MAGMA
2.3.0
Matrix Algebra for GPU and Multicore Architectures

Functions  
magma_int_t  magma_clahr2 (magma_int_t n, magma_int_t k, magma_int_t nb, magmaFloatComplex_ptr dA, magma_int_t ldda, magmaFloatComplex_ptr dV, magma_int_t lddv, magmaFloatComplex *A, magma_int_t lda, magmaFloatComplex *tau, magmaFloatComplex *T, magma_int_t ldt, magmaFloatComplex *Y, magma_int_t ldy, magma_queue_t queue) 
CLAHR2 reduces the first NB columns of a complex general nBY(nk+1) matrix A so that elements below the kth subdiagonal are zero. More...  
magma_int_t  magma_clahr2_m (magma_int_t n, magma_int_t k, magma_int_t nb, magmaFloatComplex *A, magma_int_t lda, magmaFloatComplex *tau, magmaFloatComplex *T, magma_int_t ldt, magmaFloatComplex *Y, magma_int_t ldy, struct cgehrd_data *data) 
CLAHR2 reduces the first NB columns of a complex general nBY(nk+1) matrix A so that elements below the kth subdiagonal are zero. More...  
magma_int_t  magma_dlahr2 (magma_int_t n, magma_int_t k, magma_int_t nb, magmaDouble_ptr dA, magma_int_t ldda, magmaDouble_ptr dV, magma_int_t lddv, double *A, magma_int_t lda, double *tau, double *T, magma_int_t ldt, double *Y, magma_int_t ldy, magma_queue_t queue) 
DLAHR2 reduces the first NB columns of a real general nBY(nk+1) matrix A so that elements below the kth subdiagonal are zero. More...  
magma_int_t  magma_dlahr2_m (magma_int_t n, magma_int_t k, magma_int_t nb, double *A, magma_int_t lda, double *tau, double *T, magma_int_t ldt, double *Y, magma_int_t ldy, struct dgehrd_data *data) 
DLAHR2 reduces the first NB columns of a real general nBY(nk+1) matrix A so that elements below the kth subdiagonal are zero. More...  
magma_int_t  magma_slahr2 (magma_int_t n, magma_int_t k, magma_int_t nb, magmaFloat_ptr dA, magma_int_t ldda, magmaFloat_ptr dV, magma_int_t lddv, float *A, magma_int_t lda, float *tau, float *T, magma_int_t ldt, float *Y, magma_int_t ldy, magma_queue_t queue) 
SLAHR2 reduces the first NB columns of a real general nBY(nk+1) matrix A so that elements below the kth subdiagonal are zero. More...  
magma_int_t  magma_slahr2_m (magma_int_t n, magma_int_t k, magma_int_t nb, float *A, magma_int_t lda, float *tau, float *T, magma_int_t ldt, float *Y, magma_int_t ldy, struct sgehrd_data *data) 
SLAHR2 reduces the first NB columns of a real general nBY(nk+1) matrix A so that elements below the kth subdiagonal are zero. More...  
magma_int_t  magma_zlahr2 (magma_int_t n, magma_int_t k, magma_int_t nb, magmaDoubleComplex_ptr dA, magma_int_t ldda, magmaDoubleComplex_ptr dV, magma_int_t lddv, magmaDoubleComplex *A, magma_int_t lda, magmaDoubleComplex *tau, magmaDoubleComplex *T, magma_int_t ldt, magmaDoubleComplex *Y, magma_int_t ldy, magma_queue_t queue) 
ZLAHR2 reduces the first NB columns of a complex general nBY(nk+1) matrix A so that elements below the kth subdiagonal are zero. More...  
magma_int_t  magma_zlahr2_m (magma_int_t n, magma_int_t k, magma_int_t nb, magmaDoubleComplex *A, magma_int_t lda, magmaDoubleComplex *tau, magmaDoubleComplex *T, magma_int_t ldt, magmaDoubleComplex *Y, magma_int_t ldy, struct zgehrd_data *data) 
ZLAHR2 reduces the first NB columns of a complex general nBY(nk+1) matrix A so that elements below the kth subdiagonal are zero. More...  
