/* -- MAGMA (version 1.12.0) -- Univ. of Tennessee, Knoxville Univ. of California, Berkeley Univ. of Colorado, Denver @date @precisions normal z -> s d c @author Stan Tomov @author Mark Gates */ #include "common_magmamagma_internal.h" /** Purpose ------- ZGEHRD reduces a COMPLEX_16 general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q' * A * Q = H . This version stores the triangular matrices used in the factorization so that they can be applied directly (i.e., without being recomputed) later. As a result, the application of Q is much faster. Arguments --------- @param[in] n INTEGER The order of the matrix A. N >= 0. @param[in] ilo INTEGER @param[in] ihi INTEGER It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a previous call to ZGEBAL; otherwise they should be set to 1 and N respectively. See Further Details. 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. @param[in,out] A COMPLEX_16 array, dimension (LDA,N) On entry, the N-by-N general matrix to be reduced. On exit, the upper triangle and the first subdiagonal of A are overwritten with the upper Hessenberg matrix H, and the elements below the first subdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details. @param[in] lda INTEGER The leading dimension of the array A. LDA >= max(1,N). @param[out] tau COMPLEX_16 array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to zero. @param[out] work (workspace) COMPLEX_16 array, dimension (LWORK) On exit, if INFO = 0, WORK[0] returns the optimal LWORK. @param[in] lwork INTEGER The length of the array WORK. LWORK >= N*NB, where NB is the optimal blocksize. \n If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. @param[out] dT COMPLEX_16 array on the GPU, dimension NB*N, where NB is the optimal blocksize. It stores the NB*NB blocks of the triangular T matrices used in the reduction. @param[out] info INTEGER - = 0: successful exit - < 0: if INFO = -i, the i-th argument had an illegal value. Further Details --------------- The matrix Q is represented as a product of (ihi-ilo) elementary reflectors Q = H(ilo) H(ilo+1) . . . H(ihi-1). 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) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on exit in A(i+2:ihi,i), and tau in TAU(i). The contents of A are illustrated by the following example, with n = 7, ilo = 2 and ihi = 6: @verbatim on entry, on exit, ( a a a a a a a ) ( a a h h h h a ) ( a a a a a a ) ( a h h h h a ) ( a a a a a a ) ( h h h h h h ) ( a a a a a a ) ( v2 h h h h h ) ( a a a a a a ) ( v2 v3 h h h h ) ( a a a a a a ) ( v2 v3 v4 h h h ) ( a ) ( a ) @endverbatim where a denotes an element of the original matrix A, h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i). This implementation follows the hybrid algorithm and notations described in S. Tomov and J. Dongarra, "Accelerating the reduction to upper Hessenberg form through hybrid GPU-based computing," University of Tennessee Computer Science Technical Report, UT-CS-09-642 (also LAPACK Working Note 219), May 24, 2009. This version stores the T matrices in dT, for later use in magma_zunghr. @ingroup magma_zgeev_comp ********************************************************************/ extern "C" magma_int_t magma_zgehrd( magma_int_t n, magma_int_t ilo, magma_int_t ihi, magmaDoubleComplex *A, magma_int_t lda, magmaDoubleComplex *tau, magmaDoubleComplex *work, magma_int_t lwork, magmaDoubleComplex_ptr dT, magma_int_t *info) { #define A(i_,j_) ( A + (i_) + (j_)*lda) #ifdef HAVE_clBLAS #define dA(i_,j_) dwork, ((i_) + (j_)*ldda + nb*ldda*2) #define dT(i_,j_) dT, ((i_) + (j_)*nb + dT_offset) #define dV(i_,j_) dwork, ((i_) + (j_)*ldda + nb*ldda) #define dwork(i_) dwork, ((i_)) #else #define dA(i_,j_) (dA + (i_) + (j_)*ldda) #define dT(i_,j_) (dT + (i_) + (j_)*nb) #define dV(i_,j_) (dV + (i_) + (j_)*ldda) #define dwork(i_) (dwork + (i_)) #endif // Constants const magmaDoubleComplex c_one = MAGMA_Z_ONE; const magmaDoubleComplex c_zero = MAGMA_Z_ZERO; // Local variables magma_int_t nb = magma_get_zgehrd_nb( n ); magma_int_t ldda = magma_roundup( n, 32 ); magma_int_t i, nh, iws; magma_int_t iinfo; magma_int_t lquery; *info = 0; iws = n*nb; work[0] = MAGMA_Z_MAKE( iws, 0 ); lquery = (lwork == -1); if (n < 0) { *info = -1; } else if (ilo < 1 || ilo > max(1,n)) { *info = -2; } else if (ihi < min(ilo,n) || ihi > n) { *info = -3; } else if (lda < max(1,n)) { *info = -5; } else if (lwork < iws && ! lquery) { *info = -8; } if (*info != 0) { magma_xerbla( __func__, -(*info) ); return *info; } else if (lquery) return *info; // Adjust from 1-based indexing ilo -= 1; // Quick return if possible nh = ihi - ilo; if (nh <= 1) { work[0] = c_one; return *info; } // Now requires lwork >= iws; else dT won't be computed in unblocked code. // If not enough workspace, use unblocked code //if ( lwork < iws ) { // nb = 1; //} if (nb == 1 || nb > nh) { // Use unblocked code below i = ilo; } else { // Use blocked code magma_queue_t queue; magma_device_t cdev; magma_getdevice( &cdev ); magma_queue_create( cdev, &queue ); // GPU workspace is: // nb*ldda for dwork for zlahru // nb*ldda for dV // n*ldda for dA magmaDoubleComplex *magmaDoubleComplex_ptr dwork; if (MAGMA_SUCCESS != magma_zmalloc( &dwork, 2*nb*ldda + n*ldda )) { *info = MAGMA_ERR_DEVICE_ALLOC; return *info; } magmaDoubleComplex *dV = dwork + nb*ldda; magmaDoubleComplex *dA = dwork + nb*ldda*2; magmaDoubleComplex *dTi; magmaDoubleComplex *T; magma_zmalloc_cpu( &T, nb*nb ); if ( T == NULL ) { magma_free( dwork ); *info = MAGMA_ERR_HOST_ALLOC; return *info; } // zero first block of V, which is lower triangular magmablas_zlaset( MagmaFull, nb, nb, c_zero, c_zero, dV(0,0), ldda, queue ); // Set elements 0:ILO-1 and IHI-1:N-2 of TAU to zero for (i = 0; i < ilo; ++i) tau[i] = c_zero; for (i = max(0,ihi-1); i < n-1; ++i) tau[i] = c_zero; assert( nb % 4 == 0 ); for (i=0; i < nb*nb; i += 4) T[i] = T[i+1] = T[i+2] = T[i+3] = c_zero; magmablas_zlaset( MagmaFull, nb, n, c_zero, c_zero, dT(0,0), nb, queue ); // Copy the matrix to the GPU magma_zsetmatrix( n, n-ilo, A(0,ilo), lda, dA(0,0), ldda, queue ); for (i = ilo; i < ihi-1 - nb; i += nb) { // Reduce columns i:i+nb-1 to Hessenberg form, returning the // matrices V and T of the block reflector H = I - V*T*V' // which performs the reduction, and also the matrix Y = A*V*T // Get the current panel (no need for the 1st iteration) magma_zgetmatrix( ihi-i, nb, dA(i,i-ilo), ldda, A(i,i), lda, queue ); // add 1 to i for 1-based index magma_zlahr2( ihi, i+1, nb, dA(0,i-ilo), ldda, dV(0,0), ldda, A(0,i), lda, &tau[i], T, nb, work, n, queue ); // Copy T from the CPU to dT on the GPU dTi = dT + (i - ilo)*nb; magma_zsetmatrix( nb, nb, T, nb, dTidT(0,i-ilo), nb, queue ); magma_zlahru( n, ihi, i, nb, A(0,i), lda, dA(0,i-ilo), ldda, // dA dA(i,i-ilo), ldda, // dY, stored over current panel dV(0,0), ldda, dTidT(0,i-ilo), dwork(0), queue ); } // Copy remainder to host magma_zgetmatrix( n, n-i, dA(0,i-ilo), ldda, A(0,i), lda, queue ); magma_free( dwork ); magma_free_cpu( T ); magma_queue_destroy( queue ); } // Use unblocked code to reduce the rest of the matrix // add 1 to i for 1-based index i += 1; lapackf77_zgehd2(&n, &i, &ihi, A, &lda, tau, work, &iinfo); work[0] = MAGMA_Z_MAKE( iws, 0 ); return *info; } /* magma_zgehrd */