sy/hetrd: Tridiagonal reduction

Functions
magma_int_t	magma_chetrd (magma_uplo_t uplo, magma_int_t n, magmaFloatComplex *A, magma_int_t lda, float *d, float *e, magmaFloatComplex *tau, magmaFloatComplex *work, magma_int_t lwork, magma_int_t *info)
	CHETRD reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T. More...
magma_int_t	magma_chetrd2_gpu (magma_uplo_t uplo, magma_int_t n, magmaFloatComplex_ptr dA, magma_int_t ldda, float *d, float *e, magmaFloatComplex *tau, magmaFloatComplex *A, magma_int_t lda, magmaFloatComplex *work, magma_int_t lwork, magmaFloatComplex_ptr dwork, magma_int_t ldwork, magma_int_t *info)
	CHETRD2_GPU reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T. More...
magma_int_t	magma_chetrd_gpu (magma_uplo_t uplo, magma_int_t n, magmaFloatComplex_ptr dA, magma_int_t ldda, float *d, float *e, magmaFloatComplex *tau, magmaFloatComplex *A, magma_int_t lda, magmaFloatComplex *work, magma_int_t lwork, magma_int_t *info)
	CHETRD_GPU reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T. More...
magma_int_t	magma_chetrd_mgpu (magma_int_t ngpu, magma_int_t nqueue, magma_uplo_t uplo, magma_int_t n, magmaFloatComplex *A, magma_int_t lda, float *d, float *e, magmaFloatComplex *tau, magmaFloatComplex *work, magma_int_t lwork, magma_int_t *info)
	CHETRD reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T. More...
magma_int_t	magma_dsytrd (magma_uplo_t uplo, magma_int_t n, double *A, magma_int_t lda, double *d, double *e, double *tau, double *work, magma_int_t lwork, magma_int_t *info)
	DSYTRD reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T. More...
magma_int_t	magma_dsytrd2_gpu (magma_uplo_t uplo, magma_int_t n, magmaDouble_ptr dA, magma_int_t ldda, double *d, double *e, double *tau, double *A, magma_int_t lda, double *work, magma_int_t lwork, magmaDouble_ptr dwork, magma_int_t ldwork, magma_int_t *info)
	DSYTRD2_GPU reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T. More...
magma_int_t	magma_dsytrd_gpu (magma_uplo_t uplo, magma_int_t n, magmaDouble_ptr dA, magma_int_t ldda, double *d, double *e, double *tau, double *A, magma_int_t lda, double *work, magma_int_t lwork, magma_int_t *info)
	DSYTRD_GPU reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T. More...
magma_int_t	magma_dsytrd_mgpu (magma_int_t ngpu, magma_int_t nqueue, magma_uplo_t uplo, magma_int_t n, double *A, magma_int_t lda, double *d, double *e, double *tau, double *work, magma_int_t lwork, magma_int_t *info)
	DSYTRD reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T. More...
magma_int_t	magma_ssytrd (magma_uplo_t uplo, magma_int_t n, float *A, magma_int_t lda, float *d, float *e, float *tau, float *work, magma_int_t lwork, magma_int_t *info)
	SSYTRD reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T. More...
magma_int_t	magma_ssytrd2_gpu (magma_uplo_t uplo, magma_int_t n, magmaFloat_ptr dA, magma_int_t ldda, float *d, float *e, float *tau, float *A, magma_int_t lda, float *work, magma_int_t lwork, magmaFloat_ptr dwork, magma_int_t ldwork, magma_int_t *info)
	SSYTRD2_GPU reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T. More...
magma_int_t	magma_ssytrd_gpu (magma_uplo_t uplo, magma_int_t n, magmaFloat_ptr dA, magma_int_t ldda, float *d, float *e, float *tau, float *A, magma_int_t lda, float *work, magma_int_t lwork, magma_int_t *info)
	SSYTRD_GPU reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T. More...
magma_int_t	magma_ssytrd_mgpu (magma_int_t ngpu, magma_int_t nqueue, magma_uplo_t uplo, magma_int_t n, float *A, magma_int_t lda, float *d, float *e, float *tau, float *work, magma_int_t lwork, magma_int_t *info)
	SSYTRD reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T. More...
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)
	ZHETRD reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T. More...
magma_int_t	magma_zhetrd2_gpu (magma_uplo_t uplo, magma_int_t n, magmaDoubleComplex_ptr dA, magma_int_t ldda, double *d, double *e, magmaDoubleComplex *tau, magmaDoubleComplex *A, magma_int_t lda, magmaDoubleComplex *work, magma_int_t lwork, magmaDoubleComplex_ptr dwork, magma_int_t ldwork, magma_int_t *info)
	ZHETRD2_GPU reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T. More...
magma_int_t	magma_zhetrd_gpu (magma_uplo_t uplo, magma_int_t n, magmaDoubleComplex_ptr dA, magma_int_t ldda, double *d, double *e, magmaDoubleComplex *tau, magmaDoubleComplex *A, magma_int_t lda, magmaDoubleComplex *work, magma_int_t lwork, magma_int_t *info)
	ZHETRD_GPU reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T. More...
magma_int_t	magma_zhetrd_mgpu (magma_int_t ngpu, magma_int_t nqueue, 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)
	ZHETRD reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T. More...

