barrido sistemático de , se construye una "Celda Cartesiana Miranda", en la ... Celdas Cartesianas de Poincaré, Método iterativo de barrido de detección de ...
Received: 27 02 2015 Accepted: 12 03 2015 Published: 30 04 2015 © 2015 CIRES
Adv. Mat. Sci. & Tech. Vol 9 Nº 1-3 Art 1 pp 1–20, 2015 ISSN 1316–2012 Depósito Legal pp96–0071
A MULTISECTION AND SWEEPING METHOD FOR SOLVING NONSMOOTH EQUATIONS GUILLERMO MIRANDA1, LARRY MENDOZA2 & REMIGIO MEDRANO3
1,2,3
1 Escuela de Matemática, Facultad de Ciencias, Universidad Central de Venezuela, Caracas, Venezuela. Línea de Investigación Matemática Aplicada, Departamento de Ciencias Básicas, Universidad Nacional Experimental Politécnica "Antonio José de Sucre", Vicerrectorado "Luis Caballero Mejías"; Caracas, Venezuela.
ABSTRACT A new multisection and sweeping numerical method is presented for solving the equation where only continuity is required for the mapping of an open simply connected subset of into . The bidimensional case is worked out in detail, and instead of "Poincaré Cartesian Cells" ("Poincaré-Rectangles" in we define "confluent" subdomains, where the scalar components of exhibit a constant sign. Once these confluent subdomains are detected by a systematic sweeping of , a "Miranda Cartesian Cell" where a solution " " is granted to exist, is constructed, and starting from it, a multisection numerical method (Quadrisection in ), which generalizes the bisection method based on Bolzano’s Theorem, is applied to successively approximate " ". Key Words: Nonsmooth equations, Non differentiable operators, Quadrisection, Numerical approximation, Poincaré Cartesian Cells, Iterative sweeping sign detection method.
UN MÉTODO DE BARRIDO Y MULTISECCIÓN PARA RESOLVER ECUACIONES NO SUAVES. RESUMEN Un nuevo método numérico de multisección y barrido se presenta para resolver la ecuación en la cual sólo se requiere continuidad para la aplicación de un subconjunto abierto simplemente conexo de en . El caso bidimensional es trabajado en detalle, y en lugar de "Celdas Cartesianas Poincaré" (Rectángulos Poincaré en ), definimos subdominios "confluentes", en los cuales las componentes escalares de muestran un signo constante. Una vez que estos subdominios confluentes son detectados mediante un barrido sistemático de , se construye una "Celda Cartesiana Miranda", en la cual se garantiza la existencia de una solución " ", y a partir de ella, un método numérico de multisección (Cuadrisección en ), el cual generaliza al método de bisección basado en el Teorema de Bolzano, se aplica para aproximar " " sucesivamente. Palabras Claves: Ecuaciones no suaves, Operadores no diferenciables, Cuadrisección, Aproximación numérica, Celdas Cartesianas de Poincaré, Método iterativo de barrido de detección de signos.
FICHA GUILLERMO MIRANDA, LARRY MENDOZA & REMIGIO MEDRANO, 2015.- A MULTISECTION AND SWEEPING METHOD FOR SOLVING NONSMOOTH EQUATIONS. Adv. Mat. Sci. & Tech. 9(1-3): 1-20. Phy. Div. ISSN 1316–2012.
A MULTISECTION AND SWEEPING METHOD FOR SOLVING NONSMOOTH EQUATIONS
Poincaré’s Theorem, and which can be stated as follows:
1. INTRODUCTION One of the classic problems in numerical analysis is to find suitable methods to approximate the values of " ", which satisfy the equation , since it is usually impossible to find explicit solutions because these problems result from nonlinear equations, which can be algebraic, transcendental, or combinations of both.
"If is continuous in an open set Ω and write , and if can it be found a rectangle (to be called "Poincaré rectangle") such that and keep a constant sign on the edges of in the form: ,
These methods become a powerful systematic strategy to solve many problems that frequently appear in various fields of science and engineering, in which you need to program in computer packages based on iterative procedures.
and
then = (0, 0) has a solution
More specifically, consider the equation ⃗ , where is a continuous but not necessarily differentiable function defined on Ω open and simply connected domain in . The existence of a solution ⃗ has been the object of much well-known theoretical research as well as development of numerical methods to approximate such solutions (see [1], [2], [3], [4]). In the special case of differentiability of , this has been carried out with iterative numerical methods such as Newton’s, the secant method and its modifications ([5] and [6]).
