Prof. Dr.-Ing. A. Czylwik

Transcription

1 Seminar to the lecture Computer-based Engineering Mathematics N T S Prof. Dr.-Ing. A. Czylwik Amanuel Geda, M.Sc. Room: BA 240, Tel:49(0) , geda@nts.uni-due.de Department of Communication Systems University of Duisburg-Essen

2 Contents 1 Introduction to Matlab What is Matlab? Expressions Variables Numbers Operators Functions Handling Matrices Entering Matrices and Addressing the Elements Generating Matrices Concatenation Deleting rows and columns Array Orientation Skalar-Array Mathematics Array-Array Mathematics Basic Plotting using Matlab Two-Dimensional Graphics Multiple plots on the same axis Line styles, markers and color Plotting rapidly changing mathematical functions Three-Dimensional graphics Curve plots Mesh and surface plots Multiple plots in a figure: subplot Functions Domain, Range and Symmetries

3 3.2 Transformation of Coordinates Normalization of Physical Quantities Local Minima and Maxima Poles and Zeros Curve fitting, interpolation and approximation of functions using Polynomials Polynomial curve fitting Polynomial interpolation Taylor series expansion Vectors and Matrices Creating Matrices Inverses and Determinants Vector Norms and the scalar product in Matlab Matlab functions that work on diagonals of a matrix Systems of Linear Equations Linear System, Coefficient Matrix, Augmented Matrix Solutions of Linear Systems: Existence and Uniqueness Numerical methods in Linear Algebra Matlab syntax and descriptions for LU- and Cholesky Factorizations Matlab solutions for Systems of Linear Equations Eigenvalues and Eigenvectors Eigenvalues, Eigenvectors MATLAB methods to determine eigenvalues and eigenvectors Linear independence of eigenvectors Orthonormal set, Symmetric matrices Diagonalization of a matrix, canonical form Cayley-Hamilton theorem Nonlinear Equations 73

4 8.1 Polynomial roots Solving real function using fzero Solving Equations using Symbolic Toolbox System of nonlinear equations Differentiation Polynomial derivatives Approximate derivative Symbolic derivative Integration Introduction to Numerical Integration Numerical integration using Matlab Symbolic Integration Ordinary Differential Equations Introduction Numerical methods to solve initial value problems (IVP) of ODEs Solving ODEs in Matlab Systems of First order ODEs Homogeneous Linear Systems of ODEs

5 1 Chapter 1 Introduction to Matlab 1.1 What is Matlab? Matlab 1 is a high-performance language for technical computing. It integrates computation, programming and visualisation in a user-friendly environment where problems and solutions are expressed in an easy-to-understand mathematical notation. Matlab is an interactive system whose basic data element is an array that does not require dimensioning. This allows the user to solve many technical computing problems, especially those with matrix and vektor operations, in less time than it would take to write a programm in a scalar noninteractive language such as C or Fortan. Matlab features a family of application-specific solutions which are called toolboxes. It is very important to most users of Matlab, that toolboxes allow to learn and apply specialized technology. These toolboxes are comprehensive collections of Matlab functions, so-called M-files, that extend the Matlab environment to solve particular classes of problems. Matlab is a matrix-based programming tool. Although matrices often need not to be dimensioned explicity, the user has always to look carefully for matrix dimensiones. If it is not defined otherwise, the standard matrix exhibits two dimensiones n x m. Column vectors an row vectors are represented consistently by n x 1 and 1 x n matrices, respectively. Matlab operations can be classified into the followng types of operations: arithmetic and logical operations, mathematical functions, graphical functons, and input/output operations In the following sections, individual elements of Matlab operations are explained in detail. 1 Matlab is a registered trademark of The MathWorks, Inc.

6 1.2 Expressions Like most other programming languages, Matlab provides mathematical expressions, but unlike most programming languages, these expressions involve entire matrices. The building blocks of expressions are Variables Numbers Operators Functiones Variables Matlab does not require any type declarations or dimension statements. When a new variable name is introduced, it automatically creates the variable and allocates the appropriate amount of memory. If the variable already exists, Matlab changes its contents and, if necessary, allocates new storage. For example >> books = 10 creates a 1-by-1 matrix named books and stores the value 10 in its single element. In the expression above, >> constitutes the Matlab prompt, where the commands can be entered. Variable names consist of a string, which start with a letter, followed by any number of letters, digits, or underscores. Matlab is case sensitive; it distinguishes between uppercase and lowercase letters. A and a are not the same variable. To view the matrix assigned to any variable, simply enter the variable name Numbers Matlab uses the conventional decimal notation. A decimal point and a leading plus or minus sign is optional. Scientific notation uses the letter e to specify a power-of-ten scale factor. Imaginary numbers use either i or j as a suffix. Some examples of legal numbers are: e e21 2i j 2e3i j Operators Expressions use familar arithmetic operators and precedence rules. Some examples are:

7 + Addition - Subtraction * Multiplication / Division Complex conjugate transpose ( ) Brackets to specify the evaluation order Functions Matlab provides a large number of standard elementary mathematical functions, including sin, sqrt, exp and abs. Taking the square root or logarithm of a negative number does not lead to an error; the appropriate komplex result is produced automatically. Matlab also provides a lot of advanced mathematical functions, including Bessel and Gamma functions. Most of these functions accept complex arguments. For a list of the elementary mathematical functions, type >> help elfun Some of the functions, like sqrt and sin are build-in. They are a fixed part of the Matlab core so they are very efficient. The drawback is that the computational details are not readily accessible. Other functions, like gamma, are implemented in so called M-files. You can see the code and even modify it if you want. 1.3 Handling Matrices Matlab was mainly designed to deal with matrices. In Matlab, a matrix is a rectangular array of numbers. So scalars can be interpreted to be 1-by-1 matrices and vektors are matrices with only one row or column. Matlab has other ways to store both numeric and nonnumeric data, but in the beginning of learning Matlab, it is usually best to think of everything as a matrix. The operations in Matlab are designed to be as natural as possible. Where other programming languages work only with single numbers, Matlab allows to work with entire matrices quickly and easily Entering Matrices and Addressing the Elements The elements of a matrix must be entered one-by-one in a list where the elements of a row must be separated with commas or blank spaces and the rows are divided by semicolons. The whole list must be surrounded with square brackets, e.g.: >> A = [1 2 3; 8 6 4; 3 6 9] After pressing " Enter" Matlab displays the numbers entered in the command line

