/* ////////////////////////////////////////////////////////////////////////////
-- Testing dlag2s and slag2d
*/
int main( int argc, char** argv )
{
TESTING_INIT();
real_Double_t gbytes, gpu_perf, gpu_time, cpu_perf, cpu_time;
double error, work[1];
float serror, swork[1];
double c_neg_one = MAGMA_D_NEG_ONE;
float s_neg_one = MAGMA_S_NEG_ONE;
magma_int_t ione = 1;
magma_int_t m, n, lda, ldda, size, info;
magma_int_t ISEED[4] = {0,0,0,1};
magma_int_t status = 0;
float *SA, *SR;
double *A, *R;
magmaFloat_ptr dSA;
magmaDouble_ptr dA;
magma_opts opts;
opts.parse_opts( argc, argv );
printf("%% func M N CPU GB/s (ms) GPU GB/s (ms) ||R||_F\n");
printf("%%====================================================================\n");
for( int itest = 0; itest < opts.ntest; ++itest ) {
for( int iter = 0; iter < opts.niter; ++iter ) {
m = opts.msize[itest];
n = opts.nsize[itest];
lda = m;
ldda = magma_roundup( m, opts.align ); // multiple of 32 by default
// m*n double-real loads and m*n single-real stores (and vice-versa for slag2d)
gbytes = (real_Double_t) m*n * (sizeof(double) + sizeof(float)) / 1e9;
size = ldda*n; // ldda >= lda
TESTING_MALLOC_CPU( SA, float, size );
TESTING_MALLOC_CPU( A, double, size );
TESTING_MALLOC_CPU( SR, float, size );
TESTING_MALLOC_CPU( R, double, size );
TESTING_MALLOC_DEV( dSA, float, size );
TESTING_MALLOC_DEV( dA, double, size );
lapackf77_dlarnv( &ione, ISEED, &size, A );
lapackf77_slarnv( &ione, ISEED, &size, SA );
magma_dsetmatrix( m, n, A, lda, dA, ldda, opts.queue );
magma_ssetmatrix( m, n, SA, lda, dSA, ldda, opts.queue );
/* =====================================================================
Performs operation using LAPACK dlag2s
=================================================================== */
cpu_time = magma_wtime();
lapackf77_dlag2s( &m, &n, A, &lda, SA, &lda, &info );
cpu_time = magma_wtime() - cpu_time;
cpu_perf = gbytes / cpu_time;
if (info != 0) {
printf("lapackf77_dlag2s returned error %d: %s.\n",
(int) info, magma_strerror( info ));
}
/* ====================================================================
Performs operation using MAGMA dlag2s
=================================================================== */
gpu_time = magma_sync_wtime( opts.queue );
magmablas_dlag2s( m, n, dA, ldda, dSA, ldda, opts.queue, &info );
gpu_time = magma_sync_wtime( opts.queue ) - gpu_time;
gpu_perf = gbytes / gpu_time;
if (info != 0) {
printf("magmablas_dlag2s returned error %d: %s.\n",
(int) info, magma_strerror( info ));
}
magma_sgetmatrix( m, n, dSA, ldda, SR, lda, opts.queue );
/* =====================================================================
compute error |SA_magma - SA_lapack|
should be zero if both are IEEE compliant
=================================================================== */
blasf77_saxpy( &size, &s_neg_one, SA, &ione, SR, &ione );
serror = lapackf77_slange( "Fro", &m, &n, SR, &lda, swork );
printf( "dlag2s %5d %5d %7.2f (%7.2f) %7.2f (%7.2f) %8.2e %s\n",
(int) m, (int) n,
cpu_perf, cpu_time*1000., gpu_perf, gpu_time*1000.,
serror, (serror == 0 ? "ok" : "failed") );
status += ! (serror == 0);
/* =====================================================================
Reset matrices
=================================================================== */
lapackf77_dlarnv( &ione, ISEED, &size, A );
lapackf77_slarnv( &ione, ISEED, &size, SA );
magma_dsetmatrix( m, n, A, lda, dA, ldda, opts.queue );
magma_ssetmatrix( m, n, SA, lda, dSA, ldda, opts.queue );
/* =====================================================================
Performs operation using LAPACK slag2d
=================================================================== */
cpu_time = magma_wtime();
lapackf77_slag2d( &m, &n, SA, &lda, A, &lda, &info );
cpu_time = magma_wtime() - cpu_time;
cpu_perf = gbytes / cpu_time;
if (info != 0) {
printf("lapackf77_slag2d returned error %d: %s.\n",
(int) info, magma_strerror( info ));
}
/* ====================================================================
Performs operation using MAGMA slag2d
=================================================================== */
magma_ssetmatrix( m, n, SA, lda, dSA, ldda, opts.queue );
gpu_time = magma_sync_wtime( opts.queue );
magmablas_slag2d( m, n, dSA, ldda, dA, ldda, opts.queue, &info );
gpu_time = magma_sync_wtime( opts.queue ) - gpu_time;
gpu_perf = gbytes / gpu_time;
if (info != 0) {
printf("magmablas_slag2d returned error %d: %s.\n",
(int) info, magma_strerror( info ));
}
magma_dgetmatrix( m, n, dA, ldda, R, lda, opts.queue );
/* =====================================================================
compute error |A_magma - A_lapack|
should be zero if both are IEEE compliant
=================================================================== */
blasf77_daxpy( &size, &c_neg_one, A, &ione, R, &ione );
error = lapackf77_dlange( "Fro", &m, &n, R, &lda, work );
printf( "slag2d %5d %5d %7.2f (%7.2f) %7.2f (%7.2f) %8.2e %s\n",
(int) m, (int) n,
cpu_perf, cpu_time*1000., gpu_perf, gpu_time*1000.,
error, (error == 0 ? "ok" : "failed") );
status += ! (error == 0);
TESTING_FREE_CPU( SA );
TESTING_FREE_CPU( A );
TESTING_FREE_CPU( SR );
TESTING_FREE_CPU( R );
TESTING_FREE_DEV( dSA );
TESTING_FREE_DEV( dA );
printf( "\n" );
fflush( stdout );
}
if ( opts.niter > 1 ) {
printf( "\n" );
}
}
opts.cleanup();
TESTING_FINALIZE();
return status;
}
extern "C" magma_int_t
magma_dsgesv_gpu(char trans, magma_int_t n, magma_int_t nrhs,
double *dA, magma_int_t ldda,
magma_int_t *ipiv, magma_int_t *dipiv,
double *dB, magma_int_t lddb,
double *dX, magma_int_t lddx,
double *dworkd, float *dworks,
magma_int_t *iter, magma_int_t *info)
{
/* -- MAGMA (version 1.4.0) --
Univ. of Tennessee, Knoxville
Univ. of California, Berkeley
Univ. of Colorado, Denver
August 2013
Purpose
=======
DSGESV computes the solution to a real system of linear equations
A * X = B or A' * X = B
where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
DSGESV first attempts to factorize the matrix in real SINGLE PRECISION
and use this factorization within an iterative refinement procedure
to produce a solution with real DOUBLE PRECISION norm-wise backward error
quality (see below). If the approach fails the method switches to a
real DOUBLE PRECISION factorization and solve.
The iterative refinement is not going to be a winning strategy if
the ratio real SINGLE PRECISION performance over real DOUBLE PRECISION
performance is too small. A reasonable strategy should take the
number of right-hand sides and the size of the matrix into account.
This might be done with a call to ILAENV in the future. Up to now, we
always try iterative refinement.
The iterative refinement process is stopped if
ITER > ITERMAX
or for all the RHS we have:
RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX
where
o ITER is the number of the current iteration in the iterative
refinement process
o RNRM is the infinity-norm of the residual
o XNRM is the infinity-norm of the solution
o ANRM is the infinity-operator-norm of the matrix A
o EPS is the machine epsilon returned by DLAMCH('Epsilon')
The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00 respectively.
