double-complex precision
[Eigenvalue]

Functions

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)
 ZLAHR2 reduces the first NB columns of a complex general n-BY-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero.
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 n-BY-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero.
magma_int_t magma_zlahru (magma_int_t n, magma_int_t ihi, magma_int_t k, magma_int_t nb, magmaDoubleComplex *A, magma_int_t lda, magmaDoubleComplex_ptr dA, magma_int_t ldda, magmaDoubleComplex_ptr dY, magma_int_t lddy, magmaDoubleComplex_ptr dV, magma_int_t lddv, magmaDoubleComplex_ptr dT, magmaDoubleComplex_ptr dwork)
 ZLAHRU is an auxiliary MAGMA routine that is used in ZGEHRD to update the trailing sub-matrices after the reductions of the corresponding panels.
magma_int_t magma_zlahru_m (magma_int_t n, magma_int_t ihi, magma_int_t k, magma_int_t nb, magmaDoubleComplex *A, magma_int_t lda, struct zgehrd_data *data)
 ZLAHRU is an auxiliary MAGMA routine that is used in ZGEHRD to update the trailing sub-matrices after the reductions of the corresponding panels.
magma_int_t magma_zlatrsd (magma_uplo_t uplo, magma_trans_t trans, magma_diag_t diag, magma_bool_t normin, magma_int_t n, const magmaDoubleComplex *A, magma_int_t lda, magmaDoubleComplex lambda, magmaDoubleComplex *x, double *scale, double *cnorm, magma_int_t *info)
 ZLATRSD solves one of the triangular systems with modified diagonal (A - lambda*I) * x = s*b, (A - lambda*I)**T * x = s*b, or (A - lambda*I)**H * x = s*b, with scaling to prevent overflow.

Function Documentation

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 
)

ZLAHR2 reduces the first NB columns of a complex general n-BY-(n-k+1) matrix A so that elements below the k-th 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.

Parameters:
[in] n INTEGER The order of the matrix A.
[in] k INTEGER The offset for the reduction. Elements below the k-th 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,N-K+1) On entry, the n-by-(n-k+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 n-by-nb 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,N-K+1) On entry, the n-by-(n-k+1) general matrix A. On exit, the elements on and above the k-th subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced matrix; the elements below the k-th 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 n-by-nb matrix Y.
[in] ldy INTEGER The leading dimension of the array Y. LDY >= N.

Further Details --------------- 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+k-1) = 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 (n-k+1)-by-nb 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 GPU-based computing," University of Tennessee Computer Science Technical Report, UT-CS-09-642 (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 n-BY-(n-k+1) matrix A so that elements below the k-th 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.

Parameters:
[in] n INTEGER The order of the matrix A.
[in] k INTEGER The offset for the reduction. Elements below the k-th 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,N-K+1) On entry, the n-by-(n-k+1) general matrix A. On exit, the elements on and above the k-th subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced matrix; the elements below the k-th 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 n-by-nb 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.

Further Details --------------- 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+k-1) = 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 (n-k+1)-by-nb 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 GPU-based computing," University of Tennessee Computer Science Technical Report, UT-CS-09-642 (also LAPACK Working Note 219), May 24, 2009.

magma_int_t magma_zlahru ( magma_int_t  n,
magma_int_t  ihi,
magma_int_t  k,
magma_int_t  nb,
magmaDoubleComplex *  A,
magma_int_t  lda,
magmaDoubleComplex_ptr  dA,
magma_int_t  ldda,
magmaDoubleComplex_ptr  dY,
magma_int_t  lddy,
magmaDoubleComplex_ptr  dV,
magma_int_t  lddv,
magmaDoubleComplex_ptr  dT,
magmaDoubleComplex_ptr  dwork 
)

ZLAHRU is an auxiliary MAGMA routine that is used in ZGEHRD to update the trailing sub-matrices after the reductions of the corresponding panels.

See further details below.