".
This Poincaré’s Theorem was generalized in 1940 by C. Miranda (see [9]). If a quadrisection is applied to , generally another "Poincaré rectangle" whith ¼th of the size of can be found, and then develop a "quadrisection method" generating a sequence of "Poincaré rectangles" of decreasing size in order to find a solution in the two-dimensional case. Instead of generalizing Poincaré’s Theorem for 3-D cells in Cartesian meshes with constant and alternating signs of the scalar functions , , (defining a 3-D system of equations) on opposite faces of the 3-D cell, in this work we generalize the bisection method for dimensions bigger than one, based on Bolzano’s Theorem, and in a procedure for sign investigation, which is a sweeping method not limited to detecting sign changes in the pairs of opposite cell boundary faces or sides in Cartesian meshes, cells for which no methodological indication has been given to find them, and thus we shall speak of a multisection method for any dimension, preceded by a sweeping method that does not depend on the possibility of finding "Poincaré’s cells", but only in finding "confluent" domains, to be defined in what follows.
The case assuming only continuity of , was first approached by Bolzano in 1817 (see [7]) in the onedimensional case . When the hypotheses of Bolzano’s existence Theorem are verified, i.e. when you can find an interval , such that (we'll call "Bolzano interval" such ), then the "bisection method" applied to is an iterative method which generates a succession of midpoints that are approximate solutions that converge to a solution of . While studying the "Three body problem" of Celestial Mechanics in 1883 (see [8]), Poincaré obtained a two-dimensional generalization ( ) of Bolzano’s theorem, which we shall call 2
GUILLERMO MIRANDA, LARRY MENDOZA & REMIGIO MEDRANO
2. SOLVING NONSMOOTH EQUATIONS BY SUCCESSIVE APPROXIMATIONS.
3. A SCALAR EQUATION FOR A REAL UNKNOWN "x".
While the ideas presented in this article naturally apply to any dimension "n", to illustrate then in full detail, we shall exemplify the case n = 2, when the multi-section is reduced to a quadrisection.
Suppose the equation to be solved is of the form f(x) = 0, f real and continuous in a closed interval [i0, d0] in which f changes sign at the endpoints, that is, such that:
Next, a quadrisection method is presented, which through a preliminary "sweeping" method detects "confluence subdomains", assuring the determination of an initial rectangle in which there is a solution of the system, and then, starting from that initial rectangle, a generalization of the bisection method is applied, i.e., a quadrisection method, which, as is well known, has the advantage over other conventional methods, involving differentiability or analyticity, to assume only continuity for solving systems of nonlinear scalar equations which are nonsmooth or piecewise smooth, such as often occurs in models used in physics or engineering. Various numerical examples are initially presented to illustrate the proposed method, in which explicit solution for the system are available, and in this mammer appreciate the method’s precision, which can be estimated from the size of the initial rectangle, even before the start of the iterations associated with a quadrisection.
Case I)
Figure 1. Case I) The point is
(1) Then, there exists at least one point such that . (This is Bolzano’s Theorem, which is a consequence of the Intermediate Value Theorem for Continuous Functions). This theorem guarantees the existence of an , but does not indicate how to find it, and for this purpose we shall remember the Bisection Method, which does not need the existence of . We look for the sign of at the midpoint of the initial interval, which we call . midpoint
with this midpoint, the length of the interval [i0, d0] is divided into two equal parts, forming two subintervals and . Generating three possible cases. This is shown graphically in Figure 1.
Case II)
root of root of
(2)
Case III)
; Case II) The root of , is in the interval [
3
, is in the interval ].
Case III) The
A MULTISECTION AND SWEEPING METHOD FOR SOLVING NONSMOOTH EQUATIONS
Case I: If and equation is solved, because of
which implies to take , and , this corresponds to the second midpoint, and with this point a second interval , half the initial one, is built, and then there are again three possible subcases A), B) and C) as in the case II).
, then the is a zero or root
If , then you have to test the hypothesis (1) of Bolzano’s Theorem, for both intervals and , generating the following cases:
Note 1. In both subcases B) and C) of cases II) and III), we take , which is the third midpoint.
Case II: If on the interval , then it verifies the hypothesis (1) of Bolzano’s Theorem, which implies to take , and , this corresponds to the second midpoint, and with this point a new second interval , half the initial one is built, and then, there are three possible sub cases A), B) y C), forming two subintervals and , which is illustrated in Figure 2.
