MAGMA  2.0.0
Matrix Algebra for GPU and Multicore Architectures

Functions

magma_int_t magma_ssytrf (magma_uplo_t uplo, magma_int_t n, float *A, magma_int_t lda, magma_int_t *ipiv, magma_int_t *info)
 SSYTRF computes the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. More...
 
magma_int_t magma_ssytrf_aasen (magma_uplo_t uplo, magma_int_t cpu_panel, magma_int_t n, float *A, magma_int_t lda, magma_int_t *ipiv, magma_int_t *info)
 SSYTRF_AASEN computes the factorization of a real symmetric matrix A based on a communication-avoiding variant of the Aasen's algorithm . More...
 
magma_int_t magma_ssytrf_nopiv (magma_uplo_t uplo, magma_int_t n, float *A, magma_int_t lda, magma_int_t *info)
 SSYTRF_nopiv computes the LDLt factorization of a real symmetric matrix A. More...
 
magma_int_t magma_ssytrf_nopiv_gpu (magma_uplo_t uplo, magma_int_t n, magmaFloat_ptr dA, magma_int_t ldda, magma_int_t *info)
 SSYTRF_nopiv_gpu computes the LDLt factorization of a real symmetric matrix A. More...
 
magma_int_t magma_ssytrs_nopiv_gpu (magma_uplo_t uplo, magma_int_t n, magma_int_t nrhs, magmaFloat_ptr dA, magma_int_t ldda, magmaFloat_ptr dB, magma_int_t lddb, magma_int_t *info)
 SSYTRS solves a system of linear equations A*X = B with a real symmetric matrix A using the factorization A = U * D * U**H or A = L * D * L**H computed by SSYTRF_NOPIV_GPU. More...
 

Detailed Description

Function Documentation

magma_int_t magma_ssytrf ( magma_uplo_t  uplo,
magma_int_t  n,
float *  A,
magma_int_t  lda,
magma_int_t *  ipiv,
magma_int_t *  info 
)

SSYTRF computes the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method.

The form of the factorization is

A = U*D*U^H  or  A = L*D*L^H

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

This is the blocked version of the algorithm, calling Level 3 BLAS.

Parameters
[in]uplomagma_uplo_t
  • = MagmaUpper: Upper triangle of A is stored;
  • = MagmaLower: Lower triangle of A is stored.
[in]nINTEGER The order of the matrix A. N >= 0.
[in,out]AREAL 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, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).
[in]ldaINTEGER The leading dimension of the array A. LDA >= max(1,N).
[out]ipivINTEGER array, dimension (N) Details of the interchanges and the block structure of D. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If UPLO = MagmaUpper and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = MagmaLower and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
[out]infoINTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: if INFO = i, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.

Further Details

If UPLO = MagmaUpper, then A = U*D*U', where U = P(n)*U(n)* ... P(k)U(k) ..., i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

       (   I    v    0   )   k-s

U(k) = ( 0 I 0 ) s ( 0 0 I ) n-k k-s s n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k).

If UPLO = MagmaLower, then A = L*D*L', where L = P(1)*L(1)* ... P(k)*L(k) ..., i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

       (   I    0     0   )  k-1

L(k) = ( 0 I 0 ) s ( 0 v I ) n-k-s+1 k-1 s n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

magma_int_t magma_ssytrf_aasen ( magma_uplo_t  uplo,
magma_int_t  cpu_panel,
magma_int_t  n,
float *  A,
magma_int_t  lda,
magma_int_t *  ipiv,
magma_int_t *  info 
)

SSYTRF_AASEN computes the factorization of a real symmetric matrix A based on a communication-avoiding variant of the Aasen's algorithm .

The form of the factorization is

A = U*D*U**H or A = L*D*L**H

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and banded matrix of the band width equal to the block size.

