Selcuk Journal of
Applied Mathematics
Sel¸cuk J. Appl. Math. Vol. 4, No. 2, pp. 13–22, 2003
Cauchy Solver
Ay¸se Bulgak1 and Diliaver Eminov2
1 Research Centre of Applied Mathematics, Sel¸cuk University, Konya, Turkey; e-mail: abulgak@selcuk.edu.tr
2 Spectel Ltd., Dublin, Ireland;
e-mail: Diliaver.Eminov@spectel.com Received: September 10, 2003
Summary. This article presents a dialogue program ”Cauchy Solver 2.1” that has been elaborated for computing the Cauchy problem for discrete-time and differential linear equations. The program features are: plotting graphical representation of components and Euclidean norm of solution of the Cauchy problem for any chosen vectors; check-ing practical stability of coefficient matrices; findcheck-ing certain numerical characteristics of solutions of the Cauchy problems; finding extremal initial vectors in case if coefficient matrix is practically stable; plot-ting graphical representation of components and Euclidean norm of solution of the Cauchy problem for such vectors.
Key words: Cauchy problem, differential equations, discrete-time equations, stability, practical stability
2000 Mathematics Subject Classification: 65D07,97U070
1. Introduction
The Cauchy Solver and Discrete-time Cauchy Solver have been elab-orated as a part of the Matrix-Vector Calculator MVC 1.0 [3]. We are presenting their new versions, Cauchy Solver 2.1 ( CS ) and Discrete-time Cauchy Solver 2.1 ( DTCS ) in this article. These programs support the books [1],[7] and represent a selection of computer-aided methods for mathematical education.
The programs CS and DTCS are written on MS Visual C++, us-ing intuitive dialogs of OS Windows, and work under OS WINDOWS-95 and higher.
The programs are available athttp://www.sumam.selcuk.edu.tr/software.html.
These programs are being used for mathematical studying in the Statistics Department of the Selcuk University and the Engineering Faculty of the Selcuk University.
2. Discrete-Time Cauchy Solver
DTCS is intended for solving a Discrete-time Cauchy problem
(1) x(n + 1) = Ax(n), n = m, m + 1, . . . , T ; x(m) = a,
for the given integers m and T , matrix A and vector a. A simple dialogue allows constructing interpolations of the components of the vector sequence in the question and its Euclidean norm. The users can also draw the graphic of the first degree spline function for a table
min(√ω∗(1− 1 2ω∗)
(n−m)/2,An−m), n = m, m + 1, . . . , T choosing a real number ω∗ > 1. It allows detecting practically ill-posed problems with respect to the discrete-time asymptotic stability of A (see [1], [5]).
The number ω(A) [1], [5] is suggested as a measure of quality of discrete-time asymptotic stability of A. Its definition bases on the solution of the discrete-time Lyapunov matrix equation
(2) A∗HA− H = −I
with identical matrix I. If H = H∗ > 0 exists and satisfies to (2) then ω(A) =H else ω(A) = ∞. The parameter ω(A) allows estimating the solution of (1):
x(n) < min(√ω∗(1− 1 2ω∗)
(n−m)/2,An−m)a, n = m, m + 1, . . . . Taking into account the uncertainties in data, the user can choose a number ω∗ > 1 that characterizes a level of practical discrete-time asymptotic stability. The value ω∗ can also be chosen with respect to the standard format, which is used for computing [1], [5]. The matrix A is known as a practically discrete - time asymptotically stable if
ω(A) < ω∗. In another case it is known as a practically unstable. The value ω(A) has representation
ω(A) = max
xm=0
(xm2+xm+12+ . . . +xk2+ . . .)/xm2. It is interesting that if A is a discrete-time asymptotically stable matrix, then the Cauchy problem (1) has two extremal initial vectors in the infinite interval. They are defined by the eigenvectors of matrix H, which corresponds to extremal eigenvalues of the A . Since that, if λNand λ1 are the maximal and minimal eigenvalues of a positive definite matrix H, and the vectors xmax (xmax = 1) and xmin (xmin = 1) are correspond them, then for the solutions of (1) and the following Cauchy problems
(4) U (n + 1) = AU (n), n = m, m + 1, . . . , M ; U (m) = xmax, and
(5) W (n + 1) = AW (n), n = m, m + 1, . . . , M ; W (m) = xmin the inequalities hold
Wm2+Wm+12+ . . . +Wk2+ . . . ≤ (xm2+xm+12+ . . . +xk2+ . . .)/xm2
≤ Um2+Um+12+ . . . +Uk2+ . . . .
3. Discrete-time Cauchy Solver Dialogue
The main window of DTCG comprises of the graphical area, main menu with a toolbar, control panel and status bar ( see Fig 1 ). The main menu , toolbar and status bar purpose is as common and intuitive as in any other Windows application.
3.1. The main menu
The main menu includes four submenus - File (see Table 1), Edit, Tools (see Table 2) and Help (currently only displays the about box).
New Clear the graphical area, the matrix and vectors
Open Load data from file
Save Save current data to file
Save As... Assign name to the data file and save it Save Picture As... Current graphics is saved in the .bmp file
Exit Exit the program
Fig. 1.
Run Draw graphics
Clear Clear graphics area Move Move Finish vector to Start vector Matrix Show Matrix dialog
Table 2
Menu Edit includes submenus undo, cut , copy, paste that can be used during editing elements of the Vectors table.
3.2. Matrix Dialogue
Submenu Matrix displays the Matrix dialog (see fig. 2) that allows the user to enter the Matrix A and parameter Omega - the value of parameter ω∗. Matrix range switch allows the user to choose an appropriate range of the matrix.
The dialog also allows checking the practical discrete-time asymp-totic stability of the given matrix. Click the button ”Check” to per-form the checking. If the matrix is stable, the ”Maximal” and ”Min-imal” modes become available. Otherwise, only ”Manual” mode is available and warning ”The matrix is unstable” is displayed.
Fig. 2.
The user can choose either ”Maximal” or ”Minimal” to make the maximal or minimal vector the Starting vector of the program. Click ”OK” to accept changes or ”Cancel” to discard them.
3.3. Control Panel
Control Panel comprises of: Component, Start, Stop, Colour, Scale. ”Component” allows choosing the component number, which graph-ics will be drawn ( it can be changed between 1 and N ). It also has two special positions:
- 0 corresponds to Euclidean norm of the solution vector, - (-1) corresponds to the function
min(√ω∗(1− 1 2ω∗)
(n−m)/2,An−m), n = m, m + 1, . . . , T The ”Start” is an integer number m.
The ”Stop” is an integer number T.
The ”Colour” allows choosing the graphics colour. It has six pre-defined colours - black, blue, red, green, yellow and purple.
If ”Auto” is chosen, the Xmax, Xmin, Y max, Y min are inac-tive. The program computes Xmax, Xmin by formulas Xmin = m, Xmax = T.
The values Y max, Y min depend on component k which is chosen in component box.
For k = 1, 2, . . . , N the values Y max, Y min are computed by formulas
Y min = min
j=m,m+1,...,T(A j−mx
m)k, Y max =j=m,m+1,...,Tmax (Aj−mxm)k. For k = 0 the values Y max, Y min are computed by formulas
Y min = min
j=m,m+1,...,TA j−mx
m, Y max =j=m,m+1,...,Tmax Aj−mxm. For k =−1 the values Y max, Y min are computed by formulas
Y min = min j=m,m+1,...,Tmin( √ ω∗(1− 1 2ω∗) (j−m)/2,Aj−m), Y max = max j=m,m+1,...,Tmin( √ ω∗(1− 1 2ω∗) (j−m)/2,Aj−m). If ”Manual” is chosen, Xmax, Xmin, Y max, Y min have to be entered manually and Xmax > Xmin, Y max > Y min. These reals have to be entered using the same format as the cells of the table of ”Vectors” , see below.
3.4. Table of Vectors
The table of Vectors contains two rows, ”Start” and ”Stop”, which elements are sequentially numbered.
The user fills only the ”Start” row. It includes the components of the vector xm. The order of entering the values into the table is not important. ”Tab” button allows moving to the next right cell and the combination of ”Shift+Tab” to the left cell. Any cell can be selected using the mouse.
Each value has to be entered as [−]d.dddde[sign]ddd where d is a single decimal digit, dddd is one or more decimal digits, ddd is up to three decimal digits, and sign is + or−.
The ”Start” vector can be set using the matrix dialog. If the ma-trix is stable (the user can have it checked clicking on the ”Check” button of the dialog), the user is given an option to enter the ”Start” vector manually, or take the maximal or minimal vector as a starting vector. Choosing the appropriate mode allows setting the starting vector with the values of xmax or xmin.
As the result of computing the program fills the ”Stop” row by the components of the vector AT −mxm.
Fig. 3.
3.5. Graphical area
The graphics area is organized by orthogonal co-ordinate system, x-axis and y-axis. These lines are black. The crossing point corre-sponds to the point (Xmin, Y min). The points (Xmin, Y max) and (Xmax, Y min) are marked red. Co-ordinates of the point on the graphical area under the mouse pointer are shown in the status bar panes.
