ON FANO CONFIGURATIONS OF THE LEFT HALL PLANE OF ORDER 9

Konuralp Journal of Mathematics Volume 4 No. 2 pp. 116–123 (2016) c KJM THE ANALYSIS OF THE EFFECT OF THE NORMS IN THE STEP SIZE SELECTION FOR THE NUMERICAL INTEGRATION GÜLNUR ÇELİK KIZILKAN, AHMET DUMAN, AND KEMAL AYDIN Abstract. In scientific studies involving norm calculations, the choice of the norm affects the obtained results. We have aimed to examine the behavior of the step sizes using different norms and norm inequalities in step size strategy obtained in [1] for linear Cauchy problems. 1. Introduction Selection of step size is an important concept for the convergence of the numerical solution to exact solution in numerical integration of differential equation systems. For the use constant step size, it must be investigated how should be selected the step size in the first step of numerical integration. Also, if the solution is changing slowly in some regions and it is changing rapidly in some another regions then it is not practical to use constant step size in numerical integration. So, we should use small step sizes in the region where the solution changes rapidly and we should choose larger step size in the region where the solution changes slowly. In literature, step size strategies have been given for the numerical integration. Consider the Cauchy problem X 0 = F (t, X), X(t0 ) = X0 on the region D = {(t, X) : |t − t0 | ≤ T, |xj − xj0 | ≤ bj }, where X(t) = (xj (t)), X0 = (xj0 ); xj0 = xj (t0 ), F (t, X) = (fj ); fj = fj (t, x1 , x2 , ..., xN ), F (t, X) ∈ C 1 ([t0 − T, t0 + T ] × RN ), X(t), X0 and b = (bj ) ∈ RN . In [1, 2] a step size strategy for F (t, X) = AX is proposed by (1.1) hi ≤ 2δL 1 1 √ ( )2 4 α N 5 βi−1 such that the local error ||LEi || ≤ δL . Strategy given in (1.1) is the generalization of the strategy in [3, 4]. 2010 Mathematics Subject Classification. 67F35, 67L05, 97N30. Key words and phrases. Step size strategy, linear systems, norms. 116 THE ANALYSIS OF THE EFFECT OF THE NORMS IN THE STEP SIZE ... 117 The above-mentioned step size strategies are based on matrix and vector norms. As in all the scientific studies involving norm calculations, the choice of the norm affects the obtained results in step size strategies. The aim of this paper to examine the behavior of the step sizes using different norms and norm inequalities in step size strategy obtained in [1] for linear Cauchy problems. In section 2, we have introduced the step size strategy based on error analysis for the linear systems (SSS). We have reminded commonly used vector and matrix norms. In section 3, we have investigated the effects of choice of the norms on step size strategy. Finally, we have analyzed the all strategies with numerical examples. 2. The step size strategy and norms 2.1. The Step Size Strategy (SSS). Let us consider the Cauchy problem X 0 = AX, X(t0 ) = X0 . (2.1) Following inequality is given h2i ||A||2 ||Z(τi )||, τi ∈ [ti−1 , ti ) 2 for the local error of the Cauchy problem (2.1) in i -th step of the numerical integration. According to equation (2.2), the upper bound of local error for the system (2.1) is given by √ 1 (2.3) ||LEi || ≤ ( α2 βi−1 ) N 5 h2i , 2 where ||A|| ≤ N √ √maxi,j |aij | = N α, √ ||Z|| ≤ N maxj |zj | ≤ N maxj (supτi |zj (τi )|) ≤ N βi−1 . From the inequality(2.3)in the stepi, the step size is calculated by (2.2) ||LEi || ≤ 1 2δL 1 hi ≤ ( √ )( )2 4 α N 5 βi−1 such that the local error ||LEi || ≤ δL where δL is the error level that is determined by user ([1, 