*> \brief \b STFTRI * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download STFTRI + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE STFTRI( TRANSR, UPLO, DIAG, N, A, INFO ) * * .. Scalar Arguments .. * CHARACTER TRANSR, UPLO, DIAG * INTEGER INFO, N * .. * .. Array Arguments .. * REAL A( 0: * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> STFTRI computes the inverse of a triangular matrix A stored in RFP *> format. *> *> This is a Level 3 BLAS version of the algorithm. *> \endverbatim * * Arguments: * ========== * *> \param[in] TRANSR *> \verbatim *> TRANSR is CHARACTER*1 *> = 'N': The Normal TRANSR of RFP A is stored; *> = 'T': The Transpose TRANSR of RFP A is stored. *> \endverbatim *> *> \param[in] UPLO *> \verbatim *> UPLO is CHARACTER*1 *> = 'U': A is upper triangular; *> = 'L': A is lower triangular. *> \endverbatim *> *> \param[in] DIAG *> \verbatim *> DIAG is CHARACTER*1 *> = 'N': A is non-unit triangular; *> = 'U': A is unit triangular. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix A. N >= 0. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is REAL array, dimension (NT); *> NT=N*(N+1)/2. On entry, the triangular factor of a Hermitian *> Positive Definite 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 = 'T' then RFP is *> the 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 nt *> elements of lower packed A. The LDA of RFP A is (N+1)/2 when *> TRANSR = 'T'. 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 (triangular) 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, A(i,i) is exactly zero. The triangular *> matrix is singular and its inverse can 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 realOTHERcomputational * *> \par Further Details: * ===================== *> *> \verbatim *> *> We first consider Rectangular Full Packed (RFP) 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 *> the 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 *> the transpose of the last three columns of AP lower. *> 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 = 'T'. RFP A in both UPLO cases is just the *> 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 then consider Rectangular Full Packed (RFP) 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 *> the 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 *> the transpose of the last two columns of AP lower. *> 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 = 'T'. RFP A in both UPLO cases is just the *> 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 STFTRI( TRANSR, UPLO, DIAG, 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, DIAG INTEGER INFO, N * .. * .. Array Arguments .. REAL A( 0: * ) * .. * * ===================================================================== * * .. Parameters .. REAL ONE PARAMETER ( ONE = 1.0E+0 ) * .. * .. Local Scalars .. LOGICAL LOWER, NISODD, NORMALTRANSR INTEGER N1, N2, K * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL XERBLA, STRMM, STRTRI * .. * .. 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, 'T' ) ) THEN INFO = -1 ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN INFO = -2 ELSE IF( .NOT.LSAME( DIAG, 'N' ) .AND. .NOT.LSAME( DIAG, 'U' ) ) \$ THEN INFO = -3 ELSE IF( N.LT.0 ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'STFTRI', -INFO ) RETURN END IF * * Quick return if possible * IF( N.EQ.