Lecture notes Advanced Engineering mathematics

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Advanced Mathematics for Engineers Wolfgang Ertel translated by Elias Drotle and Richard Cubek October 1, 2012Chapter 1 Linear Algebra 1.1 Video Lectures We use the excellent video lectures from G. Strang, the author of 1, available from: http:// ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010. In particular we show the following lectures: Lec Topics 1 The geometry of linear equations (lecture 01) n 2 Transposes, Permutations, Spaces R (lecture 05) 3 Column Space and Nullspace (lecture 06) 4 Solving Ax = 0: Pivot Variables, Special Solutions (lecture 07) 5 Independence, Basis, and Dimension (lecture 09) 6 The Four Fundamental Subspaces (lecture 10) 7 Orthogonal Vectors and Subspaces (lecture 14) 8 Properties of Determinants (lecture 18) 9 Determinant Formulas and Cofactors (lecture 19) 10 Cramer's rule, inverse matrix, and volume (lecture 20) 11 Eigenvalues and Eigenvectors (lecture 21) 12 Symmetric Matrices and Positive De niteness (lecture 25) 13 Linear Transformations and Their Matrices (lecture 30) 1.2 Exercises Exercise 1.1 Solve the nonsingular triangular system u +v +w =b (1.1) 1 v +w =b (1.2) 2 w =b (1.3) 3 Show that your solution gives a combination of the columns that equals the column on the right. Exercise 1.2 Explain why the system u +v +w = 2 (1.4) u + 2v + 3w = 1 (1.5) v + 2w = 0 (1.6)4 1 Linear Algebra is singular, by nding a combination of the three equations that adds up to 0 = 1. What value should replace the last zero on the right side, to allow the equations to have solutions, and what is one of the solutions? Inverses and Transposes 1 Exercise 1.3 Which properties of a matrix A are preserved by its inverse (assuming A exists)? (1) A is triangular (2) A is symmetric (3) A is tridiagonal (4) all entries are whole numbers 3 (5) all entries are fractions (including whole numbers like ) 1 Exercise 1.4 a) How many entries can be chosen independently, in a symmetric matrix of order n? b) How many entries can be chosen independently, in a skew-symmetric matrix of order n? Permutations and Elimination Exercise 1.5 a) Find a square 3 3 matrixP , that multiplied from left to any 3m matrixA exchanges rows 1 and 2. b) Find a squarenn matrixP , that multiplied from left to anynm matrixA exchanges rows i and j. Exercise 1.6 A permutation is a bijective mapping from a nite set onto itself. Applied to vectors of length n, a permutation arbitrarily changes the order of the vector compo- nents. The word \ANGSTBUDE" is a permutation of \BUNDESTAG". An example of a permutation on vectors of length 5 can be described by (3; 2; 1; 5; 4): This means component 3 moves to position 1, component 2 stays where it was, component 1 moves to position 3, component 5 moves to position 4 and component 4 moves to position 5. a) Give a 5 5 matrix P that implements this permutation. b) How can we come from a permutation matrix to its inverse? Exercise 1.7 a) Find a 3 3 matrixE, that multiplied from left to any 3m matrixA adds 5 times row 2 to row 1. b) Describe ann matrixE, that multiplied from left to anynm matrixA addsk times row i to row j. c) Based on the above answers, prove that the elimination process of a matrix can be realized by successive multiplication with matrices from left.1.2 Exercises 5 Column Spaces and NullSpaces 3 Exercise 1.8 Which of the following subsets of R are actually subspaces? a) The plane of vectors with rst component b = 0. 