Given a matrix a and a matrix b, their kronecker product c a tensor b, also called their matrix direct product, is an matrix. If a is an mbyn matrix and b is a pbyq matrix, then krona,b is an mpbynq matrix formed by taking all possible products between the elements of a and the matrix b. The kronecker product is a particular universal bilinear map on a pair of vector spaces, each of which consists of matrices of a specified size. Computes kronecker tensor product of two matrices, at least one of which is sparse. I need to take kronecker product of two matrices one of them is an identity matrix, my system size happens to be of the order of 256256 for both the matrices. Im trying to find an efficient way to compute kronecker products of matrices using intel fortran mkl. Marcus and moyls 2 dwelled upon hadamard product to be a principal submatrix of kronecker product, also visick 3 showed that a b pt na bp, where a and b are arbitrary n n matrices and p is an n2 tn selection matrix such that p p i. Horn gave a widespread information about hadamard product in 1990 1. To simplify the representation of sparse matrix factorization of a transform matrix in the compact block matrix form.
Kronecker product an overview sciencedirect topics. The kronecker product has widespread applications in signal processing, discrete wavelets, statistical physics, computer graphics, fractals, quantum mechanics and quantum computing. The kronecker product has a rich and very pleasing algebra that supports a wide range of fast, elegant, and practical algorithms. Computing a kronecker product of two matrices physics forums. In mathematics, the kronecker product, sometimes denoted by. Highlevel tensor manipulation in fortran view on github download. In olderstyle fortran, the arrays would be of some surelybigenough size, fixed at compile time, and there would be additional parameters describing the bounds in use for each invocation. It may be the case i only have 2 matrices q1 and q2, however they are very large sparse matrices such that their kronecker product surely cannot be stored. Kronecker tensor product matlab kron mathworks italia. It is claimed that many of the difficulties that are encountered in working with the algebra can be alleviated by paying close attention to the indices that are concealed beneath the conventional matrix notation. On kronecker products, tensor products and matrix differential calculus by d.
The kronecker product, also called tensor product, is a fundamental matrix algebra operation, which is widely used. Eigenkroneckerproduct kronecker tensor product helper class for dense matrices. For matrices, kroneckerproduct gives the matrix direct product. There is no user documented routine in lapack to directly compute the kronecker product, but if you look in the testingmatgen directory the routines dsczlakf2 may be of some help. In comparison to other scientific programming languages like matlab or python fortran is not as comfortable to use when dealing with high level programming features of tensor manipulation. Its probably not that hard, but i feel like my head is about to explode atm, so if you can help me out thatd be cool. This option allows users to search by publication, volume and page selecting this option will search the current publication in context. For example, the following three code snippets all result in the same operator. If a is an m by n matrix and b is a p by q matrix, then kron a,b is an mp by nq matrix formed by taking all possible products between the elements of a and the matrix b.
Pitsianis 56 built a kronecker compiler that permits the user to specify. Kronecker tensor product matlab kron mathworks deutschland. I would like to know if this is the most efficient code if the q matrices are sparse. Kronecker product project gutenberg selfpublishing. Matrix calculus, kronecker product and tensor product. In addition to stateoftheart dmrg codes, itensor features the autompo system, which is a powerful domainspecific language for creating matrix product operator mpo tensor networks as a sum of local terms, which is especially useful for quantum mechanics applications. Kronecker tensor product with identity matrix in fortran. We present a sourcetosource compiler that processes matrix formulae in the form of kronecker product factorizations.
The kronecker product of matrices a 2ri k, and b 2rl m, is denoted by a b, and yields a matrix of dimensions il km. Hi, i am looking for fortran code preferably f90f95 to do a kronecker product, maybe on multiple 2 matrices but that is not strictly necessary i can always iterate, even if that would be slower. It is more efficient to store only the nonzero elements of a sparse matrix. Several trends in scientific computing suggest that this important matrix operation will have an increasingly greater role to play in the future. Other names for the kronecker product include tensor product, direct. Kronecker product kernel provides the standard approach in the kernel methods literature for learning from pairinput data, where both data points and prediction tasks have their own feature. Tensor product and kronecker product mathematics stack. Lapackscalapack development view topic kronecker product. Besides the standard techniques for linear and multilinear algebra many advanced topics are included. Product of two levicivita tensors with contractions. Our selfcontained volume provides an accessible introduction to linear and multilinear algebra as well as tensor calculus. The kronecker product should not be confused with the usual matrix. Tensor toolbox for modern fortran ttb commercial fem software packages offer interfaces user subroutines written in fortran for custom defined user materials like umat in abaqus or hypela2 in msc. The generalized kronecker delta or multiindex kronecker delta of order 2p is a type p,p tensor that is a completely antisymmetric in its p upper indices, and also in its p lower indices.