With the midpoints, a finite sequence of points is obtained, which is formed by approximate solutions in the intervals . That is, each approximate solution belongs to a closed interval,
,
Figure 2. Case A) The point is
Case B)
root of of
(3)
where each of these closed intervals are of decreasing length halves. .
Case III: If in the interval , then the hypothesis (1) of Bolzano’s theorem is verified in the interval ,
Case A)
,…,
Case C)
; Case B) The root of , is in the interval
4
, is in the interval
; Case C) The root
GUILLERMO MIRANDA, LARRY MENDOZA & REMIGIO MEDRANO
This process is performed iteratively until the point , obtained by numerical bisection, approximately solve the equation, i.e., , with a preset accuracy, which determines the number " " necessary to satisfy it.
Suppose there is an open and simply connected domain in the plane X–Y, not necessarily convex, in which and are continuous and in which there is a single solution . If the equation describes a curve and equation , describes a curve in the Cartesian plane, then suppose that divides in two open subdomains and , such that if then and if ( ) , then . Suppose also divides ( ) into two open subdomains and , such that if ( ) then and if then , as can be seen in Figure 3.
4. TWO REAL SCALAR EQUATIONS FOR TWO UNKNOWNS "x" AND "y". Suppose the system of two equations to be solved is of the form: {
(4)
Figure 3.1. Curve
Figure 3.2. Curve Figure 3. Curves
Both curves intersection of
and and
divide the region ( is the solution point
and
) into 4 open subdomains (A, B, C, D) and the of system (4). This can be seen in Figure 4.
Figure 4. Intersection of Curves
5
and
.
A MULTISECTION AND SWEEPING METHOD FOR SOLVING NONSMOOTH EQUATIONS
Note 2. In this first example, which can be solved in a trivial way, we proceed as if we did not know how to solve algebraic equations.
The four open subdomains are denoted by: A=
and
B=
and
C=
and
D=
and
;
To solve the system numerically (for convenience and ease initially) define a square mesh of square cells (196 cells ⎕) and points (225 points), that is, on the axes of abscissae and ordinates we chose for the square a length of 7 units, ), and if we establish a spacing of 0,5 units on both axes , we can then construct a mesh in the Cartesian plane, sufficiently fine for detecting subdomains A, B, C and D, as and curves are easily noticeable in such a case.
;
The proposed idea in this paper to initially determine a rectangle in which is granted the existence of at least one solution , consists in detecting, through a systematic sweep, a region where "confluence" of subdomains B and D or also A and C is secured (see Figure 4.). Example 1. Given the system of two algebraic
equations: {
In addition, for the study of signs in the mesh, a code of symbols to denote the 4 regions that are generated when and intersect, so as to indicate the symbol of on the grid, will be established as outlined below:
(5)
find the two solutions of the system, using the sweeping and quadrisection method. Properties of
Symbols
in the mesh
Representation of:
If
and
If
and
:
Different signs.
If
and
:
Equal negative signs.
If
and
:
Different signs.
If
If
or
Equal positive signs.
The point is on a curve.
:
The point is a solution of the system.
:
Now we proceed to make the tabulation of the symbols and signs for the mesh starting from point (0,0) and varying the value of the abscissa 0.5 U at a time, and fix the ordinate, so that the trip is done horizontally, and then increase the point ordinate by 0.5 U, and so on, up to the full sweep of the mesh, and the signs obtained will indicate when the functions and change sign.
As usual, we will construct the points on the grid by means of the Cartesian Product , where: X = [0, 0.5, 1, 1.5, ... , 7] = Y, so that the abscissae are of the form , , and the ordinates also vary from 0 to 7 with a step size of 0.5.
