*> \brief \b CPFTRI * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download CPFTRI + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE CPFTRI( TRANSR, UPLO, N, A, INFO ) * * .. Scalar Arguments .. * CHARACTER TRANSR, UPLO * INTEGER INFO, N * .. Array Arguments .. * COMPLEX A( 0: * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> CPFTRI computes the inverse of a complex Hermitian positive definite *> matrix A using the Cholesky factorization A = U**H*U or A = L*L**H *> computed by CPFTRF. *> \endverbatim * * Arguments: * ========== * *> \param[in] TRANSR *> \verbatim *> TRANSR is CHARACTER*1 *> = 'N': The Normal TRANSR of RFP A is stored; *> = 'C': The Conjugate-transpose TRANSR of RFP A is stored. *> \endverbatim *> *> \param[in] UPLO *> \verbatim *> UPLO is CHARACTER*1 *> = '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 ( N*(N+1)/2 ); *> On entry, the Hermitian matrix A in RFP format. RFP format is *> described by TRANSR, UPLO, and N as follows: If TRANSR = 'N' *> then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is *> (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is *> the Conjugate-transpose of RFP A as defined when *> TRANSR = 'N'. The contents of RFP A are defined by UPLO as *> follows: If UPLO = 'U' the RFP A contains the nt elements of *> upper packed A. If UPLO = 'L' the RFP A contains the elements *> of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR = *> 'C'. When TRANSR is 'N' the LDA is N+1 when N is even and N *> is odd. See the Note below for more details. *> *> On exit, the Hermitian inverse of the original matrix, in the *> same storage format. *> \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, the (i,i) element of the factor U or L is *> zero, and the 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 December 2016 * *> \ingroup complexOTHERcomputational * *> \par Further Details: * ===================== *> *> \verbatim *> *> We first consider Standard Packed Format when N is even. *> We give an example where N = 6. *> *> AP is Upper AP is Lower *> *> 00 01 02 03 04 05 00 *> 11 12 13 14 15 10 11 *> 22 23 24 25 20 21 22 *> 33 34 35 30 31 32 33 *> 44 45 40 41 42 43 44 *> 55 50 51 52 53 54 55 *> *> *> Let TRANSR = 'N'. RFP holds AP as follows: *> For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last *> three columns of AP upper. The lower triangle A(4:6,0:2) consists of *> conjugate-transpose of the first three columns of AP upper. *> For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first *> three columns of AP lower. The upper triangle A(0:2,0:2) consists of *> conjugate-transpose of the last three columns of AP lower. *> To denote conjugate we place -- above the element. This covers the *> case N even and TRANSR = 'N'. *> *> RFP A RFP A *> *> -- -- -- *> 03 04 05 33 43 53 *> -- -- *> 13 14 15 00 44 54 *> -- *> 23 24 25 10 11 55 *> *> 33 34 35 20 21 22 *> -- *> 00 44 45 30 31 32 *> -- -- *> 01 11 55 40 41 42 *> -- -- -- *> 02 12 22 50 51 52 *> *> Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate- *> transpose of RFP A above. One therefore gets: *> *> *> RFP A RFP A *> *> -- -- -- -- -- -- -- -- -- -- *> 03 13 23 33 00 01 02 33 00 10 20 30 40 50 *> -- -- -- -- -- -- -- -- -- -- *> 04 14 24 34 44 11 12 43 44 11 21 31 41 51 *> -- -- -- -- -- -- -- -- -- -- *> 05 15 25 35 45 55 22 53 54 55 22 32 42 52 *> *> *> We next consider Standard Packed Format when N is odd. *> We give an example where N = 5. *> *> AP is Upper AP is Lower *> *> 00 01 02 03 04 00 *> 11 12 13 14 10 11 *> 22 23 24 20 21 22 *> 33 34 30 31 32 33 *> 44 40 41 42 43 44 *> *> *> Let TRANSR = 'N'. RFP holds AP as follows: *> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last *> three columns of AP upper. The lower triangle A(3:4,0:1) consists of *> conjugate-transpose of the first two columns of AP upper. *> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first *> three columns of AP lower. The upper triangle A(0:1,1:2) consists of *> conjugate-transpose of the last two columns of AP lower. *> To denote conjugate we place -- above the element. This covers the *> case N odd and TRANSR = 'N'. *> *> RFP A RFP A *> *> -- -- *> 02 03 04 00 33 43 *> -- *> 12 13 14 10 11 44 *> *> 22 23 24 20 21 22 *> -- *> 00 33 34 30 31 32 *> -- -- *> 01 11 44 40 41 42 *> *> Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate- *> transpose of RFP A above. One therefore gets: *> *> *> RFP A RFP A *> *> -- -- -- -- -- -- -- -- -- *> 02 12 22 00 01 00 10 20 30 40 50 *> -- -- -- -- -- -- -- -- -- *> 03 13 23 33 11 33 11 21 31 41 51 *> -- -- -- -- -- -- -- -- -- *> 04 14 24 34 44 43 44 22 32 42 52 *> \endverbatim *> * ===================================================================== SUBROUTINE CPFTRI( TRANSR, UPLO, N, A, INFO ) * * -- LAPACK computational routine (version 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * December 2016 * * .. Scalar Arguments .. CHARACTER TRANSR, UPLO INTEGER INFO, N * .. Array Arguments .. COMPLEX A( 0: * ) * .. * * ===================================================================== * * .. Parameters .. REAL ONE COMPLEX CONE PARAMETER ( ONE = 1.0E+0, CONE = ( 1.0E+0, 0.0E+0 ) ) * .. * .. Local Scalars .. LOGICAL LOWER, NISODD, NORMALTRANSR INTEGER N1, N2, K * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL XERBLA, CTFTRI, CLAUUM, CTRMM, CHERK * .. * .. Intrinsic Functions .. INTRINSIC MOD * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 NORMALTRANSR = LSAME( TRANSR, 'N' ) LOWER = LSAME( UPLO, 'L' ) IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'C' ) ) THEN INFO = -1 ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'CPFTRI', -INFO ) RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) \$ RETURN * * Invert the triangular Cholesky factor U or L. * CALL CTFTRI( TRANSR, UPLO, 'N', N, A, INFO ) IF( INFO.GT.0 ) \$ RETURN * * If N is odd, set NISODD = .TRUE. * If N is even, set K = N/2 and NISODD = .FALSE. * IF( MOD( N, 2 ).EQ.0 ) THEN K = N / 2 NISODD = .FALSE. ELSE NISODD = .TRUE. END IF * * Set N1 and N2 depending on LOWER * IF( LOWER ) THEN N2 = N / 2 N1 = N - N2 ELSE N1 = N / 2 N2 = N - N1 END IF * * Start execution of triangular matrix multiply: inv(U)*inv(U)^C or * inv(L)^C*inv(L). There are eight cases. * IF( NISODD ) THEN * * N is odd * IF( NORMALTRANSR ) THEN * * N is odd and TRANSR = 'N' * IF( LOWER ) THEN * * SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:N1-1) ) * T1 -> a(0,0), T2 -> a(0,1), S -> a(N1,0) * T1 -> a(0), T2 -> a(n), S -> a(N1) * CALL CLAUUM( 'L', N1, A( 0 ), N, INFO ) CALL CHERK( 'L', 'C', N1, N2, ONE, A( N1 ), N, ONE, \$ A( 0 ), N ) CALL CTRMM( 'L', 'U', 'N', 'N', N2, N1, CONE, A( N ), N, \$ A( N1 ), N ) CALL CLAUUM( 'U', N2, A( N ), N, INFO ) * ELSE * * SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:N2-1) * T1 -> a(N1+1,0), T2 -> a(N1,0), S -> a(0,0) * T1 -> a(N2), T2 -> a(N1), S -> a(0) * CALL CLAUUM( 'L', N1, A( N2 ), N, INFO ) CALL CHERK( 'L', 'N', N1, N2, ONE, A( 0 ), N, ONE, \$ A( N2 ), N ) CALL CTRMM( 'R', 'U', 'C', 'N', N1, N2, CONE, A( N1 ), N, \$ A( 0 ), N ) CALL CLAUUM( 'U', N2, A( N1 ), N, INFO ) * END IF * ELSE * * N is odd and TRANSR = 'C' * IF( LOWER ) THEN * * SRPA for LOWER, TRANSPOSE, and N is odd * T1 -> a(0), T2 -> a(1), S -> a(0+N1*N1) * CALL CLAUUM( 'U', N1, A( 0 ), N1, INFO ) CALL CHERK( 'U', 'N', N1, N2, ONE, A( N1*N1 ), N1, ONE, \$ A( 0 ), N1 ) CALL CTRMM( 'R', 'L', 'N', 'N', N1, N2, CONE, A( 1 ), N1, \$ A( N1*N1 ), N1 ) CALL CLAUUM( 'L', N2, A( 1 ), N1, INFO ) * ELSE * * SRPA for UPPER, TRANSPOSE, and N is odd * T1 -> a(0+N2*N2), T2 -> a(0+N1*N2), S -> a(0) * CALL CLAUUM( 'U', N1, A( N2*N2 ), N2, INFO ) CALL CHERK( 'U', 'C', N1, N2, ONE, A( 0 ), N2, ONE, \$ A( N2*N2 ), N2 ) CALL CTRMM( 'L', 'L', 'C', 'N', N2, N1, CONE, A( N1*N2 ), \$ N2, A( 0 ), N2 ) CALL CLAUUM( 'L', N2, A( N1*N2 ), N2, INFO ) * END IF * END IF * ELSE * * N is even * IF( NORMALTRANSR ) THEN * * N is even and TRANSR = 'N' * IF( LOWER ) THEN * * SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) ) * T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0) * T1 -> a(1), T2 -> a(0), S -> a(k+1) * CALL CLAUUM( 'L', K, A( 1 ), N+1, INFO ) CALL CHERK( 'L', 'C', K, K, ONE, A( K+1 ), N+1, ONE, \$ A( 1 ), N+1 ) CALL CTRMM( 'L', 'U', 'N', 'N', K, K, CONE, A( 0 ), N+1, \$ A( K+1 ), N+1 ) CALL CLAUUM( 'U', K, A( 0 ), N+1, INFO ) * ELSE * * SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) ) * T1 -> a(k+1,0) , T2 -> a(k,0), S -> a(0,0) * T1 -> a(k+1), T2 -> a(k), S -> a(0) * CALL CLAUUM( 'L', K, A( K+1 ), N+1, INFO ) CALL CHERK( 'L', 'N', K, K, ONE, A( 0 ), N+1, ONE, \$ A( K+1 ), N+1 ) CALL CTRMM( 'R', 'U', 'C', 'N', K, K, CONE, A( K ), N+1, \$ A( 0 ), N+1 ) CALL CLAUUM( 'U', K, A( K ), N+1, INFO ) * END IF * ELSE * * N is even and TRANSR = 'C' * IF( LOWER ) THEN * * SRPA for LOWER, TRANSPOSE, and N is even (see paper) * T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1), * T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k * CALL CLAUUM( 'U', K, A( K ), K, INFO ) CALL CHERK( 'U', 'N', K, K, ONE, A( K*( K+1 ) ), K, ONE, \$ A( K ), K ) CALL CTRMM( 'R', 'L', 'N', 'N', K, K, CONE, A( 0 ), K, \$ A( K*( K+1 ) ), K ) CALL CLAUUM( 'L', K, A( 0 ), K, INFO ) * ELSE * * SRPA for UPPER, TRANSPOSE, and N is even (see paper) * T1 -> B(0,k+1), T2 -> B(0,k), S -> B(0,0), * T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k * CALL CLAUUM( 'U', K, A( K*( K+1 ) ), K, INFO ) CALL CHERK( 'U', 'C', K, K, ONE, A( 0 ), K, ONE, \$ A( K*( K+1 ) ), K ) CALL CTRMM( 'L', 'L', 'C', 'N', K, K, CONE, A( K*K ), K, \$ A( 0 ), K ) CALL CLAUUM( 'L', K, A( K*K ), K, INFO ) * END IF * END IF * END IF * RETURN * * End of CPFTRI * END