Computer dialogue system MVC

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(1)Sel cuk J. Appl. Math. Vol. 2, No. 2, pp. 17{38, 2001. Selcuk Journal of Applied Mathematics. Computer dialogue system MVC Haydar Bulgak,1 Diliaver Eminov2 1 2. Research Centre of Applied Mathematics, Selcuk University, Konya, Turkey e-mail: hbulgak@selcuk.edu.tr Spectel Ltd., Dublin, Ireland e-mail: Diliaver.Eminov@spectel.com. Received: December 16, 2001. Summary. The computer-aided methods of mathematical educa-. tion include three main parts: computers, textbooks and software. An experimental version of this approach was elaborated in the Research Centre of Applied Mathematics of the Selcuk University. This version includes four textbooks: Analysis 2], Linear Algebra 6], Linear Di erence Equations and Stability Theory 1], and Matrix Computations with Guaranteed Accuracy in Stability Theory 12]. The version is supported by the computer dialogue system Matrix-vector calculator (MVC) 20], 21].. Key words: projections onto invariant subspaces, matrix spectrum dichotomy, matrix equations, linear algebraic equations, pseudospectrum, computer dialogue system, stability theory, condition number Mathematics Subject Classication (1991): 39A11, 65F, 65L, 65Y 1. Introduction Designing the problem solving environments with automatic result verication, which provide full control over e ects of the computational errors and the uncertainties in the data, is a new methodology for scientic computations. It is interesting to create such algorithms and programs that are stable to the inuence of rounding-o errors and that allow getting the results along with the possibility to evaluate results precision.

(2) 18. H. Bulgak and D. Eminov. or detecting the ill-conditionality of the problem. This kind of algorithms and programs are called the algorithms and programs with guaranteed accuracy. Creating the algorithms with guaranteed accuracy requires changing several classical concepts of linear algebra. In the mid 1980s, Prof. Babenko K.I. mentioned the lack of suitable textbooks, which systematically and briey represent supplementary information from the di erent elds of mathematics pertaining to numerical method applications ( see 3], p. 7 ). At present, there are a number of books that ll that vacuum ( see, for example, 3], 16], 22], 25], 27]-29], 34], 35], 43] ). A certain contribution in the system of new teaching of mathematics was made in the Selcuk University Research Centre of Applied Mathematics ( see 1], 2], 6], 12] ). A computer system - MVC 20], 21] - was created to support the books. This system is not a computer language. It is an extended version of a pocket calculator. MVC works under WINDOWS-98/NT/2000 operation system on an IBM PC compatible computer. The MVC has the following subsystems: Matrix-Vector Calculator - MVC Graphics Constructor Cauchy Solver Discrete-time Cauchy Solver Spectrum. This system is used for mathematical studying in the Statistics Department of the Selcuk University and the Engineering Faculty of the Selcuk University. Here we will briey describe the main points of our approach.. 2. Analysis Mathematical Analysis studying must begin with a "linear continuum" concept 30]. The real numbers is an interpretation of a "linear continuum". Another interpretation of it is the Euclidean line. An important part of the "linear continuum" concept is the fact that any interval is an equivalent to a line. Here we can recall a wellknown Zenon paradox that Attila will never catch a turtle. This is a reason why we can use only symbolic identications for computing "linear continuum" elements. Instead of the real numbers, which are used in numerical computing, computers use so-called computer numbers ( oating-point numbers ). This numbers is a nite subset of the rational numbers set ( see, for example, 2], 13], 40],41] )..

(3) Matrix-Vector Calculator MVC. 19. 2.1. Format. Let > 1 P; P+ k > 0 be integers. In 2] a set of computer numbers was introduced as follows k F( P; P+ k) = f0g fz 2 R j z = p P mj j =1. ;j . where p mj 2 Z p 2 P; P+ ] mj 2 0 ; 1] m1 > 0g: The sets F = F( P; P+ k) are called the formats. The number "1 (F) = P+ (1 ; ;k ) is the biggest number of F ";1 (F) = ;"1 (F) is the smallest number of F "0 (F) = P; = is the smallest positive element of F. If "1 (F) = 1;k then there are no members of F in (1 1 + "1 (F)) but the number 1 + "1 (F) belongs to F. It is clear that k = 1 ; ln "1 = ln P; = 1 ; ln "0 = ln P+ = ln "1 = ln ; ln 1 ; "1= ]= ln : If Q; P; P+ Q+ k s then F = F( P; P+ k) G = F( Q; Q+ s) , and G is called a thin format with respect to F. In this case F is called a rough format with respect to G. Before any arithmetic operations can be performed on R, we have to choose a format for approximations of reals and in the sequel F we refer to the chosen format as to the standard format. Given a real number z and the format F, we dene z ]F as an element of F which is the nearest one measured by j:j to z . To every format F we also assign F-approximations a]F (ai ]F) of a real vector a and A]F (Aij ]F) of a real matrix A. It is clear that for any z, jz j "1 , there are exist two numbers such that jz ]F ; z j "1 jz j + "0 z ]F = z + jz j + = 0 jj "1 j j "0 are true. For a matrix A ( A 2 RN N max1ij N jAij j "1 ) and a vector f (f 2 RN max1j N jfj j "1 ) the similar statements are exist p p kf ]F ; f k "1 kf k + N"0 kA]F ; Ak N"1kAk + N"0 Here kf k - Euclidean norm of vector f kAk - the spectral norm of matrix A. The concept of the format allows a student to understand how the computer engineers resolve Zenon paradox for computing..

