IntLinInc3D package User manual
Contents
1. Use Example 3 to input the data and to run the package 15 6 5 Empty set and whole space In this section we consider how one can conclude that the solution set is empty or it coincides with the whole space R after examination of the pictures and output produced by IntLinInc3D 6 5 1 Empty set Emptiness of the solution set H means that all its intersections po k 1 8 with separate orthants are empty Every polyhedron po does not contain straight lines therefore its emptiness is equivalent to the absence of vertices Overall the set H is empty if and only if it does not have orientation points When applied to the work of the package IntLinInc3D the above theoretical statement transforms to the following recommendation on how to recognize emptiness of the set H The solution set is empty if and only if the package produces no picture and outputs the message Number of orientation points 0 The solution set is empty The above recommendation does not depend on the specific values of the view arguments for which the launch function runs but relies on the fact that the package correctly understands the geometry of the solution set This is also true for the other recommendations of the manual Example 7 It is obvious that the equation 0 0 0 x 1 does not have solutions z R To see how the package IntLinInc3D processes it just type uc 000 oC uC ud 1 od ud Cxind3D uC oC ud od 0 12. 1 Example 10 The united solution set to the equation I 1 m m is comprised of eight points but for OrientPoints 0 it looks like the empty white box x oe 1 1 0 1 0 1 l 1 0 1 0 1 OrientPoints 0 OrientPoints l Use Example 3 to input the data and to run the package 19 What does empty yellow box mean We emphasize that the empty yellow box never means that the solution set is empty or that the solution set coincides with the whole space The empty yellow box at a picture indicates that the package has been called with the optional argument varargnin and the prescribed cut box from this argument lies entirely in the solution set The origin of coordinates in the empty yellow box may be marked as an orientation point when it belongs to the prescribed cut box and OrientPoints 1 Example 11 The solution set of the two sided inequality 5 lt x lt 5 includes the entire box 3 3 x 3 3 x 3 3 Let us input the initial data ue f 1 0 0 J7 oC uC ud 5 od 5 and call the function Cxind3D with the prescribed cut box 3 3 x 3 3 x 3 3 specified in varargnin Depending on the values of the argument OrientPoints we obtain the pictures x3 N o a N O x3 O ny o N w Cxind3sD uG 0 d 0d 0 0 3 3 378 3 3 4 CxindsD uC 0C ud 0d 17 0 3 3 3 3 3 3 CeindsD uC 0C 11d 00 051 53504 4505 5 53 5 Ce
3. is the whole space R with a star removed To obtain its picture input the commands 6 4 supA 1 2 2 2 1 2 2 2 1 infA supA 2 infb 2 ones 3 1 20 supb infb EqnCt13D infA supA infb supb 0 0 21 2 If the automatic cut is supplemented by transparency of the real faces i e transparency 1 this substantially extends the class of the solution sets for which one can determine from the picture either presence or absence of the cavity Example 13 Cube without mirror pyramid The united solution set to the system of interval equations 1 1 1 1 1 1 0 5 1 0 0 1 1 XC 0 1 0 1 1 0 0 1 1 1 is a cube with the mirror pyramid deleted In order to evaluate the influence of the argument transparency input the commands infA ones 1 3 eye 3 supA ones 1 3 eye 3 infb 5 ones 3 1 supb 5 ones 3 1 EqnWeak3D infA supA infb supb 0 0 Then change the last argument of the function EqnWeak3D from 0 to 1 and compare the pictures obtained transparency 0 transparency 1 22 3 Using the start values of all view arguments enables one to uniquely determine from the picture the presence or absence of the cavity for almost every solution set In particular the following rule may prove helpful if for the start values of the view arguments the origin of coordinates is marked as an orientation point at the picture then the so
4. with the automatic cut box are always transparent and have red color Unlike the prescribed cut the automatic cut does not require a preliminary work of the user and preserves information about global properties of the solu tion set Next we give recommendations that will provide the user with capability to distinguish between bounded and unbounded solution sets judging on the picture obtained by automatic cut 1 First of all we should note that the problem of recognition whether H is un bounded can be reduced to the problem of recognition whether the sets po are un bounded The set H is bounded if all poz k 1 2 8 are bounded If at least one of the components po is unbounded then H is unbounded too 2 If the set po has a red face then it is unbounded Example 2 The united solution set for the system of equations 1 1 0 0 1 0 1 1 0 c 1 0 0 1 1 1 is unbounded and consists of 8 trihedral angles To get the picture of this set input infA 1 0 0 0 10 00 1 supA 100 0 10 00 14 inba de ie J supb 1 1 1 EqnWeak3D infA supA infb supb 0 1 12 3 If the set po is bodily and all its faces have green color then the set po is bounded Example 3 The united solution set to the system of equations 1 1 0 0 0 1 1 0 0 0 Luj l 0 0 0 1 0 0 0 l is bounded and consists of 8 cubes To produce its picture input the commands infA 1 00 0 1 0 0 O 1 eye 3 J
