SUBROUTINE ZTFTRI( TRANSR, UPLO, DIAG, N, A, INFO ) * * -- LAPACK routine (version 3.2.1) -- * * -- Contributed by Fred Gustavson of the IBM Watson Research Center -- * -- April 2009 -- * * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. CHARACTER TRANSR, UPLO, DIAG INTEGER INFO, N * .. * .. Array Arguments .. COMPLEX*16 A( 0: * ) * .. * * Purpose * ======= * * ZTFTRI computes the inverse of a triangular matrix A stored in RFP * format. * * This is a Level 3 BLAS version of the algorithm. * * Arguments * ========= * * TRANSR (input) CHARACTER * = 'N': The Normal TRANSR of RFP A is stored; * = 'C': The Conjugate-transpose TRANSR of RFP A is stored. * * UPLO (input) CHARACTER * = 'U': A is upper triangular; * = 'L': A is lower triangular. * * DIAG (input) CHARACTER * = 'N': A is non-unit triangular; * = 'U': A is unit triangular. * * N (input) INTEGER * The order of the matrix A. N >= 0. * * A (input/output) COMPLEX*16 array, dimension ( N*(N+1)/2 ); * On entry, the triangular 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 nt * 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 (triangular) inverse of the original matrix, in * the same storage format. * * INFO (output) 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. * * Further Details * =============== * * 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 * * ===================================================================== * * .. Parameters .. COMPLEX*16 CONE PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) ) * .. * .. Local Scalars .. LOGICAL LOWER, NISODD, NORMALTRANSR INTEGER N1, N2, K * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL XERBLA, ZTRMM, ZTRTRI * .. * .. 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( .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( 'ZTFTRI', -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 ZTRTRI( 'L', DIAG, N1, A( 0 ), N, INFO ) IF( INFO.GT.0 ) + RETURN CALL ZTRMM( 'R', 'L', 'N', DIAG, N2, N1, -CONE, A( 0 ), + N, A( N1 ), N ) CALL ZTRTRI( 'U', DIAG, N2, A( N ), N, INFO ) IF( INFO.GT.0 ) + INFO = INFO + N1 IF( INFO.GT.0 ) + RETURN CALL ZTRMM( 'L', 'U', 'C', DIAG, N2, N1, CONE, 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 ZTRTRI( 'L', DIAG, N1, A( N2 ), N, INFO ) IF( INFO.GT.0 ) + RETURN CALL ZTRMM( 'L', 'L', 'C', DIAG, N1, N2, -CONE, A( N2 ), + N, A( 0 ), N ) CALL ZTRTRI( 'U', DIAG, N2, A( N1 ), N, INFO ) IF( INFO.GT.0 ) + INFO = INFO + N1 IF( INFO.GT.0 ) + RETURN CALL ZTRMM( 'R', 'U', 'N', DIAG, N1, N2, CONE, A( N1 ), + N, A( 0 ), N ) * 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 ZTRTRI( 'U', DIAG, N1, A( 0 ), N1, INFO ) IF( INFO.GT.0 ) + RETURN CALL ZTRMM( 'L', 'U', 'N', DIAG, N1, N2, -CONE, A( 0 ), + N1, A( N1*N1 ), N1 ) CALL ZTRTRI( 'L', DIAG, N2, A( 1 ), N1, INFO ) IF( INFO.GT.0 ) + INFO = INFO + N1 IF( INFO.GT.0 ) + RETURN CALL ZTRMM( 'R', 'L', 'C', DIAG, N1, N2, CONE, 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 ZTRTRI( 'U', DIAG, N1, A( N2*N2 ), N2, INFO ) IF( INFO.GT.0 ) + RETURN CALL ZTRMM( 'R', 'U', 'C', DIAG, N2, N1, -CONE, + A( N2*N2 ), N2, A( 0 ), N2 ) CALL ZTRTRI( 'L', DIAG, N2, A( N1*N2 ), N2, INFO ) IF( INFO.GT.0 ) + INFO = INFO + N1 IF( INFO.GT.0 ) + RETURN CALL ZTRMM( 'L', 'L', 'N', DIAG, N2, N1, CONE, + 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 ZTRTRI( 'L', DIAG, K, A( 1 ), N+1, INFO ) IF( INFO.GT.0 ) + RETURN CALL ZTRMM( 'R', 'L', 'N', DIAG, K, K, -CONE, A( 1 ), + N+1, A( K+1 ), N+1 ) CALL ZTRTRI( 'U', DIAG, K, A( 0 ), N+1, INFO ) IF( INFO.GT.0 ) + INFO = INFO + K IF( INFO.GT.0 ) + RETURN CALL ZTRMM( 'L', 'U', 'C', DIAG, K, K, CONE, 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 ZTRTRI( 'L', DIAG, K, A( K+1 ), N+1, INFO ) IF( INFO.GT.0 ) + RETURN CALL ZTRMM( 'L', 'L', 'C', DIAG, K, K, -CONE, A( K+1 ), + N+1, A( 0 ), N+1 ) CALL ZTRTRI( 'U', DIAG, K, A( K ), N+1, INFO ) IF( INFO.GT.0 ) + INFO = INFO + K IF( INFO.GT.0 ) + RETURN CALL ZTRMM( 'R', 'U', 'N', DIAG, K, K, CONE, A( K ), N+1, + A( 0 ), N+1 ) 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 ZTRTRI( 'U', DIAG, K, A( K ), K, INFO ) IF( INFO.GT.0 ) + RETURN CALL ZTRMM( 'L', 'U', 'N', DIAG, K, K, -CONE, A( K ), K, + A( K*( K+1 ) ), K ) CALL ZTRTRI( 'L', DIAG, K, A( 0 ), K, INFO ) IF( INFO.GT.0 ) + INFO = INFO + K IF( INFO.GT.0 ) + RETURN CALL ZTRMM( 'R', 'L', 'C', DIAG, K, K, CONE, 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 ZTRTRI( 'U', DIAG, K, A( K*( K+1 ) ), K, INFO ) IF( INFO.GT.0 ) + RETURN CALL ZTRMM( 'R', 'U', 'C', DIAG, K, K, -CONE, + A( K*( K+1 ) ), K, A( 0 ), K ) CALL ZTRTRI( 'L', DIAG, K, A( K*K ), K, INFO ) IF( INFO.GT.0 ) + INFO = INFO + K IF( INFO.GT.0 ) + RETURN CALL ZTRMM( 'L', 'L', 'N', DIAG, K, K, CONE, A( K*K ), K, + A( 0 ), K ) END IF END IF END IF * RETURN * * End of ZTFTRI * END