(4) 20. H. Bulgak and D. Eminov. 2.2. Regular subsets of a format. For any integer p P; p P+ , we can dene the following subset k Fp = Fp( P; P+ k) = fz 2 R j z = p P mj j =1. ;j . where mj 2 Z mj 2 0 ; 1] m1 > 0g: If p = p;k then the minimal element of Fp is equal to p;1 and the maximal element is equal to p ; p. Moreover, the distance between the neighbour elements of Fp is equal to p. The number of elements of Fp is equal to ( ; 1) k;1. Hence. F( P; P+ k) = f0g f and F has 1 + 2(P+ ; P; + 1)(. ; 1) k;1. P+. p=P;. Fpg. elements.. 2.3. Real function graphics inside format. Let y = f (x) be a real function. It Graphics is a subset of R2 :. R = f(x y) : y = f (x) x 2 Rg: It is clear that the graphics of y = f (x) inside format F is a set. F = f(x y) : y = f (x)]F x 2 Fg: For example, y = x2 function graphics inside F = F(2 ;1 1 2) = f;1:5;1;0:75;0:5;0:375;0:2500:250:3750:50:7511:5g is a set which contains 11 points inside a rectangular domain x 2 ;1:5 1:5] y 2 0 1:5]. Let G = F(2 ;2 1 3) be a thin format with respect to the format F and let H = F(2 ;2 2 10) be a thin format with respect to the format G. In this case the graphics G and H of the function y = x2 inside a rectangular domain x 2 ;1:5 1:5] y 2 0 1:5] are given in Fig. 1 and Fig. 2 respectively. 2.4. Another regular subsets of rational numbers. Let " be a positive number and let M be a natural number. The set Q" (M ) = fx : x = n" n = 0 1 2 3 : : : M g is another alternative to the linear continuum. The similar sets are used for simulating the pixel structure of a computer screen..

(5) Matrix-Vector Calculator MVC. 21. Fig. 1.. Fig. 2.. 2.5. Plotting function graphics on computer screen. Visualising a real function graphics on the computer screen, we have to remember that the pixels on the screen are represented in Decaurte coordinate system. It is given by two natural numbers: m - number of columns and n - number of rows. Ordinary, we use the resolutions with n = 640 and m = 480 n = 800 and m = 600 or n = 1024 and m = 768 for a personal computer systems. If the distance between the horizontal neighbour pixels is a real number x and between two vertical pixels is a real number y then the cortesian product Qx (n) Qy (m) is another alternative to the continuum. Let y = f (x) has F graphics inside the Format F . For its part. F \ f(x y ) : a x b A y B g where a b A B 2 F, we can draw a full screen picture if we choose x = (b ; a)=m and y = (B ; A)=n: Hence we have the sets X = fx : x = a + k (b ; a)=m k = 0 1 : : : mg.

(6) 22. H. Bulgak and D. Eminov. Y = fy : y = A + k (B ; A)=n k = 0 1 : : : ng and the Graphics plotted on the screen is the set ( F )screen = f(x y ) : y = f (x)]F]Y x 2 X g: Hence the intervals f(a + k (b ; a)=m a + (k + 1) (b ; a)=m) k = 0 1 ::: m ; 1g are the "blind intervals". It is illustrated by the following example. Let n = 300 m = 300 a = ;1:5 b = 1:5 A = 0 B = 1:5. The graphics of the functions y = x2 and y = g (x), g(x) = 300x 0 < x < 0:005 g(x) = ;299:98x + 2:9999 0:005 x < 0:01 g(x) = x2 ;1:5 x 0 0:01 x 1:5 inside a rectangle ;1:5 x 1:5 0 y 1:5 are the same. This graphics is given in the Fig. 2. It is clear that g (0:005) = 1:5 is not plotted on the screen. So the question "how accurate the graphics is plotted?" is the main issue for the screen visualisation. Solving this problem, the students are intensively learning the main topics of mathematical analysis such as the continuity, "- continuity, "- oscillation, maximal and minimal points, derivatives of a real function. 2.6. Graphics Constructor. The Graphics Constructor 20] is one of the autonomous parts of MVC. Students use it for building graphics for the real "-continuous functions. A student chooses a graphical area and a table function using the program, which approximates suciently well the given real function in the question. The student can see how the nal curve depends on chosen type of interpolation function. The Graphics Constructor draws the rst, second and a special third degree interpolation functions. It can also draw the graphical representation of the Newton interpolation polinom.. 3. Linear algebra The topics of the linear algebra are given in their natural self-relations in 6]. In the classical linear algebra we have two main problems 33], 36], 37]:.

