ࡱ>     szgx ^bjbjT~T~ z66Q&J J EEE$iiiP lil1"T+M"MMMT (^`d $T-Ei4SNTii-MMwB:::i<MEM:i::e|MPipz}X0,N"E Pb`d:bf\gPbPbPb--\PbPbPbiiiiPbPbPbPbPbPbPbPbPbJ S:  Pre-Engineering 220 Introduction to MatLab & Scientific Programming J Kiefer Gottfried W. Leibnitz: It is unworthy for excellent men to lose hours like slaves in the labour of calculation which could be safely relegated to anyone else if machines were used. 2013 Table of Contents  TOC \o "1-3" \h \z \u  HYPERLINK \l "_Toc358821468" Table of Contents  PAGEREF _Toc358821468 \h 1  HYPERLINK \l "_Toc358821469" I. Introduction  PAGEREF _Toc358821469 \h 3  HYPERLINK \l "_Toc358821470" A. Numerical Methods or Numerical Analysis  PAGEREF _Toc358821470 \h 3  HYPERLINK \l "_Toc358821471" 1. Numerical Analysis  PAGEREF _Toc358821471 \h 3  HYPERLINK \l "_Toc358821472" 2. Newtons Method for Solving a Nonlinear Equationan example  PAGEREF _Toc358821472 \h 3  HYPERLINK \l "_Toc358821473" 3. Series  PAGEREF _Toc358821473 \h 5  HYPERLINK \l "_Toc358821474" 4. Error  PAGEREF _Toc358821474 \h 5  HYPERLINK \l "_Toc358821475" B. Programming  PAGEREF _Toc358821475 \h 6  HYPERLINK \l "_Toc358821476" 1. Program Design  PAGEREF _Toc358821476 \h 6  HYPERLINK \l "_Toc358821477" 2. Branching  PAGEREF _Toc358821477 \h 6  HYPERLINK \l "_Toc358821478" 3. Loops  PAGEREF _Toc358821478 \h 6  HYPERLINK \l "_Toc358821479" 4. I/O  PAGEREF _Toc358821479 \h 6  HYPERLINK \l "_Toc358821480" 5. Precision Issues  PAGEREF _Toc358821480 \h 7  HYPERLINK \l "_Toc358821481" 6. Debugging  PAGEREF _Toc358821481 \h 7  HYPERLINK \l "_Toc358821482" II. MatLab  PAGEREF _Toc358821482 \h 8  HYPERLINK \l "_Toc358821483" A. Program Features  PAGEREF _Toc358821483 \h 8  HYPERLINK \l "_Toc358821484" 1. Commands  PAGEREF _Toc358821484 \h 8  HYPERLINK \l "_Toc358821485" 2. Arrays  PAGEREF _Toc358821485 \h 10  HYPERLINK \l "_Toc358821486" 3. Array Operations  PAGEREF _Toc358821486 \h 11  HYPERLINK \l "_Toc358821487" B. Files  PAGEREF _Toc358821487 \h 11  HYPERLINK \l "_Toc358821488" 1. m-files  PAGEREF _Toc358821488 \h 11  HYPERLINK \l "_Toc358821489" 2. Script files  PAGEREF _Toc358821489 \h 12  HYPERLINK \l "_Toc358821490" 3. Function files  PAGEREF _Toc358821490 \h 12  HYPERLINK \l "_Toc358821491" C. Plots  PAGEREF _Toc358821491 \h 13  HYPERLINK \l "_Toc358821492" 1. Two Dimensional Graphs (pp. 133-158  PAGEREF _Toc358821492 \h 13  HYPERLINK \l "_Toc358821493" 2. Three Dimensional Graphs  PAGEREF _Toc358821493 \h 13  HYPERLINK \l "_Toc358821494" D. Programs  PAGEREF _Toc358821494 \h 14  HYPERLINK \l "_Toc358821495" 1. Branches  PAGEREF _Toc358821495 \h 14  HYPERLINK \l "_Toc358821496" 2. Loops (pp. 190-200)  PAGEREF _Toc358821496 \h 16  HYPERLINK \l "_Toc358821497" 3. Input/output (pp 114-118)  PAGEREF _Toc358821497 \h 17  HYPERLINK \l "_Toc358821498" III. Numerical Solution of Nonlinear Equations  PAGEREF _Toc358821498 \h 18  HYPERLINK \l "_Toc358821499" A. Non-Linear Equationsone at a time  PAGEREF _Toc358821499 \h 18  HYPERLINK \l "_Toc358821500" 1. The Problem  PAGEREF _Toc358821500 \h 18  HYPERLINK \l "_Toc358821501" 2. Bisection  PAGEREF _Toc358821501 \h 18  HYPERLINK \l "_Toc358821502" 3. Newtons Method or the Newton-Raphson Method  PAGEREF _Toc358821502 \h 19  HYPERLINK \l "_Toc358821503" 4. Secant Method  PAGEREF _Toc358821503 \h 20  HYPERLINK \l "_Toc358821504" 5. Hybrid Methods  PAGEREF _Toc358821504 \h 20  HYPERLINK \l "_Toc358821505" B. Systems of Nonlinear Equations  PAGEREF _Toc358821505 \h 21  HYPERLINK \l "_Toc358821506" 1. Newton-Raphson  PAGEREF _Toc358821506 \h 21  HYPERLINK \l "_Toc358821507" 2. Implicit Iterative Methods  PAGEREF _Toc358821507 \h 21  HYPERLINK \l "_Toc358821508" IV. Linear Algebra  PAGEREF _Toc358821508 \h 23  HYPERLINK \l "_Toc358821509" A. Matrix Arithmetic  PAGEREF _Toc358821509 \h 23  HYPERLINK \l "_Toc358821510" 1. Matrices  PAGEREF _Toc358821510 \h 23  HYPERLINK \l "_Toc358821511" 2. Addition & Subtraction  PAGEREF _Toc358821511 \h 23  HYPERLINK \l "_Toc358821512" 3. Multiplication  PAGEREF _Toc358821512 \h 23  HYPERLINK \l "_Toc358821513" 4. Inverse Matrix  PAGEREF _Toc358821513 \h 24  HYPERLINK \l "_Toc358821514" B. Simultaneous Linear Equations  PAGEREF _Toc358821514 \h 25  HYPERLINK \l "_Toc358821515" 1. The Problem  PAGEREF _Toc358821515 \h 25  HYPERLINK \l "_Toc358821516" 2. Gaussian Elimination  PAGEREF _Toc358821516 \h 25  HYPERLINK \l "_Toc358821517" 3. Matrix Operations  PAGEREF _Toc358821517 \h 26  HYPERLINK \l "_Toc358821518" 4. Gauss-Jordan Elimination  PAGEREF _Toc358821518 \h 28  HYPERLINK \l "_Toc358821519" C. Iterative Methods  PAGEREF _Toc358821519 \h 30  HYPERLINK \l "_Toc358821520" 1. Jacobi Method  PAGEREF _Toc358821520 \h 30  HYPERLINK \l "_Toc358821521" 2. Gauss-Seidel Method  PAGEREF _Toc358821521 \h 31  HYPERLINK \l "_Toc358821522" D. Applications  PAGEREF _Toc358821522 \h 32  HYPERLINK \l "_Toc358821523" 1. Electrical Circuit  PAGEREF _Toc358821523 \h 32  HYPERLINK \l "_Toc358821524" 2. Truss System  PAGEREF _Toc358821524 \h 33  HYPERLINK \l "_Toc358821525" V. Interpolation and Curve Fitting  PAGEREF _Toc358821525 \h 34  HYPERLINK \l "_Toc358821526" A. Polynomial Interpolation  PAGEREF _Toc358821526 \h 34  HYPERLINK \l "_Toc358821527" 1. Uniqueness  PAGEREF _Toc358821527 \h 34  HYPERLINK \l "_Toc358821528" 2. Newtons Divided Difference Interpolating Polynomial  PAGEREF _Toc358821528 \h 35  HYPERLINK \l "_Toc358821529" B. Least Squares Fitting  PAGEREF _Toc358821529 \h 38  HYPERLINK \l "_Toc358821530" 1. Goodness of Fit  PAGEREF _Toc358821530 \h 38  HYPERLINK \l "_Toc358821531" 2. Least Squares Fit to a Polynomial  PAGEREF _Toc358821531 \h 38  HYPERLINK \l "_Toc358821532" 3. Least Squares Fit to Non-polynomial Function  PAGEREF _Toc358821532 \h 40  HYPERLINK \l "_Toc358821533" MatLab Sidelight Number One  PAGEREF _Toc358821533 \h 41  HYPERLINK \l "_Toc358821534" 1. Polynomials  PAGEREF _Toc358821534 \h 41  HYPERLINK \l "_Toc358821535" 2. Curve Fitting & Interpolation  PAGEREF _Toc358821535 \h 42  HYPERLINK \l "_Toc358821536" VI. Integration  PAGEREF _Toc358821536 \h 43  HYPERLINK \l "_Toc358821537" A. Newton-Cotes Formul  PAGEREF _Toc358821537 \h 43  HYPERLINK \l "_Toc358821538" 1. Trapezoid Rule  PAGEREF _Toc358821538 \h 43  HYPERLINK \l "_Toc358821539" 2. Extension to Higher Order Formul  PAGEREF _Toc358821539 \h 44  HYPERLINK \l "_Toc358821540" B. Numerical Integration by Random Sampling  PAGEREF _Toc358821540 \h 47  HYPERLINK \l "_Toc358821541" 1. Random Sampling  PAGEREF _Toc358821541 \h 47  HYPERLINK \l "_Toc358821542" 2. Samples of Random Sampling  PAGEREF _Toc358821542 \h 48  HYPERLINK \l "_Toc358821543" 3. Integration  PAGEREF _Toc358821543 \h 48  HYPERLINK \l "_Toc358821544" MatLab Sidelight Number Two  PAGEREF _Toc358821544 \h 53  HYPERLINK \l "_Toc358821545" 1. Nonlinear Equations  PAGEREF _Toc358821545 \h 53  HYPERLINK \l "_Toc358821546" 2. Integration  PAGEREF _Toc358821546 \h 53  HYPERLINK \l "_Toc358821547" VII. Ordinary Differential Equations  PAGEREF _Toc358821547 \h 55  HYPERLINK \l "_Toc358821548" A. Linear First Order Equations  PAGEREF _Toc358821548 \h 55  HYPERLINK \l "_Toc358821549" 1. One Step Methods  PAGEREF _Toc358821549 \h 55  HYPERLINK \l "_Toc358821550" 2. Error  PAGEREF _Toc358821550 \h 56  HYPERLINK \l "_Toc358821551" MatLab Sidelight Number Three  PAGEREF _Toc358821551 \h 58  HYPERLINK \l "_Toc358821552" 1. First Order Ordinary Differential Equations (ODE)  PAGEREF _Toc358821552 \h 58  HYPERLINK \l "_Toc358821553" B. Second Order Ordinary Differential Equations  PAGEREF _Toc358821553 \h 59  HYPERLINK \l "_Toc358821554" 1. Reduction to a System of First Order Equations  PAGEREF _Toc358821554 \h 59  HYPERLINK \l "_Toc358821555" 2. Difference Equations  PAGEREF _Toc358821555 \h 60  I. Introduction A. Numerical Methods or Numerical Analysis 1. Numerical Analysis a. Definition Concerned with solving mathematical problems by the operations of arithmetic. That is, we manipulate ( EMBED Equation.3 , etc.) numerical values rather than derive or manipulate analytical mathematic expressions ( EMBED Equation.3 , etc.). We will be dealing always with approximate values rather than exact formul. b. History Recall the definition of a derivative in Calculus:  EMBED Equation.3 , where  EMBED Equation.3  and  EMBED Equation.3 . We will work it backwards, using  EMBED Equation.3 . In fact, before Newton and Leibnitz invented Calculus, the numerical methods were the methods. Mathematical problems were solved numerically or geometrically, e.g., Kepler and Newton with their orbits and gravity. Many of the numerical methods still used today were developed by Newton and his predecessors and contemporaries. They, or their computers, performed numerical calculations by hand. Thats one reason it could take Kepler so many years to formulate his Laws of planetary orbits. In the 19th and early 20th centuries adding machines were used, mechanical and electric. In business, also, payroll and accounts were done by hand. Today, we use automatic machines to do the arithmetic, and the word computer no longer refers to a person, but to the machine. The machines are cheaper and faster than people; however, they still have to be told what to do, and when to do itcomputer programming. 2. Newtons Method for Solving a Nonlinear Equationan example a. Numerical solution Lets say we want to evaluate the cube root of 467. That is, we want to find a value of x such that  EMBED Equation.3 . Put another way, we want to find a root of the following equation:  EMBED Equation.3 . If f(x) were a straight line, then  EMBED Equation.3 . In fact,  EMBED Equation.3 , but lets say that  EMBED Equation.3  and solve for x1.  EMBED Equation.3 . Note that we are using  EMBED Equation.3 . Having now obtained a new estimate for the root, we repeat the process to obtain a sequence of estimated roots which we hope converges on the exact or correct root.  EMBED Equation.3   EMBED Equation.3  etc. In our example,  EMBED Equation.3  and  EMBED Equation.3 . If we take our initial guess to be  EMBED Equation.3 , then by iterating the formula above, we generate the following table: i EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 06-25110818.324109.7718207.870627.7966.8172182.331637.7590.1080.0350 EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  [Note: The pocket calculator has a (yx) button, but a computer may do  EMBED Equation.3  to get x3.] b. Analytical solution How might we solve for the cube root of 467 analytically or symbolically? Take logarithms.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3 = 7.758402264. . . We used the (ln) button on our pocket calculator, followed by the (ex) button. In earlier times, wed have used log tables. But, whence cometh those tables and how does the calculator evaluate ln 467 or e2.0488? 3. Series  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The infinite series are exact. However, in practice we always keep a finite number of terms. In principle, we can achieve arbitrary precision, if we have the necessary patience. Pocket calculators and computer programs add up enough terms in a series to achieve a specified precision, say 8 or 16 significant digits. 4. Error In this context, the term error does not refer to a mistake. Rather, it refers to the ideas of deviation or of uncertainty. Every measured value is uncertain, according to the precision of the measuring instrument. Every computed value is uncertain, according to the number of significant digits carried along or according to the number of terms retained in the summation of a series. Consequently, all numerical solutions are approximate. Oftentimes, in discussing an example problem, the correct exact solution is known, so it is possible to determine how an approximate numerical solution deviates from that exact solution. Indeed, algorithms are often tested by applying them to problems having known exact solutions. However, in real life, we dont know the correct exact solution. We cant know how far our approximate solutions deviate from the correct, exact, but unknown solution. In other words, we have to approximate the solution to a problem, but also we can only estimate the error. Fortunately, we have means of estimating error. A goodly portion of the discussion in a Numerical Methods textbook is devoted to rigorous estimation of error. In this course, we wont concern ourselves with a detailed discussion of error analysis. Nonetheless, we want to be always aware of the error issue, keeping in mind at least qualitatively the limitations of a numerical solution. From time to time in the paragraphs that follow some aspects of the error involved with a particular algorithm will be briefly discussed. B. Programming The computer carries out the tedious arithmetic, but it must be told what to do. That is the function of a computer program. A program may be written in one of any number of programming languages, however there are certain features or issues that all languages have in common. 