Arguments
=========
TRANS (input) CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A'* X = B (Transpose)
= 'C': A'* X = B (Conjugate transpose = Transpose)
N (input) INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
dA (input or input/output) DOUBLE PRECISION array on the GPU, dimension (ldda,N)
On entry, the N-by-N coefficient matrix A.
On exit, if iterative refinement has been successfully used
(info.EQ.0 and ITER.GE.0, see description below), A is
unchanged. If double precision factorization has been used
(info.EQ.0 and ITER.LT.0, see description below), then the
array dA contains the factors L and U from the factorization
A = P*L*U; the unit diagonal elements of L are not stored.
ldda (input) INTEGER
The leading dimension of the array dA. ldda >= max(1,N).
IPIV (output) INTEGER array, dimension (N)
The pivot indices that define the permutation matrix P;
row i of the matrix was interchanged with row IPIV(i).
Corresponds either to the single precision factorization
(if info.EQ.0 and ITER.GE.0) or the double precision
factorization (if info.EQ.0 and ITER.LT.0).
dIPIV (output) INTEGER array on the GPU, dimension (min(M,N))
The pivot indices; for 1 <= i <= min(M,N), row i of the
matrix was moved to row IPIV(i).
dB (input) DOUBLE PRECISION array on the GPU, dimension (lddb,NRHS)
The N-by-NRHS right hand side matrix B.
lddb (input) INTEGER
The leading dimension of the array dB. lddb >= max(1,N).
dX (output) DOUBLE PRECISION array on the GPU, dimension (lddx,NRHS)
If info = 0, the N-by-NRHS solution matrix X.
lddx (input) INTEGER
The leading dimension of the array dX. lddx >= max(1,N).
dworkd (workspace) DOUBLE PRECISION array on the GPU, dimension (N*NRHS)
This array is used to hold the residual vectors.
dworks (workspace) SINGLE PRECISION array on the GPU, dimension (N*(N+NRHS))
This array is used to store the real single precision matrix
and the right-hand sides or solutions in single precision.
iter (output) INTEGER
< 0: iterative refinement has failed, double precision
factorization has been performed
-1 : the routine fell back to full precision for
implementation- or machine-specific reasons
-2 : narrowing the precision induced an overflow,
the routine fell back to full precision
-3 : failure of SGETRF
-31: stop the iterative refinement after the 30th iteration
> 0: iterative refinement has been successfully used.
Returns the number of iterations
info (output) INTEGER
= 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = i, U(i,i) computed in DOUBLE PRECISION is
exactly zero. The factorization has been completed,
but the factor U is exactly singular, so the solution
could not be computed.
===================================================================== */
#define dB(i,j) (dB + (i) + (j)*lddb)
#define dX(i,j) (dX + (i) + (j)*lddx)
#define dR(i,j) (dR + (i) + (j)*lddr)
double c_neg_one = MAGMA_D_NEG_ONE;
double c_one = MAGMA_D_ONE;
magma_int_t ione = 1;
double *dR;
float *dSA, *dSX;
double Xnrmv, Rnrmv;
double Anrm, Xnrm, Rnrm, cte, eps;
magma_int_t i, j, iiter, lddsa, lddr;
/* Check arguments */
*iter = 0;
*info = 0;
if ( n < 0 )
*info = -1;
else if ( nrhs < 0 )
*info = -2;
else if ( ldda < max(1,n))
*info = -4;
else if ( lddb < max(1,n))
*info = -8;
else if ( lddx < max(1,n))
*info = -10;
if (*info != 0) {
magma_xerbla( __func__, -(*info) );
return *info;
}
if ( n == 0 || nrhs == 0 )
return *info;
lddsa = n;
lddr = n;
dSA = dworks;
dSX = dSA + lddsa*n;
dR = dworkd;
eps = lapackf77_dlamch("Epsilon");
Anrm = magmablas_dlange('I', n, n, dA, ldda, (double*)dworkd );
cte = Anrm * eps * pow((double)n, 0.5) * BWDMAX;
/*
* Convert to single precision
*/
//magmablas_dlag2s( n, nrhs, dB, lddb, dSX, lddsx, info ); // done inside dsgetrs with pivots
if (*info != 0) {
*iter = -2;
goto FALLBACK;
}
magmablas_dlag2s( n, n, dA, ldda, dSA, lddsa, info );
if (*info != 0) {
*iter = -2;
goto FALLBACK;
}
// factor dSA in single precision
magma_sgetrf_gpu( n, n, dSA, lddsa, ipiv, info );
if (*info != 0) {
*iter = -3;
goto FALLBACK;
}
// Generate parallel pivots
{
magma_int_t *newipiv;
magma_imalloc_cpu( &newipiv, n );
if ( newipiv == NULL ) {
*iter = -3;
goto FALLBACK;
}
swp2pswp( trans, n, ipiv, newipiv );
magma_setvector( n, sizeof(magma_int_t), newipiv, 1, dipiv, 1 );
magma_free_cpu( newipiv );
}
// solve dSA*dSX = dB in single precision
// converts dB to dSX and applies pivots, solves, then converts result back to dX
magma_dsgetrs_gpu( trans, n, nrhs, dSA, lddsa, dipiv, dB, lddb, dX, lddx, dSX, info );
// residual dR = dB - dA*dX in double precision
magmablas_dlacpy( MagmaUpperLower, n, nrhs, dB, lddb, dR, lddr );
if ( nrhs == 1 ) {
magma_dgemv( trans, n, n,
c_neg_one, dA, ldda,
dX, 1,
c_one, dR, 1 );
}
else {
magma_dgemm( trans, MagmaNoTrans, n, nrhs, n,
c_neg_one, dA, ldda,
dX, lddx,
c_one, dR, lddr );
}
// TODO: use MAGMA_D_ABS( dX(i,j) ) instead of dlange?
for( j=0; j < nrhs; j++ ) {
i = magma_idamax( n, dX(0,j), 1) - 1;
magma_dgetmatrix( 1, 1, dX(i,j), 1, &Xnrmv, 1 );
Xnrm = lapackf77_dlange( "F", &ione, &ione, &Xnrmv, &ione, NULL );
i = magma_idamax ( n, dR(0,j), 1 ) - 1;
magma_dgetmatrix( 1, 1, dR(i,j), 1, &Rnrmv, 1 );
Rnrm = lapackf77_dlange( "F", &ione, &ione, &Rnrmv, &ione, NULL );
if ( Rnrm > Xnrm*cte ) {
goto REFINEMENT;
}
}
*iter = 0;
return *info;
REFINEMENT:
for( iiter=1; iiter < ITERMAX; ) {
*info = 0;
// convert residual dR to single precision dSX
// solve dSA*dSX = R in single precision
// convert result back to double precision dR
// it's okay that dR is used for both dB input and dX output.
magma_dsgetrs_gpu( trans, n, nrhs, dSA, lddsa, dipiv, dR, lddr, dR, lddr, dSX, info );
if (*info != 0) {
*iter = -3;
goto FALLBACK;
}
// Add correction and setup residual
// dX += dR --and--
// dR = dB
// This saves going through dR a second time (if done with one more kernel).
