = 0; jj < N; ++jj ) {
vtst = magma_slapy2( VR[jj+j*lda], VR[jj+(j+1)*lda]);
if (vtst > vmx)
vmx = vtst;
if ( (VR[jj + (j+1)*lda])==0. &&
MAGMA_S_ABS( VR[jj+j*lda] ) > vrmx)
{
vrmx = MAGMA_S_ABS( VR[jj+j*lda] );
}
}
if (vrmx / vmx < 1. - ulp*2.)
result[1] = ulpinv;
}
}
result[1] *= ulp;
}
if ( opts.jobvl == MagmaVec ) {
// Do test 3: | A**T * VL - VL * W**T | / ( n |A| )
// Note this writes result[3] also
lapackf77_sget22( MagmaTransStr, MagmaNoTransStr, MagmaTransStr,
&N, h_A, &lda, VL, &lda, w1, w1i,
h_work, &result[2] );
result[2] *= ulp;
// Do test 4: | |VL(i)| - 1 | and whether largest component real
result[3] = -1.;
for( int j = 0; j < N; ++j ) {
tnrm = 1.;
if (w1i[j] == 0.)
tnrm = magma_cblas_snrm2( N, &VL[j*lda], ione );
else if (w1i[j] > 0.)
tnrm = magma_slapy2( magma_cblas_snrm2( N, &VL[j*lda], ione ),
magma_cblas_snrm2( N, &VL[(j+1)*lda], ione ));
result[3] = max( result[3], min( ulpinv, MAGMA_S_ABS(tnrm-1.)/ulp ));
if (w1i[j] > 0.) {
vmx = vrmx = 0.;
for( int jj = 0; jj < N; ++jj ) {
vtst = magma_slapy2( VL[jj+j*lda], VL[jj+(j+1)*lda]);
if (vtst > vmx)
vmx = vtst;
if ( (VL[jj + (j+1)*lda])==0. &&
MAGMA_S_ABS( VL[jj+j*lda]) > vrmx)
{
vrmx = MAGMA_S_ABS( VL[jj+j*lda] );
}
}
if (vrmx / vmx < 1. - ulp*2.)
result[3] = ulpinv;
}
}
result[3] *= ulp;
}
}
if ( opts.check == 2 ) {
// more extensive tests
// this is really slow because it calls magma_zgeev multiple times
float *LRE, DUM;
TESTING_MALLOC_PIN( LRE, float, n2 );
lapackf77_slarnv( &ione, ISEED, &n2, h_A );
lapackf77_slacpy( MagmaUpperLowerStr, &N, &N, h_A, &lda, h_R, &lda );
// ----------
// Compute eigenvalues, left and right eigenvectors
magma_sgeev( MagmaVec, MagmaVec,
N, h_R, lda, w1, w1i,
VL, lda, VR, lda,
h_work, lwork, &info );
if (info != 0)
printf("magma_zgeev (case V, V) returned error %d: %s.\n",
(int) info, magma_strerror( info ));
// ----------
// Compute eigenvalues only
// These are not exactly equal, and not in the same order, so skip for now.
//lapackf77_slacpy( MagmaUpperLowerStr, &N, &N, h_A, &lda, h_R, &lda );
//magma_sgeev( MagmaNoVec, MagmaNoVec,
// N, h_R, lda, w2, w2i,
// &DUM, 1, &DUM, 1,
// h_work, lwork, &info );
//if (info != 0)
// printf("magma_sgeev (case N, N) returned error %d: %s.\n",
// (int) info, magma_strerror( info ));
//
//// Do test 5: W(full) = W(partial, W only)
//result[4] = 1;
//for( int j = 0; j < N; ++j )
// if ( w1[j] != w2[j] || w1i[j] != w2i[j] )
// result[4] = 0;
// ----------
// Compute eigenvalues and right eigenvectors
lapackf77_slacpy( MagmaUpperLowerStr, &N, &N, h_A, &lda, h_R, &lda );
magma_sgeev( MagmaNoVec, MagmaVec,
N, h_R, lda, w2, w2i,
&DUM, 1, LRE, lda,
h_work, lwork, &info );
if (info != 0)
printf("magma_sgeev (case N, V) returned error %d: %s.\n",
(int) info, magma_strerror( info ));
// Do test 6: W(full) = W(partial, W and VR)
result[5] = 1;
for( int j = 0; j < N; ++j )
if ( w1[j] != w2[j] || w1i[j] != w2i[j] )
result[5] = 0;
// Do test 8: VR(full) = VR(partial, W and VR)
result[7] = 1;
for( int j = 0; j < N; ++j )
for( int jj = 0; jj < N; ++jj )
if ( ! MAGMA_S_EQUAL( VR[j+jj*lda], LRE[j+jj*lda] ))
result[7] = 0;
// ----------
// Compute eigenvalues and left eigenvectors
lapackf77_slacpy( MagmaUpperLowerStr, &N, &N, h_A, &lda, h_R, &lda );
magma_sgeev( MagmaVec, MagmaNoVec,
N, h_R, lda, w2, w2i,
LRE, lda, &DUM, 1,
h_work, lwork, &info );
if (info != 0)
printf("magma_sgeev (case V, N) returned error %d: %s.\n",
(int) info, magma_strerror( info ));
// Do test 7: W(full) = W(partial, W and VL)
result[6] = 1;
for( int j = 0; j < N; ++j )
if ( w1[j] != w2[j] || w1i[j] != w2i[j] )
result[6] = 0;
// Do test 9: VL(full) = VL(partial, W and VL)
result[8] = 1;
for( int j = 0; j < N; ++j )
for( int jj = 0; jj < N; ++jj )
if ( ! MAGMA_S_EQUAL( VL[j+jj*lda], LRE[j+jj*lda] ))
result[8] = 0;
TESTING_FREE_PIN( LRE );
}
/* =====================================================================
Performs operation using LAPACK
Do this after checks, because it overwrites VL and VR.