1. Program Design a. Stages Conceptiondefine the problem Develop the algorithmmap out or outline the solution Codewrite the program Debug & verifytrace the program; perform trial runs with known results; correct logical & syntax errors b. Building blocks Sequential operationsinstructions done one after the other in a specified order Branching operationsselecting alternative sequences of operations Looping operationsrepeating subsets of operations I/O operationsreading and writing data 2. Branching a. Simple yes or noselect between just 2 alternative actions b. Nested branchesa sequence of decisions or branches; decision tree c. Select casemore than two alternative actions 3. Loops a. Counted loopa section of code is executed a specified number of times b. Conditional loopa section of code is iterated until a specified condition is met c. Infinite loopthe condition for ending the loop never is encountered, so the program never ends 4. I/O a. Inputkeyboard or data file b. Outputmonitor, output file, printer; numbers, text, graphics 5. Precision Issues a. Binary The computer does its arithmetic with binary numbers, that is, base-2. E.g., 0, 1, 10, 11, 100, 101, 110, 111, etc. We are accustomed to working and thinking with base-10 numbers. In producing the machine language code (the executable) and carrying out calculations, all numerical values are translated from base-10 to base-2 then back again for output. Usually, we dont need to care about this. However, it can be a source of loss of precision in our numerical values because the machine stores values with only finite precision. b. Precision A single binary digit (0 or 1) is called a bit. Eight bits make up a byte. Within the machine, the unit of information that is transferred at one time to/from the CPU and main memory is called a word. The size of a word, or the word length, varies from one machine to another. Typically, itll be from 4 to 64 bits. A 4-byte word contains 32 bits, etc. One memory cell or memory location holds one or more words. Lets say its one word, or 4 bytes. Whatever information (number) is stored in one such memory cell must be expressible as a string of 32 bits and no more. For instance, a non-terminating binary fraction will be truncated, e.g., (0.1)10 = (0.00011001100110011. . .)2. Only 32 digits will be stored in memory. When translated back into decimal, the number will be (0.09999997)10, not (0.1)10. Similarly, the finite precision places a limit on the largest and the smallest numerical value that can be stored in a memory cell. In the back of our minds, we always remain aware of the physical limitations of the machine. 6. Debugging When syntax errors are all eliminated, the program may very well run smoothly to completion. Perhaps it produces results which are clearly absurd; perhaps the results appear quite plausible. A programmer must always take steps to convince itself that the program is working correctly; the temptation to assume must be resisted. One of the most insidious assumptions is that the program is doing what the programmer intended it to do. Perhaps, a typing error has produced a statement that has no syntax error, but does a different operation from that intended. Perhaps the logical sequence of steps written by the programmer doesnt accomplish the task intended by the programmer. This why program tracing is so important, why it is essential to insert print statements all through the program to display the intermediate values of variables, why it is essential to check and double check such things as argument lists and dimensions and the values of indiceschecking not what the programmer intended, but what the program actually does. The other, almost easier, aspect of debugging involves applying the program to a problem whose solution is already known. It also involves repeating a numerical solution with different values of various parameters such as step size and convergence tolerance. It involves comparing a numerical solution for consistency with previous experience. II. MatLab A. Program Features Work in MatLab is done in a variety of windows. The windows used most often are the Command, Figure, Editor, and Help windows. When the program is started, three windows are displayedCommand, Current Directory, and Command History windows. The first thing to do upon starting the program is to select the Desktop Menu, select Desktop Layout, select Command Window Only. 1. Commands a. Command lines (p. 9) Commands are entered at the command prompt (>>). When the enter key is pressed, the command is executed and the output (if any) is displayed at once. All commands are recorded in the Command History. Results from those previous commands are remembered. More than one command may be entered on one line, separated by commas. The commands are executed in order when enter is pressed. A command can be continued to the next line with an ellipsis followed by enter. The command history can be accessed with the up and down arrow keys. Suppress command output--If a command is ended with a semicolon, display of its output (if any) is suppressed. The product of the command is still available, just not displayed in the command window. Comments Comment lines are started with the % symbol. They are not executed when the enter key is pressed. A comment may also be attached to the end of a command, before pressing the enter key. Clearing the Command Window The clc command clears the Command Window, but does not erase the command history. b. Arithmetic operators (p. 10) OperationSymbol precedenceAddition+ 4Subtraction- 4Multiplication* 3Right division/ 3Left division\ 3Exponentiation^ 2 Notice the distinction between right & left division. Left division is right division raised to the 1 power: 3\5 = 5/3. Expressions enclosed in parentheses are evaluated first. Nested parentheses are executed from innermost outward. c. Built-in functions (pp. 13-16) Commonly used math functions are built-in. There are the usual sqrt, exp, sin, cos, etc., as shown in the tables in the text. In addition, there are so-called rounding functions. The argument, x, may be an expression. FunctionDescriptionRound(x)Round to nearest integerFix(x)Round toward zeroCeil(x)Round toward infinityFloor(x)Round toward infinityRem(x,y)Remainder of x/ySign(x)Returns the algebraic sign of x: 1, -1, or 0 d. Scalars A scalar is a numerical constant, like 5 or 8746 or 45.998, etc. A scalar variable is a name, really the label of a memory location. A numerical value is stored in a variable. That numerical value may be changed at any time. A variable name must begin with a letter, but may otherwise contain letters, digits and the underscore character. There is a limit to how many characters the name may be, but that varies with the MatLab version. Built-in scalar variables: ans, pi, eps = 2^(-52), inf (infinity), and my favorite, NaN (not a number). The variable ans is used to store the value of an expression or command that has not been assigned a variable name. Caution! The built-in scalar variables may be reassigned, whether inadvertently, or advertently. The values stored in variables are all retained until or unless they are removed from memory with the clear command. A list of variables presently in memory is obtained with the who or whos commands. e. Assignment operator Numerical values are assigned to a variable name with the assignment operator. The assignment operator is the = sign, but it does not mean equal to. It means store this value in the memory location labeled by the specified variable name. Only a single variable name can be on the left-hand side, while the right-hand side may be a single number or a computable expression including other, previously defined, variables. The initial assignment of a value to a variable serves to define that variable. There is no special declaration of variable types as is seen in some programming languages. f. Numerical display formats (p. 12-13) The format command sets the display format of numerical values. See Table 1-2 in the text. Basically, the number of digits displayed can be either 4 or 14(15) in either fixed point or exponential notation. 2. Arrays An array is a matrix, or rather a matrix is an array of numbers. An n by m matrix has n rows and m columns. All variables in Matlab are arrays, even scalars, which are 1x1 arrays. a. Vectors A vector is a one-dimensional array. A row vector has one row and n columns. A row vector is defined by listing its elements enclosed by square brackets and separated by commas or spaces. E.g., a three element row vector is defined by A = [a1 , a2 , a3]. Similarly, a column vector is defined by listing its elements enclosed by square brackets and separated by semicolons. B = [b1 ; b2 ; b3] The column vector has one column and m rows. Alternatively, row vectors may be defined by first element (zi), last element (zf) and the spacing between the elements (q). Z = [ zi : q : zf ] The linspace command creates a row vector by specifying the first and last element and the number of elements. Z = linspace(z1,zf,n) A character string is stored in MatLab as a vector, one character to one element. For instance, B = Now is the time for all creates a 23-element vector, as there are 23 characters (including spaces) in the phrase enclosed in the single quote marks. Each element may be addressed and altered/replaced/deleted individually. b. Two-dimensional arrays A = [first row ; second row ; third row ; . . .] The rows can be specified as individual row vectors. The elements can be expressions. Special arrays are zeros (elements all zero), ones (elements all ones), and eye (the identity matrix). The matrix transpose operator is the single quote mark. B = A ( B is the transpose of A. c. Addressing matrix elements Individual elements of an array are referred to by their indices. A(k) is the kth element of the vector A. B(m,n) is the element in the mth row & nth column. It may be desirable to address an entire row or column of a matrix, perhaps a subset of a row or column. In that case a colon (:) is used to indicate a range. The 3rd through 6th elements of a vector are addressed by A(3:6), etc. Likewise, all the elements of the mth row of a matrix are addressed by B(m,:). The m through n columns of all the rows of a matrix B are designated by B(:,m:n). The most general case would be a block of elements within the matrixB(m:n,p:q). d. Adding or deleting matrix elements It is possible alter the sizes of a previously defined array variables. This done simply by addressing additional vector(matrix) elements and assigning them values. Say that A is a 4-element vector. We add elements to the vector by assigning values to the extra elements. A(5)=5 , A(6)=7 , A97)=-98 , etc. Alternatively, a preexisting vector may be appended to another. C=[ A B] or C=[G ; H]. Likewise, rows, columns, or entire matrices may be appended to a matrix. Of course, the dimensions of the added rows, columns, & matrices must match the matrix being enlarged. A vector or matrix can be reduced in size, as well, by assigning nothing to some the elements, thusly: B(:,4,9)=[]. This particular example will eliminate all rows from columns 4 9. e. Built in array manipulations Some common array handling functions are built-in. These are listed on pages 41 43 of the text. 3. Array Operations a. Matrix operations Arrays are multiplied, divided, added, subtracted, etc. according to the usual rules of matrix arithmetic. Inverse A-1 = A^-1 Left & right division X = A-1B = A\B X = DC-1 = D/C Left and right division arise because matrix multiplication is not commutative. b. Element by element operations There exist also what are called element-by-element operations. In that case, an operation is carried out on every element of an array. A period is added in front of the math operator to indicate element-by-element operation. E.g., .