zpftri.f

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*> \brief \b ZPFTRI
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZPFTRI( TRANSR, UPLO, N, A, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANSR, UPLO
*       INTEGER            INFO, N
*       .. Array Arguments ..
*       COMPLEX*16         A( 0: * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZPFTRI computes the inverse of a complex Hermitian positive definite
*> matrix A using the Cholesky factorization A = U**H*U or A = L*L**H
*> computed by ZPFTRF.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANSR
*> \verbatim
*>          TRANSR is CHARACTER*1
*>          = 'N':  The Normal TRANSR of RFP A is stored;
*>          = 'C':  The Conjugate-transpose TRANSR of RFP A is stored.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \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 ( N*(N+1)/2 );
*>          On entry, the Hermitian matrix A in RFP format. RFP format is
*>          described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
*>          then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
*>          (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is
*>          the Conjugate-transpose of RFP A as defined when
*>          TRANSR = 'N'. The contents of RFP A are defined by UPLO as
*>          follows: If UPLO = 'U' the RFP A contains the nt elements of
*>          upper packed A. If UPLO = 'L' the RFP A contains the elements
*>          of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
*>          'C'. When TRANSR is 'N' the LDA is N+1 when N is even and N
*>          is odd. See the Note below for more details.
*>
*>          On exit, the Hermitian inverse of the original matrix, in the
*>          same storage format.
*> \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, the (i,i) element of the factor U or L is
*>                zero, and the inverse could not be computed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup pftri
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  We first consider Standard Packed Format when N is even.
*>  We give an example where N = 6.
*>
*>      AP is Upper             AP is Lower
*>
*>   00 01 02 03 04 05       00
*>      11 12 13 14 15       10 11
*>         22 23 24 25       20 21 22
*>            33 34 35       30 31 32 33
*>               44 45       40 41 42 43 44
*>                  55       50 51 52 53 54 55
*>
*>
*>  Let TRANSR = 'N'. RFP holds AP as follows:
*>  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
*>  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
*>  conjugate-transpose of the first three columns of AP upper.
*>  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
*>  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
*>  conjugate-transpose of the last three columns of AP lower.
*>  To denote conjugate we place -- above the element. This covers the
*>  case N even and TRANSR = 'N'.
*>
*>         RFP A                   RFP A
*>
*>                                -- -- --
*>        03 04 05                33 43 53
*>                                   -- --
*>        13 14 15                00 44 54
*>                                      --
*>        23 24 25                10 11 55
*>
*>        33 34 35                20 21 22
*>        --
*>        00 44 45                30 31 32
*>        -- --
*>        01 11 55                40 41 42
*>        -- -- --
*>        02 12 22                50 51 52
*>
*>  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
*>  transpose of RFP A above. One therefore gets:
*>
*>
*>           RFP A                   RFP A
*>
*>     -- -- -- --                -- -- -- -- -- --
*>     03 13 23 33 00 01 02    33 00 10 20 30 40 50
*>     -- -- -- -- --                -- -- -- -- --
*>     04 14 24 34 44 11 12    43 44 11 21 31 41 51
*>     -- -- -- -- -- --                -- -- -- --
*>     05 15 25 35 45 55 22    53 54 55 22 32 42 52
*>
*>
*>  We next  consider Standard Packed Format when N is odd.
*>  We give an example where N = 5.
*>
*>     AP is Upper                 AP is Lower
*>
*>   00 01 02 03 04              00
*>      11 12 13 14              10 11
*>         22 23 24              20 21 22
*>            33 34              30 31 32 33
*>               44              40 41 42 43 44
*>
*>
*>  Let TRANSR = 'N'. RFP holds AP as follows:
*>  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
*>  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
*>  conjugate-transpose of the first two   columns of AP upper.
*>  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
*>  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
*>  conjugate-transpose of the last two   columns of AP lower.
*>  To denote conjugate we place -- above the element. This covers the
*>  case N odd  and TRANSR = 'N'.