5. 0 1 0 1 0 1 It is obvious from the pictures that every solution set except the mirror pyra mid has a cavity 29 7 How to install and operate the package IntLinInc3D 1 Download the file http interval ict nsc ru Programing MCodes IntLinInc3D zip 2 Unpack it into a separate directory folder 3 Set MATLAB paths to this directory 4 In MATLAB command window input the initial data for the sys tems 1 8 and call launch functions according to this manual and to IntLinInc2D User manual 8 References 1 I A SHARAYA Boundary interval method and visualization of polyhedral sets to appear in Reliable Computing 2 I A SHARAYA Quantifier free descriptions for interval quantifier linear systems Trudy Instituta Matematiki i Mekhaniki UrO RAN Proceedings of the Institute of Mathematics and Mechanics Ural Branch of the Russian Academy of Sciences 20 2014 No 2 pp 311 323 In Russian http interval ict nsc ru sharaya Papers trIMM14 pdf 3 S P SHARY A new technique in systems analysis under interval uncer tainty and ambiguity Reliable Computing 8 2002 No 5 pp 321 418 http interval ict nsc ru shary Papers ANewTech pdf 4 J ROHN Solvability of systems of interval linear equations and inequalities In Linear optimization problems with inexact data M Fiedler J Nedoma J Ramik J Rohn K Zimmermann New York Springer 2006 P 35 77 http interval ict nsc ru Library InteBooks Inexact
6. points while es the es ations of the real faces off is a matter of taste i Let us run the function EgnTol13D once again but without visualizing the orientation points and without transparency of the real faces gt gt EqnTol13D infA supA infb supb 0 0 The new picture looks like this gt At the picture the solution set resembles a cobblestone 4 Processing the picture To make a diamond of the cobblestone we use the standard Data Exploration Tools from MATLAB we rotate using tool Rotate 3D decrease the size using tool Zoom and turn on the light using Scene Light gt The obtained picture is suitable for saving as 2D image in eps pdf jpeg and such like formats Remark In the text below all pictures have been processed by MATLAB Data Exploration Tools after being outputted from IntLinInc3D package 10 6 3 Unbounded sets The present subsection is devoted to two tools of the package IntLinInc3D designed for creating adequate pictures of unbounded sets These tools are automatic cut and plotting orientation points 6 3 1 Automatic cut A standard trick that should be employed in the course of visualizing unbounded sets by computer systems is to prescribe the range of coordinates or in other words axis limits As the result we only see a part of the set bounded by the specified range The package IntLinInc3D is also able to visualize a part of the solutio
7. supA 100 O 10 00 1 eye 3 J infb 1 1 1 2 2 2 1 supb 1 1 1 2 2 2 EqnWeak3D infA supA infb supb 0 1 4 Specification of the optional argument varargin often prevents from de termining whether the solution set is un bounded Example 4 If in Examples 2 and 3 we take the prescribed cut box as 1 5 1 5 x 1 5 1 5 x 1 5 1 5 and call the package by the command EqnWeak3D infA supA infb supb 0 1 1 5 1 5 1 5 1 5 1 5 1 5 the pictures will be identical x3 13 6 3 2 Plotting orientation points In Examples 2 and 3 we intentionally assigned the value 0 to the argument OrientPoints in order to show that sometimes when e g all the nonempty sets po are bodily we can do without drawing orientation points In the general case only combination of the automatic cut with plotting orientation points produces a boundedness criterion for the set po The set po is bounded if and only if at the picture obtained with the automatic cut and plotting orientation points either the set po does not have edges or two vertices are marked at every its edge Example 5 The united solution set to the system of relations 1 1 0 0 1 0 1 1 0 1 0 0 1 1 Z 1 7 1 0 0 1 2 0 1 0 1 2 0 0 ji 2 consists of bounded a segment two squares a cube and unbounded pieces a ray two half strips and a semi infinite square prism One can get the picture using the command
8. 16 6 5 2 Whole space The solution set coincides with the whole space R if and only if the call of the package without the optional argument varargnin results in the output message Number of orientation points 1 and transparent red cube as a picture For OrientPoints 1 the origin of coordinates is marked as an orientation point The value of the argument transparency does not affect the picture x3 O nN o a N w x3 O no a o a N w x2 x1 OrientPoints 0 OrientPoints 1 Example 8 The united solution set to the interval inequality 1 1 1 1 1 1 z gt 0 is the whole space One can obtain its various pictures by inputting infA 1 1 1 sup 1 1 1 infb 0 supb 0 GeqWeak3D infA supA infb supb 0 0 and varying the values of the last two arguments of the function GeqWeak3D 17 6 5 3 And what is it There are two pictures that can be mistakenly interpreted as either empty solution set or a solution set coinciding with the whole space These are empty white box coordinate system without any objects and empty yellow box What does empty white box mean First of all one should bear in mind that the empty white box in no way means that the solution set is empty or that the solution set coincides with the whole space If at the picture you see a coordinate system that does not have visual ized objects then you should pay attention to th
9. IntLinInc3D package User manual Contents About the package Purpose structure Properties of visualized sets Notation in figures Recommendations on preparation and analysis of figures Ooo PON 6 1 Tools of preparation and analysis 6 2 How to make a diamond 6 3 Unbounded sets 6 4 Thin meager but important 6 5 Empty set and the whole space 6 6 How to test the presence of cavity 7 How to install and operate 8 References 1 About the package Necessary software is MATLAB The package implements the boundary intervals method 1 Author of IntLinInc3D and the boundary intervals method is Irene A Sharaya Institute of Computational Technologies SB RAS Novosibirsk The package IntLinInc3D is free software Its source codes are open Date of the first release is September 1 2014 The latest release is available from http interval ict nsc ru Programing and http interval ict nsc ru sharaya 2 Purpose The package IntLinInc3D is intended to visualize various solution sets for interval and point i e noninterval systems of relations These systems and solution sets are listed below Interval systems 1 the set of formal solutions for the interval inclusion Cx Cd 1 in Kaucher arithmetic where C C C KR is an interval matrix with given endpoints C and C x R is a real vector of unknowns d d d KR is an interval vector with given endpoints d and d
10. LP pdf Irene A Sharaya September 1 2014 Institute of Computational Technologies SB RAS Novosibirsk Russia 26
11. e arguments varargnin and OrientPoints The argument varargnin determines a prescribed cut box for the solution set Therefore when the package is called with the optional ar gument varargnin the picture contains only a part of the solution set within this box The arguments OrientPoints helps either displaying or hiding the orientation points of the solution set Depending on the specific values of these arguments there exist three ways of correct interpretation of the empty white 22 box 1 If the argument varargnin is present and OrientPoints 1 then the empty white box means that the completion of the solution set contains the entire prescribed cut box Example 9 In the Diamond example see Section 5 2 Example 1 let us specify the prescibed cut box 1 2 x 1 2 x 1 2 and set OrientPoints equal to 1 in the arguments of the function EgqnTo13D gt gt EqnTol3D infA supA infb supb 1 0 1 2 1 2 1 2 We get the picture gt oe 5 18 2 If the argument varargnin is present and OrientPoints 0 then set OrientPoints 1 and rerun the package to make sure that there exit or do not exist isolated points of the solution set in the prescribed cut box 3 If the argument varargnin is not present then the empty white box means that the argument OrientPoints 0 and the solution set consist of isolated points For correct visualization of the solution set one should call the package with OrientPoints
12. e empty set to 3 for bodily polyhedron The set H can have a cavity The definition and examples of the sets with cavities are in Section 6 6 A hyperplane P in R is named supporting hyperplane of a closed set M if P and M have at least one common point and M is contained in a closed half space bounded by P By support of a polyhedron M we call the intersection of M with any its supporting hyperplane It is obvious that supports of a polyhedron are polyhedrons too Each set po as well as every polyhedron in R can have supports whose dimensions are 0 1 or 2 The support of the dimension 0 is a vertez the support of the dimension 1 is an edge the support of the dimension 2 is a face of po We say that a point of R is an orientation point of H if it is a vertex of some poz k 1 2 8 The set H is empty if and only if it has no orientation points 5 Notation in figures To work with figures of the set H in R we introduce the following definitions A cut box is a box i e rectangular parallelepiped with edges parallel to the coordinate axes such that its intersection with the set H is to be visualized by the package IntLinInc3D There exist two types of cut boxes depending on the type of the cut automatic cut box is calculated by the package IntLinInc3D while prescribed cut box is inputted by the user Let us denote by po the intersection of the set po with the cut box A real face is a face of po ly
13. equalities and two sided inequalities Aajzt b Aa Ria ba E R m N U T0 Di2 lt A a Aa paare D2 E IR m E NU 0 8 Ae b 3 A 3 eR es Di3 E R m E NU 10 Dia SA rs bi A s E Re Dia Di5 ER m E Nv 0 with Mmi me M3 m gt 0 In 2 it is shown that each solution set listed above can be represented as the set of formal solutions to the inclusion 1 Therefore the visualization of this set play a key role in the package which is reflected in the title IntLinInc3D i e Interval Linear Inclusion The last letters 3D mean that the dimension of the unknowns is 3 x R Remark The package IntLinInc3D is aimed at illustrating simple exam ples in publications education etc so it works most correctly when the initial data are integers and lie in the range 107 10 3 Structure The main function of the package is Cxind3D It is designed to visualize the set of formal solutions for the inclusion 1 The functions used in the main one are AddV ChooseDrawingBox DrawHedrons Boundarylntervals ClearRows Intervals2 Path ChangeVariables ClearZeroRows NonRepeatRows The package contains auxiliary functions for the problems equivalent to 1 The choice of the auxiliary function depends on which of the systems 2 7 is to be processed and for interval systems on the solution type The names of the auxiliary functions reflect this dependency The name
14. igure arguments OrientPoints transparency or to view its fragment argument varargin 6 2 How to make a diamond Here we describe a typical way of obtaining a good picture of the solution set Example 1 Diamond We need to obtain a figure of the tolerable solution set for the system 3 5 0 2 0 2 1 1 0 2 3 5 0 2 z 1 1 0 2 0 2 3 5 1 1 Sequence of actions 1 Inputting the initial data for the system In this example we consider the system of equations 2 in which 3 5 0 0 25 2 2 A 0 35 0 A 2 35 2 b 1 d 0 0 35 gt 2 35 1 We input the data step by step gt gt infA 3 5 O 0O 03 50 00 8 5 gt gt supa 3 5 2 2 23 5 2 22235 gt gt inib s i i 1 1 gt gt supb 1 1 1 2 Calling the launch function with the start values of the view arguments To get a 3D draft picture that adequately represents the geometry of the solution set we call the auxiliary function EqnTo13D with the start values of the view arguments gt gt OrientPoints 1 gt gt transparency 1 gt gt EgnTol3D infA supA infb supb OrientPoints transparency We get Number of orientation points 27 3 Choosing the view arguments Analyzing the picture we can conclude that the solution set is bounded see Section 6 3 it does not have a cavity see Sec tion 6 6 and every nonempty set po is a bodily polyhedron Such solution sets are better viewed without orientation
15. ind3p uC oC ud od 1 1 3 3 3 3 3 3 In this example the empty yellow box looks very much like the picture of the whole space R and differs only in color 20 6 6 How to test the presence of cavity We shall speak that the solution set H has a cavity if the origin of coordi nates does not lie in H while every ray starting from the origin of coordinates intersects H Any cavity is a bounded polyhedral set It is not necessarily convex but it is always star convex with respect to the origin of coordinates for every point x from the set it also contains the segment 0 1 As a consequence the cavity is connected The origin of coordinates is an interior point of the cavity Figuratively speaking the cavity is a house for the origin of coordinates its walls built from real faces of the solution set and there are no doors and windows In this section we discuss how to tune the view arguments to test the presence of a cavity from the information provided by the picture 1 Using the automatic cut i e absence of the optional argument varargin even for any arbitrary values of the arguments OrientPoints and transparency makes it evident whether the cavity is present or not for a sufficiently wide class of the solution sets Example 12 Whole space without a star Controllable solution set to the system of interval equations I 1 1 2 2 2 2 2 1 1 r 2 2 2 2 2 1 1 2
16. ing on the boundary of the set H An automatic cut face is a face of po lying on the boundary of the automatic cut box A prescribed cut face is a face of po lying on the boundary of the prescribed cut box A face from orthant is a face of po arising from the intersection of H with the k th orthant A face from orthant lies neither on the boundary of the set H nor on the boundary of the cut box Notation in figures e is an orientation point is a real face 4 is an automatic cut face is a prescribed cut face Faces from orthants are not visible in figures of the set A 6 Recommendations on preparation and analysis of figures How to choose and run a launch function according to the system 1 8 and their solution types is explained in IntLinInc2D User manual This is why such explanation is omitted here We mention only that in the case of three unknowns the launch functions have input arguments OrientPoints transparency and varargin but do not have output arguments P1 P2 P3 P4 The package IntLinInc3D produces the following information about the so lution set H for users 1 messages in command window in particular the message about the number of orientation points 2 the list of orientation points as output argument of the launch functions 3 a figure of the solution set in a special window The number of orientation points and their list are objective geometric char acteristics of the solu
17. lution set has no cavity Example 14 Mirror pyramid The tolerable solution set for the interval equation 1 1 L 1 1 1 1 is a mirror pyramid infA ones 1 3 supA ones 1 3 infb 1 supb Ii EqnTol3D infA supA infb supb 1 1 Example 15 Boundary of mirror pyramid AE solution set to the system ae crap pis e Gwe a is the boundary of a mirror pyramid ee oe infA ones 2 3 Q O40 supA ones 2 3 nel infb 1 1 supb L 1 1d qe Aq A pP A PE PE OR 1 ba E E J EqnAEss3D infA supA Ag infb supb bgq 1 1 23 4 Finally we have the optional argument varargin that enables one to visualize intersection of the solution set H with any desired box The prescribed cut box determined by the argument varargin allows to see the intersection of the solution set with any separate orthant O To do this we 1 put one of the box vertices in the origin of the coordinates 2 put the opposite vertex of the box into the interior of the orthant O at such a distance that the box contains with suitable excess all orientation points from the orthants the information about orientation points of the solution set within an orthant can be extract from the picture produced for the start values of the view arguments or from the list of orientation points 3 assign the chosen box to the optional argument vara
18. m N is a natural positive integer number KR z Z z z R is the set of Kaucher intervals in contrast to the set of classical intervals IR z Z z z R z lt Z the requirement z lt Z is absent for Kaucher intervals KR z Z z z R is the set of Kaucher intervals over the extended real axis R RU o oo multiplication C by x is standard for Kaucher arithmetic the inclusion C is defined by inequalities Cz gt d and Cx lt d which are understood componentwise Cx and Cx are the left and right endpoints of the interval vector Cx Cx Cx respectively 2 all possible AE solution sets for the interval system of equations Ar b Ac IR be IR meN 2 3 all possible quantifier solution sets for the interval system of inequalities Ar gt b AeIR beIR men 3 Or Ar lt b AcIR BE IR meN 4 4 various quantifier solution sets for the interval mixed system of linear equations and inequalities Arcb AcIR BE IR co gt lt meN 5 specifically we mean all those solutions for which quantifier description has AE order of quantifiers for rows with the relation Point systems 1 the solution set for the system Az B lz gt c A BeER cE R meN 6 2 the solution set for the system Ar c lt Biz d A B e R c deR meN 7 3 the solution set for system of linear equations in