* or .^ Notice that a dot product between two vectors can be carried out by an element-by-element multiplication: sum(A.*B) = a1b1 + a2b2 + a3b3 + . . . c. Analyzing arrays The built-in array functions are listed in Table 3-1, pages 64-65. These include Inv and Det. B. Files 1. m-files MatLab commands can be stored in a plain text file, and then run in the Command window. The general term for a series of commands is a script. Writing such a series of commands is called scripting. In MatLab, script files are saved with the extension .m, hence the term m-files. The m-file may be created & edited in any plain text editor, such as Notepad, or by any word processing program that is capable of storing plain text. There is also an Edit Window in MatLab itself. a. Editor Script files are created, edited, saved, and run in the Edit window. b. I/O (pp 95-117) Input Assign variables in the Command window before running the script. Use the Input command or function within the script to interactively enter data. Variable = input(message string) Output Dispwrites to the workspace Fprintfallows formatting of the printed line(s). 2. Script files a. Running Run by entering the file name at the prompt in the Command window. Run by pressing the run button in the Editor window In either case, commands previously issued and variables previously defined in the Command window are known to the script file. b. Comments & documentation There must be comments throughout a script file describing the purpose of the script, defining the variables used, describing the required input, etc. The purpose of the documentation is to make plain what is happening in the script to yourself or another programmer at some later date, not to mention to the instructor. Get in the habit early of over-commenting your scripts. c. Inline & feval These are commands to create one-liners. Functioname = inline(math expression as character string) x = functioname(arguments) variable = feval(function name,argument value) 3. Function files function command A function file differs from the general script file in that it is self-contained. Variables assigned in the work space (Command window) are not available inside the function file in general. Likewise, variables assigned within the function file are not available outside the function file. Variables have to be assigned inside the function file, or passed via the argument list in the function statement, or of course by input commands. The first line of a function file is function[arguments-out] = functioname(arguments-in) Typically, the function is saved in the file functioname.m; that is, the file name is the same as the function. The function is invoked by entering the functioname(arguments-in) Data can be passed to the function through global variables, the argument-in list, and through input commands within the function, as well as xlsread commands. The function produces output through disp, fprintf, and plot commands within the script, or through the arguments-out list. It is possible to define variables to be global variables by including the Global command in all script files, and the Command window as well. Global variable list C. Plots 1. Two Dimensional Graphs (pp. 133-158 a. Line plots Executing the plot or the fplot command automatically opens a Figure Window. Plot(X,Y) plots Y vs X, where X & Y are vectors of the same length. If no other parameters are specified, the graph is plotted in a bare-bones fashion, with a line connecting the data points, but no axis titles, or data point symbols, etc. The axes are scaled over the intervals spanned by the vectors X & Y. However, there are parameters within the plot command as well as additional commands whose purpose is to change the format of the graph. A graph can be formatted interactively within the Figure Window, as well. For plotting a function, there is the command fplot(function,xmin,xmax,ymin,ymax). The function, y = f(x), is entered as a character string, as in 45*cos(3*x^3). The drawback of fplot is that the f(x) cannot include variable names, only the dummy variable. b. Other plots There are available other plotting commands that produce log graphs, bar graphs, pie charts, etc. c. Multiple graphs It is possible to graph several curves on the same plot, using the Hold On and Hold Off commands. Alternatively, it is possible to create several separate graphs on a single page with the Subplot command. 2. Three Dimensional Graphs a. Line plots (p 323) Plot3(X,Y,Z) This one is intended to plot X(t), Y(t), & Z(t) all as functions of a fourth parameter, t. b. Surface plots (pp 324-330) Mesh(X,Y,Z) or Surf(X,Y,Z) These commands plot Z(X,Y). The mesh command creates a wire-grid surface, while the surf command adds color shading to the surface. There are variations of mesh & surf that produce surface graphs of differing appearancemeshz, meshc, surfc, etc. c. Contour plots (p 330) Contour(X,Y,Z,n) and variations. d. Special graphics (p 331) Bar3(Y) Sphere or [X,Y,Z]=Sphere(n) produces a set of (X,Y,Z) to be used by mesh or surf to plot a sphere. [X,Y,Z]=Cylinder(r) produces a set of points to be used by mesh or surf to draw a cylinder. r is a vector that specifies the profile of the cylinder. r = some f(t) e. view command The View command alters the angle at which a 3-d plot is viewed, by specifying the azimuth and elevation angles of the view point. View(az,el), with az and el specified in degrees, relative to the xz-plane and the xy-plane, respectively. D. Programs MatLab has many built-in functions and computing tools. Nonetheless, it becomes necessary to write a special-purpose solution for a specific problem. No one commercial computing package can address every possible situation, and no one lab can have every commercial product on hand. Previously, we have used assignment statements to carry out calculations, and plot commands to produce graphical output. Computer programs require also statements to make decisions, to make comparisons and to carry out repetitive operations, not to mention input and output. 1. Branches a. Relational & logical operators (p.174) OperatorDescription<Less than>Greater than<=Less than or equal to>=Greater than or equal to= =Equal to**~=Not equal to*The equal to operator consists of two equal signs, with no space between them. If two numbers are compared, the result is 1 (logical true) or 0 (logical false). Comparing two scalars yields a scalar 1 or 0. Arrays are compared element-by element. The result is a logical array of 1s and 0s. Evidently, the two arrays must be the same size if they are to be compared with each other. Similarly, a scalar is compared with an array element-by element, and the result is logical array of 1s and 0s. The elements of logical arrays can be used to address elements in ordinary arrays. Since the relational comparisons produce numerical values, relational operators can be used within mathematical expressions. In mathematical expressions, the relational operators are evaluated after all mathematical operators. Logical operators OperatorDescription& A&BA AND B =true if both A and B are true, false otherwise| A|BA OR B =true if A or B is true, false if both are false~ ~ANOT A =true if A is false, false if A is true See the order of precedence on page 178. Notice that NOT comes after exponentiation and before multiplication, etc., but that the other logical operators (AND, OR) come last. There are a number of built-in logical functions, described on pages 179 180. b. If (pp. 182-190) The IF statement is used to select between two courses of action. Several IF statements may be nested to create a binary decision tree. The decision is based on the truth or falsity of a statement or conditional expression. A conditional expression is an expression consisting of relational and/or logical operators. The expression will have the value true or false. i. if-end ( a block of commands is executed if the conditional expression is true, skipped if its false. if conditional expression Matlab commands end ii. if-else-end ( in this case, there are two blocks of MatLab commandsone is executed if the conditional expression is true, the other if it is false. if conditional expression MatLab commands else Matlab commands end iii. if-elseif-else-end ( using two conditional expressions, one of three sets of Matlab commands is executed. if conditional expression MatLab commands elseif conditional expression MatLab commands else Matlab commands end c. Case If we desire to select from among more than 2 or 3 cases, then it may be more convenient to use the switch-case statement. switch switch expression case value1 MatLab commands case value2 MatLab commands case value3 MatLab commands etc. otherwise MatLab commands end The switch expression is a scalar or string variable or an expression that can take on the values value1, value2, value3, etc. If none of the specified values occur, then the block following the otherwise command is executed. The otherwise command is optional. 2. Loops (pp. 190-200) Another thing we want a computer program to do automatically is to repeat an operation. a. Counting The for-end loop executes a block of MatLab commands a specified number of times. for k = f:s:t MatLab commands end The loop executes for k = f, f+s, f+2s, f+3s, . . ., t. The increment, s, may be omitted in which case it is assumed to be 1. b. Conditional Alternatively, a loop may be executed as long as a conditional expression remains true. while conditional expression MatLab commands end The variables in the conditional expression must have initial values assigned, and at least one of the variables must be changed within the loop. 3. Input/output (pp 114-118) a. File input variable = xlsread(filename,sheetname,range)(import data from an Excel spreadsheet b. Import Wizard. The Import Wizard is invoked by selecting Import Data in the File Menu. c. File output fprintf--writes to a plain text disk file fprint(fid,arguments) fid=open(filename) fclose(fid) xlswrite(filename,sheetname,range,variablename)--export to an Excel spreadsheet III. Numerical Solution of Nonlinear Equations A. Non-Linear Equationsone at a time There are closed form solutions for quadratic and even 3rd degree polynomial equations. Higher degree polynomials can sometimes be factored. However, in general there is no closed form analytical solution to non-linear equations. 1. The Problem a. Roots & zeroes We seek to find x such that  EMBED Equation.3  or perhaps such that  EMBED Equation.3 . In the latter case, we merely set  EMBED Equation.3 . We are looking for a root of the equation  EMBED Equation.3  or a zero of the function f(x). b. Graphical solution Plot f(x) vs. xobserve where the graph crosses the x-axis or plot f(x) and g(x) vs. x and observe where the two curves intersect. A graph wont give a precise root, but we can use the graph to choose an initial estimate of the root. 2. Bisection a. Setup For brevity, say fo = f(xo) and f1 = f(x1), etc. Say further that  EMBED Equation.3  is the desired root. The graph shows us that  EMBED Equation.3  because f(x) crosses the x-axis between [xo,x1]. b. Algorithm Let us find the midpoint of [xo,x1], and call it b. i)  EMBED Equation.3  and then  EMBED Equation.3  ii) Does  EMBED Equation.3 ? If so, quit cause  EMBED Equation.3 . iii) If not, then if  EMBED Equation.3 , then set  EMBED Equation.3  and  EMBED Equation.3  or if  EMBED Equation.3 , then set instead  EMBED Equation.3  and  EMBED Equation.3 . iv) Is  EMBED Equation.3 ? If so, quit and set  EMBED Equation.3 . v) If not, then repeat beginning with step (i). It is well also to count the iterations and to place a limit on the number of iterations that will be performed. Otherwise, the program could be trapped in an infinite loop. Also, it is well to test for the cases  EMBED Equation.3  and  EMBED Equation.3 . It may be that the function does not cross the x-axis between fo and f1, or crosses more than once. 3. Newtons Method or the Newton-Raphson Method a. Taylors series Any well-behaved function can be expanded in a Taylors series:  EMBED Equation.3 . Lets say that x is close to xo and keep just the first two terms.  