magma_int_t magma_clahr2  (  magma_int_t  n, 
magma_int_t  k,  
magma_int_t  nb,  
magmaFloatComplex_ptr  dA,  
magma_int_t  ldda,  
magmaFloatComplex_ptr  dV,  
magma_int_t  lddv,  
magmaFloatComplex *  A,  
magma_int_t  lda,  
magmaFloatComplex *  tau,  
magmaFloatComplex *  T,  
magma_int_t  ldt,  
magmaFloatComplex *  Y,  
magma_int_t  ldy,  
magma_queue_t  queue  
) 
CLAHR2 reduces the first NB columns of a complex general nBY(nk+1) matrix A so that elements below the kth subdiagonal are zero.
The reduction is performed by an orthogonal similarity transformation Q' * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I  V*T*V', and also the matrix Y = A * V. (Note this is different than LAPACK, which computes Y = A * V * T.)
This is an auxiliary routine called by CGEHRD.
[in]  n  INTEGER The order of the matrix A. 
[in]  k  INTEGER The offset for the reduction. Elements below the kth subdiagonal in the first NB columns are reduced to zero. K < N. 
[in]  nb  INTEGER The number of columns to be reduced. 
[in,out]  dA  COMPLEX array on the GPU, dimension (LDDA,NK+1) On entry, the nby(nk+1) general matrix A. On exit, the elements in rows K:N of the first NB columns are overwritten with the matrix Y. 
[in]  ldda  INTEGER The leading dimension of the array dA. LDDA >= max(1,N). 
[out]  dV  COMPLEX array on the GPU, dimension (LDDV, NB) On exit this nbynb array contains the Householder vectors of the transformation. 
[in]  lddv  INTEGER The leading dimension of the array dV. LDDV >= max(1,N). 
[in,out]  A  COMPLEX array, dimension (LDA,NK+1) On entry, the nby(nk+1) general matrix A. On exit, the elements on and above the kth subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced matrix; the elements below the kth subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors. The other columns of A are unchanged. See Further Details. 
[in]  lda  INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[out]  tau  COMPLEX array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details. 
[out]  T  COMPLEX array, dimension (LDT,NB) The upper triangular matrix T. 
[in]  ldt  INTEGER The leading dimension of the array T. LDT >= NB. 
[out]  Y  COMPLEX array, dimension (LDY,NB) The nbynb matrix Y. 
[in]  ldy  INTEGER The leading dimension of the array Y. LDY >= N. 
[in]  queue  magma_queue_t Queue to execute in. 
The matrix Q is represented as a product of nb elementary reflectors
Q = H(1) H(2) . . . H(nb).
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+k1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(i+k+1:n,i), and tau in TAU(i).
The elements of the vectors v together form the (nk+1)bynb matrix V which is needed, with T and Y, to apply the transformation to the unreduced part of the matrix, using an update of the form: A := (I  V*T*V') * (A  Y*T*V').
The contents of A on exit are illustrated by the following example with n = 7, k = 3 and nb = 2:
( a a a a a ) ( a a a a a ) ( a a a a a ) ( h h a a a ) ( v1 h a a a ) ( v1 v2 a a a ) ( v1 v2 a a a )
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 GPUbased computing," University of Tennessee Computer Science Technical Report, UTCS09642 (also LAPACK Working Note 219), May 24, 2009.
magma_int_t magma_clahr2_m  (  magma_int_t  n, 
magma_int_t  k,  
magma_int_t  nb,  
magmaFloatComplex *  A,  
magma_int_t  lda,  
magmaFloatComplex *  tau,  
magmaFloatComplex *  T,  
magma_int_t  ldt,  
magmaFloatComplex *  Y,  
magma_int_t  ldy,  
struct cgehrd_data *  data  
) 
CLAHR2 reduces the first NB columns of a complex general nBY(nk+1) matrix A so that elements below the kth subdiagonal are zero.
