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
-------
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;
}