!> SSYSV_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 SSYTRS_3.
!> 

UPLO

!>          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

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

NRHS

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

A

!>          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

!>          LDA is INTEGER
!>          The leading dimension of the array A.  LDA >= max(1,N).
!> 

E

!>          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

!>          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

!>          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

!>          LDB is INTEGER
!>          The leading dimension of the array B.  LDB >= max(1,N).
!> 

WORK

!>          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

!>          LWORK is INTEGER
!>          The length of WORK.  LWORK >= 1. For best performance
!>          of factorization stage LWORK >= 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

!>          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.
!> 

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

!>
!>  December 2016,  Igor Kozachenko,
!>                  Computer Science Division,
!>                  University of California, Berkeley
!>
!>  September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
!>                  School of Mathematics,
!>                  University of Manchester
!>
!>