1 b) The plane of vectors b with b = 1. 1 c) The vectorsb withb b = 0 (this is the union of two subspaces, the plane b = 0 and the 1 2 1 plane b = 0). 2 d) The solitary vector b = (0; 0; 0). e) All combinations of two given vectors x = (1; 1; 0) and y = (2; 0; 1). f) The vectors (b , b , b ) that satisfy b b + 3b = 0. 1 2 3 3 2 1 Exercise 1.9 Let P be the plane in 3-space with equation x + 2y +z = 6. What is the 3 equation of the planeP through the origin parallel toP ? AreP andP subspaces ofR ? 0 0 Exercise 1.10 Which descriptions are correct? The solutions x of 2 3     x 1 1 1 1 0 4 5 x Ax = = (1.7) 2 1 0 2 0 x 3 form a plane, line, point, subspace, nullspace of A, column space of A. Ax = 0 and Pivot Variables Exercise 1.11 For the matrix   0 1 4 0 A = (1.8) 0 2 8 0 determine the echelon formU, the basic variables, the free variables, and the general solution toAx = 0. Then apply elimination to Ax =b, with components b andb on the right side; 1 2 nd the conditions for Ax = b to be consistent (that is, to have a solution) and nd the general solution in the same form as Equation (3). What is the rank of A? Exercise 1.12 Write the general solution to 2 3     u 1 2 2 1 4 5 v = (1.9) 2 4 5 4 w as the sum of a particular solution toAx =b and the general solution toAx = 0, as in (3). Exercise 1.13 Find the value of c which makes it possible to solve u +v + 2w = 2 (1.10) 2u + 3vw = 5 (1.11) 3u + 4v +w =c (1.12)6 1 Linear Algebra Solving Ax =b Exercise 1.14 Is it true that if v , v , v are linearly independent, that also the vectors 1 2 3 w = v +v , w = v +v , w = v +v are linearly independent? (Hint: Assume some 1 1 2 2 1 3 3 2 3 combination c w +c w +c w = 0, and nd which c are possible.) 1 1 2 2 3 3 i Exercise 1.15 Find a counterexample to the following statement: If v ;v ;v ;v is a basis 1 2 3 4 4 for the vector space R , and if W is a subspace, then some subset of the v's is a basis for W . Exercise 1.16 Suppose V is known to have dimension k. Prove that a) any k independent vectors in V form a basis; b) any k vectors that span V form a basis. In other words, if the number of vectors is known to be right, either of the two properties of a basis implies the other. 5 Exercise 1.17 Prove that if V and W are three-dimensional subspaces of R , then V and W must have a nonzero vector in common. Hint: Start with bases of the two subspaces, making six vectors in all. The Four Fundamental Subspaces Exercise 1.18 Find the dimension and construct a basis for the four subspaces associated with each of the matrices     0 1 4 0 0 1 4 0 A = and U = (1.13) 0 2 8 0 0 0 0 0 Exercise 1.19 If the product of two matrices is the zero matrix, AB = 0, show that the column space of B is contained in the nullspace of A. (Also the row space of A is the left nullspace of B, since each row of A multiplies B to give a zero row.) 0 0 Exercise 1.20 Explain why Ax = b is solvable if and only if rank A = rank A , where A is formed from A by adding b as an extra column. Hint: The rank is the dimension of the column space; when does adding an extra column leave the dimension unchanged? Exercise 1.21 Suppose A is an m by n matrix of rank r. Under what conditions on those numbers does 1 1 a) A have a two-sided