This is then used to define the matrix calculus, culminating in things such as the derivative of a matrix with respect to a matrix and the chain rule for a derivative of a matrix. Given two arrays a and b, the tensor product of them both is a welldefined object, and i was asking whether there is a specific function in fortran, say, tensor, such that tensor a,b is what im looking for. Does anyone know an algorithm for computing kronecker products of two matrices. In some abstract treatments, this last sentence alone defines the tensor product. It is a generalization of the outer product which is denoted by the same symbol from vectors to matrices, and gives the matrix of the tensor product with respect to a standard choice of basis. Symmetric 3x3 tensors may be stored as 6x1 column vectors with the help of the well known voigt. The tensor function accepts input in many different formats, so that you may use whichever is most convenient at a particular time. The tensor product is a universal bilinear map on a pair of vector spaces of any sort. Besides the standard techniques for linear algebra, multilinear algebra and tensor calculus, many advanced topics are included where emphasis is placed on the kronecker product and tensor product. The algebra of the kronecker products of matrices is recapitulated using a. Matrix calculus, kronecker product and tensor product a.
Prove that the kronecker delta has the tensor character indicated. Without a context the first sentence might be a bit weird but to me it is still weird with a context becuase i cant figure out what it means. Module function overview tensor toolbox for modern fortran. Parallel multiplication of a vector by a kronecker tensor. The package contains functions that calculate the kronecker product of two matrices of any size. The product of two levicivita tensors is a sum of products of kronecker deltas. Given a m\timesn matrix a and a p\timesq matrix b, their kronecker product c a tensor b, also called their matrix direct product, is an mp\timesnq. Parallel multiplication of a vector by a kronecker tensor product of matrices. This module provides several useful assignments, operators and functions for the derived tensor data types. K krona,b returns the kronecker tensor product of matrices a and b.
That is, store all nonzero elements of the matrix into a linear array and provide auxiliary arrays to describe the locations of the nonzero elements in the original matrix. Sparse matrix storage formats intel math kernel library. Fortran statements for minimum, maximum, and kronecker. K kron a,b returns the kronecker tensor product of matrices a and b. Emphasis is placed on the kronecker product and tensor product. The algebra of the kronecker products of matrices is recapitulated using a notation that reveals the tensor structures of the matrices. Kronecker metrik in mathematics, the kronecker product.
Simple fortran 95 module for kronecker product and direct sum kronecker sum to be added as needed. We then move on to introducing the symmetric kronecker product, and we derive sev. You could maybe extract part of the code from there. Kronecker tensor product with identity matrix in fortran using mpi. There are a number of common storage formats used for sparse matrices, but most of them employ the same basic technique.
We can build kronecker product from tensor outer product by transposing some dimensions of the result and. This matlab function returns the kronecker tensor product of matrices a and b. Robust kroneckerdecomposable component analysis for. Robust kronecker decomposable component analysis for lowrank modeling. Kronecker sum of matrices or direct sum and kronecker product of matrices or tensor product are elegant and useful mathematical tools 3. Prove also that it is a constant or numerical tensor, that is, it has the same components in all coordinate systems. On the traces of hadamard and kronecker products of matrices. Efficient way to perform tensor products in fortran. Kronecker products and matrix calculus with applications. Warning if you want to replace a matrix by its kronecker product with some matrix, do not do this.
Kroneckerproduct works on vectors, matrices, or in general, full arrays of any depth. The author, graham, starts with matrix notation preliminaries, and then proceeds to the definition of the kronecker product, a. Implement the kronecker product of two matrices arbitrary sized resulting in a block. Selecting this option will search all publications across the scitation platform selecting this option will search all publications for the publishersociety in context.
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