!> SLASYF_RK computes a partial factorization of a real symmetric
!> matrix A using the bounded Bunch-Kaufman (rook) diagonal
!> pivoting method. The partial factorization has the form:
!>
!> A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
!>       ( 0  U22 ) (  0   D  ) ( U12**T U22**T )
!>
!> A  =  ( L11  0 ) (  D   0  ) ( L11**T L21**T )  if UPLO = 'L',
!>       ( L21  I ) (  0  A22 ) (  0       I    )
!>
!> where the order of D is at most NB. The actual order is returned in
!> the argument KB, and is either NB or NB-1, or N if N <= NB.
!>
!> SLASYF_RK is an auxiliary routine called by SSYTRF_RK. It uses
!> blocked code (calling Level 3 BLAS) to update the submatrix
!> A11 (if UPLO = 'U') or A22 (if UPLO = 'L').
!> 

UPLO

!>          UPLO is CHARACTER*1
!>          Specifies whether the upper or lower triangular part of the
!>          symmetric matrix A is stored:
!>          = 'U':  Upper triangular
!>          = 'L':  Lower triangular
!> 

N

!>          N is INTEGER
!>          The order of the matrix A.  N >= 0.
!> 

NB

!>          NB is INTEGER
!>          The maximum number of columns of the matrix A that should be
!>          factored.  NB should be at least 2 to allow for 2-by-2 pivot
!>          blocks.
!> 

KB

!>          KB is INTEGER
!>          The number of columns of A that were actually factored.
!>          KB is either NB-1 or NB, or N if N <= NB.
!> 

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, contains:
!>            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.
!> 

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 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.
!> 

IPIV

!>          IPIV is INTEGER array, dimension (N)
!>          IPIV describes the permutation matrix P in the factorization
!>          of matrix A as follows. The absolute value of IPIV(k)
!>          represents the index of row and column that were
!>          interchanged with the k-th row and column. The value of UPLO
!>          describes the order in which the interchanges were applied.
!>          Also, the sign of IPIV represents the block structure of
!>          the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
!>          diagonal blocks which correspond to 1 or 2 interchanges
!>          at each factorization step.
!>
!>          If UPLO = 'U',
!>          ( in factorization order, k decreases from N to 1 ):
!>            a) A single positive entry IPIV(k) > 0 means:
!>               D(k,k) is a 1-by-1 diagonal block.
!>               If IPIV(k) != k, rows and columns k and IPIV(k) were
!>               interchanged in the submatrix A(1:N,N-KB+1:N);
!>               If IPIV(k) = k, no interchange occurred.
!>
!>
!>            b) A pair of consecutive negative entries
!>               IPIV(k) < 0 and IPIV(k-1) < 0 means:
!>               D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
!>               (NOTE: negative entries in IPIV appear ONLY in pairs).
!>               1) If -IPIV(k) != k, rows and columns
!>                  k and -IPIV(k) were interchanged
!>                  in the matrix A(1:N,N-KB+1:N).
!>                  If -IPIV(k) = k, no interchange occurred.
!>               2) If -IPIV(k-1) != k-1, rows and columns
!>                  k-1 and -IPIV(k-1) were interchanged
!>                  in the submatrix A(1:N,N-KB+1:N).
!>                  If -IPIV(k-1) = k-1, no interchange occurred.
!>
!>            c) In both cases a) and b) is always ABS( IPIV(k) ) <= k.
!>
!>            d) NOTE: Any entry IPIV(k) is always NONZERO on output.
!>
!>          If UPLO = 'L',
!>          ( in factorization order, k increases from 1 to N ):
!>            a) A single positive entry IPIV(k) > 0 means:
!>               D(k,k) is a 1-by-1 diagonal block.
!>               If IPIV(k) != k, rows and columns k and IPIV(k) were
!>               interchanged in the submatrix A(1:N,1:KB).
!>               If IPIV(k) = k, no interchange occurred.
!>
!>            b) A pair of consecutive negative entries
!>               IPIV(k) < 0 and IPIV(k+1) < 0 means:
!>               D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
!>               (NOTE: negative entries in IPIV appear ONLY in pairs).
!>               1) If -IPIV(k) != k, rows and columns
!>                  k and -IPIV(k) were interchanged
!>                  in the submatrix A(1:N,1:KB).
!>                  If -IPIV(k) = k, no interchange occurred.
!>               2) If -IPIV(k+1) != k+1, rows and columns
!>                  k-1 and -IPIV(k-1) were interchanged
!>                  in the submatrix A(1:N,1:KB).
!>                  If -IPIV(k+1) = k+1, no interchange occurred.
!>
!>            c) In both cases a) and b) is always ABS( IPIV(k) ) >= k.
!>
!>            d) NOTE: Any entry IPIV(k) is always NONZERO on output.
!> 

W

!>          W is REAL array, dimension (LDW,NB)
!> 

LDW

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

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
!>
!>