inverse: AA =A A =I? b) Ax =b have in nitely many solutions for every b? T Exercise 1.22 IfAx = 0 has a nonzero solution, show thatA y =f fails to be solvable for some right sides f. Construct an example of A and f. Orthogonality 3 Exercise 1.23 InR nd all vectors that are orthogonal to (1, 1, 1) and (1, -1, 0). Produce from these vectors a mutually orthogonal system of unit vectors (an orthogonal system) in1.2 Exercises 7 3 R . Exercise 1.24 Show that xy is orthogonal to x +y if and only ifkxk =kyk. Exercise 1.25 LetP be the plane (not a subspace) in 3-space with equationx+2yz = 6. 0 Find the equation of a plane P parallel to P but going through the origin. Find also a 0 vector perpendicular to those planes. What matrix has the plane P as its nullspace, and 0 what matrix hast P as its row space? Projections Exercise 1.26 Suppose A is the 4 4 identity matrix with its last column removed. A is 4 3. Project b = (1; 2; 3; 4) onto the column space of A. What shape is the projection matrix P and what is P ? Determinants 2 Exercise 1.27 How are det(2A), det(A), and det(A ) related to det A, when A is n by n? Exercise 1.28 Find the determinants of: a) a rank one matrix 2 3 1   4 5 4 2 1 2 A = (1.14) 2 b) the upper triangular matrix 2 3 4 4 8 8 6 7 0 1 2 2 6 7 U = (1.15) 4 5 0 0 2 6 0 0 0 2 T c) the lower triangular matrix U ; 1 d) the inverse matrix U ; e) the \reverse-triangular" matrix that results from row exchanges, 2 3 0 0 0 2 6 7 0 0 2 6 6 7 M = (1.16) 4 5 0 1 2 2 4 4 8 8 Exercise 1.29 If every row of A adds to zero prove that det A = 0. If every row adds to 1 prove that det(AI) = 0. Show by example that this does not imply det A = 1.8 1 Linear Algebra Properties of Determinants Exercise 1.30 Suppose A is the n by n tridiagonal matrix with 1's everywhere on the n three diagonals: 2 3   1 1 0   1 1 4 5 1 1 1 1 A = ; A = ; A = ;::: (1.17) 1 2 3 1 1 0 1 1 Let D be the determinant of A ; we want to nd it. n n a) Expand in cofactors along the rst row of A to show that D =D D . n n n1 n2 b) Starting from D = 1 and D = 0 nd D ;D ;:::;D . By noticing how these numbers 1 2 3 4 8 cycle around (with what period?) nd D . 1000 Exercise 1.31 Explain why a 5 by 5 matrix with a 3 by 3 zero submatrix is sure to be a singular (regardless of the 16 nonzeros marked by x's): 2 3 x x x x x 6 7 x x x x x 6 7 6 7 the determinant of A = 0 0 0 x x is zero. (1.18) 6 7 4 5 0 0 0 x x 0 0 0 x x Exercise 1.32 If A is m by n and B is n by m, show that       0 A I 0 det = = det AB: Hint: Postmultiply by : (1.19) B I B I Do an example with m n and an example with m n. Why does the second example have det AB = 0? Cramers' rule Exercise 1.33 The determinant is a linear function of the column 1. It is zero if two columns are equal. Whenb =Ax =x a +x a +x a goes into the rst column ofA, then 1 1 2 2 3 3 the determinant of this matrix B is 1 jb a aj =jx a +x a +x a a aj =xja a aj =x detA 2 3 1 1 2 2 3 3 2 3 1 1 2 3 1 a) What formula for x comes from left side = right side? 1 b) What steps lead to the middle equation? Eigenvalues and Eigenvectors Exercise 1.34 Suppose that  is an eigenvalue of A, and x is its eigenvector: Ax =x. a) Show that this same x is an eigenvector of B =A 7I, and nd the eigenvalue. 