CLAHEF_RK computes a partial factorization of a complex Hermitian


 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**H U22**H )


 A  =  ( L11  0 ) (  D   0  ) ( L11**H L21**H )  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.


 CLAHEF_RK is an auxiliary routine called by CHETRF_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


          Hermitian 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 COMPLEX array, dimension (LDA,N)


          On entry, the Hermitian 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 Hermitian 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 COMPLEX array, dimension (N)


          On exit, contains the superdiagonal (or subdiagonal)


          elements of the Hermitian 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 Hermitian 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 COMPLEX 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

  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