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version 1.6, 2011/07/22 07:38:12 version 1.7, 2011/11/21 20:43:05
Line 1 Line 1
       SUBROUTINE DTFSM( TRANSR, SIDE, UPLO, TRANS, DIAG, M, N, ALPHA, A,  *> \brief \b DTFSM
      $                  B, LDB )  *
   *  =========== DOCUMENTATION ===========
 *  *
 *  -- LAPACK routine (version 3.3.1)                                    --  * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
 *  *
 *  -- Contributed by Fred Gustavson of the IBM Watson Research Center --  *> \htmlonly
 *  -- April 2011                                                      --  *> Download DTFSM + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtfsm.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtfsm.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtfsm.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DTFSM( TRANSR, SIDE, UPLO, TRANS, DIAG, M, N, ALPHA, A,
   *                         B, LDB )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          TRANSR, DIAG, SIDE, TRANS, UPLO
   *       INTEGER            LDB, M, N
   *       DOUBLE PRECISION   ALPHA
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   A( 0: * ), B( 0: LDB-1, 0: * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> Level 3 BLAS like routine for A in RFP Format.
   *>
   *> DTFSM  solves the matrix equation
   *>
   *>    op( A )*X = alpha*B  or  X*op( A ) = alpha*B
   *>
   *> where alpha is a scalar, X and B are m by n matrices, A is a unit, or
   *> non-unit,  upper or lower triangular matrix  and  op( A )  is one  of
   *>
   *>    op( A ) = A   or   op( A ) = A**T.
   *>
   *> A is in Rectangular Full Packed (RFP) Format.
   *>
   *> The matrix X is overwritten on B.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
 *  *
   *> \param[in] TRANSR
   *> \verbatim
   *>          TRANSR is CHARACTER*1
   *>          = 'N':  The Normal Form of RFP A is stored;
   *>          = 'T':  The Transpose Form of RFP A is stored.
   *> \endverbatim
   *>
   *> \param[in] SIDE
   *> \verbatim
   *>          SIDE is CHARACTER*1
   *>           On entry, SIDE specifies whether op( A ) appears on the left
   *>           or right of X as follows:
   *>
   *>              SIDE = 'L' or 'l'   op( A )*X = alpha*B.
   *>
   *>              SIDE = 'R' or 'r'   X*op( A ) = alpha*B.
   *>
   *>           Unchanged on exit.
   *> \endverbatim
   *>
   *> \param[in] UPLO
   *> \verbatim
   *>          UPLO is CHARACTER*1
   *>           On entry, UPLO specifies whether the RFP matrix A came from
   *>           an upper or lower triangular matrix as follows:
   *>           UPLO = 'U' or 'u' RFP A came from an upper triangular matrix
   *>           UPLO = 'L' or 'l' RFP A came from a  lower triangular matrix
   *>
   *>           Unchanged on exit.
   *> \endverbatim
   *>
   *> \param[in] TRANS
   *> \verbatim
   *>          TRANS is CHARACTER*1
   *>           On entry, TRANS  specifies the form of op( A ) to be used
   *>           in the matrix multiplication as follows:
   *>
   *>              TRANS  = 'N' or 'n'   op( A ) = A.
   *>
   *>              TRANS  = 'T' or 't'   op( A ) = A'.
   *>
   *>           Unchanged on exit.
   *> \endverbatim
   *>
   *> \param[in] DIAG
   *> \verbatim
   *>          DIAG is CHARACTER*1
   *>           On entry, DIAG specifies whether or not RFP A is unit
   *>           triangular as follows:
   *>
   *>              DIAG = 'U' or 'u'   A is assumed to be unit triangular.
   *>
   *>              DIAG = 'N' or 'n'   A is not assumed to be unit
   *>                                  triangular.
   *>
   *>           Unchanged on exit.
   *> \endverbatim
   *>
   *> \param[in] M
   *> \verbatim
   *>          M is INTEGER
   *>           On entry, M specifies the number of rows of B. M must be at
   *>           least zero.
   *>           Unchanged on exit.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>           On entry, N specifies the number of columns of B.  N must be
   *>           at least zero.
   *>           Unchanged on exit.
   *> \endverbatim
   *>
   *> \param[in] ALPHA
   *> \verbatim
   *>          ALPHA is DOUBLE PRECISION
   *>           On entry,  ALPHA specifies the scalar  alpha. When  alpha is
   *>           zero then  A is not referenced and  B need not be set before
   *>           entry.
   *>           Unchanged on exit.
   *> \endverbatim
   *>
   *> \param[in] A
   *> \verbatim
   *>          A is DOUBLE PRECISION array, dimension (NT)
   *>           NT = N*(N+1)/2. On entry, the 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 either in normal or
   *>           transpose Format. If UPLO = 'L' the RFP A contains
   *>           the NT elements of lower packed A either in normal or
   *>           transpose Format. 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 is N when is odd.
   *>           See the Note below for more details. Unchanged on exit.
   *> \endverbatim
   *>
   *> \param[in,out] B
   *> \verbatim
   *>          B is DOUBLE PRECISION array, dimension (LDB,N)
   *>           Before entry,  the leading  m by n part of the array  B must
   *>           contain  the  right-hand  side  matrix  B,  and  on exit  is
   *>           overwritten by the solution matrix  X.