1 b) Assuming  =6 0, show that x is also an eigenvector of A and nd the eigenvalue.1.2 Exercises 9 Exercise1.35 Show that the determinant equals the product of the eigenvalues by imagining that the characteristic polynomial is factored into det(AI) = ( )( ) ( ) (1.20) 1 2 n and making a clever choice of . Exercise 1.36 Show that the trace equals the sum of the eigenvalues, in two steps. First, n1 nd the coecient of () on the right side of (15). Next, look for all the terms in 2 3 a  a  a 11 12 1n 6 7 a a   a 21 22 2n 6 7 det(AI) = det (1.21) 6 7 . . . . . . 4 5 . . . a a  a  n1 n2 nn n1 which involve () . Explain why they all come from the product down the main diagonal, n1 and nd the coecient of () on the left side of (15). Compare. Diagonalization of Matrices 1 Exercise 1.37 Factor the following matrices into SS :     1 1 2 1 A = and A = : (1.22) 1 1 0 0 T Exercise 1.38 Suppose A =uv is a column times a row (a rank-one matrix). a) By multiplying A times u show that u is an eigenvector. What is ? b) What are the other eigenvalues (and why)? T c) Compute trace(A) =v u in two ways, from the sum on the diagonal and the sum of's. 1 Exercise 1.39 If A is diagonalizable, show that the determinant of A = SS is the product of the eigenvalues. Symmetric and Positive Semi-De nite Matrices p T T Exercise 1.40 If A = QQ is symmetric positive de nite, then R = Q Q is its symmetric positive de nite square root. Why doesR have real eigenvalues? ComputeR and 2 verify R =A for     2 1 10 6 A = and A = : (1.23) 1 2 6 10 Exercise 1.41 If A is symmetric positive de nite and C is nonsingular, prove that B = T C AC is also symmetric positive de nite. Exercise 1.42 If A is positive de nite and a is increased, prove from cofactors that the 11 determinant is increased. Show by example that this can fail if A is inde nite.10 1 Linear Algebra Linear Transformation Exercise 1.43 Suppose a linear mapping T transforms (1, 1) to (2, 2) and (2, 0) to (0, 0). Find T (v): (a) v = (2; 2) (b) v = (3; 1) (c) v = (1; 1) (d) v = (a;b) o Exercise 1.44 Suppose T is re ection across the 45 line, and S is re ection across the y axis. If v = (2; 1) then T (v) = (1; 2). Find S(T (v)) and T (S(v)). This shows that generally ST =6 TS. n Exercise 1.45 Suppose we have two bases v ;:::;v and w ;:::;w for R . If a vector has 1 n 1 n coecients b in one basis and c in the other basis, what is the change of basis matrix in i i b =Mc? Start from b v +::: +b v =Vb =c w +::: +c w =Wc: (1.24) 1 1 n n 1 1 n n Your answer representsT (v) =v with input basis ofv's and output basis ofw's. Because of di erent bases, the matrix is not I.Chapter 2 Computer Algebra De nition 2.1 Computer Algebra = Symbol Processing + Numerics + Graphics De nition 2.2 Symbol Processing is calculating with symbols (variables, constants, function symbols), as in Mathematics lectures. Advantages of Symbol Processing: often considerably less computational e ort compared to numerics. symbolic results (for further calculations), proofs in the strict manner possible. Disadvantages of Symbol Processing: often there is no symbolic (closed form) solution, then Numerics will be applied, e.g.: Calculation of Antiderivatives x Solving Nonlinear Equations like: (e = sinx) Example 2.1 1. symbolic:   0 lnx lim =? (asymptotic behavior) x1 x + 1   0 1 (x + 1) lnx lnx 1 lnx x = = 2 2 x + 1 (x + 1) (x + 1)x (x + 1)   0 lnx 1 lnx lnx x1 : 0 2 2 2 x + 1 x x x 2. numeric: 0 lim f (x) =? x112 2 Computer Algebra 2 Example 2.2 Numerical solution of x = 5 5 5 