Computational Methods of Scientific Programming Fall 2008

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Gwendolyn Hopkins
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1 MIT OpenCurseWare Cmputatinal Methds f Scientific Prgramming Fall 2008 Fr infrmatin abut citing these materials r ur Terms f Use, visit:

2 Hmewrk #3 Slutin Due Thursday Octber 30, 2008 Questin (1): (25-pints) (a) Write, cmpile and run a C r C++ prgram that generates tw tables f the Gamma functin. The Gamma functin satisfies the fllwing equatin $ "(z) = % t z#1 e #t dt 0 where z is the argument f the gamma functin and can be cmplex. (See (See fr infrmatin n the Gamma functin). Gamma functins are rarely cmputed by directly integrating the equatin abve. Generally they are evaluated by series expansins. Fr ne table, table will be generated fr z between 1 and 10 in increment f 1 when z is an integer. This table shuld give values f Γ(z), Γ(z+1/3), Γ(z+1/2) and Γ(z+2/3). Fr the secnd table, Γ(z) shuld be generated fr z between -1 and +1 in increments f 0.1. Results shuld be tabulated with at least 6-significant digits. The submissin shuld include (a) A discussin f the algrithms used in the prgram with ratinales fr the chices (b) The tables generated by the prgram and The C r C++ surce cde Slutin: The slutin t this prblem fllws the Frtran slutin almst exactly. The nly change made here was the tlerance fr the accuracy f the calculatin can be passed thrugh runstring and argc (number f arguments) and argv (argument strings) are used t read the tlerance value. The default tlerance is 10^-6. Only the fractinal argument (z) are effected by the tlerance. Tw results are shwn belw. The slutin is implemented in HW03_01.c and the results are: % gcc HW03_01.c - HW03_01 % HW03_01 Table f gamma functins f psitive n integers n gamma(n) gamma(n+1/3) gamma(n+1/2) gamma(n+2/3) Table f gamma functins fr nn-integer arguments HW03 Sln

3 Three methds are used here: Eulers frmula GammaEul Eulers infinite prduct GammaInf Asympttic series expansin GammaSer Tlerance n calculatin is 1.000e-06 z GammaEul(z) GammaInf(z) GammaSer(z) Case when tlerance is made smaller: TAHCmputer[157] HW03_01 1e-7 first part f results un-effected. Table f gamma functins fr nn-integer arguments Three methds are used here: Eulers frmula GammaEul Eulers infinite prduct GammaInf Asympttic series expansin GammaSer Tlerance n calculatin is 1.000e-07 z GammaEul(z) GammaInf(z) GammaSer(z) HW03 Sln

4 Questin (2): (25-pints). Write a prgram that reads a file cntaining text, determines (1) The average and rt-mean-square (RMS) scatter abut the mean f the number f characters per wrd and (2) The average and rt-mean-square (RMS) scatter abut the mean f the number f wrds per sentence. A sentence can end with a perid r questin mark. The text belw is cntained in the file Q2_text.txt. The basic analysis f spacecraft tracking data requires relating the psitin and velcity f the spacecraft t the psitin and velcity f the tracking system. The crdinate system used in spacecraft navigatin is shwn in Figure 1. The basic measurements are f r and its time derivative and sequences f these measurements, cmbined with knwledge f the tracking statin lcatin and equatins f mtins f the spacecraft, allw the psitin f the spacecraft dented here by distance r, right ascensin, a, and declinatin, d, and its velcity t be determined as a functin f time. In additin t knwing the crdinates f the spacecraft in an inertial crdinate system, the crdinates f slar system bdies are als needed in this frame. Tracking data cllected n spacecraft near planets can als be used t imprve the ephemerides the planets thrugh the gravitatinal perturbatins f the spacecraft mtins. Large cmbined analysis f tracking data and direct measurements f planets (radar and ptical psitins) are used t generate planetary ephemeredes. Hints: Remember if reads are cded as scanf r getchar then the file Q2_text.txt can be re- directed int the prgram using: Q2C < Q2text.txt where Q2C is the name f the prgram (yu can call the prgram any name yu like). Slutin Again this slutin fllws the Frtran cde quite clsely and gets the file name frm the runstring using the cntents f the argv character arrays passed t the main prgram. The slutin is implemented in HW03_02.c and the utput is: When nly letters are cunted: % gcc HW03_02.c HW03_02 % HW03_02../HW02_sln/Q2_text.txt HW 3 Q2: Reading file../hw02_sln/q2_text.txt HW03_02: In file../hw02_sln/q2_text.txt there are: HW03 Sln