6
GUILLERMO MIRANDA, LARRY MENDOZA & REMIGIO MEDRANO
N°
x
y
F1
F2
N°
x
y
F1
F2
N°
x
y
F1
F2
1 2 3 4
0.0 0.5 1.0 1.5
0.0 0.0 0.0 0.0
– – – –
+ + + +
46 47 48 49
0.0 0.5 1.0 1.5
1.5 1.5 1.5 1.5
– – – –
– –
+ +
50 51
2.0 2.5
1.5 1.5
– –
– – – – –
91 92 93 94
0.0 0.5 1.0 1.5
3.0 3.0 3.0 3.0
– – – –
– – – –
5 6
2.0 2.5
0.0 0.0
+
95 2.0 3.0 96 2.5 3.0
– –
– –
7
3.0
8 9
3.5 4.0
0.0
–
+
52
3.0
1.5
0.0 0.0
+ +
53 54
3.5 4.0
1.5 1.5
0.0
– – –
–
+
97 3.0 3.0
+ +
98 3.5 3.0 99 4.0 3.0
– –
– –
+
+
1.5
– – –
10
4.5
+
55
4.5
+
100 4.5 3.0
+
+
11 12 13 14 15
5.0
0.0
+
+
56
5.5 6.0 6.5 7.0
0.0 0.0 0.0 0.0
+ + + +
+ + + +
57 58 59 60
5.0
1.5
+
+
101 5.0 3.0
+
+
5.5 6.0 6.5 7.0
1.5 1.5 1.5 1.5
+ + + +
+ + + +
102 103 104 105
5.5 6.0 6.5 7.0
3.0 3.0 3.0 3.0
+ + + +
+ + + +
16 17 18 19 20
0.0 0.5 1.0 1.5 2.0
0.5 0.5 0.5 0.5 0.5
– – – – –
+ + + + +
61 62 63 64 65
0.0 0.5 1.0 1.5 2.0
2.0 2.0 2.0 2.0 2.0
– – – – –
106 107 108 109 110
0.0 0.5 1.0 1.5 2.0
3.5 3.5 3.5 3.5 3.5
– – – – –
– – – – –
+ +
66 67
2.5 3.0
2.0 2.0
– –
– – – – – –
21 22
2.5 3.0
0.5 0.5
– –
+
111 2.5 3.5 112 3.0 3.5
– –
–
+
68
3.5
2.0
–
+
113 3.5 3.5
– – –
23
3.5
0.5
24
4.0
0.5
+
69
4.0
2.0
114 4.0 3.5
+
– –
+
70
4.5
2.0
– –
+
0.5
– –
25
4.5
26
5.0
+
115 4.5 3.5
+
+
0.5
+
+
71
5.0
2.0
+
+
116 5.0 3.5
+
+
27 28 29 30
5.5 6.0 6.5 7.0
0.5 0.5 0.5 0.5
+ + + +
+ + + +
72 73 74 75
5.5 6.0 6.5 7.0
2.0 2.0 2.0 2.0
+ + + +
+ + + +
117 118 119 120
5.5 6.0 6.5 7.0
3.5 3.5 3.5 3.5
+ + + +
+ + + +
31 32 33 34 35
0.0 0.5 1.0 1.5 2.0
1.0 1.0 1.0 1.0 1.0
– – – – –
+ + + + +
76 77 78 79 80
0.0 0.5 1.0 1.5 2.0
2.5 2.5 2.5 2.5 2.5
– – – – –
– – – – –
121 122 123 124 125
0.0 0.5 1.0 1.5 2.0
4.0 4.0 4.0 4.0 4.0
– – – – –
36 37
2.5 3.0
1.0 1.0
– –
+ +
81 82
2.5 3.0
2.5 2.5
– –
126 2.5 4.0 127 3.0 4.0
+
– –
38
3.5
1.0
–
+
83
3.5
2.5
128 3.5 4.0
+
–
39
4.0
1.0
+
84
4.0
2.5
+
129 4.0 4.0
+
40
4.5
1.0
– –
– –
– – –
– – – – – –
+
85
4.5
2.5
+
+
130 4.5 4.0
+
– –
41
5.0
1.0
+
+
86
5.0
2.5
+
+
131 5.0 4.0
+
+
42 43 44 45
5.5 6.0 6.5 7.0
1.0 1.0 1.0 1.0
+ + + +
+ + + +
87 88 89 90
5.5 6.0 6.5 7.0
2.5 2.5 2.5 2.5
+ + + +
+ + + +
132 133 134 135
+ + + +
+ + + +
Symbols
7
Symbols
5.5 6.0 6.5 7.0
4.0 4.0 4.0 4.0
Symbols
A MULTISECTION AND SWEEPING METHOD FOR SOLVING NONSMOOTH EQUATIONS
N° 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165
x 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
y 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5
F1 – – – – –
5.5 6.0 6.5 7.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
4.5 4.5 4.5 4.5 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0
+ + + + + + + + + + + + + + + + + + +
+ + + + + +
F2 – – – – – – – – – – + + + + + – – – – – – – – – – + + + + +
Symbols
N° 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195
x 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
y 5.5 5.5 5.5 5.5 5.5 5.5 5.5 5.5 5.5 5.5 5.5
F1 + + + + + + + + + + +
F2 – – – – – – – – – –
5.5 6.0 6.5 7.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
5.5 5.5 5.5 5.5 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0
+ + + + + + + + + + + + + + + + + + +
+ + + + – – – – – – – – – –
Observations 1. When the above table is constructed, there is a study of signs for the values and according to the established code symbols to denote the regions or subdomains resulting when and intersect. To find possible solutions you start by finding the evaluated sign changes in each such function. For example, for the sign changes in items No. 10-11, 25-26, 40-41 and so on; for the sign changes in the items No. 50-51, 66-67, 83-84, and so on. Thus we proceed to identify all possible sign changes in the values of both functions. The solution is achieved
+
+ + + + +
Symbols
N° 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225
x 0.0 0.5 10 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
y 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5 6.5
F1 + + + + + + + + + + +
F2 – – – – – – – – – –
5.5 6.0 6.5 7.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
6.5 6.5 6.5 6.5 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0 7.0
+ + + + + + + + + + + + + + + + + + +
+ + + + – – – – – – – – – –
Symbols
+
+ + + + +
when both evaluated functions change sign simultaneously, corresponding to ítems No. 98, and 99 in the table above (full dot). Based on the above displayed table, we proceed to plot symbols associated with the various combinations of signs for and , with their respective sub-mesh in each of the subdomains that result when and curves associated with the system (5) intersect. See Figure 5. For brevity, in this example we shall only seek to detect a solution for . .
