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MAGMA
1.5.0
Matrix Algebra for GPU and Multicore Architectures
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Functions | |
magma_int_t | magma_clahr2 (magma_int_t n, magma_int_t k, magma_int_t nb, magmaFloatComplex *dA, magma_int_t ldda, magmaFloatComplex *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) |
CLAHR2 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. 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 n-BY-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero. More... | |
magma_int_t | magma_clahru (magma_int_t n, magma_int_t ihi, magma_int_t k, magma_int_t nb, magmaFloatComplex *A, magma_int_t lda, magmaFloatComplex *dA, magma_int_t ldda, magmaFloatComplex *dY, magma_int_t lddy, magmaFloatComplex *dV, magma_int_t lddv, magmaFloatComplex *dT, magmaFloatComplex *dwork) |
CLAHRU is an auxiliary MAGMA routine that is used in CGEHRD to update the trailing sub-matrices after the reductions of the corresponding panels. More... | |
magma_int_t | magma_clahru_m (magma_int_t n, magma_int_t ihi, magma_int_t k, magma_int_t nb, magmaFloatComplex *A, magma_int_t lda, struct cgehrd_data *data) |
CLAHRU is an auxiliary MAGMA routine that is used in CGEHRD to update the trailing sub-matrices after the reductions of the corresponding panels. More... | |
magma_int_t | magma_clatrsd (magma_uplo_t uplo, magma_trans_t trans, magma_diag_t diag, magma_bool_t normin, magma_int_t n, const magmaFloatComplex *A, magma_int_t lda, magmaFloatComplex lambda, magmaFloatComplex *x, float *scale, float *cnorm, magma_int_t *info) |
CLATRSD 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. More... | |
magma_int_t magma_clahr2 | ( | magma_int_t | n, |
magma_int_t | k, | ||
magma_int_t | nb, | ||
magmaFloatComplex * | dA, | ||
magma_int_t | ldda, | ||
magmaFloatComplex * | 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 | ||
) |
CLAHR2 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 CGEHRD.
[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 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 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 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 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 n-by-nb matrix Y. |
[in] | ldy | INTEGER The leading dimension of the array Y. LDY >= N. |
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_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 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 CGEHRD.
[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 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 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 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. |
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_clahru | ( | magma_int_t | n, |
magma_int_t | ihi, | ||
magma_int_t | k, | ||
magma_int_t | nb, | ||
magmaFloatComplex * | A, | ||
magma_int_t | lda, | ||
magmaFloatComplex * | dA, | ||
magma_int_t | ldda, | ||
magmaFloatComplex * | dY, | ||
magma_int_t | lddy, | ||
magmaFloatComplex * | dV, | ||
magma_int_t | lddv, | ||
magmaFloatComplex * | dT, | ||
magmaFloatComplex * | dwork | ||
) |
CLAHRU is an auxiliary MAGMA routine that is used in CGEHRD to update the trailing sub-matrices after the reductions of the corresponding panels.
See further details below.
[in] | n | INTEGER The order of the matrix A. N >= 0. |
[in] | ihi | INTEGER Last row to update. Same as IHI in cgehrd. |
[in] | k | INTEGER Number of rows of the matrix Am (see details below) |
[in] | nb | INTEGER Block size |
[out] | A | COMPLEX 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 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 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 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 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 array on the GPU, dimension N*NB. |
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_clahru_m | ( | magma_int_t | n, |
magma_int_t | ihi, | ||
magma_int_t | k, | ||
magma_int_t | nb, | ||
magmaFloatComplex * | A, | ||
magma_int_t | lda, | ||
struct cgehrd_data * | data | ||
) |
CLAHRU is an auxiliary MAGMA routine that is used in CGEHRD to update the trailing sub-matrices after the reductions of the corresponding panels.
See further details below.
[in] | n | INTEGER The order of the matrix A. N >= 0. |
[in] | ihi | INTEGER Last row to update. Same as IHI in cgehrd. |
[in] | k | INTEGER Number of rows of the matrix Am (see details below) |
[in] | nb | INTEGER Block size |
[out] | A | COMPLEX 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. |
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_clatrsd | ( | magma_uplo_t | uplo, |
magma_trans_t | trans, | ||
magma_diag_t | diag, | ||
magma_bool_t | normin, | ||
magma_int_t | n, | ||
const magmaFloatComplex * | A, | ||
magma_int_t | lda, | ||
magmaFloatComplex | lambda, | ||
magmaFloatComplex * | x, | ||
float * | scale, | ||
float * | cnorm, | ||
magma_int_t * | info | ||
) |
CLATRSD 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 CTRSV 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 ctrevc to compute eigenvectors. It does not modify A during the computation.
[in] | uplo | magma_uplo_t Specifies whether the matrix A is upper or lower triangular.
|
[in] | trans | magma_trans_t Specifies the operation applied to A.
|
[in] | diag | magma_diag_t Specifies whether or not the matrix A is unit triangular.
|
[in] | normin | magma_bool_t Specifies whether CNORM has been set or not.
|
[in] | n | INTEGER The order of the matrix A. N >= 0. |
[in] | A | COMPLEX 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 Eigenvalue to subtract from diagonal of A. |
[in,out] | x | COMPLEX 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 | REAL 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) REAL array, dimension (N)
|
[out] | info | INTEGER
|
A rough bound on x is computed; if that is less than overflow, CTRSV 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 CTRSV 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 A upper triangular 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 CTRSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow).