double precision
[Symmetric indefinite solve: computational]

Functions

magma_int_t magma_dsysv (magma_uplo_t uplo, magma_int_t n, magma_int_t nrhs, double *A, magma_int_t lda, magma_int_t *ipiv, double *B, magma_int_t ldb, magma_int_t *info)
 DSYSV computes the solution to a real system of linear equations A * X = B, where A is an n-by-n symmetric matrix and X and B are n-by-nrhs matrices.
magma_int_t magma_dsytrf (magma_uplo_t uplo, magma_int_t n, double *A, magma_int_t lda, magma_int_t *ipiv, magma_int_t *info)
 DSYTRF computes the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method.
magma_int_t magma_dsytrf_nopiv (magma_uplo_t uplo, magma_int_t n, double *A, magma_int_t lda, magma_int_t *info)
 DSYTRF_nopiv computes the LDLt factorization of a real symmetric matrix A.
magma_int_t magma_dsytrf_nopiv_gpu (magma_uplo_t uplo, magma_int_t n, magmaDouble_ptr dA, magma_int_t ldda, magma_int_t *info)
 DSYTRF_nopiv_gpu computes the LDLt factorization of a real symmetric matrix A.
magma_int_t magma_dsytrs_nopiv_gpu (magma_uplo_t uplo, magma_int_t n, magma_int_t nrhs, magmaDouble_ptr dA, magma_int_t ldda, magmaDouble_ptr dB, magma_int_t lddb, magma_int_t *info)
 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 DSYTRF_NOPIV_GPU.

Function Documentation

magma_int_t magma_dsysv ( magma_uplo_t  uplo,
magma_int_t  n,
magma_int_t  nrhs,
double *  A,
magma_int_t  lda,
magma_int_t *  ipiv,
double *  B,
magma_int_t  ldb,
magma_int_t *  info 
)

DSYSV computes the solution to a real system of linear equations A * X = B, where A is an n-by-n symmetric matrix and X and B are n-by-nrhs matrices.

The diagonal pivoting method is used to factor A as A = U * D * U**H, if uplo = 'U', or A = L * D * L**H, if uplo = 'L', 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. The factored form of A is then used to solve the system of equations A * X = B.

Parameters:
[in] uplo CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.
[in] n INTEGER The number of linear equations, i.e., the order of the matrix A. n >= 0.
[in] nrhs INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. nrhs >= 0.
[in,out] A DOUBLE PRECISION array, dimension (lda,n) On entry, the symmetric matrix A. If uplo = 'U', 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 = 'L', 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 block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**H or A = L*D*L**H as computed by DSYTRF.

Parameters:
[in] lda INTEGER The leading dimension of the array A. lda >= max(1,n).
[out] ipiv INTEGER array, dimension (n) Details of the interchanges and the block structure of D, as determined by DSYTRF. 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 = 'U' 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 = 'L' 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.
[in,out] B (input/output) DOUBLE PRECISION array, dimension (ldb,nrhs) On entry, the n-by-nrhs right hand side matrix B. On exit, if info = 0, the n-by-nrhs solution matrix X.
[in] ldb INTEGER The leading dimension of the array B. ldb >= max(1,n).
[out] info INTEGER = 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, so the solution could not be computed.
magma_int_t magma_dsytrf ( magma_uplo_t  uplo,
magma_int_t  n,
double *  A,
magma_int_t  lda,
magma_int_t *  ipiv,
magma_int_t *  info 
)

DSYTRF 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] UPLO CHARACTER*1

  • = 'U': Upper triangle of A is stored;
  • = 'L': 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 = 'U', 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 = 'L', 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] LDA INTEGER The leading dimension of the array A. LDA >= max(1,N).
[out] IPIV INTEGER 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 = 'U' 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 = 'L' 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] INFO INTEGER

  • = 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 = 'U', 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 = 'L', 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_dsytrf_nopiv ( magma_uplo_t  uplo,
magma_int_t  n,
double *  A,
magma_int_t  lda,
magma_int_t *  info 
)

DSYTRF_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 = 'U', or A = L * D * L^H, if UPLO = 'L', 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] UPLO CHARACTER*1

  • = 'U': Upper triangle of A is stored;
  • = 'L': 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 = 'U', 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 = 'L', 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] LDA INTEGER The leading dimension of the array A. LDA >= max(1,N).
[out] INFO INTEGER

  • = 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_dsytrf_nopiv_gpu ( magma_uplo_t  uplo,
magma_int_t  n,
magmaDouble_ptr  dA,
magma_int_t  ldda,
magma_int_t *  info 
)

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

The factorization has the form A = U^H * D * U, if UPLO = 'U', or A = L * D * L^H, if UPLO = 'L', 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] UPLO CHARACTER*1

  • = 'U': Upper triangle of A is stored;
  • = 'L': 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 (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U', 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 = 'L', 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] LDA INTEGER The leading dimension of the array A. LDA >= max(1,N).
[out] INFO INTEGER

  • = 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_dsytrs_nopiv_gpu ( magma_uplo_t  uplo,
magma_int_t  n,
magma_int_t  nrhs,
magmaDouble_ptr  dA,
magma_int_t  ldda,
magmaDouble_ptr  dB,
magma_int_t  lddb,
magma_int_t *  info 
)

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 DSYTRF_NOPIV_GPU.

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] nrhs INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
[in] dA DOUBLE_PRECISION array on the GPU, dimension (LDA,N) The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by DSYTRF_NOPIV_GPU.
[in] ldda INTEGER The leading dimension of the array A. LDA >= max(1,N).

param[in,out] dB DOUBLE_PRECISION array on the GPU, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X.

Parameters:
[in] lddb INTEGER The leading dimension of the array B. LDB >= max(1,N).
[out] info INTEGER

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

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