*>
*>         RFP A                   RFP A
*>
*>                                   -- --
*>        02 03 04                00 33 43
*>                                      --
*>        12 13 14                10 11 44
*>
*>        22 23 24                20 21 22
*>        --
*>        00 33 34                30 31 32
*>        -- --
*>        01 11 44                40 41 42
*>
*>  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
*>  transpose of RFP A above. One therefore gets:
*>
*>
*>           RFP A                   RFP A
*>
*>     -- -- --                   -- -- -- -- -- --
*>     02 12 22 00 01             00 10 20 30 40 50
*>     -- -- -- --                   -- -- -- -- --
*>     03 13 23 33 11             33 11 21 31 41 51
*>     -- -- -- -- --                   -- -- -- --
*>     04 14 24 34 44             43 44 22 32 42 52
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE zpftri( TRANSR, UPLO, N, A, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANSR, UPLO
      INTEGER            INFO, N
*     .. Array Arguments ..
      COMPLEX*16         A( 0: * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      COMPLEX*16         CONE
      parameter( one = 1.d0, cone = ( 1.d0, 0.d0 ) )
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER, NISODD, NORMALTRANSR
      INTEGER            N1, N2, K
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           lsame
*     ..
*     .. External Subroutines ..
      EXTERNAL           xerbla, ztftri, zlauum, ztrmm, zherk
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          mod
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      info = 0
      normaltransr = lsame( transr, 'N' )
      lower = lsame( uplo, 'L' )
      IF( .NOT.normaltransr .AND. .NOT.lsame( transr, 'C' ) ) THEN
         info = -1
      ELSE IF( .NOT.lower .AND. .NOT.lsame( uplo, 'U' ) ) THEN
         info = -2
      ELSE IF( n.LT.0 ) THEN
         info = -3
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'ZPFTRI', -info )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 )
     $   RETURN
*
*     Invert the triangular Cholesky factor U or L.
*
      CALL ztftri( transr, uplo, 'N', n, a, info )
      IF( info.GT.0 )
     $   RETURN
*
*     If N is odd, set NISODD = .TRUE.
*     If N is even, set K = N/2 and NISODD = .FALSE.
*
      IF( mod( n, 2 ).EQ.0 ) THEN
         k = n / 2
         nisodd = .false.
      ELSE
         nisodd = .true.
      END IF
*
*     Set N1 and N2 depending on LOWER
*
      IF( lower ) THEN
         n2 = n / 2
         n1 = n - n2
      ELSE
         n1 = n / 2
         n2 = n - n1
      END IF
*
*     Start execution of triangular matrix multiply: inv(U)*inv(U)^C or
*     inv(L)^C*inv(L). There are eight cases.
*
      IF( nisodd ) THEN
*
*        N is odd
*
         IF( normaltransr ) THEN
*
*           N is odd and TRANSR = 'N'
*
            IF( lower ) THEN
*
*              SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:N1-1) )
*              T1 -> a(0,0), T2 -> a(0,1), S -> a(N1,0)
*              T1 -> a(0), T2 -> a(n), S -> a(N1)
*
               CALL zlauum( 'L', n1, a( 0 ), n, info )
               CALL zherk( 'L', 'C', n1, n2, one, a( n1 ), n, one,
     $                     a( 0 ), n )
               CALL ztrmm( 'L', 'U', 'N', 'N', n2, n1, cone, a( n ), n,
     $                     a( n1 ), n )
               CALL zlauum( 'U', n2, a( n ), n, info )
*
            ELSE
*
*              SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:N2-1)
*              T1 -> a(N1+1,0), T2 -> a(N1,0), S -> a(0,0)
*              T1 -> a(N2), T2 -> a(N1), S -> a(0)
*
               CALL zlauum( 'L', n1, a( n2 ), n, info )
               CALL zherk( 'L', 'N', n1, n2, one, a( 0 ), n, one,
     $                     a( n2 ), n )