5 Mean characters per wrd 5.38 with RMS 3.15 in 166 wrds; Mean wrds per sentence with RMS in 6 sentences. When the numeric 1 is cunted as a character the result is: HW03_02: In file test.txt there are: Mean chacters per wrd 5.35 with RMS 3.16 in 167 wrds; Mean wrds per sentence with RMS in 6 sentences Questin (3): (50-pints) Write a C r C++ prgram that will cmpute the mtins f a set f particles underging mutual gravitatinal attractin. The prgram shuld generate the trajectries f each f the particles with an errr tlerance that is prprtinal t the separatin f the particles. The prgram shuld be able t handle large numbers f particles and thught shuld be given as t hw t input the initial psitins and velcities f particles when there are a large number f particles. As a test f yur prgram: Evaluate the trajectries f the 6 particles belw with an errr tlerance f 1.e-6. (This case is similar t the cllisin f tw Sun-Earth-Mn systems) The integratin shuld be run fr 515-days and the psitins and velcities at the end f 515 days shuld be included in the utput. The values belw give the mass (kg), X and Y psitin (km) and X and Y velcities (km/s) f the 6 particles t be evaluated. 2.0e+30 kg XY 0.0e e+00 (km) VXY 0.000e e+00 (km/sec) 8.8e+28 kg XY 1.5e e+00 (km) VXY 0.000e e+01 (km/sec) 7.3e+22 kg XY 1.5e e+07 (km) VXY e e+01 (km/sec) 2.0e+30 kg XY -1.0e e+00 (km) VXY 1.000e e+00 (km/sec) 8.8e+28 kg XY -8.5e e+00 (km) VXY 1.000e e+01 (km/sec) 7.3e+22 kg XY -8.5e e+07 (km) VXY e e+01 (km/sec) Yur answer t this questin shuld include: (a) The algrithms used and the design f yur prgram (b) The C r C++ prgram surce cde (I will cmpile and run yur prgrams). If yu prgram des nt run r takes mre than few minutes t run, let me knw s that I will treat it with cautin. The results frm the test case abve with psitins and velcities at the end f 515 days. Yu culd als explre the effects f changing the accuracy tlerance n the results. Slutin The slutin here again clsely fllws the Frtran slutin. Glbal variables and defined quantities are put in an include file declared abve the main prgram. (The statements in the include culd have just been added t file. The advantage f the include wuld be if multiple surce files were used. The basic mdules in the prgram are: read_nbdy reads the bdy definitins in the frm f mass, psitin x and y, and velcity x and y. The implementatin in this rutine nly interprets lines that start with a space. All ther lines are interpreted as cmments HW03 Sln

6 reprt_ics rutine t write ut the psitin and velcities f the bdies are specified times. A character string is als passed t anntate the type utput. int_step this subrutine advances the integratin by ne time step using the integratr discussed abve. eval_step this subrutine evaluates if the current time is adequate fr the precisin needed and determines if the step size shuld be increased r decreased. This rutine is cded as vid and the new_step size is returned by passing in the pinter t new_step. We culd have implemented this as duble functin with the new_step size returned frm the functin with the return(new_step) line. get_accel this subrutine cmpures the acceleratins f all the bdies given the psitins passed in its arguments list. Time and velcity are als passed but these are needed in the gravitatinal nly mdel. Drag type frces culd be easily added t this rutine. Animate this is simple rutine that uses VT100 escapes sequences t plt the mtins f the bdies n a 85 by 45 grid. Static declaratins are needed in this rutine because a number f lcal variables (such as scalex and scaley) need t be saved. All these rutine are declared vid because the variables, which are changed by them, are passed as pinters (remember array declaratins are autmatically pinters). All f the array manipulatins in these rutines needed t dne explicitly. The simple statements in the Frtran cde needed t be replaced with explicit fr lps. Cmmunicatin in the prgram is thrugh a cmbinatin f glbal variables (such as psi[2][max_bdy] and variables passed int and ut f rutines. Pinters need t be used frm thse variables that are updated by the functins. There was a small errr in the riginal Frtran Runge-Kutta cde and it has nw been updated. Sme additinal utput was als added t shw the clsest apprach distance. The cde is implemented in HW03_03.c with include file NBdyC.h The data file with the test case is NBdy.dat The utput f the prgram as requested in the hmewrk is: % HW03_03 NBdy.dat File NBdy.dat Run days, 1.00e-06 tl HW 03 Q 03: Initial Cnditins At time days fr 6 bdies Bdy Mass (kg) PsX (km) PsY (km) VelX (km/s) VelY (km/s) e e e e e e e e e e e e e e e e e e e e e e e e e e+01 TIME days, Step days HW03 Sln

7 animatin space remved. O HW 03 Q 03: Final cnditins At time days fr 6 bdies Bdy Mass (kg) PsX (km) PsY (km) VelX (km/s) VelY (km/s) e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e+01 Smallest step size needed days Clsest apprach distance was e+04 km Bdies 2 and 4 Ntice here that step size gets very small during the clse encunters f the bdies. If we end the integratin earlier befre the clse encunter, the step size remains much mre reasnable. % HW03_03 NBdy.dat Being utput remved O O HW 03 Q 03: Final cnditins At time days fr 6 bdies Bdy Mass (kg) PsX (km) PsY (km) VelX (km/s) VelY (km/s) e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e+00 Smallest step size needed days Clsest apprach distance was e+06 km Bdies 5 and 6 In this case, the smallest step size was 0.5 days HW03 Sln

COP2800 Homework #3 Assignment Spring 2013

COP2800 Homework #3 Assignment Spring 2013 YOUR NAME: DATE: LAST FOUR DIGITS OF YOUR UF-ID: Please Print Clearly (Blck Letters) YOUR PARTNER S NAME: DATE: LAST FOUR DIGITS OF PARTNER S UF-ID: Please Print Clearly Date Assigned: 15 February 2013