Detailed Description

Function Documentation

magma_int_t magma_chetrd	(	magma_uplo_t	uplo,
		magma_int_t	n,
		magmaFloatComplex *	A,
		magma_int_t	lda,
		float *	d,
		float *	e,
		magmaFloatComplex *	tau,
		magmaFloatComplex *	work,
		magma_int_t	lwork,
		magma_int_t *	info
	)

CHETRD reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T.

Parameters

[in]	uplo	magma_uplo_t = MagmaUpper: Upper triangle of A is stored; = MagmaLower: Lower triangle of A is stored.
[in]	n	INTEGER The order of the matrix A. N >= 0.
[in,out]	A	COMPLEX 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.
[in]	lda	INTEGER The leading dimension of the array A. LDA >= max(1,N).
[out]	d	COMPLEX array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
[out]	e	COMPLEX 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.
[out]	tau	COMPLEX array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).
[out]	work	(workspace) COMPLEX array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]	lwork	INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB is the optimal blocksize given by magma_get_chetrd_nb(). 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.
[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).

magma_int_t magma_chetrd2_gpu	(	magma_uplo_t	uplo,
		magma_int_t	n,
		magmaFloatComplex_ptr	dA,
		magma_int_t	ldda,
		float *	d,
		float *	e,
		magmaFloatComplex *	tau,
		magmaFloatComplex *	A,
		magma_int_t	lda,
		magmaFloatComplex *	work,
		magma_int_t	lwork,
		magmaFloatComplex_ptr	dwork,
		magma_int_t	ldwork,
		magma_int_t *	info
	)

CHETRD2_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.

Parameters

[in]	uplo	magma_uplo_t = MagmaUpper: Upper triangle of A is stored; = MagmaLower: Lower triangle of A is stored.
[in]	n	INTEGER The order of the matrix A. N >= 0.
[in,out]	dA	COMPLEX array on the GPU, dimension (LDDA,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.
[in]	ldda	INTEGER The leading dimension of the array A. LDDA >= max(1,N).
[out]	d	COMPLEX array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
[out]	e	COMPLEX 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.
[out]	tau	COMPLEX array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).
[out]	A	(workspace) COMPLEX array, dimension (LDA,N) On exit the diagonal, the upper part (if uplo=MagmaUpper) or the lower part (if uplo=MagmaLower) are copies of DA
[in]	lda	INTEGER The leading dimension of the array A. LDA >= max(1,N).
[out]	work	(workspace) COMPLEX array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]	lwork	INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB is the optimal blocksize given by magma_get_chetrd_nb(). 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.
[out]	dwork	(workspace) COMPLEX array on the GPU, dim (MAX(1,LDWORK))
[in]	ldwork	INTEGER The dimension of the array DWORK. LDWORK >= ldda*ceil(n/64) + 2*ldda*nb, where nb = magma_get_chetrd_nb(n), and 64 is for the blocksize of magmablas_chemv.
[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).

