How to perform vector Matrix Multiplication

I need to multiply a time series of 4x4 matrices by a corresponding Time series of 4x4 transformation matrices. I am looking for the most efficient method without a for loop.

P_new = T * P

where: P is a nx4x4 matrix, where n is the number of time epochs.

T is a nx4x4 transformation matrix, where n is the number of time steps.

P_new is a nx4x4 version of P that is transformed.

In a loop I would do for i=1:n; P_new(i,1:4,1:4) = T(i,1:4,1:4) * P(i,1:4,1:4); end

How could I do this outside a loop??

Answer 1

[James Tursa]

First I would suggest you reorder your dimensions to put the 4x4 matrices first, so that P is 4x4xn and T is 4x4xn. That way all of your 4x4 matrices are contiguous in memory. Then you have some options from the FEX (some of which are mex functions that require a C compiler):

MULTIPROD: https://www.mathworks.com/matlabcentral/fileexchange/8773-multiple-matrix-multiplications--with-array-expansion-enabled

MMX: https://www.mathworks.com/matlabcentral/fileexchange/37515-mmx-multithreaded-matrix-operations-on-n-d-matrices

MTIMESX: https://www.mathworks.com/matlabcentral/fileexchange/25977-mtimesx-fast-matrix-multiply-with-multi-dimensional-support