8
GUILLERMO MIRANDA, LARRY MENDOZA & REMIGIO MEDRANO
Figure 5. Graphic of 14 × 14 square mesh cells for System (5), with the sub-meshes associated to different symbols.
The previous mesh was favorable to discern points belonging to and curves, and even solution points of the system. What if the scales on the axes are different? Now, consider a rectangular grid of , rectangular cells (140 cells ⎕) and points (165 points), that is, in the horizontal axis we chose a length of 5 units and in the ordinate-
axis
we
chose
a
length of 7 units ), and establish a spacing of 0.5 units for the abscissa-axis and 0.7 units for the ordinate-axis and After doing a computation analogous to the one done for the above fine mesh, the sub-mesh symbols shown in Figure 6. are obtained. .
.
Figure 6. Graphic of 10 × 14 rectangular mesh cells for the system (5) and its sub-meshes.
9
A MULTISECTION AND SWEEPING METHOD FOR SOLVING NONSMOOTH EQUATIONS
In Figure 5. we can see the mesh defined by square mesh cells for System (5), implying that the mesh fits very well to the curves of the system (5) represented by and , causing some grid points match some points on the curve such as (5,0), (3,2), (3,4) and (5,6), and also a point coinciding exactly with solution (4,3) of system. In Figure 6., however, we note that the grid defined by rectangular mesh cells, does not fit well with the curves and for system (5); having only one point (5,0) on the curve and other points are very close to the solution of the system (5), but do not match it.
Example 1 only used algebraic functions and , case when the system could be solved explicitly using radicals, but the goal was then only to illustrate the graphical detection of confluences of subdomains, and its dependence upon the relative coarseness of the mesh employed. Example 2. Consider the system:
{
(6)
To solve the system, which now includes transcendental functions in and , we must remember that a phase shift of in the argument does not change the value of the tangent function, so that,
This analysis shows that the problem of detecting the confluence of sub-domains will be easier or more difficult, depending on how fine the mesh is constructed for the solution of the problem.
(7)
It follows that describes the curve
when the point defined by equation
Also .
, that is, the straight line
Furthermore, .
We see that
.
because: In Figure 7., we can see three of the infinitely many exact solutions:
. Moreover,
:
,
10
and
.
GUILLERMO MIRANDA, LARRY MENDOZA & REMIGIO MEDRANO
A) Solution
.
B) Solution
C) Solution Figure 7. Subdomains
in
and
A)
.
This equation has the obvious solution with . If
With
.
and three of the infinitely many exact solutions of the system Solution ; B) Solution and C) Solution
All exact solutions are obtained by solving first , i.e.,
.
, there is the obvious solution , and substituting the value of , , we obtain:
, then
, with
.
If we did not have the exact solutions with , then we would have to develop a strategy of successive approximations which, imitating the bisection method, goes "enclosing" the solution point, but prior to that, it is necessary to detect confluence of subdomains B and D, etc. .
. . 11
A MULTISECTION AND SWEEPING METHOD FOR SOLVING NONSMOOTH EQUATIONS
Searching to generate sign values for , and avoiding, we
,
with
and first
(
essay
)
and so the moved point
would be in the region
; then we generate at the point
(see Figure 8.).
evaluations of both functions there and we have:
Figure 8.
