*> \brief \b ZHETRI_ROOK computes the inverse of HE matrix using the factorization obtained with the bounded Bunch-Kaufman ("rook") diagonal pivoting method.
*
* =========== DOCUMENTATION ===========
*
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*
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*
* Definition:
* ===========
*
* SUBROUTINE ZHETRI_ROOK( UPLO, N, A, LDA, IPIV, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* COMPLEX*16 A( LDA, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZHETRI_ROOK computes the inverse of a complex Hermitian indefinite matrix
*> A using the factorization A = U*D*U**H or A = L*D*L**H computed by
*> ZHETRF_ROOK.
*> \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**H;
*> = 'L': Lower triangular, form is A = L*D*L**H.
*> \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 block diagonal matrix D and the multipliers
*> used to obtain the factor U or L as computed by ZHETRF_ROOK.
*>
*> 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.
*> \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 block structure of D
*> as determined by ZHETRF_ROOK.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX*16 array, dimension (N)
*> \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 November 2013
*
*> \ingroup complex16HEcomputational
*
*> \par Contributors:
* ==================
*>
*> \verbatim
*>
*> November 2013, Igor Kozachenko,
*> Computer Science Division,
*> University of California, Berkeley
*>
*> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
*> School of Mathematics,
*> University of Manchester
*> \endverbatim
*
* =====================================================================
SUBROUTINE ZHETRI_ROOK( UPLO, N, A, LDA, IPIV, WORK, INFO )
*
* -- LAPACK computational routine (version 3.5.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2013
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
COMPLEX*16 A( LDA, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
COMPLEX*16 CONE, CZERO
PARAMETER ( ONE = 1.0D+0, CONE = ( 1.0D+0, 0.0D+0 ),
$ CZERO = ( 0.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J, K, KP, KSTEP
DOUBLE PRECISION AK, AKP1, D, T
COMPLEX*16 AKKP1, TEMP
* ..
* .. External Functions ..
LOGICAL LSAME
COMPLEX*16 ZDOTC
EXTERNAL LSAME, ZDOTC
* ..
* .. External Subroutines ..
EXTERNAL ZCOPY, ZHEMV, ZSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DCONJG, MAX, DBLE
* ..
* .. 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
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZHETRI_ROOK', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Check that the diagonal matrix D is nonsingular.
*
IF( UPPER ) THEN
*
* Upper triangular storage: examine D from bottom to top
*
DO 10 INFO = N, 1, -1
IF( IPIV( INFO ).GT.0 .AND. A( INFO, INFO ).EQ.CZERO )
$ RETURN
10 CONTINUE
ELSE
*
* Lower triangular storage: examine D from top to bottom.
*
DO 20 INFO = 1, N
IF( IPIV( INFO ).GT.0 .AND. A( INFO, INFO ).EQ.CZERO )
$ RETURN
20 CONTINUE
END IF
INFO = 0
*
IF( UPPER ) THEN
*
* Compute inv(A) from the factorization A = U*D*U**H.
*
* K is the main loop index, increasing from 1 to N in steps of
* 1 or 2, depending on the size of the diagonal blocks.
*
K = 1
30 CONTINUE
*
* If K > N, exit from loop.
*
IF( K.GT.N )
$ GO TO 70
*
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 diagonal block
*
* Invert the diagonal block.
*
A( K, K ) = ONE / DBLE( A( K, K ) )
*
* Compute column K of the inverse.
*
IF( K.GT.1 ) THEN
CALL ZCOPY( K-1, A( 1, K ), 1, WORK, 1 )
CALL ZHEMV( UPLO, K-1, -CONE, A, LDA, WORK, 1, CZERO,
$ A( 1, K ), 1 )
A( K, K ) = A( K, K ) - DBLE( ZDOTC( K-1, WORK, 1, A( 1,
$ K ), 1 ) )
END IF
KSTEP = 1
ELSE
*
* 2 x 2 diagonal block
*
* Invert the diagonal block.
*
T = ABS( A( K, K+1 ) )
AK = DBLE( A( K, K ) ) / T
AKP1 = DBLE( A( K+1, K+1 ) ) / T
AKKP1 = A( K, K+1 ) / T
D = T*( AK*AKP1-ONE )
A( K, K ) = AKP1 / D
A( K+1, K+1 ) = AK / D
A( K, K+1 ) = -AKKP1 / D
*
* Compute columns K and K+1 of the inverse.
