MODELING OF BIOMEDICAL SYSTEMS
MODELING OF BIOMEDICAL SYSTEMS BMED 2200
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This 20 page Class Notes was uploaded by Joshuah Labadie on Monday October 19, 2015. The Class Notes belongs to BMED 2200 at Rensselaer Polytechnic Institute taught by Staff in Fall. Since its upload, it has received 91 views. For similar materials see /class/224826/bmed-2200-rensselaer-polytechnic-institute in Biomedical Engineering at Rensselaer Polytechnic Institute.
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Date Created: 10/19/15
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implementation of the Gauss algorithm to solve 1 2 3 4 x1 11 2 1 3 1 x2 11 4 1 5 5 x3 13 5 3 1 1 x4 4 Problem 2 Jacobi Method You are given the following system of linear equations 8 0 1 x1 1 2 9 0 x2 2 0 2 7 KxS 3 a Using the Jacobi method what are P and Q b Perform the first three iterations for the Jacobi method Use x0 1 1 1T as the starting point Feel free to use MATLAB for this problem or perform the solution by handcalculator However show your work for the three steps I A Problem 3 Solution of a 39 You are given the following nonlinear equation x lnx O for a b and c below perform iterations until the change from one iteration to the next has an absolute value of less than 001 a Use x0 1 and the method of successive substitutions to solve this problem Show the results for each iteration b Use x1 001 and x2 1and the linear interpolation method to solve this problem Show the results for each iteration c Use x0 1 and the NewtonRaphson method to solve this problem Showthe results for each iteration d List the number of iterations required to meet the desired accuracy for each of the three methods Which one requires the most and which one the fewest iterations Use MATLAB for this problem is encouraged but not required MW S AN im a g iwg ai a S E9 gig 5 gL s 39gg IgEg mggffa 5 Kl R2734 K3 f 31gt Ea Wigs E C UA5 x 5 Ba n m kwwieg ah W a ag gig ES 5 SJgtgt51 a ac M ba alCS iC 60 3 r W Mal 561 kgi z i G 3 7t 1 7 was Less it and 36 git CD Expm LE5 m wms 05 awaw Vcw gdpia5 ism ragga 5 M scvw m3 6 M le a C6633 i522 as kuwa K E1ltEET1CEQgt Eq gawk Skim 113 KLESl 131 3432 SEQ m 33 24 SEgtE 33 M 33 ESM i N N r 2 r v ac wigG W W5 a r EW MamgE aM L5x ea S w F594 w fgus a airs364 Va s g Ugim 5355 393 N THE v wax an 252 just bag kwa m Lfm 7353 lt3 kzg ii kg 2 9 w gig KM k2 Elia Kw 5 ll 7 9 6433 ESE y k egj K M ng RC5 Z 9 Es w Kng Wka g 5ng E1 3 E1quot k V lt 7 53 0 quot i mai gf K ng f chgg KECK WgJ W 9 Smi Na gig if r SS 394 346 2 Pm w U Mu Giana i x l 53 an pubg fu ag ampP ampS quotTQM Kgi ll 3quot K3 Kai 512g git ksi k CSElt 2 lt3 I M 5511 k 39 2 o I 5 1 4 K 3 r ga ygef WW iii 7 gm assigl 7 39 Aim 39 w wig t 9 g g g g 39 A if Thomas Finn Modeling of Biomedical Systems Homework 3 Problem 1 Due February 22nd A l 2 3 4 2 l 3 l 4 l 5 5 5 3 l l b lllll34 We39re modifying each row individually We can do this just by modifying A but its easier to read we make each line a Variable Al bl lrcluding b because those need to be modifed along with the rest of the line 2 RowThreeA3 b RowFourA4 b4 Zeroing RowTwo I take the current RowTwo and subtract RowOne multiplied by the first number in RowTwo over the first nunber in the row one In this case 2 over 1 RowTwoRowTwo RowOneRowTwolRowOnel Result 0 3 3 7 If the first number began as O tnen RowTwo RowOne O nappers ARowOne RowTwo RowThree RowFourReporting intermediate matrix Zeroing RowThree RowThreeRowThree RowOneRowThreelRowOnel RowThreeRowThree RowTwoRowThree2RowTwo2 ARowOne RowTwo RowThree RowFour RowFourRowFour RowOneRowFourlRowOnel RowFourRowFour RowTwoRowFour2RowTwo2 Normally RowFourRowFour RowThreeRowFour3RowThree3 We can39t make the third zero in RowFour because RowThree already has a zero there and can39t modify that spot This isn39t a big deal we just switch the two rows and get an upper triangular matrix just the same o olt o ARowOne RowTwo RowFour RowThree bA5 Final b AAl4 Final A wih b column removed v4b4A44 Solve for v4 V3b3 A3 4v4A33 Back substitute v2b2 A23v3 A24V4A22 vlbl Al2v2 Al3v3 Al4v4All xvlv2v3v4 Final Answer You can test this with the orignal A t x original b Note if you do Final A X you won39t get original b you39ll get tLe final modified b Result A 1 2 3 4 11 0 3 3 7 11 4 1 5 5 13 5 3 1 1 4 10000 20000 30000 40000 110000 0 30000 30000 70000 110000 0 0 0 53333 53333 50000 30000 10000 10000 40000 110000 110000 253333 53333 10000 20000 30000 40000 0 30000 30000 70000 0 0 70000 26667 0 0 0 53333 10000 20000 40000 10000 33 0sz FX z w my ML Z x w R ML 7X m M m 3 gx WQX39 X m 30 I 2le 3 I ya xx JNQXKZI 32 5 A Modeling in Biomedical Systems Homework 3 Problem 3 Linear Interpolation Code Thomas Finn Modeling of Biomedical Systems Homework 3 Problem 3 Linear Interpolation xlOl x2l x32 Just Lastlteration0 nl Counter optional while absx3 LastlterationgtOl fllogxlxl f2logx2x2 fractionflx2 xl f2 fl x3xl fraction Will snow results of iteration late picking a number to start while loop H Lastlterationx2 XZSign x2gt0 X3Sign x3gt0 if XZSign 7 X3Sign Are x2 x2x3 If yes r l nd x3 same sign else xlx3 lf no replace xl end Iterationsnx3 nnl if ngt20 Stops infinite loops in case of divergence n Records results of iteration pause end end n Iterations disp39Final x339 x3 RESULTS OF ITERATION n Iterations 08231 07253 06675 06318 06093 05948 05854 Final x3 Homework 3 Problem 3 Newton s Method Thomas Finn Modeling of Biomedical Systems Homework 3 Problem 3 Newton Raphson method Newton39s Method There are a few ways to solve this here is mine x3l The initial guess this will over write x0 XO9999 Just a random number to start the While loop nl Counter variable to see how many iterations while absx3 XOgtOl x0x3 f logx0 x0 The function fDerivative lXO l The function39s derivative x3 x0 ffDerivative Newton Raphson Method nnl if ngtlOO Stops infinite loop incase diverge n pause end end n disp39Final x339 x3 RESULTS x0 4 Final x3 Homew ork 1 Problem 1 k E S Es gtEI gtE P as kmetms Problem 2 used n Ha Hn P Pr eany Here 3 uxes are assumed m be pusmve System boundary a Derwe me dynamu mude desmbmg ms sys tem b What sthe simcmumemc mamx N7 e b2 and b3 uwer a sumtmnsfurthe uxes an yuu have n hat my mum ms change the magram Shawn abuve7 Problem 3 has the ququg stumhmmem mater r 0 0 D 0 0 fl 0 D I 1 1 1 o 0 o 0 I D 0 O l 71 7 l 71 0 0 0 0 o a o 1 71 o n o 71 u 0 0 0 0 0 l U 0 0 fl 4 1 1 s m resmtputenuany currem v nut than vmatvvuu d be a pusswb e resum Homework 2 Problem 1 Solution of Svstem of Linear quot 39 39 quot Consider the following system A gt B a V1 V2 b1 V3 b2 Assume that the fluxes entering and leaving the system are known and given by b1 2 and b2 1 Compute v1 v2 and v3 using the Gauss algorithm discussed in class by hand Show the intermediate steps Problem 2 39 39 39 of Gauss alqorithm Implement the Gauss algorithm in MATLAB Make sure that the algorithm will output some of the intermediate results Include a printout of the algorithm Problem 3 Using Gauss algorithm for Solution of Linear Algebraic Eguations a Use the MATLAB implementation of the Gauss algorithm to solve Problem 1 b Repeat the procedure in MATLAB only for b1 4 and b2 3 Problem 4 Dynamics and Steady State What would happen to the dynamic model if the fluxes entering and leaving the system would be given by b1 1 b2 1 and v1 1 Le all inputs and outputs are fixed at these values Homework 4 Problem 1 Derivation of central difference Derive the central difference approximation of the first derivative of yi with an error of magnitude h2 Yi39 Yio5 Yio5h 0012 Problem 2 Solution of ODE Consider the following ODE y with initial condition y0 5 a Discretize the equation using a forward difference scheme with step size of magnitude h Write the equation and the initial condition in discretized form b Solve the discretized equation for h 01 from t 0 until t 2 Present all intermediate steps yo y1 y2 etc Using MATLAB is strongly encouraged c Plot the solution of the ODE derived from analytically solving the equation and the solution derived from part b in the same plot Problem 3 Solution of nonlinear equation The following empirical relationship is available for the light propagation through biological tissues 44401431 071914 A1 Where A is the internal reflectance factor for a tissue and nrel is the ratio of the refractive indices of the tissue and the medium When nrel equals 1 A reduces to unity Find the value of nrel that results in a reflectance factor of 4 Use a method of your choice from among the ones covered in this class Illustrate your technique by showing intermediate steps 0688 0063614