(7) Matrix-Vector Calculator MVC. 23. a spectral problem systems of the linear algebraic equations Ax = f: We are going to give a modern version of the linear algebra that deals with only suciently well posed problems. Similar approach is realised in the 25], 27], 29], 34], 35], 43]. 3.1. Symmetric spectral problem. Begin with two examples. The rst example. Let. 4 1. S= 12 be apsymmetric matrix. It is clear that S has 2 eigenvalues 1 = p 3 + 2 2 = 3 ; 2 . But these numbers are irrational, since there is no format which includes this numbers. Another example. Take a standard format F and a matrix 1 + 0:5" 0 1 S= 0 1 + 0:25"1 It is clear that S ]F = I - the identical matrix and hence arbitrary vector x(kxk = 1) is an eigenvector of S ]F , but there are only two eigenvectors of S which norms are equal to 1. These two examples allow us to conclude that instead of eigenvalues we have to speak about pseudoeigenvalues and instead of eigenvector we have to speak about a maximal invariant subspace that associated to the well-separated part of the spectrum. Classical results on invariant subspaces of linear operators are contained in 17], 28]. Let S be N -dimensional symmetric matrix with ordered eigenvalues 1(S ) 2 (S ) : : : N (S ). Assume that a set of m eigenvalues are separated from the other ones to a distance ( > 0) or, more correctly, for one p p(S ) ; p+1(S ) : : : p+m(S ) p+m+1(S ) + is true. Denote Lm (S ) is the maximal invariant subspace of S that associated to the eigenvalues p+1(S ) p+2(S ) : : : p+m(S ). Let be N -dimensional symmetric matrix and 1(S + ) 2(S + ) : : : N (S + ): Denote Lm (S + ) is the maximal invariant subspace of S + that associated to the eigenvalues p+1 (S + ) p+2(S + ) : : : p+m(S + ). One can prove the following theorem..

(8) 24. H. Bulgak and D. Eminov. Theorem 1. Let G and P be orthogonal projectors to Lm(S ) and Lm (S + ) respectively. If 2k k < then the inequality 2 k k kG ; P k 2 ; k k is true. 3.2. Eigenvalue decomposition (EVD). For each symmetric matrix S we can nd an orthogonal matrix U such that U SU is a diagonal matrix. The diagonal elements of this matrix are the eigenvalues of S . Inside the standard format F we can speak about a pseudo-orthogonal matrix V such that V SV is a diagonal matrix, which diagonal elements are the pseudoeigenvalues of S . This representation of S is known as an eigenvalue decomposition (EVD) of S . Computing of EVD is a well-posed problem see, for example, 38]). 3.3. Non-symmetric spectral problem. In general case the spectral problem for non-symmetric matrices is illposed. There are two main generalisations of EVD for non-symmetric matrices. The rst one is SVD - a singular value decomposition: for any square matrix A there are exist the orthogonal matrices U and V such that UAV is a diagonal matrix with non-negative elements on the diagonal. These elements are known as the singular values of A. Another generalisation is the well-known Schur theorem: Let 1(A) 2(A) : : : m(A) be the eigenvalues of A, which are well-separated by the given closed curve ; from another part of the spectrum of A, then there is exist an orthogonal matrix U such that U AU = B0 D C is true. Here B and C are respectively m and N ; m dimension square matrices, and the eigenvalues of B are 1 (A) 2(A) : : : m(A). 3.3.1. Spectral problem in the main computer languages. At present the computer-aided education methods are based on computer languages like MATLAB, MATHEMATICA and MAPLE. All these languages can also be used as a toolbox for solving the traditional problems of the applied mathematics..

(9) Matrix-Vector Calculator MVC. 25. Unfortunately, sometimes the users have no information about the accuracy of the computed results. It is not convenient for the educational process since the users cannot perceive that they are working with the nite precision computing. In general they have no information about the quality of the problem in the question. This situation is related to the traditional textbooks of the main mathematical courses. It can be illustrated by the following examples. Take the matrices 27]. 0 289 2064 336 128 80 B 1142 30 1312 512 298 B B ;29 ;2000 756 384 1008 B 512 128 640 0 640 C=B B B B 1053 2256 ;504 ;384 ;756 B @ ;287 ;16 1712 ;128 1968. 32 128 224 512 800. 1 00 0 0 1 0 0 01 BB 0 0 0 0 1 0 0 CC 32 C CC BB 0 0 0 0 0 1 0 CC 48 C C B C 128 C P = B 0 0 0 0 0 0 1 C C B BB 1 0 0 0 0 0 0 CCC 208 C C @0 1 0 0 0 0 0A 2032 A 16. ;30 ;2166 ;287 ;1565 ;512 ;551 ;1152 ;289. 0 1 0 0 0 0 0 01 01 BB 0 1 0 0 0 0 0 CC BB 0 BB 1 0 1 0 0 0 0 CC BB ;1 B C B ; 1 ; 1 P = P L = B BB 00 00 01 10 01 00 00 CCC L = BBB 01 B@ 1 0 0 0 0 1 0 CA B@ ;1 Let. 0110101. 0. 0010000. 1 0 0C CC 0 0C C 0 0C : C 0 0C C 1 0A. 0. 0 0 0 00. 1. 0 0 0. 0. 1 0 0. 0 0 0. ;1. 0 1 0 1. ;1 0. 0 0 0 0 0. ;1 0 1. D = P ;1 CP F = P ;1DP R = LCL;1: It is clear that the spectra of the matrices C C D F , and R are the same. Here R is an upper triangular matrix. The eigenvalues that are computed by MATLAB (The language of Technical computing, Version 5.1.0.421) are represented in Table 1. (C ) (C ) (D) (F ) 7:4375 4:5759+5:1266i 4:5759;5:1266i ;1:6680+6:3482i ;1:6680;6:3482i ;6:6266+2:7764i ;6:6266;2:7764i. 6:1487 3:7394+3:9260i 3:7394;3:9260i ;1:3708+4:8607i ;1:3708;4:8607i ;5:4430+2:0861i ;5:4430;2:0861i. 6:8538+2:9390i 6:8538;2:9390i 1:6387+6:5986i 1:6387;6:5986i ;4:7175+5:2190i ;4:7175;5:2190i ;7:5501. 7:3467 4:5053+5:0464i 4:5053;5:0464i ;1:6566+6:2245i ;1:6566;6:2245i ;6:5220+2:7118i ;6:5220;2:7118i. Table 1 The eigenvalues of the matrices C C D and F computed by MATHEMATICA 3.0 are represented in Table 2..

