int main ()
{
double a[N][N], b[N][N], c[N][N], c_ref[N][N];
init(a, b, N);
work(a, b, c, N);
work_ref(a, b, c_ref, N);
check(c, c_ref);
return 0;
}
int main ()
{
double a[N], a_ref[N], b[N], res, ref, diff;
init(a, a_ref, b, N);
res = work(a, b, N);
ref = work_ref(a_ref, b, N);
diff = res - ref;
if (diff > EPS || -diff > EPS)
abort ();
return 0;
}
extern "C" magma_int_t
magma_zgeqrf2_2q_gpu(
magma_int_t m, magma_int_t n,
magmaDoubleComplex_ptr dA, size_t dA_offset, magma_int_t ldda,
magmaDoubleComplex *tau,
magma_queue_t* queues,
magma_int_t *info)
{
/* -- clMAGMA (version 1.3.0) --
Univ. of Tennessee, Knoxville
Univ. of California, Berkeley
Univ. of Colorado, Denver
@date November 2014
Purpose
=======
ZGEQRF computes a QR factorization of a complex M-by-N matrix A:
A = Q * R.
Arguments
=========
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
dA (input/output) COMPLEX_16 array on the GPU, dimension (LDDA,N)
On entry, the M-by-N matrix dA.
On exit, the elements on and above the diagonal of the array
contain the min(M,N)-by-N upper trapezoidal matrix R (R is
upper triangular if m >= n); the elements below the diagonal,
with the array TAU, represent the orthogonal matrix Q as a
product of min(m,n) elementary reflectors (see Further
Details).
LDDA (input) INTEGER
The leading dimension of the array dA. LDDA >= max(1,M).
To benefit from coalescent memory accesses LDDA must be
divisible by 16.
TAU (output) COMPLEX_16 array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
or another error occured, such as memory allocation failed.
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 complex scalar, and v is a complex 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).
===================================================================== */
#define dA(a_1,a_2) dA, (dA_offset + (a_1) + (a_2)*(ldda))
#define work_ref(a_1) ( work + (a_1))
#define hwork ( work + (nb)*(m))
magmaDoubleComplex_ptr dwork;
magmaDoubleComplex *work;
magma_int_t i, k, ldwork, lddwork, old_i, old_ib, rows;
magma_int_t nbmin, nx, ib, nb;
magma_int_t lhwork, lwork;
*info = 0;
if (m < 0) {
*info = -1;
} else if (n < 0) {
*info = -2;
} else if (ldda < max(1,m)) {
*info = -4;
}
if (*info != 0) {
magma_xerbla( __func__, -(*info) );
return *info;
}
k = min(m,n);
if (k == 0)
return MAGMA_SUCCESS;
nb = magma_get_zgeqrf_nb(m);
lwork = (m+n) * nb;
lhwork = lwork - (m)*nb;
if ( MAGMA_SUCCESS != magma_zmalloc( &dwork, n*nb )) {
*info = MAGMA_ERR_DEVICE_ALLOC;
return *info;
}
/*
if ( MAGMA_SUCCESS != magma_zmalloc_cpu( &work, lwork ) ) {
*info = MAGMA_ERR_HOST_ALLOC;
magma_free( dwork );
return *info;
}
*/
cl_mem buffer = clCreateBuffer(gContext, CL_MEM_READ_WRITE | CL_MEM_ALLOC_HOST_PTR, sizeof(magmaDoubleComplex)*lwork, NULL, NULL);
work = (magmaDoubleComplex*)clEnqueueMapBuffer(queues[0], buffer, CL_TRUE, CL_MAP_READ | CL_MAP_WRITE, 0, lwork*sizeof(magmaDoubleComplex), 0, NULL, NULL, NULL);
nbmin = 2;
nx = 2*nb;
ldwork = m;
lddwork= n;
if (nb >= nbmin && nb < k && nx < k) {
/* Use blocked code initially */
old_i = 0; old_ib = nb;
for (i = 0; i < k-nx; i += nb) {
ib = min(k-i, nb);
rows = m -i;
magma_zgetmatrix_async(rows, ib, dA(i, i), ldda, work_ref(i), ldwork, queues[0], NULL);
clFlush(queues[0]);
if (i>0){
/* Apply H' to A(i:m,i+2*ib:n) from the left */
magma_zlarfb_gpu( MagmaLeft, MagmaConjTrans, MagmaForward, MagmaColumnwise,
m-old_i, n-old_i-2*old_ib, old_ib,
dA(old_i, old_i ), ldda, dwork,0, lddwork,
dA(old_i, old_i+2*old_ib), ldda, dwork,old_ib, lddwork, queues[1]);
magma_zsetmatrix_async( old_ib, old_ib, work_ref(old_i), ldwork,
dA(old_i, old_i), ldda, queues[1], NULL);
clFlush(queues[1]);
}
magma_queue_sync(queues[0]);
lapackf77_zgeqrf(&rows, &ib, work_ref(i), &ldwork, tau+i, hwork, &lhwork, info);
/* Form the triangular factor of the block reflector
H = H(i) H(i+1) . . . H(i+ib-1) */
lapackf77_zlarft( MagmaForwardStr, MagmaColumnwiseStr,
&rows, &ib,
work_ref(i), &ldwork, tau+i, hwork, &ib);
zpanel_to_q( MagmaUpper, ib, work_ref(i), ldwork, hwork+ib*ib );
magma_zsetmatrix( rows, ib, work_ref(i), ldwork, dA(i,i), ldda, queues[0]);
zq_to_panel( MagmaUpper, ib, work_ref(i), ldwork, hwork+ib*ib );
if (i + ib < n)
{
magma_zsetmatrix( ib, ib, hwork, ib, dwork, 0, lddwork, queues[1]);
if (i+nb < k-nx){
/* Apply H' to A(i:m,i+ib:i+2*ib) from the left */
magma_zlarfb_gpu( MagmaLeft, MagmaConjTrans, MagmaForward, MagmaColumnwise,
rows, ib, ib,
dA(i, i ), ldda, dwork,0, lddwork,
dA(i, i+ib), ldda, dwork,ib, lddwork, queues[1]);
magma_queue_sync(queues[1]);
}else {
magma_zlarfb_gpu( MagmaLeft, MagmaConjTrans, MagmaForward, MagmaColumnwise,
rows, n-i-ib, ib,
dA(i, i ), ldda, dwork,0, lddwork,
dA(i, i+ib), ldda, dwork,ib, lddwork, queues[1]);
magma_zsetmatrix( ib, ib, work_ref(i), ldwork, dA(i,i), ldda, queues[1]);
clFlush(queues[1]);
}
old_i = i;
old_ib = ib;
}
}
} else {
i = 0;
}
magma_free(dwork);
/* Use unblocked code to factor the last or only block. */
if (i < k) {
ib = n-i;
rows = m-i;
magma_zgetmatrix( rows, ib, dA(i, i), ldda, work, rows, queues[0]);
lhwork = lwork - rows*ib;
lapackf77_zgeqrf(&rows, &ib, work, &rows, tau+i, work+ib*rows, &lhwork, info);
magma_zsetmatrix( rows, ib, work, rows, dA(i, i), ldda, queues[0]);
}
clEnqueueUnmapMemObject(queues[0], buffer, work, 0, NULL, NULL);
clReleaseMemObject(buffer);
// magma_free_cpu(work);
return *info;
} /* magma_zgeqrf2_gpu */
extern "C" magma_int_t
magma_cgeqrf_gpu( magma_int_t m, magma_int_t n,
magmaFloatComplex *dA, magma_int_t ldda,
magmaFloatComplex *tau, magmaFloatComplex *dT,
magma_int_t *info )
{
/* -- MAGMA (version 1.4.0) --
Univ. of Tennessee, Knoxville
Univ. of California, Berkeley
Univ. of Colorado, Denver
August 2013
Purpose
=======
CGEQRF computes a QR factorization of a complex M-by-N matrix A:
A = Q * R.
This version stores the triangular dT matrices used in
the block QR factorization so that they can be applied directly (i.e.,
without being recomputed) later. As a result, the application
of Q is much faster. Also, the upper triangular matrices for V have 0s
in them. The corresponding parts of the upper triangular R are inverted
and stored separately in dT.
Arguments
=========
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
dA (input/output) COMPLEX array on the GPU, dimension (LDDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(M,N)-by-N upper trapezoidal matrix R (R is
upper triangular if m >= n); the elements below the diagonal,
with the array TAU, represent the orthogonal matrix Q as a
product of min(m,n) elementary reflectors (see Further
Details).
LDDA (input) INTEGER
The leading dimension of the array dA. LDDA >= max(1,M).
To benefit from coalescent memory accesses LDDA must be
dividable by 16.
TAU (output) COMPLEX array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
dT (workspace/output) COMPLEX array on the GPU,
dimension (2*MIN(M, N) + (N+31)/32*32 )*NB,
where NB can be obtained through magma_get_cgeqrf_nb(M).
It starts with MIN(M,N)*NB block that store the triangular T
matrices, followed by the MIN(M,N)*NB block of the diagonal
inverses for the R matrix. The rest of the array is used as workspace.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
or another error occured, such as memory allocation failed.
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 complex scalar, and v is a complex 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).
