Assignment Fall 2014

Transcription

1 Assignment Fall 04 De: Wednesday, 0 December at 5 PM. Upload yor soltion to corse website as a zip file YOURNAME_ASSIGNMENT_5 which incldes the script for each qestion as well as all Matlab fnctions and scripts (of yor own creation) called by the scripts; both scripts and fnctions mst conform to the formats described in Instrctions and Qestions below. Instrctions Download (from the corse website Assignment 5 page) the Assignment_5_Templates folder. This folder contains a template for the script associated with each qestion (A5Qy_Template for Qestion y), as well as a template for each fnction which we as yo to create (fnc_template for a fnction fnc). The Assignment_5_Templates folder also contains the grade_o_matic files for Assignment 5 (please see Assignment for a description of grade_o_matic ) as well as all axiliary files which yo will need for Assignment 5. We indicate here several general format and performance reqirements: (a.) Yor script for Qestion y of Assignment 5 mst be a proper Matlab.m script file and mst be named A5Qy.m. In some cases the script will be trivial and yo may sbmit the template as is jst remove the _Template in yor YOURNAME_ASSIGNMENT_5 folder. Bt note that yo still mst sbmit a proper A5Qy.m script or grade_o_matic will not perform correctly. (b.) In this assignment, for each qestion y, we will specify inpts and otpts both for the script A5Qy and (as is more traditional) any reqested Matlab fnctions; we shall denote the former as script inpts and script otpts and the latter as fnction inpts and fnction otpts. For each qestion and hence each script, and also each fnction, we will identify allowable instances for the inpts the parameter vales or parameter domains for which the codes mst wor. (c.) Recall that for scripts, inpt variables mst be assigned otside yor script (of corse before the script is exected) not inside yor script in the worspace; all other variables reqired by the script mst be defined inside the script. Hence yo shold test yor scripts in the following fashion: clear the worspace; assign the inpt variables in the worspace; rn yor script. Note for Matlab fnctions yo need not tae sch precations: all inpts and otpts are passed throgh the inpt and otpt argment lists; a fnction enjoys a private worspace. (d.) We as that in the sbmitted version of yor scripts and fnctions yo sppress all display by placing a ; at the end of each line of code. (Of corse dring debgging yo will often choose to display many intermediate and final reslts.) We also reqire that before yo pload yor soltion to corse website yo rn grade_o_matic (from yor YOURNAME_ASSIGN- MENT_5 folder) for final confirmation that all of yor scripts and fnctions are in the proper format. Note that, in Assignment 5, for of the five qestions, A5Qy for y =,,,4, are mltiple choice: yo shold enter yor answers in the respective scripts A5Qy.m per standard grade_o_matic pro- Note that, for display in verbose mode, grade_o_matic will nroll arrays and present as a row vector.

2 tocol; please doble-chec the letters yo have chosen for each qestion before ploading yor YOURNAME_ASSIGNMENT_5 folder. Note that the last qestion, A5Q5, is not mltiple choice. Qestions. (0 points) Preamble: Yo are not to se Matlab for this qestion (except of corse for the mltiple-choice script file A5Q.m for grade_o_matic) either to identify or confirm the correct reslt yo shold develop yor answer withot recorse to a compter or even a calclator. The point of this qestion is to mae sre that yo nderstand the basic linear algebra. We consider in this problem the system of linear eqations A = f () where A is a given matrix, f is a given vector, and is the vector we wish to find. We introdce two matrices 0 A I = 0, () 0 0 and A II =, () which will be relevant in Parts (i),(ii) and Parts (iii),(iv) respectively. In Parts (i),(ii), A of eqation () is given by A I of eqation (). In other words, we consider the system A I = f given by for f to be specified below. 0 f 0 f =, 0 0 f A I f (i) (5 points) For f = ( ) T (recall T denotes transpose), (a) A I = f has a niqe soltion. (b) A I = f has no soltion.