(10) 26. H. Bulgak and D. Eminov. (C ). (C ). 4:30898+4:73644i. 2. 7:00739. 4:30898;4:73644i. ;1:56301+5:88134i ;1:56301;5:88134i ;6:24967+2:56836i ;6:24967;2:56836i. 4 1 0. ;1 ;2 ;4. (D). (F ). 7:85815. 7:16056+3:11114i. 4:84176;5:50039i. 1:70531+6:96541i. 4:84176+5:50039i. 7:16056;3:11114i. ;1:76531+6:80819i 1:70531;6:96541i ;1:76531;6:80819i ;4:93002+5:50205i ;7:00552+2:98424i ;4:93002;5:50205i ;7:00552;2:98424i ;7:87171. Table 2 The eigenvalues sets, which are computed by MAPLE (Maple 5, Release V, Version 5.00) for the matrices C C D F , and R, are the same and equal to f;4 ;2 ;1 0 1 2 4g. Consider a matrix. 02 B5 X=B @. 1 8C CA :. 2 3 4. 6 7 9 10 11 12 14 16 16 15. Computing the eigenvalues of X the MAPLE gives as the answer the message that the eigenvalues in the question satisfy to equation 4 + 13z ; 98z 2 ; 34z 3 + z 4 = 0. It is easy to check that the equation is the characteristic equation for X but the user wanted to get the exact values. These examples show that the spectral problem in its traditional formulation doesn't have a suciently good realisation in the popular software. 3.3.2. Spectral portrait of a matrix. The problem of eigenvalues com-. putation is not the one that can be considered as a correctly stated. From the view of the "indeniteness principle" we can only guarantee that instead of matrix A we deal with a certain matrix A0 that is close to A. We know about this matrix only that the inequality kA ; A0 k "kAk is true for a small positive value ". The " characterises either the number of digits in a computer cell or the uncertainties in data. Thereby, instead of the eigenvalues of the matrix A we deal with the spots of its ""-spectrum". Denition. ( see 23], 27], 31], 32], 39], 42] ). Let " 0 be given. A complex number is in the ""-spectrum" of A, which we denote by " (A), if one of the following equivalent conditions is satised: 1. the smallest singular value of A ; I is less than or equal to "kAk 2. a vector u is exist such that (u u) = 1 and k(A ; I )uk "kAk.

(11) Matrix-Vector Calculator MVC. 27. 3. is in the spectrum of A or k(A ; I );1k 1=("kAk) 4. is in the spectrum of A + B ( ), where the matrix B ( ) satises to kB ( )k "kAk 5. is in the spectrum of A + "kAkv2 v1 , where kv1k 1 kv2k 1 6. is in the spectrum of A ; (Av ; v )v , where kv k 1 and kAv ; v k "kAk . The elements of " (A) are known as the pseudoeigenvalues. Let S (0 z ) be a circle with the radius z and the centre in (0 0). It is easy to check that " (A)

(12) S (0 (1 + ")kAk). Hence for any " < 0:01 the ""-spectrum" of A lies inside S (0 1:01kAk). For the given numbers 1 > 2 > 3 > :::: > k > 0 we can write k (A) : : : 3 (A) 2 (A) 1 (A) S (0 (1 + 1 )kAk): Hence for all < 0:01 we can draw a set (A), which is contained inside the circle S (0 1:01kAk). If 1 > 2 and we have 1 (A) then we can rene 2 (A) and so on, up to k . The picture, which has been rened k times, is called a spectral portrait of the matrix A. Let's return to the matrix C . The Dialogue Computer System MVC allows us to choose the di erent colours for the di erent numbers i . For example, in Fig. 3 we have the numbers 1 = 10;1 2 = 10;3 3 = 10;5 4 = 10;7 5 = 10;9 6 = 10;11 7 = 10;13 8 = 10;14 and eight di erent colours. Also in Fig. 3 we have a spectral portrait of C . The MVC allows us to get the more detailed picture of a selected region that can be a subject of a particular interest for us. We can select this rectangular region using the mouse (see the result in Fig. 4 ). We see that the "10;14-spectrum" of C has a diameter greater then 30. This is the reason why the program MATLAB gives the distinct results for the matrix C and matrices C D and F , which are similar to C (see Table 1). The MVC allows either checking how many eigenvalues of A lie inside the chosen rectangular region or detecting that the bound of the chosen rectangular region overlaps with the pseudospectrum of A. 3.3.3. Dichotomy by imaginary axis. Attempts to formulate the con-. cept of quality for the eigenvalues of non-self-adjoint matrices led to an idea of pseudo-eigenvalues ( see, for example, 23], 27],31], 32], 39], 42]). Further, the parameters of the matrix spectrum dichotomy by the imaginary axis and the unit circle, and also the schemes of their computation, were stated 9], 10], 24]..