EMBED Equation.3  We want to solve for x such that f(x) = 0.  EMBED Equation.3   EMBED Equation.3  In effect we have approximated f(x) by a straight line; x is the intercept of that line with the x-axis. It may or may not be a good approximation for the root  EMBED Equation.3 .  b. Algorithm i) choose an initial estimate, xi ii) compute f(xi) and  EMBED Equation.3  iii) compute the new estimate:  EMBED Equation.3  iv) return to step (ii) with i = i + 1 c. Comments It turns out that if the initial estimate of the root is a good one, then the method is guaranteed to converge, and rapidly. Even if the estimate is not so good, the method will converge to a rootmaybe not the one we anticipated. Also, if there is a  EMBED Equation.3  point nearby the method can have trouble. Its always a good thing to graph f(x) first. 4. Secant Method a. Finite differences A finite difference is merely the difference between two numerical values.  EMBED Equation.3  or  EMBED Equation.3  Derivatives are approximated by divided differences.  EMBED Equation.3  We may regard this divided difference as an estimate of  EMBED Equation.3  at xi or at xi+1 or at the midpoint between xi and xi+1. b. The Secant method We simply replace  EMBED Equation.3  by the divided difference in the Newton-Raphson formula:  EMBED Equation.3 . Notice the indices: i + 1, i, i 1. With the Secant Method, we dont use a functional form for  EMBED Equation.3 . We do have to carry along two values of f, however. Care must be taken that  EMBED Equation.3  not be too small, which would cause an overflow error by the computer. This may occur if  EMBED Equation.3  due to the finite precision of the machine. This may also give a misleading result for the convergence test of  EMBED Equation.3 . To avoid that, we might use the relative deviation to test for convergence.  EMBED Equation.3  c. Compare and contrast Both the Newton-Raphson and Secant Methods locate just one root at a time. Newton: requires evaluation of f and of  EMBED Equation.3  at each step; converges rapidly. Secant: requires evaluation only of f at each step; converges less rapidly. 5. Hybrid Methods A hybrid method combines the use in one program of two or more specific methods. For instance, we might use bisection to locate a root roughly, then use the Secant Method to compute the root more precisely. For instance, we might use bisection to locate multiple roots of an equation, then use Newton-Raphson to refine each one. B. Systems of Nonlinear Equations Consider a system of n nonlinear equations with n unknowns.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  1. Newton-Raphson a. Matrix notation Lets write the system of equations as a matrix equation.  EMBED Equation.3  The unknowns form a column matrix also.  EMBED Equation.3 . We might write the system of equations compactly as  EMBED Equation.3 . b. The Method The Newton-Raphson method for simultaneous equations involves evaluating the derivative matrix,  EMBED Equation.3 , whose elements are defined to be  EMBED Equation.3 . If the inverse  EMBED Equation.3  exists, then we can generate a sequence of approximations for the roots of functions {fi}.  EMBED Equation.3  At each step, all the partial derivatives must be evaluated and the  EMBED Equation.3  matrix inverted. The iteration continues until all the  EMBED Equation.3 . If the inverse matrix does not exist, then the method fails. If the number of equations, n, is more than a handful, the method becomes very cumbersome and time consuming. 2. Implicit Iterative Methods The Newton-Raphson method is an iterative method in the sense that it generates a sequence of successive approximations by repeating, or iterating, the same formula. However, the term iterative method as commonly used refers to a particular class of algorithms which might more descriptively be called implicit iterative methods. Such algorithms occur in many numerical contexts as well see in subsequent sections of this course. At this point, we apply the approach to the system of simultaneous nonlinear equations. a. General form Let  EMBED Equation.3  be the solution matrix to the equation  EMBED Equation.3 . I.e.,  EMBED Equation.3 . Now, solve algebraically each  EMBED Equation.3  for xi. This creates a new set of equations,  EMBED Equation.3 , where  EMBED Equation.3  refers to the set of unknowns {xj} excluding xi. Algebraically, this looks funny, because each unknown is expressed in terms of all the other unknowns, hence the term implicit. Of course, what we really mean is  EMBED Equation.3 . Alternatively, in terms of matrix elements, the equations take the form  EMBED Equation.3 . b. Algorithm In a program, the iterative method is implemented thusly: i) choose an initial guess,  EMBED Equation.3  ii) compute  EMBED Equation.3  iii) test  EMBED Equation.3  iv) if yes, set  EMBED Equation.3  and exit v) if no, compute  EMBED Equation.3 , etc. c. Convergence We hope that  EMBED Equation.3 . For what conditions will this be true? Consider a region R in the space of {xi} such that  EMBED Equation.3  for  EMBED Equation.3  and suppose that for  EMBED Equation.3  in R there is a positive number  EMBED Equation.3  such that  EMBED Equation.3 . Then, it can be shown that if  EMBED Equation.3  lies in R, the iterative method will converge. What does this mean, practically? It means that if the initial guess,  EMBED Equation.3 , is close enough to  EMBED Equation.3 , then the method will converge to  EMBED Equation.3  after some number, k, of iterations. Big deal. IV. Linear Algebra A. Matrix Arithmetic The use of matrix notation to represent a system of simultaneous equations was introduced in section III-B-1 above, mainly for the sake of brevity. In solving simultaneous linear equations, matrix operations are central. There follows, therefore, a brief review of the salient properties of matrices. Fuller discussion of the properties of matrices may be found in various texts, particularly Linear Algebra texts. 1. Matrices A matrix is an n x m array of numbers. In these notes a matrix is symbolized by a letter with a line on top,  EMBED Equation.3 ; n is the number of rows and m is the number of columns. If n = m, the matrix is said to be a square matrix. If the matrix has only one column(row) it is said to be a column(row) matrix. The jth element in the ith row of a matrix is indicated by subscripts, bij. Mathematically, an entity like a matrix is defined by a list of properties and operations, for instance the rules for adding or multiplying two matrices. Also, matrices can be regarded as one way to represent members of a group in Group Theory.  EMBED Equation.3   EMBED Equation.3  2. Addition & Subtraction a. Definition The addition is carried out by adding the respective matrix elements.  EMBED Equation.3   EMBED Equation.3  b. Rules The sum of two matrices is also a matrix. Only matrices having the same number of rows and the same number of columns may be added. Matrix addition is commutative and associative.  EMBED Equation.3   EMBED Equation.3  3. Multiplication a. Definition  EMBED Equation.3   EMBED Equation.3  b. Rules The product of two matrices is also a matrix. The number of elements in a row of  EMBED Equation.3  must equal the number of elements in a column of  EMBED Equation.3 . Matrix multiplication is not commutative.  EMBED Equation.3  A matrix may be multiplied by a constant, thusly:  EMBED Equation.3 . The result is also a matrix. 4. Inverse Matrix a. Unit matrix The unit matrix is a square matrix with the diagonal elements equal to one and the off-diagonal elements all equal to zero. Heres a 3x3 unit matrix:  EMBED Equation.3  b. Inverse The inverse of a matrix,  EMBED Equation.3 , (denoted  EMBED Equation.3 ) is a matrix such that  EMBED Equation.3 . The inverse of a particular matrix may not exist, in which case the matrix is said to be singular. The solution of a system of simultaneous equations in effect is a problem of evaluating the inverse of a square matrix. B. Simultaneous Linear Equations 1. The Problem a. Simultaneous equations We wish to solve a system of n linear equations in n unknowns.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  where the {bij} and the {ci} are constants. b. Matrix notation The system of equations can be written as a matrix multiplication.  EMBED Equation.3 , where  EMBED Equation.3 ,  EMBED Equation.3  and  EMBED Equation.3 . When n is small ( EMBED Equation.3 , say) a direct or one-step method is used. For larger systems, iterative methods are preferred. 2. Gaussian Elimination In a one-step approach, we seek to evaluate the inverse of the  EMBED Equation.3  matrix.  EMBED Equation.3   EMBED Equation.3  The solution is obtained by carrying out the matrix multiplication  EMBED Equation.3 . a. Elimination You may have seen this in high school algebra. For brevitys sake, lets let n = 3.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  In essence, we wish to eliminate unknowns from the equations by a sequence of algebraic steps. normalization i) multiply eqn. 1 by  EMBED Equation.3  and add to eqn. 2; replace eqn. 2. reduction ii) multiply eqn 1 by  EMBED Equation.3  and add to eqn. 3; replace eqn. 3.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  iii) multiply eqn. 2 by  EMBED Equation.3  and add to eqn. 3; replace eqn. 3.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  We have eliminated x1 and x2 from eqn.3 and x1 from eqn. 2. back substitution iv) solve eqn. 3 for x3, substitute in eqn. 2 & 1. solve eqn. 2 for x2, substitute in eqn. 1. solve eqn. 1 for x1. b. Pivoting Due to the finite number of digits carried along by the machine, we have to worry about the relative magnitudes of the matrix elements, especially the diagonal elements. In other words, the inverse matrix,  EMBED Equation.3  may be effectively singular even if not actually so. To minimize this possibility, we commonly rearrange the set of equations to place the largest coefficients on the diagonal, to the extent possible. This process is called pivoting. e.g. 37x2 3x3 = 4 19x1 2x2 + 48x3 = 99 7x1 + 0.6x2 +15x3 = -9 rearrange 19x1 2x2 + 48x3 = 99 37x2 3x3 = 4 7x1 + 0.6x2 +15x3 = -9 or 7x1 + 0.6x2 +15x3 = -9 37x2 3x3 = 4 19x1 2x2 + 48x3 = 99 3. Matrix Operations In preparation for writing a computer program, well cast the elimination and back substitution in the form of matrix multiplications. a. Augmented matrix  EMBED Equation.3  b. Elementary matrices Each single step is represented by a single matrix multiplication. The elimination steps:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The first back substitution step:  EMBED Equation.3   EMBED Equation.3  This completes one cycle. Next we eliminate one unknown from the second row using  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  This completes the second cycle. The final cycle is  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  We identify the inverse matrix  EMBED Equation.3 . Notice that the order of the matrix multiplications is significant. Naturally, we want to automate this process, and generalize to n equations. 