DSYTRF_ROOK computes the factorization of a real symmetric matrix A


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


 The form of the factorization is


    A = U*D*U**T  or  A = L*D*L**T


 where U (or L) is a product of permutation and unit upper (lower)


 triangular matrices, 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.

UPLO

          UPLO is CHARACTER*1


          = 'U':  Upper triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

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, the block diagonal matrix D and the multipliers used


          to obtain the factor U or L (see below for further details).

LDA

          LDA is INTEGER


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

IPIV

          IPIV is INTEGER array, dimension (N)


          Details of the interchanges and the block structure of D.


          If UPLO = 'U':


               If IPIV(k) > 0, then rows and columns k and IPIV(k)


               were interchanged and D(k,k) is a 1-by-1 diagonal block.


               If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and


               columns k and -IPIV(k) were interchanged and rows and


               columns k-1 and -IPIV(k-1) were inerchaged,


               D(k-1:k,k-1:k) is a 2-by-2 diagonal block.


          If UPLO = 'L':


               If IPIV(k) > 0, then rows and columns k and IPIV(k)


               were interchanged and D(k,k) is a 1-by-1 diagonal block.


               If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and


               columns k and -IPIV(k) were interchanged and rows and


               columns k+1 and -IPIV(k+1) were inerchaged,


               D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

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 = -i, the i-th argument had an illegal value


          > 0:  if INFO = i, D(i,i) is exactly zero.  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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

April 2012

  If UPLO = 'U', then A = U*D*U**T, where


     U = P(n)*U(n)* ... *P(k)U(k)* ...,


  i.e., U is a product of terms P(k)*U(k), where k decreases from n to


  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1


  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as


  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such


  that if the diagonal block D(k) is of order s (s = 1 or 2), then


             (   I    v    0   )   k-s


     U(k) =  (   0    I    0   )   s


             (   0    0    I   )   n-k


                k-s   s   n-k


  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).


  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),


  and A(k,k), and v overwrites A(1:k-2,k-1:k).


  If UPLO = 'L', then A = L*D*L**T, where


     L = P(1)*L(1)* ... *P(k)*L(k)* ...,


  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to


  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1


  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as


  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such


  that if the diagonal block D(k) is of order s (s = 1 or 2), then


             (   I    0     0   )  k-1


     L(k) =  (   0    I     0   )  s


             (   0    v     I   )  n-k-s+1


                k-1   s  n-k-s+1


  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).


  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),


  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

   April 2012, Igor Kozachenko,


                  Computer Science Division,


                  University of California, Berkeley


  September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,


                  School of Mathematics,


                  University of Manchester