8 >> A = Addressing an element of a matrix is also very easy. The n-th element of the m-th column in matrix A from above is A(n,m). So typing >> A(1,3) + A(2,1) + A(3,2) will compute the answer ans = 17 Thek-th to1-th elements of them-th ton-th columns can be addressed bya(k:1,m:n),e.g. >> A(2:3,1:2) ans = Further exaples: >> A(1,1:2) addresses the first two elements of the first row. ans = 1 2 >> A(:,2) addresses all elements of the second column. ans = Generating Matrices There are different ways to generate matrices. Assigning elements explicitly was presented in the paragraph above. To create a row vector with 101 equidistant values starting at 0 and ending by π, this method would be very tedious. So two other possibilities are shown below:

9 >> x = linspace(0, pi, 101) or >> x = (0:0.01:1)*pi In the first case, the Matlab function linspace is used to create x. The function s arguments are described by: linspace(first_value, last_value, number_of_values) with the default number_of_values = 100. In the second case, the colon notation (0:0.01:1) creates an array that starts at 0, increments by 0.01 and ends at 1. Afterwards each element in this array is multiplied by π to create the desired values in x. Both of these array creation forms are common in Matlab. While the colon notation form allws to specify the increment between data elements directly, but not the number of data elements, the Matlab function linspace allows to specify the number of data elements directly, but not the increment value between these data elements. The colon notation is very often used in Matlab therefore a closer look should be taken on it. (first_value:increment:last_value) creates an array starting at first_value, ending at last_value with an increment which can be negative as well, e.g. >> v = (10:-2:0) v = If the increment is 1, then its usage is optional: >> w = (5:10) w = Matlab also provides four functions that generate basic matrices: zeros, ones, rand and randn. These functions will be discussed in chapter Concatenation Concatenation is a process of joining small matrices to make bigger ones. In fact, the first matrix A was created by concatenating its individual elements. The pair of square brackets, [ ], is the concatenation operator. For an example, start with the 3-by-3 matrix A, and form

10 >> F = [A A+10; A*2 A*4]. The result is an 6-by-6 matrix, obtained by joining the four submatrices. F = Deleting rows and columns To delete rows or columns of a matrix, just use a pair of square brackets, e.g. >> A(2,:) = [] deletes the second row of A. A = It is not possible to delete a single element of a matrix, because afterwards it would not still be a matrix. (Exception: vectors, since here deleting an element is the same as deleting a row/column.) Array Orientation The orientation of an array can be changed with the Matlab transpose operator : >> a = 0:3 a = >> b = a b = Skalar-Array Mathematics Addition, subtraction, multiplication and division by a scalar apply the operation to all elements of the array:

11 >> c = [ ; ; ] c = >> 2*c-1 multiplies each element in c by two and subtracts one from each element of the result. ans = Array-Array Mathematics When two arrays have the same dimensions, which means that they have the same number of rows and columns, addition, subtraction, multiplication and division apply on an element-by-element basis in Matlab. >> d = [1 2 3; 4 5 6] d = >> e = [2 2 2; 3 3 3] e = >> f = d+e % adds d to e on an element-by-elenemt basis f = >> g = 2*d-e % multiplies d by two and subtracts e from the result g = Element-by-element multiplication an division work similary, but the notation is slightly different: >> h = d.*e h =

12 The element-by-element multiplication uses the dot multiplication symbol.*, the elementby-element array division uses either./ or.\ >> d./e ans = >> e.\d ans = In both cases, the elements of the array in front of the slash is divided by the elements of the array behind the slash. To compute a matrix multiplication only the asterisk * must be used, e.g. >> C = A * B Therefore the number of columns of A must be equal the number of rows of B. >> A = [1 2 3; 4 5 6] A = >> B = [1 2; 3 4; 5 6] B = >> C = A * B C =

13 9 Chapter 2 Basic Plotting using Matlab 2.1 Two-Dimensional Graphics Graphs (in 2-D) are drawn with the plot statement. In its simplest form, it takes a single vector argument y as in plot(y). In this case the elements of y are plotted against their indexes, e.g. plot(rand(1, 20)) plots 20 random numbers against the integers 1 to 20, joining successive points with straight lines. Straight-line graphs are drawn by giving the x and y coordinates of the end-points in two vectors. For example, to draw a line between the point with cartesian coordinates (0, 1) and (4, 3) use the statement plot([0 4], [1 3]) Exercise 1 Draw lines joining the following points: (0, 0), (1, 1), (2, 0), (1,-1) and (0, 0). If x and y are vectors of the same length, the command plot(x,y) opens a graphics window and draws the elements of y versus the elements of x. For example, x = -4 : 0.01 : 4 ; y = sin(x) ; plot(x, y) The vector x is a partition of the domain with mesh size 0.01, and y is a vector giving the values of sine at the nodes of this partition (recall that sin(x) operates entry-wise). When plotting a curve, the plot routine is actually connecting consecutive points induced by the partition with line segments. Thus, the mesh size should be chosen sufficiently small to render the appearance of a smooth curve.

14 Exercise 2 Plot y versus x for y = exp( x 2 ) and x varies from -10 to 10. Plot y versus t for y = sin(2πft), t varies from 0 to 1, for f = 2 and f = 5. Plot y versus t for y = cos(2πft), t varies from 0 to 1, for f = 2 and f = Multiple plots on the same axis The easiest way is simply to use hold to keep the current plot on the axes. All subsequent plots are added to the axes until hold is released, either with hold off, or just hold, which toggles the hold state. The second way is to use plot with multiple arguments, e.g. plot(x1, y1, x2, y2, x3, y3,... ) plots the (vector) pairs (x1, y1), (x2, y2), etc. The advantage of this method is that the vector pairs may have different lengths. MATLAB automatically selects a different color for each pair. For example, x = -2*pi : 0.01 : 2*pi ; plot(x,cos(x),x,sin(x)) %Equivalent to this command is: plot(x,cos(x)) hold on plot(x,sin(x), -r ) hold off Exercise 3 Draw x 2,x 3,x 4, exp( x 2 ) on the same figure using both methods as described above. Add legend using Matlab command legend( x^2,x^3,x^4,exp(-x^2) ) Line styles, markers and color Line styles, markers and colors may be selected for a graph with a string argument to plot, e.g. plot(x, y, -- ) joins the plotted points with dashed lines, whereas plot(x, y, o )

15 draws circles at the data points with no lines joining them. You can specify all three properties, e.g. plot(x,sin(x), x, cos(x), --mo ) plots sin (x) in the default style and color and cos (x) with circles joined with dashes in magenta. The available colors are denoted by the symbols c, m, y, k, r, g, b, w. You can have fun trying to figure out what they mean, or you can use help plot to see the full range of possible symbols Plotting rapidly changing mathematical functions In all the graphing examples so far, the x coordinates of the points plotted have been incremented uniformly, e.g. x = 0 : 0.01 : 4. If the function being plotted changes very rapidly in some places, this can be inefficient, and can even give a misleading graph. For example, x = 0.01:.001:.1; plot(x, sin(1./x)) MATLAB has a function called fplot which uses a more elegant approach. Whereas the above method evaluates sin(1/x) at equally spaced intervals, fplot evaluates it more frequently over regions where it changes more rapidly. Here is how to use it: fplot( sin(1/x),[ ]) % 1./x not needed! Exercise 4 Now draw the functions from exercise 3 using fplot. Try also using ezplot. For more information check the MATLAB help by typing at command window help ezplot. 2.2 Three-Dimensional graphics MATLAB s primary commands for creating three dimensional graphics of numericallydefined functions are plot3, mesh, surf, and surfl Curve plots Completely analogous to plot in two dimensions, the command plot3 produces curves in three-dimensional space. If x, y, and z are three vectors of the same size, then the command plot3(x,y,z) produces a perspective plot of the piecewise linear curve in threespace passing through the points whose coordinates are the respective elements of x, y, and z. These vectors are usually defined parametrically. For example,

16 t = 0.01 : 0.01 : 20*pi ; x = cos(t) ; y = sin(t) ; z = t.^3 ; plot3(x, y, z) %produces a helix that is compressed near the x-y plane Mesh and surface plots The mesh command draws three-dimensional wire mesh surface plots.similarly, threedimensional faceted surface plots are drawn with the command surf. The following MATLAB M-File generate a mexican hut using the mesh command. % Mexican hut as an example for 3D plot. [x y] = meshgrid (-8: 0.5 : 8) ; r = sqrt(x.^2 + y.^2) ; z = sin(r)./r ; mesh(x,y,z) title( Mexican hut ) Mexican hut The following script shows how to graph the function over the disk f(x,y) = x 1 + y 2 (2.1) (x 1) 2 + (y 3) 2 = 4 (2.2) % Example for 3D Plot from the book A Matlab comapanion for a multivariable calculus % by J. Cooper, University of Maryland % First make a meshgrid in r, theta coordinates r = linspace(0,2,21); theta = linspace(0, 2*pi, 41);

17 [R,TH] = meshgrid(r,theta) % Now convert into a curvilinear meshgrid in x,y % coordinates X = 1 + R.*cos(TH); Y = 3 + R.*sin(TH); Z = X-1 + Y.^2; surf(x,y,z) hold on % add the plane z = -5 with the curvilinear % meshgrid surf(x,y,-5+0*z) hold off Plot of f(x,y) = x 1 +y 2 over the disk (x 1) 2 + (y 3) 2 = z axis y axis x axis 2 3 Exercise 5 Plot a Gaussian probability density function f x,y (x,y) with zero mean and variance 1 within the range x [ 5, 5],y [ 5, 5] : f x,y (x,y) = 1 1 2π e 2 (x2 +y 2 ) Try out the commands mesh, surf and contour to plot the graph! Exercise 6 Plot the function (2.3) over the unit disk, centered at the origin. g(r,θ) = (r/2) sin(θ) + r 3 cos(3θ) (2.4) Multiple plots in a figure: subplot You can show a number of plots in the same figure window with the subplot function. The statement subplot(m,n,p) divides the figure window into m X n small sets of axes, and selects the p th set for the current plot (numbered by row from the left of the top row).

18 Exercise 7 Plot the function from Exercise 5 in four subplots using the commands mesh in subplot(2,2,1), surf in subplot(2,2,2), surfl in subplot(2,2,3) and contour in subplot(2,2,4). Give title to each plot as for instance title( Plot of Gaussian pdf using mesh )

19 15 Chapter 3 Functions This chapter analyzes important characteristics of functions using Matlab. 3.1 Domain, Range and Symmetries Consider the functions Exercise 1 f(x) = 1 x2 4 1 g(x) =. 1 x2 4 Draw on the same figure the functions f(x) and g(x) for 5 x 5 with distinct colors, markers and legends. Determine the domain, range and symmetry points of f(x) and g(x) so that these are real-valued. Name the x- and y-axis using the Matlab commands xlabel( x \rightarrow ) ; ylabel( f(x), g(x) \rightarrow ) 3.2 Transformation of Coordinates Consider the function g(x) drawn in the interval 5 x 5: clc; clear;close all x = -5 :.05 : 5 ; g_x = 1./ sqrt( abs(1-0.25*x.^2 ) ) ; plot( x, g_x, -r. ) hold on;

