MEI Structured Mathematics. Module Summary Sheets. Numerical Methods (Version B reference to new book)

Transcription

1 MEI Matematics in Education and Industy MEI Stuctued Matematics Module Summay Seets (Vesion B efeence to new book) Topic : Appoximations Topic : Te solution of equations Topic : Numeical integation Topic 4: Appoximating functions Topic 5: Numeical diffeentiation Topic 6: Rates of convegence in numeical pocesses Pucases ave te licence to make multiple copies fo use witin a single establisment Mac, 5 MEI, Albion House, Maket Place, Westbuy BA DE Company No 6549 England and Wales Tel: Fax: Registeed wit te Caity Commission, numbe 589

2 Summay NM Topic : Appoximations Pages -6 Execise A Q 6 Page 7 Execise A Q 4 Pages - Execise B Q, 4 Pages 5- Execise C Q 4 Execise D Q, 5 Pages - Execise D Q4 Eo Let X be an appoximation to an exact value, x, ten te eo is e X x and ence X x + e Te elative eo is e X x( + ) x Recoding numbes Fixed point: decimal point in te coect position ie 4564 Floating point: (also called standad fom) in wic te numbe is witten as a b wee a < and b is an intege Rounding and copping If a numbe is to be witten to n significant figues ten te numbe is ounded if te neaest n-digit numbe is given Te numbe is copped if all te digits afte te nt ae discaded ie 46 to 4 decimal places becomes 5 if ounded and 4 if copped Inteval estimates If a numbe, n, is ecoded to p decimal places ten te maximum possible ounding eo is Te inteval witin wic te numbe lies is [n ±5 -p ] s b 5 s elative eo 5x b a Popagation of eos Te numbe of decimal places quoted in a sum sould be no moe tan te numbe quoted in te tem wit te least numbe of decimal places Te numbe of significant places in a poduct (o quotient) sould be no moe tan te numbe of significant figues in te facto wit te least significant figues Weneve two nealy numbes ae subtacted ten tee may be a loss of many significant figues in te accuacy of te answe Ill-conditioned poblems If a small cange to te input causes a lage cange in te output, ten te poblem is said to be ill-conditioned Vesion B: page Competence statements v,,, 4, 5 Eg If a eigt of 8cm is ecoded as 8 cm ten te absolute eo is 65 8 Eg Find te absolute eo wen a mass of 697 kg is ecoded to te neaest kg e ; Eg Calculate te elative eo in appoximating (i) 456 to 5 decimal places, (ii) 456 to significant places (i) To 5 dp numbe is 46, an eo of 4 Relative eo is (ii) To sf te numbe is Relative eo is Eg 6(copping) 6 eo 6, elative eo 6 Eg 7(ounding) eo, elative eo 5 6 Eg ( dp) 5 9 (coect to sf) (and tis is coect to only sf ) Eg Fo f( x) ; ( x ) f 99 5 but if te value 99 is ounded to 99 ten f 99 5

3 Summay NM Topic : Solution of Equations Capte Pages 8- Execise A Q, Initial Appoximations Te pocess of finding te numeical solution of an equation is in two pats: (i) te detemination of an appoximation to te oots; (ii) te successive impovement of tese appoximations to give te oots to any desied degee of accuacy Te poblem may suggest an appoximate oot A gapical calculato can povide an appoximate oot, by alteing anges and using te zoom facility A table of values may be calculated If te gap is continuous and tee is a cange of sign between x a and x b ten tee is at least one oot in te inteval [a,b] Eg Solve te equation x + x 7 A gap indicates tat tee is one oot in te inteval [,] f() 7, f() f() 7 demonstating a sign cange in te inteval [,] Capte Pages -6 Execise B Q Inteval Bisection Metod If a oot lies between x x and x x ten x ( x+ x) eplaces eite x o x so tat te two values staddle te oot Te inteval witin wic lies te oot (te eo bounds) is successively alved At any stage te best possible estimate is te mid-point of te inteval If te stating inteval is [ x,x] ten afte niteations te maximum possible eo is n + ( ) x x Inteval bisection metod f() -7,f() -,f() 7; inteval [,] So take x,x ; x ( + ) 5 f(5) 875; inteval [, 5]; x ( + 5 ) 5 f(5) 97; inteval [5,5]; x 4 ( ) 75 Continuing wit te pocess will give an inteval of [465, 48], giving a value of 47 ± to decimal places Capte Pages 8-4 Execise C Q Capte Page 4-47 Execise D Q Fixed Point Iteation Reaange te equation f( x) in te fom x g( x) Ten apply te iteative pocedue x+ g( x) If a is te oot and ε is te eo in x ten x a + ε x g x g a g a g' a ( + ε ) + ε ε+ g εg ε ε + ie a + a + ' a Since a g a tis gives + g' a Given a good enoug stating point, te pocedue will convege if g ' ( x) < fo some inteval including te oot You need to be competent in te use of speadseets Examples of eac of tese metods and all topics in tis module can be woked on a speadseet Vesion B: page Competence statements e, e, e Fixed Point Iteation metod 7 x + x 7 may be ewitten x ; x + 7 Tis is x g( x) wee g( x) x + 4x Taking a nea oot to be x 5, g'( x), ( x + ) g '(5) < so convegence will occu x 5 x x x 6 x poceding similaly gives x 45 andx