4. Gauss-Jordan Elimination a. Inverse matrix We might multiply all the elementary matrices together before multiplying by the augmented matrix. That is, carry out the evaluation of  EMBED Equation.3 , then perform  EMBED Equation.3 .  b. Algorithm  EMBED Equation.3   EMBED Equation.3  n = number of equations k = index of the step or cycle aij = elements of the original augmented matrix,  EMBED Equation.3 . For each value of k, do the i = k line first. c. Example n = 3 and n + 1 = 4  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  k = 0  EMBED Equation.3  e.g., for k = 1, i = 1, j = 1 & j = 4  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  k = 1  EMBED Equation.3  k = 2  EMBED Equation.3  k = 3  EMBED Equation.3   EMBED Equation.3  C. Iterative Methods For n > about 40, the one-step methods take too long and accumulate too much round-off error. 1. Jacobi Method a. Recursion formula Each equation is solved for one of the unknowns.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  In short  EMBED Equation.3 , i = 1, 2, 3, . . .,n. Of course, we cannot have bii = 0 for any i. So before starting the iterative program, we may have to reorder the equations. Further, it can be shown that if  EMBED Equation.3  for each i, then the method will converge, though it may be slowly. Heres an outline of the showing. The first iteration is:  EMBED Equation.3  After several iterations,  EMBED Equation.3  We want  EMBED Equation.3 , which will happen if  EMBED Equation.3 . b. Algorithm We need four arrays:  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3 . Firstly, select an initial guess (k = 0)  EMBED Equation.3 . Secondly, compute a new  EMBED Equation.3  (k + 1 = 1).  EMBED Equation.3  Thirdly, test for convergence.  EMBED Equation.3 . Notice that all the xi must pass the test. If all the xi do not pass the test, then repeat until they do. 2. Gauss-Seidel Method The Gauss-Seidel Method hopes to speed up the convergence by using newly computed values of xi at once, as soon as each is available. Thus, in computing xnew(12), for instance, the values of xnew(1), xnew(2), . . ., xnew(11) are used on the right hand side of the formula. We still need to keep separate sets of xnew and xold in order to perform the convergence tests.  D. Applications A couple of cases in engineering that give rise to simultaneous linear equations. 1. Electrical Circuit  (7+2+6)x1 2x2 6x3 = 300 -2x1 + (2+5+4+1)x2 4x3 x4 = 0 -6x1 4x2 + (4+9+6)x3 9x4 = 0 -x2 9x3 + (9+1+11)x4 = 0  EMBED Equation.3 ; solution:  EMBED Equation.3  2. Truss System   EMBED Equation.3 ; solution:  EMBED Equation.3  V. Interpolation and Curve Fitting Suppose one has a set of data pairs: xfx1f1x2f2x3f3 EMBED Equation.3  EMBED Equation.3 xmfmwhere fi is the measured (or known) value of f(x) at xi. We would like to find a function that will approximate f(x) for all x in a specified range. There are two basic approaches: interpolation and curve fitting. A. Polynomial Interpolation With interpolation, the approximating function passes through the data points. Commonly, the unknown f(x) is approximated by a polynomial of degree n, pn(x), which is required to pass through all the data points, or a subset thereof. 1. Uniqueness Theorem: Given {xi} and {fi}, i = 1, 2, 3, . . ., n + 1, there exists one and only one polynomial of degree n or less which reproduces f(x) exactly at the {xi}. Notes i) There are many polynomials of degree > n which also reproduce the {fi}. ii) There is no guarantee that the polynomial pn(x) will accurately reproduce f(x) for  EMBED Equation.3 . It will do so if f(x) is a polynomial of degree n or less. Proof: We require that pn(x) = fi for all i = 1, 2, 3, . . ., n+1. This leads to a set of simultaneous linear equations  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  which wed solve for the {ai}. As long as no two of the {xi} are the same, the solution to such a set of simultaneous linear equations is unique. The significance of uniqueness is that no matter how an interpolating polynomial is derived, as long as it passes through all the data points, it is the interpolating polynomial. There are many methods of deriving an interpolating polynomial. Here, well consider just one. 2. Newtons Divided Difference Interpolating Polynomial a. Divided differences The first divided difference is defined to be (notice the use of square brackets)  EMBED Equation.3 ,  EMBED Equation.3  If f(x) is differentiable in the interval [a,b], then there exists at least one point between a and b at which  EMBED Equation.3 . In practice, we would take a as close to b as we can (limited by the finite precision of the machine) and say that  EMBED Equation.3 . Higher order differences are defined as well: ordernotationdefinition0 EMBED Equation.3  EMBED Equation.3 1 EMBED Equation.3  EMBED Equation.3 2 EMBED Equation.3  EMBED Equation.3 3 EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 n EMBED Equation.3  EMBED Equation.3  b. Newtons divided difference formula Build the formula up step by step: i) two data points (x1,f1) & (x2,f2). We wish to approximate f(x) for x1 < x < x2. As a first order approximation, we use a straight line (p1(x) so that  EMBED Equation.3   EMBED Equation.3  Solve for f(x)  EMBED Equation.3  ii) Now, if f(x) is a straight line, then f(x) = p1(x). If not, there is a remainder, R1.  EMBED Equation.3  We dont know f(x), so we cannot evaluate f[x,x2,x1]. However, if we had a third data point we could approximate  EMBED Equation.3 . Then we have a quadratic  EMBED Equation.3 . iii) If f(x) is not a quadratic polynomial, then there is still a remainder, R2.  EMBED Equation.3  To estimate R2, we need a fourth data point and the next order divided difference. . .  EMBED Equation.3  iv) Jump to the generalization for n + 1 data points:  EMBED Equation.3 , where  EMBED Equation.3   EMBED Equation.3  Notice that i)  EMBED Equation.3 , etc. and ii) the (x xi) factors are also those of the previous term times one more factor. c. Inverse interpolation The NDDIP lends itself to inverse interpolation. That is, given f(x), approximate x. In effect, we are solving f(x) = 0 when f(x) is in the form of a table of data. Simply reverse the roles of the {fi} and the {xi}.  EMBED Equation.3  Set f(x) = 0 and evaluate x = pn(0). In practice, with a Fortran program, one would just reverse the data columns and use the same code. d. Example The difference table is computed thusly: for j=1:n+1 diff(j,1)=f(j) end for j=2:n+1 for i=1:n+1-j+1 diff(i,j)=( diff(i+1,j-1)-diff(i,j-1) )/(x(i+j-1)-x(i)) end end Divided Difference Table for n = 6 jxff[ , ]f[ , , ]f[ , , , ]f[ , , , , ]f[ , , , , , ]f[ , , , , , , ]11-1.50.51.667-2.5831.583-0.7270.2722-13-3.52.167-0.960.35332.50.5-0.50.833-0.2330.1430.250.750.3670.0175411.30.464.51.651.7752.5The sixth degree polynomial constructed from this table is  EMBED Equation.3 . Line by line, the script might look like this: fac = ex x(1) p0 = diff(1,1) p1 = p0 + fac*diff(1,2) fac = fac*(ex-x(2)) p2 = p1 + fac*diff(1,3) fac = fac*(ex-x(3)) p3 = p2 + fac*diff(1,4) fac = fac*(ex-x(4)) p4 = p3 + fac*diff(1,5)) fac = fac*(ex-x(5)) p5 = p4 + fac*diff(1,6) fac = fac*(ex-x(6)) p6 = p5 + fac*diff(1,7) Notice that we must use a different variable name for the argument x from the name used for the data array x(i). Of course, its more general and flexible to use a loop. fac = 1.0 p = diff(1,1) for j=1:n fac = fac*(ex-x(j)) p = p + fac*diff(1,j+1) end e. Issues with high degree polynomials If we have a large number of data points, 20 or 100 or 1000s, it does not pay to use the entire data table to create a 20 or 100 or 1000th degree polynomial. The greater the degree, the more often the pn goes up and down between the data points. Our confidence that  EMBED Equation.3  actually decreases. Its better to interpolate on subsets of the data using a p3 or a p4 using data points that surround the specified x. This process can be incorporated into the program. These low-degree segments are sometimes called splines. B. Least Squares Fitting Often, there are errors or uncertainties in the data values,  EMBED Equation.3 , for instance. Perhaps forcing the approximating function to pass through the data points is not the wisest approach. An alternative approach is to assume a functional form for the unknown f(x) and adjust it to best fit the uncertain data. A way to judge what is best is needed. 1. Goodness of Fit The method of least squares uses a particular measure of goodness of fit. a. Total squared error, E First of all, never forget that the word error in this context means uncertainty. Now, lets say {xi,fi} are the n+1 data values and f(x) is the assumed function. Then E is defined to be  EMBED Equation.3  The { EMBED Equation.3 } are weighting factors that depend on the nature of the uncertainties in the data {fi}. For measured values, the  EMBED Equation.3 , the experimental uncertainties. Often, we just take all the  EMBED Equation.3 , perhaps implying that the experimental uncertainties are all the same.. In that case,  EMBED Equation.3 . b. Least squares fit We wish to derive an f(x) which minimizes E. That means taking the derivative of E with respect to each adjustable parameter in f(x) and setting it equal to zero. We obtain a set of simultaneous linear equations with the adjustable parameters as the unknowns. These are called the normal equations. 2. Least Squares Fit to a Polynomial Assume that  EMBED Equation.3 . a. Total squared error  EMBED Equation.3  We have four adjustable parameters: a, b, c, and d. Notice that, unlike the interpolating polynomial, there may be any number of data pairs, regardless of the number of parameters. Lets take all the  EMBED Equation.3 . The partial derivative with respect to the adjustable parameters are  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  b. Normal equations Collect the like powers of xi and set the derivatives equal to zero.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  In terms of the matrix elements we used in solving simultaneous linear equations,  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3 , etc. The system is solved by any standard method, Gauss-Jordan, Gauss-Seidel, even by Cramers method. c. Accuracy of fit Wed like to have some statistical measure of how good the fit between the {fi} and f(x) is. This will depend on the relation between E and the { EMBED Equation.3 }. Lets consider a quantity called (N = n + 1)  EMBED Equation.3 . If all  EMBED Equation.3 , then  EMBED Equation.3 . Now, on another hand, if  EMBED Equation.3 , then  EMBED Equation.3 , where g is the number of adjustable parameters and N g is the number of degrees of freedom in the mathematical model for the data. Wed like to see  EMBED Equation.3  for a good fit, while  EMBED Equation.3  indicates that the quality of the fit is ambiguous (sometimes called over fitted), and  EMBED Equation.3  indicates a poor fit. 3. Least Squares Fit to Non-polynomial Function The process is similar when fitting to a function that is not a polynomial. For instance, say that  EMBED Equation.3 . We wish to fit this function to the data shown at right. In this case, N = 10 and g = 3. The adjustable parameters are a, b and c.  