The reduction is performed by an orthogonal similarity transformation Q' * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I  V*T*V', and also the matrix Y = A * V. (Note this is different than LAPACK, which computes Y = A * V * T.)
This is an auxiliary routine called by CGEHRD.
[in]  n  INTEGER The order of the matrix A. 
[in]  k  INTEGER The offset for the reduction. Elements below the kth subdiagonal in the first NB columns are reduced to zero. K < N. 
[in]  nb  INTEGER The number of columns to be reduced. 
[in,out]  A  COMPLEX array, dimension (LDA,NK+1) On entry, the nby(nk+1) general matrix A. On exit, the elements on and above the kth subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced matrix; the elements below the kth subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors. The other columns of A are unchanged. See Further Details. 
[in]  lda  INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[out]  tau  COMPLEX array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details. 
[out]  T  COMPLEX array, dimension (LDT,NB) The upper triangular matrix T. 
[in]  ldt  INTEGER The leading dimension of the array T. LDT >= NB. 
[out]  Y  COMPLEX array, dimension (LDY,NB) The nbynb matrix Y. 
[in]  ldy  INTEGER The leading dimension of the array Y. LDY >= N. 
[in,out]  data  Structure with pointers to dA, dT, dV, dW, dY which are distributed across multiple GPUs. 
The matrix Q is represented as a product of nb elementary reflectors
Q = H(1) H(2) . . . H(nb).
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+k1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(i+k+1:n,i), and tau in TAU(i).
The elements of the vectors v together form the (nk+1)bynb matrix V which is needed, with T and Y, to apply the transformation to the unreduced part of the matrix, using an update of the form: A := (I  V*T*V') * (A  Y*T*V').
The contents of A on exit are illustrated by the following example with n = 7, k = 3 and nb = 2:
( a a a a a ) ( a a a a a ) ( a a a a a ) ( h h a a a ) ( v1 h a a a ) ( v1 v2 a a a ) ( v1 v2 a a a )
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 GPUbased computing," University of Tennessee Computer Science Technical Report, UTCS09642 (also LAPACK Working Note 219), May 24, 2009.
magma_int_t magma_dlahr2  (  magma_int_t  n, 
magma_int_t  k,  
magma_int_t  nb,  
magmaDouble_ptr  dA,  
magma_int_t  ldda,  
magmaDouble_ptr  dV,  
magma_int_t  lddv,  
double *  A,  
magma_int_t  lda,  
double *  tau,  
double *  T,  
magma_int_t  ldt,  
double *  Y,  
magma_int_t  ldy,  
magma_queue_t  queue  
) 
DLAHR2 reduces the first NB columns of a real general nBY(nk+1) matrix A so that elements below the kth subdiagonal are zero.
The reduction is performed by an orthogonal similarity transformation Q' * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I  V*T*V', and also the matrix Y = A * V. (Note this is different than LAPACK, which computes Y = A * V * T.)
This is an auxiliary routine called by DGEHRD.
[in]  n  INTEGER The order of the matrix A. 
[in]  k  INTEGER The offset for the reduction. Elements below the kth subdiagonal in the first NB columns are reduced to zero. K < N. 
[in]  nb  INTEGER The number of columns to be reduced. 
[in,out]  dA  DOUBLE PRECISION array on the GPU, dimension (LDDA,NK+1) On entry, the nby(nk+1) general matrix A. On exit, the elements in rows K:N of the first NB columns are overwritten with the matrix Y. 
[in]  ldda  INTEGER The leading dimension of the array dA. LDDA >= max(1,N). 
[out]  dV  DOUBLE PRECISION array on the GPU, dimension (LDDV, NB) On exit this nbynb array contains the Householder vectors of the transformation. 
[in]  lddv  INTEGER The leading dimension of the array dV. LDDV >= max(1,N). 
[in,out]  A  DOUBLE PRECISION array, dimension (LDA,NK+1) On entry, the nby(nk+1) general matrix A. On exit, the elements on and above the kth subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced matrix; the elements below the kth subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors. The other columns of A are unchanged. See Further Details. 
