# Chapter 1. Basic Concepts

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1 Socrtes Dilecticl Process: The Þrst step is the seprtion of subject into its elements. After this, by deþning nd discovering more bout its prts, one better comprehends the entire subject Socrtes ( ) BCE Chpter 1 Bsic Concepts Numericl methods hve been round for long time. However, the usge of numericl methods ws limited due to the lengthy hnd clcultions involved in their implementtion. In our current society the ppliction of numericl nlysis nd numericl methods occurs in just bout every Þeld of science nd engineering. This is due in prt to the rpidly chnging digitl computer industry. Digitl computers hve provided fst computtionl device for the development nd implementtion of numericl methods which cn hndle vriety of difficult mthemticl problems. To understnd how numericl methods nd numericl nlysis techniques re developed the reder is required to hve knowledge of certin bckground mteril from clculus nd liner lgebr. We begin by reviewing some fundmentls which re used extensively in this text. Derivtive of Function The derivtive of continuous function y = f(x) is deþned by the limiting process f(x + h) f(x) lim h 0 h = f " (x) = dy dx, if this limit exits. This limiting process cn be represented using the lterntive nottions 1 dy dx = lim y x 0 x or dy dx x=x0 = f " (x 0 )= lim x x 0 f(x) f(x 0 ) x x 0. The nottion m = f " (x 0 ) denotes the derivtive evluted t the point x 0. This derivtive represents the slope m of the tngent line to the curve y = f(x) which psses through the point (x 0,f(x 0 )). BCE ( Before Common Er ) replces B.C. ( Before Christ ) usge.

2 2 Fundmentl Theorem of Clculus df (x) Let F (x) denote ny function such tht dx = f(x), wheref(x) is continuous function over the domin x b. Divide the domin (, b) into n equl subintervls of length x = b. This cn be done by deþning n = x 0 nd b = x n with x i = x 0 + i x for i =0, 1, 2,...,n. The resulting numbers = x 0 <x 1 <x 2 < <x n 1 <x n = b re then sid to prtition the intervl (, b) into n equl subintervls. Let c i denote number in the ith subintervl, where x i 1 c i x i for i =1, 2,...,n nd form the sum S n = n! f(c i ) x = f(c 1 ) x + f(c 2 ) x + + f(c n ) x i=1 The fundmentl theorem of clculus sttes tht lim S n = lim n n n! f(c i ) x = i=1 " b f(x) dx = F (x)] b = F (b) F () represents the re under the curve y = f(x), bovethex-xis if f(x) > 0, nd between the limits x = " nd x x = b. Note tht if G(x) = f(t) dt, then dg(x) dx = f(x). Tylor Series for Functions of Single Vrible Afunctionf(x) of single vrible x is sid to be nlytic in the neighborhood of point x = x 0 if it cn be represented in convergent power series of the form f(x) =f(x 0 )+f " (x 0 )(x x 0 )+ f "" (x 0 ) 2!! f(x) = n=0 (x x 0 ) f m (x 0 ) (x x 0 ) m + m! f (n) (1.1) (x 0 ) (x x 0 ) n n! where by deþnition 0! = 1 nd the zero derivtive denotes the function itself so tht f (0) (x 0 )=f(x 0 ). For Tylor series to exist in the neighborhood of point x 0, one must ssume tht the function f(x) hs continuous derivtives of ll orders which cn be evluted t the point x 0. Some well known Tylor series expnsions re e x =1 + x + x2 2! + x3 3! + x4 4! + x5 5! + sin x =x x3 3! + x5 5! x7 7! + cos x =1 x2 2! + x4 4! x6 6! +