===================================================================== */
#define a_ref(a_1,a_2) (dA+(a_2)*(ldda) + (a_1))
#define t_ref(a_1) (dT+(a_1)*nb)
#define d_ref(a_1) (dT+(minmn+(a_1))*nb)
#define dd_ref(a_1) (dT+(2*minmn+(a_1))*nb)
#define work_ref(a_1) ( work + (a_1))
#define hwork ( work + (nb)*(m))
magma_int_t i, k, minmn, old_i, old_ib, rows, cols;
magma_int_t ib, nb;
magma_int_t ldwork, lddwork, lwork, lhwork;
magmaFloatComplex *work, *ut;
/* check arguments */
*info = 0;
if (m < 0) {
*info = -1;
} else if (n < 0) {
*info = -2;
} else if (ldda < max(1,m)) {
*info = -4;
}
if (*info != 0) {
magma_xerbla( __func__, -(*info) );
return *info;
}
k = minmn = min(m,n);
if (k == 0)
return *info;
nb = magma_get_cgeqrf_nb(m);
lwork = (m + n + nb)*nb;
lhwork = lwork - m*nb;
if (MAGMA_SUCCESS != magma_cmalloc_pinned( &work, lwork )) {
*info = MAGMA_ERR_HOST_ALLOC;
return *info;
}
ut = hwork+nb*(n);
memset( ut, 0, nb*nb*sizeof(magmaFloatComplex));
magma_queue_t stream[2];
magma_queue_create( &stream[0] );
magma_queue_create( &stream[1] );
ldwork = m;
lddwork= n;
if ( (nb > 1) && (nb < k) ) {
/* Use blocked code initially */
old_i = 0; old_ib = nb;
for (i = 0; i < k-nb; i += nb) {
ib = min(k-i, nb);
rows = m -i;
magma_cgetmatrix_async( rows, ib,
a_ref(i,i), ldda,
work_ref(i), ldwork, stream[1] );
if (i>0){
/* Apply H' to A(i:m,i+2*ib:n) from the left */
cols = n-old_i-2*old_ib;
magma_clarfb_gpu( MagmaLeft, MagmaConjTrans, MagmaForward, MagmaColumnwise,
m-old_i, cols, old_ib,
a_ref(old_i, old_i ), ldda, t_ref(old_i), nb,
a_ref(old_i, old_i+2*old_ib), ldda, dd_ref(0), lddwork);
/* store the diagonal */
magma_csetmatrix_async( old_ib, old_ib,
ut, old_ib,
d_ref(old_i), old_ib, stream[0] );
}
magma_queue_sync( stream[1] );
lapackf77_cgeqrf(&rows, &ib, work_ref(i), &ldwork, tau+i, hwork, &lhwork, info);
/* Form the triangular factor of the block reflector
H = H(i) H(i+1) . . . H(i+ib-1) */
lapackf77_clarft( MagmaForwardStr, MagmaColumnwiseStr,
&rows, &ib,
work_ref(i), &ldwork, tau+i, hwork, &ib);
/* Put 0s in the upper triangular part of a panel (and 1s on the
diagonal); copy the upper triangular in ut and invert it. */
magma_queue_sync( stream[0] );
csplit_diag_block(ib, work_ref(i), ldwork, ut);
magma_csetmatrix( rows, ib, work_ref(i), ldwork, a_ref(i,i), ldda );
if (i + ib < n) {
/* Send the triangular factor T to the GPU */
magma_csetmatrix( ib, ib, hwork, ib, t_ref(i), nb );
if (i+nb < k-nb){
/* Apply H' to A(i:m,i+ib:i+2*ib) from the left */
magma_clarfb_gpu( MagmaLeft, MagmaConjTrans, MagmaForward, MagmaColumnwise,
rows, ib, ib,
a_ref(i, i ), ldda, t_ref(i), nb,
a_ref(i, i+ib), ldda, dd_ref(0), lddwork);
}
else {
cols = n-i-ib;
magma_clarfb_gpu( MagmaLeft, MagmaConjTrans, MagmaForward, MagmaColumnwise,
rows, cols, ib,
a_ref(i, i ), ldda, t_ref(i), nb,
a_ref(i, i+ib), ldda, dd_ref(0), lddwork);
/* Fix the diagonal block */
magma_csetmatrix( ib, ib, ut, ib, d_ref(i), ib );
}
old_i = i;
old_ib = ib;
}
}
} else {
i = 0;
}
/* Use unblocked code to factor the last or only block. */
if (i < k) {
ib = n-i;
rows = m-i;
magma_cgetmatrix( rows, ib, a_ref(i, i), ldda, work, rows );
lhwork = lwork - rows*ib;
lapackf77_cgeqrf(&rows, &ib, work, &rows, tau+i, work+ib*rows, &lhwork, info);
magma_csetmatrix( rows, ib, work, rows, a_ref(i, i), ldda );
}
magma_queue_destroy( stream[0] );
magma_queue_destroy( stream[1] );
magma_free_pinned( work );
return *info;
/* End of MAGMA_CGEQRF */
} /* magma_cgeqrf */
extern "C" magma_int_t
magma_zgeqrf2_gpu( magma_int_t m, magma_int_t n,
magmaDoubleComplex *dA, magma_int_t ldda,
magmaDoubleComplex *tau,
magma_int_t *info )
{
/* -- MAGMA (version 1.4.0) --
Univ. of Tennessee, Knoxville
Univ. of California, Berkeley
Univ. of Colorado, Denver
August 2013
Purpose
=======
ZGEQRF computes a QR factorization of a complex M-by-N matrix A:
A = Q * R.
This version has LAPACK-complaint arguments.
This version assumes the computation runs through the NULL stream
and therefore is not overlapping some computation with communication.
Other versions (magma_zgeqrf_gpu and magma_zgeqrf3_gpu) store the
intermediate T matrices.
Arguments
=========
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
dA (input/output) COMPLEX_16 array on the GPU, dimension (LDDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(M,N)-by-N upper trapezoidal matrix R (R is
upper triangular if m >= n); the elements below the diagonal,
with the array TAU, represent the orthogonal matrix Q as a
product of min(m,n) elementary reflectors (see Further
Details).
LDDA (input) INTEGER
The leading dimension of the array dA. LDDA >= max(1,M).
To benefit from coalescent memory accesses LDDA must be
dividable by 16.
TAU (output) COMPLEX_16 array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
or another error occured, such as memory allocation failed.
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 complex scalar, and v is a complex 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).
===================================================================== */
#define dA(a_1,a_2) ( dA+(a_2)*(ldda) + (a_1))
#define work_ref(a_1) ( work + (a_1))
#define hwork ( work + (nb)*(m))
magmaDoubleComplex *dwork;
magmaDoubleComplex *work;
magma_int_t i, k, ldwork, lddwork, old_i, old_ib, rows;
magma_int_t nbmin, nx, ib, nb;
magma_int_t lhwork, lwork;
/* Function Body */
*info = 0;
if (m < 0) {
*info = -1;
} else if (n < 0) {
*info = -2;
} else if (ldda < max(1,m)) {
*info = -4;
}
if (*info != 0) {
magma_xerbla( __func__, -(*info) );
return *info;
}
k = min(m,n);
if (k == 0)
return *info;
nb = magma_get_zgeqrf_nb(m);
lwork = (m+n) * nb;
lhwork = lwork - (m)*nb;
if (MAGMA_SUCCESS != magma_zmalloc( &dwork, (n)*nb )) {
*info = MAGMA_ERR_DEVICE_ALLOC;
return *info;
}
if (MAGMA_SUCCESS != magma_zmalloc_pinned( &work, lwork )) {
magma_free( dwork );
*info = MAGMA_ERR_HOST_ALLOC;
return *info;
}
magma_queue_t stream[2];
magma_queue_create( &stream[0] );
magma_queue_create( &stream[1] );
nbmin = 2;
nx = nb;
ldwork = m;
lddwork= n;
if (nb >= nbmin && nb < k && nx < k) {
/* Use blocked code initially */
old_i = 0; old_ib = nb;
for (i = 0; i < k-nx; i += nb) {
ib = min(k-i, nb);
rows = m -i;
magma_zgetmatrix_async( rows, ib,
dA(i,i), ldda,
work_ref(i), ldwork, stream[1] );
if (i>0){
/* Apply H' to A(i:m,i+2*ib:n) from the left */
magma_zlarfb_gpu( MagmaLeft, MagmaConjTrans, MagmaForward, MagmaColumnwise,
m-old_i, n-old_i-2*old_ib, old_ib,
dA(old_i, old_i ), ldda, dwork, lddwork,
dA(old_i, old_i+2*old_ib), ldda, dwork+old_ib, lddwork);
magma_zsetmatrix_async( old_ib, old_ib,
work_ref(old_i), ldwork,
dA(old_i, old_i), ldda, stream[0] );
}
magma_queue_sync( stream[1] );
lapackf77_zgeqrf(&rows, &ib, work_ref(i), &ldwork, tau+i, hwork, &lhwork, info);
/* Form the triangular factor of the block reflector
H = H(i) H(i+1) . . . H(i+ib-1) */
lapackf77_zlarft( MagmaForwardStr, MagmaColumnwiseStr,
&rows, &ib,
work_ref(i), &ldwork, tau+i, hwork, &ib);
zpanel_to_q( MagmaUpper, ib, work_ref(i), ldwork, hwork+ib*ib );
magma_zsetmatrix( rows, ib, work_ref(i), ldwork, dA(i,i), ldda );
zq_to_panel( MagmaUpper, ib, work_ref(i), ldwork, hwork+ib*ib );
if (i + ib < n) {
magma_zsetmatrix( ib, ib, hwork, ib, dwork, lddwork );
if (i+nb < k-nx) {
/* Apply H' to A(i:m,i+ib:i+2*ib) from the left */
magma_zlarfb_gpu( MagmaLeft, MagmaConjTrans, MagmaForward, MagmaColumnwise,
rows, ib, ib,
dA(i, i ), ldda, dwork, lddwork,
dA(i, i+ib), ldda, dwork+ib, lddwork);
}
else {
magma_zlarfb_gpu( MagmaLeft, MagmaConjTrans, MagmaForward, MagmaColumnwise,
rows, n-i-ib, ib,
dA(i, i ), ldda, dwork, lddwork,
dA(i, i+ib), ldda, dwork+ib, lddwork);
magma_zsetmatrix( ib, ib,
work_ref(i), ldwork,
dA(i,i), ldda );
}
old_i = i;
old_ib = ib;
}
}
} else {
i = 0;
}
magma_free( dwork );
/* Use unblocked code to factor the last or only block. */
if (i < k) {
ib = n-i;
rows = m-i;
magma_zgetmatrix( rows, ib, dA(i, i), ldda, work, rows );
lhwork = lwork - rows*ib;
lapackf77_zgeqrf(&rows, &ib, work, &rows, tau+i, work+ib*rows, &lhwork, info);
magma_zsetmatrix( rows, ib, work, rows, dA(i, i), ldda );
}
magma_free_pinned( work );
magma_queue_destroy( stream[0] );
magma_queue_destroy( stream[1] );
return *info;
} /* magma_zgeqrf2_gpu */
magma_err_t
magma_sgeqrf2_gpu( magma_int_t m, magma_int_t n,
magmaFloat_ptr dA, size_t dA_offset, magma_int_t ldda,
float *tau, magma_err_t *info,
magma_queue_t queue)
{
/* -- clMAGMA (version 1.0.0) --
Univ. of Tennessee, Knoxville
Univ. of California, Berkeley
Univ. of Colorado, Denver
April 2012
Purpose
=======
SGEQRF computes a QR factorization of a real M-by-N matrix A:
A = Q * R.
Arguments
=========
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
dA (input/output) REAL array on the GPU, dimension (LDDA,N)
On entry, the M-by-N matrix dA.
On exit, the elements on and above the diagonal of the array
contain the min(M,N)-by-N upper trapezoidal matrix R (R is
upper triangular if m >= n); the elements below the diagonal,
with the array TAU, represent the orthogonal matrix Q as a
product of min(m,n) elementary reflectors (see Further
Details).
LDDA (input) INTEGER
The leading dimension of the array dA. LDDA >= max(1,M).
