ssytri_3x.f

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*> \brief \b SSYTRI_3X
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE SSYTRI_3X( UPLO, N, A, LDA, E, IPIV, WORK, NB, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            INFO, LDA, N, NB
*       ..
*       .. Array Arguments ..
*       INTEGER            IPIV( * )
*       REAL               A( LDA, * ),  E( * ), WORK( N+NB+1, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*> SSYTRI_3X computes the inverse of a real symmetric indefinite
*> matrix A using the factorization computed by SSYTRF_RK or SSYTRF_BK:
*>
*>     A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
*>
*> where U (or L) is unit upper (or lower) triangular matrix,
*> U**T (or L**T) is the transpose of U (or L), P is a permutation
*> matrix, P**T is the transpose of P, and D is symmetric and block
*> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
*>
*> This is the blocked version of the algorithm, calling Level 3 BLAS.
*> \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 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 REAL array, dimension (LDA,N)
*>          On entry, diagonal of the block diagonal matrix D and
*>          factors U or L as computed by SYTRF_RK and SSYTRF_BK:
*>            a) ONLY diagonal elements of the symmetric 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 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] E
*> \verbatim
*>          E is REAL array, dimension (N)
*>          On entry, contains the superdiagonal (or subdiagonal)
*>          elements of the symmetric 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.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*>          IPIV is INTEGER array, dimension (N)
*>          Details of the interchanges and the block structure of D
*>          as determined by SSYTRF_RK or SSYTRF_BK.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL 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 singleSYcomputational
*
*> \par Contributors:
*  ==================
*> \verbatim
*>
*>  June 2017,  Igor Kozachenko,
*>                  Computer Science Division,
*>                  University of California, Berkeley
*>
*> \endverbatim
*
*  =====================================================================
      SUBROUTINE ssytri_3x( UPLO, N, A, LDA, E, 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( * )
      REAL              A( lda, * ), E( * ), WORK( n+nb+1, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ONE, ZERO
      parameter( one = 1.0e+0, zero = 0.0e+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            CUT, I, ICOUNT, INVD, IP, K, NNB, J, U11
      REAL               AK, AKKP1, AKP1, D, T, U01_I_J, U01_IP1_J,
     $                   u11_i_j, u11_ip1_j
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           lsame
*     ..
*     .. External Subroutines ..
      EXTERNAL           sgemm, ssyswapr, strtri, strmm, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max, mod
*     ..
*     .. 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( 'SSYTRI_3X', -info )
         RETURN
      END IF
      IF( n.EQ.0 )
     $   RETURN
*
*     Workspace got Non-diag elements of D
*
      DO k = 1, n
         work( k, 1 ) = e( k )
      END DO
*
*     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
*
*        Begin Upper
*
*        invA = P * inv(U**T) * inv(D) * inv(U) * P**T.
*
         CALL strtri( 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 ) = zero
            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 ) = work( k, invd+1 )
               k = k + 1
            END IF
            k = k + 1
         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
               icount = 0
*              count negative elements,
               DO i = cut+1-nnb, cut
                  IF( ipiv( i ).LT.0 ) icount = icount + 1
               END DO
*              need a even number for a clear cut
               IF( mod( icount, 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 ).GT.0 ) THEN
                  DO j = 1, nnb
                     work( i, j ) = work( i, invd ) * work( i, j )
                  END DO
               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 + 1
               END IF
               i = i + 1
            END DO
*
*           invD1 * U11
*
            i = 1
            DO WHILE ( i.LE.nnb )
               IF( ipiv( cut+i ).GT.0 ) THEN
                  DO j = i, nnb
                     work( u11+i, j ) = work(cut+i,invd) * work(u11+i,j)
                  END DO
               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 + 1
               END IF
               i = i + 1
            END DO
*
*           U11**T * invD1 * U11 -> U11
*
            CALL strmm( '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 sgemm( '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 strmm( '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.
*        Interchange rows and columns I and IPIV(I) in reverse order
*        from the formation order of IPIV vector for Upper case.
*
*        ( We can use a loop over IPIV with increment 1,
*        since the ABS value of IPIV(I) represents the row (column)
*        index of the interchange with row (column) i in both 1x1
*        and 2x2 pivot cases, i.e. we don't need separate code branches
*        for 1x1 and 2x2 pivot cases )
*
         DO i = 1, n
             ip = abs( ipiv( i ) )
             IF( ip.NE.i ) THEN
                IF (i .LT. ip) CALL ssyswapr( uplo, n, a, lda, i ,ip )
                IF (i .GT. ip) CALL ssyswapr( uplo, n, a, lda, ip ,i )
             END IF
         END DO
*
      ELSE
*
*        Begin Lower
*
*        inv A = P * inv(L**T) * inv(D) * inv(L) * P**T.
*
         CALL strtri( uplo, 'U', n, a, lda, info )
*
*        inv(D) and inv(D) * inv(L)
*
         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 ) = zero
            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 ) = work( k, invd+1 )
               k = k - 1
            END IF
            k = k - 1
         END DO
*
*        inv(L**T) = (inv(L))**T
*
*        inv(L**T) * inv(D) * inv(L)
*
         cut = 0
         DO WHILE( cut.LT.n )
            nnb = nb
            IF( (cut + nnb).GT.n ) THEN
               nnb = n - cut
            ELSE
               icount = 0
*              count negative elements,
               DO i = cut + 1, cut+nnb
                  IF ( ipiv( i ).LT.0 ) icount = icount + 1
               END DO
*              need a even number for a clear cut
               IF( mod( icount, 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 ).GT.0 ) THEN
                  DO j = 1, nnb
                     work( i, j ) = work( cut+nnb+i, invd) * work( i, j)
                  END DO
               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 - 1
               END IF
               i = i - 1
            END DO
*
*           invD1*L11
*
            i = nnb
            DO WHILE( i.GE.1 )
               IF( ipiv( cut+i ).GT.0 ) THEN
                  DO j = 1, nnb
                     work( u11+i, j ) = work( cut+i, invd)*work(u11+i,j)
                  END DO

               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 - 1
               END IF
               i = i - 1
            END DO
*
*           L11**T * invD1 * L11 -> L11
*
            CALL strmm( '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 sgemm( '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 strmm( '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(L**T) * inv(D) * inv(L) * P**T.
*        Interchange rows and columns I and IPIV(I) in reverse order
*        from the formation order of IPIV vector for Lower case.
*
*        ( We can use a loop over IPIV with increment -1,
*        since the ABS value of IPIV(I) represents the row (column)
*        index of the interchange with row (column) i in both 1x1
*        and 2x2 pivot cases, i.e. we don't need separate code branches
*        for 1x1 and 2x2 pivot cases )
*
         DO i = n, 1, -1
             ip = abs( ipiv( i ) )
             IF( ip.NE.i ) THEN
                IF (i .LT. ip) CALL ssyswapr( uplo, n, a, lda, i ,ip )
                IF (i .GT. ip) CALL ssyswapr( uplo, n, a, lda, ip ,i )
             END IF
         END DO
*
      END IF
*
      RETURN
*
*     End of SSYTRI_3X
*
      END