// -- not really: first time is read, second time is write.
for( j=0; j < nrhs; j++ ) {
magmablas_daxpycp( n, dR(0,j), dX(0,j), dB(0,j) );
}
// residual dR = dB - dA*dX in double precision
if ( nrhs == 1 ) {
magma_dgemv( trans, n, n,
c_neg_one, dA, ldda,
dX, 1,
c_one, dR, 1 );
}
else {
magma_dgemm( trans, MagmaNoTrans, n, nrhs, n,
c_neg_one, dA, ldda,
dX, lddx,
c_one, dR, lddr );
}
/* Check whether the nrhs normwise backward errors satisfy the
* stopping criterion. If yes, set ITER=IITER>0 and return. */
for( j=0; j < nrhs; j++ ) {
i = magma_idamax( n, dX(0,j), 1) - 1;
magma_dgetmatrix( 1, 1, dX(i,j), 1, &Xnrmv, 1 );
Xnrm = lapackf77_dlange( "F", &ione, &ione, &Xnrmv, &ione, NULL );
i = magma_idamax ( n, dR(0,j), 1 ) - 1;
magma_dgetmatrix( 1, 1, dR(i,j), 1, &Rnrmv, 1 );
Rnrm = lapackf77_dlange( "F", &ione, &ione, &Rnrmv, &ione, NULL );
if ( Rnrm > Xnrm*cte ) {
goto L20;
}
}
/* If we are here, the nrhs normwise backward errors satisfy
* the stopping criterion, we are good to exit. */
*iter = iiter;
return *info;
L20:
iiter++;
}
/* If we are at this place of the code, this is because we have
* performed ITER=ITERMAX iterations and never satisified the
* stopping criterion. Set up the ITER flag accordingly and follow
* up on double precision routine. */
*iter = -ITERMAX - 1;
FALLBACK:
/* Single-precision iterative refinement failed to converge to a
* satisfactory solution, so we resort to double precision. */
magma_dgetrf_gpu( n, n, dA, ldda, ipiv, info );
if (*info == 0) {
magmablas_dlacpy( MagmaUpperLower, n, nrhs, dB, lddb, dX, lddx );
magma_dgetrs_gpu( trans, n, nrhs, dA, ldda, ipiv, dX, lddx, info );
}
return *info;
}
/**
Purpose
-------
DSGETRS solves a system of linear equations
A * X = B, A**T * X = B, or A**H * X = B
with a general N-by-N matrix A using the LU factorization computed
by MAGMA_SGETRF_GPU. B and X are in DOUBLE PRECISION, and A is in SINGLE PRECISION.
This routine is used in the mixed precision iterative solver
magma_dsgesv.
Arguments
---------
@param[in]
trans magma_trans_t
Specifies the form of the system of equations:
- = MagmaNoTrans: A * X = B (No transpose)
- = MagmaTrans: A**T * X = B (Transpose)
- = MagmaConjTrans: A**H * X = B (Conjugate transpose)
@param[in]
n INTEGER
The order of the matrix A. N >= 0.
@param[in]
nrhs INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
@param[in]
dA SINGLE PRECISION array on the GPU, dimension (LDDA,N)
The factors L and U from the factorization A = P*L*U
as computed by CGETRF_GPU.
@param[in]
ldda INTEGER
The leading dimension of the array dA. LDDA >= max(1,N).
@param[in]
dipiv INTEGER array on the GPU, dimension (N)
The pivot indices; for 1 <= i <= N, after permuting, row i of the
matrix was moved to row dIPIV(i).
Note this is different than IPIV from DGETRF, where interchanges
are applied one-after-another.
@param[in]
dB DOUBLE PRECISION array on the GPU, dimension (LDDB,NRHS)
On entry, the right hand side matrix B.
@param[in]
lddb INTEGER
The leading dimension of the arrays X and B. LDDB >= max(1,N).
@param[out]
dX DOUBLE PRECISION array on the GPU, dimension (LDDX, NRHS)
On exit, the solution matrix dX.
@param[in]
lddx INTEGER
The leading dimension of the array dX, LDDX >= max(1,N).
@param
dSX (workspace) SINGLE PRECISION array on the GPU used as workspace,
dimension (N, NRHS)
@param[out]
info INTEGER
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value
@ingroup magma_dgesv_comp
********************************************************************/
extern "C" magma_int_t
magma_dsgetrs_gpu(
magma_trans_t trans, magma_int_t n, magma_int_t nrhs,
magmaFloat_ptr dA, magma_int_t ldda,
magmaInt_ptr dipiv,
magmaDouble_ptr dB, magma_int_t lddb,
magmaDouble_ptr dX, magma_int_t lddx,
magmaFloat_ptr dSX,
magma_int_t *info)
{
/* Constants */
float c_one = MAGMA_S_ONE;
/* Local variables */
bool notran = (trans == MagmaNoTrans);
magma_int_t inc;
magma_int_t lddsx = n;
*info = 0;
if ( (! notran) &&
(trans != MagmaTrans) &&
(trans != MagmaConjTrans) ) {
*info = -1;
} else if (n < 0) {
*info = -2;
} else if (nrhs < 0) {
*info = -3;
} else if (ldda < n) {
*info = -5;
} else if (lddb < n) {
*info = -8;
} else if (lddx < n) {
*info = -10;
}
// I think this is resolved, but it is unclear what the issue ever was.
//else if (lddx != lddb) { /* TODO: remove it when dslaswp will have the correct interface */
// *info = -10;
//}
if (*info != 0) {
magma_xerbla( __func__, -(*info) );
return *info;
}
/* Quick return if possible */
if (n == 0 || nrhs == 0) {
return *info;
}
magma_queue_t queue;
magma_device_t cdev;
magma_getdevice( &cdev );
magma_queue_create( cdev, &queue );
if (notran) {
inc = 1;
/* Get X by row applying interchanges to B and cast to single */
/*
* TODO: clean dslaswp interface to have interface closer to zlaswp
*/
magmablas_dslaswp( nrhs, dB, lddb, dSX, lddsx,
n, dipiv, inc, queue );
/* Solve L*X = B, overwriting B with SX. */
magma_strsm( MagmaLeft, MagmaLower, MagmaNoTrans, MagmaUnit,
n, nrhs, c_one, dA, ldda, dSX, lddsx, queue );
/* Solve U*X = B, overwriting B with X. */
magma_strsm( MagmaLeft, MagmaUpper, MagmaNoTrans, MagmaNonUnit,
n, nrhs, c_one, dA, ldda, dSX, lddsx, queue );
magmablas_slag2d( n, nrhs, dSX, lddsx, dX, lddx, queue, info );
}
else {
inc = -1;
/* Cast the DOUBLE PRECISION RHS to SINGLE PRECISION */
magmablas_dlag2s( n, nrhs, dB, lddb, dSX, lddsx, queue, info );
/* Solve A**T * X = B, or A**H * X = B */
magma_strsm( MagmaLeft, MagmaUpper, trans, MagmaNonUnit,
n, nrhs, c_one, dA, ldda, dSX, lddsx, queue );
magma_strsm( MagmaLeft, MagmaLower, trans, MagmaUnit,
n, nrhs, c_one, dA, ldda, dSX, lddsx, queue );
magmablas_dslaswp( nrhs, dX, lddx, dSX, lddsx,
n, dipiv, inc, queue );
}
magma_queue_destroy( queue );
return *info;
} /* magma_dsgetrs */
/**
Purpose
-------
DSPOSV computes the solution to a real system of linear equations
A * X = B,
where A is an N-by-N symmetric positive definite matrix and X and B
are N-by-NRHS matrices.
DSPOSV first attempts to factorize the matrix in real SINGLE PRECISION
and use this factorization within an iterative refinement procedure
to produce a solution with real DOUBLE PRECISION norm-wise backward error
quality (see below). If the approach fails the method switches to a
real DOUBLE PRECISION factorization and solve.
The iterative refinement is not going to be a winning strategy if
the ratio real SINGLE PRECISION performance over real DOUBLE PRECISION
performance is too small. A reasonable strategy should take the
number of right-hand sides and the size of the matrix into account.
This might be done with a call to ILAENV in the future. Up to now, we
always try iterative refinement.
The iterative refinement process is stopped if
ITER > ITERMAX
or for all the RHS we have:
RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX
where
o ITER is the number of the current iteration in the iterative
refinement process
o RNRM is the infinity-norm of the residual
o XNRM is the infinity-norm of the solution
o ANRM is the infinity-operator-norm of the matrix A
o EPS is the machine epsilon returned by DLAMCH('Epsilon')
The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00 respectively.