20 A simple coordinate transformation is obtained by shifting g(x) to g 2 (x) = g(x x 0 ). For x 0 = 1: x_0 = 1; g2_x = 1./ sqrt( abs(1 - (x-x_0).^2) ) ; g2_x = 1./ sqrt( abs(1 -.25*(x-x_0).^2) ) ; plot( x, g2_x, -g ) hold off; Coordinate transformation g(x) g(x 1) g(x), g 2 (x) x Exercise Draw on the same figure the coordinate transformation g 3 (x) = g(ax) for a = 2 (scaling of the x-axis). Draw on the same figure the coordinate transformation g 4 (x) = g(ax x 0 ) for a = 2 and x 0 = 1. Draw a new figure with the non-linear coordinate transformation g 5 (x) = g(x 2 ). 3.3 Normalization of Physical Quantities Physical quantities are normalized with the purpose of simplifying calculations and eliminate the use of the physical units of distinct variables. Consider the the transfer function of the following RLC filter: H(ω) = U o(ω) U i (ω) = jωl/r 1 ω 2 LC + jωl/r.

21 R U i (ω) L C U o (ω) The squared magnitude of the RLC filter using the technical frequency f is given by H(f) 2 = (2πfL/R) 2 [ 1 (2πf) 2 LC ] 2 + [2πfL/R] 2. Expressing: f in khz, L in mh, C in mf, R in Ω results in: H(f) 2 = [ 1 ( 2π f khz ( 2π f khz ) 2 L C mh mf ) 2 L/mH R/Ω ] 2 + [ 2π f khz ] 2. L/mH R/Ω Exercise Draw H(f) 2 in the range 0 f 20kHz for L = 100µH, C = 10µF and R = 10Ω with the corresponding labels in the x- and y-axis. 3.4 Local Minima and Maxima For a function y = f(x), local minima and maxima are determined by solving the expression dy dx = Lim f(x + x) f(x) x 0 x A graphical approximation of the derivate can be calculated in Matlab using the diff command. For a vector y, diff(y) calculates the differences between adjacent elements of y. As an example, we would like to determine graphically the local minima and maxima of the function y(x) = x cos(x) for π x π: = 0

22 clc;clear;close all Delta = 0.01; x = -pi : Delta : pi; y = x.* cos( x ); plot(x,y); grid on;hold on plot( x(1:end-1), diff(y)./diff(x), r ) xlabel( x \rightarrow ) ylabel( f(x) \rightarrow ) title( Approximate derivation ) legend( y = xcos(x), dy/dx ) 4 3 Approximate derivation y = xcos(x) dy/dx f(x) Local Minimum Local Maximum x Notice that the command diff( y ) reduces the dimension of the vector y by one element. Therefore, the figure is drawn by reducing the elements of the vector x by one element - using the command x(1:end-1). Exercise Determine the local minima and maxima of the function ( ) 2 2π f L/mH khz R/Ω H(f) 2 = [ 1 ( 2π f khz ) 2 L C mh mf ] 2 + [ 2π f khz ] 2 L/mH R/Ω for 0 f 20kHz, L = 100µH, C = 10µF and R = 10Ω with the corresponding labels in the x- and y-axis. The location of the local minima and maxima can be solved analytically in the case that a function can be expressed in a polynomial form. The procedure is described in the following.

23 First, a polynomial is defined in Matlab by a row vector that contain the coefficients of the polynomial. As an example, the polynomial p(x) = x 3 2x 5 is represented by: p = [ ]; The derivative q(x) = dp(x) dx is calculated using the Matlab command polyder: q = polyder( p ); The calculation of the roots of q(x), that is q(x) = 0, provides the location of the local minima and maxima. This takes place using the command roots: r = roots( q ); Finally, the value of the local minima and maxima of p(x) are evaluated by means of the command polyval: polyval( p, r ); 1 2 p(x) x = 0.81 y = 3.91 x = 0.81 y = x More complex functions can be represented by the ratio of two polynomials H(x) = p 1(x) p 2 (x) (as is the case of the transfer function of most linear time invariant systems, for example). The derivative of H(x) results in the ratio of two polynomials q 1 (x) and q 2 (x), that is: dh(x) dx = q 1(x) q 2 (x). The polynomials q 1 (x) and q 2 (x) are evaluated using the command polyder as follows:

24 [ q_1, q_2 ] = polyder( p_1, p_2 ); Finally, the location of the local minima and maxima can be determined by calculating the roots of q 1 (ω) using the command roots, as it was done before. Exercise Determine the local minima and maxima of the function ( ) 2 2π f L/mH khz R/Ω H(f) 2 = [ 1 ( 2π f khz ) 2 L C mh mf ] 2 + [ 2π f khz ] 2 L/mH R/Ω for L = 100µH, C = 10µF and R = 10Ω. Compare this result with the resonant frequency f = 1 2π LC. 3.5 Poles and Zeros The poles and zeros of a function H(x) are obtained by calculating the roots of the denominator and the numerator H(x), respectively: Exercise Draw the poles and zeros diagram of H(f) 2 = H(x) = k (x X N,1) (x X N,2 ) (x X P,1 ) (x X P,2 ) [ 1 ( 2π f khz for L = 100µH, C = 10µF and R = 10Ω. ( 2π f khz ) 2 L C mh mf ) 2 L/mH R/Ω ] 2 + [ 2π f khz ] 2 L/mH R/Ω

