ZHETRI_3 computes the inverse of a complex Hermitian indefinite


 matrix A using the factorization computed by ZHETRF_RK or ZHETRF_BK:


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


 where U (or L) is unit upper (or lower) triangular matrix,


 U**H (or L**H) is the conjugate of U (or L), P is a permutation


 matrix, P**T is the transpose of P, and D is Hermitian and block


 diagonal with 1-by-1 and 2-by-2 diagonal blocks.


 ZHETRI_3 sets the leading dimension of the workspace  before calling


 ZHETRI_3X that actually computes the inverse.  This is the blocked


 version of the algorithm, calling Level 3 BLAS.

UPLO

          UPLO is CHARACTER*1


          Specifies whether the details of the factorization are


          stored as an upper or lower triangular matrix.


          = '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 COMPLEX*16 array, dimension (LDA,N)


          On entry, diagonal of the block diagonal matrix D and


          factors U or L as computed by ZHETRF_RK and ZHETRF_BK:


            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


                should be provided on entry 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.


          On exit, if INFO = 0, the Hermitian inverse of the original


          matrix.


             If UPLO = 'U': the upper triangular part of the inverse


             is formed and the part of A below the diagonal is not


             referenced;


             If UPLO = 'L': the lower triangular part of the inverse


             is formed and the part of A above the diagonal is not


             referenced.

LDA

          LDA is INTEGER


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

E

          E is COMPLEX*16 array, dimension (N)


          On entry, 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) not referenced;


          If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.


          NOTE: For 1-by-1 diagonal block D(k), where


          1 <= k <= N, the element E(k) is not referenced in both


          UPLO = 'U' or UPLO = 'L' cases.

IPIV

          IPIV is INTEGER array, dimension (N)


          Details of the interchanges and the block structure of D


          as determined by ZHETRF_RK or ZHETRF_BK.

WORK

          WORK is COMPLEX*16 array, dimension (N+NB+1)*(NB+3).


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

LWORK

          LWORK is INTEGER


          The length of WORK. LWORK >= (N+NB+1)*(NB+3).


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


          the routine only calculates the optimal size of 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) = 0; the matrix is singular and its


               inverse could not be computed.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2017