magma_int_t magma_chetrd_gpu	(	magma_uplo_t	uplo,
		magma_int_t	n,
		magmaFloatComplex_ptr	dA,
		magma_int_t	ldda,
		float *	d,
		float *	e,
		magmaFloatComplex *	tau,
		magmaFloatComplex *	A,
		magma_int_t	lda,
		magmaFloatComplex *	work,
		magma_int_t	lwork,
		magma_int_t *	info
	)

CHETRD_GPU reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T.

Parameters

[in]	uplo	magma_uplo_t = MagmaUpper: Upper triangle of A is stored; = MagmaLower: Lower triangle of A is stored.
[in]	n	INTEGER The order of the matrix A. N >= 0.
[in,out]	dA	COMPLEX array on the GPU, dimension (LDDA,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.
[in]	ldda	INTEGER The leading dimension of the array A. LDDA >= max(1,N).
[out]	d	COMPLEX array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
[out]	e	COMPLEX 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.
[out]	tau	COMPLEX array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).
[out]	A	(workspace) COMPLEX array, dimension (LDA,N) On exit the diagonal, the upper part (if uplo=MagmaUpper) or the lower part (if uplo=MagmaLower) are copies of dA
[in]	lda	INTEGER The leading dimension of the array A. LDA >= max(1,N).
[out]	work	(workspace) COMPLEX array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]	lwork	INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB is the optimal blocksize given by magma_get_chetrd_nb(). 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.
[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).

magma_int_t magma_chetrd_mgpu	(	magma_int_t	ngpu,
		magma_int_t	nqueue,
		magma_uplo_t	uplo,
		magma_int_t	n,
		magmaFloatComplex *	A,
		magma_int_t	lda,
		float *	d,
		float *	e,
		magmaFloatComplex *	tau,
		magmaFloatComplex *	work,
		magma_int_t	lwork,
		magma_int_t *	info
	)

CHETRD reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T.

Parameters

[in]	ngpu	INTEGER Number of GPUs to use. ngpu > 0.
[in]	nqueue	INTEGER The number of GPU queues used for update. 10 >= nqueue > 0.
[in]	uplo	magma_uplo_t = MagmaUpper: Upper triangle of A is stored; = MagmaLower: Lower triangle of A is stored.
[in]	n	INTEGER The order of the matrix A. N >= 0.
[in,out]	A	COMPLEX 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.
[in]	lda	INTEGER The leading dimension of the array A. LDA >= max(1,N).
[out]	d	COMPLEX array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
[out]	e	COMPLEX 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.
[out]	tau	COMPLEX array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).
[out]	work	(workspace) COMPLEX array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]	lwork	INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB is the optimal blocksize given by magma_get_chetrd_nb(). 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.
[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).

magma_int_t magma_dsytrd	(	magma_uplo_t	uplo,
		magma_int_t	n,
		double *	A,
		magma_int_t	lda,
		double *	d,
		double *	e,
		double *	tau,
		double *	work,
		magma_int_t	lwork,
		magma_int_t *	info
	)

DSYTRD reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T.

