DSYTRF_RK computes the factorization of a real symmetric matrix A


 using the bounded Bunch-Kaufman (rook) diagonal pivoting method:


    A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),


 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.


 This is the blocked version of the algorithm, calling Level 3 BLAS.


 For more information see Further Details section.

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.

A

          A is DOUBLE PRECISION 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 DOUBLE PRECISION 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. For more info see Further


          Details section.


          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 matrix A(1:N,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,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 matrix A(1:N,1:N).


                  If -IPIV(k-1) = k-1, no interchange occurred.


            c) In both cases a) and b), 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 matrix A(1:N,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: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 matrix A(1:N,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 matrix A(1:N,1:N).


                  If -IPIV(k+1) = k+1, no interchange occurred.


            c) In both cases a) and b), always ABS( IPIV(k) ) >= k.


            d) NOTE: Any entry IPIV(k) is always NONZERO on output.

WORK

          WORK is DOUBLE PRECISION array, dimension ( MAX(1,LWORK) ).


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The length of WORK.  LWORK >=1.  For best performance


          LWORK >= N*NB, where NB is the block size returned


          by ILAENV.


          If LWORK = -1, then a workspace query is assumed;


          the routine only calculates the optimal size of the WORK


          array, 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

 TODO: put correct description

  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