Imitating the horizontal sweeping employed in the previous example, if we let remain fixed, to
shift in
(
you reach to
). With (
(
)
falls on
,
such that
,
(
) which
, then it would be an
(
, and before reaching point , we must go through an )
is, it must be
and
(
)
, that
, with small positive ,
and then we would have obtained a point belonging to subdomain
, and for
will not have changed sign, but
, since
canceling with from
) or a sign
, say
(
But reaching a point such as
then move horizontally, we see that if tends to the right, then adds more negativity to so that has to be moved horizontally to the left to find a sign shift in
point location.
) will:
(8) In what follows, we will call yellow region or sub domain, any of the subdomains B or D, also and , will be called green and blue regions respectively. Then the point
Which is a yellow subdomain, and we have moved from in a green region to in a yellow región (see Figure 9.).
Figure 9.
12
point location.
GUILLERMO MIRANDA, LARRY MENDOZA & REMIGIO MEDRANO
It is convenient that
be large enough so that ,
be slightly larger than , and then, moving from to the right, we change back from the blue region to the second yellow región , characterized by (the first yellow region is characterized by ), seeking a confluence of subdomains B and D (see Figure 11.). Figure 10.
(
If from the point ( change
point location.
) we move to
) with " " variable, seeking to color,
región of
we
see
that
if
,
son
and returns to the green , so " " must be greater than in to
change from yellow region to blue región will happen when,
, what
Figure 11.
let's say
Now we choose point
locating points.
:
, with small positive . with positive but such that does not reach the green region, and from descend vertically to reach from the yellow to the green region where we started, to form a cycle with
As is a second coordinate perturbation, being the first perturbation, but only in coordinate " ", now we shall consider a positive perturbation in coordenate " " for , but also a positive perturbation in coordinate " " for .
, such that
Definition:
With these coordinate perturbations, blue región (see Figure 10.).
is a rectangle.
The rectangle , is assured to contain its interior the solution point , located at the confluence of the 4 subdomains or colored regions (see Figure 12. A.). Call "Miranda rectangle" to one such as containing a confluence of subdomains B and D or also A and C.
belongs to a
13
A MULTISECTION AND SWEEPING METHOD FOR SOLVING NONSMOOTH EQUATIONS
Now one can apply the proposed method of quadrisection to said rectangle, so that at least ¼th of rectangle always contains two yellow subdomains. Thus we see that only the upper right rectangle (of the four rectangles resulting from quadrisection) contains two yellow subdomains (see Figure 12.B.), and now this "Miranda rectangle" is
A)
subjected to quadrisection again, since this is one of the four resultant rectangles which also encloses solution point by construction, being a point of confluence of both yellow subdomains. With each quadrisection, the coordinates of are being left locked in closed intervals of decreasing length.
B) Applying the method of quadrisection.
locating points..
Figure 12. Construction of the "Miranda rectangle".
The ordinate of the horizontal line which halves has the value:
and
,
having defined the subdomains according to the following table of signs: A B C D
The term being less than is the result, with not too big a choice of , which assures that this first quarter of rectangle contains .
In the three dimensional case, a solution point can be interpreted as the intersection of a curve with a surface associated with equation being the intersection of two other and surfaces, associated with equations and respectively, or as the intersection of three curves , and , each of which is the intersection of two surfaces:
5. THREE–DIMENSIONAL EXTENSION In the two-dimensional case, a solution point could be interpreted as the intersection of two curves and , which generated four subdomains A, B, C and D used to study the convergence of a sequence of decreasing rectangles towards a solution of the system of two equations: 14
GUILLERMO MIRANDA, LARRY MENDOZA & REMIGIO MEDRANO
,
and
The article by Hernandez and Rubio (see [11]) considered non differentiable equations, but requires convex domains, and in our work this requirement is removed.
.
For the confluence subdomains to be used in this three-dimensional case, one can define eight subdomains A, B, C, D, E, F, G, H:
Martinez and Qi (see [12]) required locally Lipschitz functions, while our method only requires continuity.
A B C D E F G H
Bertsekar (see [13]) considers the optimization point of view for systems.
APPENDIX I.
With this designation, we proceed similarly to detect the appropriate confluence of subdomains, similarly to what was done in the two-dimensional case. However, since only signs and not numerical values are needed, the sweeping is not as costly as it would seem at first sight. The efficiency, understood as defined in the article by Jaiswal (see [10]), for this method of sweeping and multi-section, will be the subject of further study.