3 (c) A I = f has an infinity of soltions of the form 0 = 0 + α 0 0 for any (real nmber) α. (d) A I = f has an infinity of soltions of the form = 0 + α 0 0 for any (real nmber) α. (ii) (5 points) For f = ( 0) T, (a) A I = f has a niqe soltion. (b) A I = f has no soltion. (c) A I = f has an infinity of soltions of the form 0 = 0 + α 0 0 for any (real nmber) α. (d) A I = f has an infinity of soltions of the form = 0 + α 0 0 for any (real nmber) α. Now, in Parts (iii), (iv), A of eqation () is given by A II of eqation (). In other words, we consider the system A II = f given by f f =, f A II f

4 for f to be specified below. (iii) (5 points) For f = ( ) T (recall T denotes transpose), (a) A II = f has a niqe soltion. (b) A II = f has no soltion. (c) A II = f has an infinity of soltions of the form 0 = 0 + α 0 0 for any (real nmber) α. (d) A II = f has an infinity of soltions of the form = 0 + α 0 0 for any (real nmber) α. (iv) (5 points) For f = ( 0) T, (a) A II = f has a niqe soltion. (b) A II = f has no soltion. (c) A II = f has an infinity of soltions of the form 0 = 0 + α 0 0 for any (real nmber) α. (d) A II = f has an infinity of soltions of the form = 0 + α 0 0 for any (real nmber) α. 4

5 There are no inpts or otpts for this qestion. Yor answer shold be a one-line script, with yor mltiple-choice answers, as indicated in A5Q_Template.m (as always, remove the _Template before ploading to corse website in yor folder YOURNAME_ASSIGNMENT_5).. (0 points) Preamble: Yo are not to se Matlab for this qestion (except of corse for the mltiple-choice script file A5Q.m for grade_o_matic) either to identify or confirm the correct reslt yo shold develop yor answer withot recorse to a compter or even a calclator. The point of this qestion is to mae sre that yo nderstand the basic linear algebra. We consider the system of three springs and masses shown in Figre. f = f = f = = m = m = m wall f = 0 f = f = 0 The eqilibrim displacements satisfy the linear system of three eqations in three nnowns, A = f, given by m m m = = 0 = 0. = (4) 0 0 The matrix A is SPD (Symmetric Positive 0 Definite). Note yo0 shold only consider the particlar right-hand side f (forces) indicated. We now redce the system ˆ by Gassian Elimination to the form U = f, where U is an pper trianglar matrix. We may then find by Bac Sbstittion. Note that we do not perform any partial pivoting reordering of the rows of A for stability (since the matrix is SPD there is no need), and frthermore we do not perform any reordering of the colmns of the matrix A for optimization: we wor directly on the matrix A as given by eqation (4). Figre : The spring-mass system for Qestion. A f (i) (4 points) The element U (i.e., the entry in the i = second row, j = second colmn) of U is given by (a) / (b) (c) / (d) 5/ 5

6 (ii) (4 points) The element U (i.e., the entry in the i = second row, j = third colmn) of U is given by (a) (b) (c) /5 (d) 0 (iii) (4 points) The element fˆ (i.e., the second entry in the fˆ vector) is given by (a) / (b) (c) 0 (d) 5/ (iv) (4 points) The element fˆ (i.e., the third entry in the fˆ vector) is given by (a) /5 (b) 0 (c) /5 (d) 8/5 (v) (4 points) The displacement of the third mass,, is given by (a) / (b) / (c) 5/ (d) / There are no inpts or otpts for this qestion. Yor answer shold be a one-line script, with yor mltiple-choice answers, as indicated in A5Q_Template.m (as always, remove the _Template before ploading to corse website in yor folder YOURNAME_ASSIGNMENT_5).. (0 points) Preamble: Yo are not to se Matlab for this qestion (except of corse for the mltiple-choice script file A5Q.m for grade_o_matic) either to identify or confirm the correct reslt yo shold develop yor answer withot recorse to a compter or even a 6

7 calclator. The point of this qestion is to mae sre that yo nderstand the basic linear algebra. Figre : The spring-mass system for Qestion. For the six springs in series the spring constants are = whereas for the parallel special spring which directly lins mass and mass 6 the spring constant is special = 4; yo may assme that all qantities are provided in consistent nits. Note that all the springs are described by the linear Hooe relation. We consider the system of springs and masses shown in Figre. Eqilibrim force balance on each mass and Hooe s law for the spring constittive relation leads to the system of six eqations in six nnowns, A = f, f f 0 a 0 c f ; = (5) f f c 0 b 6 f 6 A f where we will as yo to specify a, b, and c in the qestions below. Note for the correct choices of a, b, and c, the matrix A is SPD. ˆ We now redce the system A = f by Gassian Elimination to form U = f, where U is an pper trianglar matrix. Note that we do not perform any partial pivoting reordering of the rows of A for stability (since the matrix is SPD there is no need), and frthermore we do not perform any reordering of the colmns of the matrix A for optimization: we wor directly on the matrix A as given by eqation (5). Recall that since A is SPD we are sre that we will not enconter a zero pivot. (i) (4 points) The vale of a is (a) (b) 4 7

8 (c) 6 (d) 8 (e) (ii) (4 points) The vale of b is (a) (b) 4 (c) 6 (d) 8 (e) (iii) (4 points) The vale of c is (a) (b) 4 (c) 6 (d) 8 (e) (iv) (4 points) The nmber of nonzero elements in the (pper trianglar) matrix U is (a) (b) 6 (c) (d) 5 (e) 6 Hint: Consider Gassian Elimination (and the fill-in process) to dedce the only possibly correct option from the available choices. 8

9 (v) (4 points) The entry (A ) 6 6, (i.e., the entry in the i = sixth row, j = sixth colmn of the inverse matrix of A) is (a) /U 6 6 (b) /A 6 6 (c) U 6 6 (d) A 6 6 (e) (f + f + f + f 4 + f 5 + f 6 )/U 6 6 where in each case sbscript 6 6 refers to the entry in the i = sixth row, j = sixth colmn. Hint: Recall the physical interpretation of the sixth colmn of A. There are no inpts or otpts for this qestion. Yor answer shold be a one-line script, with yor mltiple-choice answers, as indicated in A5Q_Template.m (as always, remove the _Template before ploading to corse website in yor folder YOURNAME_ASSIGNMENT_5). 4. (0 points) Preamble: Yo shold develop yor responses based on theoretical considerations. However, yo may se Matlab to motivate or confirm (or perhaps rectify) yor theoretical predictions. We consider the system of n springs and masses shown in Figre. We consider the particlar case in which i =, i n, and f i =, i n; yo may assme that all qantities are f f f f n m m m n m n wall n free Figre : The spring-mass system for Qestion 4. n provided in consistent nits. The displacements of the masses,, satisfies a linear system of n n eqations in n nnowns, A = f. The Matlab script provided below forms the stiffness n matrix A ( = A in Matlab) f and force f vector f ( f = f in Matlab) and then f solves for the displacements ( = in Matlab) in three different fashions. Yo may assme that prior to n exection of the script m the worspace m containsm only n (Matlab n) whichm is a positive integer scalar. % begin script % form A and f A = spalloc(n,n,*n); 9

10 A(,) = ; A(,) = -; for i = :n- A(i,i) = ; A(i,i-) = -; A(i,i+) = -; end A(n,n) = ; A(n,n-) = -; nmnonzero_of_a = nnz(a); f = ones(n,); % solve A = f in three different ways nmtimes_compte = 0; tic for itimes = :nmtimes_compte = A\f; % REFER TO THIS LINE in Qestion 4(ii) end avg_time_first_way = toc/nmtimes_compte; A_declare_fll = fll(a); tic for itimes = :nmtimes_compte = A_declare_fll\f; end avg_time_second_way = toc/nmtimes_compte; Ainv_declare_sparse = sparse(inv(a)); % in fact if A is "declared sparse" then inv(a) will atomatically be "declared sparse" tic for itimes = :nmtimes_compte = Ainv_declare_sparse*f; end avg_time_third_way = toc/nmtimes_compte; % note that we do not inclde the time to compte inv(a) in avg_time_third_way % end script The Matlab bacslash operator will not perform any partial pivoting reordering of the rows of A for stability (since the matrix is SPD there is no need); frthermore Matlab will not perform any reordering of the colmns of the matrix A for efficiency (since the matrix 0

