Section 1.1 Algorithms. Key terms: Algorithm definition. Example from Trapezoidal Rule Outline of corresponding algorithm Absolute Error

Transcription

1 Section 1.1 Algorithms Key terms: Algorithm definition Example from Trapezoidal Rule Outline of corresponding algorithm Absolute Error Approximating square roots Iterative method Diagram of a general iterative method Stopping condition Outline of the iterative method Leslie Population Model Matrix Model Long Term Behavior Outline of the computations

2 Definition: An algorithm is a precisely defined sequence of steps for performing a specified task. We will design and implement by hand, with a calculator, or in MATLAB algorithms for computing the approximate solution of certain classes of mathematical problems. Objectives include: Determining conditions under which an algorithm is expected to work. Determining how accurate the approximate solution produced is compared to the exact solution for prototype problems. Determining how various parameter values within the algorithm affect the performance.

3 Our algorithms will consist of three basic components: 1. A list of input parameters; values supplied in order to have the algorithm perform its tasks. 2. Specific operations to be performed and the order in which they need to be performed. 3. Identify the output of the algorithm that is reported to the user. The text discusses three examples: Calculating the mean and standard deviation for a set of data. Trapezoidal Rule for approximating. Approximating a Square Root.

4 Trapezoidal Rule Suppose that function f(x) is only specified by an equispaced distinct ordered data set; recall this, means that x i < x i+1 and x i+1 - x i = h for all i. In order to approximate where a = x 0 and b = x n we construct the piecewise linear function through the data set and compute the sum of the areas under the individual line segments. The area under the line segments is computed using the formula for the area of trapezoid. We have h h h h f( x ) f( x ) f( x ) f( x ) f( x ) f( x ) f( x ) f( x ) n 1 n Which can be rearranged into the form h f(x 0 ) 2 f(x 1 ) 2 f(x 2 ) 2 2 f(x n 1 ) f(x n ) = h n 1

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6 2 1 dx x Example: Use the trapezoidal rule to approximate. Let T n be the approximate value from the trapezoidal algorithm using n subintervals. n T n

7 We know that the exact value of the integral is ln(2), so the absolute error in the trapezoidal approximation with n subinterval is given by 1 2 dx en Tn Tn ln( 2) x We enlarge our table to include this information: n T n Absolute Error e n e-005

8 We see that the error decreases as the number of subintervals n increases. To look for a specific pattern for how the error decreases as n increases here we compute the ratios of successive error e n. n T n Absolute Error e n Error Ratio e n / e n/ e Observe that the error ratio column implies that each time n is doubled, the absolute error is reduced by roughly a factor of one-quarter.

9 Approximating a Square Root Each application of the trapezoidal rule gave a single approximation to the value of the definite interval and many of the algorithms we encounter will perform similarly. However, there is another type of algorithm which generates a sequence of approximations that hopefully converge toward a solution. Such algorithms are called iterative. Generally two stopping criteria are used: 1. A test that indicates convergence. Some error expression is tested to see if it meets a standard determined by the input tolerance. Various error expressions are used, depending upon the details of the iterative method. 2. A counter to determine if a specified maximum number of iterations is exceeded is initialized at the start, incremented at each stage & tested at the end of each stage. If the counter exceeds the number of steps specified, then a message is displayed with possible options or an output routine is performed. It is generally not necessary to save every term of the sequence generated.

10 The following diagram presents the flow of logic for a general iterative method.

11 Let a be a nonnegative real number. For any positive real number x 0, the sequence generated by the iterative method x n 1 For n = 0, 1, 2, converges to 1 a xn 2 x provides an estimate for the difference. x a n 1. We will show (later) that the quantity x In this algorithm we need to retain the last two approximations in order to implement a stopping condition. n x n 1 n a

12 The algorithm looks likes the following: and stop the algorithm

13 Application: Leslie Population Model A Leslie population model[1] uses data collected about birth rates and survival rates of populations which are broken into age groups or categories. Field studies or laboratory studies must be conducted to provide the information on the rates. Once the rate information is collected a model for predicting the changes of the populations within the age groups is constructed where the rates are entries of a matrix A. The basic equation is (new population vector) = A (old population vector) [1] See Leslie, P.H., On the Use of Matrices in Certain Population Mathematics, Biometrika 33, 1945.

14 Example: A laboratory study of a blood parasite has revealed that there are three age categories that determine its effect on the host. We refer to these categories as juvenile (J), adult (A), and senior (S). The parasite is capable of reproduction in each of these stages and only a certain fraction of the organism survive to move on to the next stage. The birth and survival rates are shown in table. We interpret this information as follows.

15 The new population of juveniles is determined by the total births from each of the categories. The information regarding the changes in the population is summarized in three equations. Let x 1 be the current population of juveniles, x 2 the current population of adults, and x 3 the current population of seniors. We have the equations Next we set up a matrix equation to compute changes in the population.

16 Analysis of the Long Term Behavior of the Leslie Population Model Consider an initial population vector denoted by s 0.

17 Suppose we have initial population vector then computing s n = As n-1 for n =1, 2, 3, 4 we get s 0 s 1 s 2 s 3 s Juveniles Adults Seniors (Figures rounded to whole numbers.)

18 It happens that matrix A has three distinct eigenvalues so the corresponding eigenvectors are linearly independent and form a basis for the vector space of 3- vectors. Let ( i, p i ) i = 1,2,3 denote the three eigenpairs. The eigenvalue that is largest in absolute value is called the dominant eigenvalue. Suppose that Then it follows that The ultimate behavior of the populations is controlled by the term containing the dominant eigenvalue: Dominant eigenvector So we need to compute eigenvectors for the matrix A. Then we need to compute the coefficient c 1 for the linear combination of s 0 in terms of the eigenvectors. We have that the coefficient c ,eigenvalue eigenvector λ , and

19 To start estimate s 4 as follows: For future estimates we have approximately n=20 n=40 n= Future population totals are Thus we need to use algorithms for estimating accurately eigenvalues and corresponding eigenvectors. Such methods are iterative. Note that in more complicated models the matrix could be large and square.