2]). While formulating the step sizes (2.4), a more practical way is obtained for calculations by using the upper bound (2.3) instead of the upper bound (2.2) of the local error. The effects of the calculation errors resulting from floating point arithmetic are reduced in doing so. (2.4) 2.2. Vector and Matrix Norms and Relations between Matrix Norms. A norm is a real valued function that provides a measure of the size of vectors and matrices. For X = (xj ) ∈ RN , some commonly used norms are given below. The l2 norm (Euclidean norm) is defined by PN 1 ||X||2 = ( j=1 x2j ) 2 . The l1 norm (sum norm) is given as ||X||1 = PN j=1 |xj |. Another norm is formulated by ||X||∞ = maxj |xj |, 118 GÜLNUR ÇELİK KIZILKAN, AHMET DUMAN, AND KEMAL AYDIN which is called as l∞ norm (maximum norm). For A = (aij ) ∈ RM ×N , the most frequently used matrix norms are the l1 (maximum column) norm PM ||A||1 = maxj i=1 |aj |, the l∞ ( maximum row) norm PN ||A||∞ = maxi j=1 |aj |, the l2 (spectral) norm p ||A||2 = λmax (AT A), where λmax (AT A) is the maximum eigenvalue of the matrix AT A, Frobenius norm PM PN 1 ||A||F = ( i=1 j=1 |aij |2 ) 2 , and the maximum norm ||A||max = maxi,j |aij |. We have used in our study the relations p √ ||A||2 ≤ ||A||F , ||A||F ≤ N ||A||2 , ||A||2 ≤ N ||A||max , ||A||2 ≤ ||A||1 ||A||∞ which hold for all matrices A = (aij ) ∈ RN ×N . And we have also used the compatible norms in this study. For all information about norms in this section, you can see for example [5, 6, 7, 8, 9, 10]. 3. An Analysis on the Effect of the Norms in the Step Size Selection 3.1. The Effect of Choice of Norm to Step Size Strategy. The inequality (2.4) given in [1, 2] gives step sizes based on matrix and vector norms in the i -th step of numerical integration of the Cauchy problem (2.1) such that local error is smaller than δL error level. Different formulations are obtained for the step size according to the choice of the norms in the inequality (2.2). Changes that occur in step sizes may be significant. Now, let investigate the effect of the norms to step sizes. In calculations consider that ||Z(τi )||k ≤ supτi ||Z(τi )||k ≤ βk,i−1 , k = 1, 2, ∞. Strategy 1 (SSS1)The step sizes given by 1 2δL 1 (3.1) hi ≤ ( ) 2 , τi ∈ [ti−1 , ti ) ||A||2 β2,i−1 are obtained from the inequality (2.2) according to l2 norm. Strategy 2 (SSS2) The step sizes given by 1 2δL 1 (3.2) hi ≤ ( ) 2 , τi ∈ [ti−1 , ti ) ||A||1 β1,i−1 are obtained from the inequality (2.2) according to l1 norm. Strategy 3 (SSS3) The step sizes given by 1 2δL 1 (3.3) hi ≤ ( ) 2 , τi ∈ [ti−1 , ti ) ||A||∞ β∞,i−1 from the inequality (2.2) according to l∞ norm. Strategy 4 (SSS4) The step sizes given by 1 2δL 1 (3.4) hi ≤ ( ) 2 , τi ∈ [ti−1 , ti ) ||A||F β2,i−1 THE ANALYSIS OF THE EFFECT OF THE NORMS IN THE STEP SIZE ... 119 Table 1. Step number with the strategies in numerical integration from the inequality (2.2) according to l2 norm. √ Strategy 5 (SSS5) By using inequality ||A||2 ≤ ||A||F ≤ N ||A||2 , the step sizes are calculated by (3.5) hi ≤ 1 1 2δL ( ) 2 , τi ∈ [ti−1 , ti ) ||A||2 N β2,i−1 from the inequality (2.2) according to l2 norm. Strategy 6 (SSS6)By using inequality ||A||2 ≤ N ||A||max , the step sizes obtained by (3.6) hi ≤ 1 2δL 1 ( ) 2 , τi ∈ [ti−1 , ti ) N ||A||max β2,i−1 from the inequality (2.2). Strategy 7 (SSS7)The step sizes are given as follows 1 2δL 1 ( ) 2 , τi ∈ [ti−1 , ti ) ||A||1 ||A||∞ β2,i−1 p by considering the inequality ||A||2 ≤ ||A||1 ||A||∞ . (3.7) hi ≤ 3.2. Analysis of the S (...truncated)