=Paper=
{{Paper
|id=Vol-1490/paper30
|storemode=property
|title=Method of UNIT testing for computing software modules algorithms
|pdfUrl=https://ceur-ws.org/Vol-1490/paper30.pdf
|volume=Vol-1490
}}
==Method of UNIT testing for computing software modules algorithms==
https://ceur-ws.org/Vol-1490/paper30.pdf
Mathematical Modeling
Method of UNIT testing for algorithms of computing
software modules
Kovartsev A.N., Popova-Kovartseva D.A., Gorshkova E.E.
Samara State Aerospace University
Abstract. The method of automating Unit testing processes for computing
software modules is considered in the paper. Modern means of testing
automation, which are analyzed in many scientific studies, specialize mainly in
testing graphical user interfaces, web-interfaces, network communications,
information systems, etc., which is the result of a huge demand in the market
for software products within these spheres. Software modules of computing
character are overshadowed by such products despite the fact that these
modules have considerable scientific and practical value. They deal mostly
with high tech spheres: aerospace cluster, energy industry, defense complex,
etc. The article presents an original method of Unit testing for computing
modules based on the algorithm of global search for infinite discontinuity points
of testing function, which allows to find fatal errors in computing software
modules, as well as incorrectness in implementation of algorithms for
mathematical models. The universal method of Unit testing is offered within the
class of computing modules, which helps to minimize the time for program
debugging and to find fatal errors with less effort, as well as to organize total
module testing for all modules of the program.
Keywords: Unit testing, testing automation, computing software modules,
global optimization, fatal errors
Citation: Kovartsev A.N., Popova-Kovartseva D.A., Gorshkova E.E. Method
of unit testing for algorithms of computing software modules. Proceedings of
Information Technology and Nanotechnology (ITNT-2015), CEUR Workshop
Proceedings, 2015; 1490: 252-261. DOI: 10.18287/1613-0073-2015-1490-252-
261
Introduction
Developers of modern software (SW) face the challenge to carry out their
projects within tight deadlines and with minimal resources consumption [1, 2],
whereas software vendors strive to carry out testing appropriately, quickly, and
thoroughly. Most types of work, used in software programming, are to be supported
by automated means of testing.
Modern means of automated testing are constantly expanding at present and
include testing of graphical user interfaces, checking requirement compatibility,
download speed, code coverage, web-interface, network communications, information
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systems, etc. This results from the huge demand in the market for software products
within these spheres. Software modules of computing character are overshadowed by
such products despite the fact that these modules have considerable scientific and
practical value. They deal mostly with high tech spheres [3]: aerospace cluster,
energy industry, defense complex, etc.
Testing is the method of providing the required level of SW quality. In the
general sense, software testing is the process that allows to determine the correctness,
completeness and quality of the developed software product. Unfortunately, it is
impossible to determine unambiguously whether the analyzed program functions
correctly or not, as well as to guarantee the absence of defects in a software product
because human factor problems may appear at all SW lifecycle stages [4]. Therefore,
all existing testing methods operate within private formal methods of testing
organized for the analyzed product. The list of modern methods and approaches for
solving the problem of program testing is extensive and diverse. On the one hand, this
diversity is determined by the current practice of using a computer while solving a
variety of problems and, accordingly, by the specific features of software products
themselves.
The class of computing software modules (CSM), based on the use of
mathematical models, has some specific features. Such programs, carried out on a
computer, are sure to calculate any function that realizes the display output of input
data. This implies that a computer by means of its resources finishes the definition of
a partly determined function, which results in complete definition. Consequently, it is
possible to estimate if the results of program execution are right or wrong only by
comparing the specification of the expected function with the results of its
calculation; this is carried out in the testing process. For the class of computing
software modules, methods of Unit testing [20] and test tools based on models using
formal methods [4,6] are more suitable.
Formal methods usually allow to solve a limited range of testing software tasks
within a particular class, however, they are able to work effectively in industrial
projects and require a minimum number of special skills and knowledge so that to be
used [4]. Currently, within monitoring the formal properties of SW, methods of test
construction are thoroughly developed on the basis of finite-state automation [7, 8].
For them, accuracy characteristics and evaluation of completeness of performed tests
are known. In these techniques, computing route is analyzed not only for performing
some formulae, but also for coordination with the specified automation model of the
proper behavior. Nowadays these methods are rarely used, mainly for monitoring
small critical applications [4]. Formal methods are rather โheavy-weightโ, they
require well-qualified specialists, at least at the stage of software modeling. The weak
point of formal methods is the need to construct the model itself on the parallel basis
and to check its correctness.
The practice of industrial software production shows that the most effective
strategy is to search and correct as many errors as possible at the earliest stage of
software development. Methods of Unit testing are appropriate for this purpose. Unit
testing consists in checking the software performance on a set of input and
corresponding output data. This approach allows to reveal a significant number of
errors and locate them quickly as it is easier to find an error within a module than to
do it within the whole project. CSM can be easily interpreted with a vector
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computable function ๐ = ๐(๐) = (๐1 (๐), ๐2 (๐), โฆ , (๐))๐ , having a set of input
parameters ๐ = (๐ฅ1 , ๐ฅ2 , โฆ , ๐ฅ๐ )๐ of the module and a set of calculated data ๐ =
(๐ง1 , ๐ง2 , โฆ , ๐ง๐ )๐ .
The origin of errors for CSM is extensive and diverse. Errors occur when initial
and boundary conditions, which characterize the value and location of external
factors, are set incorrectly. They may be related with a set of limitations and
assumptions resulting from the physical nature of the object and limiting the range of
input parameters.
Among the errors of this kind for CSM, two most common groups can be
distinguished: calculation errors and logical (algorithmic) errors. Calculation errors
are mostly connected with incorrect recording or programming of mathematical
expressions and manifest themselves as arithmetic error of division by 0, the square
root of a negative number, as calculation of rational or transcendental functions, etc.
They can only be detected while executing the program and lead to program stoppage.
Logical errors are connected with the distortion of problem solving algorithm
and result from incorrect problem setting, wrong consideration of all conditions for
solving the problem, incorrect management organization within CSM, and errors in
the input of logical expressions. These errors are difficult to correct, corrections often
being made with the help of formal methods [5].
Nevertheless, there is a universal method for detecting errors of calculating and
partly logical character in computing software modules [3, 9]. This method is based
on the fact that all computing modules, implemented as a program, โworkโ within
integrity of used functions or mathematical models. Otherwise, the program canโt be
used. If we organize the search for infinite discontinuity of the function by some
means, we will be able to detect all calculating errors of any origin mentioned above.
Relatively โsimpleโ methods of global optimization (GO) of multivariable functions
are appropriate for this purpose[10].
In this paper we consider the improved algorithm of global search for points of
discontinuity of the second type, intended for detecting error situations in the software
modules of computing character. This algorithm is the basis of Unit testing for
computing software modules.
1. Problem Statement
Let the area of error search in computing module be a unit cube (in general โ
hypercube) ะ ๏ฝ [0, 1] ๏ด [0, 1] ๏ด [0, 1] , which is proportionally divided into eight
smaller cubes.
Unit testing algorithm of computing modules, formally described by vector
function ๐ = ๐(๐), can be put as the problem of global optimization
max f k2 ( X ), k ๏ฝ 1,..., m, (1)
X
which is aimed at detecting (or ensuring the absence of) discontinuity points of the
second type in the examined function. The infinite discontinuity ๏ฑ ๏ฅ is realized in
numbers marked by code NaN on a computer; in fact we can specify the upper limit
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(๐)
of acceptable values |๐๐ (๐)| โค ๐๐ ๐ข๐ or each calculating parameter of a module. To
simplify the situation, we will further consider scalar function ๐(๐) = ๐๐๐ฅ๐๐2 (๐).
Among the well-known one-parameter methods of multiextremal optimization,
R.G. Strongin statistic information method is the most effective [10]. The method is
based on the use of approximate posterior probability distribution of global extremum
location, which is formed in the process of function testing, which allows to realize a
more balanced strategy to search for function global minimum. This strategy is so
effective that it is often transferred from one-dimensional case to multivariable
function optimization.
It is shown in the paper [10] that function extremum search is realized by
maximizing a simple characteristic function:
( zi ๏ญ zi ๏ญ1 ) 2
R(i) ๏ฝ ๏ญ( xi ๏ญ xi ๏ญ1 ) ๏ซ ๏ญ 2( zi ๏ซ zi ๏ญ1 ) , (2)
๏ญ( xi ๏ญ xi ๏ญ1 )
where ๏ญ โ estimation of Lipschitz constant, which is calculated in the process of
function extremum search:
๏ฌ1 M ๏ฝ 0, z ๏ญ zi ๏ญ1
๏ญ๏ฝ๏ญ M ๏ฝ max i ,
๏ฎrM M ๏พ 0, i ( xi ๏ญ xi ๏ญ1 )
where r โ parameter.
The condition of Lipschitz for optimized function simplifies greatly the search
for function extremum as the limitation of function growth degree allows to find local
extremum vicinity quickly. However, while searching for a set of infinite
discontinuity, areas of function monotony as well as extremum vicinity are equally
useless. It is much more important to determine the criteria which will be responsive
to the fast increase and decrease of function.
Nowadays, mathematical aspects of function behavior in vicinities of
discontinuity points are not analyzed thoroughly, which complicates detecting
function discontinuity presence while analyzing its behavior on local continuous
sections. The paper [11] offers the characteristic function, created by analogy with
(2), but it is more adapted to solve the task of search for infinite discontinuities points.
The following characteristic function is offered to use:
R(i) ๏ฝ (d 2 f ( X ic ))2 Dir , (3)
where ๐๐๐ โ the centre of a cube, ๐ท๐ โ cube diagonal of search algorithm, r โ scaling
parameter.
The second differential of function can be calculated with the help of
interpolation of initial function using Newtonโs first interpolation formula [12] for full
factorial plan 3๐ [13]. Then, using ๐2,2,2 (๐) it is easy to calculate ๐ 2 ๐(๐) โ
๐ 2 ๐2,2,2 (๐).
In search algorithm, there is a proportional division of search area into smaller
parts: the unit interval โ into 2 parts; square โ into 4 parts, cube โ into 8 parts, etc. Fig.
1 shows a diagram of search area division for two-dimensional case. We will consider
the three-dimensional case further in order to present a good illustration.
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y
1
0
0 1 x
Fig. 1. โ Diagram of Search Area Division
Construction of interpolating polynomial for multidimensional case can be
realized as follows.
๐ = โ๐ง๐๐๐ โ(๐, ๐, ๐ = 0,1,2)
is the matrix of testing function values at nodal points of full factorial plan 3๐ . We
(0) (0) (0)
๐ฅ โ๐ฅ1 ๐ฅ โ๐ฅ2 ๐ฅ โ๐ฅ3
introduce new variables ๐ = 1 ;๐ = 2 ;๐ = 3 . The starting point of
โ โ โ
(0) (0) (0)
interpolation grid is ๐ (0) = (๐ฅ1 , ๐ฅ2 , ๐ฅ3 ; h
โ the step of interpolation grid.
Newtonโs interpolation polynomial is to be calculated in the form of matrix. Symbols
of multidimensional matrix and content of main operations are borrowed from the
paper [13]. To do this, we need to calculate the matrix of finite differences ๐ฬ.
z๏ก ๏1r z๏ก ๏2rr z๏ก ๏1q z๏ก ๏1qr๏ซ1 z๏ก ๏ซ2
๏1qrr z๏ก ๏2qq z๏ก ๏ซ1
๏2qqr z๏ก ๏ซ2
๏2qqrr z๏ก
~ 1๏ซ1 1๏ซ 2 1๏ซ1 1๏ซ1๏ซ1 1๏ซ1๏ซ 2 2 ๏ซ1 2 ๏ซ1๏ซ1 ๏ซ1๏ซ 2 (4)
Z ๏ฝ ๏1p z๏ก ๏ z
pr ๏ก ๏ z ๏ z
prr ๏ก qp ๏ก ๏ qpr ๏กz ๏ qprr ๏ก z ๏ z qqp ๏ก ๏ qqpr ๏ก z ๏2qqprr z๏ก
2 ๏ซ1 2๏ซ 2 1๏ซ 2 1๏ซ 2 ๏ซ1 1๏ซ 2 ๏ซ 2 2๏ซ 2 2 ๏ซ 2 ๏ซ1 ๏ซ 2๏ซ 2
๏2pp z๏ก ๏ z
ppr ๏ก ๏ z ๏ z
pprr ๏ก qpp ๏ก ๏ qppr ๏กz ๏ qpprr ๏กz ๏ z
qqpp ๏ก ๏qqppr ๏ก z ๏2qqpprr z๏ก
Here ฮฑ= 000. We introduce a vector
Q ๏ฝ (1 q q(q ๏ญ 1) / 2)๏ข;
P ๏ฝ (1 p p( p ๏ญ 1) / 2)๏ข; (5)
R ๏ฝ (1 r r (r ๏ญ 1) / 2)๏ข .
Considering (๏ฌ, ๏ญ) -convoluted product of vectors Q, P, R with ฮป = 0, ยต= 0, we get
the matrix of independent variables
๐ฬ = ๐๐๐
= โ๐๐ โ ๐๐ โ ๐๐ โ (๐, ๐, ๐ = 0,1,2 (6)
Thus, Newtonโs interpolation polynomial has the following matrix form
๏จ ๏ฉ ๏ฅ Z~
2
~ ~ ~
P2, 2, 2 ( X ) ๏ฝ3 Z X ๏ฝ c0c1c2 ๏ X c0c1c2 , (6)
c0 ,c1 ,c3 ๏ฝ0
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where c ๏ฝ (c0 , c1, c2 ) โ Caylean summation index [14].
The elements of matrix (4) can be calculated using the definition of finite
differences of corresponding orders.
To calculate partial derivatives we need vectors
๐โฒ = (0 1 ๐ โ 0.5)โฒ ; ๐ ; = (0 1 ๐ โ 0.5)โฒ ; ๐
โฒ = (0 1 ๐ โ 0.5)โฒ ; ๐โฒโฒ = ๐โฒโฒ =
๐
โฒโฒ = (0 0 1)โฒ ,
then
1 1 1
Px๏ข1 ( X ) ๏ฝ3 ( Z~ (Q๏ขPR )) ; Px๏ข2 ( X ) ๏ฝ3 ( Z~ (QP๏ขR)) ; Px๏ข3 ( X ) ๏ฝ3 ( Z~ (QPR๏ข)) ,
h h h
and the second derivatives:
1 1 1
Px๏ข๏ข1x1 ( X ) ๏ฝ3 ( Z~ (Q๏ข๏ขPR )) 2 ; Px๏ข๏ข2 x2 ( X ) ๏ฝ3 (Z~ (QP๏ข๏ขR)) 2 ; Px๏ข๏ข3 x3 ( X ) ๏ฝ3 ( Z~ (QPR๏ข๏ข)) 2 .
h h h
Now it is easy to calculate the value of the second differential in the centers of
each eight cubes of original search area partition:
d 2 f ( X ic ) ๏ป ( Px๏ข1๏ข x1 ( X ic ) ๏ซ Px๏ข๏ข2 x2 ( X ic ) ๏ซ Px๏ข๏ข3 x3 ( X ic ) ๏ซ 2( Px๏ข1๏ข x2 ( X ic ) ๏ซ Px๏ข1๏ข x3 ( X ic ) ๏ซ Px๏ข๏ข2 x3 ( X ic )))h2 .
The value of characteristic function is calculated for each of new cubes, which
are written onto the line ordered list in descending order of values. At each stage of
search algorithm for discontinuity points of testing function, the first item on the list is
chosen โ a cube with maximum value of characteristic function, which is subjected to
further division. The algorithm works till the condition of algorithm stoppage appears:
(max|๐ง๐ | > ๐๐ ๐ข๐ )ห
(๐๐๐๐ โ๐ < ๐). This condition of algorithm stoppage ensures the
completion of its work if the value of testing function is outside function domain or
the given density of viewing the original area of testing function is reached.
2. Examples of using the proposed method of Unit testing for computing modules
2.1. Method testing with model examples
The proposed testing method of CSM got its name as finite differences method
(FDM) due to the use of finite differences of the function. The efficiency of its work
can be evaluated by means of discontinuous function โ Kovartsevโs test [11],
specially developed for this case, and a set of test functions generated by the GKLS
generator [15]. The first test is characterized by a single discontinuity point (which is
difficult to detect) added to linear combination of error functions. The second one is a
continuous function with a large number of local extrema. Test functions are
presented in Table 1.
In literature, the efficiency of search algorithms is usually evaluated using the
operating characteristics machine [16]. Operating characteristic is the dependence of
error detection probability ๐๐๐๐ on the amount of calls to the tested function ๐๐ .
Since the second-order discontinuity points can be found by any of the global
optimization algorithms, the efficiency of the proposed FDM algorithm was compared
with the efficiency of direct GO method, for example, the modified bisection method
(BM) [17].
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Table 1. A set of test functions
โ Function General view
1 Kovartsev test function:
( x ๏ญa )2 ๏ซ( x2 ๏ญa2i )2 ( x ๏ญb )2 ๏ซ( x2 ๏ญb2 )2
19 ๏ญ 1 1i ๏ญ 1 1
f ( x1 , x2 ) ๏ฝ ๏ฅ (i ๏ซ 1)e 0.01 ๏ซ 1/(1 ๏ญ e 0.01 )
i ๏ฝ0
x1 , x 2 ๏ [0; 1]
20 local extrema. One second-order discontinuity
point
2 GKLS test functions.
Continuous twice differentiable function. 10 local
extrema. One global extremum. Points of
discontinuity are not observed.
The operating characteristics of FDM and BM methods are shown in Fig. 2.
