/** Purpose ------- ZHETRD reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T. Arguments --------- @param[in] uplo magma_uplo_t - = MagmaUpper: Upper triangle of A is stored; - = MagmaLower: Lower triangle of A is stored. @param[in] n INTEGER The order of the matrix A. N >= 0. @param[in,out] A COMPLEX_16 array, dimension (LDA,N) On entry, the Hermitian matrix A. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = MagmaLower, the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, if UPLO = MagmaUpper, the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = MagmaLower, the diagonal and first subdiagonal of A are over- written by the corresponding elements of the tridiagonal matrix T, 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] d COMPLEX_16 array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i). @param[out] e COMPLEX_16 array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = MagmaUpper, E(i) = A(i+1,i) if UPLO = MagmaLower. @param[out] tau COMPLEX_16 array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details). @param[out] work (workspace) COMPLEX_16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK. @param[in] lwork INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB is the optimal blocksize given by magma_get_zhetrd_nb(). \n If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. @param[out] info INTEGER - = 0: successful exit - < 0: if INFO = -i, the i-th argument had an illegal value Further Details --------------- If UPLO = MagmaUpper, the matrix Q is represented as a product of elementary reflectors Q = H(n-1) . . . H(2) H(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(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in A(1:i-1,i+1), and tau in TAU(i). If UPLO = MagmaLower, the matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) . . . H(n-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 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in TAU(i). The contents of A on exit are illustrated by the following examples with n = 5: if UPLO = MagmaUpper: if UPLO = MagmaLower: ( d e v2 v3 v4 ) ( d ) ( d e v3 v4 ) ( e d ) ( d e v4 ) ( v1 e d ) ( d e ) ( v1 v2 e d ) ( d ) ( v1 v2 v3 e d ) where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i). @ingroup magma_zheev_comp ********************************************************************/ extern "C" magma_int_t magma_zhetrd( magma_uplo_t uplo, magma_int_t n, magmaDoubleComplex *A, magma_int_t lda, double *d, double *e, magmaDoubleComplex *tau, magmaDoubleComplex *work, magma_int_t lwork, magma_int_t *info) { #define A(i_, j_) ( A + (i_) + (j_)*lda ) #define dA(i_, j_) (dA + (i_) + (j_)*ldda) const char* uplo_ = lapack_uplo_const( uplo ); magma_int_t ldda = roundup( n, 32 ); magma_int_t nb = magma_get_zhetrd_nb( n ); const magmaDoubleComplex c_zero = MAGMA_Z_ZERO; const magmaDoubleComplex c_neg_one = MAGMA_Z_NEG_ONE; const magmaDoubleComplex c_one = MAGMA_Z_ONE; const double d_one = MAGMA_D_ONE; magma_int_t kk, nx; magma_int_t i, j, i_n; magma_int_t iinfo; magma_int_t ldw, lddw, lwkopt; magma_int_t lquery; *info = 0; int upper = (uplo == MagmaUpper); lquery = (lwork == -1); if (! upper && uplo != MagmaLower) { *info = -1; } else if (n < 0) { *info = -2; } else if (lda < max(1,n)) { *info = -4; } else if (lwork < nb*n && ! lquery) { *info = -9; } /* Determine the block size. */ ldw = n; lddw = ldda; lwkopt = n * nb; if (*info == 0) { work[0] = MAGMA_Z_MAKE( lwkopt, 0 ); } if (*info != 0) { magma_xerbla( __func__, -(*info) ); return *info; } else if (lquery) return *info; /* Quick return if possible */ if (n == 0) { work[0] = c_one; return *info; } magmaDoubleComplex *dA; #ifdef FAST_HEMV magma_int_t ldwork2 = ldda*ceildiv(n,64); #else magma_int_t ldwork2 = 0; #endif if (MAGMA_SUCCESS != magma_zmalloc( &dA, ldda*n + 2*lddw*nb + ldwork2 )) { *info = MAGMA_ERR_DEVICE_ALLOC; return *info; } magmaDoubleComplex *dwork = dA + ldda*n; #ifdef FAST_HEMV magmaDoubleComplex *dwork2 = dwork + 2*lddw*nb; #endif //if (n < 2048) // nx = n; //else // nx = 512; nx = min( 128, n ); // nx <= n is required // clear out dwork in case it has NANs (used as y in zhemv) // rest of dwork (used as work in magmablas_zhemv) doesn't need to be cleared magmablas_zlaset( MagmaFull, n, nb, c_zero, c_zero, dwork, lddw ); if (upper) { /* Copy the matrix to the GPU */ magma_zsetmatrix( n, n, A(0, 0), lda, dA(0, 0), ldda ); /* Reduce the upper triangle of A. Columns 1:kk are handled by the unblocked method. */ kk = n - (n - nx + nb - 1) / nb * nb; for (i = n - nb; i >= kk; i -= nb) { /* Reduce columns i:i+nb-1 to tridiagonal form and form the matrix W which is needed to update the unreduced part of the matrix */ /* Get the current panel (no need for the 1st iteration) */ if (i != n-nb) magma_zgetmatrix( i+nb, nb, dA(0, i), ldda, A(0, i), lda ); #ifdef FAST_HEMV magma_zlatrd2( uplo, i+nb, nb, A(0, 0), lda, e, tau, work, ldw, dA(0, 0), ldda, dwork, lddw, dwork2, ldwork2 ); #else magma_zlatrd( uplo, i+nb, nb, A(0, 0), lda, e, tau, work, ldw, dA(0, 0), ldda, dwork, lddw ); #endif /* Update the unreduced submatrix A(0:i-2,0:i-2), using an update of the form: A := A - V*W' - W*V' */ magma_zsetmatrix( i + nb, nb, work, ldw, dwork, lddw ); magma_zher2k( uplo, MagmaNoTrans, i, nb, c_neg_one, dA(0, i), ldda, dwork, lddw, d_one, dA(0, 0), ldda ); /* Copy superdiagonal elements back into A, and diagonal elements into D */ for (j = i; j < i+nb; ++j) { *A(j-1,j) = MAGMA_Z_MAKE( e[j - 1], 0 ); d[j] = MAGMA_Z_REAL( *A(j, j) ); } } magma_zgetmatrix( kk, kk, dA(0, 0), ldda, A(0, 0), lda ); /* Use CPU code to reduce the last or only block */ lapackf77_zhetrd( uplo_, &kk, A(0, 0), &lda, d, e, tau, work, &lwork, &iinfo ); } else { /* Copy the matrix to the GPU */ if (1 <= n-nx) magma_zsetmatrix( n, n, A(0,0), lda, dA(0,0), ldda ); /* Reduce the lower triangle of A */ for (i = 0; i < n-nx; i += nb) { /* Reduce columns i:i+nb-1 to tridiagonal form and form the matrix W which is needed to update the unreduced part of the matrix */ /* Get the current panel (no need for the 1st iteration) */ if (i != 0) magma_zgetmatrix( n-i, nb, dA(i, i), ldda, A(i, i), lda ); #ifdef FAST_HEMV magma_zlatrd2( uplo, n-i, nb, A(i, i), lda, &e[i], &tau[i], work, ldw, dA(i, i), ldda, dwork, lddw, dwork2, ldwork2 ); #else magma_zlatrd( uplo, n-i, nb, A(i, i), lda, &e[i], &tau[i], work, ldw, dA(i, i), ldda, dwork, lddw ); #endif /* Update the unreduced submatrix A(i+ib:n,i+ib:n), using an update of the form: A := A - V*W' - W*V' */ magma_zsetmatrix( n-i, nb, work, ldw, dwork, lddw ); magma_zher2k( MagmaLower, MagmaNoTrans, n-i-nb, nb, c_neg_one, dA(i+nb, i), ldda, &dwork[nb], lddw, d_one, dA(i+nb, i+nb), ldda ); /* Copy subdiagonal elements back into A, and diagonal elements into D */ for (j = i; j < i+nb; ++j) { *A(j+1,j) = MAGMA_Z_MAKE( e[j], 0 ); d[j] = MAGMA_Z_REAL( *A(j, j) ); } } /* Use CPU code to reduce the last or only block */ if (1 <= n-nx) magma_zgetmatrix( n-i, n-i, dA(i, i), ldda, A(i, i), lda ); i_n = n-i; lapackf77_zhetrd( uplo_, &i_n, A(i, i), &lda, &d[i], &e[i], &tau[i], work, &lwork, &iinfo ); } magma_free( dA ); work[0] = MAGMA_Z_MAKE( lwkopt, 0 ); return *info; } /* magma_zhetrd */
