`); let searchUrl = `/search/`; history.forEach((elem) => { prevsearch.find('#prevsearch-options').append(`
${elem} `); }); } $('#search-pretype-options').empty(); $('#search-pretype-options').append(prevsearch); let prevbooks = $(false); [ {title:"Recently Opened Textbooks", books:previous_books}, {title:"Recommended Textbooks", books:recommended_books} ].forEach((book_segment) => { if (Array.isArray(book_segment.books) && book_segment.books.length>0 && nsegments<2) { nsegments+=1; prevbooks = $(`
${book_segment.title} `); let searchUrl = "/books/xxx/"; book_segment.books.forEach((elem) => { prevbooks.find('#prevbooks-options'+nsegments.toString()).append(`
${elem.title} ${ordinal(elem.edition)} ${elem.author} `); }); } $('#search-pretype-options').append(prevbooks); }); } function anon_pretype() { let prebooks = null; try { prebooks = JSON.parse(localStorage.getItem('PRETYPE_BOOKS_ANON')); }catch(e) {} if ('previous_books' in prebooks && 'recommended_books' in prebooks) { previous_books = prebooks.previous_books; recommended_books = prebooks.recommended_books; if (typeof PREVBOOKS !== 'undefined' && Array.isArray(PREVBOOKS)) { new_prevbooks = PREVBOOKS; previous_books.forEach(elem => { for (let i = 0; i < new_prevbooks.length; i++) { if (elem.id == new_prevbooks[i].id) { return; } } new_prevbooks.push(elem); }); new_prevbooks = new_prevbooks.slice(0,3); previous_books = new_prevbooks; } if (typeof RECBOOKS !== 'undefined' && Array.isArray(RECBOOKS)) { new_recbooks = RECBOOKS; for (let j = 0; j < new_recbooks.length; j++) { new_recbooks[j].viewed_at = new Date(); } let insert = true; for (let i=0; i < recommended_books.length; i++){ for (let j = 0; j < new_recbooks.length; j++) { if (recommended_books[i].id == new_recbooks[j].id) { insert = false; } } if (insert){ new_recbooks.push(recommended_books[i]); } } new_recbooks.sort((a,b)=>{ adate = new Date(2000, 0, 1); bdate = new Date(2000, 0, 1); if ('viewed_at' in a) {adate = new Date(a.viewed_at);} if ('viewed_at' in b) {bdate = new Date(b.viewed_at);} // 100000000: instead of just erasing the suggestions from previous week, // we just move them to the back of the queue acurweek = ((new Date()).getDate()-adate.getDate()>7)?0:100000000; bcurweek = ((new Date()).getDate()-bdate.getDate()>7)?0:100000000; aviews = 0; bviews = 0; if ('views' in a) {aviews = acurweek+a.views;} if ('views' in b) {bviews = bcurweek+b.views;} return bviews - aviews; }); new_recbooks = new_recbooks.slice(0,3); recommended_books = new_recbooks; } localStorage.setItem('PRETYPE_BOOKS_ANON', JSON.stringify({ previous_books: previous_books, recommended_books: recommended_books })); build_popup(); } } var whiletyping_search_object = null; var whiletyping_search = { books: [], curriculum: [], topics: [] } var single_whiletyping_ajax_promise = null; var whiletyping_database_initial_burst = 0; //number of consecutive calls, after 3 we start the 1 per 5 min calls function get_whiletyping_database() { //gets the database from the server. // 1. by validating against a local database value we confirm that the framework is working and // reduce the ammount of continuous calls produced by errors to 1 per 5 minutes. return localforage.getItem('whiletyping_last_attempt').then(function(value) { if ( value==null || (new Date()) - (new Date(value)) > 1000*60*5 || (whiletyping_database_initial_burst < 3) ) { localforage.setItem('whiletyping_last_attempt', (new Date()).getTime()); // 2. Make an ajax call to the server and get the search database. let databaseUrl = `/search/whiletype_database/`; let resp = single_whiletyping_ajax_promise; if (resp === null) { whiletyping_database_initial_burst = whiletyping_database_initial_burst + 1; single_whiletyping_ajax_promise = resp = new Promise((resolve, reject) => { $.ajax({ url: databaseUrl, type: 'POST', data:{csrfmiddlewaretoken: "D6w3vOt4YZGqIlUK1lHn6JUtDavY5vYh7xn6NBa0zGBWgebdewSqxe4NZtMLzlTu"}, success: function (data) { // 3. verify that the elements of the database exist and are arrays if ( ('books' in data) && ('curriculum' in data) && ('topics' in data) && Array.isArray(data.books) && Array.isArray(data.curriculum) && Array.isArray(data.topics)) { localforage.setItem('whiletyping_last_success', (new Date()).getTime()); localforage.setItem('whiletyping_database', data); resolve(data); } }, error: function (error) { console.log(error); resolve(null); }, complete: function (data) { single_whiletyping_ajax_promise = null; } }) }); } return resp; } return Promise.resolve(null); }).catch(function(err) { console.log(err); return Promise.resolve(null); }); } function get_whiletyping_search_object() { // gets the fuse objects that will be in charge of the search if (whiletyping_search_object){ return Promise.resolve(whiletyping_search_object); } database_promise = localforage.getItem('whiletyping_database').then(function(database) { return localforage.getItem('whiletyping_last_success').then(function(last_success) { if (database==null || (new Date()) - (new Date(last_success)) > 1000*60*60*24*30 || (new Date('2023-04-25T00:00:00')) - (new Date(last_success)) > 0) { // New database update return get_whiletyping_database().then(function(new_database) { if (new_database) { database = new_database; } return database; }); } else { return Promise.resolve(database); } }); }); return database_promise.then(function(database) { if (database) { const options = { isCaseSensitive: false, includeScore: true, shouldSort: true, // includeMatches: false, // findAllMatches: false, // minMatchCharLength: 1, // location: 0, threshold: 0.2, // distance: 100, // useExtendedSearch: false, ignoreLocation: true, // ignoreFieldNorm: false, // fieldNormWeight: 1, keys: [ "title" ] }; let curriculum_index={}; let topics_index={}; database.curriculum.forEach(c => curriculum_index[c.id]=c); database.topics.forEach(t => topics_index[t.id]=t); for (j=0; j
Solutions
Textbooks
`); } function build_solutions() { if (Array.isArray(solution_search_result)) { const viewAllHTML = userSubscribed ? `View All` : ''; var solutions_section = $(` Solutions ${viewAllHTML} `); let questionUrl = "/questions/xxx/"; let askUrl = "/ask/question/xxx/"; solution_search_result.forEach((elem) => { let url = ('course' in elem)?askUrl:questionUrl; let solution_type = ('course' in elem)?'ask':'question'; let subtitle = ('course' in elem)?(elem.course??""):(elem.book ?? "")+" "+(elem.chapter?"Chapter "+elem.chapter:""); solutions_section.find('#whiletyping-solutions').append(` ${elem.text} ${subtitle} `); }); $('#search-solution-options').empty(); if (Array.isArray(solution_search_result) && solution_search_result.length>0){ $('#search-solution-options').append(solutions_section); } MathJax.typesetPromise([document.getElementById('search-solution-options')]); } } function build_textbooks() { $('#search-pretype-options').empty(); $('#search-pretype-options').append($('#search-solution-options').html()); if (Array.isArray(textbook_search_result)) { var books_section = $(` Textbooks View All `); let searchUrl = "/books/xxx/"; textbook_search_result.forEach((elem) => { books_section.find('#whiletyping-books').append(` ${elem.title} ${ordinal(elem.edition)} ${elem.author} `); }); } if (Array.isArray(textbook_search_result) && textbook_search_result.length>0){ $('#search-pretype-options').append(books_section); } } function build_popup(first_time = false) { if ($('#search-text').val()=='') { build_pretype(); } else { solution_and_textbook_search(); } } var search_text_out = true; var search_popup_out = true; const is_login = false; const user_hash = null; function