Parameters

[in]	uplo	magma_uplo_t = MagmaUpper: Upper triangle of A is stored; = MagmaLower: Lower triangle of A is stored.
[in]	n	INTEGER The order of the matrix A. N >= 0.
[in,out]	A	DOUBLE PRECISION array, dimension (LDA,N) On entry, the symmetric 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.
[in]	lda	INTEGER The leading dimension of the array A. LDA >= max(1,N).
[out]	d	DOUBLE PRECISION array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
[out]	e	DOUBLE PRECISION 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.
[out]	tau	DOUBLE PRECISION array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).
[out]	work	(workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]	lwork	INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB is the optimal blocksize given by magma_get_dsytrd_nb(). 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.
[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 real scalar, and v is a real 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 real scalar, and v is a real 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).

magma_int_t magma_dsytrd2_gpu	(	magma_uplo_t	uplo,
		magma_int_t	n,
		magmaDouble_ptr	dA,
		magma_int_t	ldda,
		double *	d,
		double *	e,
		double *	tau,
		double *	A,
		magma_int_t	lda,
		double *	work,
		magma_int_t	lwork,
		magmaDouble_ptr	dwork,
		magma_int_t	ldwork,
		magma_int_t *	info
	)

DSYTRD2_GPU reduces a real symmetric 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.

Parameters

[in]	uplo	magma_uplo_t = MagmaUpper: Upper triangle of A is stored; = MagmaLower: Lower triangle of A is stored.
[in]	n	INTEGER The order of the matrix A. N >= 0.
[in,out]	dA	DOUBLE PRECISION array on the GPU, dimension (LDDA,N) On entry, the symmetric 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.
[in]	ldda	INTEGER The leading dimension of the array A. LDDA >= max(1,N).
[out]	d	DOUBLE PRECISION array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
[out]	e	DOUBLE PRECISION 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.
[out]	tau	DOUBLE PRECISION array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).
[out]	A	(workspace) DOUBLE PRECISION array, dimension (LDA,N) On exit the diagonal, the upper part (if uplo=MagmaUpper) or the lower part (if uplo=MagmaLower) are copies of DA
[in]	lda	INTEGER The leading dimension of the array A. LDA >= max(1,N).
[out]	work	(workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]	lwork	INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB is the optimal blocksize given by magma_get_dsytrd_nb(). 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.
[out]	dwork	(workspace) DOUBLE PRECISION array on the GPU, dim (MAX(1,LDWORK))
[in]	ldwork	INTEGER The dimension of the array DWORK. LDWORK >= ldda*ceil(n/64) + 2*ldda*nb, where nb = magma_get_dsytrd_nb(n), and 64 is for the blocksize of magmablas_dsymv.
[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 real scalar, and v is a real 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 real scalar, and v is a real 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).

magma_int_t magma_dsytrd_gpu	(	magma_uplo_t	uplo,
		magma_int_t	n,
		magmaDouble_ptr	dA,
		magma_int_t	ldda,
		double *	d,
		double *	e,
		double *	tau,
		double *	A,
		magma_int_t	lda,
		double *	work,
		magma_int_t	lwork,
		magma_int_t *	info
	)

DSYTRD_GPU reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T.

Parameters

[in]	uplo	magma_uplo_t = MagmaUpper: Upper triangle of A is stored; = MagmaLower: Lower triangle of A is stored.
[in]	n	INTEGER The order of the matrix A. N >= 0.
[in,out]	dA	DOUBLE PRECISION array on the GPU, dimension (LDDA,N) On entry, the symmetric 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.
[in]	ldda	INTEGER The leading dimension of the array A. LDDA >= max(1,N).
[out]	d	DOUBLE PRECISION array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
[out]	e	DOUBLE PRECISION 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.
[out]	tau	DOUBLE PRECISION array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).
[out]	A	(workspace) DOUBLE PRECISION array, dimension (LDA,N) On exit the diagonal, the upper part (if uplo=MagmaUpper) or the lower part (if uplo=MagmaLower) are copies of dA
[in]	lda	INTEGER The leading dimension of the array A. LDA >= max(1,N).
[out]	work	(workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]	lwork	INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB is the optimal blocksize given by magma_get_dsytrd_nb(). 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.
[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 real scalar, and v is a real 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 real scalar, and v is a real 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).

magma_int_t magma_dsytrd_mgpu	(	magma_int_t	ngpu,
		magma_int_t	nqueue,
		magma_uplo_t	uplo,
		magma_int_t	n,
		double *	A,
		magma_int_t	lda,
		double *	d,
		double *	e,
		double *	tau,
		double *	work,
		magma_int_t	lwork,
		magma_int_t *	info
	)

DSYTRD reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T.

