Numerical Methods Chapter 35 PDF

154 60 154 Numerical Methods 5.1 Introduction. 5.2 Significant digits and Rounding off of Numbers. E3 The limitations of analytical methods have led the engineers and scientists to evolve graphical and numerical methods. The graphical methods, though simple, give results to a low degree of accuracy....

154 60 154 Numerical Methods 5.1 Introduction. 5.2 Significant digits and Rounding off of Numbers. E3 The limitations of analytical methods have led the engineers and scientists to evolve graphical and numerical methods. The graphical methods, though simple, give results to a low degree of accuracy. Numerical methods can, however, be derived which are more accurate. ID (1) Significant digits : The significant digits in a number are determined by the following rules : (i) All non-zero digits in a number are significant. (ii) All zeros between two non-zero digits are significant. U (iii) If a number having embedded decimal point ends with a non-zero or a sequences of zeros, then all these zeros are significant digits. (iv) All zeros preceding a non-zero digit are non-significant. D YG Number Number of significant digits 3.0450 5 0.0025 2 102.030070 9 35.9200 6 0.0002050 4 20.00 4 2000 1 U (2) Rounding off of numbers : If a number is to be rounded off to n significant digits, then we follow the following rules : ST (i) Discard all digits to the right of the nth digit. (ii) If the (n+1)th digit is greater then 5 or it is 5 followed by a nonzero digit, then nth digit is increased by 1. If the (n+1)th digit is less then 5, then digit remains unchanged. (iii) If the (n+1)th digit is 5 and is followed by zero or zeros, then nth digit is increased by 1 if it is odd and it remains unchanged if it is even. 5.3 Error due to Rounding off of Numbers. If a number is rounded off according to the rules, the maximum error due to rounding does not exceed the one half of the place value of the last retained digit in the number. The difference between a numerical value X and its rounded value X1 is called round off error is given by E  X  X 1. 5.4 Truncation and Error due to Truncation of Numbers. Numerical Methods 155 Leaving out the extra digits that are not required in a number without rounding off, is called truncation or chopping off. The difference between a numerical value X and its truncated value X1 is called truncation error and is given by E  X  X 1. 60 The maximum error due to truncation of a number cannot exceed the place value of the last retained digit in the number. Remark 1 : In truncation the numerical value of a positive number is decreased and that of a negative number is increased. 2 : If we round off a large number of positive numbers to the same number of E3 Remark decimal places, then the average error due to rounding off is zero. Remark 3 : In case of truncation of a large number of positive numbers to the same number Remark ID of decimal places the average truncation error is one half of the place value of the last retained digit. 4 : If the number is rounded off and truncated to the same number of decimal Remark U places, then truncation error is greater than the round off error. 5 : Round of error may be positive or negative but truncation error is always D YG positive in case of positive numbers and negative in case of negative numbers. Number Approximated number obtained by Chopping off Rounding off 0.335217... 0.3352 0.3352 0.666666... 0.6666 0.6667 0.123451... 0.1234 0.1235 0.213450... 0.2134 0.2134 0.2139 0.2140 0.335750... 0.3357 0.3358 0.999999... 0.9999 1.0000 0.555555... 0.5555 0.5556 ST U 0.213950... 5.5 Relative and Percentage errors of Numbers. The difference between the exact value of a number X and its approximate value X1, obtained by rounding off or truncation, is known as absolute error. The quantity Thus E R  X  X1 is called the relative error and is denoted by E R. X X  X 1 X. This is a dimensionless quantity.  