Step Size Strategies Based On Error Analysis For The Linear Systems. Lineer Sistemler İçin Hata Analizi Tabanlı Adım Genişliği Stratejileri - PDF

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SDU Journal of Science (E-Journal), 0, 6 (): Step Size Strategies Based On Error Analysis For The Linear Systems Gülnur Çeli Kızılan,*, Kemal Aydın Kahramanmaraş Sütçü İmam University, Faculty of

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SDU Journal of Science (E-Journal), 0, 6 (): Step Size Strategies Based On Error Analysis For The Linear Systems Gülnur Çeli Kızılan,*, Kemal Aydın Kahramanmaraş Sütçü İmam University, Faculty of Education, Department of Primary Mathematics Education, 4600, Merez, Kahramanmaraş, Turey Selçu University, Faculty of Science, Department of Mathematics, 4075, Selçulu, Konya, Turey * Corresponding author Received: 5 August 009, Accepted: 03 June 0 Abstract: In this paper, we have obtained the step size strategies for numerical integration of the linear differential equation systems. We have given the algorithms which calculate step sizes based on the given strategies and numerical solutions. These strategies and algorithms are generalized to systems by modifying the algorithm and strategy in []. We have applied our strategies to Cauchy problem with order m. We have also give the numerical examples. Key words: Variable step size, error analysis, linear systems, numerical integration, step size strategy Lineer Sistemler İçin Hata Analizi Tabanlı Adım Genişliği Stratejileri Özet: Bu çalışmada, lineer diferensiyel denlem sistemlerinin nümeri integrasyonu için adım genişliği stratejileri elde edilmiştir. Verilen stratejilere uygun olara adım genişlileri ve nümeri çözümler hesaplayan algoritmalar verilmiştir. Bu strateji ve algoritmalar [] de verilen strateji ve algoritmanın değiştirilere sistemlere genişletilmesidir. Verilen stratejiler m. mertebeden Cauchy problemine uygulanmıştır. Ayrıca, sonuçların doğruluğunu gösterme için nümeri örneler de verilmiştir. Anahtar elimeler: Değişen adım genişliği, hata analizi, lineer sistemler, nümeri integrasyon, adım genişliği stratejisi. Introduction Choosing the step size is one of the most important concepts in numerical integration of the Cauchy problem (.) x f (t, x), x(t 0 ) x0. The use of constant step size is not practical in numerical integration. If the step size used is large in numerical integration, it provides fast convergence but also may lead to error. And the computed solution may diverge from the exact solution. On the other hand, if the step size used is small, it may give the opposite performance, i.e. the calculation time, number of the arithmetic operations, the calculation errors start to increase []. So, if the solution changes rapidly, the step size should be chosen small. Inversely, if the solution changes slowly, then step size should be chosen larger. In [,3], the step size strategies based on error analysis were given for numerical integration of the Cauchy problem (.) on the region D {(t, x) : t [t 0, T ], x x0 b} and there was also given an algorithm which calculates the step size based on error analysis and numerical solution in each step is given. For Euler method the step sizes are given by the following inequality 49 G. Çeli Kızılan, K. Aydın hi ( L ), Mt (.) i where max z ( i ) M ti and for second order Runge-Kutta method as follows, ti i ti L 3 hi ( ), M ti where max ( f tt f. f tx f x. f t f. f x f f xx )( ) M ti such as local error is [ ti,ti ) smaller than the required error level L in each step of the integration. Here L is the error level that is determined by the user and z (t ) is the solution of the Cauchy problem (.3) z f (t, z ), z (ti ) yi, t [ti, ti ), ( y0 x0 ), where yi is the numerical solution taen from numerical method the i-th step. If the existence of the solution of Cauchy problem given by equation (.) on region D = {( t, x ): t t 0 a, x x0 b } is unnown; the step size has been given by hi min{a, b0i / M i }, where y i is the numerical solution obtained in the i-th step, z (t ) is the solution of the Cauchyproblem (.3), bi is the upper bound of z y i error, b0i min{b0i, bi }, Di {(t, z ) : t t i a, z y i b0i } and M i is the upper bound of region Di [, 3, 4]. f (t, z ) on For detailed nowledge on the numerical integration of the Cauchy problem (.) the references [5-7] can be examined. In this paper, we want to investigate a step size strategy for the Cauchy problem (.4) X F (t, X ), X (t 0 ) X 0 on the region D {(t, X ) : t t 0 T, x j x j 0 b j } (.5) by generalizing the step size strategy given in [, 3] for Cauchy problem (.). Here, X (t ) ( x j (t )), X 0 ( x j 0 ) ; x j 0 x j (t 0 ), F (t, X ) ( f j ) ; f j f j (t, x, x,..., x ), F (t, X ) C m ( [t 0, T ] ) and X (t ), X 0, b (b j ). We suppose that F (t, X ) AX (t ), where A (aij ) and consider the Cauchy problem given by (.6) X (t ) AX (t ) F (t, X ), X (t 0 ) X 0. The aim of this paper is to generalize the algorithm and strategy given in [, 3] for the Cauchy problem (.6). In our study, we have used the Euler's method for simplicity. In section ; the concept of local error given in [, 3, 8] as being defined for systems of differential equations and local error analysis has been examined. In section 3, the step size strategies based on error analysis have been applied to systems and algorithms which calculate step size and numerical solution in each step have been given. In section 4; the step size strategies 50 SDU Journal of Science (E-Journal), 0, 6 (): have been given for m-th order Cauchy problem. Finally, in Section 5 numerical examples have been given as applications of the algorithms.. Preliminaries In this study, as a norm in we use Euclidean norm, which is defined as follows y = y j For every A (aij ) j, y ( y j )., we use the Frobenius norm, i.e. A a i j ij... Local Error We give the concept of local error, which is given for Cauchy problem (.) in [, 3, 8], for Cauchy problem (.4) as follows. Let us construct the Cauchy problem given as follows Z F (t, Z ), Z (ti ) Yi, Y0 X 0 t [ti, ti ), (.) where Yi ( yij ) is the numerical solution taen from numerical method the i-th step. The vector of local error LEi of a numerical method is given by yi z (ti ) LEi yi z (ti ) LEi LEi Yi Z (ti ) y z (t ) LE i i i (.) and LEi is the local error of the numerical method... Euler s Method Euler's method for Cauchy problem (.4) is defined in [9] by Yi Yi hi Fi. Here Fi ( fij ), Yi ( yij ) and hi ti ti..3. Error Analysis For Systems Let us apply error analysis in [] to Cauchy problem (.3). The component LEij of (.) is given as follows, 5 G. Çeli Kızılan, K. Aydın LEij yij z j (ti ) y(i ) j hi f (i ) j ( z j (ti ) z j (ti )(ti ti ) y(i ) j hi f (i ) j ( y(i ) j hi f (i ) j z ( ij )(ti ti ) )! z ( ij )hi )! z ( ij )hi, j,,...,, ij (ti, ti ).! The vector of local error on interval [ ti, ti ) is as follows, LEij LEi z ( i ) z ( i ) LEi LEi hi Z ( ij ).!! LE z ( ) i i (.3) 3. The Step Size Strategies For The Linear Systems Let us give the theorem which gives the upper bound of local error of the system X AX (t ) F (t, X ), X (t 0 ) X 0. Theorem. The vector of local error of Cauchy problem (.6) is h LEi i A Z ( ij ), ij [ti, ti ).! Proof. If we tae F (t, Z ) AZ (t ) in (.), then we find Z (t ) AZ (t ), (3.) where A (aij ). So it is clear that (3.) Z (t ) AZ (t ) A( AZ (t )) A Z (t ). If we substitute this result into the equation (.3), then the vector of local error will be as follows: h LEi i A Z ( ij ), ij [ti, ti ).! Theorem. The upper bound of local error for the system (.6) is (3.3) LEi ( i ) 5 hi, where max aij and max( sup z j ( i ) ) i. i, j j t t i i i Proof. If we tae the norm of the equation (3.), then we obtain LEi hi A Z ( i ) hi A Z ( i ). Since the inequalities A max aij, Z max z j i, j are valid for every A (aij ) (3.4) j and Z ( z j ) [0], taing max aij and i, j max( sup z j ( i ) ) i, the inequality (3.4) can be written as j t t i i i 5 SDU Journal of Science (E-Journal), 0, 6 (): LEi ( i ) 5 hi. Here, it is clear that max z j i. j 3.. Step Size Strategy From (3.3), the step size is computed using the inequality (3.5) h ( L ) 4 5 in the -th step such that local error LE L, where L is the error level determined by the user. Since we have considered the problem (.6) on the region D {(t, X ) : t t 0 T, x j x j 0 b j, j,,..., }, it is clear that sup z j ( ) ) b j z j (t ). t t So, the calculation of can be as follows in practice: max{b j z j (t ) } max{b j yj }. j j Corollary. For =, the inequality (3.5) is equivalent to the inequality (.), which is the step size strategy given in [] for the first order nonlinear nonhomogeneous ordinary differential equations. Proof. Let us tae f (t, x) ax in the Cauchy problem (.). Since z (t ) az and max z ( ) M t are valid, we obtain t t max z ( ) sup z ( ) sup a z ( ) t t t t a t t sup z ( ) M t t t where M t is the number in (.). It is clear that a and sup z j ( ) sup z ( ) max( sup z j ( ) ). t t t t j t t If we substitute the result above into the equation (3.5), then the step size is computed as follows: L L L h ( ) ( ) ( ). (3.6) Mt So, it is shown that the inequality (3.5) is equivalent to (.) for =. Remar. In accordance with our goal, the step size can be chosen from the equation (3.4) in the -th step. Theoretically, when step sizes are computed by (3.4), local error of the problem may be very close to the number L. Since all numerical computations on the computer are performed using floating point arithmetic, the round-off errors occur. So it may occur that LE L for some. Therefore, we have given the step size 53 G. Çeli Kızılan, K. Aydın strategy in (3.5) to avoid possible effects of errors in floating point arithmetic. It is clear that the step sizes computed by (3.5) are smaller than those obtained by (3.4). 3.. Algorithm ow, let us give the algorithm which calculates step sizes using the equation (3.5) and numerical solution in each step. This algorithm is a modification of the algorithm in []. Step 0: Give the t₀, T, b, h*, δl, X₀, A data. Step : Calculate 0 and. Step : Calculate and h ; hˆ ( L 4 5 ) Step 3: Control h with K; K:. If t hˆ T ; then.. If hˆ h* then h hˆ... If hˆ h* then h =0 and the process stops. ˆ. If t hˆ T then hˆ T t. ˆ ˆ.. If hˆ h* then h hˆ. ˆ.. If hˆ h* then h =0 and the process stops. Step 4: Calculate t t h and Y ( I h A)Y. Replace by + and go to step. Here; is the step number, T is the number given on D, h is the proposed step size by the step size strategy, h* is the practical parameter for step size, t t 0 hi, where i 0 h i i 0. In Algorithm, Algorithm K which is called Step Size Control Algorithm in [] concludes the computation procedure. Remar. For the reason mentioned in Remar, (3.5) is obtained by increasing the inequality (3.4) and step sizes obtained by (3.5) become smaller. Therefore, if the step sizes are computed by the inequality (3.5), their values increase and the local errors occur much smaller than the number δl. To compute the numerical solutions in error level δl with sufficiently large step sizes, let us give the following modified step size strategy. 54 SDU Journal of Science (E-Journal), 0, 6 (): Modified Step Size Strategy Initially, the step size is chosen by the inequality (3.5) in the -th step. For any ( ); h i i hˆ and LE i are calculated from i = to i = p such that LE p δl and LE p δl, where LE δl in the -th step. Then, the step size is computed by h p hˆ, hˆ ( L 4 5 (3.7) ) in the -th step for numerical integration of the linear system (.6). The step size (3.7) is sufficiently large to calculate the numerical solution in error level δl. Although , is chosen by the user, we suggest the user tae. The following diagram represents the modified strategy: max aij i, j max{b j y( ) j } j hˆ ( L 4 5 h l l ) Yl ( I hl A)Y hˆ l Z l e Ah Y h hˆ l l LE = Y Z l Y l : =l + LEl L l:=p h = p h Diagram. Calculation of the step size with respect to modified strategy in the -th step. 55 G. Çeli Kızılan, K. Aydın 3.4. Algorithm Step 0: Give the t₀, T, b, h*, δl, X₀, A data. Step : Calculate 0 and. Step : Calculate and h i ; hˆi ( L 4 5 ) ; Step 3: Calculate Yi, Z i and LEi for i, i ; ˆ Yi ( hˆi A)Y, Z i e Ah Y, LEi = Yi Z i i Step 4: If LEi δl then replace i by i+. Calculate hi i hˆi and go to step 3. Step 5: Calculate h = i h i, Step 6: Control h with K. Step 7: Calculate t t h and Y ( I h A)Y. Replace by + and go to step. Algorithm is obtained from Algorithm by replacing step 4 by steps 4, 5 and Application of Step Size Strategies to The m-th Order Cauchy Problem Consider the Cauchy problem as follows: x ( m ) am x ( m ) a x a0 x 0. (4.) By taing x x, x x, x ( m ) xm the equation (4.) can be written as x 0 x 0 x a m 0 0 a 0 x 0 x, a m x m x (t0 ) x0 x (t0 ) x0 x (t ) x m 0 m0 with initial condition x j (t 0 ) x j 0 (j =,,..., m). Shortly, it can be given as (4.) X CX F (t, X ), X (t 0 ) X 0. Here, it is clear that the matrix C is the companion matrix. Then, the step size strategies and algorithms which are given for the numeric integration of the Cauchy problem (.6) can be easily used for the numeric integration of m-th order Cauchy problem (4.). 5. umerical Results Example. Consider the Cauchy problem 0 x x x 0.5 x x (0) x ( 0), (5.) on the region D {(t, X ) : t [0, 5], xi xi 0 5}. Let h* =0- and δl=0-. 