               CALL ztrmm( 'R', 'U', 'C', 'N', n1, n2, cone, a( n1 ), n,
     $                     a( 0 ), n )
               CALL zlauum( 'U', n2, a( n1 ), n, info )
*
            END IF
*
         ELSE
*
*           N is odd and TRANSR = 'C'
*
            IF( lower ) THEN
*
*              SRPA for LOWER, TRANSPOSE, and N is odd
*              T1 -> a(0), T2 -> a(1), S -> a(0+N1*N1)
*
               CALL zlauum( 'U', n1, a( 0 ), n1, info )
               CALL zherk( 'U', 'N', n1, n2, one, a( n1*n1 ), n1, one,
     $                     a( 0 ), n1 )
               CALL ztrmm( 'R', 'L', 'N', 'N', n1, n2, cone, a( 1 ), n1,
     $                     a( n1*n1 ), n1 )
               CALL zlauum( 'L', n2, a( 1 ), n1, info )
*
            ELSE
*
*              SRPA for UPPER, TRANSPOSE, and N is odd
*              T1 -> a(0+N2*N2), T2 -> a(0+N1*N2), S -> a(0)
*
               CALL zlauum( 'U', n1, a( n2*n2 ), n2, info )
               CALL zherk( 'U', 'C', n1, n2, one, a( 0 ), n2, one,
     $                     a( n2*n2 ), n2 )
               CALL ztrmm( 'L', 'L', 'C', 'N', n2, n1, cone, a( n1*n2 ),
     $                     n2, a( 0 ), n2 )
               CALL zlauum( 'L', n2, a( n1*n2 ), n2, info )
*
            END IF
*
         END IF
*
      ELSE
*
*        N is even
*
         IF( normaltransr ) THEN
*
*           N is even and TRANSR = 'N'
*
            IF( lower ) THEN
*
*              SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
*              T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
*              T1 -> a(1), T2 -> a(0), S -> a(k+1)
*
               CALL zlauum( 'L', k, a( 1 ), n+1, info )
               CALL zherk( 'L', 'C', k, k, one, a( k+1 ), n+1, one,
     $                     a( 1 ), n+1 )
               CALL ztrmm( 'L', 'U', 'N', 'N', k, k, cone, a( 0 ), n+1,
     $                     a( k+1 ), n+1 )
               CALL zlauum( 'U', k, a( 0 ), n+1, info )
*
            ELSE
*
*              SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
*              T1 -> a(k+1,0) ,  T2 -> a(k,0),   S -> a(0,0)
*              T1 -> a(k+1), T2 -> a(k), S -> a(0)
*
               CALL zlauum( 'L', k, a( k+1 ), n+1, info )
               CALL zherk( 'L', 'N', k, k, one, a( 0 ), n+1, one,
     $                     a( k+1 ), n+1 )
               CALL ztrmm( 'R', 'U', 'C', 'N', k, k, cone, a( k ), n+1,
     $                     a( 0 ), n+1 )
               CALL zlauum( 'U', k, a( k ), n+1, info )
*
            END IF
*
         ELSE
*
*           N is even and TRANSR = 'C'
*
            IF( lower ) THEN
*
*              SRPA for LOWER, TRANSPOSE, and N is even (see paper)
*              T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1),
*              T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
*
               CALL zlauum( 'U', k, a( k ), k, info )
               CALL zherk( 'U', 'N', k, k, one, a( k*( k+1 ) ), k, one,
     $                     a( k ), k )
               CALL ztrmm( 'R', 'L', 'N', 'N', k, k, cone, a( 0 ), k,
     $                     a( k*( k+1 ) ), k )
               CALL zlauum( 'L', k, a( 0 ), k, info )
*
            ELSE
*
*              SRPA for UPPER, TRANSPOSE, and N is even (see paper)
*              T1 -> B(0,k+1),     T2 -> B(0,k),   S -> B(0,0),
*              T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
*
               CALL zlauum( 'U', k, a( k*( k+1 ) ), k, info )
               CALL zherk( 'U', 'C', k, k, one, a( 0 ), k, one,
     $                     a( k*( k+1 ) ), k )
               CALL ztrmm( 'L', 'L', 'C', 'N', k, k, cone, a( k*k ), k,
     $                     a( 0 ), k )
               CALL zlauum( 'L', k, a( k*k ), k, info )
*
            END IF
*
         END IF
*
      END IF
*
      RETURN
*
*     End of ZPFTRI
*
      END