(13) 28. H. Bulgak and D. Eminov. Fig. 3.. Fig. 4..

(14) Matrix-Vector Calculator MVC. 29. In this connection, the necessity in computing the maximal invariant subspaces of the non-self-adjoint matrices or, more exactly, in computing the projectors on these subspaces was emphasised. The algebraic formulation of this problem has been given in the works 9]-11]. Let G(+0 A) be a projector on the maximal invariant subspace L+ (A) of the matrix A that is associated to its eigenvalues that are situated strictly in the left-hand half-plane. We can compute it as the solution of the following system A H + HA + GG ; (I ; G) (I ; G) = 0 (1) GH ; HG = 0 G2 ; G = 0 AG ; GA = 0 on the set of the square N dimensional matrices G H = H > 0. The system (1) is a generalisation of the Lyapunov matrix equation. The theorem is proved in 10], 11]. Due to this theorem the matrix A has no eigenvalues on the imaginary axis, if there is exist a solution of (1). Moreover, in this case G = G(+0 A) is the required projector and (A) = 2kAjkH k is the dichotomy parameter of the matrix A. Further, the guaranteed accuracy algorithm of the projector G = G(+0 A) computation has been substantiated in 10]. Relations between the standard Format and the parameter (A) with respect to one variant of the orthogonal power method of computing G are discovered in 10]. It allows dening a number which depends on "1 . This number characterises a quality of separating the spectrum of A by imaginary axis. The algorithm allows either compute the matrices P and Y , kP ; Gk 2"1 kGk kY ; H k 2"1 kH k or recognised that the matrix A is a practically ill-posed matrix ((A) > ). The formulas (1) and the parameter (A) allow giving the error estimations for the computed approximations to G and H . The following theorem is true 10]. Theorem 2. Let square N -dimensional matrices A P and Y = Y > 0 satisfy inequalities kY kkY ;1 k 0:1= kAP ; PAk kAk kP 2 ; P k 0:5 kAY + Y A + P P ; (I ; P ) (I ; P )k 2 kP Y ; Y P k kY k 1 where ~ = kAkkY k < 800~ min(1 8~=kP k). Then the matrix A is exponentially dichotomous, N+ (A) = N1 N;(A) = N ; N1 ( N1 is a.

(15) 30. H. Bulgak and D. Eminov. quantity of the singular numbers of matrix P greater than 1 ; 2 ) . Moreover, the following estimations are valid: j(A) ; ~ j 92~ kP ; Gk 27~2 kY ; H k=kY k 90~ . where G and H are solutions of system (1).. The variant of this theorem allows formulating the renement algorithm of the obtained approximate solution of the system (1) in case of well-posed problem ((A) < ) . The similar approach is known for dichotomy parameter by circle, rectangular and ellipse (see 6], 19], 26] ). There are a number of algorithms that allow computing the projector to a maximal invariant eigensubspace that corresponds to the eigenvalues that are well separated by the given closed curve 9], 18], 19], 26], 29]. 3.4. Inverse Problem. The problem of computing A;1 for a regular N -dimensional matrix A is known as an inverse problem. The MVC has several inverse functions for inverting bi-diagonal, triangular, and square matrices. All functions has as a control parameter. This parameter characterises the quality of regularity for A. The functions return the inverse matrix in the question if 1 (A) < N (A). Here 1 (A) N (A) are the minimal and maximal singular values of A. The MVC has functions for computing a Penrose inverse matrix for a rectangular matrix with N rows and M columns. The left Penrose inverse matrix is computed for M < N . The function gives either approximation to the matrix in the question or inequality 1 (A) < M (A). The last inequality means that the columns of A are practically linear dependent. The right Penrose inverse matrix is computed for M > N . The function gives either approximation to the matrix in the question or inequality 1(A) < N (A). The last inequality means that the rows of A are practically linear dependent. 3.5. System of the linear algebraic equations. From a pedagogical point of view, it is important to prove an existence of a nontrivial vector that satises to Ax = 0, if A has 0 as an eigenvalue. There is exist an exact formula for x 6] in case of Jacobean symmetric matrix A. Using the Householder transformations and the plane rotations, this formula allows computing EVD for symmetric.