25 21 Chapter 4 Curve fitting, interpolation and approximation of functions using Polynomials This chapter covers the basic MATLAB tools for curve fitting, interpolation and approximation of functions using polynomials. 4.1 Polynomial curve fitting The Matlab command polyfit finds the coefficients of a polynomial that fits a set of data in a least-squares sense: p = polyfit(x, y, n) finds the best-fit n-degree polynomial that approximates the data points x and y. x and y are vectors containing the x and y data to be fitted, and n is the degree of the polynomial to return. For example, consider the x-y test data: x = [ ]; y = [ ]; %A third degree polynomial that approximately fits the data is : p = polyfit(x,y,3) p = Let us now compute the values of the polyfit estimate over a finer range, and plot the estimate over the real data values for comparison: x2 = 1:.1:5; y2 = polyval(p,x2); plot(x,y, o,x2,y2) grid on legend( measured points, polyfit of degree 3 ) title( \bf{polynomial Curve Fitting} ) Note: The command polyval(p,x2) evaluates the polynomial p for each element of the vector x2. For more information check Matlab help by writting at command window help polyval The figure on the next page shows the measured data and the curve fitting to it.

26 Polynomial Curve Fitting measured points polyfit of degree y x Exercise 1 Consider the following test data : x = [ ] y = [ ] Find a fifth degree polynomial that fits the data, compute the values of the polyfit estimate over a finer range and plot the estimate over the real data values for comparison. 4.2 Polynomial interpolation Polynomials are useful as easier-to-compute approximations of more complicated functions, via a Taylor series expansion or by a low-degree best-fit polynomial using the polyfit function. Example Let s approximate y = sin(x) between 0 and 2π using a polynomial of degree 5 and plot both on the same figure for comaprison. % Example: function approxiamtion using polynomial % the given function is y = sin(x) % the objective is to approximate the function b/n 0 and 2*pi using poly - % nomial of degree 5. close all x = 0:0.1:2*pi ;

27 y = sin(x); plot(x,y) xlabel( x \rightarrow ) ylabel( y \rightarrow ) hold on p = polyfit(x,y,5) xx = 0: 0.1: 2*pi ; yy = polyval(p,xx); plot(xx,yy, -sr ) legend( sin(x), poly. of degree 5 approx. ) Polynomial approximation of the sin(x) over one period sin(x) poly. of degree 5 approx. 0.5 y x Exercise 2 Approximate y = sin(x) between 2π and 2π using a polynomial of degree 5 and plot both on the same figure for comaprison. Now approximate y = sin(x) between 2π and 2π using a polynomial of degree 7 and plot both on the second figure for comaprison. Piecewise-polynomial interpolation is typically better than a single high-degree polynomial. The following example shows the difference between them. % comparison of piecewise-polynomial versus single high degree polynomial

28 %interpolations % first in fig1 single polynomial approximation of higher order close all n = 10; x = -5:0.1:5 ; y = 1./(x.^2 + 1) ; p = polyfit(x,y,n) figure(1) fplot( 1/(x^2 + 1), [-7 7], rs- ) hold on xx = -5:0.01:5 ; plot(xx,polyval(p,xx), g*- ) legend( f(x)=1/(x^2 + 1), Single polynomial approx. of f(x) ) title( \bf{single polynomial approximation of a function } ) % Now the piecewise polynomial interpolation using spline or spline + ppval figure(2) yy = spline(x,y,xx) ; fplot( 1/(x^2 + 1), [-7 7], rs- ) hold on plot(xx,yy, g*- ) legend( f(x)=1/(x^2 + 1), Piecewise-polynomial approx. of f(x) ) title( \bf{piecewise-polynomial approximation of a function } ) The result of the first plot using the single polynomial approximation is : Single polynomial approximation of a function f(x)=1/(x 2 + 1) Single polynomial approx. of f(x) From this figure it could be observed that as n increases the error in the center improves but increases near the endpoints.

29 The second figure, which is shown below, using the piecewise-polynomial interpolation shows a better result Piecewise polynomial approximation of a function f(x)=1/(x^2 + 1} Piecewise polynomial approx. of f(x) Exercise 3 a) Approximate the function y = sin 3 (x) between 2π and 2π using a single polynomial of degree 7 and plot both the original function and the approximated polynomial on the same figure for comaprison. b) Approximate the function y = sin 3 (x) between 2π and 2π using piecewise-polynomial interpolation and plot both the original function and the approximated polynomial on the same figure for comparison. 4.3 Taylor series expansion The Taylor series for an analytic function f(x) about the base point x = a is given by : f(x) = (x a) n. f(n) (a) (4.1) n! n=0 The Matlab syntax and its description about taylor series is available from the Matlab help under Symbolic Math Toolbox. The main commands are : taylor(f) is the fifth order Maclaurin polynomial approximation to f(x). taylor(f,n,v) returns the (n-1)-order Maclaurin polynomial approximation to f(x), where f is a symbolic expression representing a function and v specifies the independent variable in the expression. v can be a string or symbolic variable.

30 taylor(f,n,v,a) returns the Taylor series approximation to f(x) about a. The argument a can be a numeric value, a symbol, or a string representing a numeric value or an unknown. You can supply the arguments n, v, and a in any order. taylor determines the purpose of the arguments from their position and type. You can also omit any of the arguments n,v, and a. If you do not specify v, taylor uses findsym to determine the function s independent variable.the default value to n is 6. Note: The independent variable in the above descriptions is symbolic and hence should first be declared as syms x. Example The taylor series expansion of f(x) = sin(x) at x = 0 up to the order of 5 is : syms x f = sin(x) f_taylor = taylor(f) Matlab answer to these commands is : f_taylor = x^5/120 - x^3/6 + x The taylor series expansion of f(x) = sin(x) at around x = 0 up to the order of 7 is : syms x f = sin(x) f_tr_7 = taylor(f,8) Matlab answer to these commands is : f_tr_7 = x^5/120 - x^7/ x^3/6 + x We now plot the function sin(x) and the taylor series obtained above for comparison. fplot( sin(x), [-2*pi 2*pi -1 1]) hold on fplot( x^5/120 - x^7/ x^3/6 + x, [-2*pi 2*pi ], rs-- ) Observe from the figure below how well the polynomial obtained through taylor series approximates the function for small values of x ( 3 x 3).

