zsytri2x.f

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*> \brief \b ZSYTRI2X
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZSYTRI2X( UPLO, N, A, LDA, IPIV, WORK, NB, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, N, NB
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       COMPLEX*16         A( LDA, * ), WORK( N+NB+1,* )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZSYTRI2X computes the inverse of a complex symmetric indefinite matrix
*> A using the factorization A = U*D*U**T or A = L*D*L**T computed by
*> ZSYTRF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the details of the factorization are stored
*>          as an upper or lower triangular matrix.
*>          = 'U':  Upper triangular, form is A = U*D*U**T;
*>          = 'L':  Lower triangular, form is A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is COMPLEX*16 array, dimension (LDA,N)
*>          On entry, the NNB diagonal matrix D and the multipliers
*>          used to obtain the factor U or L as computed by ZSYTRF.
*>
*>          On exit, if INFO = 0, the (symmetric) 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.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          Details of the interchanges and the NNB structure of D
*>          as determined by ZSYTRF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is COMPLEX*16 array, dimension (N+NB+1,NB+3)
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*>          NB is INTEGER
*>          Block size
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          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.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2017
*
*> \ingroup complex16SYcomputational
*
*  =====================================================================
      SUBROUTINE zsytri2x( UPLO, N, A, LDA, IPIV, WORK, NB, INFO )
*
*  -- LAPACK computational routine (version 3.7.1) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     June 2017
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, N, NB
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      COMPLEX*16         A( lda, * ), WORK( n+nb+1,* )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      COMPLEX*16         ONE, ZERO
      parameter( one = ( 1.0d+0, 0.0d+0 ),
     $                   zero = ( 0.0d+0, 0.0d+0 ) )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            I, IINFO, IP, K, CUT, NNB
      INTEGER            COUNT
      INTEGER            J, U11, INVD

      COMPLEX*16         AK, AKKP1, AKP1, D, T
      COMPLEX*16         U01_I_J, U01_IP1_J
      COMPLEX*16         U11_I_J, U11_IP1_J
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           lsame
*     ..
*     .. External Subroutines ..
      EXTERNAL           zsyconv, xerbla, ztrtri
      EXTERNAL           zgemm, ztrmm, zsyswapr
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      info = 0
      upper = lsame( uplo, 'U' )
      IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
         info = -1
      ELSE IF( n.LT.0 ) THEN
         info = -2
      ELSE IF( lda.LT.max( 1, n ) ) THEN
         info = -4
      END IF
*
*     Quick return if possible
*
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'ZSYTRI2X', -info )
         RETURN
      END IF
      IF( n.EQ.0 )
     $   RETURN
*
*     Convert A
*     Workspace got Non-diag elements of D
*
      CALL zsyconv( uplo, 'C', n, a, lda, ipiv, work, iinfo )
*
*     Check that the diagonal matrix D is nonsingular.
*
      IF( upper ) THEN
*
*        Upper triangular storage: examine D from bottom to top
*
         DO info = n, 1, -1
            IF( ipiv( info ).GT.0 .AND. a( info, info ).EQ.zero )
     $         RETURN
         END DO
      ELSE
*
*        Lower triangular storage: examine D from top to bottom.
*
         DO info = 1, n
            IF( ipiv( info ).GT.0 .AND. a( info, info ).EQ.zero )
     $         RETURN
         END DO
      END IF
      info = 0
*
*  Splitting Workspace
*     U01 is a block (N,NB+1)
*     The first element of U01 is in WORK(1,1)
*     U11 is a block (NB+1,NB+1)
*     The first element of U11 is in WORK(N+1,1)
      u11 = n
*     INVD is a block (N,2)
*     The first element of INVD is in WORK(1,INVD)
      invd = nb+2