To benefit from coalescent memory accesses LDDA must be
dividable by 16.
TAU (output) REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
if INFO = -9, internal GPU memory allocation failed.
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).
===================================================================== */
#define dA(a_1,a_2) dA, (dA_offset + (a_1) + (a_2)*(ldda))
#define work_ref(a_1) ( work + (a_1))
#define hwork ( work + (nb)*(m))
magmaFloat_ptr dwork;
float *work;
magma_int_t i, k, ldwork, lddwork, old_i, old_ib, rows;
magma_int_t nbmin, nx, ib, nb;
magma_int_t lhwork, lwork;
*info = 0;
if (m < 0) {
*info = -1;
} else if (n < 0) {
*info = -2;
} else if (ldda < max(1,m)) {
*info = -4;
}
if (*info != 0) {
magma_xerbla( __func__, -(*info) );
return *info;
}
k = min(m,n);
if (k == 0)
return MAGMA_SUCCESS;
nb = magma_get_sgeqrf_nb(m);
lwork = (m+n) * nb;
lhwork = lwork - (m)*nb;
if ( MAGMA_SUCCESS != magma_malloc( &dwork, n*nb*sizeof(float))) {
*info = MAGMA_ERR_DEVICE_ALLOC;
return *info;
}
if ( MAGMA_SUCCESS != magma_malloc_host((void**)&work, lwork*sizeof(float)) ) {
*info = MAGMA_ERR_HOST_ALLOC;
magma_free( dwork );
return *info;
}
magma_event_t event[2] = {NULL, NULL};
nbmin = 2;
nx = nb;
ldwork = m;
lddwork= n;
if (nb >= nbmin && nb < k && nx < k) {
/* Use blocked code initially */
old_i = 0; old_ib = nb;
for (i = 0; i < k-nx; i += nb) {
ib = min(k-i, nb);
rows = m -i;
magma_queue_sync( queue );
magma_sgetmatrix_async(rows, ib, dA(i, i), ldda, work_ref(i), 0, ldwork, queue, &event[0]);
if (i>0){
/* Apply H' to A(i:m,i+2*ib:n) from the left */
magma_slarfb_gpu( MagmaLeft, MagmaTrans, MagmaForward, MagmaColumnwise,
m-old_i, n-old_i-2*old_ib, old_ib,
dA(old_i, old_i ), ldda, dwork,0, lddwork,
dA(old_i, old_i+2*old_ib), ldda, dwork,old_ib, lddwork, queue);
magma_ssetmatrix_async( old_ib, old_ib, work_ref(old_i), 0, ldwork,
dA(old_i, old_i), ldda, queue, &event[1]);
}
magma_event_sync(event[0]);
lapackf77_sgeqrf(&rows, &ib, work_ref(i), &ldwork, tau+i, hwork, &lhwork, info);
if (i > 0) {
magma_event_sync(event[1]);
}
/* Form the triangular factor of the block reflector
H = H(i) H(i+1) . . . H(i+ib-1) */
lapackf77_slarft( MagmaForwardStr, MagmaColumnwiseStr,
&rows, &ib,
work_ref(i), &ldwork, tau+i, hwork, &ib);
spanel_to_q( MagmaUpper, ib, work_ref(i), ldwork, hwork+ib*ib );
magma_ssetmatrix(rows, ib, work_ref(i), 0, ldwork, dA(i,i), ldda, queue);
sq_to_panel( MagmaUpper, ib, work_ref(i), ldwork, hwork+ib*ib );
if (i + ib < n)
{
magma_ssetmatrix(ib, ib, hwork, 0, ib, dwork, 0, lddwork, queue);
if (i+nb < k-nx)
/* Apply H' to A(i:m,i+ib:i+2*ib) from the left */
magma_slarfb_gpu( MagmaLeft, MagmaTrans, MagmaForward, MagmaColumnwise,
rows, ib, ib,
dA(i, i ), ldda, dwork,0, lddwork,
dA(i, i+ib), ldda, dwork,ib, lddwork, queue);
else {
magma_slarfb_gpu( MagmaLeft, MagmaTrans, MagmaForward, MagmaColumnwise,
rows, n-i-ib, ib,
dA(i, i ), ldda, dwork,0, lddwork,
dA(i, i+ib), ldda, dwork,ib, lddwork, queue);
magma_ssetmatrix(ib, ib, work_ref(i), 0, ldwork, dA(i,i), ldda, queue);
}
old_i = i;
old_ib = ib;
}
}
} else {
i = 0;
}
magma_free(dwork);
/* Use unblocked code to factor the last or only block. */
if (i < k) {
ib = n-i;
rows = m-i;
magma_sgetmatrix(rows, ib, dA(i, i), ldda, work, 0, rows, queue);
lhwork = lwork - rows*ib;
lapackf77_sgeqrf(&rows, &ib, work, &rows, tau+i, work+ib*rows, &lhwork, info);
magma_ssetmatrix(rows, ib, work, 0, rows, dA(i, i), ldda, queue);
}
magma_free_host(work);
return *info;
} /* magma_sgeqrf2_gpu */
/* Subroutine */ int sgtt01_(integer *n, real *dl, real *d__, real *du, real *
dlf, real *df, real *duf, real *du2, integer *ipiv, real *work,
integer *ldwork, real *rwork, real *resid)
{
/* System generated locals */
integer work_dim1, work_offset, i__1, i__2;
/* Local variables */
static integer i__, j;
static real anorm;
static integer lastj;
extern /* Subroutine */ int sswap_(integer *, real *, integer *, real *,
integer *), saxpy_(integer *, real *, real *, integer *, real *,
integer *);
static real li;
static integer ip;
extern doublereal slamch_(char *), slangt_(char *, integer *,
real *, real *, real *), slanhs_(char *, integer *, real *
, integer *, real *);
static real eps;
#define work_ref(a_1,a_2) work[(a_2)*work_dim1 + a_1]
/* -- LAPACK test routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
February 29, 1992
Purpose
=======
SGTT01 reconstructs a tridiagonal matrix A from its LU factorization
and computes the residual
norm(L*U - A) / ( norm(A) * EPS ),
where EPS is the machine epsilon.
Arguments
=========
N (input) INTEGTER
The order of the matrix A. N >= 0.
DL (input) REAL array, dimension (N-1)
The (n-1) sub-diagonal elements of A.
D (input) REAL array, dimension (N)
The diagonal elements of A.
DU (input) REAL array, dimension (N-1)
The (n-1) super-diagonal elements of A.
DLF (input) REAL array, dimension (N-1)
The (n-1) multipliers that define the matrix L from the
LU factorization of A.
DF (input) REAL array, dimension (N)
The n diagonal elements of the upper triangular matrix U from
the LU factorization of A.
DUF (input) REAL array, dimension (N-1)
The (n-1) elements of the first super-diagonal of U.
DU2F (input) REAL array, dimension (N-2)
The (n-2) elements of the second super-diagonal of U.
IPIV (input) INTEGER array, dimension (N)
The pivot indices; for 1 <= i <= n, row i of the matrix was
interchanged with row IPIV(i). IPIV(i) will always be either
i or i+1; IPIV(i) = i indicates a row interchange was not
required.
WORK (workspace) REAL array, dimension (LDWORK,N)
LDWORK (input) INTEGER
The leading dimension of the array WORK. LDWORK >= max(1,N).
RWORK (workspace) REAL array, dimension (N)
RESID (output) REAL
The scaled residual: norm(L*U - A) / (norm(A) * EPS)
=====================================================================
Quick return if possible
Parameter adjustments */
--dl;
--d__;
--du;
--dlf;
--df;
--duf;
--du2;
--ipiv;
work_dim1 = *ldwork;
work_offset = 1 + work_dim1 * 1;
work -= work_offset;
--rwork;
/* Function Body */
if (*n <= 0) {
*resid = 0.f;
return 0;
}
eps = slamch_("Epsilon");
/* Copy the matrix U to WORK. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
work_ref(i__, j) = 0.f;
/* L10: */
}
/* L20: */
}
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if (i__ == 1) {
work_ref(i__, i__) = df[i__];
if (*n >= 2) {
work_ref(i__, i__ + 1) = duf[i__];
}
if (*n >= 3) {
work_ref(i__, i__ + 2) = du2[i__];
}
} else if (i__ == *n) {
work_ref(i__, i__) = df[i__];
} else {
work_ref(i__, i__) = df[i__];
work_ref(i__, i__ + 1) = duf[i__];
if (i__ < *n - 1) {
work_ref(i__, i__ + 2) = du2[i__];
}
}
/* L30: */
}
/* Multiply on the left by L. */
lastj = *n;
for (i__ = *n - 1; i__ >= 1; --i__) {
li = dlf[i__];
i__1 = lastj - i__ + 1;
saxpy_(&i__1, &li, &work_ref(i__, i__), ldwork, &work_ref(i__ + 1,
i__), ldwork);
ip = ipiv[i__];
if (ip == i__) {
/* Computing MIN */
i__1 = i__ + 2;
lastj = min(i__1,*n);
} else {
i__1 = lastj - i__ + 1;
sswap_(&i__1, &work_ref(i__, i__), ldwork, &work_ref(i__ + 1, i__)
, ldwork);
}
/* L40: */
}
/* Subtract the matrix A. */
work_ref(1, 1) = work_ref(1, 1) - d__[1];
if (*n > 1) {
work_ref(1, 2) = work_ref(1, 2) - du[1];
work_ref(*n, *n - 1) = work_ref(*n, *n - 1) - dl[*n - 1];
work_ref(*n, *n) = work_ref(*n, *n) - d__[*n];
i__1 = *n - 1;
for (i__ = 2; i__ <= i__1; ++i__) {
work_ref(i__, i__ - 1) = work_ref(i__, i__ - 1) - dl[i__ - 1];
work_ref(i__, i__) = work_ref(i__, i__) - d__[i__];
work_ref(i__, i__ + 1) = work_ref(i__, i__ + 1) - du[i__];
/* L50: */
}
}
/* Compute the 1-norm of the tridiagonal matrix A. */
anorm = slangt_("1", n, &dl[1], &d__[1], &du[1]);
/* Compute the 1-norm of WORK, which is only guaranteed to be
upper Hessenberg. */
*resid = slanhs_("1", n, &work[work_offset], ldwork, &rwork[1])
;
/* Compute norm(L*U - A) / (norm(A) * EPS) */
if (anorm <= 0.f) {
if (*resid != 0.f) {
*resid = 1.f / eps;
}
} else {
*resid = *resid / anorm / eps;
}
return 0;
/* End of SGTT01 */
} /* sgtt01_ */
/**
Purpose
-------
DGEQRF computes a QR factorization of a real M-by-N matrix A:
A = Q * R.