Arguments
---------
@param[in]
uplo magma_uplo_t
- = MagmaUpper: Upper triangle of A is stored;
- = MagmaLower: Lower triangle of A is stored.
@param[in]
n INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.
@param[in]
nrhs INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
@param[in,out]
dA DOUBLE PRECISION array on the GPU, dimension (LDDA,N)
On entry, the symmetric matrix A. If UPLO = MagmaUpper, the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = MagmaLower, the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if iterative refinement has been successfully used
(INFO.EQ.0 and ITER.GE.0, see description below), then A is
unchanged, if double factorization has been used
(INFO.EQ.0 and ITER.LT.0, see description below), then the
array dA contains the factor U or L from the Cholesky
factorization A = U**T*U or A = L*L**T.
@param[in]
ldda INTEGER
The leading dimension of the array dA. LDDA >= max(1,N).
@param[in]
dB DOUBLE PRECISION array on the GPU, dimension (LDDB,NRHS)
The N-by-NRHS right hand side matrix B.
@param[in]
lddb INTEGER
The leading dimension of the array dB. LDDB >= max(1,N).
@param[out]
dX DOUBLE PRECISION array on the GPU, dimension (LDDX,NRHS)
If INFO = 0, the N-by-NRHS solution matrix X.
@param[in]
lddx INTEGER
The leading dimension of the array dX. LDDX >= max(1,N).
@param
dworkd (workspace) DOUBLE PRECISION array on the GPU, dimension (N*NRHS)
This array is used to hold the residual vectors.
@param
dworks (workspace) SINGLE PRECISION array on the GPU, dimension (N*(N+NRHS))
This array is used to store the real single precision matrix
and the right-hand sides or solutions in single precision.
@param[out]
iter INTEGER
- < 0: iterative refinement has failed, double precision
factorization has been performed
+ -1 : the routine fell back to full precision for
implementation- or machine-specific reasons
+ -2 : narrowing the precision induced an overflow,
the routine fell back to full precision
+ -3 : failure of SPOTRF
+ -31: stop the iterative refinement after the 30th iteration
- > 0: iterative refinement has been successfully used.
Returns the number of iterations
@param[out]
info INTEGER
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value
- > 0: if INFO = i, the leading minor of order i of (DOUBLE
PRECISION) A is not positive definite, so the
factorization could not be completed, and the solution
has not been computed.
@ingroup magma_dposv_driver
********************************************************************/
extern "C" magma_int_t
magma_dsposv_gpu(
magma_uplo_t uplo, magma_int_t n, magma_int_t nrhs,
magmaDouble_ptr dA, magma_int_t ldda,
magmaDouble_ptr dB, magma_int_t lddb,
magmaDouble_ptr dX, magma_int_t lddx,
magmaDouble_ptr dworkd, magmaFloat_ptr dworks,
magma_int_t *iter,
magma_int_t *info)
{
#define dB(i,j) (dB + (i) + (j)*lddb)
#define dX(i,j) (dX + (i) + (j)*lddx)
#define dR(i,j) (dR + (i) + (j)*lddr)
#define dSX(i,j) (dSX + (i) + (j)*lddsx)
// Constants
const double BWDMAX = 1.0;
const magma_int_t ITERMAX = 30;
const double c_neg_one = MAGMA_D_NEG_ONE;
const double c_one = MAGMA_D_ONE;
const magma_int_t ione = 1;
// Local variables
magmaDouble_ptr dR;
magmaFloat_ptr dSA, dSX;
double Xnrmv, Rnrmv;
double Anrm, Xnrm, Rnrm, cte, eps;
magma_int_t i, j, iiter, lddsa, lddsx, lddr;
/* Check arguments */
*iter = 0;
*info = 0;
if ( n < 0 )
*info = -1;
else if ( nrhs < 0 )
*info = -2;
else if ( ldda < max(1,n))
*info = -4;
else if ( lddb < max(1,n))
*info = -7;
else if ( lddx < max(1,n))
*info = -9;
if (*info != 0) {
magma_xerbla( __func__, -(*info) );
return *info;
}
if ( n == 0 || nrhs == 0 )
return *info;
lddsa = n;
lddsx = n;
lddr = n;
dSA = dworks;
dSX = dSA + lddsa*n;
dR = dworkd;
magma_queue_t queue;
magma_device_t cdev;
magma_getdevice( &cdev );
magma_queue_create( cdev, &queue );
eps = lapackf77_dlamch("Epsilon");
Anrm = magmablas_dlansy( MagmaInfNorm, uplo, n, dA, ldda, (double*)dworkd, n*nrhs, queue );
cte = Anrm * eps * magma_dsqrt( n ) * BWDMAX;
/*
* Convert to single precision
*/
magmablas_dlag2s( n, nrhs, dB, lddb, dSX, lddsx, queue, info );
if (*info != 0) {
*iter = -2;
goto fallback;
}
magmablas_dlat2s( uplo, n, dA, ldda, dSA, lddsa, queue, info );
if (*info != 0) {
*iter = -2;
goto fallback;
}
// factor dSA in single precision
magma_spotrf_gpu( uplo, n, dSA, lddsa, info );
if (*info != 0) {
*iter = -3;
goto fallback;
}
// solve dSA*dSX = dB in single precision
magma_spotrs_gpu( uplo, n, nrhs, dSA, lddsa, dSX, lddsx, info );
// residual dR = dB - dA*dX in double precision
magmablas_slag2d( n, nrhs, dSX, lddsx, dX, lddx, queue, info );
magmablas_dlacpy( MagmaFull, n, nrhs, dB, lddb, dR, lddr, queue );
if ( nrhs == 1 ) {
magma_dsymv( uplo, n,
c_neg_one, dA, ldda,
dX, 1,
c_one, dR, 1, queue );
}
else {
magma_dsymm( MagmaLeft, uplo, n, nrhs,
c_neg_one, dA, ldda,
dX, lddx,
c_one, dR, lddr, queue );
}
// TODO: use MAGMA_D_ABS( dX(i,j) ) instead of dlange?
for( j=0; j < nrhs; j++ ) {
i = magma_idamax( n, dX(0,j), 1, queue ) - 1;
magma_dgetmatrix( 1, 1, dX(i,j), 1, &Xnrmv, 1, queue );
Xnrm = lapackf77_dlange( "F", &ione, &ione, &Xnrmv, &ione, NULL );
i = magma_idamax( n, dR(0,j), 1, queue ) - 1;
magma_dgetmatrix( 1, 1, dR(i,j), 1, &Rnrmv, 1, queue );
Rnrm = lapackf77_dlange( "F", &ione, &ione, &Rnrmv, &ione, NULL );
if ( Rnrm > Xnrm*cte ) {
goto refinement;
}
}
*iter = 0;
goto cleanup;
//return *info;
refinement:
for( iiter=1; iiter < ITERMAX; ) {
*info = 0;
// convert residual dR to single precision dSX
magmablas_dlag2s( n, nrhs, dR, lddr, dSX, lddsx, queue, info );
if (*info != 0) {
*iter = -2;
goto fallback;
}
// solve dSA*dSX = R in single precision
magma_spotrs_gpu( uplo, n, nrhs, dSA, lddsa, dSX, lddsx, info );
// Add correction and setup residual
// dX += dSX [including conversion] --and--
// dR = dB
for( j=0; j < nrhs; j++ ) {
magmablas_dsaxpycp( n, dSX(0,j), dX(0,j), dB(0,j), dR(0,j), queue );
}
// residual dR = dB - dA*dX in double precision
if ( nrhs == 1 ) {
magma_dsymv( uplo, n,
c_neg_one, dA, ldda,
dX, 1,
c_one, dR, 1, queue );
}
else {
magma_dsymm( MagmaLeft, uplo, n, nrhs,
c_neg_one, dA, ldda,
dX, lddx,
c_one, dR, lddr, queue );
}
// TODO: use MAGMA_D_ABS( dX(i,j) ) instead of dlange?