2 x = 5; x = ; 2x =x + x x   1 5 x = x + 2 x iteration:   1 5 x = x + n+1 n 2 x n n x n 0 2 Startwert 1 2.25 2 2.236111 3 2.23606798 4 2.23606798 p 8 ) 5 = 2:23606798 10 (approximate solution) 2.1 Symbol Processing on the Computer Example 2.3 Symbolic Computing with natural numbers: Calculation rules, i.e. Axioms necessary. ) Peano Axioms e.g.: 8x;y;z :x +y = y +x (2.1) x + 0 = x (2.2) (x +y) +z = x + (y +z) (2.3) Out of these rules, e.g. 0 +x =x can be deduced: 0 +x = x + 0 = x z z (2:1) (2:2) Implementation of symbol processing on the computer by "Term Rewriting". Example 2.4 (Real Numbers) Chain Rule for Di erentiation: 0 0 0 f(g(x)) ) f (g(x))g (x) 1 0 sin(lnx + 2) = cos(lnx + 2) x Computer: (Pattern matching) 0 0 sin(Plus(lnx; 2)) = cos(Plus(lnx; 2))Plus (lnx; 2) 0 0 0 sin(Plus(lnx; 2)) = cos(Plus(lnx; 2))Plus(ln x; 2 )2.2 Short Introduction to Mathematica 13   1 0 sin(Plus(lnx; 2)) = cos(Plus(lnx; 2))Plus ; 0 x 1 0 sin(Plus(lnx; 2)) = cos(Plus(lnx; 2)) x cos(lnx + 2) 0 sin(Plus(lnx; 2)) = x E ective systems: Mathematica (S. Wolfram & Co.) Maple (ETH Zurich + Univ. Waterloo, Kanada) 2.2 Short Introduction to Mathematica Resources:  Library: Mathematica Handbook (Wolfram)  Mathematica Documentation Online: http://reference.wolfram.com  http://www.hs-weingarten.de/ertel/vorlesungen/mae/links.html 2.2.0.1 Some examples as jump start In1:= 3 + 23 Out1= 11 In2:= Sqrt10 Out2= Sqrt10 In3:= NSqrt10 Out3= 3.16228 In4:= NSqrt10,60 Out4= 3.1622776601683793319988935444327185337195551393252168268575 In5:= Integratex2 Sinx2, x 3 2 4 x - 6 x Cos2 x + 3 Sin2 x - 6 x Sin2 x Out5= 24 In7:= D%, x 2 2 12 x - 12 x Cos2 x Out7= 2414 2 Computer Algebra In8:= Simplify% 2 2 Out8= x Sinx In9:= SeriesExpx, x,0,6 2 3 4 5 6 x x x x x 7 Out9= 1 + x + + + + - + - + Ox 2 6 24 120 720 In10:= Expand(x + 2)3 + ((x - 5)2 (x + y)2)3 2 3 6 7 8 9 Out10= 8 + 12 x + 6 x + x + 15625 x - 18750 x + 9375 x - 2500 x + 10 11 12 5 6 7 375 x - 30 x + x + 93750 x y - 112500 x y + 56250 x y - 8 9 10 11 4 2 15000 x y + 2250 x y - 180 x y + 6 x y + 234375 x y - 5 2 6 2 7 2 8 2 9 2 281250 x y + 140625 x y - 37500 x y + 5625 x y - 450 x y + 10 2 3 3 4 3 5 3 6 3 15 x y + 312500 x y - 375000 x y + 187500 x y - 50000 x y + 7 3 8 3 9 3 2 4 3 4 7500 x y - 600 x y + 20 x y + 234375 x y - 281250 x y + 4 4 5 4 6 4 7 4 8 4 140625 x y - 37500 x y + 5625 x y - 450 x y + 15 x y + 5 2 5 3 5 4 5 5 5 93750 x y - 112500 x y + 56250 x y - 15000 x y + 2250 x y - 6 5 7 5 6 6 2 6 3 6 180 x y + 6 x y + 15625 y - 18750 x y + 9375 x y - 2500 x y + 4 6 5 6 6 6 375 x y - 30 x y + x y In11:= Factor% 2 3 4 2 3 2 Out11= (2 + x + 25 x - 10 x + x + 50 x y - 20 x y + 2 x y + 25 y - 2 2 2 2 3 4 5 6 10 x y + x y ) (4 + 4 x - 49 x - 5 x + 633 x - 501 x + 150 x - 7 8 2 3 4 5 20 x + x - 100 x y - 10 x y + 2516 x y - 2002 x y + 600 x y - 6 7 2 2 2 2 3 2 80 x y + 4 x y - 50 y - 5 x y + 3758 x y - 3001 x y + 4 2 5 2 6 2 3 2 3 3 32.2 Short Introduction to Mathematica 15 900 x y - 120 x y + 6 x y + 2500 x y - 2000 x y + 600 x y - 4 3 5 3 4 4 2 4 3 4 4 4 80 x y + 4 x y + 625 y - 500 x y + 150 x y - 20 x y + x y ) In12:= InputForm%7 Out12//InputForm= (12x2 - 12x2Cos2x)/24 In20:= PlotSin1/x, x,0.01,Pi Out20= -Graphics- In42:= Plot3Dx2 + y2, x,-1,1, y,0,1 Out42= -SurfaceGraphics- In43:= fx_,y_ := Sin(x2 + y3) / (x2 + y2) In44:= f2,3 Sin31 Out44= - 13 In45:= ContourPlotx2 + y2, x,-1,1, y,-1,1 