(16) Matrix-Vector Calculator MVC. 31. A and SVD for non-symmetric A. These representations of A allow giving the general theory for solving the linear algebraic equations Ax = f with the given right-hand vector f . The MVC has several solver functions for solving bi-diagonal, triangular, and square A matrices. All functions has as a control parameter. This parameter characterises the quality of regularity for A. The functions return the vector A;1 f if 1 (A) < N (A). Here 1(A) N (A) are the minimal and maximal singular values of A. The MVC has the function for computing a normal solution for a rectangular matrix with N rows and M columns A ( M > N ). The function returns false if the given matrix A is not well-posed ( 1 (A) < N (A)) or true in other case. If the function returned value is true then the vector in the question and a basis of the Ax = 0 vector space are given as the result of computing. The MVC has the function for computing a solution for a rectangular matrix with N rows and M columns A ( M < N ). The function returns false if the columns of A are practically linear dependent ( 1(A) < M (A)) or true in other case. If the function returned value is true then the vector u in the question is determined and it together with the value kAu ; f k=kf k are given as the result of computing. 3.6. MVC - Matrix-Vector Calculator. The MVC - Matrix-Vector Calculator 20], 21] was elaborated for calculating expressions that include vectors, matrices and constants. It also has a set of intrinsic functions that allows performing a wide range of algebraic operations. The MVC is a C++ toolbox. Users don't need to be familiar with the C++ they only have to call the functions that are listed in the book 6]. All concepts of the book can be correctly computed using the MVC. The MVC is a Windows application. The main window of the application is divided into two parts ( see Fig. 5). You can dene your own objects as Matrices, Vectors and Constants at the right-hand side of the window. A large white eld at the left-hand side is called the worksheet. Here you will enter expressions and get results. The le operations and the help are available via the main menu of the MVC. 3.6.1. Dening objects in MVC. The MVC works with objects. The objects in the MVC are Matrices, Vectors and Constants. You can.

(17) 32. H. Bulgak and D. Eminov. Fig. 5.. dene the objects in the right side of the main window. It has three pages named Constants, Vectors and Matrices. At the rst page, Constants, in the elds named 'Name' and 'Value' below the label 'New Constant' you can enter the name and the value of your Constant Object. After you entered the name and the value, press 'OK' button. The new Constant will appear at the last line of the list of the Constants. Now you can use it in your calculations. At the second page you can enter the Vector Object in the similar way. Under the label 'New Vector' enter the name of the vector, the size of it and the elements of the vector in the corresponding elds. After that click 'OK' button. The new vector will appear at the last line of the Vectors list. The third page serves for dening and viewing Matrix Objects. It is impossible to show all matrix objects at the same screen, therefore this page works in one of two modes. The switch at the left upper corner of the page switches between these modes. The rst mode is 'New Matrix'. In this mode the middle part of the page is active. Under the label 'New Matrix' enter the name of the matrix, rows and columns number. Then, at the bottom of the page, enter the elements of the matrix. When you nished entering the matrix, press 'OK' button..

(18) Matrix-Vector Calculator MVC. 33. The second mode is 'Matrices View'. It allows you to view existing Matrix Objects. If you switch to this mode, the right upper corner of the page becomes active. Choose the necessary matrix name under the label 'Matrix Name' and the matrix will be shown at the bottom of the page. Important note. All names that are used at all pages must be unique. 3.6.2. The worksheet. The large white eld at the left side of the. main window is called the worksheet. Here you can enter expressions and get the results. You can use in the expressions arbitrary oating-point numbers as well as all predened Constants, Vectors and Matrices by their names. You can enter an expression starting at the rst position of any free line. When you nished entering an expression, press < Enter >. The result will be shown at the last non-occupied line. If the result includes several objects they will be shown one by one, each on the new line.. 4. Discrete-time equations and stability theory The Discrete Cauchy Solver is one of the autonomous parts of the MVC. It is intended for solving a Discrete-time Cauchy problem (2) x(n + 1) = Ax(n) n = m m + 1 : : : M x(m) = a for the given integers m M , matrix A and vector a. The 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 an interpolation function for p min( !(1 ; 21! )(n;m)=2 kAkn;m)kak n = m m + 1 : : : M choosing a real number ! > 1. It allows detecting of practically illposed problems with respect to the discrete-time asymptotic stability of A ( see 1], 13] ). A number ! (A) 1], 9] is suggested as a measure of quality of discrete-time asymptotic stability of A. Its denition bases on the solution of the discrete-time Lyapunov matrix equation (3) AHA ; H = ;I: If H = H > 0 exists and satises to (3) then ! (A) = kH k else !(A) = 1. The parameter !(A) allows estimating the solution of.

(19) 34. H. Bulgak and D. Eminov. (2): kx(n)k . q. !(A)(1 ; !(1A) )(n;m)=2kak n = m m + 1 : : : :. If we have an autonomous discrete-time system (4) x(n + 1) = Ax(n) + '(x(n)) n m and for the given positive numbers q we have (5) (H'(x) '(x) q (Hx x)1+2 x 2 RN then the following theorem is true 15]. Theorem 3. Let !(A) < 1. If x(m) = q a 2 fx 2 RN : (Hx x)p g = p1q ( ! (A)kAk2 + 2!1(A) ; ! (A)kAk) then for solution to (4) with condition (5) the inequality kx(n)k . q. ! (A)(1 ; 2!1(A) )(n;m)=2 kak n = m m + 1 : : : :. is true.. Taking into account the uncertainties in data, a user can choose a number ! > 1 that characterises a level of a practical discrete-time asymptotic stability. The value ! can also be chosen with respect to the standard format, which is used for computing 13]. The given number ! > 1 and the standard format allow dening the accuracy for intermediate computations such that (2) is computing with standard format accuracy 14], 44]. There is a generalisation of ! (A) and ! parameters in case of A is an interval matrix. It allows us to formulate an algorithm with guaranteed accuracy for checking a practical discrete-time asymptotic stability of an interval matrix 5]. The MVC has the special functions for computing ! (A) with a computer precision solving a discrete-time Lyapunov matrix equation AXA ; X = ;C C = C solving a discrete-time Sylvester matrix equation BXA ; X = ;C: The functions use ! as a control parameter..