EMBED Equation.3  The normal equations are:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  When solved by the Gauss-Jordan method, these yield a = -1.041 b = -1.261 c = 0.031  EMBED Equation.3   EMBED Equation.3  The goodness of fit between these data and this function is ambiguous. A glance at a graph verifies that the fit is iffy. [Thats the technical term for it.] MatLab Sidelight Number One 1. Polynomials a. Representation In MatLab, polynomials are represented by a vector composed of the coefficients. Thusly,  EMBED Equation.3  b. Evaluate A polynomial is evaluated using the command polyval. If p is the vector of coefficients and x is the argument, polyval(p,x) or y=polyval(p,x) The roots command gives the roots of a polynomial, as elements of a vector. r=roots(p) If the roots of a polynomial are known, then the coefficient vector can be obtained by the poly command. p=poly(r) c. Add, multiply, & divide Polynomials are added by adding the vectors of their coefficients. The shorter vector has to be padded with zeros to make the two vectors the same length. Multiplication of two polynomials is done with the conv command. c(x) = a(x) * b(x) c = conv(a,b) Division is done with the deconv command. u(x)/v(x) = q(x) + r(x) [q,r] = deconv(u,v) d. derivatives  EMBED Equation.3 k = polyder(p)  EMBED Equation.3 k=polyder(a,b)  EMBED Equation.3 [n m] = polyder(u,v) followed by [k,r] = deconv(n,m) 2. Curve Fitting & Interpolation a. Least squares MatLab fits data to a polynomial using the least squares method. Fitting an nth degree polynomial to a table of (x,y) points. If the number of data points is m, then n must be m-1 or less, and greater than 0. p=polyfit(x,y,n) Fitting to functions other than polynomials is done by rewriting the function in terms of a straight line, for instance by taking the log of both sides, etc. b. Interactive fitting In the Tools menu of the Figure Window is a Basic Fitting tool. This can be used to fit a function to data interactively. See section 8.4 in the text. c. Interpolation MatLab has four interpolating techniques built-in. The command is interpl. It estimates yi = f(xi), given a set of {x,y}. yi = interpl(x,y,xi,method) The methods available are nearest returns the value of the nearest data point linear carries out linear interpolation spline carries out interpolation using a cubic polynomial based on the data points surrounding the interpolated point pchip carries out interpolation using a cubic Hermite polynomial. VI. Integration We wish to evaluate the following definite integral:  EMBED Equation.3 . We use numerical methods when i) f(x) is known analytically but is too complicated to integrate analytically or ii) f(x) is known only as a table of data. A. Newton-Cotes Formul 1. Trapezoid Rule a. Graphs Graphically, a definite integral is the area between the x-axis and the curve f(x). Areas below the axis are negative; areas above the axis are positive.  b. Trapezoids The area under the curve might be approximated most simply by a series of trapezoids and triangles.   EMBED Equation.3  Notice that x1 = a and that x8 = b. c. Interpolating polynomial In effect, we are replacing the integrand, f(x), by a straight line between each pair of points:  EMBED Equation.3 . This can be checked by integrating p1(x) analytically.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  check. d. Implementation For N data points spanning [a,b], there are N 1 trapezoids.  EMBED Equation.3  If the data are uniformly spaced, then  EMBED Equation.3  for all i, and  EMBED Equation.3 . The lines in the MatLab script might look like this: n = 10 T = 0.0 for i=2:n T = T + (x(i)-x(i-1))*(f(i)+f(i-1))/2.0 end 2. Extension to Higher Order Formul a. Forward difference interpolating polynomial Well take this opportunity to examine an alternative interpolating polynomialthe Forward Difference Polynomial. Imagine we have a table of data pairs (xi,fi) which are uniformly spaced, with spacing h. The forward differences are just the familiar deltas. first order:  EMBED Equation.3  second order:  EMBED Equation.3  Notice that the differences  EMBED Equation.3  and  EMBED Equation.3  are regarded as being evaluated at x = x1. Hence the term forward difference. Notice, too, that the forward differences are related to the divided differences simply by multiplying by the denominators.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Now, lets expand the integrand f(x) in a Taylors Series about x = x1. Further, to increase the element of confusion, let  EMBED Equation.3  so that  EMBED Equation.3 .  EMBED Equation.3  Depending on how many terms are kept, this will give a polynomial in  EMBED Equation.3  or in x. b. Simpsons rule Any number of formul may be created by replacing the integrand, f(x), with an interpolating polynomial of some specified degree. If  EMBED Equation.3 , the Trapezoid Rule is recovered. Perhaps f(x) has some curvature, so a second degree interpolating polynomial may serve better.  EMBED Equation.3   EMBED Equation.3  Expand the differences. . .  EMBED Equation.3   EMBED Equation.3  This is Simpsons Rule, which integrates over segments of three data points (or two intervals of h) in one step. c. Implementation  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Add em up. . .  EMBED Equation.3  Caveats: i) the data points must be uniformly spaced. ii) n + 1 must be odd, starting with 1 so that  EMBED Equation.3  is even. B. Numerical Integration by Random Sampling 1. Random Sampling a. Pseudorandom numbers Random numbers are a sequence of numbers,  EMBED Equation.DSMT4 , lying in the interval (0,1). There is no pattern in the progression of the numbers, nor is any number in the sequence related to any other number by a continuous function. There are statistical tests for randomness in a sequence of numbers but we wont bother with them here. The operation of a computer is deterministic, so truly random numbers cannot be generated by a computer program. However, sequences can be generated that appear to be random in that the sequence passes some of the statistical tests for randomness. Such a sequence of numbers is called pseudorandom. Here is an algorithm for generating a sequence of pseudorandom numbers:  EMBED Equation.3   EMBED Equation.3  where a, c and m are integers and mod( ) is the modulus function. The pseudorandom number uniformly distributed in the interval (0,1) is zi. In MatLab , this looks like the following: [Using the built-in remainder function (rem).] x = xo for i=1:100 x1 = rem(a*x+c,em); z = x1/em x = x1; end This process generates a sequence of numbers {zi} that have some properties of random numbers, but in fact the sequence repeats itselfits periodic. The exact sequence depends on the initial value, xo, called the seed. Usually, m is a large integer, commonly a power of 2. The numbers c and m can have no common factor (c can be zero) while a is a multiple of a prime factor of m + 1. The period of the sequence is m, which is why m needs to be large. For instance, we might take  EMBED Equation.3 , c = 0 and a = 16807. On the other hand, MatLab has built-in random number generating functions, shown in Table 3-2. b. Intervals Suppose we want our pseudorandom numbers to lie in the interval (a,b) rather than (0,1). This is easily done by scaling, or mapping onto the desired interval. Say  EMBED Equation.3 , then  EMBED Equation.3  will lie in the interval (a,b). c. Distributions The example random number generator mentioned above produces numbers uniformly distributed in (0,1). This means that is (0,1) were divided into equal subintervals, an equal number of random numbers is expected in each of those subintervals. The probability of the next random number in the sequence falling in a particular subinterval is the same for all the subintervals spanning (0,1). It is possible to form sequences of pseudorandom numbers which obey some other distribution function, such as Poisson or Gaussian, etc. We wont get into that here. 2. Samples of Random Sampling a. Coin toss We have two outcomes for each toss, of equal probability. Well generate an integer, either 1 or 2, using a pseudorandom number generator. zi = a uniformly distributed pseudorandom number in (0,1) j = int(2*zi) + 1 = 1 or 2 Well say that if j = 1, its heads, if j = 2 its tails. b. Roll of a die In this case we have six outcomes, of equal probability (we hope). So we need to produce an integer from 1 to 6. j = int(6*zi)+1 = 1, 2, 3, 4, 5 or 6 Now, if it is known that the die is loaded, we use a different scheme, creating subintervals in (0,1) whose lengths reflect the relative probabilities of the faces of the die coming up. For instance, we might say that zij EMBED Equation.3 1 EMBED Equation.3 2 EMBED Equation.3 3 EMBED Equation.3 4 EMBED Equation.3 5 EMBED Equation.3 6 3. Integration Thinking again of the definite integral as an area under a curve, we envision a rectangle whose area is equal to the total area under the curve f(x). The area of that equivalent rectangle is just the length of the integration interval (a,b) times the average value of the integrand over that interval. How to take that average? One way is to sample the integrand at randomly selected points. a. One dimensional definite integrals  EMBED Equation.3 , where the {xi} form a pseudorandom sequence uniformly distributed in (0,1). Over some other interval,  EMBED Equation.3 , where  EMBED Equation.3 . Since we are just averaging over a list of numbers, the error is O[ EMBED Equation.3 ], just like the deviation of the mean. example:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The exact result is 0.460. b. Multi-dimension integrals The random sampling approach is particularly useful with 2- and 3-dimensional integrals. The other methods of numerical integration quickly become too messy to set up.  EMBED Equation.3 , where (xi,yi,zi) is an ordered triple, each member uniformly distributed on (0,1). We may use three separate sequences of pseudorandom numbers or simply take numbers from one sequence three at a time. c. Alternate integration regions i)  EMBED Equation.3  ii) Suppose the integration region is not rectangular. Then an extra step is needed, to test for and discard random points that fall outside the integration region. e.g., a circlediscard points for which  EMBED Equation.3 , as shown in the following diagram.  Why do it this way; to ensure that the points are uniformly distributed in all directions. If points are taken uniformly distributed in the radius, the points will be more widely spread the further out from the center they lie, not uniformly spread over the area of the circle. example: compute the volume of a sphere of radius R. In this situation, the integrand is 1.  EMBED Equation.3  Numerically,  EMBED Equation.3 . Notice this: the total number of random points generated is n. However, only m of those lie within the spherical volume. The spherical volume we obtain is equal to  EMBED Equation.3  times the volume of a cube whose side is 2R. Its interesting to see what this fraction is.  EMBED Equation.3 . The ratio  EMBED Equation.3  should approach this constant as we generate more points and include them in the summation. Another way to look at this  EMBED Equation.3  issue is to say that f(x) = 1 when  EMBED Equation.3  and 0 when  EMBED Equation.3 . Then there is no distinction between n and m, and the summation is a sum of n m zeros and m ones. d. Example Evaluate  EMBED Equation.3 , where  EMBED Equation.3  is the region  EMBED Equation.3 .  EMBED Equation.3  [If you want to try it, for r = 0.5, I = 0.57.] This is equivalent to averaging the integrand over a circular area, thusly  EMBED Equation.3 . Of course, often the shape of the region of integration isnt a simple rectangle or circle. e. Example script % Script to carry out 2-dim integration via random sampling f=inline('sin(sqrt(log(x+y+1)))') n=100; r=0.5; r2=r*r; sum=0; ax=0.5-r; ay=0.5-r; bx=r+0.5; by=r+0.5; m=0 for i=1:n ex=rand*(bx-ax)+ax; why=rand*(by-ay)+ay; are=(ex-0.5)^2+(why-0.5)^2; if are<=r2 sum=sum+f(ex,why); m=m+1; end end sum=sum*(by-ay)*(bx-ax)/m; fprintf('Integral of f(x,y) over the circle = %g',sum) MatLab Sidelight Number Two 1. Nonlinear Equations a. Fzero The built-in MatLab command for solving individual equations is named fzero. x = fzero(function,x0) The initial guess is x0; function is either a mathematical expression typed as a string, or the name of a user defined function. The function has to entered in standard form: f(x) = 0. function is the f(x). If entered as a string, the function cannot include redefined variables. The initial guess can be entered as a single value, or as a 2-element vector such that the function crosses the x-axis between x0(1) and x0(2). (As in the bisection or secant methods.) b. Maximun/minimum A function to find the minimum of a function is fminbnd. [x fval] = fminbnd(function,x1,x2) The command finds the minimum of the function, if any, lying in the interval (x1,x2). 