Out45= -SurfaceGraphics- In46:= Plot3Dfx,y, x,-Pi,Pi, y,-Pi,Pi, PlotPoints - 30, PlotLabel - "Sin(x2 + y3) / (x2 + y2)", PlotRange - -1,1 Out46= -SurfaceGraphics- Sin(x2 + y3) / (x2 + y2) Sin(x2 + y3) / (x2 + y2) 2 1 1 0.5 0 2 0 -0.5 1 -1 -1 0 -2 -2 -1 -1 -1 -2 00 11 -2 -2 -1 0 1 2 22 In47:= ContourPlotfx,y, x,-2,2, y,-2,2, PlotPoints - 30, ContourSmoothing - True, ContourShading - False, PlotLabel - "Sin(x2 + y3) / (x2 + y2)" Out47= -ContourGraphics-16 2 Computer Algebra In52:= Tablex2, x, 1, 10 Out52= 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 In53:= Tablen, n2, n, 2, 20 Out53= 2, 4, 3, 9, 4, 16, 5, 25, 6, 36, 7, 49, 8, 64, 9, 81, 10, 100, 11, 121, 12, 144, 13, 169, 14, 196, 15, 225, 16, 256, 17, 289, 18, 324, 19, 361, 20, 400 In54:= Transpose% Out54= 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400 In60:= ListPlotTableRandom+Sinx/10, x,0,100 Out60= -Graphics- 1.5 1 0.5 20 40 60 80 100 -0.5 In61:= x = Tablei, i,1,6 Out61= 1, 2, 3, 4, 5, 6 In62:= A = Tableij, i,1,5, j,1,6 Out62= 1, 2, 3, 4, 5, 6, 2, 4, 6, 8, 10, 12, 3, 6, 9, 12, 15, 18, 4, 8, 12, 16, 20, 24, 5, 10, 15, 20, 25, 30 In63:= A.x Out63= 91, 182, 273, 364, 455 In64:= x.x Out64= 91 In71:= B = A.TransposeA Out71= 91, 182, 273, 364, 455, 182, 364, 546, 728, 910, 273, 546, 819, 1092, 1365, 364, 728, 1092, 1456, 1820, 455, 910, 1365, 1820, 2275 In72:= B - IdentityMatrix5 Out72= 90, 182, 273, 364, 455, 182, 363, 546, 728, 910,2.2 Short Introduction to Mathematica 17 273, 546, 818, 1092, 1365, 364, 728, 1092, 1455, 1820, 455, 910, 1365, 1820, 2274 % last command %n nth last command ?f help for function f ??f more help for f fx_,y_ := x2 Cosy de ne function f(x;y) a = 5 assign a constant to variable a f = x2 Cosy assign an expression to variable f (f is only a placeholder for the expression, not a function) Dfx,y,x derivative of f with respect to x Integratefx,y,y antiderivative of f with respect to x Simplifyexpr simpli es an expression Expandexpr expand an expression Solvefx==gx solves an equation C cancel InputFormExpr converts into mathematica input form A TeXFormExpr converts into the LT Xform E FortranFormExpr converts into the Fortran form CFormExpr converts into the C form ReadList"daten.dat", Number, Number reads 2-column table from le Tablefn, n, n_min, n_max generates a list f(n );:::;f(n ) min max Plotfx,x,x_min,x_max generates a plot of f ListPlotListe plots a list Plot3Dfx,y,x,x_min,x_max,y,y_min,y_max generates a three-dim. plot of f ContourPlotfx,y,x,x_min,x_max,y,y_min,y_max generates a contour plot of f Display"Dateiname",%,"EPS" write to the le in PostScript format Table 2.2: Mathematica some inportant commands Example 2.5 (Calculation of Square Roots) ( square root iterative ) sqrta_,genauigk_ := Modulex, xn, delta, n, Fordelta=9999999; n = 1; x=a, delta 10(-accuracy), n++, xn = x; x = 1/2(x + a/x); delta = Absx - xn; Print"n = ", n, " x = ", Nx,2accuracy, " delta = ", Ndelta; ; Nx,genauigk sqrt::usage = "sqrta,n computes the square root of a to n digits." Tablesqrti,10, i,1,2018 2 Computer Algebra ( square root recursive ) xn_,a_ := 1/2 (xn-1,a + a/xn-1,a) x1,a_ := a 2.3 Gnuplot, a professional Plotting Software Gnuplot is a powerful plotting programm with a command line interface and a batch inter- face. Online documentation can be found on www.gnuplot.info. 1 sin(x) 0.8 0.6 On the command line we can input 0.4 0.2 0 plot 0:10 