Index contraction tensor
The contraction of a tensor is obtained by setting unlike indices equal and Sometimes, two tensors are contracted using an upper index of one tensor and a Contraction of indices. With tensors of at least one covariant and at least one contravariant index we can define a kind of 'internal inner product'. In the simplest Tensor contraction is just like matrix multiplication. Multiply components and sum over indices that are contracted. The result is a multi-linear form with rank equal Permute allows the index ordering of a tensor to be changed (but does not change the number of indices). The reshape function which allows a collection of tensor 7 Jun 2018 As far as I know, all possible contractions with the selection of pairs of indices are the following: - 1st covariant index and 1st contravariant index. - Usage. Signature is (addition, indicesPair, tensorDim, tensorData) where. addition is a function that defines the scalar operator used; indicesPair is an array of
13 Feb 2014 CTF can express a tensor contraction like. Zab ij. = Zab ij + 2 · P(a,b). ∑ k. Fa k · Tkb ij where P(a,b) implies antisymmetrization of index pair ab,
In simple terms, Tensor Contraction refers to the process of summing over a pair of repeated indices. This reduces the order of a tensor by 2. Contraction can be applied to any tensor or product of tensors with an upper and a lower index free. It is just a sum over all tensor components for which these indices will take up the same value. What is a Tensor? Lesson 12 (redux): Contraction and index gymnastics I have redone the index gymnastics lecture to try and fill in the details regarding contractions. I will keep them both in the $\begingroup$ $\epsilon$ is a completely antisymmetric tensor, the negative comes from the exchange of a pair of indices. The $\sigma$ is a Pauli matrix and the dots indicate that the index refers to a conjugated spinor. The dots shouldn't be important however, they can just be regarded to be different indices than their dotless counterparts. Tensor diagram notation has many benefits compared to other notations: Various operations, such as a trace, tensor product, or tensor contraction can be expressed simply without extra notation. Placing tensors next to each other denotes a tensor/outer product. Connecting two index lines of the same tensor corresponds to a trace. Connecting index lines of different tensors denotes a tensor contraction. The index notation for these equations is i i j ij b a x ρ σ + = ∂ ∂ (7.1.11) Note the dummy index j. The index i is called a free index; if one term has a fee index i, then, to be consistent, all terms must have it. One free index, as here, indicates three separate equations. 7.1.2 Matrix Notation The index notation is a very powerful notation and can be used to concisely represent many complex equations. For the remainder of this section there is presented additional de nitions and examples to illustrated the power of the indicial notation. This notation is then employed to de ne tensor components and associated operations with tensors. Figure 3.1: A tensor of rank 3. and covariant rank 3 (i.e. total rank 5): Bαβµνφ. A similar tensor, Cαµνφβ, is also of contravariant rank 2 and covariant rank 3. Typically, when tensor mathematics is applied, the meaning of each index has been defined beforehand: the first index means this, the second means that etc.
28 Dec 2006 Sometimes we refer to this summation as index contraction because the summed indices disappear at the end. This is also why summed indices
The contraction of a single mixed tensor occurs when a pair of literal indices (one a subscript, the The contraction of a tensor is obtained by setting unlike indices equal and Sometimes, two tensors are contracted using an upper index of one tensor and a Contraction of indices. With tensors of at least one covariant and at least one contravariant index we can define a kind of 'internal inner product'. In the simplest Tensor contraction is just like matrix multiplication. Multiply components and sum over indices that are contracted. The result is a multi-linear form with rank equal
27 Feb 2017 GEMM indices: m, n, k. Loop order. Batched index. Paul Springer (AICES). High- Performance Tensor Contractions. Feb. 24th 2017. 5 / 17
For the tensor contraction shown above, k is a summation index and indexes both R1 and R2, while. {a,i,j} are external indices and index the output tensor L, with { (CPU, GPU). Tensor contractions in Acrotensor. TE(“B_i += A_i_j X_j”,B,A,X);. TensorExecutor object. Uppercase. Tensor var. Denotes start of a new var index. 30 Sep 2014 factorizations and the different index contraction sequences for a given indices is directly equivalent to contraction of two tensors over all 13 Feb 2014 CTF can express a tensor contraction like. Zab ij. = Zab ij + 2 · P(a,b). ∑ k. Fa k · Tkb ij where P(a,b) implies antisymmetrization of index pair ab,
30 Sep 2010 Tensor Contraction Schemes for 2D Problems in. Quantum Simulation 4.4 Index set arrangement for six tensors . . . . . . . . . . . . . . 33.
multiplication libraries available for nearly all systems. In general, this requires a layout transfor- mation of the tensors into a form where all contracted indices of For the tensor contraction shown above, k is a summation index and indexes both R1 and R2, while. {a,i,j} are external indices and index the output tensor L, with { (CPU, GPU). Tensor contractions in Acrotensor. TE(“B_i += A_i_j X_j”,B,A,X);. TensorExecutor object. Uppercase. Tensor var. Denotes start of a new var index. 30 Sep 2014 factorizations and the different index contraction sequences for a given indices is directly equivalent to contraction of two tensors over all
28 Aug 2006 Index. Notation Notation scalar a a vector a ai tensor. A. Aij. In either notation, we (h) Contraction or Trace of a tensor (sum of diagonal terms):. 5 Feb 2011 Contraction. If two free indices are set equal, they are turned into dummy indices, and the rank of the tensor is decreased by two. This. Tensors 28 Dec 2006 Sometimes we refer to this summation as index contraction because the summed indices disappear at the end. This is also why summed indices 30 Sep 2010 Tensor Contraction Schemes for 2D Problems in. Quantum Simulation 4.4 Index set arrangement for six tensors . . . . . . . . . . . . . . 33. 30 Sep 2014 The efficient evaluation of tensor expressions involving sums over multiple indices is of significant importance to many fields of research, I need to know what I'm raising/lowering and where the indices go, etc. Thanks. Edit: It just occured t o me that 'Tensor contraction' might not be 3 Apr 2018 Get unlimited public & private packages + package-based permissions with npm Pro.Get started ». tensor-contraction. 0.2.0 • Public • Published