This version has LAPACK-complaint arguments.
If the current stream is NULL, this version replaces it with a new
stream to overlap computation with communication.
Other versions (magma_dgeqrf_gpu and magma_dgeqrf3_gpu) store the
intermediate T matrices.
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]
dA DOUBLE_PRECISION array on the GPU, dimension (LDDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(M,N)-by-N upper trapezoidal matrix R (R is
upper triangular if m >= n); the elements below the diagonal,
with the array TAU, represent the orthogonal matrix Q as a
product of min(m,n) elementary reflectors (see Further
Details).
@param[in]
ldda INTEGER
The leading dimension of the array dA. LDDA >= max(1,M).
To benefit from coalescent memory accesses LDDA must be
divisible by 16.
@param[out]
tau DOUBLE_PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
@param[out]
info INTEGER
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value
or another error occured, such as memory allocation failed.
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_dgeqrf_comp
********************************************************************/
extern "C" magma_int_t
magma_dgeqrf2_gpu( magma_int_t m, magma_int_t n,
double *dA, magma_int_t ldda,
double *tau,
magma_int_t *info )
{
#define dA(a_1,a_2) ( dA+(a_2)*(ldda) + (a_1))
#define work_ref(a_1) ( work + (a_1))
#define hwork ( work + (nb)*(m))
double *dwork;
double *work;
magma_int_t i, k, ldwork, lddwork, old_i, old_ib, rows;
magma_int_t nbmin, nx, ib, nb;
magma_int_t lhwork, lwork;
/* Function Body */
*info = 0;
if (m < 0) {
*info = -1;
} else if (n < 0) {
*info = -2;
} else if (ldda < max(1,m)) {
*info = -4;
}
if (*info != 0) {
magma_xerbla( __func__, -(*info) );
return *info;
}
k = min(m,n);
if (k == 0)
return *info;
nb = magma_get_dgeqrf_nb(m);
lwork = (m+n) * nb;
lhwork = lwork - (m)*nb;
if (MAGMA_SUCCESS != magma_dmalloc( &dwork, n*nb )) {
*info = MAGMA_ERR_DEVICE_ALLOC;
return *info;
}
if (MAGMA_SUCCESS != magma_dmalloc_pinned( &work, lwork )) {
magma_free( dwork );
*info = MAGMA_ERR_HOST_ALLOC;
return *info;
}
/* Define user stream if current stream is NULL */
magma_queue_t stream[2];
magma_queue_t orig_stream;
magmablasGetKernelStream( &orig_stream );
magma_queue_create( &stream[0] );
if (orig_stream == NULL) {
magma_queue_create( &stream[1] );
magmablasSetKernelStream(stream[1]);
}
else {
stream[1] = orig_stream;
}
nbmin = 2;
nx = nb;
ldwork = m;
lddwork= n;
if (nb >= nbmin && nb < k && nx < k) {
/* Use blocked code initially */
old_i = 0; old_ib = nb;
for (i = 0; i < k-nx; i += nb) {
ib = min(k-i, nb);
rows = m -i;
/* download i-th panel */
magma_queue_sync( stream[1] );
magma_dgetmatrix_async( rows, ib,
dA(i,i), ldda,
work_ref(i), ldwork, stream[0] );
if (i > 0) {
/* Apply H' to A(i:m,i+2*ib:n) from the left */
magma_dlarfb_gpu( MagmaLeft, MagmaConjTrans, MagmaForward, MagmaColumnwise,
m-old_i, n-old_i-2*old_ib, old_ib,
dA(old_i, old_i ), ldda, dwork, lddwork,
dA(old_i, old_i+2*old_ib), ldda, dwork+old_ib, lddwork);
magma_dsetmatrix_async( old_ib, old_ib,
work_ref(old_i), ldwork,
dA(old_i, old_i), ldda, stream[1] );
}
magma_queue_sync( stream[0] );
lapackf77_dgeqrf(&rows, &ib, work_ref(i), &ldwork, tau+i, hwork, &lhwork, info);
/* Form the triangular factor of the block reflector
H = H(i) H(i+1) . . . H(i+ib-1) */
lapackf77_dlarft( MagmaForwardStr, MagmaColumnwiseStr,
&rows, &ib,
work_ref(i), &ldwork, tau+i, hwork, &ib);
dpanel_to_q( MagmaUpper, ib, work_ref(i), ldwork, hwork+ib*ib );
/* download the i-th V matrix */
magma_dsetmatrix_async( rows, ib, work_ref(i), ldwork, dA(i,i), ldda, stream[0] );
/* download the T matrix */
magma_queue_sync( stream[1] );
magma_dsetmatrix_async( ib, ib, hwork, ib, dwork, lddwork, stream[0] );
magma_queue_sync( stream[0] );
if (i + ib < n) {
if (i+nb < k-nx) {
/* Apply H' to A(i:m,i+ib:i+2*ib) from the left */
magma_dlarfb_gpu( MagmaLeft, MagmaConjTrans, MagmaForward, MagmaColumnwise,
rows, ib, ib,
dA(i, i ), ldda, dwork, lddwork,
dA(i, i+ib), ldda, dwork+ib, lddwork);
dq_to_panel( MagmaUpper, ib, work_ref(i), ldwork, hwork+ib*ib );
}
else {
magma_dlarfb_gpu( MagmaLeft, MagmaConjTrans, MagmaForward, MagmaColumnwise,
rows, n-i-ib, ib,
dA(i, i ), ldda, dwork, lddwork,
dA(i, i+ib), ldda, dwork+ib, lddwork);
dq_to_panel( MagmaUpper, ib, work_ref(i), ldwork, hwork+ib*ib );
magma_dsetmatrix_async( ib, ib,
work_ref(i), ldwork,
dA(i,i), ldda, stream[1] );
}
old_i = i;
old_ib = ib;
}
}
} else {
i = 0;
}
magma_free( dwork );
/* Use unblocked code to factor the last or only block. */
if (i < k) {
ib = n-i;
rows = m-i;
magma_dgetmatrix_async( rows, ib, dA(i, i), ldda, work, rows, stream[1] );
magma_queue_sync( stream[1] );
lhwork = lwork - rows*ib;
lapackf77_dgeqrf(&rows, &ib, work, &rows, tau+i, work+ib*rows, &lhwork, info);
magma_dsetmatrix_async( rows, ib, work, rows, dA(i, i), ldda, stream[1] );
}
magma_free_pinned( work );
magma_queue_destroy( stream[0] );
if (orig_stream == NULL) {
magma_queue_destroy( stream[1] );
}
magmablasSetKernelStream( orig_stream );
return *info;
} /* magma_dgeqrf2_gpu */
/* Subroutine */ int sget03_(integer *n, real *a, integer *lda, real *ainv,
integer *ldainv, real *work, integer *ldwork, real *rwork, real *
rcond, real *resid)
{
/* System generated locals */
integer a_dim1, a_offset, ainv_dim1, ainv_offset, work_dim1, work_offset,
i__1;
/* Local variables */
static integer i__;
extern /* Subroutine */ int sgemm_(char *, char *, integer *, integer *,
integer *, real *, real *, integer *, real *, integer *, real *,
real *, integer *);
static real anorm;
extern doublereal slamch_(char *), slange_(char *, integer *,
integer *, real *, integer *, real *);
static real ainvnm, eps;
#define work_ref(a_1,a_2) work[(a_2)*work_dim1 + a_1]
/* -- LAPACK test routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
June 30, 1999
Purpose
=======
SGET03 computes the residual for a general matrix times its inverse:
norm( I - AINV*A ) / ( N * norm(A) * norm(AINV) * EPS ),
where EPS is the machine epsilon.
Arguments
==========
N (input) INTEGER
The number of rows and columns of the matrix A. N >= 0.
A (input) REAL array, dimension (LDA,N)
The original N x N matrix A.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AINV (input) REAL array, dimension (LDAINV,N)
The inverse of the matrix A.
LDAINV (input) INTEGER
The leading dimension of the array AINV. LDAINV >= max(1,N).
WORK (workspace) REAL array, dimension (LDWORK,N)
LDWORK (input) INTEGER
The leading dimension of the array WORK. LDWORK >= max(1,N).
RWORK (workspace) REAL array, dimension (N)
RCOND (output) REAL
The reciprocal of the condition number of A, computed as
( 1/norm(A) ) / norm(AINV).
RESID (output) REAL
norm(I - AINV*A) / ( N * norm(A) * norm(AINV) * EPS )
=====================================================================
Quick exit if N = 0.
Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
ainv_dim1 = *ldainv;
ainv_offset = 1 + ainv_dim1 * 1;
ainv -= ainv_offset;
work_dim1 = *ldwork;
work_offset = 1 + work_dim1 * 1;
work -= work_offset;
--rwork;
/* Function Body */
if (*n <= 0) {
*rcond = 1.f;
*resid = 0.f;
return 0;
}
/* Exit with RESID = 1/EPS if ANORM = 0 or AINVNM = 0. */
eps = slamch_("Epsilon");
anorm = slange_("1", n, n, &a[a_offset], lda, &rwork[1]);
ainvnm = slange_("1", n, n, &ainv[ainv_offset], ldainv, &rwork[1]);
if (anorm <= 0.f || ainvnm <= 0.f) {
*rcond = 0.f;
*resid = 1.f / eps;
return 0;
}
*rcond = 1.f / anorm / ainvnm;
/* Compute I - A * AINV */
sgemm_("No transpose", "No transpose", n, n, n, &c_b7, &ainv[ainv_offset],
ldainv, &a[a_offset], lda, &c_b8, &work[work_offset], ldwork);
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
work_ref(i__, i__) = work_ref(i__, i__) + 1.f;
/* L10: */
}
/* Compute norm(I - AINV*A) / (N * norm(A) * norm(AINV) * EPS) */
*resid = slange_("1", n, n, &work[work_offset], ldwork, &rwork[1]);
*resid = *resid * *rcond / eps / (real) (*n);
return 0;
/* End of SGET03 */
} /* sget03_ */
/* Subroutine */ int ztrsna_(char *job, char *howmny, logical *select,
integer *n, doublecomplex *t, integer *ldt, doublecomplex *vl,
integer *ldvl, doublecomplex *vr, integer *ldvr, doublereal *s,
doublereal *sep, integer *mm, integer *m, doublecomplex *work,
integer *ldwork, doublereal *rwork, integer *info)
{
/* -- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
September 30, 1994
Purpose
=======
ZTRSNA estimates reciprocal condition numbers for specified
eigenvalues and/or right eigenvectors of a complex upper triangular
matrix T (or of any matrix Q*T*Q**H with Q unitary).