[in]  lda  INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[out]  tau  DOUBLE PRECISION array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details. 
[out]  T  DOUBLE PRECISION array, dimension (LDT,NB) The upper triangular matrix T. 
[in]  ldt  INTEGER The leading dimension of the array T. LDT >= NB. 
[out]  Y  DOUBLE PRECISION array, dimension (LDY,NB) The nbynb matrix Y. 
[in]  ldy  INTEGER The leading dimension of the array Y. LDY >= N. 
[in]  queue  magma_queue_t Queue to execute in. 
The matrix Q is represented as a product of nb elementary reflectors
Q = H(1) H(2) . . . H(nb).
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+k1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(i+k+1:n,i), and tau in TAU(i).
The elements of the vectors v together form the (nk+1)bynb matrix V which is needed, with T and Y, to apply the transformation to the unreduced part of the matrix, using an update of the form: A := (I  V*T*V') * (A  Y*T*V').
The contents of A on exit are illustrated by the following example with n = 7, k = 3 and nb = 2:
( a a a a a ) ( a a a a a ) ( a a a a a ) ( h h a a a ) ( v1 h a a a ) ( v1 v2 a a a ) ( v1 v2 a a a )
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 GPUbased computing," University of Tennessee Computer Science Technical Report, UTCS09642 (also LAPACK Working Note 219), May 24, 2009.
magma_int_t magma_dlahr2_m  (  magma_int_t  n, 
magma_int_t  k,  
magma_int_t  nb,  
double *  A,  
magma_int_t  lda,  
double *  tau,  
double *  T,  
magma_int_t  ldt,  
double *  Y,  
magma_int_t  ldy,  
struct dgehrd_data *  data  
) 
DLAHR2 reduces the first NB columns of a real general nBY(nk+1) matrix A so that elements below the kth subdiagonal are zero.
The reduction is performed by an orthogonal similarity transformation Q' * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I  V*T*V', and also the matrix Y = A * V. (Note this is different than LAPACK, which computes Y = A * V * T.)
This is an auxiliary routine called by DGEHRD.
[in]  n  INTEGER The order of the matrix A. 
[in]  k  INTEGER The offset for the reduction. Elements below the kth subdiagonal in the first NB columns are reduced to zero. K < N. 
[in]  nb  INTEGER The number of columns to be reduced. 
[in,out]  A  DOUBLE PRECISION array, dimension (LDA,NK+1) On entry, the nby(nk+1) general matrix A. On exit, the elements on and above the kth subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced matrix; the elements below the kth subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors. The other columns of A are unchanged. See Further Details. 
[in]  lda  INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[out]  tau  DOUBLE PRECISION array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details. 
[out]  T  DOUBLE PRECISION array, dimension (LDT,NB) The upper triangular matrix T. 
[in]  ldt  INTEGER The leading dimension of the array T. LDT >= NB. 
[out]  Y  DOUBLE PRECISION array, dimension (LDY,NB) The nbynb matrix Y. 
[in]  ldy  INTEGER The leading dimension of the array Y. LDY >= N. 
[in,out]  data  Structure with pointers to dA, dT, dV, dW, dY which are distributed across multiple GPUs. 
The matrix Q is represented as a product of nb elementary reflectors
Q = H(1) H(2) . . . H(nb).
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+k1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(i+k+1:n,i), and tau in TAU(i).
The elements of the vectors v together form the (nk+1)bynb matrix V which is needed, with T and Y, to apply the transformation to the unreduced part of the matrix, using an update of the form: A := (I  V*T*V') * (A  Y*T*V').
