SYSV_RK - 5.2 English - 68552

SSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices.

Purpose:

    SYSV_RK 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 bounded Bunch-Kaufman (rook) diagonal pivoting method is used
    to factor A as
       A = P*U*D*(U**T)*(P**T),  if UPLO = 'U', or
       A = P*L*D*(L**T)*(P**T),  if UPLO = 'L',
    where U (or L) is unit upper (or lower) triangular matrix,
    U**T (or L**T) is the transpose of U (or L), P is a permutation
    matrix, P**T is the transpose of P, and D is symmetric and block
    diagonal with 1-by-1 and 2-by-2 diagonal blocks.

    SSYTRF_RK is called to compute the factorization of a real
    symmetric matrix.  The factored form of A is then used to solve
    the system of equations A * X = B by calling BLAS3 routine SYTRS_3.

Parameters:

• UPLO – [in]

  UPLO is CHARACTER*1

  Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:

  = ‘U’: Upper triangle of A is stored;

  = ‘L’: Lower triangle of A is stored.

• N – [in]

  N is INTEGER

  The number of linear equations, i.e., the order of the matrix A. N >= 0.

• NRHS – [in]

  NRHS is INTEGER

  The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.

• A – [inout]

  A is REAL 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, diagonal of the block diagonal matrix D and factors U or L as computed by SSYTRF_RK:

  a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and

  b) If UPLO = ‘U’: factor U in the superdiagonal part of A. If UPLO = ‘L’: factor L in the subdiagonal part of A.

  For more info see the description of DSYTRF_RK routine.

• LDA – [in]

  LDA is INTEGER

  The leading dimension of the array A. LDA >= fla_max(1,N).

• E – [out]

  E is REAL array, dimension (N)

  On exit, contains the output computed by the factorization routine DSYTRF_RK, i.e. the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where

  If UPLO = ‘U’: E(i) = D(i-1,i), i=2:N, E(1) is set to 0;

  If UPLO = ‘L’: E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.

  NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = ‘U’ or UPLO = ‘L’ cases.

  For more info see the description of DSYTRF_RK routine.

• IPIV – [out]

  IPIV is INTEGER array, dimension (N)

  Details of the interchanges and the block structure of D, as determined by SSYTRF_RK.

  For more info see the description of DSYTRF_RK routine.

• B – [inout]

  B is REAL 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.

• LDB – [in]

  LDB is INTEGER

  The leading dimension of the array B. LDB >= fla_max(1,N).

• WORK – [out]

  WORK is REAL array, dimension ( MAX(1,LWORK) ).

  Work array used in the factorization stage. On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

• LWORK – [in]

  LWORK is INTEGER

  The length of WORK. LWORK >= 1. For best performance of factorization stage LWORK >= fla_max(1,N*NB), where NB is the optimal blocksize for DSYTRF_RK.

  If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array for factorization stage, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

• INFO – [out]

  INFO is INTEGER

  = 0: successful exit

  < 0: If INFO = -k, the k-th argument had an illegal value

  \ > 0: If INFO = k, the matrix A is singular, because: If UPLO = ‘U’: column k in the upper triangular part of A contains all zeros. If UPLO = ‘L’: column k in the lower triangular part of A contains all zeros.

  Therefore D(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L ) are all zeros. 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.

  NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes.