Two nondeterministic positive definiteness tests for unidiagonal integral matrices
Andrzej Mr\'oz
TL;DR
This paper introduces two new nondeterministic algorithms for testing positive definiteness of unidiagonal integral matrices, which are more efficient in practice and provide additional insights into the associated quadratic forms.
Contribution
The paper presents two novel algorithms with complexities of O(n^3) and O(n^4) for positive definiteness testing, utilizing edge transformations on bipartite graphs, and includes randomized variants with detailed step bounds.
Findings
Algorithms perform well in practice, often faster than standard tests.
Algorithms provide additional information on quadratic forms.
Randomized variants have well-defined maximum steps.
Abstract
For standard algorithms verifying positive definiteness of a matrix based on Sylvester's criterion, the computationally pessimistic case is this when is positive definite. We present two algorithms realizing the same task for , for which the case when is positive definite is the optimistic one. The algorithms have pessimistic computational complexities and and they rely on performing certain edge transformations, called inflations, on the edge-bipartite graph (=bigraph) associated with . We provide few variants of the algorithms, including Las Vegas type randomized ones with precisely described maximal number of steps. The algorithms work very well in practice, in many cases with a better speed than the standard tests. Moreover, the algorithms yield some…
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Random Matrices and Applications · Advanced Combinatorial Mathematics
Two nondeterministic positive definiteness tests
for unidiagonal integral matrices ††thanks: Identify applicable funding agency here. If none, delete this.
Andrzej Mróz
2016 (2019)⋆
Faculty of Mathematics and Computer Science
Nicolaus Copernicus University
ul. Chopina 12/18, 87-100 Toruń, Poland
Email: [email protected]
Abstract
For standard algorithms verifying positive definiteness of a matrix based on Sylvester’s criterion, the computationally pessimistic case is this when is positive definite. We present two algorithms realizing the same task for , for which the case when is positive definite is the optimistic one. The algorithms have pessimistic computational complexities and and they rely on performing certain edge transformations, called inflations, on the edge-bipartite graph (=bigraph) associated with . We provide few variants of the algorithms, including Las Vegas type randomized ones with precisely described maximal number of steps. The algorithms work very well in practice, in many cases with a better speed than the standard tests. Moreover, the algorithms yield some additional information on the properties on the quadratic form associated with a matrix . On the other hand, our results provide an interesting example of an application of symbolic computing methods originally developed for different purposes, with a big potential for further generalizations in matrix problems.
This is an extended version of the article [21] in which we discussed the algorithm of the complexity .
††publicationid: pubid:
978-1-5386-0869-2/17/$31.00 ©2017 IEEE
4th International Conference on Advances in Electrical Engineering 28-30 September, 2017, Dhaka, Bangladesh
⋆ Written in 2016 (and put in a drawer). Updated and published in arXiv in 2019. The update relies on: few remarks at the end, new test results after a slight optimisation of the codes, updates of the literature.
1 Introduction
Fix an integer and a matrix , i.e., is a square matrix with real coefficients. Recall that is positive definite if one of the following equivalent conditions holds (see [17, Section 7.6]):
for every non-zero , where is the quadratic form associated with , defined by the formula , for every (column vector) . 2. 2.
All eigenvalues of the matrix are positive ( is called the symmetrization of ). 3. 3.
All leading principal minors of are positive. 4. 4.
A non-positive pivot does not emerge during row reduction of to upper triangular form with “Type III” operations only (i.e., operations of adding th row multiplied by scalar to th row, with ).
We recall that the positive definiteness of a matrix is an important property exploited in many branches of theoretical and applied mathematics (see [17] and [19, 20, 27, 7, 11, 1]) and algorithmic methods related to this concept are of big importance.
Observe that the condition 4) yields quite efficient obvious algorithm for testing whether a matrix is positive definite, with pessimistic complexity (as standard Gaussian elimination). Moreover, it is clear that the pessimistic case for this algorithm is the case when is positive definite (if it is not, the row reduction process can be interrupted when first non-positive pivot emerges).
In the present paper we provide few variants of a new algorithmic positive definiteness test for so-called unidiagonal triangle-integral matrices (certain subset of real matrices containing the set all matrices with integer coefficients and the unity on the diagonal, see Section 2), for which the case when a matrix is positive definite is the optimistic one. The pessimistic complexities of our algorithms are and (see Algorithms 5.1 and 5.2, and Theorem 5.5). However, for some classes of matrices, their performance is in practice better than the standard tests as this mentioned above. The proposed algorithms can have a useful application e.g. when there is a need to perform the test for positive definiteness for a large set of large, dense integral matrices, for which one knows a priori that “most of them” are positive definite.
Our results are an application of the inflation techniques for bigraphs from [19], inspired by the recent studies of Simson [27] (see Section 2, see also [13, 14, 30]). These ideas extend older known results of Ovsienko [24], Dräxler et al. [7] and Barot-de la Peña [1] concerning combinatorial, algebraic and algorithmic analysis of integral quadratic forms and their roots. We refer to [19] for more complete bibliography and more detailed introduction to the topic. On the other hand, inflations of bigraphs, matrices and integral quadratic forms are an important ingredient of the new lively area – the Coxeter spectral analysis of bigraphs and their morsifications [27, 8, 11, 12, 13, 18, 28, 29, 30, 19, 20], cf. [1], as well as the Coxeter type study of finite dimensional algebras and their bounded derived categories, see [22, 23, 27, 20]. In the present paper we introduce slightly different than in [27, 19, 20, 28, 29] conventions concerning an approach to interrelations between matrices and bigraphs (more elementary but at the same time more general).
The paper can be treated as a continuation of a computer algebra approach to matrix problems presented by the author at SYNASC 2012 [10], see also [4, 5, 2, 3, 6, 9]. The paper is also an extended version of [21] where we discussed Algorithm 5.1 (of the complexity ) only. Algorithm 5.2 (of the complexity ) is the main contribution of the present paper. Implementations of the algorithms discussed here are included as a part of Maple packages [31] (see also author’s homepage [32] to download different projects, including C++ implementation of the inflation algorithm with the graphical presentation).
2 Preliminary notions and facts
By a real quadratic form of rank we mean a mapping defined by a homogeneous polynomial of second degree
[TABLE]
where and are fixed real numbers, for . We say that is a unit quadratic form if , for each . If all the coefficients are integers then we treat as the mapping and call it an integral quadratic form. Note that for every non-zero if and only if for every non-zero , in case is an integral quadratic form (see [26, Section 1.1]).
Each matrix defines a quadratic form by the following formula
[TABLE]
(we always treat as a column vector in this context).
Given a matrix , we set ; moreover, we denote by , the (upper) triangularisation of , i.e., is an upper triangular matrix such that for ; for ; and for (see [27, 20], cf. Example 2.6).
We often use the following simple observation.
Lemma 2.1
Fix and a matrix . Then
;
the following conditions are equivalent:
is positive definite,
is positive definite,
is positive definite.
Proof. (a) Since , we have
[TABLE]
for any , so . The equality follows from (2). The assertion (b) is an immediate consequence of (a) (see Introduction).
We introduce the following technical notions to characterize the input matrices for our algorithms.
Definition 2.2
Fix and a matrix .
- (a)
We say that is a unidiagonal111Note that the term “unidiagonal” appears in the literature with various different meanings. matrix, if has 1’s on the diagonal (equivalently, is a unit quadratic form).
- (b)
We say that is a triangle-integral matrix, if its triangularisation has integral coefficients (equivalently, is an integral quadratic form).
Clearly, each integral matrix is triangle-integral. If is a unidiagonal and triangle-integral matrix, we say that is a UTIM, for short.
The following fact provides useful restrictions for the coefficients of a positive definite matrix in terms of its diagonal.
Lemma 2.3
Let , , be a positive definite matrix. Then
, for each ;
, for each ;
if is a UTIM then each coefficient of the triangularisation of belongs to the set .
Proof. Since is positive definite, we have
[TABLE]
for each , by the formula (2), where denote the standard basis of . This implies (a) and (b).
The assertion (c) follows by (b) and Definition 2.2 (cf. [14, Lemma 4.1(a)]).
The following specialization of a (signed) multigraph from [27] is an important combinatorial tool exploited in our algorithms.
Definition 2.4
An edge-bipartite graph (or, a bigraph, for short), is a pair , where is a finite non-empty set of vertices and is a finite multiset of edges (i.e., unordered pairs of vertices) equipped with a disjoint union bipartition such that the (multi)set of edges connecting the vertices and splits into a disjoint union with and , for each pair of vertices , and either or .
In other words, a bigraph is an undirected multigraph equipped with two kinds of edges: these from (called solid edges) and (called dotted edges); and two vertices can be connected by only one kind of these two kinds of edges. We define a bigraph to be loop-free if has no loops, that is, for each vertex . We say that a bigraph is connected, if the underlying graph , obtained from by replacing all dotted edges by the continuous ones is connected. Note that a bigraph with , that is, a bigraph without dotted edges, is a usual (multi)graph; and each undirected (multi)graph can be viewed as a bigraph in this way. Later on, we always assume that each bigraph with has fixed labeling of vertices by first natural numbers, i.e., .
Given and a UTIM , we associate with the (unique) loop-free bigraph such that
- •
;
- •
if and ;
- •
if and ,
where is the triangularisation of . We say that a UTIM is connected if the bigraph is connected.
Remark 2.5
(a) In case a triangle-integral matrix is not unidiagonal one can associate with certain bigraph with loops (cf. [20, 11, 12, 13, 29]). However, in this paper we restrict to unidiagonal matrices and loop-free bigraphs. The general case is technically more complex and more delicate; we plan to study it in the future paper.
(b) Lemma 2.3(c) implies that if a UTIM is positive definite then the bigraph does not have multiple edges.
(c) The notion of a triangle-integral matrix we introduced above can be treated as a slight generalization of Simson’s concept of a rational morsification (see [28, Section 3], cf. [27, 20]). More precisely, one checks that if is a rational morsification of a bigraph , then is a triangle-integral matrix. Moreover, if is loop-free, then is unidiagonal (see [28] for the details).
Note that iff , for a pair of UTIM’s . Moreover, given a loop-free bigraph with , there exists a unique upper-triangular unidiagonal matrix such that . We call the (upper-triangular) Gram matrix of (this definition is equivalent to Simson’s original definition of the upper-triangular Gram matrix of from [27]). Note that if , for a UTIM , then . We say that a loop-free bigraph is positive (and we write ) if the Gram matrix is positive definite.
Example 2.6
Let be the following UTIM
[TABLE]
Then its triangularization and the associated bigraph look as follows:
[TABLE]
Note that is not positive (equivalently, is not positive definite, see Lemma 2.1), since has multiple edges (see Remark 2.5(b)).
3 Generalities on inflations of bigraphs
We recall from [27, 14] and [19] the following concepts of inflations of bigraphs.
Definition 3.1
Fix an integer and a loop-free bigraph with and the Gram matrix
[TABLE]
- (a)
An inflation at a vertex is an operation associating with the new bigraph such that and the edges of are obtained from by replacing any solid edge (resp. dotted edge) in incident with by a dotted one (resp. solid one) in ; the remaining edges in stay unchanged in .
- (b)
Given a pair of distinct vertices such that the set of dotted edges between and is non-empty, an inflation at the pair is an operation associating with the new bigraph with and the edges of defined as follows:
replace each of the dotted edges in by a solid one in ;
the multiset of edges is defined by setting , for each , where we set and if ;
each of the remaining edges in stay unchanged in .
Note that is defined if and only if so is , but in general (see [19, Example 4.5(a)]). Observe that a bigraph is a (multi)graph (i.e., does not have dotted edges) if and only if none of inflations at a pair is defined for .
Example 3.2
Let be the following bigraph:
[TABLE]
Then and look as follows
[TABLE]
We refer to [19] and [14, 27] for more examples and the discussion on other properties of inflations.
Following [19] we formalize an execution of inflation algorithm as follows.
Definition 3.3
Fix a loop-free bigraph . The sequence ) is called an execution of inflation algorithm on resulting in if
- •
each of its terms has the form , with , or , with ,
- •
* is defined, for any ,*
- •
.
The integer is called the length of the execution . We admit also an empty execution on of length , resulting in .
Clearly, the inflation algorithm in the sense of Definition 3.3 is not deterministic (see also Section 4 and [19]).
Proposition 3.4
Let be a loop-free bigraph and an execution of inflation algorithm on resulting in .
If is connected then so is ;
is positive if and only if so is .
Proof. Both assertions follow from [19, Corollary 3.4] (see also [19, Lemma 3.3], cf. [14, Lemma 4.1]).
It was proved in [27, Theorem 3.1] (see also [19, Theorem 4.4] and [14]) that a connected loop-free bigraph is positive if and only if there exists an execution on resulting in one of the following Dynkin graphs (see [27, 19]):
[TABLE]
In [19, Theorem 4.4] we enhanced this result by describing precise bounds for the lengths of few variants of such executions on positive bigraphs. The further discussion of the paper, concerning the announced positive definiteness tests, is based on some conclusions from [19] (cf. [19, Remark 4.6(b)]).
4 Executions of inflation algorithm
One of the crucial steps in our algorithmic solutions is the following “generic” inflation procedure, which is a specialization of randomized Algorithm 4.3 from [19]:
Algorithm 4.1
Input:* a loop-free bigraph with vertices and an integer (called a strategy).*
Output (if it stops):* a loop-free (multi)graph with vertices such that is connected if so is , and if and only if .*
InflationsAtPair**
[TABLE]
The strategy can be one of the following:
[TABLE]
By first (resp. last) dotted edge above we mean a dotted edge such that the pair is the smallest (resp. the greatest) element in with respect to the lexicographic order, among all dotted edges.
In general Algorithm 4.1 does not have a stop property, but its partial correctness, i.e., the properties of the output graph in case the algorithm stops, follow from Proposition 3.4. Note that in case the strategy equals [math] or (resp. or ) then Algorithm 4.1 is a deterministic (resp. nondeterministic randomized) algorithm.
Fix an integer and an integral quadratic form . A vector is called a root of if (see [19, 26]). We say that a vector is positive (resp. sincere) if and (resp. ), for all .
The following conclusion from [19, Theorem 4.4] provides the most important properties of Algorithm 4.1 in context of our goals.
Theorem 4.2
Let be a connected loop-free bigraph with vertices and , an arbitrary fixed strategy. Then the following hold:
The bigraph is positive if and only if Algorithm 4.1, applied for and , stops after performing at most inflations, and the returned graph is a Dynkin graph, where
[TABLE]
Assume that the integral quadratic form admits a positive sincere root. Then is positive if and only if Algorithm 4.1, applied for and , stops after performing at most inflations, and the returned graph is a Dynkin graph, where
[TABLE]
Proof. The assertion (a) (resp. (b)) follows from [19, Theorem 4.4(d)] (resp. [19, Theorem 4.4(b)]). We only recall that the proof of [19, Theorem 4.4] relies on a careful analysis of changes of root systems of bigraphs under inflations, and known properties of roots of Dynkin graphs (cf. [24, 27, 14]).
Observe that Theorem 4.2 provides bounds for the number of inflations depending only on the number of vertices of , independently on the chosen strategy . In particular, it shows that Algorithm 4.1 is a Las Vegas type randomized algorithm in case and .
Algorithm 4.3
Input:* a loop-free connected bigraph with , .*
Output:* a loop-free connected bigraph with vertices such the quadratic form admits a positive sincere root, and if and only .*
InflationsToPosSincereRoot**
[TABLE]
Theorem 4.4
Algorithm 4.3 is correct and applied to a loop-free connected bigraph with vertices, it performs at most inflations at a vertex and at most inflations at a pair.**
Proof. First note that a pair such that as in lines 3-4 always exists by the connectedness of (see Proposition 3.4(a)). Next, the instruction in the line 5 guarantees that the inflation in 6 is defined. The stop property follows from the line 7 and the construction of the loop.
Let be the bigraph returned by the algorithm applied for . Proposition 3.4(b) implies that if and only if . For the proof of the fact that admits a positive sincere root we refer to [19, Lemma 3.3] and the proof of [19, Theorem 4.4(c)] (we only outline the main idea: is constructed from the “trivial” root of the input bigraph by consecutive changes induced by inflations from the line 6).
The numbers of performed inflations at a vertex and at a pair follow obviously from the construction of the main loop (note that ).
5 Positive definiteness tests by inflations
We use the notation from the previous section. Let InflationsAtPairPos (resp. InflationsAtPairPoss) denotes the procedure which works exactly as InflationsAtPair for a bigraph with vertices and a strategy , but breaks the main loop after performing (resp. ) inflations from the line 4, see Theorem 4.2.
Now we have everything to formulate the main algorithms of the paper. Note that for the simplicity of the presentation we restrict to connected UTIM’s. Extending these ideas for the general case (i.e., for arbitrary UTIM) is not a very hard task.
Algorithm 5.1
Input:* a connected UTIM with , and a strategy (as in Algorithm 4.1).*
Output:* true if is positive definite, or false otherwise.*
PosDefTestByInflations**
[TABLE]
Correctness of the algorithm (for arbitrary strategy ) follows from Theorem 4.2(a) and the properties of Algorithm 4.1 (see also Lemma 2.1).
Algorithm 5.2
Input:* a connected UTIM with , and a strategy (as in Algorithm 4.1).*
Output:* true if is positive definite, or false otherwise.*
PosDefTestByRootInflations**
[TABLE]
Correctness of the algorithm (for arbitrary strategy ) follows from Theorem 4.2(b) and the properties of Algorithm 4.1 and Algorithm 4.3 (see also Lemma 2.1).
Remark 5.3
(a) A simple optimization of Algorithm 5.2 is possible. Namely, a graph (i.e., a bigraph without dotted edges) may appear already during the execution of InflationsToPosSincereRoot in the line 2. If this is the case, the algorithm can break the execution of the line 2 and jump (with ) to the line 4.
(b) Depending on applications and implementation purposes, one can add at the beginning of Algorithms 5.1 and 5.2 the test if satisfies the trivial necessary condition for positive definiteness from Lemma 2.3(c), i.e., if the coefficients of the triangularization of belong to the set .
(c) From the construction of the loop in InflationsAtPair it is clear, that the optimistic case for Algorithms 5.1 and 5.2 is the case when a graph emerges much before the maximal number of inflations (or ) is achieved. And this happens most often when the input matrix is positive definite (see Theorem 4.2).
(d) Implementations of all the procedures presented in the paper are available in Maple packages [31] (under the same names as the pseudocodes).
(e) An important side effect of Algorithms 5.1 and 5.2 is the computation of the Dynkin graph for a positive definite UTIM . The graph is the Dynkin type of the integral quadratic form associated with (cf. [1, 28]). It provides certain non-trivial additional information on , e.g., on its root systems (see [1, 28, 27, 19, 20] for more details).
In the rest of the section we prove that the pessimistic complexity of Algorithm 5.1 (resp. Algorithm 5.2) is (resp. ), independently on the strategy . In particular, this shows that Algorithms 5.1 and 5.2 are Las Vegas type randomized algorithms in case , for arbitrary connected UTIM .
We start with the following technical fact, which also provides hints for implementations of the algorithms. Given a bigraph with vertices, we operate on its non-symmetric Gram matrix ; we count the complexity with respect to arithmetic operations performed on the coefficients of .
Proposition 5.4
Let be a loop-free bigraph with , , and the Gram matrix .
Inflation , for , can be performed with operations.
Inflation , for , can be performed with operations.
Test whether is a Dynkin graph can be performed with operations.
Proof. The assertions (a) and (b) follow easily from Definition 3.1 (in both cases we change coefficients of corresponding to edges incident with one fixed vertex).
(c) The test whether is a Dynkin graph can be performed by applying the following steps:
check if is a simple graph, i.e., if , for every , where are the coefficients of as in Definition 3.1; 2. 2.
check if has precisely edges; 3. 3.
check if is connected (e.g., by applying Depth First Search); 4. 4.
find the list of ramifications (= vertices of degree ); 5. 5.
if then ; if () or ( and ) then is not a Dynkin graph; 6. 6.
assume that and (i.e., is a star graph with three arms); find the star type of (i.e., is a list of lengths of all three paths outgoing from ):
- (a)
if then ; 2. (b)
if then ; 3. (c)
if then ; 4. (d)
if then ; 5. (e)
otherwise, is not a Dynkin graph.
It is easy to observe that each of these 6 steps can be realized with the complexity at most .
Note that the assertion (c) of the proposition is a special case of the (bi)graph isomorphism problem, for which the polynomial-time general algorithm is not known. Moreover, carefully implemented Dynkin graph recognition test based on the above 6 steps is very efficient, it appears that its performance time is negligible in Algorithms 5.1 and 5.2.
Theorem 5.5
Let , , be a connected UTIM and , an arbitrary fixed strategy. Then the following hold:
The pessimistic complexity of Algorithm 5.1, applied to and , is .
The pessimistic complexity of Algorithm 5.2, applied to and , is .
Proof. (a) Each execution of the body of the loop in InflationsAtPair relies on: finding a dotted edge with respect to the strategy , which costs operations; applying the inflation , which costs by Proposition 5.4(b). The loop body is executed at most times (see Theorem 4.2(a)). Therefore, the execution of the line 2 in Algorithm 5.1 costs operations. Dynkin graph recognition test in the line 3 costs by Proposition 5.4(c), hence it does not affect the general complexity.
(b) First we estimate the complexity of the line 2 of Algorithm 5.2, i.e., the execution of InflationsToPosSincereRoot. By Theorem 4.4 it performs at most inflations, and this costs operations by Proposition 5.4. Additionally, Algorithm 4.3 performs times the edge search in its lines 3-4, each of them of the cost . Therefore, the complexity of the line 2 of Algorithm 5.2 is .
The execution of InflationsAtPairPoss in the line 3 costs , by Theorem 4.2(b) (cf. the proof of (a) above). Similarly as above, Dynkin graph recognition does not affect the general complexity .
6 Experiments and conclusions
In the experiments below we use our implementations of Algorithms 5.1 and 5.2 in the computer algebra system Maple [31].
We perform the test on an example coming from “nature”, i.e., from practical application of Coxeter spectral analysis in representation theory of algebras, see [19, 20]. We consider the following family of dense integral matrices. Let be the upper-triangular unidiagonal matrix whose coefficients are defined as follows:
[TABLE]
for any and (and for ).
Example 6.1
For , the matrices and the corresponding bigraphs look as follows:
[TABLE]
Note that the underlying graph of , is a complete graph, for every ; in particular, is connected. Moreover, one shows that each is the so-called Nakayama bigraph of type introduced in [20]. These bigraphs encode the -theory of Nakayama algebras, an interesting class of finite-dimensional algebras recently studied by many authors, see [20, Section 5] for details. It follows from general theory that each is positive definite of Dynkin type (cf. [20, Lemma 5.3]), i.e., our inflation algorithms, applied to , return the Dynkin graph , for every , cf. Remark 5.3(e).
We test the matrix (i.e., matrix) for positive definiteness in Maple ver. 15 on PC computer with Intel Core i5-7500 CPU 3.4GHz. First we apply to an efficient implementation of the standard positive definiteness test based on Sylvester criterion and Gaussian elimination, briefly outlined in Introduction (the procedure PosDefTestByGaussElim in [31]). It returns true after
[TABLE]
We note that the execution time of the standard Maple routine gausselim (for Gaussian elimination) on is similar: sec. The procedure PosDefTestByRootInflations (Algorithm 5.2), returns true after
[TABLE]
for every strategy , since for , the Dynkin graph emerges already after the line 2 (cf. Remark 5.3(a)). Whereas Algorithm 5.1 works with the speed depending on the strategy as follows:
[TABLE]
We presented two different executions for , and also for , since recall that these two strategies yield nondeterministic executions, with numbers of inflations (and hence the execution times) depending on the current behavior of the random number generator, see Section 4. Note that the strategy appears to not be very efficient for the matrix ; however, the strategies work very nicely, not much slower than Algorithm 5.2 above. A bottle-neck of our implementation of the strategy is the random selection of an edge among all dotted edges (see Section 4). We are working on a more efficient solution. On the other hand, it is a challenging task to construct a different, more sophisticated strategy for choosing dotted edges in Algorithm 4.1, than our four simple strategies .
The results of the above test are perhaps not breath-taking, but observe that our algorithms are competing with Gaussian elimination, one of the simplest and most efficient basic matrix algorithms. And our algorithms are conceptually quite complex and they potentially have several technical bottle-necks, not as easy to implement efficiently. The reader is referred to the file PosDefTestExperiments.mws in [31] for the details of this and other tests, including positive definite matrices of other Dynkin types and non-positive definite ones (the readers having no access to Maple are referred to the PDF version of the worksheet, also available in [31]).
Concluding, we have constructed algorithmic positive definiteness tests of the following properties:
- •
their optimistic case is a positive definite matrix on the input (in contrast to the standard tests, cf. Introduction and Remark 5.3(c));
- •
they work noticeably faster than the standard Gaussian elimination test on certain classes of matrices;
- •
they can have nice applications, e.g. for the sets of integral matrices for which it is a priori known that many of them are positive definite;
- •
provide an example of a Las Vegas randomized algorithm (i.e., the behavior varies from execution to execution on the same data, but the number of performed operations is always bounded by the quantity depending on the size of the input only); such algorithms are very interesting from the point of view of theoretical computer science;
- •
a side effect of the algorithms is the computation of the Dynkin type of the quadratic form of a positive definite matrix , encoding several additional information on , see Remark 5.3(e);
- •
they show a non-trivial practical application of deep theoretical techniques originally developed for different purposes (see [19, 20, 27, 14, 24, 1]), and having a potential for further algorithmic studies.
Recently in [25] the authors presented an algorithm of the complexity to compute the Dynkin types for a larger (than our UTIM’s) class of integral matrices, called symmetrizable quasi-Cartan matrices, having origins in Lie theory. This algorithm can also be used as a positive definiteness test, it also uses inflations, but it is essentially different than ours. One can then say that the “natural” lower bound for the pessimistic complexity of such algorithms is , the complexity of the standard Gaussian elimination. Surprisingly, recent results from [15] on some new properties of the root systems allow to consider an algorithm to compute the Dynkin type and to test the positive definiteness for quasi-Cartan matrices, which has the complexity (that is, the same as simply reading the input matrix!). We present this algorithm as well as some new applications of inflation techniques in the subsequent paper [16].
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