2. Integration a. Integrand as function The quad command evaluates a definite integral using an elaborated version of Simpsons Rule. The method adjusts the step size as it goes along.  EMBED Equation.3  q = quad(function,a,b,tol) The parameter tol is an optional tolerance. If tol is not specified, MatLab assumes  EMBED Equation.3 . The quadl command carries out the integration using another method, the adaptive Lobatto method. Thats quad-L. b. Integrand as data table When the integrand is available as a table of data pairs, MatLab uses the Trapezoid Rule. q = trapz(x,y) c. Random numbers For generating uniformly distributed pseudorandom numbers, MatLab has the command rand. A single random number in (0,1): z = rand A vector of n random numbers in (0,1): z =zrand(1,n) An nxn matrix of random numbers in (0,1): z = rand(n) An mxn matrix of random numbers in (0,1): z = rand(m,n) A row vector with n elements consisting of a random permutation of integers 1 n: m = randperm(n) VII. Ordinary Differential Equations A. Linear First Order Equations We seek to solve the following equation for x(t):  EMBED Equation.3 . There are analytical methods of solution: integration, separation of variables, infinite series, etc. In practice these may not be convenient or even possible. In such cases we resort to a numerical solution. The x(t) takes the form of a table of data pairs {ti,xi}, rather than a function. 1. One Step Methods a. Taylors Series Many numerical solutions derive from the Taylors series expansion  EMBED Equation.3 . We are given  EMBED Equation.3 , so we could substitute this into the series thusly:  EMBED Equation.3 . However, to obtain  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 , etc., we have to use the chain rule.  EMBED Equation.3   EMBED Equation.3  Its easy to see that this gets very messy rather quickly. b. Eulers Method Lets keep just the first two terms of the Taylors series:  EMBED Equation.3 , where the To is the sum of all the terms were droppingcall it the truncation error. In what follows, we will have to distinguish between the correct or exact solution, x(t), and our approximate solution, xi. We hope  EMBED Equation.3 . With the Euler Method, our algorithm is [given to, x(to) = xo and f(x,t)]  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  example:  EMBED Equation.3 , with to = 0 and xo = 4 and  EMBED Equation.3 . The algorithm is:  EMBED Equation.3 . The first few steps in the numerical solution are shown in the following table. itx0041.54217.2531.513.754223.5 EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  example: Problem 9-35  EMBED Equation.3  i = 0, 1, 2, 3, 4, 5, . . .  EMBED Equation.3   EMBED Equation.3  it (sec)v (m/s)0083.31.180.62.278.03.375.64.473.35.571.16.669.07.767.1 2. Error a. Truncation error  EMBED Equation.3  Not only do we not know what the exact solution is, we dont know how far the numerical solution deviates from the exact solution. In the case of a truncated Taylors series, we can estimate the truncation error by evaluating the first term that is dropped. For Eulers formula, thats the third term of the series.  EMBED Equation.3  Heres a graph of both the exact (but unknown) and the numerical solutions.  The deviation from the exact x(t) may tend to increase as the total truncation error accumulates from step to step, the further we get from the initial values (to,xo). The lesson ismake h small. b. Round-off error Since the values are stored in finite precision, round-off error accumulates from step to step also. Therefore, in traversing an interval  EMBED Equation.3 , wed like to have as few steps as possible. In other words, we want h to be large. Consequently, the two sources of error put competing pressure on our choice of step size, h. If we have some knowledge of x(t), we may be able to achieve a balance between large and small step size. Otherwise, its trial and error. c. Higher order methods The many numerical algorithms that have been developed over the years for solving differential equation seek to reduce the effect of truncation error by using more terms from the Taylors series, or in some way correcting for the truncation error at each step. In that way, fewer, larger steps can be used. MatLab Sidelight Number Three 1. First Order Ordinary Differential Equations (ODE) In standard form,  EMBED Equation.3  We want to solve for y(x). In Physics & Engineering, of course, we often have  EMBED Equation.3 , in which case we wish to solve for y(t); or perhaps we have  EMBED Equation.3 , etc. MatLab has several built-in ODE solvers, which all have the form of [t,y]=solver_name(ODEfunc,tspan,y0) ODEfunc is the name of the function file which defines the differential equation, the f(t,y). tspan is a vector that specifies the interval of the independent variable spanned by the solution. y0 is the initial value of y. [t,y] is the output, in the form of two column vectors. Subsequently, we would plot(t,y). a. Function file The function file calculates  EMBED Equation.3  for given values of y & t. That is, t & y are input arguments to the function, and the value of f(t,y) is returned. b. Solvers Table 9-1 lists some of the MatLab initial-value ODE solvers. Some are more sophisticated than others; some are adapted to problems in which the solution is not smooth, or is rapidly varying, etc. In most physical and engineering applications, things are smooth and not too-rapidly varying, so most times ode45 should suffice. [t,y] = ode45(function,[0:0.1:10],100) plot(t,y) MatLab also has boundary value and partial differential equation solvers, but those are not discussed in the introductory text, nor in this class. example: Problem 9-35  EMBED Equation.3  v0 = 300/60/60*1000 [t,v] = ode45(drag,[0:0.1:15],v0) plot(t,v) function dvdt = drag(t,v) dvdt = -0.0035*v*v-3; B. Second Order Ordinary Differential Equations  EMBED Equation.3 , with initial conditions  EMBED Equation.3  and  EMBED Equation.3 . 1. Reduction to a System of First Order Equations a. New Variables We start by introducing new variable names:  EMBED Equation.3 ;  EMBED Equation.3 ;  EMBED Equation.3 ;  EMBED Equation.3 . The first three variables are the solutions to the following differential equations:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  These form a set of three simultaneous first order differential equations,  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  with the initial conditions  EMBED Equation.3 ,  EMBED Equation.3  and  EMBED Equation.3  respectively. b. Solution Any method, such as Eulers, may now be applied to each first order equation in turn. Thusly:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3 . The MatLab code might look like this: z(1) = 0.0 z(2) = xo z(3) = vo h = 0.01 for i=1:100 z(1) = z(1) + h z(2) = z(2) +h*z(3) z(3) = z(3) + h*f(z(1),z(2),z(3)) end c. Example  EMBED Equation.3   EMBED Equation.3 ,  EMBED Equation.3  In this case,  EMBED Equation.3 , so the algorithm looks like  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3 . 2. Difference Equations An alternative approach to second order ordinary differential equations is to replace the derivatives with finite differences. The differential equation is replaced by a difference equation. a. Difference equation Using forward divided differences, we obtain  EMBED Equation.3  and  EMBED Equation.3 . Lets say that we have the second order differential equation  EMBED Equation.3 . The corresponding difference equation is  EMBED Equation.3 . The next step is to solve for the latest x.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The initial conditions are applied by setting to = 0, x0 = xo and  EMBED Equation.3 . b. Examples i)  EMBED Equation.3  Here,  EMBED Equation.3  and d = -g.  EMBED Equation.3  ii)  EMBED Equation.3  This time, a = -1, b = 0, c = 0 and d = -g.  EMBED Equation.3  c. Discretization error Replacing continuous derivatives with finite differences introduces what is known as discretization error. Implicitly, we are assuming a straight line between xi and xi+1 and between  EMBED Equation.3  and  EMBED Equation.3  as well. There will always be some  EMBED Equation.3  at each step which will then accumulate over the sequence of steps in the numerical solution.     PAGE  PAGE 52    xifi.240.23.65-0.26.95-1.101.24-0.451.730.272.010.102.23-0.292.520.242.770.562.991.00    % Script to implement Newton's Method del=1 x=3 k=0 func=inline('x^3-587') deriv=inline('3*x^2') while del >= 0.00005 k=k+1 if k <= 10 xnew=x-func(x)/deriv(x); del=abs((xnew-x)/x); disp(x);disp(func(x));disp(del) x=xnew; end end  EMBED Excel.Chart.8 \s  % Script to implement Gauss-Jordan Elimination b=[4 1 2;1 3 1;1 2 5]; c=[16;10;12]; a=[b c]; np=size(b); n=np(1); for k=1:n for m=k+1:n+1 a(k,m)=a(k,m)/a(k,k); end a(k,k)=1; for l=1:n if l~=k for m=k+1:n+1 a(l,m)=a(l,m)-a(l,k)*a(k,m); end a(l,k)=0; end end end a % Script to implement Gauss-Seidel b=[4 1 2;1 3 1;1 2 5]; c=[16;10;12]; xold=[1;1;1]; xnew=xold; np=size(b); n=np(1); flag=1; while flag > 0 for k=1:n sum=0; for l=1:n if k~=l sum=sum+b(k,l)*xnew(l); end end xnew(k)=(c(k)-sum)/b(k,k); end for k=1:n if abs((xnew(k)-xold(k))/xold(k)) > 0.0005 xold=xnew; break else flag=0; end end end xnew  67R" ) + > ? 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"d2ll;{a9 ;H7U`!@l;{a9 ;  xcdd``fed``ba_1147947999F`ә`әOle nCompObjofObjInfoquation Equation.39qA(-C 2x FMicrosoft Equation 3.0 DS Equation Equation.39qA П Equation Native r-_1147948027zF`ә`әOle sCompObjtfObjInfovEquation Native w)_1432561314F`ә`әOle x FMicrosoft Equation 3.0 DS Equation Equation.39q@XD "F i (2x )"x j  j=1n " d"CompObjyfObjInfo{Equation Native |_1432561322 F`ә`әOle CompObj fObjInfo Equation Native : FMicrosoft Equation 3.0 DS Equation Equation.39q0Դ 2x  o FMicrosoft Equation 3.0 DS Equation Equation.39q_1432561335F`ә`әOle CompObj fObjInfoEquation Native :_1432561346F`ә`әOle CompObjf 2x  o FMicrosoft Equation 3.0 DS Equation Equation.39qHtF "  FMicrosoft Equation 3.0 DS EqObjInfoEquation Native -_1432561355 F`ә`әOle CompObjfObjInfoEquation Native -_1149153493tF`ә`әuation Equation.39q@X  "  FMicrosoft Equation 3.0 DS Equation Equation.39qy 2BOle CompObjfObjInfoEquation Native -_1432561370"F`ә`әOle CompObj!#fObjInfo$ FMicrosoft Equation 3.0 DS Equation Equation.39qz^lW 2B =b 11 b 21 b 31 b 12 b 22 b 32 b 13Equation Native _1432561381'F`ә`әOle CompObj&(f b 23 b 33 b 14 b 24 b 34 [] FMicrosoft Equation 3.0 DS Equation Equation.39qi`M 2x =xObjInfo)Equation Native _1432561390,F`ә`әOle  1 x 2 x 3 [] FMicrosoft Equation 3.0 DS Equation Equation.39q)HFD 2C =2A +2B CompObj+-fObjInfo.Equation Native E_1149153833QW1F`ә`әOle CompObj02fObjInfo3Equation Native n FMicrosoft Equation 3.0 DS Equation Equation.39qyR c ij =a ij +b ij FMicrosoft Equation 3.0 DS Eq_1432561411*96F`ә`әOle CompObj57fObjInfo8uation Equation.39q5 2A +2B =2B +2A  FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native Q_1432561430;F`ә`әOle CompObj:<fObjInfo=Equation Native y_14325614574M@F`ә`әOle ]( (2A +2B )+2C =2A +(2B +2C ) FMicrosoft Equation 3.0 DS Equation Equation.39q%p  2C =2A 2B $CompObj?AfObjInfoBEquation Native A_1149154477EF`ә`әOle CompObjDFfObjInfoGEquation Native  FMicrosoft Equation 3.0 DS Equation Equation.39qy0 c ij =a ik b kjk " =a i1 b 1j +a i2 b 2j +a i3 b 3j +" FMicrosoft Equation 3.0 DS Equation Equation.39qܺ 2A  FMicrosoft Equation 3.0 DS Eq_1432561484JF`ә`әOle CompObjIKfObjInfoLEquation Native -_1432561493HROF`ә`әOle CompObjNPfuation Equation.39qGz 2B  FMicrosoft Equation 3.0 DS Equation Equation.39q-Դ 2A 2B `"2B 2ObjInfoQEquation Native -_1432561506TF`ә`әOle CompObjSUfObjInfoVEquation Native I_1149154746CXYF`ә`әA  FMicrosoft Equation 3.0 DS Equation Equation.39qy@ c ij =q"a ijOle CompObjXZfObjInfo[Equation Native \_1432561526>^F`ә`әOle CompObj]_fObjInfo` FMicrosoft Equation 3.0 DS Equation Equation.39qc J 2U =100010001[] FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native _1432561560cF`ә`әOle CompObjbdfObjInfoeEquation Native -_1432561569akhF`ә`әOle I 2B  FMicrosoft Equation 3.0 DS Equation Equation.39q"lW 2B  "1CompObjgifObjInfojEquation Native >_1432561579mF`ә`әOle CompObjlnfObjInfooEquation Native y FMicrosoft Equation 3.0 DS Equation Equation.39q]8ID 2B 2B  "1 =2B  "1 2B =2U  FMicrosoft Equation 3.0 DS Equation Equation.39q_1148391499rF`ә`әOle CompObjqsfObjInfot   "#&)*-012347<ADGJKNQTWX[^_befiloruxy|A@ b 11 x 1 +b 12 x 2 +"b 1n x n =c 1 FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native _1148391595pwF`ә`әOle CompObjvxfObjInfoyEquation Native _1148391647u|F`ә`әOle  A0Nl b 21 x 1 +b 22 x 2 +"b 2n x n =c 2 FMicrosoft Equation 3.0 DS Equation Equation.39qA :  "CompObj{} fObjInfo~Equation Native )_1148391620F`ә`әOle CompObjfObjInfoEquation Native  FMicrosoft Equation 3.0 DS Equation Equation.39qAxO b n1 x 1 +b n2 x 2 +"b nn x n =c n_1432561611fF`ә`әOle CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39q%(tF 2B 2x =2c  FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native A_1432561629F`ә`әOle CompObjfObjInfo Equation Native !