Procedure

     MTXV ( Matrix transpose times vector, 3x3 )

     SUBROUTINE MTXV ( M, VIN, VOUT )

Abstract

     Multiply the transpose of a 3x3 matrix on the left with a vector
     on the right.

Required_Reading

     None.

Keywords

     MATRIX
     VECTOR

Declarations

     IMPLICIT NONE

     DOUBLE PRECISION  M      ( 3,3 )
     DOUBLE PRECISION  VIN    (   3 )
     DOUBLE PRECISION  VOUT   (   3 )

Brief_I/O

     VARIABLE  I/O  DESCRIPTION
     --------  ---  --------------------------------------------------
     M          I   3X3 double precision matrix.
     VIN        I   3-dimensional double precision vector.
     VOUT       O   3-dimensional double precision vector. VOUT is
                    the product M**T * VIN.

Detailed_Input

     M        is an arbitrary 3x3 double precision matrix.
              Typically, M will be a rotation matrix since
              then its transpose is its inverse (but this is NOT
              a requirement).

     VIN      is an arbitrary 3-dimensional double precision
              vector.

Detailed_Output

     VOUT     is a 3-dimensional double precision vector. VOUT is
              the product VOUT = (M**T)  x (VIN).

Parameters

     None.

Exceptions

     Error free.

Files

     None.

Particulars

     The code reflects precisely the following mathematical expression

        For each value of the subscript I from 1 to 3:

                        3
                     .-----
                      \
           VOUT(I) =   )  M(K,I) * VIN(K)
                      /
                     '-----
                       K=1

     Note that the reversal of the K and I subscripts in the left-hand
     matrix M is what makes VOUT the product of the TRANSPOSE of
     and not simply of M itself.

Examples

     The numerical results shown for this example may differ across
     platforms. The results depend on the SPICE kernels used as
     input, the compiler and supporting libraries, and the machine
     specific arithmetic implementation.

     1) Given a 3x3 matrix and a 3-vector, multiply the transpose of
        the matrix by the vector.


        Example code begins here.


              PROGRAM MTXV_EX1
              IMPLICIT NONE

        C
        C     Local variables.
        C
              DOUBLE PRECISION      M    ( 3, 3 )
              DOUBLE PRECISION      VIN  ( 3    )
              DOUBLE PRECISION      VOUT ( 3    )

              INTEGER               I
              INTEGER               J

        C
        C     Define M and VIN.
        C
              DATA                  M    /  1.0D0, -1.0D0,  0.0D0,
             .                              1.0D0,  1.0D0,  0.0D0,
             .                              0.0D0,  0.0D0,  1.0D0  /

              DATA                  VIN  /  5.0D0, 10.0D0, 15.0D0  /

        C
        C     Multiply the transpose of M by VIN.
        C
              CALL MTXV ( M, VIN, VOUT )

              WRITE(*,'(A)') 'Transpose of M times VIN:'
              WRITE(*,'(3F10.3)') VOUT

              END


        When this program was executed on a Mac/Intel/gfortran/64-bit
        platform, the output was:


        Transpose of M times VIN:
            -5.000    15.000    15.000


        Note that typically the matrix M will be a rotation matrix.
        Because the transpose of an orthogonal matrix is equivalent to
        its inverse, applying the rotation to the vector is
        accomplished by multiplying the vector by the transpose of the
        matrix.

        Let

               -1
              M   * VIN = VOUT

        If M is an orthogonal matrix, then (M**T) * VIN = VOUT.

Restrictions

     1)  The user is responsible for checking the magnitudes of the
         elements of M and VIN so that a floating point overflow does
         not occur.

Literature_References

     None.

Author_and_Institution

     N.J. Bachman       (JPL)
     J. Diaz del Rio    (ODC Space)
     W.M. Owen          (JPL)
     W.L. Taber         (JPL)

Version

    SPICELIB Version 1.1.0, 25-AUG-2021 (JDR)

        Changed input argument name MATRIX to M for consistency with
        other routines.

        Added IMPLICIT NONE statement.

        Edited the header to comply with NAIF standard.
        Added complete code example based on the existing example.

    SPICELIB Version 1.0.2, 23-APR-2010 (NJB)

        Header correction: assertions that the output
        can overwrite the input have been removed.

    SPICELIB Version 1.0.1, 10-MAR-1992 (WLT)

        Comment section for permuted index source lines was added
        following the header.

    SPICELIB Version 1.0.0, 31-JAN-1990 (WMO)