19. n set within a predefined coordinate range We refer to such a coordinate range as prescribed cut box It is defined by an additional argument varargin In pictures the faces that arise from the prescribed cut are transparent and have yellow color in contrast to real faces of the solution set that have green color and varied transparency The standard trick has two drawbacks 1 the user himself must carry out a preliminary analysis of the set under visualization and choose a suitable coordinate range 2 the picture within a predetermined coordinate range does not allow one to make definite conclusions on global properties of the set in particular on whether it is bounded or unbounded The advantage of the package IntLinInc3D in visualization of unbounded sets is that one does not need to specify the prescribed cut box If the pre scribed cut box is not inputted then the package IntLinInc3D itself chooses the coordinate range for drawing the solution set and produced picture con tains all the essential information about the solution set We refer to such a coordinate range as automatic cut box In case of automatic cut the package IntLinInc3D takes the following actions e finds the set of all orientation points e chooses the cut box wider than the interval hull of the set of orientation points e draws intersection of the solution set with the cut box 11 At the picture the faces arising from intersection of unbounded solution set
20. of R k 1 9 eH Oy A non empty set M C R is said to be bounded if there exists A R such that for each point x of the set M the distance from x to the origin is not greater than A The empty set is considered as bounded Each set po as H in whole may be bounded or not depending on the input data of the corresponding system 1 8 By polyhedron in R we call a subset of R that may be represented as a solution set to a system of linear inequalities Ax gt b Ae R ce R bE R mneN A polytope is a bounded polyhedron and a polyhedral set is a union of finite number of polyhedrons Note that the space R is a polyhedron while the empty set is a polytope All the sets po k 1 8 are polyhedrons The set H is a polyhedral set A set M is said to be connected if any two points from it can be joined by a path lying in M A connected component of the set is its connected subset that is maximal by inclusion Each po is connected because it is convex The set H may be connected or disconnected and can have up to 8 connected components H has 8 connected components if all poz k 1 8 are nonempty and pairwise disjoint Dimension of a nonempty polyhedron is dimension of its affine hull The dimension of the empty set is assumed to be 1 We call a polyhedron bodily if it has inner points and thin or meager otherwise The dimension of a separate piece po of the set H may be from 1 for th
21. rgin If a picture obtained for the start values of the view arguments does not allow one to definitely conclude whether there is is not a cavity in the solution set we recommend the following way of actions that will resolve this uncertainty In the launch function set OrientPoints 1 and vary the value of the optional argument varargin to look at the intersection of the solution set with each orthant The solution set H has a cavity if and only if for each k 1 8 at the picture of the intersection of the solution set H with the orthant O e the origin of coordinates is not marked as an orientation point e and every coordinate axis contains an orientation point Example 16 In Examples 12 15 every column of the matrix is symmetric with respect to zero i e is a balanced interval vector therefore the solution set is symmetric with respect to every coordinate plane To check whether such a set has a cavity or not it suffices to see its intersection with only one orthant For definiteness we take the positive orthant In the launch function for each of Examples 12 15 we set OrientPoints 1 and specify a suitable prescribed cut box varargin 24 Whole space without a star Cube without mirror pyramid EqnCt13D infA supA infb supb 1 0 0 3 0 3 0 3 EqnWeak3D infA supA infb supb 1 1 0 1 0 1 0 1 Mirror pyramid Boundary of mirror pyramid EqnTol3D CinfA supA intb supb 1 1 0 1 0 1 0 1 EqnAEss3D infA supA Ag infb supb bq 1 1
22. s infA eye 3 eye 3 supA eye 3 eye 3 infb 1 1 1 1 1 2 supb 1 1 17 2 2 2 relat ions 4S a a MixWeak3D infA supA infb supb relations 1 1 amp To sum up in order to determine boundedness of the solution set from its picture we have to permit the package i to choose the cut box automatically to achieve this it is sufficient just not to specify the argument varargin and ii to draw the orientation points assign 1 to the argument OrientPoints 14 6 4 Thin meager but important To ensure that at the picture created by the package IntLinInc3D thin polyhe drons po are shown correctly and unambiguously the argument OrientPoints must have the value 1 Otherwise we will not see isolated points that are sep arate connected components of the set H and also we will not be able to distinguish bounded meager sets po from unbounded ones a segment from a ray or a straight line a plane angle from a triangle and so on Example 6 The united solution set to the system of equations is composed of a point three segments three squares and a cube At the left picture the point is not visible and the segments and squares are depicted so that their boundedness is doubtful At the right hand picture all the compo nents of the solution set are depicted correctly and interpreted unequivocally l 0 1 0 l 1 i 0 0 OrientPoints 0 OrientPoints l