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: 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 STRTRI( 'L', DIAG, N1, A( 0 ), N, INFO ) IF( INFO.GT.0 ) \$ RETURN CALL STRMM( 'R', 'L', 'N', DIAG, N2, N1, -ONE, A( 0 ), \$ N, A( N1 ), N ) CALL STRTRI( 'U', DIAG, N2, A( N ), N, INFO ) IF( INFO.GT.0 ) \$ INFO = INFO + N1 IF( INFO.GT.0 ) \$ RETURN CALL STRMM( 'L', 'U', 'T', DIAG, N2, N1, ONE, A( N ), N, \$ A( N1 ), N ) * 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 STRTRI( 'L', DIAG, N1, A( N2 ), N, INFO ) IF( INFO.GT.0 ) \$ RETURN CALL STRMM( 'L', 'L', 'T', DIAG, N1, N2, -ONE, A( N2 ), \$ N, A( 0 ), N ) CALL STRTRI( 'U', DIAG, N2, A( N1 ), N, INFO ) IF( INFO.GT.0 ) \$ INFO = INFO + N1 IF( INFO.GT.0 ) \$ RETURN CALL STRMM( 'R', 'U', 'N', DIAG, N1, N2, ONE, A( N1 ), \$ N, A( 0 ), N ) * END IF * ELSE * * N is odd and TRANSR = 'T' * IF( LOWER ) THEN * * SRPA for LOWER, TRANSPOSE and N is odd * T1 -> a(0), T2 -> a(1), S -> a(0+n1*n1) * CALL STRTRI( 'U', DIAG, N1, A( 0 ), N1, INFO ) IF( INFO.GT.0 ) \$ RETURN CALL STRMM( 'L', 'U', 'N', DIAG, N1, N2, -ONE, A( 0 ), \$ N1, A( N1*N1 ), N1 ) CALL STRTRI( 'L', DIAG, N2, A( 1 ), N1, INFO ) IF( INFO.GT.0 ) \$ INFO = INFO + N1 IF( INFO.GT.0 ) \$ RETURN CALL STRMM( 'R', 'L', 'T', DIAG, N1, N2, ONE, A( 1 ), \$ N1, A( N1*N1 ), N1 ) * ELSE * * SRPA for UPPER, TRANSPOSE and N is odd * T1 -> a(0+n2*n2), T2 -> a(0+n1*n2), S -> a(0) * CALL STRTRI( 'U', DIAG, N1, A( N2*N2 ), N2, INFO ) IF( INFO.GT.0 ) \$ RETURN CALL STRMM( 'R', 'U', 'T', DIAG, N2, N1, -ONE, \$ A( N2*N2 ), N2, A( 0 ), N2 ) CALL STRTRI( 'L', DIAG, N2, A( N1*N2 ), N2, INFO ) IF( INFO.GT.0 ) \$ INFO = INFO + N1 IF( INFO.GT.0 ) \$ RETURN CALL STRMM( 'L', 'L', 'N', DIAG, N2, N1, ONE, \$ A( N1*N2 ), N2, A( 0 ), N2 ) 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 STRTRI( 'L', DIAG, K, A( 1 ), N+1, INFO ) IF( INFO.GT.0 ) \$ RETURN CALL STRMM( 'R', 'L', 'N', DIAG, K, K, -ONE, A( 1 ), \$ N+1, A( K+1 ), N+1 ) CALL STRTRI( 'U', DIAG, K, A( 0 ), N+1, INFO ) IF( INFO.GT.0 ) \$ INFO = INFO + K IF( INFO.GT.0 ) \$ RETURN CALL STRMM( 'L', 'U', 'T', DIAG, K, K, ONE, A( 0 ), N+1, \$ A( K+1 ), N+1 ) * 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 STRTRI( 'L', DIAG, K, A( K+1 ), N+1, INFO ) IF( INFO.GT.0 ) \$ RETURN CALL STRMM( 'L', 'L', 'T', DIAG, K, K, -ONE, A( K+1 ), \$ N+1, A( 0 ), N+1 ) CALL STRTRI( 'U', DIAG, K, A( K ), N+1, INFO ) IF( INFO.GT.0 ) \$ INFO = INFO + K IF( INFO.GT.0 ) \$ RETURN CALL STRMM( 'R', 'U', 'N', DIAG, K, K, ONE, A( K ), N+1, \$ A( 0 ), N+1 ) END IF ELSE * * N is even and TRANSR = 'T' * 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 STRTRI( 'U', DIAG, K, A( K ), K, INFO ) IF( INFO.GT.0 ) \$ RETURN CALL STRMM( 'L', 'U', 'N', DIAG, K, K, -ONE, A( K ), K, \$ A( K*( K+1 ) ), K ) CALL STRTRI( 'L', DIAG, K, A( 0 ), K, INFO ) IF( INFO.GT.0 ) \$ INFO = INFO + K IF( INFO.GT.0 ) \$ RETURN CALL STRMM( 'R', 'L', 'T', DIAG, K, K, ONE, A( 0 ), K, \$ A( K*( K+1 ) ), K ) 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 STRTRI( 'U', DIAG, K, A( K*( K+1 ) ), K, INFO ) IF( INFO.GT.0 ) \$ RETURN CALL STRMM( 'R', 'U', 'T', DIAG, K, K, -ONE, \$ A( K*( K+1 ) ), K, A( 0 ), K ) CALL STRTRI( 'L', DIAG, K, A( K*K ), K, INFO ) IF( INFO.GT.0 ) \$ INFO = INFO + K IF( INFO.GT.0 ) \$ RETURN CALL STRMM( 'L', 'L', 'N', DIAG, K, K, ONE, A( K*K ), K, \$ A( 0 ), K ) END IF END IF END IF * RETURN * * End of STFTRI * END