Procedure

     MTXM  ( Matrix transpose times matrix, 3x3 )

     SUBROUTINE MTXM ( M1, M2, MOUT )

Abstract

     Multiply the transpose of a 3x3 matrix and a 3x3 matrix.

Required_Reading

     None.

Keywords

     MATRIX

Declarations

     IMPLICIT NONE

     DOUBLE PRECISION   M1   ( 3,3 )
     DOUBLE PRECISION   M2   ( 3,3 )
     DOUBLE PRECISION   MOUT ( 3,3 )

Brief_I/O

     VARIABLE  I/O  DESCRIPTION
     --------  ---  --------------------------------------------------
     M1         I   3x3 double precision matrix.
     M2         I   3x3 double precision matrix.
     MOUT       O   3x3 double precision matrix which is the product
                    (M1**T) * M2.

Detailed_Input

     M1       is any 3x3 double precision matrix. Typically,
              M1 will be a rotation matrix since then its
              transpose is its inverse (but this is NOT a
              requirement).

     M2       is any 3x3 double precision matrix.

Detailed_Output

     MOUT     is s 3x3 double precision matrix. MOUT is the
              product MOUT = (M1**T) x M2.

Parameters

     None.

Exceptions

     Error free.

Files

     None.

Particulars

     The code reflects precisely the following mathematical expression

        For each value of the subscripts I and J from 1 to 3:

        MOUT(I,J) = Summation from K=1 to 3 of  ( M1(K,I) * M2(K,J) )

     Note that the reversal of the K and I subscripts in the left-hand
     matrix M1 is what makes MOUT the product of the TRANSPOSE of M1
     and not simply of M1 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 two 3x3 matrices, multiply the transpose of the first
        matrix by the second one.


        Example code begins here.


              PROGRAM MTXM_EX1
              IMPLICIT NONE

        C
        C     Local variables.
        C
              DOUBLE PRECISION      M1   ( 3, 3 )
              DOUBLE PRECISION      M2   ( 3, 3 )
              DOUBLE PRECISION      MOUT ( 3, 3 )

              INTEGER               I
              INTEGER               J

        C
        C     Define M1 and M2.
        C
              DATA                  M1 /  1.0D0,  4.0D0,  7.0D0,
             .                            2.0D0,  5.0D0,  8.0D0,
             .                            3.0D0,  6.0D0,  9.0D0  /

              DATA                  M2 /  1.0D0, -1.0D0,  0.0D0,
             .                            1.0D0,  1.0D0,  0.0D0,
             .                            0.0D0,  0.0D0,  1.0D0  /

        C
        C     Multiply the transpose of M1 by M2.
        C
              CALL MTXM ( M1, M2, MOUT )

              WRITE(*,'(A)') 'Transpose of M1 times M2:'
              DO I = 1, 3

                 WRITE(*,'(3F10.3)') ( MOUT(I,J), J=1,3)

              END DO

              END


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


        Transpose of M1 times M2:
            -3.000     5.000     7.000
            -3.000     7.000     8.000
            -3.000     9.000     9.000

Restrictions

     1)  The user is responsible for checking the magnitudes of the
         elements of M1 and M2 so that a floating point overflow does
         not occur. (In the typical use where M1 and M2 are rotation
         matrices, this not a risk at all.)

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, 04-JUL-2021 (JDR)

        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)