31 1.5 1 Taylor series of a sin function sin(x) x 7 / x 5 /120 x 3 /6 + x Exercise 4 Determine the taylor series expansion of the following functions at around 0 up to the order of 9 and plot both the function and the approximate taylor series on the same figure like the above example. exp(x) cosh(x) sinh(x) Taylor series approximation using the Matlab command taylortool Syntax taylortool taylortool( f ) Description taylortool initiates a GUI that graphs a function against the N th partial sum of its Taylor series about a base point x = a. The default function, value of N, base point, and interval of computation for taylortool are f = x cos(x), N = 7, a = 0, and [ 2π, 2π], respectively. taylortool( f ) initiates the GUI for the given expression f. Example taylortool( exp(x*sin(x)) ), this command opens the following GUI:

32 Taylor Series Approximation T N (x) = x 6 /120 + x 4 /3 + x Exercise 5 Now use the command taylortool, like the example above, and determine the taylor series approximations near the origin for the following functions: tan(x) sin(x) cosh(x) sinh(x) sin(tan(x)) tan(sin(x))

33 29 Chapter 5 Vectors and Matrices This chapter covers the basic opration on Vectors and Matrices in the MATLAB environment. 5.1 Creating Matrices The MATLAB environment uses the term matrix to indicate a variable containing real or complex numbers arranged in a two-dimensional grid. An array is, more generally, a vector, matrix, or higher-dimensional grid of numbers. All arrays in MATLAB are rectangular, in the sense that the component vectors along any dimension are all the same length. MATLAB has a number of functions that create different kinds of matrices. The functions shown in the table below create matrices for more general use. Function ones zeros eye diag magic rand randn Description Create a matrix or array of all ones. Create a matrix or array of all zeros. Create a matrix with ones on the diagonal and zeros elsewhere. Create a diagonal matrix from a vector. Create a square matrix with rows, columns, and diagonals that add up to the same number. Create a matrix or array of uniformly distributed random numbers. Create a matrix or array of normally distributed random numbers and arrays. For more information check help matfun. Examples Here are some examples of how you can use these functions. Creating a Random Matrix. The rand function creates a matrix or array with elements uniformly distributed between zero and one. This example multiplies each element by 20:

34 A = rand(5)*20 A = Creating a Diagonal Matrix. Use diag to create a diagonal matrix from a vector. You can place the vector along the main diagonal of the matrix, or on a diagonal that is above or below the main one, as shown here. The -1 input places the vector one row below the main diagonal: >> A = [ ] A = >> diag(a) ans = >> diag(a,-1) ans = Creating a Magic Square Matrix. A magic square is a matrix in which the sum of the elements in each column, or each row, or each main diagonal is the same. To create

35 a 5-by-5 magic square matrix, use the magic function as shown. >> A = magic(5) A = Note that the elements of each row, each column, and each main diagonal add up to the same value: 65. Concatenating Matrices Matrix concatenation is the process of joining one or more matrices to make a new matrix. The brackets [] operator which we have used often serves not only as a matrix constructor, but also as the MATLAB concatenation operator. The expression C = [A B] horizontally concatenates matrices A and B. The expression C = [A; B] vertically concatenates them. >> A = ones(2, 5) * 6; % 2-by-5 matrix of 6 s >> B = rand(3, 5); >> C = [A; B] % Vertically concatenate A and B C = Accessing Single Elements We have already discussed in the previous sessions how to reference single elements of a matrix. The syntax is : A(row, column), where A is a variable matrix. Linear Indexing You can refer to the elements of a MATLAB matrix with a single subscript, A(k). MAT- LAB stores matrices and arrays not in the shape that they appear when displayed in the MATLAB Command Window, but as a single column of elements. This single column is composed of all of the columns from the matrix, each appended to the last. So, matrix A >> A = [2 6 9; 4 2 8; 3 5 1]

36 A = is actually stored in memory as the sequence 2, 4, 3, 6, 2, 5, 9, 8, 1 The element at row 3, column 2 of matrix A (value = 5) can also be identified as element 6 in the actual storage sequence. To access this element, you have a choice of using the standard A(3,2) syntax, or you can use A(6), which is referred to as linear indexing. Accessing Multiple Elements For the 4-by-4 matrix A shown below, it is possible to compute the sum of the elements in the fourth column of A by typing >> A = magic(4); >> A(1,4) + A(2,4) + A(3,4) + A(4,4) ans = 34 You can reduce the size of this expression using the colon operator. Subscript expressions involving colons refer to portions of a matrix. The expression A(1:m, n) refers to the elements in rows 1 through m of column n of matrix A. Using this notation, you can compute the sum of the fourth column of A more succinctly: sum(a(1:4, 4)) Nonconsecutive Elements To refer to nonconsecutive elements in a matrix, use the colon operator with a step value. The m:3:n in this expression means to make the assignment to every third element in the matrix. Note that this example uses linear indexing: >> B = A; >> B(1:3:16) = -10 B =

37 MATLAB supports a type of array indexing that uses one array as the index into another array. You can base this type of indexing on either the values or the positions of elements in the indexing array. Here is an example of value-based indexing where array B indexes into elements 1, 3, 6, 7, and 10 of array A. In this case, the numeric values of array B designate the intended elements of A: >> A = 5:5:50 A = >> B = [ ]; >> A(B) ans = The end Keyword MATLAB provides the keyword end to designate the last element in a particular dimension of an array. This keyword can be useful in instances where your program does not know how many rows or columns there are in a matrix. You can replace the expression in the previous example with B(1:3:end) = -10 Specifying All Elements of a Row or Column The colon by itself refers to all the elements in a row or column of a matrix. Using the following syntax, you can compute the sum of all elements in the second column of a 4-by-4 magic square A: >> A = magic(4); >> sum(a(:,2)) ans = 34 By using the colon with linear indexing, you can refer to all elements in the entire matrix. The command A(:) displays all the elements of matrix A, returning them in a columnwise order. Check it! Using Logicals in Array Indexing