11 is tri-diagonal the strctre is already optimal). In short, Matlab bacslash wors directlyˆ on the matrix A as given Gassian Elimination to obtain U ( = U in Matlab) and f followed by Bac Sbstittion to obtain. We now rn the script. In the qestions below yo shold assme that the comptational time to perform the operations is proportional to the nmber of FLOPs. (In actal practice, comptational time and FLOPs is not synonymos since the former is affected by memory access, competition for cores, networ speed, and other real-life considerations; frthermore, these real-life considerations become more important for the larger n of interest in this qestion. Inasmch, yor comptational times shold serve to gide, bt not dictate, yor answers.) (i) (5 points) For n = 5000 the script will set the vale of nmnonzero_of_a to (a) 4,998 (b) 0,000 (c) 5,000 (d) 5,000,000 (ii) (5 points) For n = 5000 the nmber of nonzero elements of U will be (a) 5,000 (b),507,50 (c) 9,999 (d) 5,000 Note yo do not see U explicitly in the script of the previos page. Here U is the pper trianglar matrix U formed internally as part of the bacslash operation = A\f on the line of the script with comment % REFER TO THIS LINE in Qestion 4(ii). Hint: Recall Gassian Elimination for tridiagonal matrices. (iii) (5 points) The ratio avg_time_second_way avg_time_first_way will behave asymptotically as Cn ρ for n, where C is a constant independent of n, and ρ is given by (a) (b) (c)

12 (d) 0 (e) - (f ) - (g) - Note we as here for the vale of ρ, not for the vale of C. (iv) (5 points) The ratio avg_time_third_way avg_time_first_way will behave asymptotically as Cn ρ for n, where C is a constant independent of n, and ρ is given by (a) (b) (c) (d) 0 (e) - (f ) - (g) - Note we as here for the vale of ρ, not for the vale of C. There are no inpts or otpts for this qestion. Yor answer shold be a one-line script, with yor mltiple-choice answers, as indicated in A5Q4_Template.m (as always, remove the _Template before ploading to corse website in yor folder YOURNAME_ASSIGNMENT_5). 5. (0 points ) We wold lie yo to write a Matlab fnction with signatre fnction [root] = root_finder(alpha) which, given a parameter α (Matlab alpha) sch that α 0, finds a root z (Matlab root) of a fnction f wiggly (z, α) sch that f (z wiggly ; α) = 0 and 0 z ; (6) the vale of z will of corse depend on the vale of the parameter α. Two comments: we have chosen f wiggly and the allowable instances of α to ensre that there will always be at

