Numerical Analysis Lecture 2 PDF

Numerical Analysis Lecture 2 Introduction Solution of Non Linear Equations Solution of Linear System of Equations Approximation of Eigen Values Interpolation and Polynomial Approximation Numerical Differentiation Numerical Integration Numerical Solution of Ordinary Differential Equations  Error...

Numerical Analysis Lecture 2 Introduction Solution of Non Linear Equations Solution of Linear System of Equations Approximation of Eigen Values Interpolation and Polynomial Approximation Numerical Differentiation Numerical Integration Numerical Solution of Ordinary Differential Equations  Errors in Computations Numerically, computed solutions are subject to certain errors. It may be fruitful to identify the error sources and their growth while classifying the errors in numerical computation. These are Inherent errors, local round-off errors local truncation errors Inherent errors It is that quantity of error which is present in the statement of the problem itself, before finding its solution. It arises due to the simplified assumptions made in the mathematical modeling of a problem. It can also arise when the data is obtained from certain physical measurements of the parameters of the problem. Local round-off errors Every computer has a finite word length and therefore it is possible to store only a fixed number of digits of a given input number. Since computers store information in binary form, storing an exact decimal number in its binary form into the computer memory gives an error. This error is computer dependent. At the end of computation of a particular problem, the final results in the computer, which is obviously in binary form, should be converted into decimal form-a form understandable to the user- before their print out. Therefore, an additional error is committed at this stage too. This error is called local round- off error.  (0.7625)10 (0.11000011(0011)) 2 If a particular computer system has a word length of 12 bits only, then the decimal number 0.7625 is stored in the computer memory in binary form as 0.110000110011. However, it is equivalent to 0.76245. Thus, in storing the number 0.7625, we have committed an error equal to 0.00005, which is the round-off error; inherent with the computer system considered. Thus, we define the error as Error = True value – Computed value Absolute error, denoted by |Error|, while, the relative error is defined as Error Relative error  True value Local truncation error It is generally easier to expand a function into a power series using Taylor series expansion and evaluate it by retaining the first few terms. For example, we may approximate the function f (x) = cos x by the series 2 4 2n x x n x cos x 1      ( 1)  2! 4! (2n)! If we use only the first three terms to compute cos x for a given x, we get an approximate answer. Here, the error is due to truncating the series. Suppose, we retain the first n terms, the truncation error (TE) is given by 2 n 2 x TE  (2n  2)! The TE is independent of the computer used. If we wish to compute cos x for accurate with five significant digits, the question is, how many terms in the expansion are to be included? In this situation 2 n 2 x 5 6 .5 10 5 10 (2n  2)! Taking logarithm on both sides, we get (2n  2) log x  log[(2n  2)!]  log10 5  6 log10 10 0.699  6  5.3 or log[(2n  2)!]  (2n  2) log x  5.3 We can observe that, the above inequality is satisfied for n = 7. Hence, seven terms in the expansion are required to get the value of cos x, with the prescribed accuracy The truncation error is given by 16 x TE  16! Numerical Analysis Lecture 2