/* Check whether the nrhs normwise backward errors satisfy the
* stopping criterion. If yes, set ITER=IITER > 0 and return. */
for( j=0; j < nrhs; j++ ) {
i = magma_idamax( n, dX(0,j), 1, queue ) - 1;
magma_dgetmatrix( 1, 1, dX(i,j), 1, &Xnrmv, 1, queue );
Xnrm = lapackf77_dlange( "F", &ione, &ione, &Xnrmv, &ione, NULL );
i = magma_idamax( n, dR(0,j), 1, queue ) - 1;
magma_dgetmatrix( 1, 1, dR(i,j), 1, &Rnrmv, 1, queue );
Rnrm = lapackf77_dlange( "F", &ione, &ione, &Rnrmv, &ione, NULL );
if ( Rnrm > Xnrm*cte ) {
goto L20;
}
}
/* If we are here, the nrhs normwise backward errors satisfy
* the stopping criterion, we are good to exit. */
*iter = iiter;
goto cleanup;
//return *info;
L20:
iiter++;
}
/* If we are at this place of the code, this is because we have
* performed ITER=ITERMAX iterations and never satisified the
* stopping criterion. Set up the ITER flag accordingly and follow
* up on double precision routine. */
*iter = -ITERMAX - 1;
fallback:
/* Single-precision iterative refinement failed to converge to a
* satisfactory solution, so we resort to double precision. */
magma_dpotrf_gpu( uplo, n, dA, ldda, info );
if (*info == 0) {
magmablas_dlacpy( MagmaFull, n, nrhs, dB, lddb, dX, lddx, queue );
magma_dpotrs_gpu( uplo, n, nrhs, dA, ldda, dX, lddx, info );
}
cleanup:
magma_queue_destroy( queue );
return *info;
}
/**
Purpose
-------
DSGEQRSV solves the least squares problem
min || A*X - B ||,
where A is an M-by-N matrix and X and B are M-by-NRHS matrices.
DSGEQRSV first attempts to factorize the matrix in real SINGLE PRECISION
and use this factorization within an iterative refinement procedure
to produce a solution with real DOUBLE PRECISION norm-wise backward error
quality (see below). If the approach fails the method switches to a
real DOUBLE PRECISION factorization and solve.
The iterative refinement is not going to be a winning strategy if
the ratio real SINGLE PRECISION performance over real DOUBLE PRECISION
performance is too small. A reasonable strategy should take the
number of right-hand sides and the size of the matrix into account.
This might be done with a call to ILAENV in the future. Up to now, we
always try iterative refinement.
The iterative refinement process is stopped if
ITER > ITERMAX
or for all the RHS we have:
RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX
where
o ITER is the number of the current iteration in the iterative
refinement process
o RNRM is the infinity-norm of the residual
o XNRM is the infinity-norm of the solution
o ANRM is the infinity-operator-norm of the matrix A
o EPS is the machine epsilon returned by DLAMCH('Epsilon')
The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00 respectively.
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. M >= N >= 0.
@param[in]
nrhs INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
@param[in,out]
dA DOUBLE PRECISION array on the GPU, dimension (LDDA,N)
On entry, the M-by-N coefficient matrix A.
On exit, if iterative refinement has been successfully used
(info.EQ.0 and ITER.GE.0, see description below), A is
unchanged. If double precision factorization has been used
(info.EQ.0 and ITER.LT.0, see description below), then the
array dA contains the QR factorization of A as returned by
function DGEQRF_GPU.
@param[in]
ldda INTEGER
The leading dimension of the array dA. LDDA >= max(1,M).
@param[in,out]
dB DOUBLE PRECISION array on the GPU, dimension (LDDB,NRHS)
The M-by-NRHS right hand side matrix B.
May be overwritten (e.g., if refinement fails).
@param[in]
lddb INTEGER
The leading dimension of the array dB. LDDB >= max(1,M).
@param[out]
dX DOUBLE PRECISION array on the GPU, dimension (LDDX,NRHS)
If info = 0, the N-by-NRHS solution matrix X.
@param[in]
lddx INTEGER
The leading dimension of the array dX. LDDX >= max(1,N).
@param[out]
iter INTEGER
- < 0: iterative refinement has failed, double precision
factorization has been performed
+ -1 : the routine fell back to full precision for
implementation- or machine-specific reasons
+ -2 : narrowing the precision induced an overflow,
the routine fell back to full precision
+ -3 : failure of SGEQRF
+ -31: stop the iterative refinement after the 30th iteration
- > 0: iterative refinement has been successfully used.
Returns the number of iterations
@param[out]
info INTEGER
- = 0: successful exit
- < 0: if info = -i, the i-th argument had an illegal value
@ingroup magma_dgels_driver
********************************************************************/
extern "C" magma_int_t
magma_dsgeqrsv_gpu(
magma_int_t m, magma_int_t n, magma_int_t nrhs,
magmaDouble_ptr dA, magma_int_t ldda,
magmaDouble_ptr dB, magma_int_t lddb,
magmaDouble_ptr dX, magma_int_t lddx,
magma_int_t *iter,
magma_int_t *info)
{
#define dB(i,j) (dB + (i) + (j)*lddb)
#define dX(i,j) (dX + (i) + (j)*lddx)
#define dR(i,j) (dR + (i) + (j)*lddr)
#define dSX(i,j) (dSX + (i) + (j)*lddsx)
double c_neg_one = MAGMA_D_NEG_ONE;
double c_one = MAGMA_D_ONE;
magma_int_t ione = 1;
double *hworkd;
float *hworks;
double *tau;
float *stau;
magmaDouble_ptr dworkd;
magmaFloat_ptr dworks;
magmaDouble_ptr dR, dT;
magmaFloat_ptr dSA, dSX, dST;
double Xnrmv, Rnrmv;
double Anrm, Xnrm, Rnrm, cte, eps;
magma_int_t i, j, iiter, lddsa, lddsx, lddr, nb, lhwork, minmn, size, ldworkd;
/* Check arguments */
*iter = 0;
*info = 0;
if ( m < 0 )
*info = -1;
else if ( n < 0 || n > m )
*info = -2;
else if ( nrhs < 0 )
*info = -3;
else if ( ldda < max(1,m))
*info = -5;
else if ( lddb < max(1,m))
*info = -7;
else if ( lddx < max(1,n))
*info = -9;
if (*info != 0) {
magma_xerbla( __func__, -(*info) );
return *info;
}
if ( m == 0 || n == 0 || nrhs == 0 )
return *info;
nb = magma_get_sgeqrf_nb(m);
minmn= min(m, n);
/* dSX contains both B and X, so must be max(m or lddb,n). */
lddsa = ldda;
lddsx = max(lddb,n);
lddr = lddb;
/*
* Allocate temporary buffers
*/
/* dworks(dSA + dSX + dST) */
size = lddsa*n + lddsx*nrhs + ( 2*minmn + ((n+31)/32)*32 )*nb;
if (MAGMA_SUCCESS != magma_smalloc( &dworks, size )) {
fprintf(stderr, "Allocation of dworks failed (%d)\n", (int) size);
*info = MAGMA_ERR_DEVICE_ALLOC;
return *info;
}
dSA = dworks;
dSX = dSA + lddsa*n;
dST = dSX + lddsx*nrhs;
/* dworkd(dR) = lddr*nrhs */
ldworkd = lddr*nrhs;
if (MAGMA_SUCCESS != magma_dmalloc( &dworkd, ldworkd )) {
magma_free( dworks );
fprintf(stderr, "Allocation of dworkd failed\n");
*info = MAGMA_ERR_DEVICE_ALLOC;
return *info;
}
dR = dworkd;
/* hworks(workspace for cgeqrs + stau) = min(m,n) + lhworks */
lhwork = (m - n + nb)*(nrhs + nb) + nrhs*nb;
size = lhwork + minmn;
magma_smalloc_cpu( &hworks, size );
if ( hworks == NULL ) {
magma_free( dworks );
magma_free( dworkd );
fprintf(stderr, "Allocation of hworks failed\n");
*info = MAGMA_ERR_HOST_ALLOC;
return *info;
}
stau = hworks + lhwork;
eps = lapackf77_dlamch("Epsilon");
Anrm = magmablas_dlange(MagmaInfNorm, m, n, dA, ldda, (double*)dworkd );
cte = Anrm * eps * pow((double)n, 0.5) * BWDMAX;
/*
* Convert to single precision
*/
magmablas_dlag2s( m, nrhs, dB, lddb, dSX, lddsx, info );
if (*info != 0) {
*iter = -2;
goto FALLBACK;
}
magmablas_dlag2s( m, n, dA, ldda, dSA, lddsa, info );
if (*info != 0) {
*iter = -2;
goto FALLBACK;
}
// factor dSA in single precision
magma_sgeqrf_gpu( m, n, dSA, lddsa, stau, dST, info );
if (*info != 0) {
*iter = -3;
goto FALLBACK;
}
// solve dSA*dSX = dB in single precision
magma_sgeqrs_gpu( m, n, nrhs, dSA, lddsa, stau, dST, dSX, lddsx, hworks, lhwork, info );
if (*info != 0) {
*iter = -3;
goto FALLBACK;
}
// residual dR = dB - dA*dX in double precision
magmablas_slag2d( n, nrhs, dSX, lddsx, dX, lddx, info );
magmablas_dlacpy( MagmaUpperLower, m, nrhs, dB, lddb, dR, lddr );
if ( nrhs == 1 ) {
magma_dgemv( MagmaNoTrans, m, n,
c_neg_one, dA, ldda,
dX, 1,
c_one, dR, 1 );
}
else {
magma_dgemm( MagmaNoTrans, MagmaNoTrans, m, nrhs, n,
c_neg_one, dA, ldda,
dX, lddx,
c_one, dR, lddr );
}
// TODO: use MAGMA_D_ABS( dX(i,j) ) instead of dlange?
