Section A assesses the Units Numerical Analysis 1 and 2 Section B assesses the Unit Mathematics for Applied Mathematics

Transcription

1 X0/70 NATIONAL QUALIFICATIONS 005 MONDAY, MAY.00 PM 4.00 PM APPLIED MATHEMATICS ADVANCED HIGHER Numerical Aalysis Read carefully. Calculators may be used i this paper.. Cadidates should aswer all questios. Sectio A assesses the Uits Numerical Aalysis ad Sectio B assesses the Uit Mathematics for Applied Mathematics. Full credit will be give oly where the solutio cotais appropriate workig. 4. Numerical Aalysis Formulae ca be foud o pages two ad three of this Questio Paper. LIB X0/70 6/670 *X0/70*

2 NUMERICAL ANALYSIS FORMULAE Taylor polyomials For a fuctio f, defied ad times differetiable for values of x close to a, the Taylor polyomial of degree is f () a f () a f( a) + f ( a)( x a) + ( x a) ( x a)!! ( ) f () a f () a ad fa ( + h) fa ( ) + f ( ah ) + h h!! ( ) Newto forward differece iterpolatio formula p p p fp = f0 + f0 + f0+ f p ( x) = L ( x) y ( ) ( ) ( ) Lagrage iterpolatio formula i i i= 0 ( x x0)( x x)... ( x xi )( x xi+ )... ( x x) where Li ( x) = ( x x )( x x )... ( x x )( x x )... ( x x ) i 0 i i i i i+ i Newto-Raphso formula For a equatio f(x) = 0, with x 0 give, + = fx ( ) x x f ( x ) Composite trapezium rule h fxdx ( ) = { f + f + ( f+ f f )} + E b 0 a a with h = ad fk = f( a+ kh) where E is (approximately) bouded by a (i) hm with f ( x ) M for a x b a or (ii) D with f D for a x b [X0/70] Page two

3 Simpso s composite rule h fxdx ( ) = { f + f + 4( f+ f f ) + ( f + f f )} + E b 0 4 a with h = a ad fk = f( a+ kh) where E is (approximately) bouded by (i) a hm 4 (iv) with f ( x ) M for a x b 80 or (ii) a D 4 with f 80 D for a x b Richardso s formula Trapezium rule: Simpso s rule: I I + ( I I ) I I + ( I I ) 5 Euler s method For a equatio = fxy (, ) with ( x0, y0) give, dx y = y + hf( x, y ) + Predictor-Corrector method: Euler-Trapezium Rule y = y + hf( x, y ) + y+ = y + h ( f ( x, y ) + f ( x+, y+ )) [Tur over [X0/70] Page three

4 Sectio A (Numerical Aalysis ad ) Aswer all the questios. A. I the usual otatio for forward differeces of fuctio values f(x) tabulated at equally spaced values of x, f i = f i + f i where f i = f(x i ) ad i =...,,, 0,,,.... Show that f 0 = f f + f f 0. If each value of f i is subject to a error whose magitude is less tha or equal to ε, determie the magitude of the maximum possible roudig error i f 0. A. The followig data (accurate to the degree implied) ad differece table are available for a fuctio f. i x i f(x i ) f f f (a) (b) (c) Idetify the value of f i this table. State a feature of the table which suggests that a polyomial of degree two would be a suitable approximatio to f. Usig the Newto forward differece formula of degree two, ad workig to three decimal places, estimate f(0. 7). A. The followig data are available for a fuctio f. x 4 f(x) Use the Lagrage iterpolatio formula to obtai a quadratic approximatio to f(x), simplifyig your aswer. A4. Obtai the Taylor polyomial of degree two for the fuctio cos x ear x = π/4. Estimate the value of cos 46 usig the first degree approximatio. State your aswer to the umber of figures you would expect to be correct, givig a reaso for your choice. [X0/70] Page four