(20) Matrix-Vector Calculator MVC. 35. 5. Di

(21) erential equations and stability theory The Cauchy Solver is another autonomous parts of the MVC. It is intended for solving a Cauchy problem (6) x_ (t) = Ax(t) m t M x(m) = a for the given reals m M , matrix A and vector a. The simple dialogue allows constructing the interpolations of the components of vector function in the question and its Euclidean norm. The users also can draw a graphic of a function kAk. p. min( e;(t;m) e(t;m)kAk)kak m t M choosing a real number > 1. It also allows detecting of practically ill-posed problems with respect to asymptotic stability of A ]. A number (A) 7], 8] is suggested as a measure of the quality of asymptotic stability of A. Its denition bases on the solution of the Lyapunov matrix equation (7) AH + HA = ;I: If H = H > 0 exists and satises to (7) then (A) = 2kAkkH k else (A) = 1. The parameter (A) allows estimating the solution of (6):. q. kAk. (A)e;(t;m) (A) kak t m: Taking into account the uncertainties in data, a user can choose a number > 1 that characterises a level of the practical asymptotic stability. The value can also be chosen with respect to the standard format, which is used for computing 13]. The algorithms with the guaranteed accuracy for computing (A) and solving the (7) is exist ( see 12] ). There is a generalisation of (A) and parameters in case of an interval matrix A. It allows us to formulate an algorithm with the guaranteed accuracy for checking of the practical asymptotic stability of an interval matrix 4]. The MVC has the special functions for computing (A) with a computer precision solving a Lyapunov matrix equation A X + XA = ;C C = C solving a Sylvester matrix equation BX + XA = ;C: The functions use as a control parameter. kx(t)k .

(22) 36. H. Bulgak and D. Eminov. 6. Conclusion The textbooks, which give the evolution of some basic concepts of the mathematical education, are prepared in the Selcuk University Research Center of Applied Mathematics 1], 2],6],12]. The ideas of nite digital computing are incorporated into the process of the classic mathematical education. The computer dialogue systems Graphics Constructor 20] and the MVC 21] support these books. The MVC is an easy to use tool that helps studying the systems of discrete time and di erential equations in linear algebra. It is very helpful in the eld of stability theory.. Acknowledgment. An invaluable help in the preparation of the. paper were lent by A. Bulgak, A.O. C!"b"kdiken, G. Demidenko, I. Matveeva, V. Vaskevich. The test examples for Matlab, Mathematica and Maple were computed by V. Ganzha. The authors are very grateful to each of the contributors mentioned above.. References 1. Akn, O . and Bulgak, H. (1998): Linear Dierence Equations and Stability Theory in Turkish], SelUn Basim Evi, Konya. 2. Aydn, K., Bulgak, A. and Bulgak, H. (2000): Analysis in Turkish], SelUn Basim Evi, Konya. 3. Babenko, K. I. (1986) : Base of Numerical Analysis. in Russian], Nauka, Moscow. 4. Bulgak, A. (2001): Checking a practical asymptotic stability of an interval matrix, Selcuk Journal of Applied Mathematics, 2, No. 1, 17{26. 5. Bulgak, A. (2001): Checking a practical discrete -time asymptotic stability of an interval matrix, Selcuk Journal of Applied Mathematics, 2, No. 2, 11{16. 6. Bulgak, A. and Bulgak, H. (2001): Linear Algebra in Turkish], SelUn Basim Evi, Konya. 7. Bulgakov, A. Ya. (1980) : An eectively calculable parameter for the stability quality of systems of linear dierential equations with constant coecients. Siberian Math J. 21,339-347. 8. Bulgakov, A. Ya. Godunov, S. K. (1982): Dicultes calculatives dans le probleme de Hurwitz et methodes pour les surmonter ( aspect calculatif du probleme Hurwitz) in French],in:Analysis and optimization of systems, Proc 5th Int. Conf., Versailles, Fr. 1982, Lect. Notes Control Inf. Sci. 44, 846 - 851, Springer- Verlag. 9. Bulgakov, A. Ya. , Godunov, S. K. (1988): Circle dichotomy of the matrix spectrum. Siberian Math. J. 29, 734{744..