3 3 If the Tylor series expnsion given by eqution (1.1) is truncted fter the mth derivtive term, then one cn write f(x) =f(x 0 )+f " (x 0 )(x x 0 )+ f "" (x 0 ) 2! where the error term is given by (x x 0 ) f m (x 0 ) (x x 0 ) m + Error (1.2) m! Error = f (m+1) (ξ) (m +1)! (x x 0) m+1, x 0 < ξ <x. (1.3) Tylor series expnsions of the form f(x 0 + h) =f(x 0 )+f " (x 0 )h + f "" (x 0 ) h2 2! + + f (m) (x 0 ) hm m! + f (m+1) (ξ 1 ) hm+1 (m +1)! f(x 0 h) =f(x 0 ) f " (x 0 )h + f "" (x 0 ) h2 2! + +( 1)m f (m) (x 0 ) hm m! +( 1)m+1 f (m+1) (ξ 2 ) hm+1 (m +1)! re used extensively in lter chpters. A continuous function f(x) is sid to hve root of multiplicity m if f(x 0 )=f " (x 0 )=f "" (x 0 )= = f (m 1) (x 0 )=0 but f (m) (x 0 ) #= 0 (1.4) Tht is, root of multiplicity m is such tht the function nd its Þrst (m 1) derivtives re zero t x = x 0. If m =1, then the root is clled simple root. For exmple, if, f(x 0 )=0, nd f " (x 0 ) #= 0, then x 0 is simple root. In contrst, the conditions f(x 0 )=0, f " (x 0 )=0, nd f "" (x 0 ) #= 0, imply x 0 is root of multiplicity 2. Note tht function f(x) which hs root x 0 of multiplicity m hs the Tylor series expnsion bout x 0 of the form f(x 0 + h) =f (m) (x 0 ) hm m! f (m+1) (x 0 ) hm+1 (m +1)! + since the function nd its Þrst (m 1) derivtives re zero t x = x 0. The Lndu Symbol O The Lndu symbol O, sometimes referred to s big Oh, is used to compre the behvior of one function f(h) with nother function g(h) s h 0. One writes f(h) =O(g(h)) if f(h) C g(h), C is positive constnt, for ll h sufficiently smll such tht lim C<. Forexmple,considerthe g(h) Tylor series expnsion for sin x. One cn write h 0 f(h) sin x = x x3 3! + O(x5 )

4 4 since sin x x x3 3! lim x 0 x 5 = 1 5! = Constnt The Lndu symbol O is used in perturbtionl methods nd numericl methods nd is sometimes referred to s n order reltion. It will be used throughout this text in the trunction of inþnite series to denote the order of the error terms. For exmple, the Tylor series expnsion for f(x 0 + h) when truncted fter the second term cn be expressed f(x 0 + h) =f(x 0 )+f " (x 0 )h + O(h 2 ) to indicte tht the error term is proportionl to h 2. One cn write Error C h 2 for ny constnt C stisfying f "" (ξ) 2! C. The nottion O(h n ) is used to denote the error being smll nd behving like Ch n,sh gets smll, where C is constnt. The sttement Error = O(h n ) is red the error is of order h n ndmenslim h 0 (Error) =Ch n for some positive constnt C. Tylor Series for Functions of Two Vribles A Tylor series expnsion of function of two vribles f(x, y) in the neighborhood of point (x 0,y 0 ) cn be written in the form f(x, y) =f(x 0,y 0 )+ f x (x x 0)+ f y (y y 0) + 1 # 2 f 2! x 2 (x x 0) f x y (x x 0)(y y 0 )+ 2 \$ f y 2 (y y 0) 2 + where ll prtil derivtives re to be evluted t the point (x 0,y 0 ). The bove Tylor series expnsion cn lso be written in n opertor nottion. DeÞne the prtil derivtive opertors D x f = f x, D yf = f y, D2 x = 2 f x 2, D xd y = 2 f x y, D2 y f = 2 f y 2, nd write the Tylor series expnsion given by eqution (1.5) in the specil cse where x = x 0 + h nd y = y 0 + k. One cn then write eqution (1.5) in the opertor form f(x 0 + h, y 0 + k) =f(x 0,y 0 )+! n=1 1 n! (hd x + kd y ) n f(x, y) f(x 0 + h, y 0 + k) =f(x 0,y 0 )+(hd x + kd y )f + 1 2! (hd x + kd y ) 2 f + 1 3! (hd x + kd y ) 3 f + 1 4! (hd x + kd y ) 4 f + etc. (1.5) (1.6)

5 5 where ll prtil derivtives re to be evluted t the point (x 0,y 0 ). Note tht the opertor terms (hd x + kd y ) m f cn be evluted by using the binomil expnsion. Exmple 2-1. (Tylor series.) If f(x, y) nd ll of its prtil derivtives through the nth order re deþned nd continuous over the rectngulr region R deþned by R = { x b, c y d} nd the Tylor series is truncted fter the nth derivtive terms, then the error term cn be clculted from knowledge of Tylor series expnsions of single vrible. Tht is, one cn replce x by x 0 + t(x x 0 ) nd y by y 0 + t(y t 0 ) in f(x, y) to obtin function of the single vrible t. One cn deþne φ(t) =f(x 0 + t(x x 0 ),y 0 + t(y t 0 )) so tht with (x, y) nd (x 0,y 0 ) Þxed, the function φ(t) is function of single vrible t. Expnding φ(t) bout t =0gives where φ(t) =φ(0) + φ " (0)t + φ "" (0) t2 2! + + φ(n) (0) tn n! + Error t n+1 Error = φ (n+1) (t ) (n +1)!, 0 <t <t. Note tht t t =1we hve φ(1) = f(x, y) nd when t =0we hve φ(0) = f(x 0,y 0 ) so tht one cn write φ(1) = f(x, y) =φ(0) + φ " (0) + φ"" (0) 2! where for x x 0 = h nd y y 0 = k we hve φ(0) =f(x 0,y 0 ) % f φ " (0) = x (x x 0)+ f φ "" (0) =. & y (y y 0) =(hd x + kd y )f + + φ(n) (0) n! % 2 f x 2 (x x 0) f x y (x x 0)(y y 0 )+ 2 f y 2 (y y 0) 2 φ (n) (0) =(hd x + kd y ) n f + Error & =(hd x + kd y ) 2 f where ll prtil derivtives re to be evluted t the point (x 0,y 0 ). The error term is given by Error t=1 = φ(n+1) (t ) (n +1)! =(hd x + kd y ) n+1 f x=ξ,y=η for 0 <t < 1, where ξ = x 0 + t h, η = y 0 + t k represent some point within the region R.