   *> \endverbatim
   *>
   *> \param[in] LDB
   *> \verbatim
   *>          LDB is INTEGER
   *>           On entry, LDB specifies the first dimension of B as declared
   *>           in  the  calling  (sub)  program.   LDB  must  be  at  least
   *>           max( 1, m ).
   *>           Unchanged on exit.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup doubleOTHERcomputational
   *
   *> \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 DTFSM( TRANSR, SIDE, UPLO, TRANS, DIAG, M, N, ALPHA, A,
        $                  B, LDB )
   *
   *  -- LAPACK computational routine (version 3.4.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
   *     November 2011
 *  *
 *     ..  
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          TRANSR, DIAG, SIDE, TRANS, UPLO        CHARACTER          TRANSR, DIAG, SIDE, TRANS, UPLO
       INTEGER            LDB, M, N        INTEGER            LDB, M, N
Line 19 Line 291
       DOUBLE PRECISION   A( 0: * ), B( 0: LDB-1, 0: * )        DOUBLE PRECISION   A( 0: * ), B( 0: LDB-1, 0: * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  Level 3 BLAS like routine for A in RFP Format.  
 *  
 *  DTFSM  solves the matrix equation  
 *  
 *     op( A )*X = alpha*B  or  X*op( A ) = alpha*B  
 *  
 *  where alpha is a scalar, X and B are m by n matrices, A is a unit, or  
 *  non-unit,  upper or lower triangular matrix  and  op( A )  is one  of  
 *  
 *     op( A ) = A   or   op( A ) = A**T.  
 *  
 *  A is in Rectangular Full Packed (RFP) Format.  
 *  
 *  The matrix X is overwritten on B.  
 *  
 *  Arguments  
 *  ==========  
 *  
 *  TRANSR  (input) CHARACTER*1  
 *          = 'N':  The Normal Form of RFP A is stored;  
 *          = 'T':  The Transpose Form of RFP A is stored.  
 *  
 *  SIDE    (input) CHARACTER*1  
 *           On entry, SIDE specifies whether op( A ) appears on the left  
 *           or right of X as follows:  
 *  
 *              SIDE = 'L' or 'l'   op( A )*X = alpha*B.  
 *  
 *              SIDE = 'R' or 'r'   X*op( A ) = alpha*B.  
 *  
 *           Unchanged on exit.  
 *  
 *  UPLO    (input) CHARACTER*1  
 *           On entry, UPLO specifies whether the RFP matrix A came from  
 *           an upper or lower triangular matrix as follows:  
 *           UPLO = 'U' or 'u' RFP A came from an upper triangular matrix  
 *           UPLO = 'L' or 'l' RFP A came from a  lower triangular matrix  
 *  
 *           Unchanged on exit.  
 *  
 *  TRANS   (input) CHARACTER*1  
 *           On entry, TRANS  specifies the form of op( A ) to be used  
 *           in the matrix multiplication as follows:  
 *  
 *              TRANS  = 'N' or 'n'   op( A ) = A.  
 *  
 *              TRANS  = 'T' or 't'   op( A ) = A'.  
 *  
 *           Unchanged on exit.  
 *  
 *  DIAG    (input) CHARACTER*1  
 *           On entry, DIAG specifies whether or not RFP A is unit  
 *           triangular as follows:  
 *  
 *              DIAG = 'U' or 'u'   A is assumed to be unit triangular.  
 *  
 *              DIAG = 'N' or 'n'   A is not assumed to be unit  
 *                                  triangular.  
 *  
 *           Unchanged on exit.  
 *  
 *  M       (input) INTEGER  
 *           On entry, M specifies the number of rows of B. M must be at  
 *           least zero.  
 *           Unchanged on exit.  
 *  
 *  N       (input) INTEGER  
 *           On entry, N specifies the number of columns of B.  N must be  
 *           at least zero.  
 *           Unchanged on exit.  
 *  
 *  ALPHA   (input) DOUBLE PRECISION  
 *           On entry,  ALPHA specifies the scalar  alpha. When  alpha is  
 *           zero then  A is not referenced and  B need not be set before  
 *           entry.  
 *           Unchanged on exit.  
 *  
 *  A       (input) DOUBLE PRECISION array, dimension (NT)  
 *           NT = N*(N+1)/2. On entry, the 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 either in normal or  
 *           transpose Format. If UPLO = 'L' the RFP A contains  
 *           the NT elements of lower packed A either in normal or  
 *           transpose Format. 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 is N when is odd.  
 *           See the Note below for more details. Unchanged on exit.  
 *  
 *  B       (input/output) DOUBLE PRECISION array,  dimension (LDB,N)  
 *           Before entry,  the leading  m by n part of the array  B must  
 *           contain  the  right-hand  side  matrix  B,  and  on exit  is  
 *           overwritten by the solution matrix  X.  
 *  
 *  LDB     (input) INTEGER  
 *           On entry, LDB specifies the first dimension of B as declared  
 *           in  the  calling  (sub)  program.   LDB  must  be  at  least  
 *           max( 1, m ).  
 *           Unchanged on exit.  
 *  
 *  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 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  
 *  
 *  Reference  
 *  =========  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     ..  *     ..
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