Parameters:
[in] n INTEGER The order of the matrix A. N >= 0.
[in] ihi INTEGER Last row to update. Same as IHI in zgehrd.
[in] k INTEGER Number of rows of the matrix Am (see details below)
[in] nb INTEGER Block size
[out] A COMPLEX_16 array, dimension (LDA,N-K) On entry, the N-by-(N-K) general matrix to be updated. The computation is done on the GPU. After Am is updated on the GPU only Am(1:NB) is transferred to the CPU - to update the corresponding Am matrix. See Further Details below.
[in] lda INTEGER The leading dimension of the array A. LDA >= max(1,N).
[in,out] dA COMPLEX_16 array on the GPU, dimension (LDDA,N-K). On entry, the N-by-(N-K) general matrix to be updated. On exit, the 1st K rows (matrix Am) of A are updated by applying an orthogonal transformation from the right Am = Am (I-V T V'), and sub-matrix Ag is updated by Ag = (I - V T V') Ag (I - V T V(NB+1:)' ) where Q = I - V T V' represent the orthogonal matrix (as a product of elementary reflectors V) used to reduce the current panel of A to upper Hessenberg form. After Am is updated Am(:,1:NB) is sent to the CPU. See Further Details below.
[in] ldda INTEGER The leading dimension of the array dA. LDDA >= max(1,N).
[in,out] dY (workspace) COMPLEX_16 array on the GPU, dimension (LDDY, NB). On entry the (N-K)-by-NB Y = A V. It is used internally as workspace, so its value is changed on exit.
[in] lddy INTEGER The leading dimension of the array dY. LDDY >= max(1,N).
[in,out] dV (workspace) COMPLEX_16 array on the GPU, dimension (LDDV, NB). On entry the (N-K)-by-NB matrix V of elementary reflectors used to reduce the current panel of A to upper Hessenberg form. The rest K-by-NB part is used as workspace. V is unchanged on exit.
[in] lddv INTEGER The leading dimension of the array dV. LDDV >= max(1,N).
[in] dT COMPLEX_16 array on the GPU, dimension (NB, NB). On entry the NB-by-NB upper trinagular matrix defining the orthogonal Hessenberg reduction transformation matrix for the current panel. The lower triangular part are 0s.
dwork (workspace) COMPLEX_16 array on the GPU, dimension N*NB.

Further Details --------------- This implementation follows the algorithm and notations described in:

S. Tomov and J. Dongarra, "Accelerating the reduction to upper Hessenberg form through hybrid GPU-based computing," University of Tennessee Computer Science Technical Report, UT-CS-09-642 (also LAPACK Working Note 219), May 24, 2009.

The difference is that here Am is computed on the GPU. M is renamed Am, G is renamed Ag.

magma_int_t magma_zlahru_m ( magma_int_t  n,
magma_int_t  ihi,
magma_int_t  k,
magma_int_t  nb,
magmaDoubleComplex *  A,
magma_int_t  lda,
struct zgehrd_data *  data 
)

ZLAHRU is an auxiliary MAGMA routine that is used in ZGEHRD to update the trailing sub-matrices after the reductions of the corresponding panels.

See further details below.

Parameters:
[in] n INTEGER The order of the matrix A. N >= 0.
[in] ihi INTEGER Last row to update. Same as IHI in zgehrd.
[in] k INTEGER Number of rows of the matrix Am (see details below)
[in] nb INTEGER Block size
[out] A COMPLEX_16 array, dimension (LDA,N-K) On entry, the N-by-(N-K) general matrix to be updated. The computation is done on the GPU. After Am is updated on the GPU only Am(1:NB) is transferred to the CPU - to update the corresponding Am matrix. See Further Details below.
[in] lda INTEGER The leading dimension of the array A. LDA >= max(1,N).
[in,out] data Structure with pointers to dA, dT, dV, dW, dY which are distributed across multiple GPUs.

Further Details --------------- This implementation follows the algorithm and notations described in:

S. Tomov and J. Dongarra, "Accelerating the reduction to upper Hessenberg form through hybrid GPU-based computing," University of Tennessee Computer Science Technical Report, UT-CS-09-642 (also LAPACK Working Note 219), May 24, 2009.

The difference is that here Am is computed on the GPU. M is renamed Am, G is renamed Ag.

magma_int_t magma_zlatrsd ( magma_uplo_t  uplo,
magma_trans_t  trans,
magma_diag_t  diag,
magma_bool_t  normin,
magma_int_t  n,
const magmaDoubleComplex *  A,
magma_int_t  lda,
magmaDoubleComplex  lambda,
magmaDoubleComplex *  x,
double *  scale,
double *  cnorm,
magma_int_t *  info 
)

ZLATRSD solves one of the triangular systems with modified diagonal (A - lambda*I) * x = s*b, (A - lambda*I)**T * x = s*b, or (A - lambda*I)**H * x = s*b, with scaling to prevent overflow.

Here A is an upper or lower triangular matrix, A**T denotes the transpose of A, A**H denotes the conjugate transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine ZTRSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned.

This version subtracts lambda from the diagonal, for use in ztrevc to compute eigenvectors. It does not modify A during the computation.

Parameters:
[in] uplo magma_uplo_t Specifies whether the matrix A is upper or lower triangular.

