*> \brief \b CSYTRS_3 * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download CSYTRS_3 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE CSYTRS_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 *> CSYTRS_3 solves a system of linear equations A * X = B with a complex *> symmetric matrix A using the factorization computed *> by CSYTRF_RK or CSYTRF_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 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**T)*(P**T); *> = 'L': Lower triangular, form is A = P*L*D*(L**T)*(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 CSYTRF_RK and CSYTRF_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. *> \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 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 CSYTRF_RK or CSYTRF_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. * *> \date June 2017 * *> \ingroup complexSYcomputational * *> \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 CSYTRS_3( UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, \$ 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, 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 COMPLEX AK, AKM1, AKM1K, BK, BKM1, DENOM * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL CSCAL, CSWAP, CTRSM, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX * .. * .. 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( 'CSYTRS_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**T. * * 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 CALL CSCAL( NRHS, ONE / A( I, I ), B( I, 1 ), LDB ) ELSE IF ( I.GT.1 ) THEN AKM1K = E( I ) AKM1 = A( I-1, I-1 ) / AKM1K AK = A( I, I ) / AKM1K DENOM = AKM1*AK - ONE DO J = 1, NRHS BKM1 = B( I-1, J ) / AKM1K BK = B( I, J ) / 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**T \ B) -> B [ U**T \ (D \ (U \P**T * B) ) ] * CALL CTRSM( 'L', 'U', 'T', 'U', N, NRHS, ONE, A, LDA, B, LDB ) * * P * B [ P * (U**T \ (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**T. * * 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 CALL CSCAL( NRHS, ONE / A( I, I ), B( I, 1 ), LDB ) ELSE IF( I.LT.N ) THEN AKM1K = E( I ) AKM1 = A( I, I ) / AKM1K AK = A( I+1, I+1 ) / AKM1K DENOM = AKM1*AK - ONE DO J = 1, NRHS BKM1 = B( I, J ) / 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**T \ B) -> B [ L**T \ (D \ (L \P**T * B) ) ] * CALL CTRSM('L', 'L', 'T', 'U', N, NRHS, ONE, A, LDA, B, LDB ) * * P * B [ P * (L**T \ (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 CSYTRS_3 * END