dtfsm.f(3)

NAME
dtfsm.f

SYNOPSIS

  Functions/Subroutines
subroutine dtfsm (TRANSR, SIDE, UPLO, TRANS, DIAG, M, N, ALPHA, A, B, LDB)
DTFSM solves a matrix equation (one operand is a triangular matrix in RFP format).

Function/Subroutine Documentation

  subroutine dtfsm (character TRANSR, character SIDE, character UPLO, character TRANS, character DIAG, integer M, integer N, double precision ALPHA, double precision, dimension( 0: * ) A, double precision, dimension( 0: ldb-1, 0: * ) B, integer LDB)

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.

Parameters
    TRANSR

              TRANSR is CHARACTER*1
              = 'N':  The Normal Form of RFP A is stored;
              = 'T':  The Transpose Form of RFP A is stored.

    SIDE

              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.

    UPLO

              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.

    TRANS

              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.

    DIAG

              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.

    M

              M is INTEGER
               On entry, M specifies the number of rows of B. M must be at
               least zero.
               Unchanged on exit.

    N

              N is INTEGER
               On entry, N specifies the number of columns of B.  N must be
               at least zero.
               Unchanged on exit.

    ALPHA

              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.

    A

              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.

    B

              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.

    LDB

              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.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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

Definition at line 275 of file dtfsm.f.

Author

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