*X203/701* X203/701. APPLIED MATHEMATICS ADVANCED HIGHER Numerical Analysis. Read carefully

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1 X0/70 NATIONAL QUALIFICATIONS 006 MONDAY, MAY.00 PM.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.. Numerical Aalysis Formulae ca be foud o pages two ad three of this Questio Paper. LI 0/70 6/70 *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 Lagrage iterpolatio formula p ( x) = L ( x) y 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 + ( f+ f f ) + ( f + f f )} + E b 0 a with h = a ad fk = f( a+ kh) where E is (approximately) bouded by (i) a hm (iv) with f ( x ) M for a x b 80 or (ii) a D with f 80 D for a x b Richardso s formula Trapezium rule: Simpso s rule: I I + ( I I ) I I + ( I I ) Euler s method dy 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. The followig data is available for a fuctio f: x f(x) Use the Lagrage iterpolatio formula to obtai a quadratic approximatio to f(x), simplifyig your aswer. Marks A. The polyomial p is the Taylor polyomial of degree three for the fuctio f ear x = π/. For the fuctio f(x) = cos x, express p(π/ + h) i the form c 0 + c h + c h + c h. Estimate the value of cos 6 usig the secod degree approximatio, statig your aswer rouded to four decimal places. Write dow the pricipal trucatio error term for this secod degree approximatio ad calculate its value. Hece state the secod degree approximatio to a appropriate degree of accuracy. A. Derive the Newto forward differece formula of degree two to fit a polyomial through the poits (x 0, f(x 0 )), (x, f(x )), (x, f(x )). 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 f (a) Calculate the maximum roudig error i f 0. (b) (c) (d) Idetify the value of f i this table. State a feature of the table which suggests that a third degree polyomial would be a good approximatio for this fuctio. Use the Newto forward differece formula of degree three to estimate f(. 6), workig to four decimal places. [X0/70] Page four

5 A. Use the Jacobi iterative procedure with x = x = x = 0 as a first approximatio to solve the followig equatios correct to two decimal places. 8x + 0 x + 0 8x = 0 6x + 9x 0 6x = 7 0 x + 0x = 79 Marks A6. Use sythetic divisio to obtai (without roudig) the quotiet Q(x) ad remaider R whe the polyomial f(x) = x + x x +. is divided by g(x) = x Give 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 two decimal places the maximum possible value of R. (You may assume that f(x) is icreasig o the iterval [0. 7, 0. 9].) dy x A7. The differetial equatio = e ( x y ) with y() = is to be solved umerically. dx Use the predictor-corrector method with Euler s method as predictor ad the trapezium rule as corrector to obtai a solutio of this equatio at x =.. Use a step size h = 0. ad oe applicatio of the corrector o each step. Perform the calculatios usig four decimal place accuracy. 6 A8. Gaussia elimiatio with partial pivotig is used to obtai the iverse of the matrix A ( ) 6 7 = 6 0. Durig the process, the followig augmeted matrix is obtaied. ( ) Complete the determiatio of the iverse of A, givig all elemets i the iverse matrix to two decimal places. Explai what is meat by a ill-coditioed matrix. Is A ill-coditioed? Give a reaso for your aswer. Why is the use of partial pivotig importat i dealig with a ill-coditioed matrix? 6 [Tur over [X0/70] Page five

6 A9. The equatio f(x) = 0 has a root close to x = x 0. Use the Taylor series expasio of f(x) about x = x 0 to derive the Newto-Raphso method of solutio of f(x) = 0. It is give that the equatio x x + = 0, has three distict real roots. Use the Newto-Raphso method to determie the root ear x = correct to two decimal places. Show that the iterative scheme may be suitable to determie the root ear x = x x + = Usig x = 0. as a startig value ad recordig successive iterates to three decimal places, use Simple Iteratio to determie this root to two decimal places. The third root lies i the iterval [.,. 0]. Use three applicatios of the bisectio method to determie a more accurate estimate of the iterval i which this root lies. Marks A0. The fuctio f is defied by f(x) = x l( + x), x 0. (a) Use Simpso s rule with two strips ad the composite Simpso s rule with four strips to obtai two estimates I ad I respectively for the itegral I = f( x) dx. Perform the calculatios usig four decimal places. (iv) ( x + x+ 6) (b) It is give that for this fuctio f ( x) = ad that f (v) (x) ( x + ) has o zero o the iterval [, ]. Use this iformatio to obtai a estimate of the maximum trucatio error i I. Hece state the value of I to a suitable accuracy. (c) Establish Richardso s formula to improve the accuracy of Simpso s rule by iterval halvig. Use Richardso extrapolatio to obtai a improved estimate for I based o the values of I ad I obtaied i part (a) of the questio. [END OF SECTION A] [X0/70] Page six

7 Sectio B (Mathematics for Applied Mathematics) Aswer all the questios. Marks B. Calculate A where A = ( 0 ). Hece solve the system of equatios x + y = x + y + z = x + y + z =. B. Give that y = l( + si x), d y where 0 < x < π, show that = dx + si. x B. Defie S = r,. Write dow formulae for S ad S +. r= Obtai a formula for (). B. Solve the differetial equatio dy cos x y, dx = give that y > 0 ad that y = whe x = 0. B. Use the substitutio + x = u to obtai x + x dx. B6. (a) (b) (c) 0 x Evaluate xe dx. x Use part ( a) to evaluate x e dx. 0 x Hece obtai (x + x) e dx. 0 [X0/70] [END OF SECTION B] [END OF QUESTION PAPER] Page seve

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