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