38 A logical array index designates the elements of an array A based on their position in the indexing array, B, not their value. In this masking type of operation, every true element in the indexing array is treated as a positional index into the array being accessed. In the following example, B is a matrix of logical ones and zeros. The position of these elements in B determines which elements of A are designated by the expression A(B): >> A = [1 2 3; 4 5 6; 7 8 9] A = >> B = logical([0 1 0; 1 0 1; 0 0 1]); >> A(B) ans = The find function can be useful with logical arrays as it returns the linear indices of nonzero elements in B, and thus helps to interpret A(B): >> find(b) ans =

39 5.2 Inverses and Determinants If A and B are square matrices, each of order n n, which satisfy the equations AB = BA = I n then B is called the inverse of A. We write B = A 1. Since the definition is symmetric, A is the inverse of B, that is A = B 1. Let us consider a general 2 2 matrix, and determine the inverse matrix and also look for the conditions for the existance of the inverse matrix. Ax = d where [ ] [ ] [ ] a11 a A = 12 x1 d1, x =, d = a 21 a 22 x 2 d 2 Thus a 11 x 1 + a 12 x 2 = d 1 a 21 x 1 + a 22 x 2 = d 2 These are linear equations in the unknowns x 1 and x 2. x 1 and x 2 are given by, provided that a 11 a 22 a 21 a 12 0, x 1 = a 22d 1 a 12 d 2 a 11 a 22 a 21 a 12, x 2 = a 21d 1 + a 11 d 2 a 11 a 22 a 21 a 12 x = [ ] [ ] 1 a22 a 12 d1 a 11 a 22 a 21 a 12 a 21 + a 11 d 2 The number a 11 a 22 a 21 a 12 is known as the determinat of the matrix A, det(a). [ ] Ax = d x = A 1 d A 1 1 a22 a = 12 det(a) a 21 a 11 The Matlab command to find the determinant of the square matrix A is det(a) and the inverse of A can be obtained using the command inv(a) if it exists, i.e. if A is non-singular. If A is square and nonsingular, then, without roundoff error, X = inv(a)*d is theoretically the same as X = A\d and Y = d*inv(a) is theoretically the same as Y = d/a. But the computations involving the backslash and slash operators are preferable because they require less computer time, less memory, and have better error-detection properties. Exercise A) Decide [ whether ] A, where 1 3 A =, 1 4

40 is singular or not. If it is non-singular, find its inverse. B)Given [ are] the matrices [ : ] A =, B = Find A 1, B 1, (AB) 1, and B 1 A C) Find A 1, where A = Find the determinants of A T, A and A 1 D) Given that A = [ ] , B = 2 1, C = verify the distributive law A(B + C) = AB + AC for the three matrices. E)Given are the matrices : A = [ ] 4 2, B = 2 1 [ ] Find the inverse of (BA) and (AB) if they exist. F)Given is the matrix : A = Find a matrix C such that A + C is the identity matrix I 3, i.e. 3 3 identity matrix. Determine AC, CA and A 2 + C 2. Check whether AC = CA. G)For a general n n matrix it could be shown that A +A T is a symmetric matrix, and A A T is a skew-symmetric matrix. Now, express the matrix : A = as the sum of a symmetric matrix and a skew-symmetric matrix.

41 Pseudoinverses Rectangular matrices do not have inverses or determinants. At least one of the equations AX = I and XA = I does not have a solution. A partial replacement for the inverse is provided by the Moore-Penrose pseudoinverse, which is computed by the pinv function. The Matalab syntax is: B = pinv(a) The Moore-Penrose pseudoinverse is a matrix B of the same dimensions as A satisfying the four conditions: A B A = A B A B = B A B is Hermitian B A is Hermitian Example >> rand( state, 0); >> A = rand(3,2) A = >> B = pinv(a) B = >> A*B ans = >> (A*B) ans =

42 >> B*A ans = >> (B*A) ans = B*A is the 2-by-2 identity matrix, but A*B is not the 3-by-3 identity matrix. However, A*B acts like an identity on a portion of the space in the sense that A*B is symmetric, A*B*A is equal to A, and B*A*B is equal to B. Exercise Let now A is given by : A = 10*rand(2,3) Determine the Moore-Penrose pseudoinverse B, proof the four conditions described earlier. 5.3 Vector Norms and the scalar product in Matlab The p-norm of a vector x ( ) 1/p x p = xi p is computed by norm(x,p). This is defined by any value of p > 1, but the most common values of p are 1, 2 and. The default value is p = 2, which corresponds to Euclidean length: Example >> v = [2 0-1]; >> [norm(v,1) norm(v) norm(v,inf)] ans = Exercise Compute the 1 st, 2 nd, 3 rd, 10 th and the infinity norms of the following vectors :

43 w = [ ] x = [ ] y = [ ] z = [ ] The scalar product of two vectors Suppose that two vectors a and b are given in component form in 3D as : a = (a 1,a 2,a 3 ) and b = (b 1,b 2,b 3 ) The scalar product is defined as : a b = a 1 b 1 + a 2 b 2 + a 3 b 3 The Matlab equivalent to this is element by element multiplication of two vectors of the same size. Exercise Let a be a vector on x-z plane with values ( 1, 0, 1) and b = (2, 3, 2) Determine the scalar product a b Check the results of (a b) (a b) and a 2 b Matlab functions that work on diagonals of a matrix Function trace diag tril triu Description Compute the sum of the elements on the main diagonal. Return the diagonals of a matrix. Return the lower triangular part of a matrix. Return the upper triangular part of a matrix. Exercise 5.4 The Matlab command pascal(3) creates a 3 3 symmetric matrix. Now, create 4 4 symmetric matrix using the function pascal and then determine : The lower triangular part of this matrix. The upper triangular part of this matrix. A vector with the main diagnoal elements of this matrix. The sum of the elements on the main diagonal.