      IF( upper ) THEN
*
*        invA = P * inv(U**T)*inv(D)*inv(U)*P**T.
*
        CALL ztrtri( uplo, 'U', n, a, lda, info )
*
*       inv(D) and inv(D)*inv(U)
*
        k=1
        DO WHILE ( k .LE. n )
         IF( ipiv( k ).GT.0 ) THEN
*           1 x 1 diagonal NNB
             work(k,invd) = 1/  a( k, k )
             work(k,invd+1) = 0
            k=k+1
         ELSE
*           2 x 2 diagonal NNB
             t = work(k+1,1)
             ak = a( k, k ) / t
             akp1 = a( k+1, k+1 ) / t
             akkp1 = work(k+1,1)  / t
             d = t*( ak*akp1-one )
             work(k,invd) = akp1 / d
             work(k+1,invd+1) = ak / d
             work(k,invd+1) = -akkp1 / d
             work(k+1,invd) = -akkp1 / d
            k=k+2
         END IF
        END DO
*
*       inv(U**T) = (inv(U))**T
*
*       inv(U**T)*inv(D)*inv(U)
*
        cut=n
        DO WHILE (cut .GT. 0)
           nnb=nb
           IF (cut .LE. nnb) THEN
              nnb=cut
           ELSE
              count = 0
*             count negative elements,
              DO i=cut+1-nnb,cut
                  IF (ipiv(i) .LT. 0) count=count+1
              END DO
*             need a even number for a clear cut
              IF (mod(count,2) .EQ. 1) nnb=nnb+1
           END IF

           cut=cut-nnb
*
*          U01 Block
*
           DO i=1,cut
             DO j=1,nnb
              work(i,j)=a(i,cut+j)
             END DO
           END DO
*
*          U11 Block
*
           DO i=1,nnb
             work(u11+i,i)=one
             DO j=1,i-1
                work(u11+i,j)=zero
             END DO
             DO j=i+1,nnb
                work(u11+i,j)=a(cut+i,cut+j)
             END DO
           END DO
*
*          invD*U01
*
           i=1
           DO WHILE (i .LE. cut)
             IF (ipiv(i) > 0) THEN
                DO j=1,nnb
                    work(i,j)=work(i,invd)*work(i,j)
                END DO
                i=i+1
             ELSE
                DO j=1,nnb
                   u01_i_j = work(i,j)
                   u01_ip1_j = work(i+1,j)
                   work(i,j)=work(i,invd)*u01_i_j+
     $                      work(i,invd+1)*u01_ip1_j
                   work(i+1,j)=work(i+1,invd)*u01_i_j+
     $                      work(i+1,invd+1)*u01_ip1_j
                END DO
                i=i+2
             END IF
           END DO
*
*        invD1*U11
*
           i=1
           DO WHILE (i .LE. nnb)
             IF (ipiv(cut+i) > 0) THEN
                DO j=i,nnb
                    work(u11+i,j)=work(cut+i,invd)*work(u11+i,j)
                END DO
                i=i+1
             ELSE
                DO j=i,nnb
                   u11_i_j = work(u11+i,j)
                   u11_ip1_j = work(u11+i+1,j)
                work(u11+i,j)=work(cut+i,invd)*work(u11+i,j) +
     $                      work(cut+i,invd+1)*work(u11+i+1,j)
                work(u11+i+1,j)=work(cut+i+1,invd)*u11_i_j+
     $                      work(cut+i+1,invd+1)*u11_ip1_j
                END DO
                i=i+2
             END IF
           END DO
*
*       U11**T*invD1*U11->U11
*
        CALL ztrmm('L','U','T','U',nnb, nnb,
     $             one,a(cut+1,cut+1),lda,work(u11+1,1),n+nb+1)
*
         DO i=1,nnb
            DO j=i,nnb
              a(cut+i,cut+j)=work(u11+i,j)
            END DO
         END DO
*
*          U01**T*invD*U01->A(CUT+I,CUT+J)
*
         CALL zgemm('T','N',nnb,nnb,cut,one,a(1,cut+1),lda,
     $              work,n+nb+1, zero, work(u11+1,1), n+nb+1)
*
*        U11 =  U11**T*invD1*U11 + U01**T*invD*U01
*
         DO i=1,nnb
            DO j=i,nnb
              a(cut+i,cut+j)=a(cut+i,cut+j)+work(u11+i,j)
            END DO
         END DO
*
*        U01 =  U00**T*invD0*U01
*
         CALL ztrmm('L',uplo,'T','U',cut, nnb,
     $             one,a,lda,work,n+nb+1)

*
*        Update U01
*
         DO i=1,cut
           DO j=1,nnb
            a(i,cut+j)=work(i,j)
           END DO
         END DO
*
*      Next Block
*
       END DO
*
*        Apply PERMUTATIONS P and P**T: P * inv(U**T)*inv(D)*inv(U) *P**T
*
            i=1
            DO WHILE ( i .LE. n )
               IF( ipiv(i) .GT. 0 ) THEN
                  ip=ipiv(i)
                 IF (i .LT. ip) CALL zsyswapr( uplo, n, a, lda, i ,ip )
                 IF (i .GT. ip) CALL zsyswapr( uplo, n, a, lda, ip ,i )
               ELSE
                 ip=-ipiv(i)
                 i=i+1
                 IF ( (i-1) .LT. ip)
     $                  CALL zsyswapr( uplo, n, a, lda, i-1 ,ip )
                 IF ( (i-1) .GT. ip)
     $                  CALL zsyswapr( uplo, n, a, lda, ip ,i-1 )
              ENDIF
               i=i+1
            END DO
      ELSE
*
*        LOWER...
*
*        invA = P * inv(U**T)*inv(D)*inv(U)*P**T.
*
         CALL ztrtri( uplo, 'U', n, a, lda, info )
*
*       inv(D) and inv(D)*inv(U)
*
        k=n
        DO WHILE ( k .GE. 1 )
         IF( ipiv( k ).GT.0 ) THEN
*           1 x 1 diagonal NNB
             work(k,invd) = 1/  a( k, k )
             work(k,invd+1) = 0
            k=k-1
         ELSE
*           2 x 2 diagonal NNB
             t = work(k-1,1)
             ak = a( k-1, k-1 ) / t
             akp1 = a( k, k ) / t
             akkp1 = work(k-1,1) / t
             d = t*( ak*akp1-one )
             work(k-1,invd) = akp1 / d
             work(k,invd) = ak / d
             work(k,invd+1) = -akkp1 / d
             work(k-1,invd+1) = -akkp1 / d
            k=k-2
         END IF
        END DO
*
*       inv(U**T) = (inv(U))**T
*
*       inv(U**T)*inv(D)*inv(U)
*
        cut=0
        DO WHILE (cut .LT. n)
           nnb=nb
           IF (cut + nnb .GE. n) THEN
              nnb=n-cut
           ELSE
              count = 0
*             count negative elements,
              DO i=cut+1,cut+nnb
                  IF (ipiv(i) .LT. 0) count=count+1
              END DO
*             need a even number for a clear cut
              IF (mod(count,2) .EQ. 1) nnb=nnb+1
           END IF
*      L21 Block
           DO i=1,n-cut-nnb
             DO j=1,nnb
              work(i,j)=a(cut+nnb+i,cut+j)
             END DO
           END DO
*     L11 Block
           DO i=1,nnb
             work(u11+i,i)=one
             DO j=i+1,nnb
                work(u11+i,j)=zero
             END DO
             DO j=1,i-1
                work(u11+i,j)=a(cut+i,cut+j)
             END DO
           END DO
*
*          invD*L21
*
           i=n-cut-nnb
           DO WHILE (i .GE. 1)
             IF (ipiv(cut+nnb+i) > 0) THEN
                DO j=1,nnb
                    work(i,j)=work(cut+nnb+i,invd)*work(i,j)
                END DO
                i=i-1
             ELSE
                DO j=1,nnb
                   u01_i_j = work(i,j)
                   u01_ip1_j = work(i-1,j)
                   work(i,j)=work(cut+nnb+i,invd)*u01_i_j+
     $                        work(cut+nnb+i,invd+1)*u01_ip1_j
                   work(i-1,j)=work(cut+nnb+i-1,invd+1)*u01_i_j+
     $                        work(cut+nnb+i-1,invd)*u01_ip1_j
                END DO
                i=i-2
             END IF
           END DO
*
*        invD1*L11
*
           i=nnb
           DO WHILE (i .GE. 1)
             IF (ipiv(cut+i) > 0) THEN
                DO j=1,nnb
                    work(u11+i,j)=work(cut+i,invd)*work(u11+i,j)
                END DO
                i=i-1
             ELSE
                DO j=1,nnb
                   u11_i_j = work(u11+i,j)
                   u11_ip1_j = work(u11+i-1,j)
                work(u11+i,j)=work(cut+i,invd)*work(u11+i,j) +
     $                      work(cut+i,invd+1)*u11_ip1_j
                work(u11+i-1,j)=work(cut+i-1,invd+1)*u11_i_j+
     $                      work(cut+i-1,invd)*u11_ip1_j
                END DO
                i=i-2
             END IF
           END DO
*
*       L11**T*invD1*L11->L11
*
        CALL ztrmm('L',uplo,'T','U',nnb, nnb,
     $             one,a(cut+1,cut+1),lda,work(u11+1,1),n+nb+1)
*
         DO i=1,nnb
            DO j=1,i
              a(cut+i,cut+j)=work(u11+i,j)
            END DO
         END DO
*

