*> \brief \b CHETRS_3
*
* =========== DOCUMENTATION ===========
*
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*
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*
* Definition:
* ===========
*
* SUBROUTINE CHETRS_3( UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, LDB, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* COMPLEX A( LDA, * ), B( LDB, * ), E( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*> CHETRS_3 solves a system of linear equations A * X = B with a complex
*> Hermitian matrix A using the factorization computed
*> by CHETRF_RK or CHETRF_BK:
*>
*> A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
*>
*> where U (or L) is unit upper (or lower) triangular matrix,
*> U**H (or L**H) is the conjugate of U (or L), P is a permutation
*> matrix, P**T is the transpose of P, and D is Hermitian and block
*> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
*>
*> This algorithm is using 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 triangular, form is A = P*U*D*(U**H)*(P**T);
*> = 'L': Lower triangular, form is A = P*L*D*(L**H)*(P**T).
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is COMPLEX array, dimension (LDA,N)
*> Diagonal of the block diagonal matrix D and factors U or L
*> as computed by CHETRF_RK and CHETRF_BK:
*> a) ONLY diagonal elements of the Hermitian 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.
*> \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 COMPLEX array, dimension (N)
*> On entry, contains the superdiagonal (or subdiagonal)
*> elements of the Hermitian 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 CHETRF_RK or CHETRF_BK.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is COMPLEX array, dimension (LDB,NRHS)
*> On entry, the right hand side matrix B.
*> On exit, the solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complexHEcomputational
*
*> \par Contributors:
* ==================
*>
*> \verbatim
*>
*> June 2017, 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 CHETRS_3( UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB,
$ 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 UPLO
INTEGER INFO, LDA, LDB, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
COMPLEX A( LDA, * ), B( LDB, * ), E( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
COMPLEX ONE
PARAMETER ( ONE = ( 1.0E+0,0.0E+0 ) )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I, J, K, KP
REAL S
COMPLEX AK, AKM1, AKM1K, BK, BKM1, DENOM
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL CSSCAL, CSWAP, CTRSM, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, CONJG, MAX, REAL
* ..
* .. Executable Statements ..
*
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( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CHETRS_3', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 )
$ RETURN
*
IF( UPPER ) THEN
*
* Begin Upper
*
* Solve A*X = B, where A = U*D*U**H.
*
* P**T * B
*
* Interchange rows K and IPIV(K) of matrix B in the same order
* that the formation order of IPIV(I) vector for Upper case.
*
* (We can do the simple loop over IPIV with decrement -1,
* since the ABS value of IPIV(I) represents the row index
* of the interchange with row i in both 1x1 and 2x2 pivot cases)
*
DO K = N, 1, -1
KP = ABS( IPIV( K ) )
IF( KP.NE.K ) THEN
CALL CSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
END IF
END DO
*
* Compute (U \P**T * B) -> B [ (U \P**T * B) ]
*
CALL CTRSM( 'L', 'U', 'N', 'U', N, NRHS, ONE, A, LDA, B, LDB )
*
* Compute D \ B -> B [ D \ (U \P**T * B) ]
*
I = N
DO WHILE ( I.GE.1 )
IF( IPIV( I ).GT.0 ) THEN
S = REAL( ONE ) / REAL( A( I, I ) )
CALL CSSCAL( NRHS, S, B( I, 1 ), LDB )
ELSE IF ( I.GT.1 ) THEN
AKM1K = E( I )
AKM1 = A( I-1, I-1 ) / AKM1K
AK = A( I, I ) / CONJG( AKM1K )
DENOM = AKM1*AK - ONE
DO J = 1, NRHS
BKM1 = B( I-1, J ) / AKM1K
BK = B( I, J ) / CONJG( AKM1K )
B( I-1, J ) = ( AK*BKM1-BK ) / DENOM
B( I, J ) = ( AKM1*BK-BKM1 ) / DENOM
END DO
I = I - 1
END IF
I = I - 1
END DO
*
* Compute (U**H \ B) -> B [ U**H \ (D \ (U \P**T * B) ) ]
*
CALL CTRSM( 'L', 'U', 'C', 'U', N, NRHS, ONE, A, LDA, B, LDB )
*
* P * B [ P * (U**H \ (D \ (U \P**T * B) )) ]
*
* Interchange rows K and IPIV(K) of matrix B in reverse order
* from the formation order of IPIV(I) vector for Upper case.
*
* (We can do the simple loop over IPIV with increment 1,
* since the ABS value of IPIV(I) represents the row index
* of the interchange with row i in both 1x1 and 2x2 pivot cases)
*
DO K = 1, N, 1
KP = ABS( IPIV( K ) )
IF( KP.NE.K ) THEN
CALL CSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
END IF
END DO
*
ELSE
*
* Begin Lower
*
* Solve A*X = B, where A = L*D*L**H.
*
* P**T * B
* Interchange rows K and IPIV(K) of matrix B in the same order
* that the formation order of IPIV(I) vector for Lower case.
*
* (We can do the simple loop over IPIV with increment 1,
* since the ABS value of IPIV(I) represents the row index
* of the interchange with row i in both 1x1 and 2x2 pivot cases)
*
DO K = 1, N, 1
KP = ABS( IPIV( K ) )
IF( KP.NE.K ) THEN
CALL CSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
END IF
END DO
*
* Compute (L \P**T * B) -> B [ (L \P**T * B) ]
*
CALL CTRSM( 'L', 'L', 'N', 'U', N, NRHS, ONE, A, LDA, B, LDB )
*
* Compute D \ B -> B [ D \ (L \P**T * B) ]
*
I = 1
DO WHILE ( I.LE.N )
IF( IPIV( I ).GT.0 ) THEN
S = REAL( ONE ) / REAL( A( I, I ) )
CALL CSSCAL( NRHS, S, B( I, 1 ), LDB )
ELSE IF( I.LT.N ) THEN
AKM1K = E( I )
AKM1 = A( I, I ) / CONJG( AKM1K )
AK = A( I+1, I+1 ) / AKM1K
DENOM = AKM1*AK - ONE
DO J = 1, NRHS
BKM1 = B( I, J ) / CONJG( AKM1K )
BK = B( I+1, J ) / AKM1K
B( I, J ) = ( AK*BKM1-BK ) / DENOM
B( I+1, J ) = ( AKM1*BK-BKM1 ) / DENOM
END DO
I = I + 1
END IF
I = I + 1
END DO
*
* Compute (L**H \ B) -> B [ L**H \ (D \ (L \P**T * B) ) ]
*
CALL CTRSM('L', 'L', 'C', 'U', N, NRHS, ONE, A, LDA, B, LDB )
*
* P * B [ P * (L**H \ (D \ (L \P**T * B) )) ]
*
* Interchange rows K and IPIV(K) of matrix B in reverse order
* from the formation order of IPIV(I) vector for Lower case.
*
* (We can do the simple loop over IPIV with decrement -1,
* since the ABS value of IPIV(I) represents the row index
* of the interchange with row i in both 1x1 and 2x2 pivot cases)
*
DO K = N, 1, -1
KP = ABS( IPIV( K ) )
IF( KP.NE.K ) THEN
CALL CSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
END IF
END DO
*
* END Lower
*
END IF
*
RETURN
*
* End of CHETRS_3
*
END