pretype_setup() { $('#search-text').focusin(function() { $('#search-popup').addClass('show'); resize_popup(); search_text_out = false; }); $( window ).resize(function() { resize_popup(); }); $('#search-text').focusout(() => { search_text_out = true; if (search_text_out && search_popup_out) { $('#search-popup').removeClass('show'); } }); $('#search-popup').mouseenter(() => { search_popup_out = false; }); $('#search-popup').mouseleave(() => { search_popup_out = true; if (search_text_out && search_popup_out) { $('#search-popup').removeClass('show'); } }); $('#search-text').on("keyup", delay(() => { build_popup(); }, 200)); build_popup(true); let prevbookUrl = `/search/pretype_books/`; let prebooks = null; try { prebooks = JSON.parse(localStorage.getItem('PRETYPE_BOOKS_'+(is_login?user_hash:'ANON'))); }catch(e) {} if (prebooks && 'previous_books' in prebooks && 'recommended_books' in prebooks) { if (is_login) { previous_books = prebooks.previous_books; recommended_books = prebooks.recommended_books; if (prebooks.time && new Date().getTime()-prebooks.time<1000*60*60*6) { build_popup(); return; } } else { anon_pretype(); return; } } $.ajax({ url: prevbookUrl, method: 'POST', data:{csrfmiddlewaretoken: "D6w3vOt4YZGqIlUK1lHn6JUtDavY5vYh7xn6NBa0zGBWgebdewSqxe4NZtMLzlTu"}, success: function(response){ previous_books = response.previous_books; recommended_books = response.recommended_books; if (is_login) { localStorage.setItem('PRETYPE_BOOKS_'+user_hash, JSON.stringify({ previous_books: previous_books, recommended_books: recommended_books, time: new Date().getTime() })); } build_popup(); }, error: function(response){ console.log(response); } }); } $( document ).ready(pretype_setup); $( document ).ready(function(){ $('#search-popup').on('click', '.search-view-item', function(e) { e.preventDefault(); let autoCompleteSearchViewUrl = `/search/autocomplete_search_view/`; let objectUrl = $(this).attr('href'); let selectedId = $(this).data('objid'); let searchResults = []; $("#whiletyping-solutions").find("a").each(function() { let is_selected = selectedId === $(this).data('objid'); searchResults.push({ objectId: $(this).data('objid'), contentType: $(this).data('contenttype'), category: $(this).data('category'), selected: is_selected }); }); $("#whiletyping-books").find("a").each(function() { let is_selected = selectedId === $(this).data('objid'); searchResults.push({ objectId: $(this).data('objid'), contentType: $(this).data('contenttype'), category: $(this).data('category'), selected: is_selected }); }); $.ajax({ url: autoCompleteSearchViewUrl, method: 'POST', data:{ csrfmiddlewaretoken: "D6w3vOt4YZGqIlUK1lHn6JUtDavY5vYh7xn6NBa0zGBWgebdewSqxe4NZtMLzlTu", query: $('#search-text').val(), searchObjects: JSON.stringify(searchResults) }, dataType: 'json', complete: function(data){ window.location.href = objectUrl; } }); }); });
FAQs
Computation of Selected Eigenvalues and Eigenvectors
If a real matrix has a simple eigenvalue of largest magnitude, the sequence x k = A x k – 1 converges to the eigenvector corresponding to the largest eigenvalue, where x0 is a normalized initial approximation, and all subsequent xk are normalized.
How to find eigenvalues by power method? ›
To apply the Power Method to a square matrix A, begin with an initial guess u 0 for the eigenvector of the dominant eigenvalue. Then, for i ⩾ 1 , calculate u i = Au i − 1 / | | Au i − 1 | | , until consecutive vectors u i are either identical or opposite.
What do you mean by power method? ›
The Power Method is a simple method that under mild conditions yields a vector corresponding to the eigenvalue that is largest in magnitude. Throughout this section we will assume that a given matrix A ∈ Cm×m is nondeficient: there exists a. nonsingular matrix X and diagonal matrix Λ such that A = XΛX.
