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invort_c

 Procedure Abstract Required_Reading Keywords Brief_I/O Detailed_Input Detailed_Output Parameters Exceptions Files Particulars Examples Restrictions Literature_References Author_and_Institution Version Index_Entries

Procedure

invort_c ( Invert nearly orthogonal matrices )

void invort_c ( ConstSpiceDouble   m  ,
SpiceDouble        mit )

Abstract

Construct the inverse of a 3x3 matrix with orthogonal columns and
non-zero column norms using a numerically stable algorithm. The
rows of the output matrix are the columns of the input matrix
divided by the length squared of the corresponding columns.

None.

MATRIX

Brief_I/O

VARIABLE  I/O  DESCRIPTION
--------  ---  --------------------------------------------------
m          I   A 3x3 matrix.
mit        O   m after transposition and scaling of rows.

Detailed_Input

m           is a 3x3 matrix.

Detailed_Output

mit         is the matrix obtained by transposing m and dividing
the rows by squares of their norms.

None.

Exceptions

1)  If any of the columns of `m' have zero length, the error
SPICE(ZEROLENGTHCOLUMN) is signaled by a routine in the call
tree of this routine.

2)  If any column is too short to allow computation of the
reciprocal of its length without causing a floating point
overflow, the error SPICE(COLUMNTOOSMALL) is signaled by a
routine in the call tree of this routine.

None.

Particulars

Suppose that m is the matrix

.-                      -.
|   A*u    B*v     C*w   |
|      1      1       1  |
|                        |
|   A*u    B*v     C*w   |
|      2      2       2  |
|                        |
|   A*u    B*v     C*w   |
|      3      3       3  |
`-                      -'

where the vectors (u , u , u ),  (v , v , v ),  and (w , w , w )
1   2   3      1   2   3          1   2   3

are unit vectors. This routine produces the matrix:

.-                      -.
|   a*u    a*u     a*u   |
|      1      2       3  |
|                        |
|   b*v    b*v     b*v   |
|      1      2       3  |
|                        |
|   c*w    c*w     c*w   |
|      1      2       3  |
`-                      -'

where a = 1/A, b = 1/B, and c = 1/C.

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 double precision 3x3 matrix with mutually orthogonal
rows of arbitrary length, compute its inverse. Check that the
original matrix times the computed inverse produces the
identity matrix.

Example code begins here.

/.
Program invort_ex1
./
#include <stdio.h>
#include "SpiceUsr.h"

int main( )
{

/.
Local variables.
./
SpiceDouble          imat   ;
SpiceDouble          mout   ;

SpiceInt             i;

/.
Define a matrix to invert.
./
SpiceDouble          m       = { {0.0, -1.0, 0.0},
{0.5,  0.0, 0.0},
{0.0,  0.0, 1.0} };

printf( "Original Matrix:\n" );
for ( i = 0; i < 3; i++ )
{

printf( "%16.7f %15.7f %15.7f\n", m[i], m[i], m[i] );

}

/.
Invert the matrix, then output.
./
invort_c ( m, mout );

printf( " \n" );
printf( "Inverse Matrix:\n" );
for ( i = 0; i < 3; i++ )
{

printf( "%16.7f %15.7f %15.7f\n",
mout[i], mout[i], mout[i] );

}

/.
Check the `m' times `mout' produces the identity matrix.
./
mxm_c ( m, mout, imat );

printf( " \n" );
printf( "Original times inverse:\n" );
for ( i = 0; i < 3; i++ )
{

printf( "%16.7f %15.7f %15.7f\n",
imat[i], imat[i], imat[i] );

}

return ( 0 );
}

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

Original Matrix:
0.0000000      -1.0000000       0.0000000
0.5000000       0.0000000       0.0000000
0.0000000       0.0000000       1.0000000

Inverse Matrix:
0.0000000       2.0000000       0.0000000
-1.0000000       0.0000000       0.0000000
0.0000000       0.0000000       1.0000000

Original times inverse:
1.0000000       0.0000000       0.0000000
0.0000000       1.0000000       0.0000000
0.0000000       0.0000000       1.0000000

None.

None.

Author_and_Institution

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

Version

-CSPICE Version 1.0.1, 01-NOV-2021 (JDR)

Edited the header to comply with NAIF standard. Fixed I/O type
of argument "mit" in -Brief_I/O table. Extended -Abstract
section.