for( j=0; j < nrhs; j++ ) {
i = magma_idamax( n, dX(0,j), 1) - 1;
magma_dgetmatrix( 1, 1, dX(i,j), 1, &Xnrmv, 1 );
Xnrm = lapackf77_dlange( "F", &ione, &ione, &Xnrmv, &ione, NULL );
i = magma_idamax ( m, dR(0,j), 1 ) - 1;
magma_dgetmatrix( 1, 1, dR(i,j), 1, &Rnrmv, 1 );
Rnrm = lapackf77_dlange( "F", &ione, &ione, &Rnrmv, &ione, NULL );
if ( Rnrm > Xnrm*cte ) {
goto REFINEMENT;
}
}
*iter = 0;
/* Free workspaces */
magma_free( dworks );
magma_free( dworkd );
magma_free_cpu( hworks );
return *info;
REFINEMENT:
/* TODO: this iterative refinement algorithm works only for compatibile
* systems (B in colspan of A).
* See Matrix Computations (3rd ed) p. 267 for correct algorithm. */
for( iiter=1; iiter < ITERMAX; ) {
*info = 0;
// convert residual dR to single precision dSX
magmablas_dlag2s( m, nrhs, dR, lddr, dSX, lddsx, info );
if (*info != 0) {
*iter = -2;
goto FALLBACK;
}
// solve dSA*dSX = R in single precision
magma_sgeqrs_gpu( m, n, nrhs, dSA, lddsa, stau, dST, dSX, lddsx, hworks, lhwork, info );
if (*info != 0) {
*iter = -3;
goto FALLBACK;
}
// Add correction and setup residual
// dX += dSX [including conversion] --and--
// dR[1:n] = dB[1:n] (only n rows, not whole m rows! -- useless if m > n)
for( j=0; j < nrhs; j++ ) {
magmablas_dsaxpycp( n, dSX(0,j), dX(0,j), dB(0,j), dR(0,j) );
}
// dR = dB (whole m rows)
magmablas_dlacpy( MagmaUpperLower, m, nrhs, dB, lddb, dR, lddr );
// residual dR = dB - dA*dX in double precision
if ( nrhs == 1 ) {
magma_dgemv( MagmaNoTrans, m, n,
c_neg_one, dA, ldda,
dX, 1,
c_one, dR, 1 );
}
else {
magma_dgemm( MagmaNoTrans, MagmaNoTrans, m, nrhs, n,
c_neg_one, dA, ldda,
dX, lddx,
c_one, dR, lddr );
}
/* Check whether the nrhs normwise backward errors satisfy the
* stopping criterion. If yes, set ITER=IITER > 0 and return. */
for( j=0; j < nrhs; j++ ) {
i = magma_idamax( n, dX(0,j), 1) - 1;
magma_dgetmatrix( 1, 1, dX(i,j), 1, &Xnrmv, 1 );
Xnrm = lapackf77_dlange( "F", &ione, &ione, &Xnrmv, &ione, NULL );
i = magma_idamax ( m, dR(0,j), 1 ) - 1;
magma_dgetmatrix( 1, 1, dR(i,j), 1, &Rnrmv, 1 );
Rnrm = lapackf77_dlange( "F", &ione, &ione, &Rnrmv, &ione, NULL );
if ( Rnrm > Xnrm*cte ) {
goto L20;
}
}
/* If we are here, the nrhs normwise backward errors satisfy
* the stopping criterion, we are good to exit. */
*iter = iiter;
/* Free workspaces */
magma_free( dworks );
magma_free( dworkd );
magma_free_cpu( hworks );
return *info;
L20:
iiter++;
}
/* If we are at this place of the code, this is because we have
* performed ITER=ITERMAX iterations and never satisified the
* stopping criterion. Set up the ITER flag accordingly and follow
* up on double precision routine. */
*iter = -ITERMAX - 1;
FALLBACK:
/* Single-precision iterative refinement failed to converge to a
* satisfactory solution, so we resort to double precision. */
magma_free( dworks );
magma_free_cpu( hworks );
/*
* Allocate temporary buffers
*/
/* dworkd = dT for dgeqrf */
nb = magma_get_dgeqrf_nb( m );
size = (2*min(m, n) + (n+31)/32*32 )*nb;
if ( size > ldworkd ) {
magma_free( dworkd );
if (MAGMA_SUCCESS != magma_dmalloc( &dworkd, size )) {
fprintf(stderr, "Allocation of dworkd2 failed\n");
*info = MAGMA_ERR_DEVICE_ALLOC;
return *info;
}
}
dT = dworkd;
/* hworkd(dtau + workspace for dgeqrs) = min(m,n) + lhwork */
size = lhwork + minmn;
magma_dmalloc_cpu( &hworkd, size );
if ( hworkd == NULL ) {
magma_free( dworkd );
fprintf(stderr, "Allocation of hworkd2 failed\n");
*info = MAGMA_ERR_HOST_ALLOC;
return *info;
}
tau = hworkd + lhwork;
magma_dgeqrf_gpu( m, n, dA, ldda, tau, dT, info );
if (*info == 0) {
// if m > n, then dB won't fit in dX, so solve with dB and copy n rows to dX
magma_dgeqrs_gpu( m, n, nrhs, dA, ldda, tau, dT, dB, lddb, hworkd, lhwork, info );
magmablas_dlacpy( MagmaUpperLower, n, nrhs, dB, lddb, dX, lddx );
}
magma_free( dworkd );
magma_free_cpu( hworkd );
return *info;
}
/**
Purpose
-------
DSGESV computes the solution to a real system of linear equations
A * X = B, A**T * X = B, or A**H * X = B,
where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
DSGESV first attempts to factorize the matrix in real SINGLE PRECISION
and use this factorization within an iterative refinement procedure
to produce a solution with real DOUBLE PRECISION norm-wise backward error
quality (see below). If the approach fails the method switches to a
real DOUBLE PRECISION factorization and solve.
