Fixed size matrix/vector multiplication

Gandalf supports matrix/vector multiplication with the matrix optionally being (implicitly) transposed. If the matrix is triangular, Gandalf also supports multiplication by the inverse of the matrix, computed implicitly as described in the introduction to this chapter. These operations can be written as

$${\bf y} =\!\! A{\bf x}\;\;\;\;\mbox{OR}\;\;\;\; {\bf y}= A^{\top}{\bf x}\;\;\;\mbox{OR}$$

$${\bf y} =\!\! A^{-1}{\bf x}\;\;\;\;\mbox{OR}\;\;\;\; {\bf y}= A^{-\top}{\bf x}\;\;\;\mbox{(triangular $A$ only)}$$

The Gandalf routines involving a $3\times 4$ matrix for the first of these (no transpose of $A$) is

      Gan_Vector4 v4x;
      Gan_Vector3 v3y;
      Gan_Matrix34 m34A;

      /* ... set up m34A and v4x ... */
      gan_mat34_multv4_q ( &m34A, &v4x, &v3y ); /* macro, or */
      v3y = gan_mat34_multv4_s ( &m34A, &v4x ); /* function call */

while if $A$ is to be transposed then the routines are

      Gan_Vector3 v3x;
      Gan_Vector4 v4y;
      Gan_Matrix34 m34A;

      /* ... set up m34A and v3x ... */
      gan_mat34T_multv3_q ( &m34A, &v3x, &v4y ); /* macro, or */
      v4y = gan_mat34T_multv3_s ( &m34A, &v3x ); /* function call */

There are similar routines gan_mat33_multv3_[qs]() and gan_mat33T_multv3_[qs]() for $A$ being a general $3\times 3$ matrix. For symmetric matrices, there is only one pair of routines:

      Gan_Vector3 v3x, v3y;
      Gan_SquMatrix33 sm33S;

      /* ... set up sm33S using e.g. gan_symmat33_fill_q(), and v3x ... */
      gan_symmat33_multv3_q ( &sm33S, &v3x, &v3y ); /* macro, or */
      v3y = gan_symmat33_multv3_s ( &sm33S, &v3x ); /* function call */

In the case that matrix $A$ is triangular, Gandalf also supports multiplication of vectors by the inverse of $A$. In this case the result may be computed in-place in the input vector, So the complete set of matrix/vector multiplication routines for triangular matrices is

      Gan_Vector3 v3x, v3y;
      Gan_SquMatrix33 sm33L;

      /* ... set up sm33L using e.g. gan_ltmat33_fill_q(), and v3x ... */

      /* multiply vector by lower triangular matrix */
      gan_ltmat33_multv3_q ( &sm33L, &v3x, &v3y ); /* macro, or */
      v3y = gan_ltmat33_multv3_s ( &sm33L, &v3x ); /* function call, or */
      gan_ltmat33_multv3_i ( &sm33L, &v3x ); /* macro, in-place in v3x */

      /* multiply vector by upper triangular matrix */
      gan_ltmat33T_multv3_q ( &sm33L, &v3x, &v3y ); /* macro, or */
      v3y = gan_ltmat33T_multv3_s ( &sm33L, &v3x ); /* function call, or */
      gan_ltmat33T_multv3_i ( &sm33L, &v3x ); /* macro, in-place in v3x */

      /* multiply vector by inverse of lower triangular matrix */
      gan_ltmat33I_multv3_q ( &sm33L, &v3x, &v3y ); /* macro, or */
      v3y = gan_ltmat33I_multv3_s ( &sm33L, &v3x ); /* function call, or */
      gan_ltmat33I_multv3_i ( &sm33L, &v3x ); /* macro, in-place in v3x */

      /* multiply vector by inverse of upper triangular matrix */
      gan_ltmat33IT_multv3_q ( &sm33L, &v3x, &v3y ); /* macro, or */
      v3y = gan_ltmat33IT_multv3_s ( &sm33L, &v3x ); /* function call, or */
      gan_ltmat33IT_multv3_i ( &sm33L, &v3x ); /* macro, in-place in v3x */

Error detection: If implicit inverse is used (the ...I_multv...() or ...IT_multv...() routines), the matrix must be non-singular, which for triangular matrices means that none of the diagonal elements should be zero. If the matrix was produced by successful Cholesky factorisation of a symmetric matrix (see Section 3.1.2.12) the matrix is guaranteed to be non-singular. This is the normal method of creating a triangular matrix, and Gandalf uses assert() to check for the singularity of the matrix.