csytri2x.f

*> \brief \b CSYTRI2X
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE CSYTRI2X( UPLO, N, A, LDA, IPIV, WORK, NB, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, N, NB
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       COMPLEX            A( LDA, * ), WORK( N+NB+1,* )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> CSYTRI2X computes the inverse of a real symmetric indefinite matrix
*> A using the factorization A = U*D*U**T or A = L*D*L**T computed by
*> CSYTRF.
*> \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 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 CSYTRF.
*>
*>          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 CSYTRF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is COMPLEX 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 complexSYcomputational
*
*  =====================================================================
      SUBROUTINE CSYTRI2X( 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            A( LDA, * ), WORK( N+NB+1,* )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      COMPLEX              ONE, ZERO
      PARAMETER          ( ONE = ( 1.0E+0, 0.0E+0 ),
     $                   ZERO = ( 0.0E+0, 0.0E+0 ) )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            I, IINFO, IP, K, CUT, NNB
      INTEGER            COUNT
      INTEGER            J, U11, INVD

      COMPLEX   AK, AKKP1, AKP1, D, T
      COMPLEX   U01_I_J, U01_IP1_J
      COMPLEX   U11_I_J, U11_IP1_J
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           CSYCONV, XERBLA, CTRTRI
      EXTERNAL           CGEMM, CTRMM, CSYSWAPR
*     ..
*     .. 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( 'CSYTRI2X', -INFO )
         RETURN
      END IF
      IF( N.EQ.0 )
     $   RETURN
*
*     Convert A
*     Workspace got Non-diag elements of D
*
      CALL CSYCONV( 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 CTRTRI( 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) = ONE /  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 CTRMM('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 CGEMM('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 CTRMM('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 CSYSWAPR( UPLO, N, A, LDA, I ,IP )
                 IF (I .GT. IP) CALL CSYSWAPR( UPLO, N, A, LDA, IP ,I )
               ELSE
                 IP=-IPIV(I)
                 I=I+1
                 IF ( (I-1) .LT. IP)
     $                  CALL CSYSWAPR( UPLO, N, A, LDA, I-1 ,IP )
                 IF ( (I-1) .GT. IP)
     $                  CALL CSYSWAPR( 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 CTRTRI( 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) = ONE /  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 CTRMM('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 CGEMM('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
*
*        L01 =  L22**T*invD2*L21
*
         CALL CTRMM('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 CSYSWAPR( UPLO, N, A, LDA, I ,IP  )
                 IF (I .GT. IP) CALL CSYSWAPR( UPLO, N, A, LDA, IP ,I )
               ELSE
                 IP=-IPIV(I)
                 IF ( I .LT. IP) CALL CSYSWAPR( UPLO, N, A, LDA, I ,IP )
                 IF ( I .GT. IP) CALL CSYSWAPR( UPLO, N, A, LDA, IP ,I )
                 I=I-1
               ENDIF
               I=I-1
            END DO
      END IF
*
      RETURN
*
*     End of CSYTRI2X
*
      END