Parameters

[in]	ngpu	INTEGER Number of GPUs to use. ngpu > 0.
[in]	nqueue	INTEGER The number of GPU queues used for update. 10 >= nqueue > 0.
[in]	uplo	magma_uplo_t = MagmaUpper: Upper triangle of A is stored; = MagmaLower: Lower triangle of A is stored.
[in]	n	INTEGER The order of the matrix A. N >= 0.
[in,out]	A	DOUBLE PRECISION array, dimension (LDA,N) On entry, the symmetric 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.
[in]	lda	INTEGER The leading dimension of the array A. LDA >= max(1,N).
[out]	d	DOUBLE PRECISION array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
[out]	e	DOUBLE PRECISION 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.
[out]	tau	DOUBLE PRECISION array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).
[out]	work	(workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]	lwork	INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB is the optimal blocksize given by magma_get_dsytrd_nb(). 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.
[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 real scalar, and v is a real 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 real scalar, and v is a real 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).

magma_int_t magma_ssytrd	(	magma_uplo_t	uplo,
		magma_int_t	n,
		float *	A,
		magma_int_t	lda,
		float *	d,
		float *	e,
		float *	tau,
		float *	work,
		magma_int_t	lwork,
		magma_int_t *	info
	)

SSYTRD reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T.

Parameters

[in]	uplo	magma_uplo_t = MagmaUpper: Upper triangle of A is stored; = MagmaLower: Lower triangle of A is stored.
[in]	n	INTEGER The order of the matrix A. N >= 0.
[in,out]	A	REAL array, dimension (LDA,N) On entry, the symmetric 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.
[in]	lda	INTEGER The leading dimension of the array A. LDA >= max(1,N).
[out]	d	REAL array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
[out]	e	REAL 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.
[out]	tau	REAL array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).
[out]	work	(workspace) REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]	lwork	INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB is the optimal blocksize given by magma_get_ssytrd_nb(). 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.
[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 real scalar, and v is a real 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 real scalar, and v is a real 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).

magma_int_t magma_ssytrd2_gpu	(	magma_uplo_t	uplo,
		magma_int_t	n,
		magmaFloat_ptr	dA,
		magma_int_t	ldda,
		float *	d,
		float *	e,
		float *	tau,
		float *	A,
		magma_int_t	lda,
		float *	work,
		magma_int_t	lwork,
		magmaFloat_ptr	dwork,
		magma_int_t	ldwork,
		magma_int_t *	info
	)

SSYTRD2_GPU reduces a real symmetric 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.

Parameters

[in]	uplo	magma_uplo_t = MagmaUpper: Upper triangle of A is stored; = MagmaLower: Lower triangle of A is stored.
[in]	n	INTEGER The order of the matrix A. N >= 0.
[in,out]	dA	REAL array on the GPU, dimension (LDDA,N) On entry, the symmetric 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.
[in]	ldda	INTEGER The leading dimension of the array A. LDDA >= max(1,N).
[out]	d	REAL array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
[out]	e	REAL 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.
[out]	tau	REAL array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).
[out]	A	(workspace) REAL array, dimension (LDA,N) On exit the diagonal, the upper part (if uplo=MagmaUpper) or the lower part (if uplo=MagmaLower) are copies of DA
[in]	lda	INTEGER The leading dimension of the array A. LDA >= max(1,N).
[out]	work	(workspace) REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]	lwork	INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB is the optimal blocksize given by magma_get_ssytrd_nb(). 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.
[out]	dwork	(workspace) REAL array on the GPU, dim (MAX(1,LDWORK))
[in]	ldwork	INTEGER The dimension of the array DWORK. LDWORK >= ldda*ceil(n/64) + 2*ldda*nb, where nb = magma_get_ssytrd_nb(n), and 64 is for the blocksize of magmablas_ssymv.
[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 real scalar, and v is a real 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 real scalar, and v is a real 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).

