📗 Enter your ID (the wisc email ID without @wisc.edu) here: and click (or hit enter key) 1,2,3,4,5,6,7,8,9,10p7
📗 The due date is May 4, late submissions within a week will be accepted without penalty, but please submit a regrade request form: Link.
📗 The same ID should generate the same set of questions. Your answers are not saved when you refresh the page or close the browser.
📗 To input your answers, you could (1) copy and paste your answers to the text boxes for individual questions, or (2) print your answers to text files and load them into the text boxes for individual questions, or (3) print all your outputs to a single text file and load it using the button at the bottom of the page under the Submission section. More details at the bottom of page W1.
📗 Please report any bugs on Piazza.
📗 Use Newton's method and secant method to find the solution of the equation \(f\left(x\right)\) = ???. Use the initial guess and the stopping condition specified in each question. Enter the sequence of \(x\) as a vector in the text box below each question.
📗 Note: if \(f'\left(x\right) = 0\) (for Newton's method) or \(f\left(x\right) = f\left(x'\right)\) (for Secant method) at some point in the sequence, return the same \(x\) in the next iteration.
📗 Note: you can compute the derivative by hand or use the polyder (or diff) method, see Doc. You can modify Newton's formula and use numerical derivatives too, but it is not recommended for this question.
📗 Aside: polynomials with degree five or more do not have analytical solutions in general so they have to be solved using numerical methods. Details see: Wikipedia.
📗 [1 point] Create the sequence of \(x\) using Newton's method with initial guess ???, and stop after ??? iterations. There should be ??? numbers in the vector including the initial guess.
📗 Answer:
📗 [1 point] Create the sequence of \(x\) using Newton's method with initial guess ???, and stop after ??? iterations. There should be ??? numbers in the vector including the initial guess.
📗 Answer:
📗 [1 point] Create the sequence of \(x\) using Newton's method with initial guess ???, and stop when the value of the expression is less than ??? (or \(f'\left(x\right) = 0\) or \(f\left(x\right) > 10^{7}\)). The last \(x\) should have \(f\left(x\right)\) strictly less than , and include the initial guess in the vector.
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📗 [1 point] Create the sequence of \(x\) using secant method with initial guess ???, and stop after ??? iterations. There should be ??? numbers in the vector including the two initial guesses.
📗 Answer:
📗 [1 point] Create the sequence of \(x\) using secant method with initial guess ???, and stop after ??? iterations. There should be ??? numbers in the vector including the two initial guesses.
📗 Answer:
📗 [1 point] Create the sequence of \(x\) using secant method with initial guess ???, and stop when the value of the expression is less than ??? (or \(f\left(x\right) = f\left(x'\right)\) or \(f\left(x\right) > 10^{7}\)). The last \(x\) should have \(f\left(x\right)\) strictly less than , and include the two initial guesses in the vector.
📗 Answer:
📗 Approximate the area under the curve \(\varphi\left(x\right)\) = \(\dfrac{1}{\sigma \sqrt{2 \pi}} e^{- \dfrac{1}{2} \left(\dfrac{x - \mu}{\sigma}\right)^{2}}\), with \(\mu\) = ??? and \(\sigma\) = ???, between \(x = a\) = ??? and \(x = b\) = ??? using the midpoint rule. Create a matrix with 2 rows, the first row contains the list of sampled \(x\) values, and the second row contains the approximated area between the current point and the previous point. (The first number in the second row should be 0.) Use the discretization specified in each question. Enter the matrix you created in the text box below each question.
📗 Aside: the area represent the probability that a Gaussian random variable with mean \(\mu\) and standard deviation \(\sigma\) is between \(x = a\) and \(x = b\). The error function (erf) can be used to compute the same quantity. Details see: Wikipedia.
📗 [1 point] Create the matrix by sampling ??? evenly spaced points. This means the only two points are \(a\) and \(b\). The matrix should look like \(\begin{bmatrix} a & b \\ 0 & v \end{bmatrix}\) where \(v\) is the approximate area under the curve using only function value at the midpoint.
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📗 [1 point] Create the matrix by sampling ??? random points (please do not use the same sample as Question 8). You can use the sort function on a list of random numbers to put them in order, see Doc.
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📗 [1 point] Please enter any comments and suggestions including possible mistakes and bugs with your version of the questions and the auto-grading, and materials relevant to solving the question that you think are not covered well during the lectures. If you have no comments, please enter "None": do not leave it blank.
📗 Answer: .
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📗 You could submit multiple times (but please do not submit too often): the submission with the highest grade will be counted.
📗 Please also save the text in the above text box to a file using the button or copy and paste it into a file yourself .
📗 You could also load your answers from the text (or txt file) in the text box below using the button . The first two lines should be "##p: 7" and "##id: your id", and the format of the remaining lines should be "##1: your answer to question 1" newline "##2: your answer to question 2", etc. Please make sure that your answers are loaded correctly before submitting them.
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📗 Please submit your code (.m, .txt, or .pdf are all acceptable) on Canvas Assignment P7.
📗 Your coding style and technique will not be graded, but please consider the following:
(1) Add comments to describe what the code is doing.
(2) Indent "for" loops and "if" blocks.
(3) Use descriptive but short variable names.
(4) Add white spaces and/or blank lines for readability.
📗 It is highly recommended that you use the commands and functions from the lectures, but you are allowed to use any built-in MATLAB commands and functions.
📗 Example solution: .m File, .txt File.pdf File.
(1) Please do not use the example solution as the starter code. You should write all the code yourself and only use the example solution as a reference when you get stuck.
(2) Please do not use the example solution as a guide for coding style. It is written so that it only solves a specific version of the problems. You code should be more general: you should define variables for constants, define functions for repetitive actions, and use descriptive variable names.
📗 Please report possible mistakes in the solution, and if you have better (shorter or more efficient) ways of solving the same problem, you can share your code on Piazza (but please do so after the due dates).
📗 If you use one or more lines of code from the example solution, other students in the class, or code you found on the Internet, you must give attribution by putting a comment at the beginning of your code submission, for example:
% Code attribution: P7 example solution.
% Code attribution: (student name)'s P7 solution.
% Code attribution: (student name)'s answer on Piazza: (link to Piazza post).
% Code attribution: (person name)'s answer on Stack Overflow: (link to page).