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LES10A230 Engineering Mathematics IV

Project Assignment / Spring 2023

Assignment. (Grade:1-5) Look more deeply lectures delivered by Jukka Paatero. Also summarize how the application of this method has since evolved. Then, demonstrate the use of the method with reflections on the use of numerical methods, as described below. As your main task, you seek the minimum for the function below:

You are given personal values for the constants , , and . Also, you are given personal initial values  to begin your iteration.

Expected steps in solution:

1. Identify the minimum for the function using the vectoral application of Newton’s method while applying dynamic . Thus, you need to optimize  for each iteration step to improve performance and avoid overshoot. Use iterations to find the solution with the accuracy of five significant numbers.  

2. Identify the minimum again using the Newton’s method with dynamic . However, use this time numerical derivatives instead of . When using numerical derivatives, only one of the constants is being varied as with partial derivatives. Apply in this case the forward numerical derivative, . Here  equals some very small number. For each step , solve first the  and optimal  using the condition . When taking the derivative of , please remember to consider the inner derivatives for each of the coordinate axes that results as dot product with the main function. In this work it is enough that only the second term in the dot product is analyzed using numerical derivatives. Thus, the function takes the form .

3. Compare the results obtained using the numerical method with the use of accurate . Discuss at least the following:

- What values of  allow you to find an equally accurate solution as with the accurate method. Report here the values you have tested for  (for example ).

- How does the value of  influence the needed number of iterations in the Newton’s Method to reach the same minimum with the same number of significant numbers as when using the accurate ? Discuss and demonstrate this with examples.

Please submit the details of your analysis as a supplement to your report, in particular for the steps 1 & 2. You should carry out your analysis using a tool, where the logic of your solution can be followed in detail. In practice, tools like spreadsheet calculation environment (like Microsoft Excel) or scripting environment (like Matlab) are excellent for this purpose. When using such tools, please document the optimization routine in detail so that it can be understood line by line.

Please note that built-in ready optimization routines are not allowed to be used in this assignment. Instead, you should write any applied optimization scripts and/or calculations yourself. While you add full details of your analysis as a supplement, please discuss the main points of your analysis as part of your report.

Discuss challenges and lessons learned concerning the project. Discuss any difficulties you experienced with the project, and personal reflection about the project.

Instructions: Please use PDF-format for the report itself and the related reflection. If you want to submit some hand-written graphs as part of your submission, please scan them or take well readable photos of the content and include it into your report / submission. Please consider the overall submission file size and try to keep your graphics in reasonable accuracies and sizes.

This assignment is due 30.04.2023 while earlier submissions are also welcomed. This assignment is completed individually.

Max page limit 6 pages, minimum 4 pages.

Submit the assignment in a single .pdf format