6 6 Men Vlue Theorem The men vlue theorem sttes tht if f(x) is continuous function on the closed intervl [, b], then there exists point ξ, stisfying <ξ <b, such tht the slope m s of the secnt line through the points (, f()) nd (b, f(b)) equls the slope of the curve f(x) t x = ξ. This cn be written nd illustrted s follows. m s = f(b) f() b = f " (ξ), < ξ <b. (1.7) Men Vlue Theorem for Integrls The men vlue theorem for integrls sttes the if f(x) is continuous function nd integrble over n intervl [, b], then there exists vlue ξ stisfying <ξ <bsuch tht the verge vlue of the function times the length of the intervl from to b must equl the re under the curve f(x) between nd b. This cn be written nd illustrted s follows. f(ξ)(b ) = " b f(x) dx, < ξ <b (1.8) The extended men vlue theorem for integrls sttes tht if f(x) nd g(x) re continuous functions on the closed intervl [, b] nd g(x) does not chnge sign throughout the intervl, then there exits point ξ such tht " b f(x)g(x) dx = f(ξ) " b g(x) dx, < ξ <b. (1.9) Other forms of this men vlue theorem re for the conditions f(x) is positive nd monotonic over the intervl (,b) nd g(x) is integrble, then one cn sy there exists t lest one vlue for ξ such tht " b f(x)g(x) dx = f() " ξ g(x) dx, ξ b.

7 7 Extreme Vlue Theorem The extreme vlue theorem sttes tht if f(x) is continuous function over the closed intervl [, b], then there will exist points ξ nd η such tht f(ξ) is mximum vlue of f(x) over the intervl nd f(η) is minimum vlue of f(x) over theintervl.onecnthenwrite minimum = f(η) f(x) f(ξ) =mximum, for x [, b]. Rolle s Theorem The Rolle s theorem ssumes tht f(x) is continuous nd differentible on the closed intervl [, b]. One form of Rolle s theorem sttes tht if f() =0nd f(b) =0,then there must exist t lest one point ξ in the intervl such tht f " (ξ) =0, < ξ <b. Intermedite Vlue Theorem The intermedite vlue theorem sttes tht if f(x) is continuous on the closed intervl [, b] nd there exists vlue f 0 such tht f() <f 0 <f(b), then there exists t lest one vlue ξ such tht f(ξ) =f 0. In the ccompnying Þgure note tht for the f 0 selected there exists more thn one vlue for ξ such tht f(ξ) =f 0. Number Representtion A bse 10 (deciml) number system represents number N in terms of vrious powers of 10 in series hving the form N = + α n (10) n + α n 1 (10) n α 3 (10) 3 + α 2 (10) 2 + α 1 (10) 1 + α 0 (10) 0 + β 1 (10) 1 + β 2 (10) 2 + β 3 (10) 3 + (1.10) where...,α n, α n 1,...,α 3, α 2, α 1, α 0, β 1, β 2, β 3,... re coefficients representing one of the digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Leding zeros nd triling zeros re not written.

