DSYSV_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.


 DSYTRF_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 DSYTRS_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 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, if INFO = 0, diagonal of the block diagonal


          matrix D and factors U or L  as computed by DSYTRF_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 DOUBLE PRECISION 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 DSYTRF_RK.


          For more info see the description of DSYTRF_RK routine.

B

          B is DOUBLE PRECISION 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 DOUBLE PRECISION 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

  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