Parameters
[in]uplomagma_uplo_t
  • = MagmaUpper: Upper triangle of A is stored;
  • = MagmaLower: Lower triangle of A is stored.
[in]cpu_panelINTEGER If cpu_panel =0, panel factorization is done on GPU.
[in]nINTEGER The order of the matrix A. N >= 0.
[in,out]AREAL 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, the banded matrix D and the triangular factor U or L.
[in]ldaINTEGER The leading dimension of the array A. LDA >= max(1,N).
[out]ipivINTEGER array, dimension (N) Details of the interchanges.
[out]infoINTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: if INFO = i, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.
magma_int_t magma_ssytrf_nopiv ( magma_uplo_t  uplo,
magma_int_t  n,
float *  A,
magma_int_t  lda,
magma_int_t *  info 
)

SSYTRF_nopiv computes the LDLt factorization of a real symmetric matrix A.

This version does not require work space on the GPU passed as input. GPU memory is allocated in the routine.

The factorization has the form A = U^H * D * U, if UPLO = MagmaUpper, or A = L * D * L^H, if UPLO = MagmaLower, where U is an upper triangular matrix, L is lower triangular, and D is a diagonal matrix.

This is the block version of the algorithm, calling Level 3 BLAS.

Parameters
[in]uplomagma_uplo_t
  • = MagmaUpper: Upper triangle of A is stored;
  • = MagmaLower: Lower triangle of A is stored.
[in]nINTEGER The order of the matrix A. N >= 0.
[in,out]AREAL 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 INFO = 0, the factor U or L from the Cholesky factorization A = U^H D U or A = L D L^H.
Higher performance is achieved if A is in pinned memory.
[in]ldaINTEGER The leading dimension of the array A. LDA >= max(1,N).
[out]infoINTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value if INFO = -6, the GPU memory allocation failed
  • > 0: if INFO = i, the leading minor of order i is not positive definite, and the factorization could not be completed.
magma_int_t magma_ssytrf_nopiv_gpu ( magma_uplo_t  uplo,
magma_int_t  n,
magmaFloat_ptr  dA,
magma_int_t  ldda,
magma_int_t *  info 
)

SSYTRF_nopiv_gpu computes the LDLt factorization of a real symmetric matrix A.

The factorization has the form A = U^H * D * U, if UPLO = MagmaUpper, or A = L * D * L^H, if UPLO = MagmaLower, where U is an upper triangular matrix, L is lower triangular, and D is a diagonal matrix.

This is the block version of the algorithm, calling Level 3 BLAS.

Parameters
[in]uplomagma_uplo_t
  • = MagmaUpper: Upper triangle of A is stored;
  • = MagmaLower: Lower triangle of A is stored.
[in]nINTEGER The order of the matrix A. N >= 0.
[in,out]dAREAL 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 INFO = 0, the factor U or L from the Cholesky factorization A = U^H D U or A = L D L^H.
Higher performance is achieved if A is in pinned memory, e.g. allocated using cudaMallocHost.
[in]lddaINTEGER The leading dimension of the array A. LDDA >= max(1,N).
[out]infoINTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value if INFO = -6, the GPU memory allocation failed
  • > 0: if INFO = i, the leading minor of order i is not positive definite, and the factorization could not be completed.
magma_int_t magma_ssytrs_nopiv_gpu ( magma_uplo_t  uplo,
magma_int_t  n,
magma_int_t  nrhs,
magmaFloat_ptr  dA,
magma_int_t  ldda,
magmaFloat_ptr  dB,
magma_int_t  lddb,
magma_int_t *  info 
)

SSYTRS solves a system of linear equations A*X = B with a real symmetric matrix A using the factorization A = U * D * U**H or A = L * D * L**H computed by SSYTRF_NOPIV_GPU.

Parameters
[in]uplomagma_uplo_t
  • = MagmaUpper: Upper triangle of A is stored;
  • = MagmaLower: Lower triangle of A is stored.
[in]nINTEGER The order of the matrix A. N >= 0.
[in]nrhsINTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
[in]dAREAL array on the GPU, dimension (LDDA,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by SSYTRF_NOPIV_GPU.
[in]lddaINTEGER The leading dimension of the array A. LDDA >= max(1,N).
[in,out]dBREAL array on the GPU, dimension (LDDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X.
[in]lddbINTEGER The leading dimension of the array B. LDDB >= max(1,N).
[out]infoINTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value