Tensorberäkningar

Tensorberäkningar, or tensor computations, refers to the mathematical operations performed on tensors. Tensors are multidimensional arrays that generalize scalars (0-dimensional tensors), vectors (1-dimensional tensors), and matrices (2-dimensional tensors) to higher dimensions. These computations are fundamental to many fields, including physics, engineering, computer science, and machine learning.

The core of tensor computations involves manipulating these multidimensional arrays through operations such as addition, subtraction, multiplication, and contraction. Tensor addition and subtraction are element-wise operations, similar to vector and matrix operations, requiring tensors of compatible shapes. Tensor multiplication can take various forms, including element-wise product (Hadamard product) and more complex products like the tensor product (Kronecker product) and the Einstein product, which involves summing over shared indices.

Tensor contraction is a particularly important operation, often likened to matrix multiplication but generalized to higher dimensions. It involves summing over one or more pairs of indices, effectively reducing the order (number of dimensions) of the resulting tensor. This operation is crucial for tasks like calculating dot products and performing matrix multiplications in a unified framework.

The development of efficient algorithms and specialized hardware, such as GPUs, has been driven by the need to perform complex tensor computations, especially in areas like deep learning where neural networks rely heavily on tensor manipulations for processing and learning from data. Libraries like TensorFlow and PyTorch are specifically designed to facilitate these computations.