extern "C" magma_int_t magma_zhetrd2_gpu(char uplo, magma_int_t n, magmaDoubleComplex *da, magma_int_t ldda, double *d, double *e, magmaDoubleComplex *tau, magmaDoubleComplex *wa, magma_int_t ldwa, magmaDoubleComplex *work, magma_int_t lwork, magmaDoubleComplex *dwork, magma_int_t ldwork, magma_int_t *info) { /* -- MAGMA (version 1.4.1) -- Univ. of Tennessee, Knoxville Univ. of California, Berkeley Univ. of Colorado, Denver December 2013 Purpose ======= ZHETRD2_GPU reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T. This version passes a workspace that is used in an optimized GPU matrix-vector product. 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. DA (device input/output) COMPLEX_16 array, dimension (LDA,N) On entry, the Hermitian matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, if UPLO = 'U', the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over- written by the corresponding elements of the tridiagonal matrix T, 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. LDDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,N). D (output) COMPLEX_16 array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i). E (output) COMPLEX_16 array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. TAU (output) COMPLEX_16 array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details). WA (workspace/output) COMPLEX_16 array, dimension (LDA,N) On exit the diagonal, the upper part (UPLO='U') or the lower part (UPLO='L') are copies of DA LDWA (input) INTEGER The leading dimension of the array WA. LDWA >= max(1,N). WORK (workspace/output) COMPLEX_16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) INTEGER The dimension of the array WORK. LWORK >= 1. For optimum performance LWORK >= N*NB, where NB is the optimal blocksize. 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. DWORK (workspace/output) COMPLEX_16 array on the GPU, dim (MAX(1,LDWORK)) LDWORK (input) INTEGER The dimension of the array DWORK. LDWORK >= (n*n+64-1)/64 + 2*n*nb, where nb = magma_get_zhetrd_nb(n) INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value Further Details =============== If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors Q = H(n-1) . . . H(2) H(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(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in A(1:i-1,i+1), and tau in TAU(i). If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) . . . H(n-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 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in TAU(i). The contents of A on exit are illustrated by the following examples with n = 5: if UPLO = 'U': if UPLO = 'L': ( d e v2 v3 v4 ) ( d ) ( d e v3 v4 ) ( e d ) ( d e v4 ) ( v1 e d ) ( d e ) ( v1 v2 e d ) ( d ) ( v1 v2 v3 e d ) where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i). ===================================================================== */ char uplo_[2] = {uplo, 0}; magma_int_t nb = magma_get_zhetrd_nb(n); magmaDoubleComplex c_neg_one = MAGMA_Z_NEG_ONE; magmaDoubleComplex c_one = MAGMA_Z_ONE; double d_one = MAGMA_D_ONE; magma_int_t kk, nx; magma_int_t i, j, i_n; magma_int_t iinfo; magma_int_t ldw, lddw, lwkopt; magma_int_t lquery; *info = 0; int upper = lapackf77_lsame(uplo_, "U"); lquery = lwork == -1; if (! upper && ! lapackf77_lsame(uplo_, "L")) { *info = -1; } else if (n < 0) { *info = -2; } else if (ldda < max(1,n)) { *info = -4; } else if (ldwa < max(1,n)) { *info = -9; } else if (lwork < 1 && ! lquery) { *info = -11; } /* Determine the block size. */ ldw = lddw = n; lwkopt = n * nb; if (*info == 0) { work[0] = MAGMA_Z_MAKE( lwkopt, 0 ); } if (*info != 0) { magma_xerbla( __func__, -(*info) ); return *info; } else if (lquery) return *info; /* Quick return if possible */ if (n == 0) { work[0] = c_one; return *info; } if (n < 1024) nx = n; else nx = 300; if (ldwork<(ldw*n+64-1)/64 + 2*ldw*nb) { *info = MAGMA_ERR_DEVICE_ALLOC; return *info; } if (upper) { /* Reduce the upper triangle of A. Columns 1:kk are handled by the unblocked method. */ kk = n - (n - nx + nb - 1) / nb * nb; for (i = n - nb; i >= kk; i -= nb) { /* Reduce columns i:i+nb-1 to tridiagonal form and form the matrix W which is needed to update the unreduced part of the matrix */ /* Get the current panel */ magma_zgetmatrix( i+nb, nb, dA(0, i), ldda, A(0, i), ldwa ); magma_zlatrd2(uplo, i+nb, nb, A(0, 0), ldwa, e, tau, work, ldw, dA(0, 0), ldda, dwork, lddw, dwork + 2*ldw*nb, ldwork - 2*ldw*nb); /* Update the unreduced submatrix A(0:i-2,0:i-2), using an update of the form: A := A - V*W' - W*V' */ magma_zsetmatrix( i + nb, nb, work, ldw, dwork, lddw ); magma_zher2k(uplo, MagmaNoTrans, i, nb, c_neg_one, dA(0, i), ldda, dwork, lddw, d_one, dA(0, 0), ldda); /* Copy superdiagonal elements back into A, and diagonal elements into D */ for (j = i; j < i+nb; ++j) { *A(j-1,j) = MAGMA_Z_MAKE( e[j - 1], 0 ); d[j] = MAGMA_Z_REAL( *A(j, j) ); } } magma_zgetmatrix( kk, kk, dA(0, 0), ldda, A(0, 0), ldwa ); /* Use CPU code to reduce the last or only block */ lapackf77_zhetrd(uplo_, &kk, A(0, 0), &ldwa, d, e, tau, work, &lwork, &iinfo); magma_zsetmatrix( kk, kk, A(0, 0), ldwa, dA(0, 0), ldda ); } else { /* Reduce the lower triangle of A */ for (i = 0; i < n-nx; i += nb) { /* Reduce columns i:i+nb-1 to tridiagonal form and form the matrix W which is needed to update the unreduced part of the matrix */ /* Get the current panel */ magma_zgetmatrix( n-i, nb, dA(i, i), ldda, A(i, i), ldwa ); magma_zlatrd2(uplo, n-i, nb, A(i, i), ldwa, &e[i], &tau[i], work, ldw, dA(i, i), ldda, dwork, lddw, dwork + 2*ldw*nb, ldwork - 2*ldw*nb); /* Update the unreduced submatrix A(i+ib:n,i+ib:n), using an update of the form: A := A - V*W' - W*V' */ magma_zsetmatrix( n-i, nb, work, ldw, dwork, lddw ); magma_zher2k(MagmaLower, MagmaNoTrans, n-i-nb, nb, c_neg_one, dA(i+nb, i), ldda, &dwork[nb], lddw, d_one, dA(i+nb, i+nb), ldda); /* Copy subdiagonal elements back into A, and diagonal elements into D */ for (j = i; j < i+nb; ++j) { *A(j+1,j) = MAGMA_Z_MAKE( e[j], 0 ); d[j] = MAGMA_Z_REAL( *A(j, j) ); } } /* Use unblocked code to reduce the last or only block */ magma_zgetmatrix( n-i, n-i, dA(i, i), ldda, A(i, i), ldwa ); i_n = n-i; lapackf77_zhetrd(uplo_, &i_n, A(i, i), &ldwa, &d[i], &e[i], &tau[i], work, &lwork, &iinfo); magma_zsetmatrix( n-i, n-i, A(i, i), ldwa, dA(i, i), ldda ); } work[0] = MAGMA_Z_MAKE( lwkopt, 0 ); return *info; } /* zhetrd2_gpu */