The contents of A on exit are illustrated by the following example with n = 7, k = 3 and nb = 2:
( a a a a a ) ( a a a a a ) ( a a a a a ) ( h h a a a ) ( v1 h a a a ) ( v1 v2 a a a ) ( v1 v2 a a a )
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 GPUbased computing," University of Tennessee Computer Science Technical Report, UTCS09642 (also LAPACK Working Note 219), May 24, 2009.
magma_int_t magma_slahr2  (  magma_int_t  n, 
magma_int_t  k,  
magma_int_t  nb,  
magmaFloat_ptr  dA,  
magma_int_t  ldda,  
magmaFloat_ptr  dV,  
magma_int_t  lddv,  
float *  A,  
magma_int_t  lda,  
float *  tau,  
float *  T,  
magma_int_t  ldt,  
float *  Y,  
magma_int_t  ldy,  
magma_queue_t  queue  
) 
SLAHR2 reduces the first NB columns of a real general nBY(nk+1) matrix A so that elements below the kth subdiagonal are zero.
The reduction is performed by an orthogonal similarity transformation Q' * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I  V*T*V', and also the matrix Y = A * V. (Note this is different than LAPACK, which computes Y = A * V * T.)
This is an auxiliary routine called by SGEHRD.
[in]  n  INTEGER The order of the matrix A. 
[in]  k  INTEGER The offset for the reduction. Elements below the kth subdiagonal in the first NB columns are reduced to zero. K < N. 
[in]  nb  INTEGER The number of columns to be reduced. 
[in,out]  dA  REAL array on the GPU, dimension (LDDA,NK+1) On entry, the nby(nk+1) general matrix A. On exit, the elements in rows K:N of the first NB columns are overwritten with the matrix Y. 
[in]  ldda  INTEGER The leading dimension of the array dA. LDDA >= max(1,N). 
[out]  dV  REAL array on the GPU, dimension (LDDV, NB) On exit this nbynb array contains the Householder vectors of the transformation. 
[in]  lddv  INTEGER The leading dimension of the array dV. LDDV >= max(1,N). 
[in,out]  A  REAL array, dimension (LDA,NK+1) On entry, the nby(nk+1) general matrix A. On exit, the elements on and above the kth subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced matrix; the elements below the kth subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors. The other columns of A are unchanged. See Further Details. 
[in]  lda  INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[out]  tau  REAL array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details. 
[out]  T  REAL array, dimension (LDT,NB) The upper triangular matrix T. 
[in]  ldt  INTEGER The leading dimension of the array T. LDT >= NB. 
[out]  Y  REAL array, dimension (LDY,NB) The nbynb matrix Y. 
[in]  ldy  INTEGER The leading dimension of the array Y. LDY >= N. 
[in]  queue  magma_queue_t Queue to execute in. 
The matrix Q is represented as a product of nb elementary reflectors
Q = H(1) H(2) . . . H(nb).
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+k1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(i+k+1:n,i), and tau in TAU(i).
The elements of the vectors v together form the (nk+1)bynb matrix V which is needed, with T and Y, to apply the transformation to the unreduced part of the matrix, using an update of the form: A := (I  V*T*V') * (A  Y*T*V').
The contents of A on exit are illustrated by the following example with n = 7, k = 3 and nb = 2:
( a a a a a ) ( a a a a a ) ( a a a a a ) ( h h a a a ) ( v1 h a a a ) ( v1 v2 a a a ) ( v1 v2 a a a )
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 GPUbased computing," University of Tennessee Computer Science Technical Report, UTCS09642 (also LAPACK Working Note 219), May 24, 2009.
magma_int_t magma_slahr2_m  (  magma_int_t  n, 
magma_int_t  k,  
magma_int_t  nb,  
float *  A,  
magma_int_t  lda,  
float *  tau,  
float *  T,  
magma_int_t  ldt,  
float *  Y,  
magma_int_t  ldy,  
struct sgehrd_data *  data  
) 
SLAHR2 reduces the first NB columns of a real general nBY(nk+1) matrix A so that elements below the kth subdiagonal are zero.
The reduction is performed by an orthogonal similarity transformation Q' * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I  V*T*V', and also the matrix Y = A * V. (Note this is different than LAPACK, which computes Y = A * V * T.)
This is an auxiliary routine called by SGEHRD.
[in]  n  INTEGER The order of the matrix A. 