Arguments
=========
JOB (input) CHARACTER*1
Specifies whether condition numbers are required for
eigenvalues (S) or eigenvectors (SEP):
= 'E': for eigenvalues only (S);
= 'V': for eigenvectors only (SEP);
= 'B': for both eigenvalues and eigenvectors (S and SEP).
HOWMNY (input) CHARACTER*1
= 'A': compute condition numbers for all eigenpairs;
= 'S': compute condition numbers for selected eigenpairs
specified by the array SELECT.
SELECT (input) LOGICAL array, dimension (N)
If HOWMNY = 'S', SELECT specifies the eigenpairs for which
condition numbers are required. To select condition numbers
for the j-th eigenpair, SELECT(j) must be set to .TRUE..
If HOWMNY = 'A', SELECT is not referenced.
N (input) INTEGER
The order of the matrix T. N >= 0.
T (input) COMPLEX*16 array, dimension (LDT,N)
The upper triangular matrix T.
LDT (input) INTEGER
The leading dimension of the array T. LDT >= max(1,N).
VL (input) COMPLEX*16 array, dimension (LDVL,M)
If JOB = 'E' or 'B', VL must contain left eigenvectors of T
(or of any Q*T*Q**H with Q unitary), corresponding to the
eigenpairs specified by HOWMNY and SELECT. The eigenvectors
must be stored in consecutive columns of VL, as returned by
ZHSEIN or ZTREVC.
If JOB = 'V', VL is not referenced.
LDVL (input) INTEGER
The leading dimension of the array VL.
LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.
VR (input) COMPLEX*16 array, dimension (LDVR,M)
If JOB = 'E' or 'B', VR must contain right eigenvectors of T
(or of any Q*T*Q**H with Q unitary), corresponding to the
eigenpairs specified by HOWMNY and SELECT. The eigenvectors
must be stored in consecutive columns of VR, as returned by
ZHSEIN or ZTREVC.
If JOB = 'V', VR is not referenced.
LDVR (input) INTEGER
The leading dimension of the array VR.
LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.
S (output) DOUBLE PRECISION array, dimension (MM)
If JOB = 'E' or 'B', the reciprocal condition numbers of the
selected eigenvalues, stored in consecutive elements of the
array. Thus S(j), SEP(j), and the j-th columns of VL and VR
all correspond to the same eigenpair (but not in general the
j-th eigenpair, unless all eigenpairs are selected).
If JOB = 'V', S is not referenced.
SEP (output) DOUBLE PRECISION array, dimension (MM)
If JOB = 'V' or 'B', the estimated reciprocal condition
numbers of the selected eigenvectors, stored in consecutive
elements of the array.
If JOB = 'E', SEP is not referenced.
MM (input) INTEGER
The number of elements in the arrays S (if JOB = 'E' or 'B')
and/or SEP (if JOB = 'V' or 'B'). MM >= M.
M (output) INTEGER
The number of elements of the arrays S and/or SEP actually
used to store the estimated condition numbers.
If HOWMNY = 'A', M is set to N.
WORK (workspace) COMPLEX*16 array, dimension (LDWORK,N+1)
If JOB = 'E', WORK is not referenced.
LDWORK (input) INTEGER
The leading dimension of the array WORK.
LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.
RWORK (workspace) DOUBLE PRECISION array, dimension (N)
If JOB = 'E', RWORK is not referenced.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Further Details
===============
The reciprocal of the condition number of an eigenvalue lambda is
defined as
S(lambda) = |v'*u| / (norm(u)*norm(v))
where u and v are the right and left eigenvectors of T corresponding
to lambda; v' denotes the conjugate transpose of v, and norm(u)
denotes the Euclidean norm. These reciprocal condition numbers always
lie between zero (very badly conditioned) and one (very well
conditioned). If n = 1, S(lambda) is defined to be 1.
An approximate error bound for a computed eigenvalue W(i) is given by
EPS * norm(T) / S(i)
where EPS is the machine precision.
The reciprocal of the condition number of the right eigenvector u
corresponding to lambda is defined as follows. Suppose
T = ( lambda c )
( 0 T22 )
Then the reciprocal condition number is
SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )
where sigma-min denotes the smallest singular value. We approximate
the smallest singular value by the reciprocal of an estimate of the
one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is
defined to be abs(T(1,1)).
An approximate error bound for a computed right eigenvector VR(i)
is given by
EPS * norm(T) / SEP(i)
=====================================================================
Decode and test the input parameters
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
/* System generated locals */
integer t_dim1, t_offset, vl_dim1, vl_offset, vr_dim1, vr_offset,
work_dim1, work_offset, i__1, i__2, i__3, i__4, i__5;
doublereal d__1, d__2;
doublecomplex z__1;
/* Builtin functions */
double z_abs(doublecomplex *), d_imag(doublecomplex *);
/* Local variables */
static integer kase, ierr;
static doublecomplex prod;
static doublereal lnrm, rnrm;
static integer i__, j, k;
static doublereal scale;
extern logical lsame_(char *, char *);
extern /* Double Complex */ VOID zdotc_(doublecomplex *, integer *,
doublecomplex *, integer *, doublecomplex *, integer *);
static doublecomplex dummy[1];
static logical wants;
static doublereal xnorm;
extern /* Subroutine */ int dlabad_(doublereal *, doublereal *);
extern doublereal dznrm2_(integer *, doublecomplex *, integer *), dlamch_(
char *);
static integer ks, ix;
extern /* Subroutine */ int xerbla_(char *, integer *);
static doublereal bignum;
static logical wantbh;
extern /* Subroutine */ int zlacon_(integer *, doublecomplex *,
doublecomplex *, doublereal *, integer *);
extern integer izamax_(integer *, doublecomplex *, integer *);
static logical somcon;
extern /* Subroutine */ int zdrscl_(integer *, doublereal *,
doublecomplex *, integer *);
static char normin[1];
extern /* Subroutine */ int zlacpy_(char *, integer *, integer *,
doublecomplex *, integer *, doublecomplex *, integer *);
static doublereal smlnum;
static logical wantsp;
extern /* Subroutine */ int zlatrs_(char *, char *, char *, char *,
integer *, doublecomplex *, integer *, doublecomplex *,
doublereal *, doublereal *, integer *), ztrexc_(char *, integer *, doublecomplex *, integer *,
doublecomplex *, integer *, integer *, integer *, integer *);
static doublereal eps, est;
#define work_subscr(a_1,a_2) (a_2)*work_dim1 + a_1
#define work_ref(a_1,a_2) work[work_subscr(a_1,a_2)]
#define t_subscr(a_1,a_2) (a_2)*t_dim1 + a_1
#define t_ref(a_1,a_2) t[t_subscr(a_1,a_2)]
#define vl_subscr(a_1,a_2) (a_2)*vl_dim1 + a_1
#define vl_ref(a_1,a_2) vl[vl_subscr(a_1,a_2)]
#define vr_subscr(a_1,a_2) (a_2)*vr_dim1 + a_1
#define vr_ref(a_1,a_2) vr[vr_subscr(a_1,a_2)]
--select;
t_dim1 = *ldt;
t_offset = 1 + t_dim1 * 1;
t -= t_offset;
vl_dim1 = *ldvl;
vl_offset = 1 + vl_dim1 * 1;
vl -= vl_offset;
vr_dim1 = *ldvr;
vr_offset = 1 + vr_dim1 * 1;
vr -= vr_offset;
--s;
--sep;
work_dim1 = *ldwork;
work_offset = 1 + work_dim1 * 1;
work -= work_offset;
--rwork;
/* Function Body */
wantbh = lsame_(job, "B");
wants = lsame_(job, "E") || wantbh;
wantsp = lsame_(job, "V") || wantbh;
somcon = lsame_(howmny, "S");
/* Set M to the number of eigenpairs for which condition numbers are
to be computed. */
if (somcon) {
*m = 0;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
if (select[j]) {
++(*m);
}
/* L10: */
}
} else {
*m = *n;
}
*info = 0;
if (! wants && ! wantsp) {
*info = -1;
} else if (! lsame_(howmny, "A") && ! somcon) {
*info = -2;
} else if (*n < 0) {
*info = -4;
} else if (*ldt < max(1,*n)) {
*info = -6;
} else if (*ldvl < 1 || wants && *ldvl < *n) {
*info = -8;
} else if (*ldvr < 1 || wants && *ldvr < *n) {
*info = -10;
} else if (*mm < *m) {
*info = -13;
} else if (*ldwork < 1 || wantsp && *ldwork < *n) {
*info = -16;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZTRSNA", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
if (*n == 1) {
if (somcon) {
if (! select[1]) {
return 0;
}
}
if (wants) {
s[1] = 1.;
}
if (wantsp) {
sep[1] = z_abs(&t_ref(1, 1));
}
return 0;
}
/* Get machine constants */
eps = dlamch_("P");
smlnum = dlamch_("S") / eps;
bignum = 1. / smlnum;
dlabad_(&smlnum, &bignum);
ks = 1;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
if (somcon) {
if (! select[k]) {
goto L50;
}
}
if (wants) {
/* Compute the reciprocal condition number of the k-th
eigenvalue. */
zdotc_(&z__1, n, &vr_ref(1, ks), &c__1, &vl_ref(1, ks), &c__1);
prod.r = z__1.r, prod.i = z__1.i;
rnrm = dznrm2_(n, &vr_ref(1, ks), &c__1);
lnrm = dznrm2_(n, &vl_ref(1, ks), &c__1);
s[ks] = z_abs(&prod) / (rnrm * lnrm);
}
if (wantsp) {
/* Estimate the reciprocal condition number of the k-th
eigenvector.
