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In this paper, a numerical method is proposed for canonical polyadic (CP)decomposition of small size tensors. The focus is primarily on decomposition oftensors that correspond to small matrix multiplications. Here, rank of thetensors is equal to the smallest number of scalar multiplications that arenecessary to accomplish the matrix multiplication. The proposed method is basedon a constrained Levenberg-Marquardt optimization. Numerical results indicatethe rank and border ranks of tensors that correspond to multiplication ofmatrices of the size 2x3 and 3x2, 3x3 and 3x2, 3x3 and 3x3, and 3x4 and 4x3.The ranks are 11, 15, 23 and 29, respectively. In particular, a novel algorithmfor multiplying the matrices of the sizes 3x3 and 3x2 with 15 multiplicationsis presented.

Numerical CP Decomposition of Some Difficult Tensors