What is the power method for a symmetric matrix? ›
The symmetric Power method
When the matrix A is symmetric, the error term can be reduced further with the help of the fact that ‖Ax‖22=xTATAx=xA2x. For the symmetric Power method we perform the following: Generate a random vector x, set x=x/‖x‖2. Set y=Ax.
What algorithm finds the largest eigenvalue? ›
Power iteration finds the largest eigenvalue in absolute value, so even when λ is only an approximate eigenvalue, power iteration is unlikely to find it a second time. Conversely, inverse iteration based methods find the lowest eigenvalue, so μ is chosen well away from λ and hopefully closer to some other eigenvalue.
How do you find the largest value in a matrix? ›
Description. M = max( A ) returns the maximum elements of an array. If A is a vector, then max(A) returns the maximum of A . If A is a matrix, then max(A) is a row vector containing the maximum value of each column of A .
How to find the largest eigenvalue of a matrix in Python? ›
Find the largest eigenvalue
In this case, we can use the power method - a iterative method that will converge to the largest eigenvalue. Let's see the following how the power method works. Consider an n×n matrix A that has n linearly independent real eigenvalues λ1,λ2,…,λn and the corresponding eigenvectors v1,v2,…,vn.
How to find dominant eigenvalue of matrix? ›
We use the inverse power method with an LU decomposition of A − cI to calculate the dominant eigenvalue λ and its corresponding eigenvector x for (A − cI)−1. Then 1/λ and x are an eigenvalue and eigenvector for A − cI, while 1/λ + c and x are an eigenvalue and eigenvector for A.
What is the power iteration method in PCA? ›
The power iteration method is a simple and intuitive algorithm used to find the dominant eigenvector and eigenvalue of a square matrix. It's particularly useful in the context of data science and machine learning when dealing with large, high-dimensional datasets, if only the first few Eigenvectors are needed.
Does power method work for complex eigenvalues? ›
Example: Complex eigenvalue
and the power method will not work if the matrix has dominant eigenvalues which are complex conjugate to each other and our starting guess has all real entries. When we change our starting guess to the vectors which have complex entries, the power method should work as usual.
Definition: Power of a Matrix
If 𝐴 is a square matrix and 𝑘 is a positive integer, the 𝑘 t h power of 𝐴 is given by 𝐴 = 𝐴 × 𝐴 × ⋯ × 𝐴 , where there are 𝑘 copies of matrix 𝐴 .
How do you solve a power matrix? ›
For a square matrix 𝐴 and positive integr 𝑘, the 𝑘th power of 𝐴 is defined by multiplying this matrix by itself repeatedly. That is, 𝐴 to the 𝑘th power is equal to 𝐴 multiplied by 𝐴 multiplied by 𝐴, and so on, multiplied by 𝐴, where there are 𝑘 instances of the matrix 𝐴.
What is the largest eigenvalue distribution? ›
The known exact distribution of the largest eigenvalue in the null case follows from a more general result of Constantine (1963) and is expressed in terms of hyper- geometric function of a matrix argument. This hypergeometric function represents a zonal polynomial series.
What is a large eigenvalue? ›
The largest eigenvalue (in absolute value) of a normal matrix is equal to its operator norm. So, for instance, if A is a square matrix with largest eigenvalue λmax, and x is a vector, you know that ‖Ax‖≤|λmax|‖x‖, and this is sharp (here ‖⋅‖ is the usual Euclidean norm).
How do you find the second largest eigenvalue? ›
The traditional way to obtain this information is to subtract the contribution of the largest eigenvalue from the matrix, followed by an estimate of the largest eigenvalue of the remaining matrix.
![Use the power method algorithm to compute the eigenvalue with the largest magnitude of the following matrix: A=[
2 3
3 -4
] | Numerade (2024)](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mP8/h8AAvMB+NzmkbcAAAAASUVORK5CYII=)