*
IF( K.GT.1 ) THEN
CALL ZCOPY( K-1, A( 1, K ), 1, WORK, 1 )
CALL ZHEMV( UPLO, K-1, -CONE, A, LDA, WORK, 1, CZERO,
$ A( 1, K ), 1 )
A( K, K ) = A( K, K ) - DBLE( ZDOTC( K-1, WORK, 1, A( 1,
$ K ), 1 ) )
A( K, K+1 ) = A( K, K+1 ) -
$ ZDOTC( K-1, A( 1, K ), 1, A( 1, K+1 ), 1 )
CALL ZCOPY( K-1, A( 1, K+1 ), 1, WORK, 1 )
CALL ZHEMV( UPLO, K-1, -CONE, A, LDA, WORK, 1, CZERO,
$ A( 1, K+1 ), 1 )
A( K+1, K+1 ) = A( K+1, K+1 ) -
$ DBLE( ZDOTC( K-1, WORK, 1, A( 1, K+1 ),
$ 1 ) )
END IF
KSTEP = 2
END IF
*
IF( KSTEP.EQ.1 ) THEN
*
* Interchange rows and columns K and IPIV(K) in the leading
* submatrix A(1:k,1:k)
*
KP = IPIV( K )
IF( KP.NE.K ) THEN
*
IF( KP.GT.1 )
$ CALL ZSWAP( KP-1, A( 1, K ), 1, A( 1, KP ), 1 )
*
DO 40 J = KP + 1, K - 1
TEMP = DCONJG( A( J, K ) )
A( J, K ) = DCONJG( A( KP, J ) )
A( KP, J ) = TEMP
40 CONTINUE
*
A( KP, K ) = DCONJG( A( KP, K ) )
*
TEMP = A( K, K )
A( K, K ) = A( KP, KP )
A( KP, KP ) = TEMP
END IF
ELSE
*
* Interchange rows and columns K and K+1 with -IPIV(K) and
* -IPIV(K+1) in the leading submatrix A(k+1:n,k+1:n)
*
* (1) Interchange rows and columns K and -IPIV(K)
*
KP = -IPIV( K )
IF( KP.NE.K ) THEN
*
IF( KP.GT.1 )
$ CALL ZSWAP( KP-1, A( 1, K ), 1, A( 1, KP ), 1 )
*
DO 50 J = KP + 1, K - 1
TEMP = DCONJG( A( J, K ) )
A( J, K ) = DCONJG( A( KP, J ) )
A( KP, J ) = TEMP
50 CONTINUE
*
A( KP, K ) = DCONJG( A( KP, K ) )
*
TEMP = A( K, K )
A( K, K ) = A( KP, KP )
A( KP, KP ) = TEMP
*
TEMP = A( K, K+1 )
A( K, K+1 ) = A( KP, K+1 )
A( KP, K+1 ) = TEMP
END IF
*
* (2) Interchange rows and columns K+1 and -IPIV(K+1)
*
K = K + 1
KP = -IPIV( K )
IF( KP.NE.K ) THEN
*
IF( KP.GT.1 )
$ CALL ZSWAP( KP-1, A( 1, K ), 1, A( 1, KP ), 1 )
*
DO 60 J = KP + 1, K - 1
TEMP = DCONJG( A( J, K ) )
A( J, K ) = DCONJG( A( KP, J ) )
A( KP, J ) = TEMP
60 CONTINUE
*
A( KP, K ) = DCONJG( A( KP, K ) )
*
TEMP = A( K, K )
A( K, K ) = A( KP, KP )
A( KP, KP ) = TEMP
END IF
END IF
*
K = K + 1
GO TO 30
70 CONTINUE
*
ELSE
*
* Compute inv(A) from the factorization A = L*D*L**H.
*
* K is the main loop index, decreasing from N to 1 in steps of
* 1 or 2, depending on the size of the diagonal blocks.
*
K = N
80 CONTINUE
*
* If K < 1, exit from loop.
*
IF( K.LT.1 )
$ GO TO 120
*
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 diagonal block
*
* Invert the diagonal block.
*
A( K, K ) = ONE / DBLE( A( K, K ) )
*
* Compute column K of the inverse.
*
IF( K.LT.N ) THEN
CALL ZCOPY( N-K, A( K+1, K ), 1, WORK, 1 )
CALL ZHEMV( UPLO, N-K, -CONE, A( K+1, K+1 ), LDA, WORK,
$ 1, CZERO, A( K+1, K ), 1 )
A( K, K ) = A( K, K ) - DBLE( ZDOTC( N-K, WORK, 1,
$ A( K+1, K ), 1 ) )
END IF
KSTEP = 1
ELSE
*
* 2 x 2 diagonal block
*
* Invert the diagonal block.