[in]  k  INTEGER The offset for the reduction. Elements below the kth subdiagonal in the first NB columns are reduced to zero. K < N. 
[in]  nb  INTEGER The number of columns to be reduced. 
[in,out]  A  REAL array, dimension (LDA,NK+1) On entry, the nby(nk+1) general matrix A. On exit, the elements on and above the kth subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced matrix; the elements below the kth subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors. The other columns of A are unchanged. See Further Details. 
[in]  lda  INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[out]  tau  REAL array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details. 
[out]  T  REAL array, dimension (LDT,NB) The upper triangular matrix T. 
[in]  ldt  INTEGER The leading dimension of the array T. LDT >= NB. 
[out]  Y  REAL array, dimension (LDY,NB) The nbynb matrix Y. 
[in]  ldy  INTEGER The leading dimension of the array Y. LDY >= N. 
[in,out]  data  Structure with pointers to dA, dT, dV, dW, dY which are distributed across multiple GPUs. 
The matrix Q is represented as a product of nb elementary reflectors
Q = H(1) H(2) . . . H(nb).
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+k1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(i+k+1:n,i), and tau in TAU(i).
The elements of the vectors v together form the (nk+1)bynb matrix V which is needed, with T and Y, to apply the transformation to the unreduced part of the matrix, using an update of the form: A := (I  V*T*V') * (A  Y*T*V').
The contents of A on exit are illustrated by the following example with n = 7, k = 3 and nb = 2:
( a a a a a ) ( a a a a a ) ( a a a a a ) ( h h a a a ) ( v1 h a a a ) ( v1 v2 a a a ) ( v1 v2 a a a )
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 GPUbased computing," University of Tennessee Computer Science Technical Report, UTCS09642 (also LAPACK Working Note 219), May 24, 2009.
magma_int_t magma_zlahr2  (  magma_int_t  n, 
magma_int_t  k,  
magma_int_t  nb,  
magmaDoubleComplex_ptr  dA,  
magma_int_t  ldda,  
magmaDoubleComplex_ptr  dV,  
magma_int_t  lddv,  
magmaDoubleComplex *  A,  
magma_int_t  lda,  
magmaDoubleComplex *  tau,  
magmaDoubleComplex *  T,  
magma_int_t  ldt,  
magmaDoubleComplex *  Y,  
magma_int_t  ldy,  
magma_queue_t  queue  
) 
ZLAHR2 reduces the first NB columns of a complex general nBY(nk+1) matrix A so that elements below the kth subdiagonal are zero.
The reduction is performed by an orthogonal similarity transformation Q' * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I  V*T*V', and also the matrix Y = A * V. (Note this is different than LAPACK, which computes Y = A * V * T.)
This is an auxiliary routine called by ZGEHRD.
[in]  n  INTEGER The order of the matrix A. 
[in]  k  INTEGER The offset for the reduction. Elements below the kth subdiagonal in the first NB columns are reduced to zero. K < N. 
[in]  nb  INTEGER The number of columns to be reduced. 
[in,out]  dA  COMPLEX_16 array on the GPU, dimension (LDDA,NK+1) On entry, the nby(nk+1) general matrix A. On exit, the elements in rows K:N of the first NB columns are overwritten with the matrix Y. 
[in]  ldda  INTEGER The leading dimension of the array dA. LDDA >= max(1,N). 
[out]  dV  COMPLEX_16 array on the GPU, dimension (LDDV, NB) On exit this nbynb array contains the Householder vectors of the transformation. 
[in]  lddv  INTEGER The leading dimension of the array dV. LDDV >= max(1,N). 
[in,out]  A  COMPLEX_16 array, dimension (LDA,NK+1) On entry, the nby(nk+1) general matrix A. On exit, the elements on and above the kth subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced matrix; the elements below the kth subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors. The other columns of A are unchanged. See Further Details. 
[in]  lda  INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[out]  tau  COMPLEX_16 array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details. 