8 8 For exmple, the number N= in the bse ten number system is relly shorthnd representtion for the number N = 8(10) 3 + 3(10) 2 + 2(10) 1 + 6(10) 0 + 4(10) 1 +3(10) 2 + 2(10) 3. A bse 16 (hexdeciml) number system represents number N in terms of vrious powers of 16 in series hving the form N = + α n (16) n + α n 1 (16) n α 3 (16) 3 + α 2 (16) 2 + α 1 (16) 1 + α 0 (16) 0 + β 1 (16) 1 + β 2 (16) 2 + β 3 (16) 3 + (1.11) where...,α n, α n 1,...,α 3, α 2, α 1, α 0, β 1, β 2, β 3,... re coefficients representing one of the digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9,A,B,C,D,E,F}. (When the bse is lrger thn 10 it is customry to use the letters A Z to represent the needed digits.) Numbers in the bse b number system re represented using subscript b. Some exmples of bse 10 numbers represented in the bse 16 number system re listed for reference. 10 =A =E =1E = =B =F = = =C = = =5A =D = =3C =64 16 A bse 8 (octl) number system represents number N in terms of vrious powers of 8 in series hving the form N = + α n (8) n + α n 1 (8) n α 3 (8) 3 + α 2 (8) 2 + α 1 (8) 1 + α 0 (8) 0 + β 1 (8) 1 + β 2 (8) 2 + β 3 (8) 3 + (1.12) where...,α n, α n 1,...,α 3, α 2, α 1, α 0, β 1, β 2, β 3,... re coefficients representing one of the digits {0, 1, 2, 3, 4, 5, 6, 7}. A bse 2 (binry) number system represents number N in terms of vrious powers of 2 in series hving the form N = + α n (2) n + α n 1 (2) n α 3 (2) 3 + α 2 (2) 2 + α 1 (2) 1 + α 0 (2) 0 + β 1 (2) 1 + β 2 (2) 2 + β 3 (2) 3 + (1.13) where...,α n, α n 1,...,α 3, α 2, α 1, α 0, β 1, β 2, β 3,... re coefficients representing one of the digits {0, 1}.

9 9 Note tht bse b number system requires b-digits to be used s coefficients in representing the numbers. For exmple, the Bbylonins of long go used bse 60 number system which requires 60 symbols to be used s digits. Some conventions from this system tht hve survived the mny centuries is the fct tht we hve 60 seconds in minute, 60 minutes in hour, nd 360 degrees in circle. Some other nmes ssocited with number systems re the following. A bse 3 number system is clled ternry system, bse 4 system is clled quternry system, bse 5 system is clled quinry system, bse 6 number system is clled senry system, bse 7 number system is clled septenry system, bse 9 number system is clled nonry system, bse 11 number system is clled undenry number system, nd bse 12 number system is clled duodenry number system. Number Conversion To convert number N from deciml bse to bse b it is necessry to clculte the coefficients α n, α n 1,...,α 1, α 0, β 1, β 2,...inthebsebnumbersystem, where N = α n b n + α n 1 b n α 2 b 2 + α 1 b 1 + α 0 b 0 + β 1 b 1 + β 2 b 2 +. The integer prt of N is denoted I[N] nd cn be expressed in the fctored form I[N] =α 0 + b(α 1 + b(α 2 + b(α bα n ) )) from which one cn observe tht if the integer prt of N is divided by b, then α 0 is the reminder nd the quotient is Q 1 = α 1 +b(α 2 +b(α 3 + bα n ) )). If Q 1 is divided by b, then the reminder is α 1 nd the new quotient is Q 2 = α 2 +b(α 3 + α n b) )). Continuing this process nd sving the reminders α 0, α 1,...,α n the coefficients fortheintegerprtofn cn be determined. The frctionl prt of N is denoted F [N] nd cn be expressed in the form F [N] =β 1 b 1 + β 2 b 2 + β 3 b 3 + fromwhichonecnobservethtiff [N] is multiplied by b, then there results bf [N] =β 1 + β 2 b 1 + β 3 b 2 + so tht β 1 is the integer prt of bf [N] nd the term β 2 b 1 +β 3 b 2 + represents the frctionl prt of bf [N]. Hence if one continues to multiply the resulting frctionl prts by b, then one cn clculte the coefficients β 1, β 2, β 3,... ssocited with the frctionl prt of the bse b representtion of N.

10 10 Exmple 2-2. (Number conversion.) Convert the number N = to bse 2 representtion. Solution: We strt with the integer prt of N nd write I[N] = 123 = n! α i (2) i. Now divide 123 by 2 nd sve the reminder R. Continuetodividetheresulting quotients nd sve the reminders s the reminders give us the coefficients α 0, α 1,...,α n in the bse 2 representtion. One cn construct the following tble to Þnd the coefficients N I[N/2] =Q R 123 I[123/2]=61 1=α 0 61 I[61/2]=30 1=α 1 30 I[30/2]=15 0=α 2 15 I[15/2]=7 1=α 3 7 I[7/2]=3 1=α 4 3 I[3/2]=1 1=α 5 1 I[1/2]=0 1=α 6 The integer prt of N cn now be represented I[N] = 123 = The! frctionl prt of N is written in the form F [N] = = β i (2) i. Now continue to multiply the frctionl prt by 2 nd sve the integer prt ech time. These integer prts represent the coefficients β 1, β 2,... One cn construct the following tble for determining the coefficients N 2N F[2N] I[2N] =β =β =β =β =β =β 6 i=1 i=0 The frctionl prt of N cn be represented F [N] = = nd the originl number N hs the bse 2 representtion N = =

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