PC5215 Numerical Recipes with Applications - Review Problems

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Roger Wilson
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1 PC55 Numerical Recipes with Applicatios - Review Problems Give the IEEE 754 sigle precisio bit patter (biary or he format) of the followig umbers: Note that it has 8 bits for the epoet, 4 bits precisio (matissa) with the leadig oe omitted i the represetatio, ad a sig bit The epoet is biased by 7 [Read the article What Every Computer Scietist Should Kow about Floatig- Poit Arithmetic As: 3F E DCCCCCD BC3D70A Solve the followig fiite differece equatio a+ + b + c = 0 The solutio depeds o two iitial values, say, 0 ad Discuss the advatage ad disadvatage of usig the fial solutio to compute versus recursio [Hit: assumig = cost λ 3 How to make a dyamic memory allocatio for D array i C, say a[[m? What is machie ε? Ru program machar() o page 89 to determie machie epsilo for your machie, what is roughly ε for sigle precisio, double precisio, or quadruple precisio? 4 Do LU decompositio (without pivotig) of the followig matri by Crout s algorithm: [As: 0 0 L= 0 0, U 0 = 5/ 0 0 7/ 5 Give the computatioal compleity O(N k ) of the followig algorithms (also state what is N), LU, Det(A) (by LU), A - (by LU) A=b by Gaussia elimiatio, Neville s iterpolatio, Trapezoidal rule, FFT, cojugate gradiet for liear system, quick sort, heap sort

2 [As: N 3, N 3, N 3, N 3, N, N, N log N, N 3, N log N, N log N 6 What is the iverse of the followig matri? a 0 0 b 0 [Use LU decompositio or otherwise directly by AA - =I As a 0 0 b 0 7 Classify each of the followig matrices as well-coditioed or ill-coditioed Note that the coditio umber of a matri is defied as A A (a) (b) (c) (d) [As: the -orm coditio umbers are, respectively, 0 0,,, Large coditio umber meas ill-coditioig 8 What are the coditios required to the polyomials for cubic splies? [As: fuctio ad its first derivative are cotiuous at the meetig poits of each segmet 9 Apply Neville s algorithm to determie the value f() at = The iterpolatio poits are (0,), (,3), (3, 5) Determie also the polyomial (usig Lagrage formula) i epaded form [As: f() = 5/3, f( ) = + /3 + /3 0 Prove the ope formula: 0 f d = h f + O h ( ) ( ), 3 f d = h f f + O h 0 [Hit: use Taylor epasio 3 ( ) ( ),

3 What is the basic idea of Guassia itegratio? Derive a -poit formula for the Guassia itegral i the iterval [, with a costat weight W()= [As: = = / 3, w = w = Write out the steps for quick sort ad heap sort, for the followig iput data: [, 5, 3, 6, 7,, 9, 0, 4, 8 3 What is the computatioal compleity of the Newto-Raphso method for the root (such that F() =0) i N dimesios? Derive the formula for the iteratio Discuss issue o stability [As: N 3 J F, J = F/ 4 Compute the solutio of = 0 umerically usig Newto s method, startig from = Need a calculator for this [As: Iterate /+ /, after three iteratios, oe gets 44 5 How to bracket a zero, bracket a miimum, or maimum? 6 Cosider the fuctio f(,y) = + y+y Use the cojugate gradiet method to fid the miimum of the above fuctio, startig from the poit ( 0,y 0 )=(,) 7 Show that the error at i-th step i the cojugate gradiet method is of the form e N = δ d, where d( j) is the search directio i the j-th step () i j ( j) j= i [Read the article by J R Shewchuk, A itroductio to the cojugate gradiet method without the agoizig pai 8 Prove that the optimal coditio to stop i a liear search (i higher dimesios) is that g = 0, where is search directio, g is the gradiet at the ew locatio [Hit: derivative with respect to λ of f(+λ) is zero at mi or ma 9 (a) Solve the system of equatios (i least squares sese): + y = 3 + 3y = 5 3 y = (b) Solve the same problem by cojugate gradiet method (as a miimizatio problem) [As: =0507, y=075 0 Do the FFT steps for the followig iput: [,,,,,,, A set of data poits is give as followig: (0, 00), (, 0), (, 98), (3, 30), (4, 4) 3

4 Determie a straight lie (least-squares) fit f() = a+b, give also the error estimates of the fittig parameters a ad b Is there ay relatio betwee the curret problem ad Prob 9? [As: a = 05 ± 00, b= 0034 ± 0050 Cosider discretized versio of the equatio dy/d= y usig forward differece ad backward differece: y+ = y hy y = y hy Solve the differece equatios eactly ad compare them with eact solutio of the differetial equatio Which versio is preferred? [As: forward differece ( ) y = h y0, backward differece y ( ) = + h y0 Eact solutio is y(h) = y 0 ep(-h) The secod backward differece method is preferred, due to its stability (errors do ot blow up) for ay step size h 3 Cosider the Hamiltoia H( pq, ) = ( p + q) Give a secod order symplectic algorithm for solvig this system Show that the resultig update viewed as a trasformatio i the phase space (p,q) preserves the T phase space area Show eplicitly that it is symplectic ( D JD = J or dp dq is ivariat with respect to the trasformatio, where D is Jacobia matri of the trasformatio ad the matri 0 J = 0 ) [As: The Hamilto s equatios of motio are dp / dt = H / q = q, dq / dt = H / p = p, so q = q + hp h q / p = p hq ( + q )/ We ca show that the Jacobia of the above trasformatio from (p,q) to (p,q ) is, so area is preserved That is, we ca verify that Det(D)=: p p 3 h h h + p q Det( D) 4 = = = q q h h p q Ad 3 dp dq = d[( h / ) p + ( h + h / 4) q d[( h / ) q + hp 3 = ( h / )( h / ) dp dq + ( h + h / 4) h dq dp = dp dq (Sice dp dp = 0, dq dq = 0, dp dq = dq dp ) 4

5 4 Geerate poits distributed uiformly o a uit sphere ( +y +z =) 5 Give that ξ ad ξ are idepedet, uiformly distributed radom variables betwee 0 ad, what are the probability distributios of the radom variables ξ + ξ ad ξ ξ? [As: for ξ + ξ case, cosider P(ξ + ξ <) = F(), p() = df()/d = for 0 < <, - for > > 6 Write dow the trasitio matri W for a oe-dimesioal 4-spi Isig model with periodic boudary coditio usig Metropolis flip rate 7 (a) Let assume W i has ivariat distributio P for all i, ie, P = P W i, i=,,,n Show that both W s = Σλ i W i ad W p =ΠW i has ivariat distributio P, where Σλ i = ad λ i >0 How to implemet W s ad W p o computer? (b) If W i satisfies detailed balace with respect to P, does W s ad/or W p satisfy detailed balace? 5

CS321. Numerical Analysis and Computing

CS321. Numerical Analysis and Computing CS Numerical Aalysis ad Computig Lecture Locatig Roots o Equatios Proessor Ju Zhag Departmet o Computer Sciece Uiversity o Ketucky Leigto KY 456-6 September 8 5 What is the Root May physical system ca