4. Examples
Example 1. Take Cauchy problem (1) with the following data
A = ⎛ ⎝1 0.5 0.72 0.5 0.8 3 6 1 ⎞ ⎠, N = 3, m = 5, T = 10, x(5) = ⎛ ⎝−21 3 ⎞ ⎠.ω∗ = 106
The ”CHECK” mode gives the information ”The matrix is unsta-ble”. x(10) = ⎛ ⎝187.738245.592 640.965 ⎞ ⎠.
The graphic of the Euclidean norm of the solution is given in Fig. 3 ( the top curve ). It is drawn by blue color in ”AUTO” mode. Repeat it in mode ”MANUAL”. Run the system again in manual mode with component = 2 and choosing color as ”red”. We will see the graphics, which is given in fig 3.
Fig. 4.
Example 2. Take the Cauchy problem (1) with the following data
A = ⎛ ⎝0.6 0.005 0.0072 0.5 0.008 3 6 0.4 ⎞ ⎠, N = 3, m = 0, T = 10, x(0) = ⎛ ⎝01 0 ⎞ ⎠.ω∗ = 103
The ”CHECK” mode gives the information ”The matrix is unsta-ble”. For ω∗ = 106 the program gives us an opportunity to activate either the maximal or minimal mode.
x(10) = ⎛ ⎝0.06797510.330347 3.6465 ⎞ ⎠.
In ”AUTO” mode the graphics of the Euclidean norm of the solution which correspond to the maximal initial vector gives us information about the convenient frames for the figure. Using this information in ”MANUAL” mode choose Xmin = 0, Xmax = 10, Y min = 0, Y max = 21. The graphics of the Euclidean norms of the solutions which correspond to start vector (0, 1, 0)∗ ( in black ), maximal initial vector ( in red ) and minimal initial vector ( in blue ) are given in Fig. 4.
Using the ”AUTO” mode, draw a graphics of the Euclidean norm of the solution for T = 40 and initial vector (0, 1, 0)∗. Activate the ”MANUAL” mode. Clear the graphical area. Choose Xmax = 40, start = 0 and stop = 20 and color as blue. Draw the graphic again.
Click on submenu MOVE. Choose and draw the graphic again . start = 20 and stop = 40 and color as red. Draw the graphic again. The result graphic is given in Fig. 5.
Fig. 5.
5. Cauchy Solver
The program CS is organized very similarly to the program DTCS. CS is intended for solving the Cauchy problem
(6) dx(t)
dt = Ax(t), m≤ t ≤ T, x(m) = a,
for the given real m and T , matrix A and vector a. The simple di-alogue allows constructing interpolations of the components of the vector - the function in the question and its Euclidean norm. The users can also draw the graphic of the third degree spline - approxi-mation for the function
min(
(κ∗)e−(t−m)A/κ∗, e(t−m)A), m≤ t ≤ T
choosing a real number κ∗ > 1. It allows detecting of practically ill-posed problems with respect to the asymptotic stability of A ( see [2], [5]–[7] ). In addition to the ”Start” and ”Stop” values, the user must enter the value of ”Step”, which is a parameter for calculation of the solution of (6).
Let h be a small positive number - value of a step. It is convenient to choose h = T −mM < 2A1 with a natural M . Using Taylor approx-imation the program computes B = ehA. This leads us to Cauchy problem
x((n + 1)h) = Bx(nh), n = m, m + 1, . . . , M ; x(m) = a. It is clear that
dx
This allows to use the special 3-rd degree spline [3] for drawing graph-ics of the approximation to solutions of (6).
The program computes the vectors xmax,xmax = 1 and xmin,xmin = 1 such that for the solutions of the following Cauchy problems (t≥ m)
W (t) dt = AW (t), W (m) = xmin, x(t) dt = Ax(t), x(m) = a, U (t) dt = AU (t), U (m) = xmax the inequalities hold
∞ m W (t) 2dt≤ ∞ m x(t) 2dt/x(m)2 ≤ ∞ m U(t) 2dt. Another difference from DTCS is the parameter KAPPA. It repre-sents a numerical bound between well-posed and ill-posed matrices for asymptotic stability problem (it is given by the value of κ∗).
Conclusion. The CS and DTCS is a tool, which is easy to use,
that can help studying the systems of discrete-time and differential equations. It is very helpful in the field of stability theory.
Acknowledgment. We would like to thank Haydar Bulgak
(Sel-cuk University) for a number of productive discussions about computer-aided methods of studying mathematics.
References
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