*
T = ABS( A( K, K-1 ) )
AK = DBLE( A( K-1, K-1 ) ) / T
AKP1 = DBLE( A( K, K ) ) / T
AKKP1 = A( K, K-1 ) / T
D = T*( AK*AKP1-ONE )
A( K-1, K-1 ) = AKP1 / D
A( K, K ) = AK / D
A( K, K-1 ) = -AKKP1 / D
*
* Compute columns K-1 and K of the inverse.
*
IF( K.LT.N ) THEN
CALL ZCOPY( N-K, A( K+1, K ), 1, WORK, 1 )
CALL ZHEMV( UPLO, N-K, -CONE, A( K+1, K+1 ), LDA, WORK,
$ 1, CZERO, A( K+1, K ), 1 )
A( K, K ) = A( K, K ) - DBLE( ZDOTC( N-K, WORK, 1,
$ A( K+1, K ), 1 ) )
A( K, K-1 ) = A( K, K-1 ) -
$ ZDOTC( N-K, A( K+1, K ), 1, A( K+1, K-1 ),
$ 1 )
CALL ZCOPY( N-K, A( K+1, K-1 ), 1, WORK, 1 )
CALL ZHEMV( UPLO, N-K, -CONE, A( K+1, K+1 ), LDA, WORK,
$ 1, CZERO, A( K+1, K-1 ), 1 )
A( K-1, K-1 ) = A( K-1, K-1 ) -
$ DBLE( ZDOTC( N-K, WORK, 1, A( K+1, K-1 ),
$ 1 ) )
END IF
KSTEP = 2
END IF
*
IF( KSTEP.EQ.1 ) THEN
*
* Interchange rows and columns K and IPIV(K) in the trailing
* submatrix A(k:n,k:n)
*
KP = IPIV( K )
IF( KP.NE.K ) THEN
*
IF( KP.LT.N )
$ CALL ZSWAP( N-KP, A( KP+1, K ), 1, A( KP+1, KP ), 1 )
*
DO 90 J = K + 1, KP - 1
TEMP = DCONJG( A( J, K ) )
A( J, K ) = DCONJG( A( KP, J ) )
A( KP, J ) = TEMP
90 CONTINUE
*
A( KP, K ) = DCONJG( A( KP, K ) )
*
TEMP = A( K, K )
A( K, K ) = A( KP, KP )
A( KP, KP ) = TEMP
END IF
ELSE
*
* Interchange rows and columns K and K-1 with -IPIV(K) and
* -IPIV(K-1) in the trailing submatrix A(k-1:n,k-1:n)
*
* (1) Interchange rows and columns K and -IPIV(K)
*
KP = -IPIV( K )
IF( KP.NE.K ) THEN
*
IF( KP.LT.N )
$ CALL ZSWAP( N-KP, A( KP+1, K ), 1, A( KP+1, KP ), 1 )
*
DO 100 J = K + 1, KP - 1
TEMP = DCONJG( A( J, K ) )
A( J, K ) = DCONJG( A( KP, J ) )
A( KP, J ) = TEMP
100 CONTINUE
*
A( KP, K ) = DCONJG( A( KP, K ) )
*
TEMP = A( K, K )
A( K, K ) = A( KP, KP )
A( KP, KP ) = TEMP
*
TEMP = A( K, K-1 )
A( K, K-1 ) = A( KP, K-1 )
A( KP, K-1 ) = TEMP
END IF
*
* (2) Interchange rows and columns K-1 and -IPIV(K-1)
*
K = K - 1
KP = -IPIV( K )
IF( KP.NE.K ) THEN
*
IF( KP.LT.N )
$ CALL ZSWAP( N-KP, A( KP+1, K ), 1, A( KP+1, KP ), 1 )
*
DO 110 J = K + 1, KP - 1
TEMP = DCONJG( A( J, K ) )
A( J, K ) = DCONJG( A( KP, J ) )
A( KP, J ) = TEMP
110 CONTINUE
*
A( KP, K ) = DCONJG( A( KP, K ) )
*
TEMP = A( K, K )
A( K, K ) = A( KP, KP )
A( KP, KP ) = TEMP
END IF
END IF
*
K = K - 1
GO TO 80
120 CONTINUE
END IF
*
RETURN
*
* End of ZHETRI_ROOK
*
END