Distance invariant method for normalization of indexed differentials

Abstract

A distance from free to dummy indices is defined. The distance is invariant with respect to both monoterm symmetries and bottom antisymmetry. Using the distance invariant, we present an index-replacement algorithm. We then develop two normalization algorithms. One is with respect to monoterm symmetries and has complexity lower than known algorithms; the other allows one to determine the equivalence of indexed polynomials in the Einstein summation ring ${\mathcal{R}}_2[\partial ̸]$.

Introduction

Symbolic manipulation of indexed expressions, e.g. tensor expressions, is one of the oldest research topics in computer algebra (Fulling et al., 1992; Christensen and Parker, 1994; Ilyin and Kryukov, 1996; Portugal, 1998, Portugal, 1999, Portugal, 2000; Manssur et al., 2002; Martín-García et al., 2007, Martín-García et al., 2008; Liu et al., 2013). It remains a challenging problem for the following reasons: In order to compute the canonical form of an indexed polynomial, we need to find a finite Gröbner basis for the ideal generated by the basic syzygies. Unfortunately, the ideal is not finitely generated, mainly due to the property of renaming dummy index. Various efforts have been made to develop algorithms for simplifying tensor expressions. Portugal, 1998, Portugal, 1999 presented algorithms to normalize tensor expressions with respect to monoterm symmetries and cyclic symmetry. However, those algorithms are unable to deal with indexed differential expressions, because index elimination is indispensable for simplifying them.

A natural problem in symbolic computation is: Given two indexed differential expressions, how can we judge whether they are equal or not (what is the normalization algorithm)?

In differential geometry, massive calculation involving indexed differential expressions arises from tensor verification problem and the problem of finding transformation rules of indexed functions under coordinate transformation. Therefore, normalization algorithms are indispensable, as evinced by the following problems.

Problem 1.1

The curvature tensor determined by the connection on differentiable manifold $M^n$ is an indexed function $R_{jkl}^i$ defined by

$$R_{jkl}^i=\frac{\partial {\mathrm{\Gamma }}_{lj}^i}{\partial x^k}-\frac{\partial {\mathrm{\Gamma }}_{kj}^i}{\partial x^l}+{\mathrm{\Gamma }}_{lj}^h{\mathrm{\Gamma }}_{kh}^i-{\mathrm{\Gamma }}_{kj}^h{\mathrm{\Gamma }}_{lh}^i.$$

Show that R is a tensor, i.e., it satisfies the following transformation rule of ($1,3$)-typed tensor:

$$R_{j^{\prime }{k}^{\prime }{l}^{\prime }}^{i^{\prime }}=R_{jkl}^i\frac{\partial x^{i^{\prime }}}{\partial x^i}\frac{\partial x^j}{\partial x^{j^{\prime }}}\frac{\partial x^k}{\partial x^{k^{\prime }}}\frac{\partial x^l}{\partial x^{l^{\prime }}}.$$

Problem 1.2

A Riemannian manifold $M^n$ admits an almost complex structure $J_j^i$. Let $H_{jk}^i$ be 1/8 times the Nijenhuis tensor of $J_j^i$, i.e.,

$$H_{jk}^i=-\frac{1}{8}(J_j^p{\partial }_p{J}_k^i-J_k^p{\partial }_p{J}_j^i)+\frac{1}{8}(J_p^i{\partial }_j{J}_k^p-J_p^i{\partial }_k{J}_j^p).$$

The “torsional derivative” of a tensor field on $M^n$ is defined in (Willmore, 1993) as follows:

$$T_{j_1...j_b||rs}^{i_1...i_a}=H_{rs}^p\frac{\partial T_{j_1...j_b}^{i_1...i_a}}{\partial x^p}+\sum _{u=1}^a{T}_{j_1...j_b}^{i_1...i_{u-1}p{i}_{u+1}...i_a}{h}_{prs}^{i_u}-\sum _{v=1}^b{T}_{j_1...j_{v-1}p{j}_{v+1}...j_b}^{i_1...i_a}{h}_{j_{v}rs}^p,$$

where $h_{jrs}^i$ has the following “rather strange” definition:

$$2h_{jrs}^i=J_p^i(J_j^q\frac{\partial H_{rs}^p}{\partial x^q}-H_{rs}^q\frac{\partial J_j^p}{\partial x^q}+H_{qs}^p\frac{\partial J_j^q}{\partial x^r}-H_{qr}^p\frac{\partial J_j^q}{\partial x^s})-\frac{\partial H_{rs}^i}{\partial x^j}.$$

What are the transformation rules of $H_{jk}^i$ and $h_{jrs}^i$ under coordinate transformations?