4 Summay NM Topic : Solution of Equations Capte Pages 49-5 Execise E Q 5 Newton-Rapson Metod Te tangent at x to te cuve will cut te x-axis at a point wic is, geneally, close to te oot tan x x + x f( x ) f '( x ) Tis is a eaangement of te equation of te fom x+ g( x) as above It can be sown tat in tis case g '( α) It can also be sown tat ε+ ε Tis is called second ode convegence If x is coect to p decimal places ten x + will be coect to appoximately p decimal places Newton - Rapson metod f'( x) x f( x ) wit f 7 x+ x x x + x f'( x ) + x + x 7 x + 7 giving x x x x x x x 5 x 4564 x x Capte Pages 5-54 Execise E Q 9 Secant Metod Tis metod appoximates te deivative So, instead of using te deivative in te Newton- Rapson fomula te gadient of a cod is used f( x) f( x) x x appoximates to te gadient of te cuve at te point x Te iteative fomula ten becomes x f f x x f x x x x + x x f( x ) f( x ) f( x ) f( x ) Secant metod x x x x f( x ) f( x+ ) f( x) wit f 7 x x + x x, x f ( x ), f ( x ) 7 x 7 7, 7 ( ) f () 9 x , 9 7 f (8) 8 x4 8 ( ) 477, ( 9) f (477) 77 Continuing wit te pocess gives x 465 Capte Pages Execise F Q (iii) False Position (also called Linea Intepolation) Tis metod divides te inteval between x and x, given f(x ) < and f(x ) > but takes account of te magnitude of f(x ) and f(x ) Te metod uses simila tiangles; note tat te size of one side is f(x ) xf( x) xf( x) x f( x) f( x) Vesion B: page 4 Competence statements e, e, e, e4 Linea Intepolation metod Take a, b, f ( a),f ( b) 7 6 x, f() -9 So take a, f( a) -9, b, f( b)7 9 x 8,f (8) 9 So take a 8, f( a) -, b, f( b)7 8 x 4,f (4) 57 So take a 4, f( a) -57, b, f( b) x 447,f (4) 9 57 Repeating te pocess will give x 46 Tis as been obtained wit a numbe of iteations until te values ae consistent

5 Summay NM Topic : Numeical Integation Capte Pages 6-67 Execise A Q Capte Pages 68-7 Execise B Q 4 Capte Pages Execise C Q Capte Pages 7-78 Execise C Q 7 Mid-point Rule Te aea unde te cuve is split into sections wit equal widts Te midpoints of tese sections ae taken and ectangles wit tese eigts ae used to eplace te sections b Ten f d [f f (x) x ( m ) + ( m ) + + ( mn ) a Tapezium Rule Te aea unde te cuve is split into tapezia wit equal widts Ten A ( y + y) + ( y+ y ) + + ( yn + yn) ( y + ( y+ y + yn ) + yn) Simpson s Rule y + y + + y Eac stip is appoximated by a paabola toug tee points on te cuve Tis metod teefoe equies an odd numbe of points Ove te inteval [ a,b] te aea is divided into n stips of widt Ten A ( y+ 4y+ y) + ( y+ 4y+ y 4) + + ( yn + 4yn + yn) y ( + 4y+ y + + 4yn + yn) y ( + yn + 4( sum of odd odinates ) + ( sum of even odinates )) Te elationsip between te metods If T n and T n ae te values by te tapezium ule wit n and n subintevals espectively, M n te value using te midpoint ule wit n subintevals, and S n te value by using Simpson s ule wit n applications, ten Vesion B: page 5 Competence statements c, c4 n T T + M S T + M f ], ( ) n n n n n n Waning! S n is sometimes defined as Simpson's Rule applied wit n stips and sometimes wit n applications So, fo instance, S can be defined as one application wit stips, o two applications, equiing 4 stips In any examination question te definition will be explicitly given so tat tee is no confusion Example: Find an estimate fo by te tee metods wit (i) stips, (ii) 4 stips Values equied Tapezium Rule (i) stips: I ( y + y 5 + y ) 6 (ii) 4 stips: I y + ( y + y + y ) + y 5475 Simpson's Rule (i) application: I ( y + 4 y 5+ y ) 477 (ii) applications: I ( y + 4( y 5 + y 75) + y 5 + y ) x dx x y x y Midpoint metod (i) stips: A 5[y 5 +y 75 ] 44 (ii) 4 stips: A 5[y 5 +y 75+ y 65 +y 875 ] Eos Fo te Midpoint Rule and te Tapezium Rule ε Fo Simpson s Rule ε ie same numbe of steps means geate accuacy Same accuacy means fewe steps 4