Copy the matrix T to the array WORK and swap the k-th
diagonal element to the (1,1) position. */
zlacpy_("Full", n, n, &t[t_offset], ldt, &work[work_offset],
ldwork);
ztrexc_("No Q", n, &work[work_offset], ldwork, dummy, &c__1, &k, &
c__1, &ierr);
/* Form C = T22 - lambda*I in WORK(2:N,2:N). */
i__2 = *n;
for (i__ = 2; i__ <= i__2; ++i__) {
i__3 = work_subscr(i__, i__);
i__4 = work_subscr(i__, i__);
i__5 = work_subscr(1, 1);
z__1.r = work[i__4].r - work[i__5].r, z__1.i = work[i__4].i -
work[i__5].i;
work[i__3].r = z__1.r, work[i__3].i = z__1.i;
/* L20: */
}
/* Estimate a lower bound for the 1-norm of inv(C'). The 1st
and (N+1)th columns of WORK are used to store work vectors. */
sep[ks] = 0.;
est = 0.;
kase = 0;
*(unsigned char *)normin = 'N';
L30:
i__2 = *n - 1;
zlacon_(&i__2, &work_ref(1, *n + 1), &work[work_offset], &est, &
kase);
if (kase != 0) {
if (kase == 1) {
/* Solve C'*x = scale*b */
i__2 = *n - 1;
zlatrs_("Upper", "Conjugate transpose", "Nonunit", normin,
&i__2, &work_ref(2, 2), ldwork, &work[
work_offset], &scale, &rwork[1], &ierr);
} else {
/* Solve C*x = scale*b */
i__2 = *n - 1;
zlatrs_("Upper", "No transpose", "Nonunit", normin, &i__2,
&work_ref(2, 2), ldwork, &work[work_offset], &
scale, &rwork[1], &ierr);
}
*(unsigned char *)normin = 'Y';
if (scale != 1.) {
/* Multiply by 1/SCALE if doing so will not cause
overflow. */
i__2 = *n - 1;
ix = izamax_(&i__2, &work[work_offset], &c__1);
i__2 = work_subscr(ix, 1);
xnorm = (d__1 = work[i__2].r, abs(d__1)) + (d__2 = d_imag(
&work_ref(ix, 1)), abs(d__2));
if (scale < xnorm * smlnum || scale == 0.) {
goto L40;
}
zdrscl_(n, &scale, &work[work_offset], &c__1);
}
goto L30;
}
sep[ks] = 1. / max(est,smlnum);
}
L40:
++ks;
L50:
;
}
return 0;
/* End of ZTRSNA */
} /* ztrsna_ */
/* Subroutine */ int dpbtrf_(char *uplo, integer *n, integer *kd, doublereal *
ab, integer *ldab, integer *info)
{
/* -- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
March 31, 1993
Purpose
=======
DPBTRF computes the Cholesky factorization of a real symmetric
positive definite band matrix A.
The factorization has the form
A = U**T * U, if UPLO = 'U', or
A = L * L**T, if UPLO = 'L',
where U is an upper triangular matrix and L is lower triangular.
Arguments
=========
UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) INTEGER
The order of the matrix A. N >= 0.
KD (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'. KD >= 0.
AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
On exit, if INFO = 0, the triangular factor U or L from the
Cholesky factorization A = U**T*U or A = L*L**T of the band
matrix A, in the same storage format as A.
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
INFO (output) 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 is not
positive definite, and the factorization could not be
completed.
Further Details
===============
The band storage scheme is illustrated by the following example, when
N = 6, KD = 2, and UPLO = 'U':
On entry: On exit:
* * a13 a24 a35 a46 * * u13 u24 u35 u46
* a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
Similarly, if UPLO = 'L' the format of A is as follows:
On entry: On exit:
a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
a31 a42 a53 a64 * * l31 l42 l53 l64 * *
Array elements marked * are not used by the routine.
Contributed by
Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989
=====================================================================
Test the input parameters.
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
static integer c_n1 = -1;
static doublereal c_b18 = 1.;
static doublereal c_b21 = -1.;
static integer c__33 = 33;
/* System generated locals */
integer ab_dim1, ab_offset, i__1, i__2, i__3, i__4;
/* Local variables */
static doublereal work[1056] /* was [33][32] */;
static integer i__, j;
extern /* Subroutine */ int dgemm_(char *, char *, integer *, integer *,
integer *, doublereal *, doublereal *, integer *, doublereal *,
integer *, doublereal *, doublereal *, integer *);
extern logical lsame_(char *, char *);
extern /* Subroutine */ int dtrsm_(char *, char *, char *, char *,
integer *, integer *, doublereal *, doublereal *, integer *,
doublereal *, integer *);
static integer i2, i3;
extern /* Subroutine */ int dsyrk_(char *, char *, integer *, integer *,
doublereal *, doublereal *, integer *, doublereal *, doublereal *,
integer *), dpbtf2_(char *, integer *, integer *,
doublereal *, integer *, integer *), dpotf2_(char *,
integer *, doublereal *, integer *, integer *);
static integer ib, nb, ii, jj;
extern /* Subroutine */ int xerbla_(char *, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
#define work_ref(a_1,a_2) work[(a_2)*33 + a_1 - 34]
#define ab_ref(a_1,a_2) ab[(a_2)*ab_dim1 + a_1]
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1 * 1;
ab -= ab_offset;
/* Function Body */
*info = 0;
if (! lsame_(uplo, "U") && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*kd < 0) {
*info = -3;
} else if (*ldab < *kd + 1) {
*info = -5;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DPBTRF", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Determine the block size for this environment */
nb = ilaenv_(&c__1, "DPBTRF", uplo, n, kd, &c_n1, &c_n1, (ftnlen)6, (
ftnlen)1);
/* The block size must not exceed the semi-bandwidth KD, and must not
exceed the limit set by the size of the local array WORK. */
nb = min(nb,32);
if (nb <= 1 || nb > *kd) {
/* Use unblocked code */
dpbtf2_(uplo, n, kd, &ab[ab_offset], ldab, info);
} else {
/* Use blocked code */
if (lsame_(uplo, "U")) {
/* Compute the Cholesky factorization of a symmetric band
matrix, given the upper triangle of the matrix in band
storage.
Zero the upper triangle of the work array. */
i__1 = nb;
for (j = 1; j <= i__1; ++j) {
i__2 = j - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
work_ref(i__, j) = 0.;
/* L10: */
}
/* L20: */
}
/* Process the band matrix one diagonal block at a time. */
i__1 = *n;
i__2 = nb;
for (i__ = 1; i__2 < 0 ? i__ >= i__1 : i__ <= i__1; i__ += i__2) {
/* Computing MIN */
i__3 = nb, i__4 = *n - i__ + 1;
ib = min(i__3,i__4);
/* Factorize the diagonal block */
i__3 = *ldab - 1;
dpotf2_(uplo, &ib, &ab_ref(*kd + 1, i__), &i__3, &ii);
if (ii != 0) {
*info = i__ + ii - 1;
goto L150;
}
if (i__ + ib <= *n) {
/* Update the relevant part of the trailing submatrix.
If A11 denotes the diagonal block which has just been
factorized, then we need to update the remaining
blocks in the diagram:
A11 A12 A13
A22 A23
A33
The numbers of rows and columns in the partitioning
are IB, I2, I3 respectively. The blocks A12, A22 and
A23 are empty if IB = KD. The upper triangle of A13
lies outside the band.
Computing MIN */
i__3 = *kd - ib, i__4 = *n - i__ - ib + 1;
i2 = min(i__3,i__4);
/* Computing MIN */
i__3 = ib, i__4 = *n - i__ - *kd + 1;
i3 = min(i__3,i__4);
if (i2 > 0) {
/* Update A12 */
i__3 = *ldab - 1;
i__4 = *ldab - 1;
dtrsm_("Left", "Upper", "Transpose", "Non-unit", &ib,
&i2, &c_b18, &ab_ref(*kd + 1, i__), &i__3, &
ab_ref(*kd + 1 - ib, i__ + ib), &i__4);
/* Update A22 */
i__3 = *ldab - 1;
i__4 = *ldab - 1;
dsyrk_("Upper", "Transpose", &i2, &ib, &c_b21, &
ab_ref(*kd + 1 - ib, i__ + ib), &i__3, &c_b18,
&ab_ref(*kd + 1, i__ + ib), &i__4);
}
if (i3 > 0) {
/* Copy the lower triangle of A13 into the work array. */
i__3 = i3;
for (jj = 1; jj <= i__3; ++jj) {
i__4 = ib;
for (ii = jj; ii <= i__4; ++ii) {
work_ref(ii, jj) = ab_ref(ii - jj + 1, jj +
i__ + *kd - 1);
/* L30: */
}
/* L40: */
}
/* Update A13 (in the work array). */
i__3 = *ldab - 1;
dtrsm_("Left", "Upper", "Transpose", "Non-unit", &ib,
&i3, &c_b18, &ab_ref(*kd + 1, i__), &i__3,
work, &c__33);
/* Update A23 */
if (i2 > 0) {
i__3 = *ldab - 1;
i__4 = *ldab - 1;
dgemm_("Transpose", "No Transpose", &i2, &i3, &ib,
&c_b21, &ab_ref(*kd + 1 - ib, i__ + ib),
&i__3, work, &c__33, &c_b18, &ab_ref(ib +
1, i__ + *kd), &i__4);
}
/* Update A33 */
i__3 = *ldab - 1;
dsyrk_("Upper", "Transpose", &i3, &ib, &c_b21, work, &
c__33, &c_b18, &ab_ref(*kd + 1, i__ + *kd), &
i__3);
/* Copy the lower triangle of A13 back into place. */
i__3 = i3;
for (jj = 1; jj <= i__3; ++jj) {
i__4 = ib;
for (ii = jj; ii <= i__4; ++ii) {
ab_ref(ii - jj + 1, jj + i__ + *kd - 1) =
work_ref(ii, jj);
/* L50: */
}
/* L60: */
}
}
}
/* L70: */
}
} else {
/* Compute the Cholesky factorization of a symmetric band
matrix, given the lower triangle of the matrix in band
storage.
Zero the lower triangle of the work array. */
i__2 = nb;
for (j = 1; j <= i__2; ++j) {
i__1 = nb;
for (i__ = j + 1; i__ <= i__1; ++i__) {
work_ref(i__, j) = 0.;
/* L80: */
}
/* L90: */
}
/* Process the band matrix one diagonal block at a time. */
i__2 = *n;
i__1 = nb;
for (i__ = 1; i__1 < 0 ? i__ >= i__2 : i__ <= i__2; i__ += i__1) {
/* Computing MIN */
i__3 = nb, i__4 = *n - i__ + 1;
ib = min(i__3,i__4);
/* Factorize the diagonal block */
i__3 = *ldab - 1;
dpotf2_(uplo, &ib, &ab_ref(1, i__), &i__3, &ii);
if (ii != 0) {
*info = i__ + ii - 1;
goto L150;
}
if (i__ + ib <= *n) {
/* Update the relevant part of the trailing submatrix.
If A11 denotes the diagonal block which has just been
factorized, then we need to update the remaining
blocks in the diagram:
A11
A21 A22
A31 A32 A33
The numbers of rows and columns in the partitioning
are IB, I2, I3 respectively. The blocks A21, A22 and
A32 are empty if IB = KD. The lower triangle of A31
lies outside the band.
