*> \brief \b CSYTRI_3X * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download CSYTRI_3X + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE CSYTRI_3X( UPLO, N, A, LDA, E, IPIV, WORK, NB, INFO ) * * .. Scalar Arguments .. * CHARACTER UPLO * INTEGER INFO, LDA, N, NB * .. * .. Array Arguments .. * INTEGER IPIV( * ) * COMPLEX*16 A( LDA, * ), E( * ), WORK( N+NB+1, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> CSYTRI_3X computes the inverse of a complex symmetric indefinite *> 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 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 COMPLEX array, dimension (LDA,N) *> On entry, 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. *> *> 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 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[out] WORK *> \verbatim *> WORK is COMPLEX 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 complexSYcomputational * *> \par Contributors: * ================== *> \verbatim *> *> June 2017, Igor Kozachenko, *> Computer Science Division, *> University of California, Berkeley *> *> \endverbatim * * ===================================================================== SUBROUTINE CSYTRI_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( * ) COMPLEX A( LDA, * ), E( * ), WORK( N+NB+1, * ) * .. * * ===================================================================== * * .. Parameters .. COMPLEX CONE, CZERO PARAMETER ( CONE = ( 1.0E+0, 0.0E+0 ), \$ CZERO = ( 0.0E+0, 0.0E+0 ) ) * .. * .. Local Scalars .. LOGICAL UPPER INTEGER CUT, I, ICOUNT, INVD, IP, K, NNB, J, U11 COMPLEX 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 CGEMM, CSYSWAPR, CTRTRI, CTRMM, 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( 'CSYTRI_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.CZERO ) \$ 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.CZERO ) \$ 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 CTRTRI( 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 ) = CONE / A( K, K ) WORK( K, INVD+1 ) = CZERO 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-CONE ) 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 ) = CONE DO J = 1, I-1 WORK( U11+I, J ) = CZERO 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 CTRMM( 'L', 'U', 'T', 'U', NNB, NNB, \$ CONE, 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 CGEMM( 'T', 'N', NNB, NNB, CUT, CONE, A( 1, CUT+1 ), \$ LDA, WORK, N+NB+1, CZERO, 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 CTRMM( 'L', UPLO, 'T', 'U', CUT, NNB, \$ CONE, 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 CSYSWAPR( UPLO, N, A, LDA, I ,IP ) IF (I .GT. IP) CALL CSYSWAPR( 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 CTRTRI( 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 ) = CONE / A( K, K ) WORK( K, INVD+1 ) = CZERO 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-CONE ) 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) = CONE DO J = I+1, NNB WORK( U11+I, J ) = CZERO 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 CTRMM( 'L', UPLO, 'T', 'U', NNB, NNB, CONE, \$ 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 CGEMM( 'T', 'N', NNB, NNB, N-NNB-CUT, CONE, \$ A( CUT+NNB+1, CUT+1 ), LDA, WORK, N+NB+1, \$ CZERO, 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 CTRMM( 'L', UPLO, 'T', 'U', N-NNB-CUT, NNB, CONE, \$ 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 CSYSWAPR( UPLO, N, A, LDA, I ,IP ) IF (I .GT. IP) CALL CSYSWAPR( UPLO, N, A, LDA, IP ,I ) END IF END DO * END IF * RETURN * * End of CSYTRI_3X * END