[out]  T  COMPLEX_16 array, dimension (LDT,NB) The upper triangular matrix T. 
[in]  ldt  INTEGER The leading dimension of the array T. LDT >= NB. 
[out]  Y  COMPLEX_16 array, dimension (LDY,NB) The nbynb matrix Y. 
[in]  ldy  INTEGER The leading dimension of the array Y. LDY >= N. 
[in]  queue  magma_queue_t Queue to execute in. 
The matrix Q is represented as a product of nb elementary reflectors
Q = H(1) H(2) . . . H(nb).
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+k1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(i+k+1:n,i), and tau in TAU(i).
The elements of the vectors v together form the (nk+1)bynb matrix V which is needed, with T and Y, to apply the transformation to the unreduced part of the matrix, using an update of the form: A := (I  V*T*V') * (A  Y*T*V').
The contents of A on exit are illustrated by the following example with n = 7, k = 3 and nb = 2:
( a a a a a ) ( a a a a a ) ( a a a a a ) ( h h a a a ) ( v1 h a a a ) ( v1 v2 a a a ) ( v1 v2 a a a )
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 GPUbased computing," University of Tennessee Computer Science Technical Report, UTCS09642 (also LAPACK Working Note 219), May 24, 2009.
magma_int_t magma_zlahr2_m  (  magma_int_t  n, 
magma_int_t  k,  
magma_int_t  nb,  
magmaDoubleComplex *  A,  
magma_int_t  lda,  
magmaDoubleComplex *  tau,  
magmaDoubleComplex *  T,  
magma_int_t  ldt,  
magmaDoubleComplex *  Y,  
magma_int_t  ldy,  
struct zgehrd_data *  data  
) 
ZLAHR2 reduces the first NB columns of a complex general nBY(nk+1) matrix A so that elements below the kth subdiagonal are zero.
The reduction is performed by an orthogonal similarity transformation Q' * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I  V*T*V', and also the matrix Y = A * V. (Note this is different than LAPACK, which computes Y = A * V * T.)
This is an auxiliary routine called by ZGEHRD.
[in]  n  INTEGER The order of the matrix A. 
[in]  k  INTEGER The offset for the reduction. Elements below the kth subdiagonal in the first NB columns are reduced to zero. K < N. 
[in]  nb  INTEGER The number of columns to be reduced. 
[in,out]  A  COMPLEX_16 array, dimension (LDA,NK+1) On entry, the nby(nk+1) general matrix A. On exit, the elements on and above the kth subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced matrix; the elements below the kth subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors. The other columns of A are unchanged. See Further Details. 
[in]  lda  INTEGER The leading dimension of the array A. LDA >= max(1,N). 
[out]  tau  COMPLEX_16 array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details. 
[out]  T  COMPLEX_16 array, dimension (LDT,NB) The upper triangular matrix T. 
[in]  ldt  INTEGER The leading dimension of the array T. LDT >= NB. 
[out]  Y  COMPLEX_16 array, dimension (LDY,NB) The nbynb matrix Y. 
[in]  ldy  INTEGER The leading dimension of the array Y. LDY >= N. 
[in,out]  data  Structure with pointers to dA, dT, dV, dW, dY which are distributed across multiple GPUs. 
The matrix Q is represented as a product of nb elementary reflectors
Q = H(1) H(2) . . . H(nb).
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+k1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(i+k+1:n,i), and tau in TAU(i).
The elements of the vectors v together form the (nk+1)bynb matrix V which is needed, with T and Y, to apply the transformation to the unreduced part of the matrix, using an update of the form: A := (I  V*T*V') * (A  Y*T*V').
The contents of A on exit are illustrated by the following example with n = 7, k = 3 and nb = 2:
( a a a a a ) ( a a a a a ) ( a a a a a ) ( h h a a a ) ( v1 h a a a ) ( v1 v2 a a a ) ( v1 v2 a a a )
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 GPUbased computing," University of Tennessee Computer Science Technical Report, UTCS09642 (also LAPACK Working Note 219), May 24, 2009.