_1432561636F`ә`әOle $pKa 2x =x 1 x 2 "x n [] FMicrosoft Equation 3.0 DS Equation Equation.39qCompObj%fObjInfo'Equation Native (_1432561643F`ә`әppn 2c =c 1 c 2 "c n [] FMicrosoft Equation 3.0 DS Equation Equation.39q1\ 2B =Ole +CompObj,fObjInfo.Equation Native /Mb 11 b 12 "b 1n b 21 b 22 "b 2n """"b n1 b n2 "b nn [] FMicrosoft Equation 3.0 DS Equation Equation.39q_1148392073F`ә`әOle 5CompObj6fObjInfo8A9 nd"40 FMicrosoft Equation 3.0 DS Equation Equation.39q` 2B Equation Native 95_1432561655\F`ә`әOle :CompObj;fObjInfo=Equation Native >-_1432561694F`ә`әOle ?CompObj@fObjInfoBEquation Native CA_1432561663F`ә`ә FMicrosoft Equation 3.0 DS Equation Equation.39q%T 2B 2x =2c  FMicrosoft Equation 3.0 DS Equation Equation.39qOle ECompObjFfObjInfoHEquation Native Ie\ 2B  "1 2B 2x =2x =2B  "1 2c  FMicrosoft Equation 3.0 DS Equation Equation.39q_1432561715F`ә`әOle LCompObjMfObjInfoOEquation Native PG_1148392430F`+֙`+֙Ole RCompObjSf+, 2B  "1 2c  FMicrosoft Equation 3.0 DS Equation Equation.39qAPI| b 11 x 1 +b 12 x 2 +bObjInfoUEquation Native V_1148392444F`+֙`+֙Ole Y 13 x 3 =c 1 FMicrosoft Equation 3.0 DS Equation Equation.39qA0A b 21 x 1 +b 22 x 2 +b 23 x 3 =c 2CompObjZfObjInfo\Equation Native ]_1148392478BF`+֙`+֙Ole `CompObjafObjInfocEquation Native d FMicrosoft Equation 3.0 DS Equation Equation.39qAVB b 31 x 1 +b 32 x 2 +b 33 x 3 =c 3_1148392576F`+֙`+֙Ole gCompObjhfObjInfoj FMicrosoft Equation 3.0 DS Equation Equation.39qAA "b 21 b 11 FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native k]_1148392656F`+֙`+֙Ole mCompObjnfObjInfopEquation Native q]_1148392722F`+֙`+֙Ole sAA/4 "b 31 b 11 FMicrosoft Equation 3.0 DS Equation Equation.39qA~h 2b 22 xCompObjtfObjInfovEquation Native w_1148392775F`+֙`+֙ 2 +2b 23 x 3 =2c 2 FMicrosoft Equation 3.0 DS Equation Equation.39qA~P 40 2b 32 x 2 +2b 33 x 3 Ole zCompObj{fObjInfo}Equation Native ~=2c 3 FMicrosoft Equation 3.0 DS Equation Equation.39qAIئ/ "2b 32 2b 22_1148392851F`+֙`+֙Ole CompObjfObjInfoEquation Native e_1148392918F`+֙`+֙Ole CompObjf FMicrosoft Equation 3.0 DS Equation Equation.39qA~@7 2b 22 x 2 +2b 23 x 3 =2c 2 FMicrosoft Equation 3.0 DS EqObjInfoEquation Native _1148392960F`+֙`+֙Ole CompObjfObjInfoEquation Native j_1432561768F`+֙`+֙uation Equation.39qAN@ 2b 33 x 3 =2c 3 FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObjfObjInfoEquation Native >" 2B  "1 FMicrosoft Equation 3.0 DS Equation Equation.39q(z 2A =2B :2c []=b 11 b 12 b 13 c _1432561787F`+֙`+֙Ole CompObjfObjInfoEquation Native _1432561812F`+֙`+֙Ole CompObjf1 b 21 b 22 b 23 c 2 b 31 b 32 b 33 c 3 [] FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoEquation Native _1432561918 F`+֙`+֙Ole FlW 2S  1 =100"b 21 b 11 10001[] FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjfObjInfoEquation Native _1432561862F`+֙`+֙HLlW 2S  2 =100010"b 31 b 11 01[] FMicrosoft Equation 3.0 DS Equation Equation.39q     { !"#%&K(')*+,-./013245678:9;=<>@?ABCEDFGHIJLMsONPQRTSUVWXYZ[]\^_`badcfeghikjlnmoqprtuwvxy|}~V d,FYzP1n:&B@?b u ʎ ㆪaM,,He` 01d++&1D_RČ`u ,@RF\ sX@2y0~Un>k@(\PPgJn>#dO 3Hb=<>{HF= \P,D.v0o8.+KRsA<.,#c3XT_Dd J  C A? 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 1 2S  3 2S  2 2S  1 2A =b 11 b 12 b 13 c 1 02b 22 2b 23 2c 2 001  x 3 [] FMicrosoft Equation 3.0 DS Equation Equation.39q 2S  4 =10001"2b 23 001[]Ole CompObj fObjInfoEquation Native _1432562062F`+֙`+֙Ole CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39q 2S  4 2Q  1 2S  3 2S  2 2S  1 2A =b 11 b 12 b 13 cEquation Native _1432562097F`+֙`+֙Ole CompObjf 1 02b 22 02c 2 001x 3 [] FMicrosoft Equation 3.0 DS Equation Equation.39q(ܺ 2Q  2 =ObjInfoEquation Native _1432562105F`+֙`+֙Ole 100012b 22 0001[] FMicrosoft Equation 3.0 DS Equation Equation.39q}lR 2Q  2 2S  4 2Q  1 2S  3 2SCompObjfObjInfoEquation Native _1432562146`!F`+֙`+֙  2 2S  1 2A =b 11 b 12 b 13 c 1 010x 2 001x 3 [] FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObj "fObjInfo#Equation Native    "#$%(-036789:=@CHKNQTWZ]^_befilmpstwz{|}Դ 2S  5 =10"b 13 010001[] FMicrosoft Equation 3.0 DS Equation Equation.39q_1432562153&F`+֙`+֙Ole CompObj%'fObjInfo(Equation Native _1432562160$.+F`+֙`+֙Ole  CompObj*, flW 2S  6 =1"b 12 0010001[] FMicrosoft Equation 3.0 DS Equation Equation.39q\ 2Q  3 =ObjInfo-Equation Native _14325621660F`+֙`+֙Ole 1b 11 00010001[] FMicrosoft Equation 3.0 DS Equation Equation.39qHFlX 2Q  3 2S  6 2S  5 2Q  2 2SCompObj/1fObjInfo2Equation Native _1432562203)=5F`+֙`+֙  4 2Q  1 2S  3 2S  2 2S  1 =100x 1 010x 2 001x 3 [] FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObj46fObjInfo7 Equation Native !D 2B  "1 =2Q  3 2S  6 2S  5 2Q  2 2S  4 2Q  1 2S  3 2S  2 2S  1 FMicrosoft Equation 3.0 DS Eq_1432562253:F`+֙`+֙Ole &CompObj9;'fObjInfo<)uation Equation.39q"a 2B  "1 FMicrosoft Equation 3.0 DS Equation Equation.39q+  2B  "1 2AEquation Native *>_14325622618L?F`+֙`+֙Ole +CompObj>@,fObjInfoA.Equation Native /G_1148396096[DF`+֙`+֙Ole 1  FMicrosoft Equation 3.0 DS Equation Equation.39qA3x: a kjk =a kjk"1 a kkk"1 i=ka ijCompObjCE2fObjInfoF4Equation Native 5O_1148396274IF`+֙`+֙k =a ijk"1 "a ikk"1 "a kjk i`"k} FMicrosoft Equation 3.0 DS Equation Equation.39qA\ k=1,nOle 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Equation 3.0 DS EqCompObjfObjInfoEquation Native )_1148462521F`+֙`+֙uation Equation.39qA @IJ " FMicrosoft Equation 3.0 DS Equation Equation.39qAHD  x n =Ole CompObjfObjInfoEquation Native c n "b n1 x 1 "b n2 x 2 """b nn"1 x n"1 ()1b nn FMicrosoft Equation 3.0 DS Equation Equation.39q_1148462624F`+֙`+֙Ole CompObjfObjInfoEquation Native _1148464259F`+֙`+֙Ole CompObjfAQ x i =c i "b ij x jj=1i`"jn " ()1b ii FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoEquation Native c_1432562353yF`+֙PuؙOle AG; b ii e"b ij FMicrosoft Equation 3.0 DS Equation Equation.39qQ8I$> 2x  1 ="2A 2x  0 +2V CompObjfObjInfoEquation Native m_1432562384FPuؙPuؙOle CompObjfObjInfoEquation Native  FMicrosoft Equation 3.0 DS Equation Equation.39q^ 2x  k+1 ="2A 2x  k +2V  k ="'2A k+1 "'2A 3 '2A 2 '2A 1 2x  0 +'2A k+1 "'2A 3 '2A 2 2V =2A  k+1 2x  0 +2A  k 2V  FMicrosoft Equation 3.0 DS Equation Equation.39q_1432562466FPuؙPuؙOle CompObjfObjInfoEquation Native _1148465228FPuؙPuؙOle CompObjfiIt 'lim k!" 2A  k+1 2x  0 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"2(!MCsʼn{G`!(!MCsʼn{` 0xCompObj!kfObjInfo"mEquation Native n1_1148557674%FPuؙPuؙOle oCompObj$&pfObjInfo'rEquation Native sm FMicrosoft Equation 3.0 DS Equation Equation.39qQ$ df()dx=fa,b[] FMicrosoft Equation 3.0 DS Equation Equation.39q_11485577742*FPuؙPuؙOle uCompObj)+vfObjInfo,xGCl 2f()H"fa,b[] FMicrosoft Equation 3.0 DS Equation Equation.39q.(T fx 1 Equation Native yc_1148557940U/FPuؙPuؙOle {CompObj.0|fObjInfo1~Equation Native J_1148557965-K4FPuؙPuؙOle [] FMicrosoft Equation 3.0 DS Equation Equation.39q'HF$ f(x 1 )CompObj35fObjInfo6Equation Native C_11485580409FPuؙPuؙOle CompObj8:fObjInfo;Equation Native ` FMicrosoft Equation 3.0 DS Equation Equation.39qD( fx 2 ,x 1 [] FMicrosoft Equation 3.0 DS Equation Equation.39q_1148558195(d>FPuؙPuؙOle CompObj=?fObjInfo@& fx 2 []"fx 1 []x 2 "x 1 FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native _1148558133ZCFPuؙPuؙOle CompObjBDfObjInfoEEquation Native J_1148558261HFPuؙPuؙOle .(T fx 1 [] FMicrosoft Equation 3.0 DS Equation Equation.39qĐ fx 3 ,x 2 []"fx 2CompObjGIfObjInfoJEquation Native _11485581327AMFPuؙPuؙ ,x 1 []x 3 "x 1 FMicrosoft Equation 3.0 DS Equation Equation.39q.(T fx 1 []Ole CompObjLNfObjInfoOEquation Native J_1148558295F_RFPuؙPuؙOle CompObjQSfObjInfoT FMicrosoft Equation 3.0 DS Equation Equation.39qJl fx 4 ,x 3 ,x 2 []"fx 3 ,x 2 ,x 1 []x 4 "x 1Equation Native _1148557916WFPuؙPuؙOle CompObjVXf FMicrosoft Equation 3.0 DS Equation Equation.39q h " FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoYEquation Native )_1148558146\FPuؙPuؙOle CompObj[]fObjInfo^Equation Native _1148558343aFPuؙPuؙ fx n+1 ,x n ,",x 2 ,x 1 [] FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObj`bfObjInfocEquation Native O3hl fx n+1 ,x n ,",x 3 ,x 2 []"fx n ,x n"1 ,",x 2 ,x 1 []x n+1 "x 1 FMicrosoft Equation 3.0 DS Eq_1148727470PnfFPuؙPuؙOle CompObjegfObjInfohuation Equation.39qg fx,x 1 []E"fx 2 ,x[] FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native _1148727517kFPuؙPuؙOle CompObjjlfObjInfomEquation Native _1148727582ispFPuؙPuؙOle J  f(x)"f 1 x"x 1 E"f 2 "f(x)x 2 "x FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjoqfObjInforEquation Native _1148727851uFPڙPڙ;4 f(x)E"f 1 +(x"x 1 )fx 2 ,x 1 []=p 1 (x) FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObjtvfObjInfowEquation Native s  R 1 (x)=f(x)"p 1 (x)=f(x)"f 1 "(x"x 1 )fx 2 ,x 1 []=(x"x 1 )(x"x 2 )fx,x 2 ,x 1 []_1148728060l,zFPڙPڙOle CompObjy{fObjInfo| FMicrosoft Equation 3.0 DS Equation Equation.39q fx,x 2 ,x 1 []E"fx 3 ,x 2 ,x 1 [] FMicrosoft Equation 3.0 DS EqEquation Native _1148728124FPڙPڙOle CompObj~fuation Equation.39qKH) f(x)E"f 1 +(x"x 1 )fx 2 ,x 1 []+(x"x 1 )(x"x 2 )fx 3 ,x 2 ,x 1 []=p 2 (x)ObjInfoEquation Native g_1148728319}FPڙPڙOle  FMicrosoft Equation 3.0 DS Equation Equation.39q]`L< R 2 (x)=f(x)"p 2 (x) FMicrosoft Equation 3.0 DS EqCompObjfObjInfoEquation Native y_1148728428FPڙPڙOle CompObjfObjInfoEquation Native  !$%&'*-./014789:=@CFILMNQVY\adehknqrstw|uation Equation.39qp( fx,x 3 ,x 2 ,x 1 []E"fx 4 ,x 3 ,x 2 ,x 1 [] FMicrosoft Equation 3.0 DS Eq_1148728534FPڙPڙOle CompObjfObjInfo uation Equation.39q] f(x)=p n (x)+R n (x) FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native  y_1148728791FPڙPڙOle  CompObj fObjInfoEquation Native j_1148728974FPڙPڙOle N8-\ p n (x)=fx 1 []+(x"x 1 )fx 2 ,x 1 []+(x"x 1 )(x"x 2 )fx 3 ,x 2 ,x 1 []+ FMicrosoft Equation 3.0 DS EqCompObjfObjInfoEquation Native 0_1148730241FPڙPڙuation Equation.39qG "+(x"x 1 )(x"x 2 )(x"x 3 )"(x"x n )fx n+1 ,x n ,",x 2 ,x 1 []Ole CompObj fObjInfo"Equation Native # FMicrosoft Equation 3.0 DS Equation Equation.39qO p 3 =p 2 +(x"x 1 )(x"x 2 )(x"x 3 )fx 4 ,x 3 ,x 2 ,x 1 []_1148729164FPڙPڙOle (CompObj)fObjInfo+ FMicrosoft Equation 3.0 DS Equation Equation.39q<; x=p n (f)=ff 1 ,f 2 ,",f i []f(x)"f j ()+ff 1Equation Native ,X_1148729978FPڙPڙOle 2CompObj3f [] j=1i"1 " i=2n+1 " FMicrosoft Equation 3.0 DS Equation Equation.39q; p 6 (x)=fx 1 []+fxObjInfo5Equation Native 66_1148731777FPڙPڙOle ; 1 ,x 2 ,",x i [] i=27 " (x"x j ) j=1i"1 " FMicrosoft Equation 3.0 DS Equation Equation.39q;; f(x)E"p CompObj<fObjInfo>Equation Native ?W_1148732156FPڙPڙn (x) FMicrosoft Equation 3.0 DS Equation Equation.39qA0 10.070.005secOle ACompObjBfObjInfoDEquation Native E]_1148732501FPڙPڙOle GCompObjHfObjInfoJ FMicrosoft Equation 3.0 DS Equation Equation.39q; E=1 i2 f i "fx i ()() 2i=1n+1 "Equation Native K_1148732550FPڙPڙOle OCompObjPf FMicrosoft Equation 3.0 DS Equation Equation.39q-  i FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoREquation Native S6_1148732608FPڙPڙOle TCompObjUfObjInfoWEquation Native XP_1148732657FPڙPڙ4x:   i =f i FMicrosoft Equation 3.0 DS Equation Equation.39q#.  i =1Ole ZCompObj[fObjInfo]Equation Native ^?_1148732693FPڙPڙOle _CompObj`fObjInfob FMicrosoft Equation 3.0 DS Equation Equation.39q:. 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