magma_int_t magma_ssytrd_gpu	(	magma_uplo_t	uplo,
		magma_int_t	n,
		magmaFloat_ptr	dA,
		magma_int_t	ldda,
		float *	d,
		float *	e,
		float *	tau,
		float *	A,
		magma_int_t	lda,
		float *	work,
		magma_int_t	lwork,
		magma_int_t *	info
	)

SSYTRD_GPU reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T.

Parameters

[in]	uplo	magma_uplo_t = MagmaUpper: Upper triangle of A is stored; = MagmaLower: Lower triangle of A is stored.
[in]	n	INTEGER The order of the matrix A. N >= 0.
[in,out]	dA	REAL array on the GPU, dimension (LDDA,N) On entry, the symmetric 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.
[in]	ldda	INTEGER The leading dimension of the array A. LDDA >= max(1,N).
[out]	d	REAL array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
[out]	e	REAL 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.
[out]	tau	REAL array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).
[out]	A	(workspace) REAL array, dimension (LDA,N) On exit the diagonal, the upper part (if uplo=MagmaUpper) or the lower part (if uplo=MagmaLower) are copies of dA
[in]	lda	INTEGER The leading dimension of the array A. LDA >= max(1,N).
[out]	work	(workspace) REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]	lwork	INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB is the optimal blocksize given by magma_get_ssytrd_nb(). 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.
[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 real scalar, and v is a real 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 real scalar, and v is a real 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).

magma_int_t magma_ssytrd_mgpu	(	magma_int_t	ngpu,
		magma_int_t	nqueue,
		magma_uplo_t	uplo,
		magma_int_t	n,
		float *	A,
		magma_int_t	lda,
		float *	d,
		float *	e,
		float *	tau,
		float *	work,
		magma_int_t	lwork,
		magma_int_t *	info
	)

SSYTRD reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T.

Parameters

[in]	ngpu	INTEGER Number of GPUs to use. ngpu > 0.
[in]	nqueue	INTEGER The number of GPU queues used for update. 10 >= nqueue > 0.
[in]	uplo	magma_uplo_t = MagmaUpper: Upper triangle of A is stored; = MagmaLower: Lower triangle of A is stored.
[in]	n	INTEGER The order of the matrix A. N >= 0.
[in,out]	A	REAL array, dimension (LDA,N) On entry, the symmetric 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.
[in]	lda	INTEGER The leading dimension of the array A. LDA >= max(1,N).
[out]	d	REAL array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
[out]	e	REAL 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.
[out]	tau	REAL array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).
[out]	work	(workspace) REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]	lwork	INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB is the optimal blocksize given by magma_get_ssytrd_nb(). 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.
[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 real scalar, and v is a real 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 real scalar, and v is a real 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).

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
	)

ZHETRD reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T.

Parameters

[in]	uplo	magma_uplo_t = MagmaUpper: Upper triangle of A is stored; = MagmaLower: Lower triangle of A is stored.
[in]	n	INTEGER The order of the matrix A. N >= 0.
[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.
[in]	lda	INTEGER The leading dimension of the array A. LDA >= max(1,N).
[out]	d	COMPLEX_16 array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
[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.
[out]	tau	COMPLEX_16 array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).
[out]	work	(workspace) COMPLEX_16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]	lwork	INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB is the optimal blocksize given by magma_get_zhetrd_nb(). 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.
[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).

magma_int_t magma_zhetrd2_gpu	(	magma_uplo_t	uplo,
		magma_int_t	n,
		magmaDoubleComplex_ptr	dA,
		magma_int_t	ldda,
		double *	d,
		double *	e,
		magmaDoubleComplex *	tau,
		magmaDoubleComplex *	A,
		magma_int_t	lda,
		magmaDoubleComplex *	work,
		magma_int_t	lwork,
		magmaDoubleComplex_ptr	dwork,
		magma_int_t	ldwork,
		magma_int_t *	info
	)

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.