Computing MIN */
i__3 = *kd - ib, i__4 = *n - i__ - ib + 1;
i2 = min(i__3,i__4);
/* Computing MIN */
i__3 = ib, i__4 = *n - i__ - *kd + 1;
i3 = min(i__3,i__4);
if (i2 > 0) {
/* Update A21 */
i__3 = *ldab - 1;
i__4 = *ldab - 1;
dtrsm_("Right", "Lower", "Transpose", "Non-unit", &i2,
&ib, &c_b18, &ab_ref(1, i__), &i__3, &ab_ref(
ib + 1, i__), &i__4);
/* Update A22 */
i__3 = *ldab - 1;
i__4 = *ldab - 1;
dsyrk_("Lower", "No Transpose", &i2, &ib, &c_b21, &
ab_ref(ib + 1, i__), &i__3, &c_b18, &ab_ref(1,
i__ + ib), &i__4);
}
if (i3 > 0) {
/* Copy the upper triangle of A31 into the work array. */
i__3 = ib;
for (jj = 1; jj <= i__3; ++jj) {
i__4 = min(jj,i3);
for (ii = 1; ii <= i__4; ++ii) {
work_ref(ii, jj) = ab_ref(*kd + 1 - jj + ii,
jj + i__ - 1);
/* L100: */
}
/* L110: */
}
/* Update A31 (in the work array). */
i__3 = *ldab - 1;
dtrsm_("Right", "Lower", "Transpose", "Non-unit", &i3,
&ib, &c_b18, &ab_ref(1, i__), &i__3, work, &
c__33);
/* Update A32 */
if (i2 > 0) {
i__3 = *ldab - 1;
i__4 = *ldab - 1;
dgemm_("No transpose", "Transpose", &i3, &i2, &ib,
&c_b21, work, &c__33, &ab_ref(ib + 1,
i__), &i__3, &c_b18, &ab_ref(*kd + 1 - ib,
i__ + ib), &i__4);
}
/* Update A33 */
i__3 = *ldab - 1;
dsyrk_("Lower", "No Transpose", &i3, &ib, &c_b21,
work, &c__33, &c_b18, &ab_ref(1, i__ + *kd), &
i__3);
/* Copy the upper triangle of A31 back into place. */
i__3 = ib;
for (jj = 1; jj <= i__3; ++jj) {
i__4 = min(jj,i3);
for (ii = 1; ii <= i__4; ++ii) {
ab_ref(*kd + 1 - jj + ii, jj + i__ - 1) =
work_ref(ii, jj);
/* L120: */
}
/* L130: */
}
}
}
/* L140: */
}
}
}
return 0;
L150:
return 0;
/* End of DPBTRF */
} /* dpbtrf_ */
/* Subroutine */ int dlarfb_(char *side, char *trans, char *direct, char *
storev, integer *m, integer *n, integer *k, doublereal *v, integer *
ldv, doublereal *t, integer *ldt, doublereal *c__, integer *ldc,
doublereal *work, integer *ldwork)
{
/* -- LAPACK auxiliary routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
February 29, 1992
Purpose
=======
DLARFB applies a real block reflector H or its transpose H' to a
real m by n matrix C, from either the left or the right.
Arguments
=========
SIDE (input) CHARACTER*1
= 'L': apply H or H' from the Left
= 'R': apply H or H' from the Right
TRANS (input) CHARACTER*1
= 'N': apply H (No transpose)
= 'T': apply H' (Transpose)
DIRECT (input) CHARACTER*1
Indicates how H is formed from a product of elementary
reflectors
= 'F': H = H(1) H(2) . . . H(k) (Forward)
= 'B': H = H(k) . . . H(2) H(1) (Backward)
STOREV (input) CHARACTER*1
Indicates how the vectors which define the elementary
reflectors are stored:
= 'C': Columnwise
= 'R': Rowwise
M (input) INTEGER
The number of rows of the matrix C.
N (input) INTEGER
The number of columns of the matrix C.
K (input) INTEGER
The order of the matrix T (= the number of elementary
reflectors whose product defines the block reflector).
V (input) DOUBLE PRECISION array, dimension
(LDV,K) if STOREV = 'C'
(LDV,M) if STOREV = 'R' and SIDE = 'L'
(LDV,N) if STOREV = 'R' and SIDE = 'R'
The matrix V. See further details.
LDV (input) INTEGER
The leading dimension of the array V.
If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M);
if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N);
if STOREV = 'R', LDV >= K.
T (input) DOUBLE PRECISION array, dimension (LDT,K)
The triangular k by k matrix T in the representation of the
block reflector.
LDT (input) INTEGER
The leading dimension of the array T. LDT >= K.
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the m by n matrix C.
On exit, C is overwritten by H*C or H'*C or C*H or C*H'.
LDC (input) INTEGER
The leading dimension of the array C. LDA >= max(1,M).
WORK (workspace) DOUBLE PRECISION array, dimension (LDWORK,K)
LDWORK (input) INTEGER
The leading dimension of the array WORK.
If SIDE = 'L', LDWORK >= max(1,N);
if SIDE = 'R', LDWORK >= max(1,M).
=====================================================================
Quick return if possible
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
static doublereal c_b14 = 1.;
static doublereal c_b25 = -1.;
/* System generated locals */
integer c_dim1, c_offset, t_dim1, t_offset, v_dim1, v_offset, work_dim1,
work_offset, i__1, i__2;
/* Local variables */
static integer i__, j;
extern /* Subroutine */ int dgemm_(char *, char *, integer *, integer *,
integer *, doublereal *, doublereal *, integer *, doublereal *,
integer *, doublereal *, doublereal *, integer *);
extern logical lsame_(char *, char *);
extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *,
doublereal *, integer *), dtrmm_(char *, char *, char *, char *,
integer *, integer *, doublereal *, doublereal *, integer *,
doublereal *, integer *);
static char transt[1];
#define work_ref(a_1,a_2) work[(a_2)*work_dim1 + a_1]
#define c___ref(a_1,a_2) c__[(a_2)*c_dim1 + a_1]
#define v_ref(a_1,a_2) v[(a_2)*v_dim1 + a_1]
v_dim1 = *ldv;
v_offset = 1 + v_dim1 * 1;
v -= v_offset;
t_dim1 = *ldt;
t_offset = 1 + t_dim1 * 1;
t -= t_offset;
c_dim1 = *ldc;
c_offset = 1 + c_dim1 * 1;
c__ -= c_offset;
work_dim1 = *ldwork;
work_offset = 1 + work_dim1 * 1;
work -= work_offset;
/* Function Body */
if (*m <= 0 || *n <= 0) {
return 0;
}
if (lsame_(trans, "N")) {
*(unsigned char *)transt = 'T';
} else {
*(unsigned char *)transt = 'N';
}
if (lsame_(storev, "C")) {
if (lsame_(direct, "F")) {
/* Let V = ( V1 ) (first K rows)
( V2 )
where V1 is unit lower triangular. */
if (lsame_(side, "L")) {
/* Form H * C or H' * C where C = ( C1 )
( C2 )
W := C' * V = (C1'*V1 + C2'*V2) (stored in WORK)
W := C1' */
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
dcopy_(n, &c___ref(j, 1), ldc, &work_ref(1, j), &c__1);
/* L10: */
}
/* W := W * V1 */
dtrmm_("Right", "Lower", "No transpose", "Unit", n, k, &c_b14,
&v[v_offset], ldv, &work[work_offset], ldwork);
if (*m > *k) {
/* W := W + C2'*V2 */
i__1 = *m - *k;
dgemm_("Transpose", "No transpose", n, k, &i__1, &c_b14, &
c___ref(*k + 1, 1), ldc, &v_ref(*k + 1, 1), ldv, &
c_b14, &work[work_offset], ldwork);
}
/* W := W * T' or W * T */
dtrmm_("Right", "Upper", transt, "Non-unit", n, k, &c_b14, &t[
t_offset], ldt, &work[work_offset], ldwork);
/* C := C - V * W' */
if (*m > *k) {
/* C2 := C2 - V2 * W' */
i__1 = *m - *k;
dgemm_("No transpose", "Transpose", &i__1, n, k, &c_b25, &
v_ref(*k + 1, 1), ldv, &work[work_offset], ldwork,
&c_b14, &c___ref(*k + 1, 1), ldc);
}
/* W := W * V1' */
dtrmm_("Right", "Lower", "Transpose", "Unit", n, k, &c_b14, &
v[v_offset], ldv, &work[work_offset], ldwork);
/* C1 := C1 - W' */
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
c___ref(j, i__) = c___ref(j, i__) - work_ref(i__, j);
/* L20: */
}
/* L30: */
}
} else if (lsame_(side, "R")) {
/* Form C * H or C * H' where C = ( C1 C2 )
W := C * V = (C1*V1 + C2*V2) (stored in WORK)
W := C1 */
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
dcopy_(m, &c___ref(1, j), &c__1, &work_ref(1, j), &c__1);
/* L40: */
}
/* W := W * V1 */
dtrmm_("Right", "Lower", "No transpose", "Unit", m, k, &c_b14,
&v[v_offset], ldv, &work[work_offset], ldwork);
if (*n > *k) {
/* W := W + C2 * V2 */
i__1 = *n - *k;
dgemm_("No transpose", "No transpose", m, k, &i__1, &
c_b14, &c___ref(1, *k + 1), ldc, &v_ref(*k + 1, 1)
, ldv, &c_b14, &work[work_offset], ldwork);
}
/* W := W * T or W * T' */
dtrmm_("Right", "Upper", trans, "Non-unit", m, k, &c_b14, &t[
t_offset], ldt, &work[work_offset], ldwork);
/* C := C - W * V' */
if (*n > *k) {
/* C2 := C2 - W * V2' */
i__1 = *n - *k;
dgemm_("No transpose", "Transpose", m, &i__1, k, &c_b25, &
work[work_offset], ldwork, &v_ref(*k + 1, 1), ldv,
&c_b14, &c___ref(1, *k + 1), ldc);
}
/* W := W * V1' */
dtrmm_("Right", "Lower", "Transpose", "Unit", m, k, &c_b14, &
v[v_offset], ldv, &work[work_offset], ldwork);
/* C1 := C1 - W */
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
i__2 = *m;
for (i__ = 1; i__ <= i__2; ++i__) {
c___ref(i__, j) = c___ref(i__, j) - work_ref(i__, j);
/* L50: */
}
/* L60: */
}
}
} else {
/* Let V = ( V1 )
( V2 ) (last K rows)
where V2 is unit upper triangular. */