44 40 Chapter 6 Systems of Linear Equations 6.1 Linear System, Coefficient Matrix, Augmented Matrix A linear system of equations in n unknowns x 1,x 2,,x n is a set of equations of the form a 11 x a 1n x n = b 1 a 21 x a 2n x n = b 2.. (6.1) a m1 x a mn x n = b m Thus, a system of two equations in three unknown is a 11 x 1 + a 12 x 2 + a 13 x 3 = b 1 a 21 x 1 + a 22 x 2 + a 23 x 3 = b 2 The a jk are given numbers, which are called the coefficients of the system. If the b i are all zero, then (6.1) is called a homogeneous system.if at least one b i is not zero, then (6.1) is called a nonhomogeneous system. A solution of (6.1) is a set of numbers x 1,,x n that satisfies all the m equations. A solution vector of (6.1) is a verctor x whose components constitute a solution of (6.1). If the system is homogeneous, it has at least the trivial solution x 1 = 0,,x n = 0. Matrix form of the Linear System (6.1). The m equations of (6.1) can be written as a single vector equation Ax = b (6.2) where the coefficient matrix A = [a jk ] is the m n matrix a 11 a 12 a x 1 1n a 21 a 22 a 2n A =......, x = a m1 a m2 a mn x n b 1 and b = b n

45 are column verctors. It is assumed that A is not a zero matrix, which means not all the coefficients a jk are zero.the augumented matrix à of (6.1) is a 11 a 12 a 1n b 1 à = [ A b ] a 21 a 22 a 2n b 2 = a m1 a m2 a mn b n The method of solving such a system by determinants(cramer s rule) is not practical, but the Gauss elimination methos is often used and discussed brifely below. 6.2 Solutions of Linear Systems: Existence and Uniqueness The necessary and sufficient conditions for the existence of solutions and for the uniqueness can be given using the rank of the coefficient matrix A and Ã. (a) Existence A linear system of m equations in n unknowns(6.1) has solutions if and only if the coefficient matrix A and the augmented matrix à have the same rank. (b) Uniqueness The system(6.1) has precisely one solution if and only if this common rank r of A and à equals n. (c) Infinitely many solutions If this rank r is less than n, the system has infinitely many solutions. All of these are obtained by determinig r suitable unknowns in terms of the remaining n - r unknowns, to which arbitrary values can be assigned. (d) Gauss elimination If solutions exist, they can be obtained by the Gauss elimination. A homogeneous linear system has always the trivial solutionx 1 = 0,,x n = 0. Nontrivial solutions exist if and only if rank A < n. Exercise 6a Determine the existance of the solutions to the following systems of linear equations and proof also the uniqueness of the solutions 1. 2x 1 + 2x 2 + 4x 3 = 2 4x 1 + 5x 2 +13x 3 = 7 10x 1 +14x 2 +43x 3 = 25

46 2. 3x 1 4x 2 = 2 x 1 +5x 2 = 4 5x 1 +2x 2 = x 1 +3x 2 + 2x 3 = 2 x 1 x 2 3x 3 = 5 3x 1 +5x 2 + 5x 3 = 3 4. x 1 + x 2 + x 3 = 1 x 1 2x 2 + x 3 = 2 x 1 x 2 + 5x 3 = 0 5. x 1 +x 2 + x 3 + 3x 4 =0 2x 2 + 2x 4 =5 x 1 x 2 2x 3 2x 4 =4 2x 1 +4x 2 + 2x 3 +8x 4 =5 6.3 Numerical methods in Linear Algebra Gauss Elimlination This standard method for solving linear systems (6.1) is a systematic process of elimination that reduces (6.1) to triangular form because the sytem can easily solved by back substitution. For instance, a triangular system is 3x 1 + 5x 2 + 2x 3 = 8 8x 2 + 2x 3 = 7 6x 3 = 3 LU-Factorization A is now assumed to be a square matrix, n n. Below is briefly discussed three related methods that are modifications of the Gauss elimination. They are named after Doolittle, Crout, and Cholesky and use the idea of LU-Factorization, which will be expalined first. An LU-factorization of a given square matrix A is of the form A = LU (6.3)

47 where L is lower triangular and U is upper triangular matrices. For example, [ ] [ ][ ] A = = LU = It could be proved that for any nonsingular matrix (interested reader can refer to Sec.6 of advanced engineering mathematics by Kreyszig,E.)the rows can be reordered so that the resulting matrix A has an LU-factrorization (6.3) in which U turns out to be the triangular system at the end of the Gauss elimination and L is the matrix of the multipliers m jk of the Gauss elimination, with diagonal 1,, 1. The main idea here is that L and U in (6.3) can be computed directly, without solving the simultaneous equations(thus, without using the Gauss elimination. Once we have (6.3), Ax = b can be solved in two steps, simply by noting that Ax = LUx = b can be written as : Ly = b (6.4a) where Ux = y (6.4b) and solving first (6.4a) for y and then (6.4b) for x. This is called Doolittle s method. Both systems (6.4a) and (6.4b) are triangular, so their solution is the same as back substitution in the Gausss elimination. A similar method, Crout s method, is obtained from (6.3) if U(instead of L) is required to have main diagonal 1,, 1. In either case the factorization (6.3) is unique. Example A system is given by a matrix A and the vector b as follows : A = b = [ ] Solve the system using Doolittle s method (LU factorization with main diagonal elements of L are 1 ). Solution u 11 u 12 u 13 A = = m u 22 u m 31 m u 33 u 11 = 3, m 21 u 11 = 0 m 21 = 0, A = =