  • = MagmaUpper: Upper triangular
  • = MagmaLower: Lower triangular
[in] trans magma_trans_t Specifies the operation applied to A.

  • = MagmaNoTrans: Solve (A - lambda*I) * x = s*b (No transpose)
  • = MagmaTrans: Solve (A - lambda*I)**T * x = s*b (Transpose)
  • = MagmaConjTrans: Solve (A - lambda*I)**H * x = s*b (Conjugate transpose)
[in] diag magma_diag_t Specifies whether or not the matrix A is unit triangular.

  • = MagmaNonUnit: Non-unit triangular
  • = MagmaUnit: Unit triangular
[in] normin magma_bool_t Specifies whether CNORM has been set or not.

  • = MagmaTrue: CNORM contains the column norms on entry
  • = MagmaFalse: CNORM is not set on entry. On exit, the norms will be computed and stored in CNORM.
[in] n INTEGER The order of the matrix A. N >= 0.
[in] A COMPLEX_16 array, dimension (LDA,N) The triangular matrix A. If UPLO = MagmaUpper, the leading n by n upper triangular part of the array A contains the upper triangular matrix, and the strictly lower triangular part of A is not referenced. If UPLO = MagmaLower, the leading n by n lower triangular part of the array A contains the lower triangular matrix, and the strictly upper triangular part of A is not referenced. If DIAG = MagmaUnit, the diagonal elements of A are also not referenced and are assumed to be 1.
[in] lda INTEGER The leading dimension of the array A. LDA >= max (1,N).
[in] lambda COMPLEX_16 Eigenvalue to subtract from diagonal of A.
[in,out] x COMPLEX_16 array, dimension (N) On entry, the right hand side b of the triangular system. On exit, X is overwritten by the solution vector x.
[out] scale DOUBLE PRECISION The scaling factor s for the triangular system A * x = s*b, A**T * x = s*b, or A**H * x = s*b. If SCALE = 0, the matrix A is singular or badly scaled, and the vector x is an exact or approximate solution to A*x = 0.
[in,out] cnorm (input or output) DOUBLE PRECISION array, dimension (N)

  • If NORMIN = MagmaTrue, CNORM is an input argument and CNORM(j) contains the norm of the off-diagonal part of the j-th column of A. If TRANS = MagmaNoTrans, CNORM(j) must be greater than or equal to the infinity-norm, and if TRANS = MagmaTrans or MagmaConjTrans, CNORM(j) must be greater than or equal to the 1-norm.
  • If NORMIN = MagmaFalse, CNORM is an output argument and CNORM(j) returns the 1-norm of the offdiagonal part of the j-th column of A.
[out] info INTEGER

  • = 0: successful exit
  • < 0: if INFO = -k, the k-th argument had an illegal value

Further Details --------------- A rough bound on x is computed; if that is less than overflow, ZTRSV is called, otherwise, specific code is used which checks for possible overflow or divide-by-zero at every operation.

A columnwise scheme is used for solving A*x = b. The basic algorithm if A is lower triangular is

x[1:n] := b[1:n] for j = 1, ..., n x(j) := x(j) / A(j,j) x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j] end

Define bounds on the components of x after j iterations of the loop: M(j) = bound on x[1:j] G(j) = bound on x[j+1:n] Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.

Then for iteration j+1 we have M(j+1) <= G(j) / | A(j+1,j+1) | G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] | <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )

where CNORM(j+1) is greater than or equal to the infinity-norm of column j+1 of A, not counting the diagonal. Hence

G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | ) 1 <= i <= j and

|x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| ) 1 <= i < j

Since |x(j)| <= M(j), we use the Level 2 BLAS routine ZTRSV if the reciprocal of the largest M(j), j=1,..,n, is larger than max(underflow, 1/overflow).

The bound on x(j) is also used to determine when a step in the columnwise method can be performed without fear of overflow. If the computed bound is greater than a large constant, x is scaled to prevent overflow, but if the bound overflows, x is set to 0, x(j) to 1, and scale to 0, and a non-trivial solution to A*x = 0 is found.

Similarly, a row-wise scheme is used to solve A**T *x = b or A**H *x = b. The basic algorithm for upper triangular A is:

for j = 1, ..., n x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j) end

We simultaneously compute two bounds G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1 <= i <= j M(j) = bound on x(i), 1 <= i <= j

The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1. Then the bound on x(j) is

M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |

<= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| ) 1 <= i <= j

and we can safely call ZTRSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow).


Generated on 3 May 2015 for MAGMA by  doxygen 1.6.1