13 least one z which satisfies (6); we as yo only to find a root in the interval [0, ], not all roots in the interval [0, ]. Yor fnction mst be named root_finder and frthermore mst be stored in a file named root_finder.m. Yor fnction taes a single inpt: the vale of the parameter α (Matlab alpha); allowable instances mst satisfy α 0. Yor fnction yields a single otpt: a z (Matlab root) which satisfies a nmerical version of (6), f wiggly (z ; α) e- and 0 z. (7) We provide yo with the Matlab fnction f_wiggly.p with signatre fnction [f_vale] = f_wiggly(z,alpha) sch that f_wiggly(z,alpha) retrns the otpt f_vale = f wiggly (z; α). The Matlab fnction f_wiggly will accept an inpt M array z to yield otpt M array f_vale sch that f_vale(i) = f wiggly (z(i), α), i =,..., M; this cold prove convenient if yo wish (electively) to plot the z-dependence of f wiggly. Note however, that inpt alpha mst be a scalar. We as that yor fnction root_finder call the Matlab bilt-in nonlinear eqation solver fsolve. Please read the Appendix to familiarize yorself with this Matlab rotine. Three comments: althogh fsolve will tae care of all the details related to the soltion procedre, yo (in yor root_finder fnction) mst chec and ltimately ensre (7); for reasons explained in the Appendix, yor fnction root_finder will need to consider a seqence of initial gesses for fsolve to ensre that sccess a z which satisfies (7) is ltimately realized ; yo shold se the defalt vales for the varios fsolve tolerances. We provide yo with a script A5Q5_Template.m which yo shold not modify (bt as always, please remove the _Template before ploading to corse website). The deliverables for this qestion are the script A5Q5.m and most importantly yor fnction root_finder (and any other scripts or fnctions of yor own creation which are called by root_finder); please pload these deliverables to corse website in yor folder YOURNAME_ASSIGNMENT_5. Appendix: Matlab fsolve We consider the problem of finding a real root (or zero ) of a nivariate fnction: given a fnction f(z), a z b, we wish to find a real nmber z sch that f(z ) = 0; there cold be no (real) roots, one root, or many roots. (The ostensibly more general problem of solving a nonlinear eqation is in fact eqivalent to the problem of finding a root.) Althogh in this assignment we consider only nivariate fnctions, fsolve can readily treat the general mltivariate case. In the textboo we discss Newton iteration for root finding. The Matlab fnction fsolve implements an alternative, thogh often related, rote: application of optimization procedres to a nonlinear least-sqares re-formlation of the root-finding problem. It is simple to describe the framewor. If f(z ) = 0, then clearly f (z ) = 0, and frthermore since f (z) 0 for all z z is a minimizer of f(z). Hence to find a root of f(z) we can instead search for a minimizer of f (z); For each inpt instance of the parameter α, grade_o_matic shall test yor root_finder code 0 times (for this same vale of α), to mae sre that sccess is not a matter of good lc.

14 the latter is an easier problem than the former, as we can simply proceed downhill ntil we arrive at the minimm. There is one sbtlety, however: thogh a root of f(z) is necessarily a minimizer of f (z), a minimizer of f (z) is not necessarily a root of f(z). A simple example of the latter is f(z) = + z : z = 0 is clearly a minimizer of f (z), bt f(0) =, not zero, and hence z = 0 is not a root of f(z). The approach mst ths be formlated in two steps: in the first step we loo for a minimizer z of f (z); in the second step we evalate f (z ) and we accept z as a root in other words, we identify z = z only if f (z ) = 0 (or, in actal practice, sfficiently small). The syntax for fsolve is very simple: [zstarstar,fval,exitflag] = fsolve(@fnc, z_0) where zstarstar is the alleged root, fval is the vale of fnc at zstarstar, exitflag is an exit flag which provides information on the minimization process and alleged root, fnc is the Matlab embodiment of the fnction for which we see a root, and z_0 is the initial gess. In more detail, fnc is a ser-defined fnction with signatre fnction [fnc_vale] = fnc(z) which retrns fnc_vale = f(z) for given z. We note that, as is typical in the (iterative) soltion of nonlinear eqations, fsolve which effectively goes downhill to find a minimizer of f (z) (or the norm sqared of f in the case of a system of nonlinear eqations) will typically find the minimizer at the bottom of the valley to which the initial gess z 0 belongs. Ths, first, if the minimm of this valley does not constitte a root, then fsolve will not find a root, and second, in any event fsolve will only find the one root associated with the particlar valley implicitly selected by z 0. In actal practice, the initial gess often several initial gesses will be reqired plays an important role in the soltion of nonlinear eqations. Yo can visalize this reslt with the small script fsolve_demo_04.m inclded in the Assignment_5_Templates folder. As already indicated, the otpt zstarstar is z, a minimizer of f (z). In addition, fsolve indicates whether z corresponds to Eqation solved in the sense that f (z ) is zero or very small, or No soltion fond in the sense that f (z ) is far from zero. Yo can and shold also obtain this information yorself directly from fval (or fnc(zstarzstar)). In the Eqation solved case fval sfficiently close to zero we have fond a minimizer of f (z) which is indeed a root of f(z); in the No soltion fond case we have fond an extremm of f (z), typically thogh not always a minimm of f (z), which is not a zero of f(z) fval is not close to zero. 4

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