(23) Matrix-Vector Calculator MVC. 37. 10. Bulgakov, A. Ya. (1989): The basis of guaranteed accuracy in the problem of separation of invariant subspaces for non-self-adjoint matrices in Russian], Trudy Inst. Mat. 15, 12{93 English transl. in Sib. Adv. Math. 1, No. 1, 64{108 (1991) and Sib. Adv. Math. 1, No. 2, 1{56 (1991). 11. Bulgakov, A. Ya. (1989): Generalisation of the matrix Lyapunov equation, Siberian Math. J. 30, 525{532. 12. Bulgak(ov), A. Ya. (1995): Matrix Computations with Guaranteed Accuracy in Stability Theory , Selcuk University Press, Konya. 13. Bulgak, H. (1999): Pseudoeigenvalues, spectral portrait of a matrix and their connections with dierent criteria of stability, in: Error Control and Adaptivity in Scientic Computing, H. Bulgak and C. Zenger, eds., NATO Science Series, Kluwer Academic Publishers, 95{124. 14. Bulgak H. Bulgak A. Vaskevich V. (2000) : Computing an Initial value Problem for Systems of Linear Dierence Equations with Error Estimates, in: Cubature Formulae and Their Applications, ( Noskov M. V. - eds.), Krasnoyarsk, 238 - 258. 15. Bulgak, H. , Demidenko, G. V. (2001): Estimation for the region of attraction of nonlinear dierence equations. Num. Math. DOI 10.1007/s002110100353 URL: http://dx.doi.org/10.1007/s002110100353. 16. Chaitin-Chatelin, F. and Fraysse, V. (1996): Lectures on Finite Precision Computations, SIAM, Philadelphiya. 17. Daleckii, Ju. L. ,Krein, M. G. (1974) : Stability of solutions of dierential equations in banach space., Translations of Mathematical Monographs 43, Amer. Math. Soc., Providence. 18. Demidenko, G. V. (2001): On a functional approach to spectral problems of linear algebra, Selcuk Journal of Applied Mathematics, 2, No. 2, 39{52. 19. Demidenko, G. V. (2001): On constructing approximate projections onto invariant subspaces of linear operators, Intern. J. Di. Eq. Appl., 3, No. 2, 135{146. 20. Eminov, D. (2000): Graphycs Constructor, in Turkish], in: Analysis, K. Aydin, A.. Bulgak and H. Bulgak, auths., Konya, SelUn Basim Evi, 342{358. 21. Eminov, D. (2001): MVC1.0 - Matrix Vector Calculator ( C++ version), in Turkish], in: Linear Algebra, A.. Bulgak and H. Bulgak, auths., Konya, SelUn Basim Evi, 265{309. 22. Forsythe, G. E. , Malcolm M. A. and Moler C. B. (1977) : Computer Methods for Mathematical Computations. Englewood Clis, N.J., Prentice-Hall. 23. Godunov, S. K. and Ryaben'kii, V. S. (1964) : Theory of Dierence Schemes: An Introduction, North-Holland, Amsterdam. 24. Godunov, S. K. (1986) : A problem on matrix spectrum dichotomy. Siberian Math J. 27(5),24-37. 25. Godunov, S. K., Antonov, A. G., Kiriluk, O. P. and Kostin, V. I. (1993): Guaranteed accuracy in numerical linear algebra. Kluwer Academic Publishers, Dordrecht, The Netherlands. 26. Godunov, S. K. and Sadkane, M. (1996): Elliptic dichotomy of a matrix spectrum, Linear Algebra Appl. 248: 205-232. 27. Godunov, S. K. (1998): Modern Aspects of Linear Algebra, Translations of Mathematical Monographs. 175. Providence, RI: American Mathematical Society. 28. Gohberg, I. C. , Lancaster, P. and Rodman, L. (1986) : Invariant subspaces of matrices with applications, Wiley, John & Sons, Inc..

(24) 38. H. Bulgak and D. Eminov. 29. Golub, G. and Van Loan, C. (1989) : Matrix Computations, The Joan Hopkins University Press. 30. Hinchin A. (1977): Eight Lectures on Mathematical Analysis in Russian], Nauka, Moscow. 31. Kostin, V. I. , Razzakov, Sh. I. , (1985): On the convergence of the orthogonalpower method for calculating the spectrum in Russian], Trudy Inst. Mat. 6, 55-84. 32. Kostin, V. I. , (1991): On denition of matrices' spectra, High Perfomance Computing II. 33. Kostrikin, A. I. , Manin, Y. I. (1980): Linear Algebra and Geometry, in Russian], Moscow University Press, Moscow. 34. Kulisch, U. and Miranker W. L. (1981) : Computer Numeric in theory and practice . New-York, Academic Press. 35. Hammer, R. , Hocks, M. , Kulisch, U. and Ratz D. (1995) : C++ Toolbox for Veried Computing. Basic Numerical Problems, Springer. 36. Kurosch, A. G. (1975) : High Algebra Course. in Russian], Nauka, Moscow. 37. Malt'sev, A. I. (1970) : Foundations of Linear Algebra. in Russian], Nauka, Moscow. 38. Parlett, B. N. (1980) : The symmetric eigenvalue problem, Prentice Hall Inc., Englewood, Clis, N.J. 39. Riedel K.S. (1994): Generalized Epsilon-Pseudospectra. SIAM J. Numer. Anal., vol. 31, No. 4, 1219{1225. 40. A Standard for Binary Floating Point Arithmetic. (1985) ANSI/IEEE Std. 854{1985, New-York. 41. A Standard for Radix-Independent Floating Point Arithmetic. (1987) ANSI/IEEE Std. 854{1987, New-York. 42. Trefethen, L. N. (1992): Pseudospectra of matrices, in: Griths, D. F., and Watson, G. A. , eds., Proceedings of the 14 Dundee Biennial Conference on Numerical Analysis, Longman Scientic and Tech. Publ., Harlow, U.K., 234{266. 43. Trefethen, L. N. (2000): Spectral methods in MATLAB, SIAM. 44. Vaskevich, Vl. , Bulgak, H. and C nar, C. (2000): Algorithm with Guaranteed Accuracy for Computing a Solution to Linear Dierence Equations, Selcuk Journal of Applied Mathematics, 1, No. 1, 90{96..

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