Parameters

[in]	uplo	magma_uplo_t = MagmaUpper: Upper triangle of A is stored; = MagmaLower: Lower triangle of A is stored.
[in]	n	INTEGER The order of the matrix A. N >= 0.
[in,out]	dA	COMPLEX_16 array on the GPU, dimension (LDDA,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.
[in]	ldda	INTEGER The leading dimension of the array A. LDDA >= max(1,N).
[out]	d	COMPLEX_16 array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
[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.
[out]	tau	COMPLEX_16 array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).
[out]	A	(workspace) COMPLEX_16 array, dimension (LDA,N) On exit the diagonal, the upper part (if uplo=MagmaUpper) or the lower part (if uplo=MagmaLower) are copies of DA
[in]	lda	INTEGER The leading dimension of the array A. LDA >= max(1,N).
[out]	work	(workspace) COMPLEX_16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]	lwork	INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB is the optimal blocksize given by magma_get_zhetrd_nb(). 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.
[out]	dwork	(workspace) COMPLEX_16 array on the GPU, dim (MAX(1,LDWORK))
[in]	ldwork	INTEGER The dimension of the array DWORK. LDWORK >= ldda*ceil(n/64) + 2*ldda*nb, where nb = magma_get_zhetrd_nb(n), and 64 is for the blocksize of magmablas_zhemv.
[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).

magma_int_t magma_zhetrd_gpu	(	magma_uplo_t	uplo,
		magma_int_t	n,
		magmaDoubleComplex_ptr	dA,
		magma_int_t	ldda,
		double *	d,
		double *	e,
		magmaDoubleComplex *	tau,
		magmaDoubleComplex *	A,
		magma_int_t	lda,
		magmaDoubleComplex *	work,
		magma_int_t	lwork,
		magma_int_t *	info
	)

ZHETRD_GPU reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T.

Parameters

[in]	uplo	magma_uplo_t = MagmaUpper: Upper triangle of A is stored; = MagmaLower: Lower triangle of A is stored.
[in]	n	INTEGER The order of the matrix A. N >= 0.
[in,out]	dA	COMPLEX_16 array on the GPU, dimension (LDDA,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.
[in]	ldda	INTEGER The leading dimension of the array A. LDDA >= max(1,N).
[out]	d	COMPLEX_16 array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
[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.
[out]	tau	COMPLEX_16 array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).
[out]	A	(workspace) COMPLEX_16 array, dimension (LDA,N) On exit the diagonal, the upper part (if uplo=MagmaUpper) or the lower part (if uplo=MagmaLower) are copies of dA
[in]	lda	INTEGER The leading dimension of the array A. LDA >= max(1,N).
[out]	work	(workspace) COMPLEX_16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]	lwork	INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB is the optimal blocksize given by magma_get_zhetrd_nb(). 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.
[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).

magma_int_t magma_zhetrd_mgpu	(	magma_int_t	ngpu,
		magma_int_t	nqueue,
		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
	)

ZHETRD reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T.

Parameters

[in]	ngpu	INTEGER Number of GPUs to use. ngpu > 0.
[in]	nqueue	INTEGER The number of GPU queues used for update. 10 >= nqueue > 0.
[in]	uplo	magma_uplo_t = MagmaUpper: Upper triangle of A is stored; = MagmaLower: Lower triangle of A is stored.
[in]	n	INTEGER The order of the matrix A. N >= 0.
[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.
[in]	lda	INTEGER The leading dimension of the array A. LDA >= max(1,N).
[out]	d	COMPLEX_16 array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
[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.
[out]	tau	COMPLEX_16 array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).
[out]	work	(workspace) COMPLEX_16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]	lwork	INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB is the optimal blocksize given by magma_get_zhetrd_nb(). 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.
[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).