5 A5. The sequece {a r } is defied by the recurrece relatio 7a + + 4a =, with a 0 =. 5. Trace the sequece as far as the term a 4. (Work to two decimal places where appropriate.) Display the covergece of the sequece usig a cobweb diagram, ie a sketch based o drawig the lie segmets betwee the poits with coordiates (a 0, a 0 ), (a 0, a ), (a, a ), (a, a ), (a, a ), etc. Determie the fixed poit of the sequece. A6. Use sythetic divisio to obtai (without roudig) the quotiet Q(x) ad remaider R whe the polyomial f(x) = x + x 5x +. is divided by g(x) = x It is give that f is decreasig i the eighbourhood of x = 0. 7 ad that all coefficiets are exact except for the costat terms i f(x) ad g(x), which are rouded to the degree of accuracy implied. Determie to three decimal places the maximum possible value of R. A7. Derive Euler s method for the approximate solutio of the differetial equatio fxy (, ) dx = subject to the iitial coditio y(x 0 ) = y 0 ad show that the global trucatio error i the process is of first order. The solutio of the differetial equatio l( x y), y() dx = + = is required at x =.. Obtai a approximatio to this solutio usig Euler s method with step size 0.. Perform the calculatio usig four decimal place accuracy. 4 [Tur over [X0/70] Page five

6 A8. (a) Usig a Taylor polyomial of degree two, or otherwise, derive the trapezium rule over a sigle strip ad the correspodig pricipal error term. Use the composite trapezium rule with two strips ad with four strips to obtai two estimates I ad I respectively for the itegral x I = xe dx. Perform the calculatios usig five decimal places. Use Richardso extrapolatio to obtai a improved estimate I for I based o the values of I ad I. 5 (b) Values of a fuctio f at five poits x 0, x, x, x, x 4, such that x i + = x i + h (0 i ), ad spaig a iterval, are f 0, f, f, f, f 4. Usig Richardso extrapolatio o the approximatios obtaied with the trapezium rule with two ad four strips, obtai a estimate of the itegral of f o this iterval. Show that this result is the same as that obtaied usig Simpso s rule with four strips. A9. I the solutio of a problem, the followig two systems of liear equatios, A x = B ad A x = B have arise: ( ) ( ) System S : A = , B = ( ) ( ) System S : A = , B = (a) Use Gaussia elimiatio with partial pivotig to solve System S. Icorporate a row check, show three decimal places i the calculatio ad roud your aswer to oe decimal place. (b) Use the Gauss-Seidel iterative method with iitial values x = x = x = 0 to solve System S, correct to two decimal places. Show three decimal places i the calculatio. 6 4 (c) State, with a brief explaatio, whether there is evidece that either system is ill-coditioed. [X0/70] Page six

7 A0. The equatio x = g(x) has a root at x = a i a iterval I ad g (x) satisfies 0 < g (x) < for x I. A iterative process is defied by x + = g(x ) with a suitable startig value. Explai with the aid of a diagram whether the process will coverge whe a iitial approximatio x = x 0 withi I is chose. Explai briefly how the process would differ for the case where g (x) > ear x = a. The positive roots of the equatio x 7x + = 0 are kow to lie withi the itervals [0, ] ad [,. 5]. For this equatio a iterative scheme of the form ( x + ) x + = 7 is proposed for use. Show that this scheme is defiitely usuitable o [,. 5], but that it may be suitable o [0, ]. Usig x = 0. 5 as a startig value ad recordig successive iterates to three decimal places, use Simple Iteratio to determie this root to three decimal places. Use three applicatios of the bisectio method to determie a more accurate estimate of the iterval cotaiig the larger positive root. [END OF SECTION A] [Tur over for Sectio B o Page eight [X0/70] Page seve

8 Sectio B (Mathematics for Applied Mathematics) Aswer all the questios. B. Differetiate, ad simplify as appropriate, (a) fx ( ) = exp(ta x), where π < x < π, (b) g(x) = (x + ) l (x + ), where x > 0. ( ) B. Give that A =, show that A A = ki for a suitable value of k, where 0 I is the uit matrix. B. A curve is defied by the parametric equatios x = 5t 5, y = t. Fid the value of t correspodig to the poit (0, ) ad calculate the gradiet of the curve at this poit. B4. Expad ad simplify ( )4 a. a, x + B5. Express i partial fractios. x + x Hece obtai ( ) / x x + ( + x ) dx. B6. (a) Give the differetial equatio si x ycos x 0, dx = fid the geeral solutio, expressig y explicitly i terms of x. 4 (b) Fid the geeral solutio of dx = si x ycos x si x. 5 [END OF SECTION B] [END OF QUESTION PAPER] [X0/70] Page eight