        IF ( (cut+nnb) .LT. n ) THEN
*
*          L21**T*invD2*L21->A(CUT+I,CUT+J)
*
         CALL zgemm('T','N',nnb,nnb,n-nnb-cut,one,a(cut+nnb+1,cut+1)
     $             ,lda,work,n+nb+1, zero, work(u11+1,1), n+nb+1)

*
*        L11 =  L11**T*invD1*L11 + U01**T*invD*U01
*
         DO i=1,nnb
            DO j=1,i
              a(cut+i,cut+j)=a(cut+i,cut+j)+work(u11+i,j)
            END DO
         END DO
*
*        U01 =  L22**T*invD2*L21
*
         CALL ztrmm('L',uplo,'T','U', n-nnb-cut, nnb,
     $             one,a(cut+nnb+1,cut+nnb+1),lda,work,n+nb+1)

*      Update L21
         DO i=1,n-cut-nnb
           DO j=1,nnb
              a(cut+nnb+i,cut+j)=work(i,j)
           END DO
         END DO
       ELSE
*
*        L11 =  L11**T*invD1*L11
*
         DO i=1,nnb
            DO j=1,i
              a(cut+i,cut+j)=work(u11+i,j)
            END DO
         END DO
       END IF
*
*      Next Block
*
           cut=cut+nnb
       END DO
*
*        Apply PERMUTATIONS P and P**T: P * inv(U**T)*inv(D)*inv(U) *P**T
*
            i=n
            DO WHILE ( i .GE. 1 )
               IF( ipiv(i) .GT. 0 ) THEN
                  ip=ipiv(i)
                 IF (i .LT. ip) CALL zsyswapr( uplo, n, a, lda, i ,ip  )
                 IF (i .GT. ip) CALL zsyswapr( uplo, n, a, lda, ip ,i )
               ELSE
                 ip=-ipiv(i)
                 IF ( i .LT. ip) CALL zsyswapr( uplo, n, a, lda, i ,ip )
                 IF ( i .GT. ip) CALL zsyswapr( uplo, n, a, lda, ip ,i )
                 i=i-1
               ENDIF
               i=i-1
            END DO
      END IF
*
      RETURN
*
*     End of ZSYTRI2X
*
      END