6 Summay NM Topic 4; Appoximating Functions Capte 4 Pages 8-89 Example 4 Page 87 Execise 4A Q 4 Fowad Diffeence metod Fo a set of equally spaced values x, x, x, te fowad diffeence opeato is defined by f f(x + ) f(x ), f ( f ), etc Fo a polynomial of ode n, n f(x i ) constant A table of fowad diffeences may be used: to intepolate values (ie values of f(x) witin te ange of values given) to extapolate values (ie values of f(x) beyond tose given) to find te gadient of te cuve at a point Eg Given te finite diffeence table fo y f(x), find f(4) by extapolation Wat assumptions must you make? f f f x x Δ Δ Fom te table it can be seen tat nd diffeences ae te same, so assume constant (ie f(x) is a quadatic) f(4) 5 Capte 4 Pages 9-94 Example 44 Page 9 Execise 4B Q, 6 Te Newton Intepolation fomula Te fowad diffeence opeato is defined as f i f i+ f i f f f : f f + f ( + )f f f f : f f + f ( + )f ( + ) f And in geneal f p ( + ) p f Using te Binomial expansion gives p( p ) p fp f x + p +Δ f f + pδ f + Δ f +! x x x x If x x + p ten p, p, etc giving x x ( x x)( x x) f ( x) f + Δ f + Δ f +! If te function is of degee n te expansion stops n at Δ f It is ten usual to eaange te polynomial in descending powes of x Note, oweve, tat if intepolation is equied (ie a value fo f ( x) witin te ange of given values) tis value fo x may be substituted and te value fo f ( x) found witout te need fo eaangement Eg Given te following values of a function, find te value of f(75) x 5 5 f( x) Fo f(75), x x + p wit x, x 75 and 5 p 5 f (75) f + pδ f + p( p ) Δ f + x f(x) Δf Δ f Δ f (NB Te values in te Δ f column being te same is an indication tat te points of te function fit a cubic, toug tis does not pove tat te function is a cubic ) To find f(75), x x+ pwit x, x 75 and 5 p f(75) Note tat tis metod equies te given set of data to to ave equal intevals of x Vesion B: page 6 Competence statements: f

7 Summay NM Topic 4; Appoximating Functions Capte 4 Page 94 Execise 4B Q 47 Capte 4 Pages 96- Execise 4C Q4, 5 Te Lagange Intepolating polynomial Te Lagange metod is an altenative metod to find a polynomial tat fits a set of data It as te advantage tat te set of data given do not ave to be at equally spaced intevals Te staigt line toug te points ( aa, ) and ( bb, ) as A( x b) B( x a) equation y + ( a b) ( b a) Te quadatic function wose gap passes toug te points ( aa, ), ( bb, ) and ( cc, ) as equation A x b x c B( x c)( x a) C( x a)( x b) y + + b c b a c a c b ( a b)( a c) Similaly a cubic passing toug fou points would ave fou tems Tis fom may be quite easily witten down It only needs to be simplified to give te polynomial in descending powes in an examination if tat is specifically demanded If you ae asked find a value of f(x) you will not be equied to simplify te polynomial Tuncating Newton s Intepolating polynomial If data enties in te table ae appoximations ten no diffeence column will be constant If you look at te table ten it sould be possible to find te fist column tat is nealy constant, meaning tat te next diffeence column enties will be vey small An appoximating polynomial can ten be constucted Vesion B: page 7 Competence statements: f, f Eg Find te quadatic polynomial passing toug te points (, ), (, ) and (, 9) Te equation is ( ) ( ) y + ( x )( x ) ( x )( x ) ( )( ) ( )( ) 9( x )( x ) + ( )( ) ( x )( x ) x( x ) 9x( x ) y + + y ( x )( x ) + x( x ) + x( x ) y x + 4x + x x+ x x y x + x Eg An expeiment yields te following values of two vaiables, x and y Estimate a value fo y wen x 5 x 6 y Te equation is 64( x )( x 6) 7( x )( x 6) y + ( )( 6) ( )( 6) 88( x )( x ) + ( 6 )( 6 ) Wen x 5 64( 5 )( 5 6) 7( 5 )( 5 6) y + ( )( 6) ( )( 6) 88( 5 )( 5 ) + ( 6 ) ( 6 ) 64( )( ) 7( 4)( ) 88( 4)( ) y + + ( )( 5) ( )( ) ( 5)( ) y Eg Use te data in te table on te pevious page to find te cubic polynomial wic fits te data 5, x, x 5, x f, Δ f 5, Δ f 75, Δ f 75 x x( x 5) f( x) xx ( 5)( x ) x+ 5x( x- 5) + x( x- 5)( x-) x + 5x 75x+ x 5x + 5x x + 4x

