### Optimizing a Solution to a Modelling Problem

Added on - 13 Sep 2019

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U08079 Coursework 2017Optimizing a solution to a modelling problemFor the coursework you are required to design, implement and optimise aprogram to solve resistor networks.A basic example Jacobi program is provided but this program is inefficient andyou are required to improve it. You should make use of a range of codeoptimisation techniques and verify their effectiveness using benchmarking.You may need to create and solve some reasonably large resistor networks todemonstrate the effectiveness of some of the optimisations.In addition to optimising the program on a single PC, you are also required toimplement a version of your program on the department HAC cluster usingmessage passing with MPI in order to investigate possible speedup and otheroptimisations by employing multiple processes running on multipleprocessors.The ProblemA resistor is an electronic component that reduces the current and voltage ofan electrical signal. The diagram below shows a number of resistors formingan electronic circuit. The green boxes represent resistors and the linesrepresent wires (or nodes) connecting them together. Each resistor has avalue, in ohms, showing how powerful a resistor it is. More ohms means amore powerful resistor, which resists more, and thus lowers voltage further.The problem is to calculate the voltages at nodes v2 and v3. Note the leftnode is connected to a fixed 6 volt supply and the right node is connected tofixed 0 volts. There are two relevant electrical laws we can use to solve thisproblem:Kirchhoff’s Law:the net current at a node is zero, so current in must equalcurrent out at each circuit node. (Note thatcurrentis not the same asvoltage.)Ohm’s Law:If a resistor has a resistance of R ohms, and the voltage acrossthe resistor (i.e. the difference between the input and output voltages) is Vvolts, then the current flowing through the resistor is I=V/R amps.You can also utilise the fact that current flows from points of high voltage topoints of low voltage.C Cox 21.2.201716336vV2V30v

Using these laws, we can derive a system of equations which we can thensolve to give the voltages v2 and v3.Hand calculation for the above example:Node v2 has an input from the 1 ohm resistor.By Ohm’s law the current in from this resistor is (6-v2)/1 (6 is the inputvoltage, v2 is the node voltage which we want to find, and 1 is the ohm ratingof the resistor.)Node V2 outputs to the 6 ohm and the 3 ohm resistor.By Ohm’s law the current out to the 6 ohm resistor is (v2-v3)/6,and the current out to the 3 ohm resistor is (v2-v3)/3.By Kirchhoff’s law the current at the inputs and outputs of V2 must be thesame, so (6-v2)/1 = (v2-v3)/6 + (v2-v3)/3At node v3 we consider the inputs and outputs similarly:The two input resistors will have currents of (v2-v3)/6 and (v2-v3)/3and the output resistor will have a current of (v3-0)/3.So (v2-v3)/6 + (v2-v3)/3 = (v3-0)/3.Now we have two equations to solve simultaneously:(6-v2)/1 = (v2-v3)/6 + (v2-v3)/3(v2-v3)/6 + (v2-v3)/3 = (v3-0)/3.Simplifying the first equation:Start with(6-v2)/1 = (v2-v3)/6 + (v2-v3)/3Dividing by one does nothing(6-v2) = (v2-v3)/6 + (v2-v3)/3Multiply all by 6 to remove /66(6-v2) = (v2-v3) + 2(v2-v3)Carry out multiplications36 – 6v2 = v2-v3 + 2v2 – 2v3Collect terms36 – 6v2 = 3v2 – 3v3Add 6v2 to both sides, to move it36 = 9v2 – 3v3Divide through by 3 to lower numbers12 = 3v2 – v3Same with the second equation:Start with(v2-v3)/6 + (v2-v3)/3 = (v3-0)/3Subtracting 0 does nothing(v2-v3)/6 + (v2-v3)/3 = v3/3Multiply all by 6 to remove /6(v2-v3) + 2(v2-v3) = 2v3Carry out multiplicationsv2-v3 + 2v2 – 2v3 = 2v3Collect terms3v2 – 3v3 = 2v3Subtract 2v3 from both sides3v2 – 5v3 = 0We now have two equations of the form:3v2 – 1v3 = 123v2 – 5v3 = 0Which we can easily solve as a simultaneous equation as v2=5 and v3=3Notice that for a general resister network:We will end up with one equation per node;Each of these equations will have one v term for each node voltage.C Cox 21.2.2017

So if we had 4 nodes, we would obtain four equations, each of the form?v2 + ?v3 + ?v4 + ?v5 = ?.This would be much more difficult to solve manually, hence the need for theJacobi algorithm.Thus the complexity of solving the equations increases with each extra nodein the system, which is why this problem requires optimization.The Jacobi algorithm can be used to solve sets of equations of this type. Touse it we first represent the equations as a matrix multiplication, actually tomultiply an nxn matrix by a vector of size n. In our example n=2:012531332vvNote that the elements of the matrix correspond to the coefficients in theequations.3v2 – 1v3 = 123v2 – 5v3 = 0The Jacobi algorithm is based on guessing an initial solution (all nodes set to1 volt in the example program) and then refining it over a number of iterations.Here is the C source code for a simple implementation. Note the coefficientsfrom the above equations as values of matrix m and vector b:Note we use a 2-D n x n array for the matrix m and a 1-D array size n for thevector b.#include <stdio.h>// Number of nodes#define n 2int main(){ // Matrix of coefficientsconst double m[n][n]={{3.0,-1.0}, {3.0,-5.0}};// Vector of resultsconst double b[n]={12.0,0.0};// Arrays used for storing guessesdouble x[n], new_x[n];double sum;// How many iterations to do – the more the better the answerconst int MaxIts=30;int Its,i,j;// Start with all guesses at 1for(i=0;i<n;i++) x[i]=1.0;// Perform MaxIts iterationsfor(Its=1;Its<=MaxIts;Its++){ // Print out the iteration we are onprintf("Its=%d",Its);// This is the actual Jacobi algorithm// Calculate our next round of guesses in max_i[]for(i=0; i<n; i++){ sum=0.0;for(j=0;j<n;j++)C Cox 21.2.2017

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