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View Full Version : The MULTIPROD toolbox for MATLAB (2nd posting with correctedformat)

Paolo De Leva
03-01-2006, 07:54 PM
Dear subscribers,

my previous posting with the same subject had extra blank lines which made
it hard to read. The problem was due to a bad conversion, performed by the
LISTSERV software, from my HTML submission to the required plain text
version. I apologize for the inconvenience. Here is the correct version:

The MULTIPROD toolbox for the MATLAB programming language is freely
available on the following web server:

http://www.mathworks.com/matlabcentral/fileexchange/

FILE EXCHANGE CATHEGORY:
- MATLAB Central > File Exchange > Mathematics > Linear Algebra
TITLE:
- Multiplying two N-D arrays of matrices, vectors or scalars

----------------------------------------------------------------
MULTIPROD TOOLBOX
Multiplying two N-D arrays of matrices, vectors or scalars
By Paolo de Leva
----------------------------------------------------------------

KEYWORDS
N-D arrays, Multidimensional arrays, Matrix, Vector, Scalar, Product,
Multiplication, Inner product, Outer product, Matrix expansion,
Transposition.

SUMMARY
Any kind of multiple products between two block arrays of matrices, vectors
or scalars.

DESCRIPTION
MULTIPROD is a powerful generalization for N-D arrays of the MATLAB function
MTIMES and the matrix multiplication operator (*). While MTIMES works only
with 2-D arrays, MULTIPROD works also with multidimensional arrays.

Besides the element-wise multiplication operator (.*), MATLAB 7 includes
only two functions which can perform products between N-D arrays: DOT and
CROSS. However, these functions can only perform two kinds of product: the
dot product and the cross product, respectively. MULTIPROD can perform any
kind of linear algebraic and vectorial products, except for cross products:

1) Both outer and inner products between arrays of vectors
2) Arrays of scalars by arrays of scalars, vectors or matrices
3) Arrays of vectors by arrays of scalars, vectors or matrices
4) Arrays of matrices by arrays of scalars, vectors or matrices

Synthetically, it can be stated that, with MULTIPROD, the "matrix
laboratory" MATLAB evolves becoming ARRAYLAB, an "array laboratory".

Multidimensional arrays may contain matrices or vectors or even scalars
along one or two of their dimensions. For instance, a 2x5x6 array A contains
six 2x5 matrices along its first and second dimension. Thus, array A can be
described as a block array of matrices, and its size can be indicated as
follows: (2x5)x6.

Let's say that:
A is (2x5)x6
B is (5x3)x6
C is (2x3)x6

With MULTIPROD, the six matrices in A can be multiplied by the six matrices
in B in a single step:

C = MULTIPROD(A, B)

Any multidimensional array can be described as a block array. Here are a few
examples:

A is a 2x5x(6x3) array containing matrices along dim. 3 and 4
B is a 2x5x(3x4) array containing matrices along dim. 3 and 4
C is a 2x5x(1x1) array containing scalars along dim. 3 and 4
D is a 2x5x(6) array containing vectors along dim. 3
E is a 2x5x(3) array containing vectors along dim. 3
F is a 2x5x(1) array containing scalars along dim. 3

For instance, MULTIPROD can:

- Multiply the 10 matrices in A by the 10 matrices, vectors or
scalars occupying the same position in B, C, E, or F
- Multi-multiply D by A
- Multi-multiply the arrays of scalars C and F by any of the other arrays
- Perform multiple outer products between D and E

MULTIPROD has a broad field of potential applications. By calling MULTIPROD,
multiple geometrical transformations such as rotations or roto-translations
can be performed on large arrays of vectors in a single step and with no
loops. Multiple operations such as normalizing an array of vectors, or
finding their projection along the axes indicated by another array of
vectors can be performed easily, with no loops and with two or three rows of
code.

Sample functions performing some of these tasks by calling MULTIPROD or by
using parts of its code are included in the separate toolbox "Matlab Central
> File Exchange > Vector algebra for multidimensional arrays of vectors".

A sample function (LOC2LOC) performing a multiple roto-translation by
calling MULTIPROD is included in this package. This function uses 3-element
translation vectors and 3x3 rotation matrices. If you prefer to work with
homogeneous coordinates and 4x4 roto-translation matrices, you can obtain
the same result just by calling MULTIPROD twice.

If you have a single matrix R, and you want to multiply it by an array of
vectors MultiV, you need to use the function MATEXP to replicate R and
create an array of matrices MultiR which can be multi-multiplied by MultiV
with MULTIPROD:

MultiV2 = MULTIPROD(MultiR, MultiV, ...)

MATEXP is included in this package.

MULTIPROD calls, in some cases, the function MULTITRANSP, performing matrix
transpositions in arrays of matrices. I wrote it as a separate function
because I discovered it can be useful for several different purposes.
MULTITRANSP is included in this package.

The help text in the function was written with extreme care, and should be
enough for those who want to quickly understand and use the function.
Further information is given in a short manual, written in Adobe PDF format.
For those who want to understand the function code, a document written with
MS Word and explaining in detail and schematically the meaning of the
implemented syntaxes is included.

Since I wanted to be of service to as many people as possible, MULTIPROD
itself was written, debugged, refined and in several parts even optimized
for speed with extreme care. I hope it or a future version of it, modified
according to your suggestions, will be included in the next version of
MATLAB. The code ("testMULTIPROD.m") I used to test the function output is
included in this package.

Examples:

1) If A is a 5x(6x3)x2 array,
and B is a 5x(3x4)x2 array,
C = MULTIPROD(A,B,[2 3]) is a 5x(6x4)x2 array.

2) If A is a 5x(6x3)x2 array,
and B is a 5x(1x1)x2 array,
C = MULTIPROD(A,B,[2 3]) is a 5x(6x3)x2 array.

3) If A is a 5x(6x3)x2 4-D array,
and B is a 5x(3)x2 3-D array,
C = MULTIPROD(A,B,[2 3],) is a 5x(6)x2 3-D array.

4) If both A and B are 5x(6)x2 3-D arrays,
C = MULTIPROD(A,B,[2 0],[0 2]) is a 5x(6x6)x2 4-D array.

-----------------------------------------------------------

With my kindest regards,

Paolo de Leva
Department of Human Movement and Sport Sciences
University Institute of Movement Science
Rome, ITALY