X X The quantity X X  100 is known as percentage error and is denoted by E p , i.e. E p   100. X X 156 Numerical Methods Remark 1 : If a number is rounded off to n decimal digits, then | ER |  0.5  10 n1 Remark 2 : If a number is truncated to n decimal places, then | E R |  10 n1 The number of significant digits in 0.0001 is (a) 5 (b) 4 (c) 1 Solution: (c) 0.0001 has only one significant digit 1. Example: 2 exceed [DCE 1998] When a number is rounded off to n decimal places, then the magnitude of relative error does not (a) 10  n Solution: (c) (b) 10 n 1 (c) 0.5  10 n 1 E3 When the number 2.089 is rounded off to three significant digits, then the absolute error is (a) 0.01 (b) –0.01 (c) 0.001 (d) –0.001 When the number 2.089 is rounded off to three significant digits, it becomes 2.09. Hence, the absolute error that occurs is 2.089  2.09  0.001 ID Solution: (d) (d) None of these When a number is rounded off to n decimal places, then the magnitude of relative error i.e. | E R | does not exceed 0.5  10 n 1. Example: 3 (d) None of these 60 Example: 1 5.6 Algebraic and Transcendental Equation. U An equation of the form f(x)=0, is said to an algebraic or a transcendental equation according as f(x) is a polynomial or a transcendental function respectively. e.g. ax 2  bx  c  0 , ax 3  bx 2  cx  d  0 etc., where a, b, c, d  Q, are algebraic equations whereas ae x  b sin x  0 ; a log x  bx  3 etc. are transcendental equations. D YG 5.7 Location of real Roots of an Equation. By location of a real root of an equation, we mean finding an approximate value of the root graphically or otherwise. (1) Graphical Method : It is often possible to write f (x )  0 in the form f1 (x )  f2 (x ) and then plot the graphs of the functions y  f1 (x ) and y  f2 (x ). ST U Y X y=f2(x) O Y y=f1(x) P A real root of f1(x) = f2(x) X The abscissae of the points of intersection of these two graphs are the real roots of f (x )  0. (2) Location Theorem : Let y  f (x ) be a real-valued, continuous function defined on [a, b]. If f (a) and f(b) Y y= fx f(a ) X O Y a A real root of f(x) = 0 b f(b ) X Numerical Methods 157 have opposite signs i.e. f(a).f(b) < 0, then the equation f(x)=0 has at least one real root between a and b. 5.8 Position of Real Roots 60 If f (x )  0 be a polynomial equation and x 1 , x 2............ x k are the consecutive real roots of f ( x )  0 , then positive or negative sign of the values of f (), f (x ).............. f (x k ), f() will determine the intervals in which the root of f (x )  0 will lie whenever there is a change of sign from f ( x r ) to f (x r 1 ) the root lies in the interval [ x r , x r 1 ]. If all roots of equation x 3  3 x  k  0 are real, then range of value of k (a) (–2, 2) (b) [–2, 2] (c) Both Solution: (a) Let f (x )  x 3  3 x  k , then f (x )  3 x 2  3 and so f (x )  0  x  1. The values of f(x) at x  ,  1, 1,  are : E3 Example: 4 x :  1 1  f (x ) :   k  2 k  2  (d) None of these ID If all roots of given equation are real, then k  2  0 and k  2  0  2  k  2. Hence the range of k is (–2, 2) For the smallest positive root of transcendental equation x  e  x  0 , interval is (a) (0, 1) (b) (–1, 0) (c) (1, 2) (d) (2, 3) Solution: (a) Let f (x )  x  e  x  0  xe x  1  0 but f (0 )  ive and f (1)  ive. Therefore root lie in (0,1). Example: 6 The maximum number of real roots of the equation x 2 n  1  0 is (a) 2 (b) 3 (c) n [MP PET 1996] [MP PET 2001] (d) 2n Let f (x )  x 2n  1 , then f ' (x )  x 2n 1  0  x  0 D YG Solution: (a) U Example: 5 Sign of f (x ) at x  , 0,   are x:  0  f (x ) :  ive  ive  ive This show that there are two real roots of f (x )  0 which lie in the interval (,0 ) and (0,). Hence maximum number of real roots are 2. 5.9 Solution of Algebraic and Transcendental Equations ST U There are many numerical methods for solving algebraic and transcendental equations. Some of these methods are given below. After locating root of an equation, we successively approximate it to any desired degree of accuracy. (1) Iterative method: If the equation f(x) = 0 can be expressed as x = g(x) (certainly g(x) is non-constant), the value g(x0) of g(x) at x  x 0 is the next approximation to the root . Let g( x 0 )  x 1 , then x 2  g(x 1 ) is a third approximation to . This process is repeated until a number, whose absolute difference from  is as small as we please, is obtained. This number is the required root of f (x )  0 , calculated upto a desired accuracy. Thus, if x i is an approximation to  , then the next approximation x i1  g(x i )........(i) The relation (i) is known as Iterative formula or recursion formula and this method of approximating a real root of an equation f (x )  0 is called Y iterative method. (2) Successive bisection method : This method consists in locating the root of the equation f (x )  0 between a and b. If f(x) is continuous between a and b, and f(a) and f(b) are of opposite y=f(x) O a x2 x3 x1 b X 158 Numerical Methods signs i.e. f(a).f(b)