sin(x) -0.2 -0.4 to obtain the graph -0.6 -0.8 -1 0 2 4 6 8 10 Almost arbitrary customization of plots is possible via the batch interface. A simple batch le may contain the lines set terminal postscript eps color enhanced 26 set label "/Symbol a=0.01, /Symbol g=5" at 0.5,2.2 set output "bucket3.eps" plot b=0.01:1 a=0.01, c= 5, (a-b-c)/(log(a) - log(b)) \ title "(/Symbol a-/Symbol b-/Symbol g)/(ln/Symbol a - ln/Symbol b)" 8 (α-β-γ)/(lnα - lnβ) 7 6 5 producing a EPS le with the graph 4 3 α=0.01, γ=5 2 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 γ 3-dimensional plotting is also possible, e.g. with the sin((x2 + y3) / (x2 + y2)) commands 1 0.8 0.6 0.4 0.2 set isosamples 50 0 -0.2 -0.4 -0.6 -0.8 splot -pi:pi-pi:pi sin((x2 + y3) / (x2 -1 3 + y2)) 2 1 0 -3 -2 -1 -1 0 -2 1 2 -3 3 which produces the graph t tot2.4 Short Introduction to MATLAB 19 2.4 Short Introduction to MATLAB E ective systems: MATLAB & SIMULINK (MathWorks) 2.4.0.2 Some examples as jump start Out(1)=3+23 ans = 11 Out(2)=sqrt(10) ans = 3.1623 Out(3)=vpa(sqrt(10),60) = 3.16227766016837933199889354443271853371955513932521682685750 syms x syms y y=x2sin(x)2 2 2 x sin(x) z=int(y,x) 2 2 3 x (- 1/2 cos(x) sin(x) + 1/2 x) - 1/2 x cos(x) + 1/4 cos(x) sin(x) + 1/4 x - 1/3 x Der=diff(z,x) 2 2 2 2 x (- 1/2 cos(x) sin(x) + 1/2 x) + x (1/2 sin(x) - 1/2 cos(x) + 1/2) 2 2 2 - 1/4 cos(x) + x cos(x) sin(x) - 1/4 sin(x) + 1/4 - x Simple=simplify(Der) 2 2 x sin(x) Series=Taylor(exp(x),6,x,0) 2 3 4 5 1 + x + 1/2 x + 1/6 x + 1/24 x + 1/120 x (x+2)2+((x+5)2(x+y)2)3 2 6 6 (x + 2) + (x - 5) (x + y) Exp_Pol=expand(Pol) 2 6 5 4 2 3 3 4 + 4 x + x + 15625 x + 93750 x y + 234375 x y + 312500 x y 2 4 5 11 10 2 9 3 + 234375 x y + 93750 x y + 6 x y + 15 x y + 20 x y 8 4 7 5 6 6 10 9 2 8 3 + 15 x y + 6 x y + x y - 180 x y - 450 x y - 600 x y 7 4 6 5 6 12 11 10 9 - 450 x y - 180 x y + 15625 y + x - 30 x + 375 x - 2500 x20 2 Computer Algebra 8 7 5 6 9 8 2 7 3 + 9375 x - 18750 x - 30 x y+ 2250 x y + 5625 x y + 7500 x y 6 4 5 5 4 6 8 7 2 + 5625 x y + 2250 x y + 375 x y - 15000 x y - 37500 x y 6 3 5 4 4 5 3 6 7 - 50000 x y - 37500 x y - 15000 x y - 2500 x y + 56250 x y 6 2 5 3 4 4 3 5 + 140625 x y + 187500 x y + 140625 x y + 56250 x y 2 6 6 5 2 4 3 + 9375 x y - 112500 x y - 281250 x y - 375000 x y 3 4 2 5 6 - 281250 x y - 112500 x y - 18750 x y t=0:0.01:pi plot(sin(1./t)) Plot Mode- X,Y=meshgrid(-1:0.01:1,-1:0.01:1) Z=sin(X.2+Y.3)/(X.2+Y.2) surf(X,Y,Z) x=1:1:10 y(1:10)=x.2 y = 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 A_1=1 2 4; 5 6 100; -10.1 23 56 A_1 = 1.0000 2.0000 4.0000 5.0000 6.0000 100.0000 -10.1000 23.0000 56.0000 A_2=rand(3,4)2.4 Short Introduction to MATLAB 21 A_2 = 0.2859 0.7157 0.4706 0.7490 0.5437 0.8390 0.5607 0.5039 0.9848 0.4333 0.2691 0.6468 A_2'= 0.3077 0.1387 0.4756 0.3625 0.7881 0.7803 0.6685 0.1335 0.0216 0.5598 0.3008 0.9394 A_1.A_2= 3.1780 5.9925 5.0491 3.0975 43.5900 94.5714 92.6770 29.3559 26.3095 57.1630 58.7436 17.5258 U L=lu(A_1) U = -0.0990 0.2460 1.0000 -0.4950 1.0000 0 1.0000 0 0 L = -10.1000 23.0000 56.0000 0 17.3861 127.7228 0 0 -21.8770 Q R=qr(A_1) Q = -0.0884 -0.2230 0.9708 -0.4419 -0.8647 -0.2388 0.8927 -0.4501 -0.0221 R = -11.3142 17.7035 5.4445 0 -15.9871 -112.5668 0 0 -21.2384 b=1;2;3 x=A_1\b b = 1 2 3 x = 0.3842 0.3481 -0.0201 A_3=1 2 3; -1 0 5; 8 9 23 A_3 = 1 2 3 -1 0 5

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