Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3 - 3 BLAS like routine for A in RFP Format

SYNOPSIS

       SUBROUTINE CTFSM( TRANSR, SIDE, UPLO, TRANS, DIAG, M, N, ALPHA, A, B, LDB )

           CHARACTER     TRANSR, DIAG, SIDE, TRANS, UPLO

           INTEGER       LDB, M, N

           COMPLEX       ALPHA

           COMPLEX       A( 0: * ), B( 0: LDB-1, 0: * )

PURPOSE

       Level 3 BLAS like routine for A in RFP Format.
        CTFSM 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**H.
        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;
                = 'C':  The Conjugate-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  = 'C' or 'c'   op( A ) = conjg( 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) COMPLEX
                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) COMPLEX array, dimension (N*(N+1)/2)
                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 = '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 either in normal or
                conjugate-transpose Format. If UPLO = 'L' the RFP A contains
                the NT elements of lower packed A either in normal or
                conjugate-transpose Format. 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 is N when is odd.
                See the Note below for more details. Unchanged on exit.

        B       (input/output) COMPLEX 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 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

 LAPACK routine (version 3.3.1)             April 2011                             CTFSM(3lapack)