*> \brief \b SPFTRI
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SPFTRI( TRANSR, UPLO, N, A, INFO )
*
* .. Scalar Arguments ..
* CHARACTER TRANSR, UPLO
* INTEGER INFO, N
* .. Array Arguments ..
* REAL A( 0: * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SPFTRI 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 SPFTRF.
*> \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': 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 REAL 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.
*> \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.
*
*> \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 SPFTRI( TRANSR, UPLO, N, A, INFO )
*
* -- LAPACK computational routine --
* -- 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 ..
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, STFTRI, SLAUUM, STRMM, SSYRK
* ..
* .. 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( 'SPFTRI', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Invert the triangular Cholesky factor U or L.
*
CALL STFTRI( 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 SLAUUM( 'L', N1, A( 0 ), N, INFO )
CALL SSYRK( 'L', 'T', N1, N2, ONE, A( N1 ), N, ONE,
$ A( 0 ), N )
CALL STRMM( 'L', 'U', 'N', 'N', N2, N1, ONE, A( N ), N,
$ A( N1 ), N )
CALL SLAUUM( '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 SLAUUM( 'L', N1, A( N2 ), N, INFO )
CALL SSYRK( 'L', 'N', N1, N2, ONE, A( 0 ), N, ONE,
$ A( N2 ), N )
CALL STRMM( 'R', 'U', 'T', 'N', N1, N2, ONE, A( N1 ), N,
$ A( 0 ), N )
CALL SLAUUM( '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 SLAUUM( 'U', N1, A( 0 ), N1, INFO )
CALL SSYRK( 'U', 'N', N1, N2, ONE, A( N1*N1 ), N1, ONE,
$ A( 0 ), N1 )
CALL STRMM( 'R', 'L', 'N', 'N', N1, N2, ONE, A( 1 ), N1,
$ A( N1*N1 ), N1 )
CALL SLAUUM( '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 SLAUUM( 'U', N1, A( N2*N2 ), N2, INFO )
CALL SSYRK( 'U', 'T', N1, N2, ONE, A( 0 ), N2, ONE,
$ A( N2*N2 ), N2 )
CALL STRMM( 'L', 'L', 'T', 'N', N2, N1, ONE, A( N1*N2 ),
$ N2, A( 0 ), N2 )
CALL SLAUUM( '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 SLAUUM( 'L', K, A( 1 ), N+1, INFO )
CALL SSYRK( 'L', 'T', K, K, ONE, A( K+1 ), N+1, ONE,
$ A( 1 ), N+1 )
CALL STRMM( 'L', 'U', 'N', 'N', K, K, ONE, A( 0 ), N+1,
$ A( K+1 ), N+1 )
CALL SLAUUM( '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 SLAUUM( 'L', K, A( K+1 ), N+1, INFO )
CALL SSYRK( 'L', 'N', K, K, ONE, A( 0 ), N+1, ONE,
$ A( K+1 ), N+1 )
CALL STRMM( 'R', 'U', 'T', 'N', K, K, ONE, A( K ), N+1,
$ A( 0 ), N+1 )
CALL SLAUUM( '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 SLAUUM( 'U', K, A( K ), K, INFO )
CALL SSYRK( 'U', 'N', K, K, ONE, A( K*( K+1 ) ), K, ONE,
$ A( K ), K )
CALL STRMM( 'R', 'L', 'N', 'N', K, K, ONE, A( 0 ), K,
$ A( K*( K+1 ) ), K )
CALL SLAUUM( '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 SLAUUM( 'U', K, A( K*( K+1 ) ), K, INFO )
CALL SSYRK( 'U', 'T', K, K, ONE, A( 0 ), K, ONE,
$ A( K*( K+1 ) ), K )
CALL STRMM( 'L', 'L', 'T', 'N', K, K, ONE, A( K*K ), K,
$ A( 0 ), K )
CALL SLAUUM( 'L', K, A( K*K ), K, INFO )
*
END IF
*
END IF
*
END IF
*
RETURN
*
* End of SPFTRI
*
END