Efficient Anti-Triangular Matrix Multiplications

[mwl]

Does anyone know of efficient software for the multiplication with an anti-triangular matrix? These are matrices whose triangular structures goes from the lower left corner to the upper right corner. Are there codes similar to the Level 3 BLAS routines for the multiplication of these matrices? They come up in light scattering computations in high dimensions.
Thanks.

 

[AlephZero]

I don't know of any codes for this, but you probably don't need to do anything more complicated than get the Level 3 BLAS code for triangular matrices from LAPACK (or from your computer supplier if you are doing high performace computing) and reverse the order of some of the loops.

 

[mwl]

Hi AlephZero,
What a great idea! Thank you so much.
mwl

 

[AlephZero]

Actually, you might not need to change the BLAS routines, if you can store some of the matrices and vectors "upside down". For example, if
$$\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} \\ a_{31} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} y_1 \\ y_2 \\ y_3 \end{bmatrix}$$
then
$$\begin{bmatrix} a_{31} \\ a_{21} & a_{22} \\ a_{11} & a_{12} & a_{13} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} y_3 \\ y_2 \\ y_1 \end{bmatrix}$$

 

[blue_raver22]

There are a few options for efficient software for the multiplication of anti-triangular matrices. One option is to use the Level 3 BLAS routines, specifically the DGEMM function, which can handle matrices with arbitrary triangular structures. Another option is to use specialized libraries such as LAPACK, which have optimized routines for working with triangular and anti-triangular matrices. Additionally, there are open-source software packages such as SciPy and NumPy which also have efficient implementations for working with these types of matrices. It is important to consider the specific needs and dimensions of your light scattering computations to determine which software would be most efficient for your specific application.