The iterative refinement is not going to be a winning strategy if
the ratio real SINGLE PRECISION performance over real DOUBLE PRECISION
performance is too small. A reasonable strategy should take the
number of right-hand sides and the size of the matrix into account.
This might be done with a call to ILAENV in the future. Up to now, we
always try iterative refinement.
The iterative refinement process is stopped if
ITER > ITERMAX
or for all the RHS we have:
RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX
where
o ITER is the number of the current iteration in the iterative
refinement process
o RNRM is the infinity-norm of the residual
o XNRM is the infinity-norm of the solution
o ANRM is the infinity-operator-norm of the matrix A
o EPS is the machine epsilon returned by DLAMCH('Epsilon')
The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00 respectively.
Arguments
---------
@param[in]
trans magma_trans_t
Specifies the form of the system of equations:
- = MagmaNoTrans: A * X = B (No transpose)
- = MagmaTrans: A**T * X = B (Transpose)
- = MagmaConjTrans: A**H * X = B (Conjugate transpose)
@param[in]
n INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.
@param[in]
nrhs INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
@param[in,out]
dA DOUBLE PRECISION array on the GPU, dimension (ldda,N)
On entry, the N-by-N coefficient matrix A.
On exit, if iterative refinement has been successfully used
(info.EQ.0 and ITER.GE.0, see description below), A is
unchanged. If double precision factorization has been used
(info.EQ.0 and ITER.LT.0, see description below), then the
array dA contains the factors L and U from the factorization
A = P*L*U; the unit diagonal elements of L are not stored.
@param[in]
ldda INTEGER
The leading dimension of the array dA. ldda >= max(1,N).
@param[out]
ipiv INTEGER array, dimension (N)
The pivot indices that define the permutation matrix P;
row i of the matrix was interchanged with row IPIV(i).
Corresponds either to the single precision factorization
(if info.EQ.0 and ITER.GE.0) or the double precision
factorization (if info.EQ.0 and ITER.LT.0).
@param[out]
dipiv INTEGER array on the GPU, dimension (N)
The pivot indices; for 1 <= i <= N, after permuting, row i of the
matrix was moved to row dIPIV(i).
Note this is different than IPIV, where interchanges
are applied one-after-another.
@param[in]
dB DOUBLE PRECISION array on the GPU, dimension (lddb,NRHS)
The N-by-NRHS right hand side matrix B.
@param[in]
lddb INTEGER
The leading dimension of the array dB. lddb >= max(1,N).
@param[out]
dX DOUBLE PRECISION array on the GPU, dimension (lddx,NRHS)
If info = 0, the N-by-NRHS solution matrix X.
@param[in]
lddx INTEGER
The leading dimension of the array dX. lddx >= max(1,N).
@param
dworkd (workspace) DOUBLE PRECISION array on the GPU, dimension (N*NRHS)
This array is used to hold the residual vectors.
@param
dworks (workspace) SINGLE PRECISION array on the GPU, dimension (N*(N+NRHS))
This array is used to store the real single precision matrix
and the right-hand sides or solutions in single precision.
@param[out]
iter INTEGER
- < 0: iterative refinement has failed, double precision
factorization has been performed
+ -1 : the routine fell back to full precision for
implementation- or machine-specific reasons
+ -2 : narrowing the precision induced an overflow,
the routine fell back to full precision
+ -3 : failure of SGETRF
+ -31: stop the iterative refinement after the 30th iteration
- > 0: iterative refinement has been successfully used.
Returns the number of iterations
@param[out]
info INTEGER
- = 0: successful exit
- < 0: if info = -i, the i-th argument had an illegal value
- > 0: if info = i, U(i,i) computed in DOUBLE PRECISION is
exactly zero. The factorization has been completed,
but the factor U is exactly singular, so the solution
could not be computed.
@ingroup magma_dgesv_driver
********************************************************************/
extern "C" magma_int_t
magma_dsgesv_gpu(magma_trans_t trans, magma_int_t n, magma_int_t nrhs,
double *dA, magma_int_t ldda,
magma_int_t *ipiv, magma_int_t *dipiv,
double *dB, magma_int_t lddb,
double *dX, magma_int_t lddx,
double *dworkd, float *dworks,
magma_int_t *iter, magma_int_t *info)
{
#define dB(i,j) (dB + (i) + (j)*lddb)
#define dX(i,j) (dX + (i) + (j)*lddx)
#define dR(i,j) (dR + (i) + (j)*lddr)
double c_neg_one = MAGMA_D_NEG_ONE;
double c_one = MAGMA_D_ONE;
magma_int_t ione = 1;
double *dR;
float *dSA, *dSX;
double Xnrmv, Rnrmv;
double Anrm, Xnrm, Rnrm, cte, eps;
magma_int_t i, j, iiter, lddsa, lddr;
/* Check arguments */
*iter = 0;
*info = 0;
if ( n < 0 )
*info = -1;
else if ( nrhs < 0 )
*info = -2;
else if ( ldda < max(1,n))
*info = -4;
else if ( lddb < max(1,n))
*info = -8;
else if ( lddx < max(1,n))
*info = -10;
if (*info != 0) {
magma_xerbla( __func__, -(*info) );
return *info;
}
if ( n == 0 || nrhs == 0 )
return *info;
lddsa = n;
lddr = n;
dSA = dworks;
dSX = dSA + lddsa*n;
dR = dworkd;
eps = lapackf77_dlamch("Epsilon");
Anrm = magmablas_dlange(MagmaInfNorm, n, n, dA, ldda, (double*)dworkd );
cte = Anrm * eps * pow((double)n, 0.5) * BWDMAX;
/*
* Convert to single precision
*/
//magmablas_dlag2s( n, nrhs, dB, lddb, dSX, lddsx, info ); // done inside dsgetrs with pivots
if (*info != 0) {
*iter = -2;
goto FALLBACK;
}
magmablas_dlag2s( n, n, dA, ldda, dSA, lddsa, info );
if (*info != 0) {
*iter = -2;
goto FALLBACK;
}
// factor dSA in single precision
magma_sgetrf_gpu( n, n, dSA, lddsa, ipiv, info );
if (*info != 0) {
*iter = -3;
goto FALLBACK;
}
// Generate parallel pivots
{
magma_int_t *newipiv;
magma_imalloc_cpu( &newipiv, n );
if ( newipiv == NULL ) {
*iter = -3;
goto FALLBACK;
}
swp2pswp( trans, n, ipiv, newipiv );
magma_setvector( n, sizeof(magma_int_t), newipiv, 1, dipiv, 1 );
magma_free_cpu( newipiv );
}
// solve dSA*dSX = dB in single precision
// converts dB to dSX and applies pivots, solves, then converts result back to dX
magma_dsgetrs_gpu( trans, n, nrhs, dSA, lddsa, dipiv, dB, lddb, dX, lddx, dSX, info );
// residual dR = dB - dA*dX in double precision
magmablas_dlacpy( MagmaUpperLower, n, nrhs, dB, lddb, dR, lddr );
if ( nrhs == 1 ) {
magma_dgemv( trans, n, n,
c_neg_one, dA, ldda,
dX, 1,
c_one, dR, 1 );
}
else {
magma_dgemm( trans, MagmaNoTrans, n, nrhs, n,
c_neg_one, dA, ldda,
dX, lddx,
c_one, dR, lddr );
}
// TODO: use MAGMA_D_ABS( dX(i,j) ) instead of dlange?
for( j=0; j < nrhs; j++ ) {
i = magma_idamax( n, dX(0,j), 1) - 1;
magma_dgetmatrix( 1, 1, dX(i,j), 1, &Xnrmv, 1 );
Xnrm = lapackf77_dlange( "F", &ione, &ione, &Xnrmv, &ione, NULL );
i = magma_idamax ( n, dR(0,j), 1 ) - 1;
magma_dgetmatrix( 1, 1, dR(i,j), 1, &Rnrmv, 1 );
Rnrm = lapackf77_dlange( "F", &ione, &ione, &Rnrmv, &ione, NULL );
if ( Rnrm > Xnrm*cte ) {
goto REFINEMENT;
}
}
*iter = 0;
return *info;
REFINEMENT:
for( iiter=1; iiter < ITERMAX; ) {
*info = 0;
// convert residual dR to single precision dSX
// solve dSA*dSX = R in single precision
// convert result back to double precision dR
// it's okay that dR is used for both dB input and dX output.
magma_dsgetrs_gpu( trans, n, nrhs, dSA, lddsa, dipiv, dR, lddr, dR, lddr, dSX, info );
if (*info != 0) {
*iter = -3;
goto FALLBACK;
}
// Add correction and setup residual
// dX += dR --and--
// dR = dB
// This saves going through dR a second time (if done with one more kernel).