They are created for test function 1 which has a local discontinuity point of the second
type (see Table 1). As we can see from fig. 2, the efficiency of the proposed algorithm
is much greater than the efficiency of the bisection method. In Fig. 2 the solid line
indicates operating characteristics of FDM algorithm, the dashed one indicates
characteristics of BM algorithm. It happens because the bisection method is focused on
the optimization of continuous functions, which leads to a more detailed analysis of the
function areas when Lipschitz constant evaluation increases. This situation occurs
every time when the function is calculated near the points of its discontinuity. By
contrast, FDM method is oriented on looking for areas of rapid growth of the test
function.
The situation changes if FDM method โworksโ with continuous function. Figure
3 shows the operating characteristics of these methods, created for continuous GKLS
test function.
The figure illustrates the fact that the efficiency of FDM algorithm for continuous
functions is much lower than the efficiency of BM algorithm. If we have continuous
functions with no discontinuity points of the second type (test software module has no
errors), the finite difference method is forced to examine thoroughly the space of the
optimized variables.
2.2. Testing of software modules for calculating acoustic characteristics of gas
pressure regulator
This part presents the results of Unit testing for computing models included in
the program that realizes the optimization of gas pressure regulator (GPR) parameters
with use of orifice plates [21].Significant changes in pressure during orificing and
speed acceleration generate the noise which accompanies the work of these machines.
This noise exceeds the established health standards. Rational choice of orifice flow
area (and their quantity) can significantly reduce the noise level of this device [18].
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Fig. 2. โ Operating characteristics of FDM and BM methods for function โ1
Fig. 3. โ Operating characteristics of FDM and BM methods for function โ2
Output parameters of GPR for the stationary case have been calculated by
solving the system of nonlinear equations that describe gas motion in its specific
sections: the valve mechanism and orifice package:
๏ฌGx ๏ญ G1 ๏ฝ 0,
๏ฏG ๏ญ G ๏ฝ 0,
๏ฏ 1 2
๏ญ (7)
๏ฏ....
๏ฏ๏ฎGn ๏ญ1 ๏ญ Gn ๏ฝ 0,
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๐บ๐ฅ โ gas flow through the valve, ๐บ๐ โ gas flow through i-orifice plate. When the
input parameters of module are given, for example, ๐๐ , ๐๐+1 โ pressure before and after
orifice plate correspondingly; ๐๐ โ areas of i-orifice flow, etc. we can calculate gas
mass flux through orifice and as a result โ acoustic power generated by orifice plate
[19].
We used a rather simple module for calculating the orifice flow capacity (๐ถ๐ฃ๐ ) in
order to calculate the gas mass flux through orifice. Testing the module by FDM
method with approximately 200 calls to the module revealed methodological error in
the algorithm model. It turned out that if ๐๐ โ ๐๐+1 โ ๐ถ๐ฃ๐ โ โ, and ๐๐ < ๐๐+1 , flow
capacity is indefinite (error code NaN occurs).
Certainly, it would be possible to require the calculation with this module
๐ถ๐ฃ๐ (๐๐ , ๐๐+1 ) to be carried out only in accordance with the condition ๐๐ > ๐๐+1 , which
is natural for this type of device. But how can it be realized when numerical method
for solving systems of nonlinear equations (8) generates the values of independent
variables at each iteration on its own, not taking into consideration the above
mentioned circumstances? The easiest way to solve the problem is the artificial
replacement of function infinite discontinuity ๐ถ๐ฃ๐ (๐๐ , ๐๐+1 ) with a larger but finite
discontinuity, and extension of definition by โpenaltyโ value where it is indefinite. In
this case, the algorithm for solving systems of nonlinear equations is to find solutions
on its own, โstarting withโ โdysfunctionalโ combinations of independent variables.
High speed of detecting fallible combinations of FDM independent variables in
this example can be explained by the fact that at the first stage of work, when the area
of rapid function growth is not detected, FDM distributes test points of testing
function in the search area. Since function ๐ถ๐ฃ๐ (๐๐ , ๐๐+1 ) has significant areas of
uncertainty, FDM finds them quickly.
Conclusion
The paper offers an original method of Unit testing for computing modules,
based on the algorithm of global search for infinite discontinuity of the testing
function, which allows to detect fatal errors in software computing modules, as well
as incorrectness in implementation of mathematical models of algorithms.
The proposed scheme of accelerating algorithms for global optimization applied
to the search for points of discontinuity of the second order has confirmed completely
its viability with model and real examples. The basic idea of FDM algorithm is to
introduce a new heuristic characteristic function to the classical algorithm of global
optimization. The new function is based on the analysis of Strongin characteristic
function and takes into account the problem matters being solved. FDM algorithm is
the universal method of Unit testing for the class of computing modules. The
application of this method leads to the reduction of time for debugging, helps to find
fatal errors with less effort, and, as a result, to organize total testing of all program
modules.
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