We want to illustrate with simple examples the difficulties you might encounter for detection of a "Poincaré cell". Consider the case , and the simplest case where the two curves and separating subdomains where and exhibit a constant and different sign, are simply two intersecting straight lines forming an angle . By varying the angle we can appreciate the effect that the angle has in the size of a possible "Poincaré rectangle", since by reducing the angle, these rectangles should shrink.
6. CONCLUSIONS
To better illustrate the feasibility of detecting a "Poincaré rectangle" enclosing a point of intersection of two curves and , we will take a few simple examples, including the case where both curves are simply straight, and for better viewing, we use a color code based on green and red, so that the parallel sides on which the sign of the function associated with the curve changes, are colored green, and the parallel sides on which the sign of the function associated with the curve changes, are colored red. Despite its simplicity, using rectilinear curves Γ can illustrate the difficulty in locating a "Poincaré rectangle", which encloses the point of intersection, as exhibited in the last two drawings of Figure 13. below.
The sweeping method to detect confluence of subdomains that have specific signs for and , is a generalization of the investigation of signs in the Theorem of Poincaré-Miranda (see references [8] and [9]). This generalization is important because although the subdomains that are swept to investigate evidence of signs for functions and , have a larger number of network points than have pairs of opposite sides of the "Poincaré cell", the fact remains that to detect a "Poincaré cell" can be very difficult, unless the swept area is already very close around a solution point, and to get that close, no method or algorithm has been proposed so far in the literature to our knowledge. 15
A MULTISECTION AND SWEEPING METHOD FOR SOLVING NONSMOOTH EQUATIONS
Rules for constructing a "Poincaré rectangle":
3) A green side (either green) cannot cross the curve, and a red side (either red) cannot cross the curve.
1) A pair of green parallel sides should be on different sides of a curve. 2) A pair of red parallel sides should be on different sides of a curve.
Figure 13. "Poincaré rectangles" that meet the construction rules.
and and different rectangles that include it inside, one of which, the first, does not meet the rules, and the second does.
Influence of size for a rectangle to be "Poincaré" The following Figure 14. shows the positional relationship of an intersection point of two
Large rectangle that violates rules 1), 2) and 3) needed to meet Shrunk "Poincaré rectangle" the conditions of Poincaré. Figure 14. Shrinking rectangle becoming a "Poincaré rectangle"
The difficulties noted for detection of "Poincaré rectangles", partly motivated the design of our method presented here, based on confluence of subdomains.
into two subdomains in which For example, if
has a different sign.
, then vanishes on the circumference equation is
APPENDIX II There are special cases where a curve on whose points vanishes, does not separate the domain
, 16
whose
GUILLERMO MIRANDA, LARRY MENDOZA & REMIGIO MEDRANO
values in the Cartesian grid points, so that Γ-points will be in the neighborhood of a row of dots in which the absolute values of exhibit a minimal aspect.
but such that . In this case, Γ-points coincide with the points of minimum null value for F. The same would apply in the case of a maximum value of zero, as in the example
In
below, the surfasse shows tangency with the plane along Γ, which is the curve described by equation .
As an exceptional case, Γ-points do not generate "confluence" subdomains, but in such cases, Γ-points may be detected in the sweep, by including not only the sign of , but also their numerical .
Figure 15. Special case where
Figure
15.
does not generate subdomains of different signs for .
Another exceptional case occurs when the solution point does not appear as the intersection of and , but as a point of tangency. There remains a confluency of subdomains and , but only one of the subdomains and survives, and it is split into two disjoint open sets.
over a layer of a moving bed with a packing concentration is mainly controlled by a dimensionless parameter called Peclet Number " ". In turn, depends on the concentration of the suspension, and on the height " " of the bed, since by definition
APPENDIX III The vertical turbulent dispersion found in the horizontal flow of a suspension layer " " flowing
17
A MULTISECTION AND SWEEPING METHOD FOR SOLVING NONSMOOTH EQUATIONS
being the particle terminal velocity, is the inside diameter of the pipe, and the turbulent dispersion coefficient is given by the empirical expression:
where
occupied by the suspension layer, , density and viscosity of the suspension.
