SUBROUTINE DPFTRI( TRANSR, UPLO, 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 INTEGER INFO, N * .. Array Arguments .. DOUBLE PRECISION A( 0: * ) * .. * * Purpose * ======= * * DPFTRI computes the inverse of a (real) symmetric positive definite * matrix A using the Cholesky factorization A = U**T*U or A = L*L**T * computed by DPFTRF. * * Arguments * ========= * * TRANSR (input) CHARACTER * = 'N': The Normal TRANSR of RFP A is stored; * = 'T': The Transpose TRANSR of RFP A is stored. * * UPLO (input) CHARACTER * = 'U': Upper triangle of A is stored; * = 'L': Lower triangle of A is stored. * * N (input) INTEGER * The order of the matrix A. N >= 0. * * A (input/output) DOUBLE PRECISION array, dimension ( N*(N+1)/2 ) * On entry, the symmetric 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 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 symmetric 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, the (i,i) element of the factor U or L is * zero, and the inverse could not be computed. * * Further Details * =============== * * 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 first 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 * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) * .. * .. Local Scalars .. LOGICAL LOWER, NISODD, NORMALTRANSR INTEGER N1, N2, K * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL XERBLA, DTFTRI, DLAUUM, DTRMM, DSYRK * .. * .. 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( N.LT.0 ) THEN INFO = -3 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPFTRI', -INFO ) RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) + RETURN * * Invert the triangular Cholesky factor U or L. * CALL DTFTRI( 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 DLAUUM( 'L', N1, A( 0 ), N, INFO ) CALL DSYRK( 'L', 'T', N1, N2, ONE, A( N1 ), N, ONE, + A( 0 ), N ) CALL DTRMM( 'L', 'U', 'N', 'N', N2, N1, ONE, A( N ), N, + A( N1 ), N ) CALL DLAUUM( '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 DLAUUM( 'L', N1, A( N2 ), N, INFO ) CALL DSYRK( 'L', 'N', N1, N2, ONE, A( 0 ), N, ONE, + A( N2 ), N ) CALL DTRMM( 'R', 'U', 'T', 'N', N1, N2, ONE, A( N1 ), N, + A( 0 ), N ) CALL DLAUUM( 'U', N2, A( N1 ), N, INFO ) * 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 DLAUUM( 'U', N1, A( 0 ), N1, INFO ) CALL DSYRK( 'U', 'N', N1, N2, ONE, A( N1*N1 ), N1, ONE, + A( 0 ), N1 ) CALL DTRMM( 'R', 'L', 'N', 'N', N1, N2, ONE, A( 1 ), N1, + A( N1*N1 ), N1 ) CALL DLAUUM( '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 DLAUUM( 'U', N1, A( N2*N2 ), N2, INFO ) CALL DSYRK( 'U', 'T', N1, N2, ONE, A( 0 ), N2, ONE, + A( N2*N2 ), N2 ) CALL DTRMM( 'L', 'L', 'T', 'N', N2, N1, ONE, A( N1*N2 ), + N2, A( 0 ), N2 ) CALL DLAUUM( '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 DLAUUM( 'L', K, A( 1 ), N+1, INFO ) CALL DSYRK( 'L', 'T', K, K, ONE, A( K+1 ), N+1, ONE, + A( 1 ), N+1 ) CALL DTRMM( 'L', 'U', 'N', 'N', K, K, ONE, A( 0 ), N+1, + A( K+1 ), N+1 ) CALL DLAUUM( '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 DLAUUM( 'L', K, A( K+1 ), N+1, INFO ) CALL DSYRK( 'L', 'N', K, K, ONE, A( 0 ), N+1, ONE, + A( K+1 ), N+1 ) CALL DTRMM( 'R', 'U', 'T', 'N', K, K, ONE, A( K ), N+1, + A( 0 ), N+1 ) CALL DLAUUM( 'U', K, A( K ), N+1, INFO ) * 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 DLAUUM( 'U', K, A( K ), K, INFO ) CALL DSYRK( 'U', 'N', K, K, ONE, A( K*( K+1 ) ), K, ONE, + A( K ), K ) CALL DTRMM( 'R', 'L', 'N', 'N', K, K, ONE, A( 0 ), K, + A( K*( K+1 ) ), K ) CALL DLAUUM( '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 DLAUUM( 'U', K, A( K*( K+1 ) ), K, INFO ) CALL DSYRK( 'U', 'T', K, K, ONE, A( 0 ), K, ONE, + A( K*( K+1 ) ), K ) CALL DTRMM( 'L', 'L', 'T', 'N', K, K, ONE, A( K*K ), K, + A( 0 ), K ) CALL DLAUUM( 'L', K, A( K*K ), K, INFO ) * END IF * END IF * END IF * RETURN * * End of DPFTRI * END