8 Summay NM Topic 5: Numeical Diffeentiation Capte 5 Pages -5 Example 5 Page 4 Capte 5 Pages 5-7 Example 5 Page 6 Execise 5A Q Capte 5 Pages 8-9 Execise 5A Q Numeical Diffeentiation Given two points on a cuve, x and x wee x x +, ten te gadient of te tangent at x may be appoximated by te gadient of te cod f f f f Gadient Δ ie f '( x) Δ x x In pactice, you would take a sequence of values of and look fo convegence in te values of f ' As gets small tis fomula will become vey inaccuate Cental Diffeence Metod A bette appoximation fo te gadient at x would be to take values on bot sides of x f f- f( x+ ) f( x ) Gadient x x f( x+ ) f( x ) ie f ' ( x) As above, in pactice, you would take a sequence of values of and look fo convegence in te values of f ' As gets vey small tis fomula will become inaccuate Calculating te eo in f(x) wen tee is an eo in x If tee is an eo of in te value of x ten X x + and f(x) f(x) f '(x) ie if te eo I x is ten te eo in f(x) is appoximately f '(x) Vesion B: page 8 Competence statements: f Eg Obtain an appoximation to te deivative of f( x) tan x at x 4 Wit, tan( + ), tan 4 4 f '( x) Wit 5, tan( + 5) 5, tan f'( x) 7 5 Eg Obtain an appoximation to te deivative of f( x) tan x at x 4 Wit, tan( + ), tan( ) f '( x) 5 Wit 5, tan( + 5) 5, tan( 5) f'( x) 5 Eg Obtain an appoximation to te deivative of f( x) by te fowad diffeence and cental diffeence metods fo x, and 4, given te values of te function in te table ( ) x y FD CD dy/dx In fact te function is y x + wic can be diffeentiated by tecniques in C to give dy x dx x + giving te values in te last column fo compaison

9 Summay NM Topic 6: Rates of convegence in numeical pocesses Capte 6 Pages - Example 6 Page 7 Execise 6A Q, 4(i) Capte 6 Pages -7 Example 65 Page 7 Execise 6B Q, Capte 6 Pages 8- Execise 6C Q, Rates of convegence of sequences A sequence as fist ode convegence if ε ε ie ε kε + + fo all wee k < A sequence as second ode convegence if ε+ ε ie ε+ kε fo all We also equie kε < ε is te eo in te t tem It is not possible to calculate tis eo as we need te exact value (fo wic we ae looking!) to calculate it We often take te most accuate value we ave to be te exact value fo te puposes of calculating te eos Altenatively we conside te atio of diffeences of iteated values x+ x+ Te atio of diffeences fo a sequence x+ x Fo a fist ode conveging sequence, te atio of diffeences will be equal to k Fo a second ode conveging sequence te atio equies te use of te limit, wic fo a numeical pocess is te best you can do Ten te atio of te eo in one step wit te squae of te eo at te pevious step Rates of convegence in numeical integation Midpoint ule: Te absolute eo is popotional to Tis is a second ode metod Simpson s ule: Te absolute eo is popotional to 4 Tis is a fout ode metod Extapolated estimates Tn Tn 4Tn Tn T T n + S S 6S S S Sn n n n n Rates of convegence in numeical diffeentiation Fo fowad diffeence appoximations, te eo is popotional to It is a fist ode metod Te atio of diffeences between estimates obtained by epeatedly alving is aound 5 Fo cental diffeence appoximations te eo is popotional to It is a second ode metod Te atio of diffeences wen is alved is aound 5 Vesion B: page 9 Competence statements: e4, c, c4 Eg to solve te equation x + x 9 f (x) > fo all x so only one oot f() 6, f() so oot in inteval [, ] (i) Fixed point iteation: Use te iteative fomula x 9 + x wit x A speadseet tabulation gives te following: x x + x x + (x + x )/(x x ) E E E E fom wic it can be seen tat te atio of diffeences is appoximately constant (ii) Te Newton-Rapson iteative fomula gives x + 9 x + wit x x + A speadseet tabulation gives te following: x x + x α (x α)/(x + α) E E tus demonstating second ode convegence Eg Values of f( x) ae given in te table below Using tese values, estimate f( x) d xusing extapolation x y T ( + 77 ) 8555 Similaly T 85, T4 8 4T4 T T 8846 Similaly S 65769, S S S S