, are the
The average horizontal velocity in the suspension, , depends on and :
is given by Walton’s formula (see [14]),
being the injection flow of the mixture that enters the pipe with a solids concentration , and is the sectional area of the pipe occupied by the suspension layer. Assuming suspension particles to be spherical, then " " denotes their diameter. and should be obtained by solving a system of two equations and two unknowns, which is continuous but only piecewise- differentiable.
and then it turns out that , and even more so , are continuous but piecewise- differentiable functions of the variable in the suspensión, and is defined by
The ordinary differential equation describing turbulent dispersion in the vertical direction to the flow of two layers described above, once solved leads to the first equation of the system:
, being the hydraulic diameter of the pipe section
where
The second equation of the system is obtained by equalizing the pressure gradients in the suspension layer S and the layer B of the moving bed:
The numerical results obtained by applying the method of sweeping to detect domains of confluence in the box
where
,
will be presented later in a specialized journal for the oil industry. .
and 18
GUILLERMO MIRANDA, LARRY MENDOZA & REMIGIO MEDRANO
ACKNOWLEDGMENTS The authors wish to specially thank Dr. José Castillo (San Diego State University, California, USA) for his valuable suggestions in order to improve this article, and to Research & Development Engineers, José Pineda and Leonardo Cáliz (PDVSA - Intevep, Venezuela), for posing a challenging petroleum industry problem. Also we thank Professors José Machado (Basic Sciences Departament’s Head), Raquel Centeno (Research and Post-Graduate Director) and Manuel Serafin (Industrial Engineering Master’s Program Coordinator), all from Universidad Nacional Experimental Politécnica "Antonio José de Sucre", Vicerrectorado "Luis Caballero Mejías", Caracas, Venezuela, for all the logistical support provided for the present work’s development. REFERENCES. [1]
BURDEN, R. L., & FAIRES, J. D. 2005."Numerical Analysis", 8th ed., PWS Publishing, Boston,
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ORTEGA, J. M. 1972.- "Numerical Analisis A Second course". Academic Press, New York.
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RAHAL, MOHAMED & GUETTAL DJAOUIDA. 2014.- "Modified Sequential Covering Algorithem for Functions and Applications" Gen. Math. Notes, Vol. 22, N°. 1, pp. 100–115.
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ORTEGA, J. M. & RHEINBOLDT, W. C. 1970. "Iterative Solution of nonlinear equations in several variables". Academic Press, New York,
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BOLZANO, B. 1817.- "Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwei Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reele Wurzel der Gleichung liege, Gottlieb Hass", Prague,
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POINCARÉ, H. 1883.- "Sur certaines solutions particulieres du probléme des trois corps". C.R. Acad. Sci. Paris 97 (1883), 251 – 252.
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MIRANDA, C. 1940.- "Una osservazione su una teorema de Brouwer". Boll. Unione Mat. Ital. 3 (1940), 525 – 527.
[10] JAISWAL, J. P. 2014."Solving Nondifferentiable Nonlinear Equations by New Steffensen-type Iterative Methods whith Memory". Mathematical Problems in Engineering, Volumen 2014 (2014) Article ID795628, 6 pages. [11] HERNÁNDEZ, M. A. & RUBIO M. J. 2004.- "A modification of Newton’s method for nondifferentiable equations". Journal of Computational and Applied Mathematics, 164 – 165. (2004), 409 – 417.
AXELSSON, OWE. 1996.- "Iterative solution methods". Cambridge University Press.
[12] MARTÍNEZ, JOSÉ MARIO & QI, L. 1995.- "Inexact Newton methods for solving nonsmooth equations". Journal of Computational and Applied Mathematics, 60(1995), 127 – 145. .
RHEINBOLDT, W. C. 1968.- "A unifield convergence theory for a class of iterative process". SIAM J. Numer Anal., 5 (1968), 42–63. 19
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[13] BERTSEKAR, DIMITRI P. 1975."Nondifferentiable optimization via approximation". Mathematical Programming Study, 3(1975), 1–25.
[14] WALTON, I. C. 1995.- "Computer simulator of coiled tubing wellbore cleanouts in deviated wells recommends optimum pump rate and fluid viscosity", paper SPE 29491, Presented at the Production Operations Symposium, Oklahoma City, O.K., 2-4 april (1995), 471– 481. .
.
ADRESS
GUILLERMO MIRANDA1 1
Escuela de Matemática, Facultad de Ciencias, Universidad Central de Venezuela, Caracas, Venezuela.
LARRY MENDOZA2, REMIGIO MEDRANO3 1,2,3
Línea de Investigación Matemática Aplicada, Departamento de Ciencias Básicas, Universidad Nacional Experimental Politécnica "Antonio José de Sucre", Vicerrectorado "Luis Caballero Mejías"; Caracas, Venezuela.
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