=================================================================== */
if ( opts.lapack ) {
cpu_time = magma_wtime();
lapackf77_sgeev( lapack_vec_const(opts.jobvl), lapack_vec_const(opts.jobvr),
&N, h_A, &lda, w2, w2i,
VL, &lda, VR, &lda,
h_work, &lwork, &info );
cpu_time = magma_wtime() - cpu_time;
if (info != 0)
printf("lapackf77_sgeev returned error %d: %s.\n",
(int) info, magma_strerror( info ));
// check | W_magma - W_lapack | / | W |
// need to sort eigenvalues first
// copy them into complex vectors for ease
for( int j=0; j < N; ++j ) {
w1copy[j] = MAGMA_C_MAKE( w1[j], w1i[j] );
w2copy[j] = MAGMA_C_MAKE( w2[j], w2i[j] );
}
std::sort( w1copy, &w1copy[N], lessthan );
std::sort( w2copy, &w2copy[N], lessthan );
// adjust sorting to deal with numerical inaccuracy
// search down w2 for eigenvalue that matches w1's eigenvalue
for( int j=0; j < N; ++j ) {
for( int j2=j; j2 < N; ++j2 ) {
magmaFloatComplex diff = MAGMA_C_SUB( w1copy[j], w2copy[j2] );
float diff2 = magma_szlapy2( diff ) / max( magma_szlapy2( w1copy[j] ), tol );
if ( diff2 < 100*tol ) {
if ( j != j2 ) {
std::swap( w2copy[j], w2copy[j2] );
}
break;
}
}
}
blasf77_caxpy( &N, &c_neg_one, w2copy, &ione, w1copy, &ione );
error = magma_cblas_scnrm2( N, w1copy, 1 );
error /= magma_cblas_scnrm2( N, w2copy, 1 );
printf("%5d %7.2f %7.2f %8.2e %s\n",
(int) N, cpu_time, gpu_time,
error, (error < tolulp ? "ok" : "failed"));
status += ! (error < tolulp);
}
else {
printf("%5d --- %7.2f\n",
(int) N, gpu_time);
}
if ( opts.check ) {
// -1 indicates test was not run
if ( result[0] != -1 ) { printf(" | A * VR - VR * W | / ( n |A| ) = %8.2e %s\n", result[0], (result[0] < tol ? "ok" : "failed")); }
if ( result[1] != -1 ) { printf(" | |VR(i)| - 1 | = %8.2e %s\n", result[1], (result[1] < tol ? "ok" : "failed")); }
if ( result[2] != -1 ) { printf(" | A'* VL - VL * W'| / ( n |A| ) = %8.2e %s\n", result[2], (result[2] < tol ? "ok" : "failed")); }
if ( result[3] != -1 ) { printf(" | |VL(i)| - 1 | = %8.2e %s\n", result[3], (result[3] < tol ? "ok" : "failed")); }
if ( result[4] != -1 ) { printf(" W (full) == W (partial, W only) %s\n", (result[4] == 1. ? "ok" : "failed")); }
if ( result[5] != -1 ) { printf(" W (full) == W (partial, W and VR) %s\n", (result[5] == 1. ? "ok" : "failed")); }
if ( result[6] != -1 ) { printf(" W (full) == W (partial, W and VL) %s\n", (result[6] == 1. ? "ok" : "failed")); }
if ( result[7] != -1 ) { printf(" VR (full) == VR (partial, W and VR) %s\n", (result[7] == 1. ? "ok" : "failed")); }
if ( result[8] != -1 ) { printf(" VL (full) == VL (partial, W and VL) %s\n", (result[8] == 1. ? "ok" : "failed")); }
int newline = 0;
if ( result[0] != -1 ) { status += ! (result[0] < tol); newline = 1; }
if ( result[1] != -1 ) { status += ! (result[1] < tol); newline = 1; }
if ( result[2] != -1 ) { status += ! (result[2] < tol); newline = 1; }
if ( result[3] != -1 ) { status += ! (result[3] < tol); newline = 1; }
if ( result[4] != -1 ) { status += ! (result[4] == 1.); newline = 1; }
if ( result[5] != -1 ) { status += ! (result[5] == 1.); newline = 1; }
if ( result[6] != -1 ) { status += ! (result[6] == 1.); newline = 1; }
if ( result[7] != -1 ) { status += ! (result[7] == 1.); newline = 1; }
if ( result[8] != -1 ) { status += ! (result[8] == 1.); newline = 1; }
if ( newline ) {
printf( "\n" );
}
}
TESTING_FREE_CPU( w1copy );
TESTING_FREE_CPU( w2copy );
TESTING_FREE_CPU( w1 );
TESTING_FREE_CPU( w2 );
TESTING_FREE_CPU( w1i );
TESTING_FREE_CPU( w2i );
TESTING_FREE_CPU( h_A );
TESTING_FREE_PIN( h_R );
TESTING_FREE_PIN( VL );
TESTING_FREE_PIN( VR );
TESTING_FREE_PIN( h_work );
fflush( stdout );
}
if ( opts.niter > 1 ) {
printf( "\n" );
}
}
TESTING_FINALIZE();
return status;
}