Liu et al. (2009) presented a normalization algorithm for polynomials in ${\mathcal{R}}_2[\partial ̸]$, which consists of two parts. In the first part, a polynomial is rewritten modulo monoterm symmetries. In the second part, we compute the canonical form of a polynomial by collecting equivalent terms with respect to monoterm symmetries. Then, two polynomials are equal if and only if they have the same canonical forms. By using this algorithm, one can prove that $R_{jkl}^i$ and $H_{jk}^i$ in Problem 1.1, Problem 1.2 are tensors, and can derive that $h_{jrs}^i$ has the following transformation rule:

$$h_{j^{\prime }{r}^{\prime }{s}^{\prime }}^{i^{\prime }}=\frac{\partial x^r}{\partial x^{r^{\prime }}}\frac{\partial x^s}{\partial x^{s^{\prime }}}(\frac{\partial x^{i^{\prime }}}{\partial x^i}\frac{\partial x^{j^{\prime }}}{\partial x^j}{h}_{jrs}^i-\frac{{\partial }^2{x}^{i^{\prime }}}{\partial x^{j^{\prime }}\partial x^i}{H}_{rs}^i).$$

However, the algorithm with respect to monoterm symmetries in (Liu et al., 2009) has a factorial complexity. More precisely, suppose that f is a monomial indexed with i, j, $i_1$, $i_2$, where i is the number of functions, j is the number of pairs of dummy indices, $i_1$ is the number of functions of the same name without free indices, and $i_2$ is the number of functions with the same free indices and the same name. Since we need to compare all the monomials equivalent to f, the complexity is at least $O(i!j!\prod _{l=1}^k{n}_l!)$, where k is the number of groups of pairwise interchangeable indices, and $n_l$ ($l=1,2,...,k$) is the number of indices in each group. Other representative methods for computing canonical forms with respect to monoterm symmetries are due to Portugal (1999) and Manssur et al. (2002). The algorithm in (Portugal, 1999) has four steps. First, list all possible permutations according to commutativity of multiplication (symmetry under the interchange of functions). Second, for each permutation, replace the indices with integers according to the name of tensors, the index positions and index classes. Third, sort the indices inside each function. At last, find the smallest element in the equivalence class. So the complexity is at least $O(i_1!i_2!\prod _{l=1}^k\left(\begin{array}{c}{n}_l\\ 2\end{array}\right))$. Computational group theory is applied in (Manssur et al., 2002). Its complexity for some special Riemann indexed monomials (with all indices contracted) is at least $O({(j+j_1)}^4)$, where $j_1$ is the number of free indices.

The reason for such factorial complexity in some of these algorithms is that dummy indices within an indexed polynomial f are described either by original letters or by integers according to the order of their appearance. Neither of the descriptions is an invariant. In other words, they vary when we rewrite f. Consequently, to find the canonical form, we need to list and compare all the elements in the equivalence class of f.

At this point two questions arise: (1) What is an invariant?, and (2) how can we provide an efficient normalization algorithm with respect to monoterm symmetries of the invariant?

Besides, the normalization algorithm in (Liu et al., 2009) depends on a skillful and tricky classification of 2nd-order partial differential functions according to their connections (e.g. circle, chain, maximum lower tree and so on). But the classification is no longer valid for higher order. It appears infeasible to classify partial differential functions for each order.

Then another question arises: How can we provide a normalization algorithm that is independent of function classifications?

In (Liu, 2017), we only partly answered these questions based on function distances, as the proofs of validness of some algorithms are not given. Besides, the complexity of the normalization algorithm with respect to monoterm symmetries in (Liu, 2017) is at least cubic, higher than the one in this paper, as we will see below.

In this paper, we first define the distance from free indices to dummy indices, and prove that it is an invariant (with respect to certain symmetries). Then, we present an index replacement algorithm which is uniquely determined by the distance, and use it to develop a normalization algorithm with respect to monoterm symmetries for polynomials in $\mathcal{R}[\partial ̸]$. The complexity is at most $O(m^2)$, lower than known algorithms, where m is the sum of the number of free indices and the number of pairs of dummy indices. Finally, by the method of index replacement, and by choosing the monomial associated with the smallest numerical list as the canonical form, a normalization algorithm is provided for polynomials in ${\mathcal{R}}_2[\partial ̸]$, which is independent of function classifications.