// -- not really: first time is read, second time is write.
for( j=0; j < nrhs; j++ ) {
magmablas_daxpycp( n, dR(0,j), dX(0,j), dB(0,j) );
}
// residual dR = dB - dA*dX in double precision
if ( nrhs == 1 ) {
magma_dgemv( trans, n, n,
c_neg_one, dA, ldda,
dX, 1,
c_one, dR, 1 );
}
else {
magma_dgemm( trans, MagmaNoTrans, n, nrhs, n,
c_neg_one, dA, ldda,
dX, lddx,
c_one, dR, lddr );
}
/* Check whether the nrhs normwise backward errors satisfy the
* stopping criterion. If yes, set ITER=IITER > 0 and return. */
for( j=0; j < nrhs; j++ ) {
i = magma_idamax( n, dX(0,j), 1) - 1;
magma_dgetmatrix( 1, 1, dX(i,j), 1, &Xnrmv, 1 );
Xnrm = lapackf77_dlange( "F", &ione, &ione, &Xnrmv, &ione, NULL );
i = magma_idamax ( n, dR(0,j), 1 ) - 1;
magma_dgetmatrix( 1, 1, dR(i,j), 1, &Rnrmv, 1 );
Rnrm = lapackf77_dlange( "F", &ione, &ione, &Rnrmv, &ione, NULL );
if ( Rnrm > Xnrm*cte ) {
goto L20;
}
}
/* If we are here, the nrhs normwise backward errors satisfy
* the stopping criterion, we are good to exit. */
*iter = iiter;
return *info;
L20:
iiter++;
}
/* If we are at this place of the code, this is because we have
* performed ITER=ITERMAX iterations and never satisified the
* stopping criterion. Set up the ITER flag accordingly and follow
* up on double precision routine. */
*iter = -ITERMAX - 1;
FALLBACK:
/* Single-precision iterative refinement failed to converge to a
* satisfactory solution, so we resort to double precision. */
magma_dgetrf_gpu( n, n, dA, ldda, ipiv, info );
if (*info == 0) {
magmablas_dlacpy( MagmaUpperLower, n, nrhs, dB, lddb, dX, lddx );
magma_dgetrs_gpu( trans, n, nrhs, dA, ldda, ipiv, dX, lddx, info );
}
return *info;
}
extern "C" magma_int_t
magma_dsgetrs_gpu(char trans, magma_int_t n, magma_int_t nrhs,
float *dA, magma_int_t ldda,
magma_int_t *ipiv,
double *dB, magma_int_t lddb,
double *dX, magma_int_t lddx,
float *dSX,
magma_int_t *info)
{
/* -- MAGMA (version 1.4.0) --
Univ. of Tennessee, Knoxville
Univ. of California, Berkeley
Univ. of Colorado, Denver
August 2013
Purpose
=======
DSGETRS solves a system of linear equations
A * X = B or A' * X = B
with a general N-by-N matrix A using the LU factorization computed
by MAGMA_SGETRF_GPU. B and X are in DOUBLE PRECISION, and A is in SINGLE PRECISION.
This routine is used in the mixed precision iterative solver
magma_dsgesv.
Arguments
=========
TRANS (input) CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A'* X = B (Transpose)
= 'C': A'* X = B (Conjugate transpose = Transpose)
N (input) INTEGER
The order of the matrix A. N >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
dA (input) SINGLE PRECISION array on the GPU, dimension (LDDA,N)
The factors L and U from the factorization A = P*L*U
as computed by CGETRF_GPU.
LDDA (input) INTEGER
The leading dimension of the array dA. LDDA >= max(1,N).
IPIV (input) INTEGER array on the GPU, dimension (N)
The pivot indices from CGETRF_GPU; Row i of the
matrix was moved to row IPIV(i).
dB (input) DOUBLE PRECISION array on the GPU, dimension (LDDB,NRHS)
On entry, the right hand side matrix B.
LDDB (input) INTEGER
The leading dimension of the arrays X and B. LDDB >= max(1,N).
dX (output) DOUBLE PRECISION array on the GPU, dimension (LDDX, NRHS)
On exit, the solution matrix dX.
LDDX (input) INTEGER
The leading dimension of the array dX, LDDX >= max(1,N).
dSX (workspace) SINGLE PRECISION array on the GPU used as workspace,
dimension (N, NRHS)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
===================================================================== */
float c_one = MAGMA_S_ONE;
char trans_[2] = {trans, 0};
int notran = lapackf77_lsame(trans_, "N");
magma_int_t inc;
*info = 0;
if ( (! notran) &&
(! lapackf77_lsame(trans_, "T")) &&
(! lapackf77_lsame(trans_, "C")) ) {
*info = -1;
} else if (n < 0) {
*info = -2;
} else if (nrhs < 0) {
*info = -3;
} else if (ldda < n) {
*info = -5;
} else if (lddb < n) {
*info = -8;
} else if (lddx < n) {
*info = -10;
} else if (lddx != lddb) { /* TODO: remove it when dslaswp will have the correct interface */
*info = -10;
}
if (*info != 0) {
magma_xerbla( __func__, -(*info) );
return *info;
}
/* Quick return if possible */
if (n == 0 || nrhs == 0) {
return *info;
}
if (notran) {
inc = 1;
/* Get X by row applying interchanges to B and cast to single */
/*
* TODO: clean dslaswp interface to have interface closer to zlaswp
*/
//magmablas_dslaswp(nrhs, dB, lddb, dSX, lddbx, 1, n, ipiv);
magmablas_dslaswp(nrhs, dB, lddb, dSX, n, ipiv, inc);
/* Solve L*X = B, overwriting B with SX. */
magma_strsm( MagmaLeft, MagmaLower, MagmaNoTrans, MagmaUnit,
n, nrhs, c_one, dA, ldda, dSX, n);
/* Solve U*X = B, overwriting B with X. */
magma_strsm( MagmaLeft, MagmaUpper, MagmaNoTrans, MagmaNonUnit,
n, nrhs, c_one, dA, ldda, dSX, n);
magmablas_slag2d( n, nrhs, dSX, n, dX, lddx, info );
}
else {
inc = -1;
/* Cast the DOUBLE PRECISION RHS to SINGLE PRECISION */
magmablas_dlag2s( n, nrhs, dB, lddb, dSX, n, info );
/* Solve A' * X = B. */
magma_strsm( MagmaLeft, MagmaUpper, MagmaTrans, MagmaNonUnit,
n, nrhs, c_one, dA, ldda, dSX, n );
magma_strsm( MagmaLeft, MagmaLower, MagmaTrans, MagmaUnit,
n, nrhs, c_one, dA, ldda, dSX, n );
magmablas_dslaswp( nrhs, dX, lddx, dSX, n, ipiv, inc );
}
return *info;
} /* magma_dsgetrs */