if (lsame_(side, "L")) {
/* Form H * C or H' * C where C = ( C1 )
( C2 )
W := C' * V = (C1'*V1 + C2'*V2) (stored in WORK)
W := C2' */
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
dcopy_(n, &c___ref(*m - *k + j, 1), ldc, &work_ref(1, j),
&c__1);
/* L70: */
}
/* W := W * V2 */
dtrmm_("Right", "Upper", "No transpose", "Unit", n, k, &c_b14,
&v_ref(*m - *k + 1, 1), ldv, &work[work_offset],
ldwork);
if (*m > *k) {
/* W := W + C1'*V1 */
i__1 = *m - *k;
dgemm_("Transpose", "No transpose", n, k, &i__1, &c_b14, &
c__[c_offset], ldc, &v[v_offset], ldv, &c_b14, &
work[work_offset], ldwork);
}
/* W := W * T' or W * T */
dtrmm_("Right", "Lower", transt, "Non-unit", n, k, &c_b14, &t[
t_offset], ldt, &work[work_offset], ldwork);
/* C := C - V * W' */
if (*m > *k) {
/* C1 := C1 - V1 * W' */
i__1 = *m - *k;
dgemm_("No transpose", "Transpose", &i__1, n, k, &c_b25, &
v[v_offset], ldv, &work[work_offset], ldwork, &
c_b14, &c__[c_offset], ldc)
;
}
/* W := W * V2' */
dtrmm_("Right", "Upper", "Transpose", "Unit", n, k, &c_b14, &
v_ref(*m - *k + 1, 1), ldv, &work[work_offset],
ldwork);
/* C2 := C2 - W' */
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
c___ref(*m - *k + j, i__) = c___ref(*m - *k + j, i__)
- work_ref(i__, j);
/* L80: */
}
/* L90: */
}
} else if (lsame_(side, "R")) {
/* Form C * H or C * H' where C = ( C1 C2 )
W := C * V = (C1*V1 + C2*V2) (stored in WORK)
W := C2 */
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
dcopy_(m, &c___ref(1, *n - *k + j), &c__1, &work_ref(1, j)
, &c__1);
/* L100: */
}
/* W := W * V2 */
dtrmm_("Right", "Upper", "No transpose", "Unit", m, k, &c_b14,
&v_ref(*n - *k + 1, 1), ldv, &work[work_offset],
ldwork);
if (*n > *k) {
/* W := W + C1 * V1 */
i__1 = *n - *k;
dgemm_("No transpose", "No transpose", m, k, &i__1, &
c_b14, &c__[c_offset], ldc, &v[v_offset], ldv, &
c_b14, &work[work_offset], ldwork);
}
/* W := W * T or W * T' */
dtrmm_("Right", "Lower", trans, "Non-unit", m, k, &c_b14, &t[
t_offset], ldt, &work[work_offset], ldwork);
/* C := C - W * V' */
if (*n > *k) {
/* C1 := C1 - W * V1' */
i__1 = *n - *k;
dgemm_("No transpose", "Transpose", m, &i__1, k, &c_b25, &
work[work_offset], ldwork, &v[v_offset], ldv, &
c_b14, &c__[c_offset], ldc)
;
}
/* W := W * V2' */
dtrmm_("Right", "Upper", "Transpose", "Unit", m, k, &c_b14, &
v_ref(*n - *k + 1, 1), ldv, &work[work_offset],
ldwork);
/* C2 := C2 - W */
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
i__2 = *m;
for (i__ = 1; i__ <= i__2; ++i__) {
c___ref(i__, *n - *k + j) = c___ref(i__, *n - *k + j)
- work_ref(i__, j);
/* L110: */
}
/* L120: */
}
}
}
} else if (lsame_(storev, "R")) {
if (lsame_(direct, "F")) {
/* Let V = ( V1 V2 ) (V1: first K columns)
where V1 is unit upper triangular. */
if (lsame_(side, "L")) {
/* Form H * C or H' * C where C = ( C1 )
( C2 )
W := C' * V' = (C1'*V1' + C2'*V2') (stored in WORK)
W := C1' */
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
dcopy_(n, &c___ref(j, 1), ldc, &work_ref(1, j), &c__1);
/* L130: */
}
/* W := W * V1' */
dtrmm_("Right", "Upper", "Transpose", "Unit", n, k, &c_b14, &
v[v_offset], ldv, &work[work_offset], ldwork);
if (*m > *k) {
/* W := W + C2'*V2' */
i__1 = *m - *k;
dgemm_("Transpose", "Transpose", n, k, &i__1, &c_b14, &
c___ref(*k + 1, 1), ldc, &v_ref(1, *k + 1), ldv, &
c_b14, &work[work_offset], ldwork);
}
/* W := W * T' or W * T */
dtrmm_("Right", "Upper", transt, "Non-unit", n, k, &c_b14, &t[
t_offset], ldt, &work[work_offset], ldwork);
/* C := C - V' * W' */
if (*m > *k) {
/* C2 := C2 - V2' * W' */
i__1 = *m - *k;
dgemm_("Transpose", "Transpose", &i__1, n, k, &c_b25, &
v_ref(1, *k + 1), ldv, &work[work_offset], ldwork,
&c_b14, &c___ref(*k + 1, 1), ldc);
}
/* W := W * V1 */
dtrmm_("Right", "Upper", "No transpose", "Unit", n, k, &c_b14,
&v[v_offset], ldv, &work[work_offset], ldwork);
/* C1 := C1 - W' */
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
c___ref(j, i__) = c___ref(j, i__) - work_ref(i__, j);
/* L140: */
}
/* L150: */
}
} else if (lsame_(side, "R")) {
/* Form C * H or C * H' where C = ( C1 C2 )
W := C * V' = (C1*V1' + C2*V2') (stored in WORK)
W := C1 */
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
dcopy_(m, &c___ref(1, j), &c__1, &work_ref(1, j), &c__1);
/* L160: */
}
/* W := W * V1' */
dtrmm_("Right", "Upper", "Transpose", "Unit", m, k, &c_b14, &
v[v_offset], ldv, &work[work_offset], ldwork);
if (*n > *k) {
/* W := W + C2 * V2' */
i__1 = *n - *k;
dgemm_("No transpose", "Transpose", m, k, &i__1, &c_b14, &
c___ref(1, *k + 1), ldc, &v_ref(1, *k + 1), ldv, &
c_b14, &work[work_offset], ldwork);
}
/* W := W * T or W * T' */
dtrmm_("Right", "Upper", trans, "Non-unit", m, k, &c_b14, &t[
t_offset], ldt, &work[work_offset], ldwork);
/* C := C - W * V */
if (*n > *k) {
/* C2 := C2 - W * V2 */
i__1 = *n - *k;
dgemm_("No transpose", "No transpose", m, &i__1, k, &
c_b25, &work[work_offset], ldwork, &v_ref(1, *k +
1), ldv, &c_b14, &c___ref(1, *k + 1), ldc);
}
/* W := W * V1 */
dtrmm_("Right", "Upper", "No transpose", "Unit", m, k, &c_b14,
&v[v_offset], ldv, &work[work_offset], ldwork);
/* C1 := C1 - W */
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
i__2 = *m;
for (i__ = 1; i__ <= i__2; ++i__) {
c___ref(i__, j) = c___ref(i__, j) - work_ref(i__, j);
/* L170: */
}
/* L180: */
}
}
} else {
/* Let V = ( V1 V2 ) (V2: last K columns)
where V2 is unit lower triangular. */
if (lsame_(side, "L")) {
/* Form H * C or H' * C where C = ( C1 )
( C2 )
W := C' * V' = (C1'*V1' + C2'*V2') (stored in WORK)
W := C2' */
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
dcopy_(n, &c___ref(*m - *k + j, 1), ldc, &work_ref(1, j),
&c__1);
/* L190: */
}
/* W := W * V2' */
dtrmm_("Right", "Lower", "Transpose", "Unit", n, k, &c_b14, &
v_ref(1, *m - *k + 1), ldv, &work[work_offset],
ldwork);
if (*m > *k) {
/* W := W + C1'*V1' */
i__1 = *m - *k;
dgemm_("Transpose", "Transpose", n, k, &i__1, &c_b14, &
c__[c_offset], ldc, &v[v_offset], ldv, &c_b14, &
work[work_offset], ldwork);
}
/* W := W * T' or W * T */
dtrmm_("Right", "Lower", transt, "Non-unit", n, k, &c_b14, &t[
t_offset], ldt, &work[work_offset], ldwork);
/* C := C - V' * W' */
if (*m > *k) {
/* C1 := C1 - V1' * W' */
i__1 = *m - *k;
dgemm_("Transpose", "Transpose", &i__1, n, k, &c_b25, &v[
v_offset], ldv, &work[work_offset], ldwork, &
c_b14, &c__[c_offset], ldc);
}
/* W := W * V2 */
dtrmm_("Right", "Lower", "No transpose", "Unit", n, k, &c_b14,
&v_ref(1, *m - *k + 1), ldv, &work[work_offset],
ldwork);
/* C2 := C2 - W' */
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
c___ref(*m - *k + j, i__) = c___ref(*m - *k + j, i__)
- work_ref(i__, j);
/* L200: */
}
/* L210: */
}
} else if (lsame_(side, "R")) {
/* Form C * H or C * H' where C = ( C1 C2 )
W := C * V' = (C1*V1' + C2*V2') (stored in WORK)
W := C2 */
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
dcopy_(m, &c___ref(1, *n - *k + j), &c__1, &work_ref(1, j)
, &c__1);
/* L220: */
}
/* W := W * V2' */
dtrmm_("Right", "Lower", "Transpose", "Unit", m, k, &c_b14, &
v_ref(1, *n - *k + 1), ldv, &work[work_offset],
ldwork);
if (*n > *k) {
/* W := W + C1 * V1' */
i__1 = *n - *k;
dgemm_("No transpose", "Transpose", m, k, &i__1, &c_b14, &
c__[c_offset], ldc, &v[v_offset], ldv, &c_b14, &
work[work_offset], ldwork);
}
/* W := W * T or W * T' */
dtrmm_("Right", "Lower", trans, "Non-unit", m, k, &c_b14, &t[
t_offset], ldt, &work[work_offset], ldwork);
/* C := C - W * V */
if (*n > *k) {
/* C1 := C1 - W * V1 */
i__1 = *n - *k;
dgemm_("No transpose", "No transpose", m, &i__1, k, &
c_b25, &work[work_offset], ldwork, &v[v_offset],
ldv, &c_b14, &c__[c_offset], ldc);
}
/* W := W * V2 */
dtrmm_("Right", "Lower", "No transpose", "Unit", m, k, &c_b14,
&v_ref(1, *n - *k + 1), ldv, &work[work_offset],
ldwork);
/* C1 := C1 - W */
i__1 = *k;
for (j = 1; j <= i__1; ++j) {
i__2 = *m;
for (i__ = 1; i__ <= i__2; ++i__) {
c___ref(i__, *n - *k + j) = c___ref(i__, *n - *k + j)
- work_ref(i__, j);
/* L230: */
}
/* L240: */
}
}
}
}
return 0;
/* End of DLARFB */
} /* dlarfb_ */