23. s of the auxiliary functions for the interval systems solution type as tolerable controllable strong quantifier 2 Ax b EqnWeak3D EqnTol3D EqnCt13D EqnStrong3D EqnAEss3D 3 Ar gt b GeqWeak3D GeqTo13D GeqCt13D GegStrong3D GeqQtr3D U Ar lt b Leqieak30 LeqTo130 Leqotiab LeqStronga LeqitraD G Aro b wixveaxap wixro1ap MixCtiap wixSerongaD wixgeraD The solution types from the table above except the quantifier type are de fined in IntLinInc2D User manual A complete set of definitions is in 2 and it generalizes the terminology from 3 4 Note that the auxiliary functions EqnAEss3D and MixQtr3D are designed only for such quantifier solutions which 66 29 have AE order of quantifiers in rows with the relation For point systems there are two auxiliary functions the function Abs13D is intended for the system 6 with one absolute value operation the function Abs23D is designed for the system 7 which contains two such operations We shall refer to the main and auxiliary functions of the package as launch functions Arguments of the launch functions are described in comments within their bodies To see these descriptions in MATLAB command window use command help for example gt gt help EqnWeak3D 4 Properties of visualized sets Let us denote by H the visualized solution set and by po piece in k th orthant the intersection of H with the k th orthant
24. tion set They do not depend on what part of the solution set 1s visualized in the figure We do not explain how to get the list of orien tation points because this is standard MATLAB way of accessing the output arguments The situation with figures is much more complicated In the case of two un knowns the figure produced by the package IntLinInc2D gives full and explicit information about the geometric structure of the solution set On the contrary in the case of three unknowns the figure produced by the package IntLinInc3D is only a rough draft for analysis of the geometric structure of the solution set and for getting the final figure If someone wants e to make the rough draft properly e to analyse the structure of the solution set relying on the rough draft e and to prepare a final figure that gives exact representation of the geometric structure of the solution set and is suitable for saving as 2D image he should use not only standard MATLAB tools but a special knowledge about the package IntLinInc3D too Therefore recommendations on preparation and analysis of figures in IntLinInc3D package take the greater part of this User manual The most important of them are marked with P 6 1 Tools for preparation and analysis of figures A user of IntLinInc3D package has two toolkits for preparation and analysis of figures e Data Exploration Tools of MATLAB Zoom Rotate 3D Scene Light etc e view arguments of the launch f
25. unctions in IntLinInc3D package which include input arguments OrientPoints and transparency as well as optional input argument varargin The input argument OrientPoints is a parameter that controls drawing of the orientation points if it has the value 0 the program does not plot these points if it has the value 1 the orientation points are plotted The input argument transparency is a parameter responsible for the trans parency of the real faces its values may be 0 or 1 0 means the absence of the transparency 1 means that the real faces are transparent The optional input argument varargin allows a user to prescribe the cut box This argument is inputted as 6 numbers denoted as xb xe yb ye zb ze If such six numbers are present at the list of input arguments of the launch function then the cut box is xb xe x yb ye x zb ze otherwise the package IntLinInc3D itself calculates the cut box At the first visualization of the solution set it is necessary to assign the following values to the view arguments of the launch function gt gt OrientPoints E gt gt transparency 1 the optional input argument varargin is absent These are the start values of the view arguments In the general case only these values allow to receive full and explicit information about the geometric structure of the solution set At the subsequent calls of the launch functions you can change view arguments to get more good looking f
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