PLASMA 2.8.0 PLASMA - Parallel Linear Algebra for Scalable Multi-core Architectures

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int PLASMA_dsposv | ( | PLASMA_enum | uplo,
|  | int | N,
|  | int | NRHS,
|  | double * | A,
|  | int | LDA,
|  | double * | B,
|  | int | LDB,
|  | double * | X,
|  | int | LDX,
|  | int * | ITER
| ) |  |

PLASMA_dsposv - Computes the solution to a system of linear equations A * X = B, where A is an N-by-N symmetric positive definite (or Hermitian positive definite in the complex case) matrix and X and B are N-by-NRHS matrices. The Cholesky decomposition is used to factor A as

A = U**H * U, if uplo = PlasmaUpper, or A = L * L**H, if uplo = PlasmaLower,

where U is an upper triangular matrix and L is a lower triangular matrix. The factored form of A is then used to solve the system of equations A * X = B.

PLASMA_dsposv first attempts to factorize the matrix in COMPLEX and use this factorization within an iterative refinement procedure to produce a solution with COMPLEX*16 normwise backward error quality (see below). If the approach fails the method switches to a COMPLEX*16 factorization and solve.

The iterative refinement is not going to be a winning strategy if the ratio COMPLEX performance over COMPLEX*16 performance is too small. A reasonable strategy should take the number of right-hand sides and the size of the matrix into account. This might be done with a call to ILAENV in the future. Up to now, we always try iterative refinement.

The iterative refinement process is stopped if ITER > ITERMAX or for all the RHS we have: RNRM < N*XNRM*ANRM*EPS*BWDMAX where:

- ITER is the number of the current iteration in the iterative refinement process
- RNRM is the infinity-norm of the residual
- XNRM is the infinity-norm of the solution
- ANRM is the infinity-operator-norm of the matrix A
- EPS is the machine epsilon returned by DLAMCH('Epsilon').

Actually, in its current state (PLASMA 2.1.0), the test is slightly relaxed.

The values ITERMAX and BWDMAX are fixed to 30 and 1.0D+00 respectively.

Parameters

  [in] | uplo | Specifies whether the matrix A is upper triangular or lower triangular: = PlasmaUpper: Upper triangle of A is stored; = PlasmaLower: Lower triangle of A is stored.
  [in] | N | The number of linear equations, i.e., the order of the matrix A. N >= 0.
  [in] | NRHS | The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
  [in] | A | The N-by-N symmetric positive definite (or Hermitian) coefficient matrix A. If uplo = PlasmaUpper, 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. This matrix is not modified.
  [in] | LDA | The leading dimension of the array A. LDA >= max(1,N).
  [in] | B | The N-by-NRHS matrix of right hand side matrix B.
  [in] | LDB | The leading dimension of the array B. LDB >= max(1,N).
  [out] | X | If return value = 0, the N-by-NRHS solution matrix X.
  [in] | LDX | The leading dimension of the array B. LDX >= max(1,N).
  [out] | ITER | The number of the current iteration in the iterative refinement process

Returns

Return values

  PLASMA_SUCCESS | successful exit
  <0 | if -i, the i-th argument had an illegal value
  >0 | if i, the leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution has not been computed.

See also
  PLASMA_dsposv_Tile
  PLASMA_dsposv_Tile_Async
  PLASMA_dsposv
  PLASMA_dposv