56 SDU Journal of Science (E-Journal), 0, 6 (): To calculate the step sizes and the numerical solutions of the Cauchy problem (5.), the Maple procedure has been used. The results obtained from the procedure for the solution of Example have been summarized in Table and Table. Table. The values of h and LE Table. The values of calculated by Algorithm h[] e e e e e e e e- LE[] e e e e e e e e- h and LE calculated by Algorithm h[] e e e- LE[] e e e e e e e e-3 Figure and Figure illustrate the values of h and LE calculated by Algorithm and Algorithm. For the same example, when Figure and Figure are examined it is seen that the step sizes in Figure are larger than those in Figure and as a result of this, the local errors in Figure are closer to the error level δl than those in Figure. Figure. h and LE values calculated by Algorithm Figure. h and LE values calculated by Algorithm, =. Example. Consider the Cauchy problem (5.) x x x 0, x(0), x (0) on the region D {(t, x) : t [0, 5], x x0 5}. We can write the Cauchy problem (5.) as the first order Cauchy problem 0 X (t ), X (0). X (t ) (5.3) Calculate the step sizes of Cauchy problem (5.) by using Algorithm and Algorithm (tae h* =0- and δl=0-). The results have been summarized in Table 3 and Table 4. 57 G. Çeli Kızılan, K. Aydın Table 3. The values of h and LE Table 4. The values of calculated by Algorithm h[] e e e e e e e e- LE[] e e e e e e e e-3 h and LE calculated by Algorithm h[] e e e- LE[] e e e e e e e e- The tables above have been obtained by using Maple procedure. The results obtained from the procedure for the solution of Example have been summarized in Figure 3 and Figure 4. Figure 3 and Figure 4 illustrate the values of h and LE calculated by Algorithm and Algorithm. Figure 3. h and LE values calculated by Algorithm Figure 4. h and LE values calculated by Algorithm, =.0 6. Conclusion In this wor, the new step size strategies and the new algorithms have been given for the Cauchy problem (.6). Cauchy problem (.6) arises in many applications such as spring-mass systems, LRC circuits and the simple pendulum. First order series and parallel chemical reactions and process control models are also usually represented by Cauchy problem (.6). The strategies and algorithms given in this wor are the generalization to systems by modifying the algorithm and strategy in []. The algorithms calculate the step sizes based on our strategies and the numerical solution of the Cauchy problem (.6) such that local error LE L in the step -th step, where 58 SDU Journal of Science (E-Journal), 0, 6 (): L is the error level determined by the user. The strategies and algorithms have been applied to m-th order Cauchy problem. The numerical examples have also been constructed using the algorithms. The algorithms are suitable to write the computer procedure. To compute the step sizes and the numerical solutions, the Maple procedure has been used. As it can be seen from the examples, local errors occurred in numerical solutions computed by Algorithm are smaller than δl, so the number of the step sizes occurs more and numerical solutions are quite close to the exact solution. Local errors in numerical solutions computed by Algorithm are quite close to δl, so the number of the step sizes occur less and numerical solutions are close enough to the exact solution. Algorithm is an adaptive algorithm in this aspect. So, if the number of the step sizes is desired to be less, Algorithm is preferable. Otherwise, both Algorithm and can be also used. References [] Çeli Kızılan G., 004. On the finding of step size in the numerical integration of initial value problem, Master thesis, Graduate atural and Applied Sciences, Selcu University, Konya (in Turish). [] Ban S.J., Lee C.L., Cho H., Kim S.W., 00. A variable step-size adaptive algorithm for direct frequency estimation, Signal Processing, 90: [3] Çeli Kızılan G., Aydın K., 005. Step Size Strategy Based on Error Analysis, Selcu University Science and Art Faculty Journal of Science, 5: (in Turish). [4] Çeli Kızılan G., Aydın K., 006. A new variable step size algorithm for Cauchy problem, Applied Mathematics and Computation, 83: [5] Gear C.W., 97. umerical initial value problems in ordinary differential equations, Prentice- Hall, ew Jersey, 97. [6] Miraner W.L., 98. umerical methods for stiff equations and singular perturbation problems, D. Rediel Publishing Company, Holland. [7] Shampine L.F., Allen R.C., Pruess S.,996. Fundamentals of numerical computing, John Wiley&Sons, IC, ew Yor. [8] Heath M.T., 00. Scientific Computing an Introductory Survey, Second edition, McGraw-Hill, ew Yor. [9] Loan C.F.V., 000. Introduction to Scientific Computing, Prentice Hill, United States of America. [0] Gu M., 998. Stable and Efficient Algorithms for Structured Systems of Linear Equations, SIMAX, 9(), Kemal Aydın 59
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